Input–Output Analysis
This essential reference for students and scholars in the input–output
research and applications community has been fully revised and
updated to reflect important developments in the field. Expanded
coverage includes construction and application of multiregional and
interregional models, including international models and their application to global economic issues such as climate change and international trade; structural decomposition and path analysis; linkages
and key sector identification and hypothetical extraction analysis;
the connection of national income and product accounts to input–
output accounts; supply and use tables for commodity-by-industry
accounting and models; social accounting matrices; non-survey
estimation techniques; and energy and environmental applications.
Input–Output Analysis is an ideal introduction to the subject for
advanced undergraduate and graduate students in many scholarly
fields including economics, regional science, regional economics,
city, regional and urban planning, environmental planning, public
policy analysis, and public management.
ronald e. miller is Professor Emeritus of Regional Science at
the University of Pennsylvania. A pioneer in the development of
interregional input–output models, his research providing key
insights about interregional feedback effects and many other features of regional economic models spans five decades.
peter d. blair is Distinguished Senior Fellow in the Schar
School of Policy and Government, George Mason University.
Published widely in many fields, his career includes management,
research, and teaching at the National Academy of Sciences, the
Congressional Office of Technology Assessment, Technecon
Analytic Research, and the University of Pennsylvania.
Published online by Cambridge University Press
Published online by Cambridge University Press
Input–Output Analysis
Foundations and Extensions
Third Edition
Ronald E. Miller
University of Pennsylvania
Peter D. Blair
George Mason University
Published online by Cambridge University Press
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DOI: 10.1017/9781108676212
© Ronald E. Miller and Peter D. Blair 2022
This publication is in copyright. Subject to statutory exception
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no reproduction of any part may take place without the written
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First published 2022
A catalogue record for this publication is available from the British Library.
Library of Congress Cataloging-in-Publication Data
Names: Miller, Ronald E., author. | Blair, Peter D., author.
Title: Input–output analysis : foundations and extensions / Ronald E. Miller, University
of Pennsylvania, Peter D. Blair, George Mason University, Virginia.
Description: Third edition. | Cambridge ; New York : Cambridge University Press,
2021. | Includes bibliographical references and index.
Identifiers: LCCN 2020046839 (print) | LCCN 2020046840 (ebook) | ISBN
9781108484763 (hardback) | ISBN 9781108723534 (paperback) | ISBN
9781108676212 (epub)
Subjects: LCSH: Input-output analysis.
Classification: LCC HB142 .M55 2021 (print) | LCC HB142 (ebook) | DDC 339.2/3–dc23
LC record available at https://lccn.loc.gov/2020046839
LC ebook record available at https://lccn.loc.gov/2020046840
ISBN 978-1-108-48476-3 Hardback
ISBN 978-1-108-72353-4 Paperback
Cambridge University Press has no responsibility for the persistence or accuracy
of URLs for external or third-party internet websites referred to in this publication
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
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Contents
List of Figures
List of Tables
Preface
page xxiv
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1 Introduction and Overview
1.1
Introduction
1.2
Input–Output Analysis: The Basic Framework
1.3
Outline for This Text
1.4
Internet Website and Text Locations of Real Datasets
References
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2 Foundations of Input–Output Analysis
2.1
Introduction
2.2
Notation and Fundamental Relationships
2.2.1 Input–Output Transactions and National Accounts
2.2.2 Production Functions and the Input–Output Model
2.3
An Illustration of Input–Output Calculations
2.3.1 Numerical Example: Hypothetical Figures – Approach I
Impacts on Industry Outputs
Other Impacts
2.3.2 Numerical Example: Hypothetical Figures – Approach II
2.3.3 Numerical Example: Mathematical Observations
2.3.4 Numerical Example: The US 2003 Data
2.4
The Power Series Approximation of ðI AÞ1
2.4.1 A Note on Computer Speeds and Capacities
2.4.2 The Power Series Approximation
2.5
Open Models and Closed Models
2.6
The Price Model
2.6.1 Overview
2.6.2 Physical versus Monetary Transactions
2.6.3 The Price Model Based on Monetary Data
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2.6.4
Numerical Examples Using the Price Model Based on
Monetary Data
Example 2.1: Base Year Prices
Example 2.2: Changed Base Year Prices
2.6.5 Applications
2.6.6 The Price Model Based on Physical Data
Introduction of Prices
Relationship between A and C
2.6.7 Numerical Examples Using the Price Model Based on
Physical Data
Example 2.3: Base Year Prices
Example 2.4: Changed Base Year Prices
2.6.8 The Quantity Model Based on Physical Data
2.6.9 A Basic National Income Identity
2.7
Summary
Appendix 2.1 The Relationship between Approaches I and II
Appendix 2.2 The Hawkins–Simon Conditions
References
3 Input–Output Models at the Regional Level
3.1
Introduction
3.2
Single-Region Models
3.2.1 National Coefficients
3.2.2 Regional Coefficients
3.2.3 Closing a Regional Model with Respect to Households
3.3
Many-Region Models: The Interregional Approach
3.3.1 Basic Structure of Two-Region Interregional
Input–Output Models
3.3.2 Interregional Feedbacks in the Two-Region Model
3.3.3 Numerical Example: Hypothetical Two-Region
Interregional Case
3.3.4 Interregional Models with More than Two Regions
3.3.5 Implementation of the IRIO Model
3.4
Many-Region Models: The Multiregional Approach
3.4.1 The Regional Tables
3.4.2 The Interregional Tables
3.4.3 The Multiregional Model
3.4.4 Numerical Example: Hypothetical Two-Region
Multiregional Case
3.4.5 The US MRIO Models
3.4.6 Numerical Example: The Chinese Multiregional Model
for 2012
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The Balanced Regional Model
3.5.1 Structure of the Balanced Regional Model
3.5.2 Numerical Example
3.6
The Spatial Scale of Regional and Interregional Models
3.6.1 Cities or Smaller Areas
3.6.2 States or Other National Subdivisions
3.6.3 Multicountry (or Multinational) Areas
3.7
Summary
Appendix 3.1 Basic Relationships in the Multiregional Input–
Output Model
Appendix 3.2 Sectoral and Regional Aggregation in the 2012 Chinese
Multiregional Model
Appendix 3.3 The Balanced Regional Model and the Inverse of a
Partitioned (I A) Matrix
References
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4 Organization of Basic Data for Input–Output Models
4.1 Introduction
4.2 Observations on Ad Hoc Survey-Based Input–Output Tables
4.3 Observations on Common Methods for Generating Input–
Output Tables
4.4 A System of National Economic Accounts
4.4.1 The Circular Flow of Income and Consumer Expenditure
4.4.2 Savings and Investment
4.4.3 Adding Overseas Transactions: Imports, Exports, and
Other Transactions
4.4.4 The Government Sector
4.4.5 The Consolidated Balance Statement for National Accounts
4.4.6 Expressing Net Worth
4.5 National Income and Product Accounting Conventions
4.6 Assembling the Input–Output Accounts: The US Case
4.7 Additional Considerations
4.7.1 Secondary Production: Method of Reallocation
Example 4.1: Reallocation of Secondary Production
4.7.2 Secondary Production: Commodity-by-Industry Accounting
Example 4.2: Commodity-by-Industry Accounts
4.7.3 Reconciling with the National Accounts
4.7.4 Producers’ and Consumers’ Prices
Example 4.3: Trade and Transportation Margins
4.7.5 Accounting for Imports and Exports
Valuation of Imports
Example 4.4: Competitive and Non-competitive Imports
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Approach A: Classifying Imports by Commodity
Approach B: Classifying Imports by Purchaser
Approach C: Explicit Representation of Noncompetitive Imports
Approach D: Maintaining Separate Tables of Imported and
Domestic Products
Approach E: Total Supply
The Choice among Approaches to Representing Imports
4.7.6 Removing Competitive Imports from Total
Transactions Tables
Approximation Method I
Approximation Method II
Example 4.5: Import Scrubbing
Implications of the Estimating Assumptions
4.7.7 Adjustments for Inventory Change
4.7.8 Adjustments for Scrap
4.7.9 Special Considerations for Regional and
Multiregional Models
4.7.10 Summary
4.8 Valuation and Double Deflation
Example 4.6: Double Deflation
4.9 The Aggregation Problem: Level of Detail in Input–Output Tables
4.9.1 Investigating Aggregation Bias
4.9.2 The Aggregation Matrix
Example 4.7: Sectoral Aggregation
4.9.3 Measures of Aggregation Bias
Theorem 4.1
Theorem 4.2
4.9.4 Spatial Aggregation Bias
4.10 Harmonization of Input–Output Data
4.11 Summary
Appendix 4.1 Supplemental Discussion of Aggregation Bias
References
5 The Commodity-by-Industry Approach in Input–Output Models
5.1
Introduction
5.1.1 The Use Matrix
5.1.2 The Make Matrix
5.2
The Basic Accounting Relationships
5.3
Technology and Total Requirement Matrices in the Commodity–
Industry Approach
5.3.1 Industry Source of Commodity Outputs
5.3.2 Commodity Composition of Industry Outputs
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Generating Total Requirements Matrices
Using D
Using C
5.3.4 “Industry-Based” Technology
5.3.5 “Commodity-Based” Technology
5.3.6 Direct Requirements (Technical Coefficients) Matrices
Derived from Basic Data
5.3.7 Total Requirements Matrices
Approach I: Starting with Technical Coefficients
Approach II: Avoiding C ‒1 in Commodity Technology Cases
Is Singularity Likely to be a Problem in Real-World Models?
5.3.8 Commodity-by-Industry Configurations for
Multiregional Models
Numerical Examples of Alternative Direct and Total
Requirements Matrices
5.4.1 Direct Requirements Matrices
5.4.2 Total Requirements Matrices
Commodity-Demand-Driven Models
Industry-Demand-Driven Models
Negative Elements in the Commodity–Industry Framework
5.5.1 Commodity Technology
Direct Requirements Matrices
Transactions Matrices
Total Requirements Matrices
5.5.2 Industry Technology
Direct Requirements Matrices
Total Requirements Matrices
5.5.3 Making a Model Choice
Which Model to Choose?
Dealing with Negative Values
Non-square Commodity–Industry Systems
5.6.1 Commodity Technology
5.6.2 Industry Technology
Direct Requirements Matrices
Total Requirements Matrices
Mixed Technology in the Commodity–Industry Framework
5.7.1 Commodity Technology in V1
5.7.2 Industry Technology in V1
5.7.3 Numerical Examples with Mixed Technology Assumptions
Example 5.1: Commodity Technology in V1
Example 5.2: Industry Technology in V1
5.7.4 Additional Mixed Technology Variants
Summary
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5.4
5.5
5.6
5.7
5.8
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Appendix 5.1
A5.1.1
A5.1.2
Appendix 5.2
A5.2.1
A5.2.2
A5.2.3
Appendix 5.3
References
Alternative Approaches to the Derivation of
Transactions Matrices
Industry Technology
Commodity-by-Commodity Requirements
Industry-by-Industry Requirements
Commodity Technology
Commodity-by-Commodity Requirements
Industry-by-Industry Requirements
Elimination of Negatives in Commodity Technology Models
The Problem
3 3 Example
4 4 Example
5 5 Example (from Almon, 2000)
Approaches to Elimination of Negative Elements
Results of the Iterative Procedure
3 3 Example
4 4 Example
5 5 Example
Left and Right Inverses in Non-square Input–Output Systems
6 Multipliers in the Input–Output Model
6.1
Introduction
6.2
General Structure of Multiplier Analysis
6.2.1 Output Multipliers
Simple Output Multipliers
Total Output Multipliers
Example 6.1: The US Input–Output Model for 2003
Output Multipliers in Commodity–Industry Models
Commodity-Demand-Driven Models
Industry-Demand-Driven Models
6.2.2 Income/Employment Multipliers
Income Multipliers
Type I and Type II Income Multipliers
Relationship between Simple and Total Income Multipliers or
between Type I and Type II Income Multipliers
Which Multiplier to Use?
Even More Income Multipliers
6.2.3 Additional (Generalized) Multipliers
6.2.4 Summary
6.3
Multipliers in Regional Models
6.3.1 Regional Multipliers
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6.3.2
Interregional Input–Output Multipliers
Intraregional Effects
Interregional Effects
National Effects
Sectoral Effects
More than Two Regions
6.3.3 Multiregional Input–Output Multipliers
Intraregional Effects
Interregional Effects
National Effects
Sectoral Effects
Final Demand for Goods Made in a Particular Region
More than Two Regions
6.4
Miyazawa Multipliers
6.4.1 Disaggregated Household Income Groups
6.4.2 Miyazawa’s Derivation
6.4.3 Numerical Example
6.4.4 Adding a Spatial Dimension
6.5
Gross and Net Multipliers in Input–Output Models
6.5.1 Introduction
6.5.2 Multipliers in the Net Input–Output Model
Numerical Example
6.5.3 Additional Multiplier Variants
(Indirect Effects)/(Direct Effects)
“Growth Equalized” Multipliers
Another Kind of Net Multiplier
6.6
Multipliers and Elasticities
6.6.1 Output Elasticity
6.6.2 Output-to-Output Multipliers and Elasticities
Direct Effects
Total Effects
6.7
Summary
Appendix 6.1 The Equivalence of Total Household Income Multipliers and
Þ1
the Elements in the Bottom Row of ðI A
Appendix 6.2 Relationship between Type I and Type II Income Multipliers
References
7 Supply-Side Models, Linkages, and Important Coefficients
7.1
Supply-Side Input–Output Models
7.1.1 The Early Interpretation
Numerical Illustration (Hypothetical Data)
Numerical Application (US Data)
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7.1.2
7.1.3
7.1.4
7.2
7.3
7.4
Relationships between A and B and between L and G
Comments on the Early Interpretation
Joint Stability
The Issue
Conditions under Which Both A and B Will Be Stable
7.1.5 Reinterpretation as a Price Model
Connection to the Leontief Price Model (Algebra)
Connection to the Leontief Price Model
(Numerical Illustration)
A Ghosh Quantity Model
Linkages and Key Sectors in Input–Output Models
7.2.1 The Early Measures
Backward Linkage
Forward Linkage
7.2.2 Classifying Backward and Forward Linkage Results
7.2.3 Spatial Linkages
7.2.4 “Net” Backward and Forward Linkages
Net Backward Linkage
Net Forward Linkage
7.2.5 Hypothetical Extraction
Complete Extraction
Partial Extraction: Backward Linkage
Partial Extraction: Forward Linkage
7.2.6 Generalized Linkage Measures
7.2.7 Which Measures to Use?
7.2.8 Illustration Using US Data
Identifying Important Coefficients
7.3.1 Mathematical Background
7.3.2 Relative Sizes of Elements in the Leontief Inverse
Observation 1
Observation 2
Observation 3
7.3.3 “Inverse-Important” Coefficients
7.3.4 Numerical Example
7.3.5 Impacts on Gross Outputs
7.3.6 Fields of Influence
7.3.7 Additional Measures of Coefficient Importance
Converting Output to Employment, Income, etc.
Elasticity Coefficient Analysis
Relative Changes in All Gross Outputs
Impacts of Changes in More than One Element of the
A Matrix
Summary
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Appendix 7.1 The Sherman–Morrison–Woodbury Formulation
A7.1.1 Introduction
A7.1.2 Application to Leontief Inverses
Appendix 7.2 Hypothetical Extractions with Partitioned Matrices
References
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8 Decomposition Approaches
8.1
Introduction
8.2
Structural Decompositon (Additive)
8.2.1 Initial Decompositions: Changes in Gross Outputs
Numerical Example
8.2.2 Next-Level Decompositions: Digging Deeper into
Δf and ΔL
Additive Decompositions with Products of More than
Two Terms
Changes in Final Demand
8.2.3 Numerical Examples
One Category of Final Demand (p ¼ 1)
Two Categories of Final Demand (p ¼ 2)
8.2.4 Changes in the Direct Inputs Matrix
Decomposition of ΔL
Decomposition of ΔA
Numerical Illustration (continued)
8.2.5 Decompositions of Changes in Some Function of x
8.2.6 Summary for Δx
8.2.7 ASDA in a Multiregional Input–Output (MRIO) Model
8.2.8 Empirical Examples
Studies Using National Models
Early Studies with a Spatial Dimension
8.3
Structural Decomposition (Multiplicative)
8.3.1 Initial Decompositions: Relative Changes in Total
Gross Output
8.3.2 A Note on Arithmetic and Geometric Means (Averages)
8.3.3 Numerical Example (Reexamined)
8.3.4 Multiplicative Decomposition of Changes in Some
Function of x
A Three-Factor Example
A Four-Factor Example
8.4
Decomposition of Multipliers
8.4.1 Multiplier Decompositions (Multiplicative)
8.4.2 Multiplier Decompositions in an Interregional Context
8.4.3 Multiplier Decompositions (Additive)
8.4.4 A Note on Interregional Feedbacks
8.4.5 Numerical Illustration
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8.5
Paths
8.5.1 Structural Path Analysis
Incorporation of the Power Series Results
Numerical Illustration
8.5.2 Structural Path Decomposition
8.6
Summary
Appendix 8.1 Alternative Additive Decompositions of x ¼ LBf
Appendix 8.2 Additional Early Additive Structural Decomposition Studies
Appendix 8.3 The Approximate Economy-wide Equivalence of
Additive and Multiplicative SDA Effects
Appendix 8.4 A Note on Growth Rates
References
9 Nonsurvey and Partial-Survey Methods: Fundamentals
9.1
Introduction
9.2
The Question of Stability of Input–Output Data
9.2.1 Stability of National Coefficients
Comparisons of Direct-Input Coefficients
Comparisons of Leontief Inverse Matrices
Other Summary Measures
Data for the US Economy
9.2.2 Constant versus Current Prices
9.2.3 Stability of Regional Coefficients
9.2.4 Summary
9.3
Updating and Projecting Coefficients: Trends, Marginal
Coefficients, and Best Practice Methods
9.3.1 Trends and Extrapolation
9.3.2 Marginal Input Coefficients
9.3.3 “Best Practice” Firms
9.4
Updating and Projecting Coefficients: The RAS Approach and
Hybrid Methods
9.4.1 The RAS Technique
9.4.2 Example of the RAS Procedure
9.4.3 Updating Coefficients versus Transactions
Numerical Illustration
9.4.4 An Economic Interpretation of the RAS Procedure
9.4.5 Incorporating Additional Exogenous Information in an
RAS Calculation
9.4.6 Modified Example: One Coefficient Known in Advance
9.4.7 Hybrid Models: RAS with Additional Information
9.4.8 The Constrained Optimization Context
RAS as a Distance Minimization Problem
Alternative Measures of Matrix Distance
9.4.9 Infeasible Problems
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Summary
References
10 Nonsurvey and Partial-Survey Methods: Extensions
10.1
Introduction
10.2
Location Quotients and Related Techniques
10.2.1 Simple Location Quotients
10.2.2 Variations on Simple Location Quotients
Purchases-Only Location Quotients
Cross-Industry Quotients
10.2.3 Supply–Demand Pool (Commodity
Balance) Approaches
10.2.4 Fabrication Effects
10.2.5 Addressing the Cross-Hauling Issue
The Semilogarithmic Quotient
The Flegg Modification, FLQ
Augmented FLQ (AFLQ)
The Cross-Hauling Adjusted Regionalization
Method (CHARM)
Case 1: The Region Is a Net-Exporter of Good i (bi > 0)
Case 2: The Region Is a Net-Importer of Good i (bi < 0)
10.2.6 Regional Purchase Coefficients
10.2.7 “Community” Input–Output Models
10.2.8 Summary
10.3
RAS in a Regional Setting
10.4
Numerical Illustration
10.5
Exchanging Coefficients Matrices
10.6
Estimating Interregional Flows
10.6.1 Gravity Model Formulations
10.6.2 Two-Region Interregional Models
10.6.3 Two-Region Logic with More than Two Regions
10.6.4 Estimating Commodity Inflows to a Substate Region
10.6.5 Additional Studies
Commodity Flows among US States
An Optimization Model for Interregional Flows
10.7
Selected Examples
10.7.1 Generation of Regional Input–Output Tables (GRIT)
10.7.2 Double-Entry Bi-regional Input–Output
Tables (DEBRIOT)
10.7.3 The Multiregional Input–Output Model for China,
2000 (CMRIO)
10.8
International Input–Output Models
10.8.1 Introduction
10.8.2 Asian International Input–Output Tables
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“Hybrid” Many-Region Models for the
European Union
10.8.4 China–Japan “Transnational Interregional” Input–Output
(TIIO) Model, 2000
Chinese Exports to Japan for Intermediate Demand
Applications
10.8.5 Leontief’s World Model
10.8.6 Global Models
10.9
The Reconciliation Issue
10.10
Summary
Appendix 10.1 Geographical Classifications in the World
Input–Output Model
Appendix 10.2 Detailed Results for the Numerical Illustration in Section 10.4
Appendix 10.3 Brief History of Leontief Inverses with Errors in the
Coefficients of A
References
10.8.3
11 Social Accounting Matrices
11.1
Introduction
11.2
Social Accounting Matrices: Background
11.3
Social Accounting Matrices: Basic Concepts
11.4
The Households Account
11.5
The Value-Added Account
11.6
Interindustry Transactions and the Input–Output Framework
11.7
Expanding the Social Accounts
11.8
Additional Social Accounting Variables
11.9
A “Fully Articulated” SAM
11.10
SAM Multipliers
11.10.1 SAM Multipliers: Basic Structure
11.10.2 Decomposition of SAM Multipliers
Example 11.1: Reduced Form Case
11.10.3 Multipliers in an Expanded SAM
Example 11.2: The Expanded Case
11.10.4 Additive Multipliers
11.11
The Relationship between Input–Output and SAM Multipliers
11.12
Balancing SAM Accounts
11.12.1 Example: Balancing a SAM
11.12.2 Example: Balancing a SAM with
Additional Information
11.13
Some Applications of SAMs
11.13.1 Example: A Multiregional North American MacroEconomic SAM
11.13.2 Example: A SAM for India
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11.14
Summary
References
12 Energy Input–Output Analysis
12.1 Introduction
12.1.1 The Origins of Energy Input–Output Analysis
12.1.2 Early Approaches to Energy Input–Output Analysis
12.1.3 Origins of Contemporary Approaches to Energy
Input–Output Analysis
12.2 Energy Input–Output Analysis: Conceptual Overview
12.2.1 The Basic Energy Input–Output Formulation
12.2.2 The Total Energy Requirements Matrix
Example 12.1: Two-Sector Illustration of Hybrid Units
Input–Output Analysis
Example 12.2: Generalization to Several Energy Types
12.2.3 The Hybrid Units Formulation and Energy
Conservation Conditions
Example 12.2 (Revisited): Generalization to Several
Energy Types
12.2.4 Traditional Formulation of the Energy Input–Output Model
Example 12.3: Illustration of the Traditional Energy
Input–Output Formulation
12.2.5 Limitations of the Traditional Approach
12.3 Further Methodological Considerations
12.3.1 Adjusting for Energy Conversion Efficiencies
Example 12.4: Adjusting for Energy
Conversion Efficiencies
12.3.2 Commodity-by-Industry Energy Models
12.3.3 Accounting for Imports
12.3.4 Interregional and Multiregional Extensions
12.3.5 Energy Input–Output and Econometrics
12.4 Applications
12.4.1 Net Energy Analysis
Example 12.5: Net Energy Analysis
12.4.2 Energy Cost of Goods and Services
Example 12.6: Embodied Energy Example
12.4.3 Impacts of New Energy Technologies
12.4.4 An Energy Tax
12.4.5 Energy and Structural Change
Example 12.7: Structural Energy Decomposition
12.4.6 Energy Embodied in International Trade
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Example 12.8: Embodied Energy in Imports for the US
Economy, 1997–2002
12.4.7 Other Applications
12.5 Summary
Appendix 12.1 Earlier Formulation of Energy Input–Output Models
References
13 Environmental Input–Output Analysis
13.1
Introduction
13.2
The Augmented Leontief Model
13.2.1 Pollution Generation
Example 13.1: Pollution Generation – Augmented
Leontief Model
13.2.2 Pollution Elimination
Example 13.2: Pollution Elimination – Augmented
Leontief Model
13.2.3 Existence of Non-negative Solutions
Example 13.3: Pollution Activity – Augmented
Leontief Model
13.3
Physical Input–Output Tables
Example 13.4: Waste Generation as an “Input”
for Production
13.4
Life Cycle Assessment and Input–Output (LCA-IO) Models
13.4.1 An LCA-IO Example: Hybrid-Units Material
Flows – Lead in the US Economy
13.4.2 Other Approaches
13.5
Multiregional Product Supply Chains
13.5.1 The Special Case of Carbon Dioxide Emissions
13.5.2 Measuring a Carbon Footprint
Mathematical Formulation
Example 13.5: Attribution of Emissions to Production
or Consumption
13.5.3 Global MRIO Models for Assessing Carbon Footprints
Example 13.6: MRIO Consumption-Based Attribution
of Emissions
13.6
Input–Output Models with Expanded Environmental Accounts
13.7
Generalized Input–Output Analysis: General Framework
13.7.1 Accounting for Pollution Impacts
13.7.2 Generalized Impacts
Example 13.7: Generalized Input–Output Analysis
13.7.3 Summary: Generalized Input–Output Formulations
13.8
Generalized Input–Output Analysis: Extensions of the
Planning Approach
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Linear Programming: A Brief Introduction by Means of the
Leontief Model
13.8.2 Multiple Objectives
13.8.3 Conflicting Objectives and Linear Goal Programming
13.8.4 Additional Observations on Goal Programming
Specifying Objectives
Tightly Constrained Problems
Solution Methods
13.8.5 Application to the Generalized Input–Output
Planning Problem
13.8.6 Policy Programming
Policy Programming Example (Example 13.1 revisited)
13.8.7 Applications of Input–Output and Multiobjective
Decision-Making Models
13.9
Ecological Commodities
13.10
Environmentally Extended Input–Output Models
13.10.1 Fully Integrated Models
Toward a Circular Economy
Analyzing Strategies for a Materials-Circular Economy
Assessing a Materials Footprint
13.10.2 Limited Economic–Ecologic Models
Economic Subsystem
Ecologic Subsystem
Commodity-by-Industry Formulation
Example 13.8: Limited Economic–Ecologic Models
13.10.3 Illustrative EEIO Applications
13.11
Pollution Dispersion
13.11.1 Gaussian Dispersion Models
13.11.2 Coupling Pollution Dispersion with Input–
Output Models
Example 13.9: Coupling Input–Output and Pollution
Dispersion Models
13.12
Other Applications of Environmental Input–Output Analysis
13.12.1 Assessing Environmental Policy Initiatives
13.12.2 Assessing Impacts of Environmental Disasters
13.13
Summary
References
xix
13.8.1
14 Mixed and Dynamic Models
14.1 Introduction
14.2 Mixed Models
14.2.1 Exogenous Specification of One Sector’s Output
Rearranging the Basic Equations
“Extracting” the Sector
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An Alternative Approach When f 1 , , f n1 and xn Are
Exogenously Specified
14.2.3 Examples with xn Exogenous
Example 14.1: f1 ¼ 100,000, f2 ¼ 200,000, x3 ¼ 150,000
Example 14.2: f1 ¼ f2 = 0, x3 ¼ 150,000
Example 14.3: f1 ¼ 100,000, f2 ¼ 200,000, x3 ¼ 100,000
Example 14.4: The Critical Value of x3
14.2.4 Exogenous Specification of f 1 , . . . , f k , xkþ1 , . . . , xn
Example 14.5 (Example 14.2 expanded with xn1 and xn
Exogenous)
14.3 New Industry Impacts in the Input–Output Model
14.3.1 New Industry: The Final-Demand Approach
14.3.2 New Industry: Complete Inclusion in the Technical
Coefficients Matrix
14.3.3 A New Firm in an Existing Industry
14.3.4 Other Structural Changes
14.4 Dynamic Considerations in Input–Output Models
14.4.1 General Relationships
14.4.2 A Three-Period Example
Terminal Conditions
Initial Conditions
14.4.3 Numerical Example 1
Terminal Conditions
Initial Conditions
14.4.4 Numerical Example 2
Terminal Conditions
Initial Conditions
14.4.5 “Dynamic” Multipliers
14.4.6 Turnpike Growth and Dynamic Models
Example 14.6: Turnpike Growth
14.4.7 Alternative Input–Output Dynamics
14.5 Summary
Appendix 14.1 Exogenous Specification of Some Elements of x
A14.1.1 The General Case: An n-sector Model with k
Endogenous Outputs
A14.1.2 The Output-to-Output Multiplier
Matrix
A14.1.3 The Inverse of a Partitioned I AðnÞ Matrix
A14.1.4 The Case of k ¼ 2, n ¼ 3
A14.1.5 The Case of k ¼ 1, n ¼ 3
A14.1.6 “Extracting” the Last (n – k) Sectors
References
14.2.2
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15 Additional Topics
15.1 Introduction
15.2 Input–Output and Measuring Economic Productivity
15.2.1 Total Factor Productivity
The Physical Input–Output Model
A Measure of Total Factor Productivity Using
Input–Output Data
15.2.2 Numerical Example
References
15.3 Modeling Economic Impacts of Disasters
15.3.1 The Inoperability Input–Output Model
15.3.2 Other Approaches
References
15.4 Alternative Technology Models
15.4.1 Background
15.4.2 Alternative Technologies
15.4.3 Numerical Illustration
References
15.5 Graph Theory and Qualitative Input–Output Analysis
References
15.6 Fundamental Economic Structure
References
15.7 Variable Input–Output, Econometrics, and Computable General
Equilibrium Models
15.7.1 The Variable Input–Output Model
15.7.2 Regional Input–Output, Econometric, and Computable
General Equilibrium Models
References
15.8 Additional Resources for Input–Output Extensions and Applications
15.8.1 Edited Collections
15.8.2 Collections of Printed Articles
References
Appendix 15.1 More on the Derivation of Total Factor
Productivity Measures
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723
723
723
Postscript
751
Appendix A Matrix Algebra for Input–Output Models
A.1
Introduction
A.2
Matrix Operations: Addition and Subtraction
A.2.1 Addition
A.2.2 Subtraction
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753
754
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A.2.3 Equality
A.2.4 The Null Matrix
Matrix Operations: Multiplication
A.3.1 Multiplication of a Matrix by a Number
A.3.2 Multiplication of a Matrix by Another Matrix
A.3.3 The Identity Matrix
Matrix Operations: Transposition
Representation of Linear Equation Systems
Matrix Operations: Division
Diagonal Matrices
Summation Vectors
Matrix Inequalities
Partitioned Matrices
A.10.1 Multiplying Partitioned Matrices
A.10.2 The Inverse of a Partitioned Matrix
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754
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756
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761
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763
764
764
Appendix B Guide to Online Input–Output Data Resources for This Text
B.1
US Input–Output Data
B.2
Other Data Included in the Online Resource
B.3
Exercise Problems and Solutions
Selected References for US Input–Output Tables (1919–2018)
767
767
768
769
769
A.3
A.4
A.5
A.6
A.7
A.8
A.9
A.10
Appendix C
Historical Notes on the Development of Leontief’s
Input–Output Analysis
C.1
Conceptual Foundations
C.2
Quesnay and the Physiocrats
C.3
Mathematical Formalization
C.4
Leontief and the “Economy as a Circular Flow”
C.5
Development of Input–Output Analysis
References
Author Index
Subject Index
786
799
The following Supplementary Appendices, which are referenced
in this text, are available online at http://www.cambridge.org/millerandblair
Appendix SA2.1
Appendix SA3.1
Appendix SA4.1
Appendix SA5.3
Appendix SA7.2
Appendix SA8.1
771
771
772
775
776
778
783
The Relationship between Approaches I and II
Basic Relationships in the Multiregional Input–Output Model
Supplemental Discussion of Aggregation Bias
Left and Right Inverses in Nonsquare Input–Output Systems
Hypothetical Extractions with Partitioned Matrices
Alternative Additive Decompositions of x = LBf
Published online by Cambridge University Press
Contents
Appendix SA8.2
Appendix SA8.3
Additional Early Additive Structural Decomposition Studies
The Approximate Economy-wide Equivalence of Additive and
Multiplicative SDA Effects
Appendix SA10.2 Detailed Results for the Numerical Illustration in Section 10.4
Appendix SA10.3 Brief History of Leontief Inverses with Errors in the
Coefficients of A
Appendix SA12.1 Earlier Formulation of Energy Input–Output Models
Appendix SP1
Exercise Problems
Appendix SP2
Exercise Problem Solutions
Appendix SP3
Computational Exercises Workbook
Appendix SD1
US Input-Output Data (1919–2018)
Appendix SD2
Other Real World Input-Output Data Tables
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xxiii
Figures
2.1
Production functions in input space. (a) Linear production function.
(b) Classical production function. (c) Leontief production function.
(d) Activity analysis production function
A2.2.1a Solution space representation of (A2.2.2); a12 > 0 and a21 > 0
A2.2.1b Solution space representation of (A2.2.2); a21 ¼ 0
A2.2.1c Solution space representation of (A2.2.2); a12 ¼ 0
3.1
Increases in Washington final demands affecting Washington outputs
via Connecticut
A3.2.1 Regional aggregation in the 2012 Chinese multiregional model
4.1
The circular flow of income and expenditures
4.2
Circular flow example: Point of departure
4.3
Introduction of savings and investment into the circular flow of income
and expenditures
4.4
Introduction of depreciation into the circular flow of income
and expenditures
4.5
Addition of the rest of world account
4.6
Addition of the government account
4.7
Net worth
11.1
Circular flow of income, expenditure, and market
12.1
US energy use for 2019 (quadrillions of Btus)
12.2
Hudson-Jorgenson model
12.3
Net energy analysis
12.4
Changes in US energy consumption: 1972–1985
13.1
Two-sector Leontief model
13.2
Input–output and linear programming: Example 13.1
13.3
Different values of GNP for Example 13.1
13.4
Linear programming solution
13.5
Goal programming solution: Objective 1
13.6
Goal programming solution: Objective 2
13.7
Goal programming solution: Objective 3
13.8
Goal programming solution: Objective 4
13.9
Goal programming generalized input–output initial solution
13.10
Generalized input–output goal programming: Example 13.1
(Objective 1)
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58
59
75
105
117
118
119
119
121
123
126
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587
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633
634
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640
641
641
642
644
646
List of Figures
13.11
13.12
13.13
13.14
13.15
13.16
C.1
Generalized input–output goal programming: Example 13.1
(Objective 2)
Generalized input–output goal programming: Example 13.1
(Objective 3)
Generalized input–output goal programming: Example 13.1
(Objective 4)
Generalized input–output goal programming: Example 13.1
(Objective 5)
Generalized input–output goal programming: Example 13.1
(Objective 6)
Location of air pollution sources and receptor points
François Quesnay’s Tableau Économique
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648
648
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774
Tables
1.1
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
A3.2.1
Input–output transactions table
Input–output table of interindustry flows of goods
Expanded flow table for a two-sector economy
Flows (zij) for the hypothetical example
Technical coefficients (the A matrix) for the hypothetical example
Flows (zij) for the hypothetical example associated with xnew
Round-by-round impacts (in dollars) of f11 ¼ $600 and f21 ¼ $1,500
The 2003 US domestic direct requirements matrix, A
The 2003 US domestic total requirements matrix, L ¼ (I ‒ A)‒1
Input–output table of interindustry flows with households endogenous
Flows (zij) for hypothetical example, with households endogenous
Transactions in physical units
Transactions in monetary units (see Table 2.3)
Transactions in revised physical units
Transactions in monetary terms
The Leontief quantity and price models
Transactions for hypothetical example with one primary input
Flows in physical units
Transactions in physical terms (Germany, 1990) (millions of tons)
Alternative input–output price and quantity models
Interindustry, interregional flows of goods
Flow data for a hypothetical two-region interregional case
Data needed for conversion of national to regional coefficients via the
product-mix approach
Interregional shipments of commodity i
Flow data for a hypothetical two-region multiregional case
Interregional commodity shipments for the hypothetical two-region
multiregional case
Chinese interregional and intraregional transactions, 2012 (in ¥ billions)
Direct input coefficients for the Chinese multiregional economy, 2012
Leontief inverse matrix for the Chinese multiregional economy, 2012
Region- and sector-specific effects (in ¥ millions) of a ¥100 million
increase in final demand for manufacturing goods, China, 2012
Regional classifications in the 2012 Chinese multiregional model
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13
13
21
22
23
28
30
30
37
39
43
44
44
45
46
47
48
53
55
72
77
84
85
88
89
93
94
95
97
105
List of Tables
A3.2.2
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
4.19
4.20
4.21
4.22
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
A5.2.1
A5.2.2
6.1
6.2
Sectoral aggregation in the 2012 Chinese multiregional model
Basic national accounts: Example economy
Basic national accounts including rest of world
Basic national accounts including the government sector
Balance statement for the basic national accounts
Basic national accounts balance statement in matrix form
Matrix of national accounts including net-worth calculations
The commodity-by-industry use table
The industry-by-commodity supply table
Input–output transactions: Example 4.1 (millions of dollars)
Example 4.2, use and supply accounts (millions of dollars)
Production account allocated to individual products and sectors
Commodity by industry supply matrix: Running example
Consolidated commodity-by-industry input–output accounts:
Running example
Example trade and transportation margins: Example 2
Example 4.4, basic information
Example 4.4a, approach A, classifying imports by commodity
Example 4.4b, approach B, classifying imports by purchaser (use)
Example 4.4c, approach C, explicit representation of noncompetitive imports
Example 4.4d, approach D, separate imported and domestic
product tables
Example 4.4e, approach E, total supply
Approximation methods for “scrubbing” interindustry transactions of
competitive imports: Example 4.5
Double deflation: Example 6
The use matrix (U) and other data for a two-commodity, two-industry
hypothetical example (in dollars)
The make matrix (V) and other data for a two-commodity, two-industry
hypothetical example (in dollars)
The complete set of commodity–industry data
Total requirements matrices, commodity-demand driven models
Total requirements matrices, industry-demand driven models
Rewritten forms of total requirements matrices
Alternative classifications, total requirements matrices,
commodity-demand driven models
Examples of negative elements in real-world commodity–technology
direct requirements matrices [AC ¼ BC‒1]
A three-commodity, two-industry example
Share of secondary product output in total industry output (European
Union countries, 60-sector level)
Summary of two commodity/two industry results
Steps in the iterative procedure for the 3 3 example
Total requirements matrices in commodity–industry models
Model closures with respect to households
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120
122
123
124
125
127
132
133
135
137
138
139
140
141
145
146
147
148
149
150
156
159
179
179
180
190
191
192
203
204
205
217
223
232
244
253
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List of Tables
6.3
6.4
6.5
Input–output multipliers
General multiplier formulas
Leontief inverse matrix, L, for the Chinese multiregional
economy, 2012
Simple intra- and interregional output multipliers for the Chinese
multiregional input–output system, 2012
Sector-specific simple output multipliers for the Chinese multiregional
input–output system, 2012
Interrelational interregional income multipliers
Overview of the Leontief and Ghosh quantity and price models
Basic linkage measures
Classification of backward and forward linkage results
Summary of spatial/sectoral linkage measures (two-region example)
Hypothetical extraction linkages
Generalized linkage measures
Sector j backward linkage results, US 2003 data (using the output metric,
z0c ¼ i0 )
Sector i forward linkage results, US 2003 data (using the output metric,
z0c ¼ i0 )
Sector j backward linkage results, US 2003 data (alternative
normalizations)
Sector i forward linkage results, US 2003 data (alternative
normalizations)
Classification of hypothetical extraction results, US 2003 data
Number of important transactions in the 2000 China MRIO model
(l ri =xr ) 106 for the 2003 US seven-sector model
Percentage change in x resulting from Δa12 ¼ ð0:2Þa12
Upper threshold on Δaij =aij for γ ¼ 1 percent
Average values in US total requirements matrices
Column sums of |F[i, j]| for numerical P
example
Sum of all elements in F[i, j]| (kFk ¼ ij f ij )
Alternative structural decompositions
Sector-specific and economy-wide decomposition results
[equation (8.7)]
Sector-specific and economy-wide decomposition results
(with two-factor final-demand decomposition detail)
Sector-specific and economy-wide decomposition results
(with three-factor final-demand decomposition detail)
Sector-specific and economy-wide decomposition results
(with additional technology and final-demand decomposition detail)
Selected early empirical structural decompositions at a regional,
interregional, or multiregional level
First-order paths
Second-order paths
Third-order paths
Rank-ordered paths for the numerical illustration
6.6
6.7
6.8
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
7.11
7.12
7.13
7.14
7.15
7.16
7.17
7.18
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
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256
265
266
268
275
301
305
306
308
315
318
320
320
320
321
322
323
329
330
331
332
334
334
351
352
356
357
361
369
390
390
392
393
List of Tables
A8.4.1
9.1
9.2
9.3
9.4
10.1
10.2
10.3
10.4
10.5
10.6
11.1
11.2
11.3
11.4
11.5
11.6
11.7
11.8
11.9
11.10
11.11
11.12
11.13
11.14
11.15
11.16
11.17
11.18
11.19
11.20
11.21
12.1
Annual growth rate estimates for xt ¼ 100 to xt+5 ¼ 105
Values of a11 and a23 at each step in the RAS adjustment procedure
Differences from row and column margins at each step in the RAS
adjustment procedure
Elements in the diagonal matrices ^r k and ^s k , for k ¼ 1, . . . ,7
MAD and MAPE when one coefficient is known in advance in an
RAS estimate
ESA 95 table format for input–output data (type E table)
Total intraregional intermediate inputs and intraregional output
multipliers for region 1 (East) calculated from several regionalization
techniques (2012)
Components in the DEBRIOT approach
Structure of the TIIO model
China-to-Japan intermediate transactions in TIIO
Summary of prominent GMRIO databases
The basic national accounts balance statement in matrix form
(Table 4.5 revisited) The basic national accounts balance statement in
matrix form: Example
The basic national accounts balance statement in matrix form expanded
to include the households account
The basic national accounts balance statement in matrix form: Example,
expanded to include the households account
The basic national accounts balance statement in matrix form expanded
to include the value-added account
The basic national accounts balance statement in matrix form: Example,
expanded to include the value-added account
SAM framework example: Input–output representation
SAM framework example using social accounting conventions
Input–output accounts for Table 11.6 revisited
Expanded value-added accounts
Expanded final-demand accounts
Sources of value-added income
Expanded input–output accounts
SAM representation of expanded input–output accounts
Basic national accounts balance statement in matrix form expanded to
include additional macro transactions
Expanded national accounts balance statement in matrix form: Example
expanded to include additional macro transactions
Reduced form fully articulated SAM: Example 11.1
Expanded form fully articulated SAM: Example 11.1
SAM framework example using social accounting conventions (revised
to include endogenous final demand)
Comparative input–output and SAM multipliers
Unbalanced SAM: Examples 11.12.1 and 11.12.2
Typical efficiencies of direct use and of conversion of traditional fossil
fuels to electricity
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481
481
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List of Tables
12.2
12.3
Energy and dollar flows: Example 12.1
Interindustry economic transactions: Example 12.2
(millions of dollars)
Energy flows: Example 12.2 (quadrillion Btus)
Dollar transactions for Example 12.3 (millions of dollars)
Energy flows for Example 12.3 (quadrillion Btus)
Energy flows for Example 12.3 revised (quadrillion Btus)
Summary of energy input–output relationships
Interindustry transactions in hybrid units: Example 12.4
Input–output transactions for the US economy in hybrid units (1967)
Technical coefficients: Example 12.4
Leontief inverse: Example 12.4
Total primary energy intensities: Example 12.4
Power plant inputs: Example 12.4
Embodied energy in $1 million worth of automobiles delivered to final
demand (USA in 2002)
Selected empirical structural decompositions of changes in energy use
or pollution emissions
Calculated total primary energy intensities (kBtu/$1997) for 1997 and
2002 and percentage improvement from 1997 to 2002 (adapted from
Weber, 2009)
Sources of change in total energy intensity for 26 nations (1995–2007)
Summary of energy input–output relationships: Initial formulation
Pollution-generation example: Dollar transactions
Input–output transactions: Pollution-expanded model example
A typical physical input–output table: Conceptual definitions
Physical input–output table (revised)
PIOT for Italy (1995): Waste defined as negative input
Hybrid unit transactions – Industrial lead material flows in the
US economy
Three-sector economy, Example 13.5
Input–output transactions (millions of dollars) for Example 13.7
Direct impact coefficients for Example 13.7
Policy programming: Composite scenario weights
Economic–ecologic commodity flows: Matrix definitions
Economic–ecologic commodity flows
Basic structure of EEIOA models
Limited commodity-by-industry economic–ecologic model
Economic–ecologic models: Example 13.8
Illustrative real input–output data referenced in this text
Selected international conferences on input–output analysis
12.4
12.5
12.6
12.7
12.8
12.9
12.10
12.11
12.12
12.13
12.14
12.15
12.16
12.17
12.18
12.19
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13.2
13.3
13.4
13.5
13.6
13.7
13.8
13.9
13.10
13.11
13.12
13.13
13.14
13.15
B.1
C.1
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Preface
We started working on the first edition of this book (Miller and Blair, 1985) in the late
1970s. At that time, input–output as an academic topic (outside Wassily Leontief’s
Harvard research group) was a little more than 25 years old – approximately
1952–1979. We use 1952 because that was when the first author was introduced to
input–output analysis in a sophomore-year economics class at Harvard taught by
Robert Kuenne, who later claimed that was the first time input–output had been
included (anywhere) in an undergraduate economics course.
In 1962, the first author joined the faculty of the Regional Science Department at the
University of Pennsylvania (Penn). He was asked by then department chair Walter
Isard to teach the graduate course in linear models for regional analysis; this was to
include a strong input–output component. At that time, coverage of the topic in texts
was to be found primarily in two chapters of Dorfman, Samuelson and Solow (1958),
in Chenery and Clark (1959), in Stone (1961), and in a long chapter on input–output at
the regional level in Isard et al. (1960); later there were texts by Miernyk (1965), Yan
(1969), and Richardson (1972).
The second author of the current text began teaching an applied course covering
extensions of the input–output approach to energy, environmental, and other contemporary policy issues of the time in that same regional science program at Penn in the
early 1970s, and by the end of that decade the need for a comprehensive and up-to-date
textbook became apparent to us. The first edition of this book very much reflected our
shared experiences with students (primarily graduate or undergraduate submatriculants) in mostly regional science and public policy courses at Penn during the 1960s
and 1970s. In addition to the basics (“foundations”), many of the additional topics we
included (“extensions”) reflected our research interests at that time – interregional
feedbacks for one of us, energy and environmental applications for the other, and
spatial aggregation in many-region models as a joint interest.
As input–output analysis developed as a discipline, both in research and practice, the
need for an update to the original 1985 text materialized and the second edition was
published in 2009 (Miller and Blair, 2009). Over the past decade, applications of
input–output to contemporary economic issues matured quickly, and with the pace of
development it became clear that we should consider another new edition to chronicle
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Preface
these developments. We began to take this notion seriously around 2017 – now more
than six decades into the input–output timeline.
In this new edition, we have updated, expanded, and/or refined the discussion of the
following:
• The construction and application of multiregional (MRIO) and interregional (IRIO)
models, including international models and their application to global economic
issues such as climate change and international trade;
• Structural decomposition analysis (SDA), including additive and multiplicative
forms, multiplier matrix decompositions, and structural path analysis;
• Linkages and key sector identification;
• National income and product accounts (NIPAs) and their connection to input–output
accounts;
• Supply and use tables for commodity-by-industry accounting and models;
• Social accounting matrices (SAMs) and their connection to input–output data;
• Location quotients and related techniques for estimating regional technology coefficients with available sources of data;
• Energy and environmental applications including applications to contemporary
issues such as global climate change, embodied energy and pollution in international
trade, and the concept of an environmental footprint using input–output analysis;
• The hypothetical extraction approach to linkage analysis;
• Estimating interregional flows;
• Hybrid methods for estimation of transactions and trade flows; and
• The historical background and context for Leontief’s work.
The historical material on US input–output data has been reworked and updated,
especially to reflect the international movement toward commodity supply and
industry use formulations, and is included as part of the supplementary website
accompanying this text at two levels of aggregation for the convenience of students
and practitioners.
With appreciation we acknowledge many helpful conversations, face-to-face and
electronic, with many colleagues around the world over the years, including Takahiro
Akita, William Beyers, Anne Carter, Dick Conway, Faye Duchin, Geoffrey Hewings,
Takeo Ihara, Satoshi Inomata, Andrew Isserman, Randall Jackson, Louis de Mesnard,
Jan Oosterhaven, Mark Planting, Karen Polenske, Joseph Richter, Jeffery Round, José
Rueda-Cantuche, and Guy West.
Finally, we single out three colleagues with whom we have had almost continuous
interaction for years: Erik Dietzenbacher, with whom we have had literally hundreds of
discussions and from whom we have had as many suggestions; Michael Lahr, who has
been a constant source of critical observations and has recommended and helped us
track down countless important references; and Umed Temursho, to whom we have
turned countless times for suggestions in general and for particular help with decompositions and some of the newer hypothetical extraction results.
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Preface
xxxiii
References
Chenery, Hollis B. and Paul G. Clark. 1959. Interindustry Economics. New York: John Wiley and
Sons.
Dorfman, Robert, Paul A. Samuelson and Robert M. Solow. 1958. Linear Programming and
Economic Analysis. New York: McGraw-Hill.
Isard, Walter, David F. Bramhall, Gerald A. P. Carrothers, John H. Cumberland, Leon N. Moses,
Daniel O. Price and Eugene W. Schooler. 1960. Methods of Regional Analysis: An Introduction
to Regional Science. New York: The Technology Press of MIT and John Wiley and Sons.
Miernyk, William. 1965. The Elements of Input–Output Analysis. New York: Random House.
Miller, Ronald E. and Peter D. Blair. 1985. Input–Output Analysis: Foundations and Extensions.
Englewood Cliffs, NJ: Prentice-Hall.
2009. Input–Output Analysis: Foundations and Extensions (Second Edition). Cambridge:
Cambridge University Press.
Richardson, Harry W. 1972. Input–Output and Regional Economics. New York: John Wiley and
Sons (Halsted Press).
Stone, Richard. 1961. Input–Output and National Accounts. Paris: Organization for Economic
Cooperation and Development.
Yan, Chiou-Shuang. 1969. Introduction to Input–Output Economics. New York: Holt, Rinehart and
Winston.
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1
1.1
Introduction and Overview
Introduction
Input–output analysis is the name given to an analytical framework developed by
Professor Wassily Leontief in the late 1930s, in recognition of which he received the
Nobel Prize in Economic Science in 1973 (Leontief, 1936, 1941). One often speaks of
a Leontief model when referring to input–output. The term interindustry analysis is
also used since the fundamental purpose of the input–output framework is to analyze
the interdependence of industries in an economy. Today the basic concepts set forth by
Leontief are key components of many types of economic analysis and, indeed, input–
output analysis is one of the most widely applied methods in economics (Baumol,
2000). This book develops the framework set forth by Leontief and explores the many
extensions that have been developed over the last 85 years.
In its most basic form, an input–output model consists of a system of linear
equations, each one of which describes the distribution of an industry’s product
throughout the economy. Most of the extensions to the basic input–output framework
are introduced to incorporate additional detail of economic activity, such as over time
or space, to accommodate limitations of available data or to connect input–output
models to other kinds of economic analysis tools. This book is an updated and
considerably expanded edition of our earlier versions of this textbook (Miller and
Blair, 1985, 2009).
In this chapter we introduce the basic input–output analysis framework and outline
the topics to be covered in the balance of the text. Appendix C provides a historical
account of the work leading up to Leontief’s formulation and its subsequent development and refinement. More detailed historical accounts of the early development of
input–output analysis and input–output accounts are given in Polenske and Skolka
(1976, chapter 1) and Stone (1984). An extensive history of applications of input–
output analysis since Leontief’s introduction of it is provided in Rose and Miernyk
(1989). In the present text we cover many of the developments in input–output since its
widespread application as an analysis tool began in the early 1950s. Leontief himself
participated in a number of these developments and applications, as will be evident
throughout this text (see also Polenske, 1999, 2004).
1
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Introduction and Overview
The widespread availability of high-speed digital computers has made Leontief’s
input–output analysis a widely applied and useful tool for economic analysis at many
geographic levels – local, regional, national, and even international. Prior to the
appearance of modern computers, the computational requirements of input–output
models made them difficult and even impractical to implement. Today, in the USA
alone, input–output is routinely applied in national economic analysis by the US
Department of Commerce, and in regional economic planning and analysis by states,
industry, and the research community. The model is widely applied throughout the
world; the United Nations has promoted input–output as a practical planning tool for
developing countries and has sponsored a standardized system of economic accounts
for constructing input–output tables. Also, interactions among nations of the world are
promoted by various international organizations such as the Organization for
Economic Co-operation and Development, the Asian Development Bank, the
Institute of Development Economies, IDE-JETRO, and research consortia, such as
the World Input-Output Database, the Global Multi-region input–output Database
known as EORA, the global multi-regional environmentally extended supply and
use / input–output database known as EXIOBASE, or the emerging collection of
intercountry input–output tables, Full International and Global Accounts for
Research in Input–Output Analysis known as FIGARO, being prepared by the
European Commission agency, Eurostat.
Input–output has also been extended to be part of an integrated framework of
employment and social accounting metrics associated with industrial production and
other economic activity, as well as to accommodate more explicitly such topics as
international and interregional flows of products and services or accounting for energy
consumption and environmental pollution associated with interindustry activity. In this
text, we present the foundations of the input–output model as originally developed by
Leontief, as well as the evolution of many methodological extensions to the basic
framework. In addition, we illustrate many of the applications of input–output and its
usefulness for practical policy questions. Throughout the text, we will review some of
the current research frontiers.
1.2
Input–Output Analysis: The Basic Framework
The basic Leontief input–output model is generally constructed from observed economic data for a specific geographic region (nation, state, county, etc.). One is
concerned with the activity of a group of industries that both produce goods (outputs)
and consume goods from other industries (inputs) in the process of producing each
industry’s own output. In practice, the number of industries considered may vary from
only a few to hundreds or even thousands. For instance, an industrial sector title might
read “manufactured products,” or that same sector might be broken down into many
different specific products.
The fundamental information used in input–output analysis concerns the flows of
products from each industrial sector, considered as a producer, to each of the sectors,
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1.3 Outline for This Text
3
itself and others, considered as consumers. This basic information from which an
input–output model is developed is contained in an interindustry transactions table.
The rows of such a table describe the distribution of a producer’s output throughout the
economy. Each column describes the composition of inputs required by a particular
industry to produce its output. These interindustry exchanges of goods constitute the
shaded portion of Table 1.1. The additional columns, labeled Final Demand, record the
sales by each sector to final markets for their production, such as personal consumption
purchases and sales to the federal government. For example, electricity is sold to
businesses in other sectors as an input to production (an interindustry transaction), and
also to residential consumers (a final-demand sale). The additional rows, labeled Value
Added, account for the other (non-industrial) inputs to production, such as labor,
depreciation of capital, indirect business taxes, and imports.
The formulation of analytical models using the basic input–output data as just
described is the principal purpose of this text. There is considerable literature devoted
to assembling the basic data used in input–output models from surveys or interpretation of other primary and secondary sources of economic data. Some of this literature
is referenced in Chapter 4, but, for the most part, in this text we focus on the
formulation of models using available data or on methods to compensate for the lack
of available data.
1.3
Outline for This Text
This text is organized into 15 chapters, beginning with the theory and assumptions of
the basic input–output framework, then exploring many of the extensions developed
over the last six decades. The text deals mostly with methodological developments, but
also covers some of the practical issues associated with implementation of input–
output models, including many references to the applied literature. Chapters 2–6 cover
the main methodological considerations in input–output analysis. Chapters 7–14 cover
many issues associated with the application of input–output analysis to practical
problems. The concluding chapter, Chapter 15, sketches a range of relevant topics
for which available space did not permit a more detailed treatment or that were
beyond the scope of this text. The following describes the main topics covered in each
chapter:
• Chapter 2 introduces Leontief’s conceptual input–output framework and explains
how to develop the fundamental mathematical relationships from the interindustry
transactions table. The key assumptions associated with the basic Leontief model
and implications of those assumptions are recounted and the economic interpretation
of the basic framework is explored. The basic framework is illustrated with a highly
aggregated model of the US economy. In addition, the “price model” formulation of
the input–output framework is introduced to explore the role of prices in input–
output models. Appendices (some of which appear in the supplementary online
resources for this text, http://www.cambridge.org/millerandblair) to this chapter
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Table 1.1 Input–output transactions table
Industry Producers as Consumers
Agric.
Industry
Producers
Mining
Const.
Manuf.
Trade
Transp.
Final Demand for Goods and Services
Services
Other
Personal
Consumption
Expenditures
Gross
Private
Domestic
Investment
Government
Purchases of
Goods &
Services
Agriculture
Mining
Construction
Manufacturing
Trade
Transportation
Services
Other Industry
Value
Added
Employees
Employee Compensation
Business Owners
and Capital
Government
Profit-type income and capital consumptions allowances
Indirect Business Taxes
Gross Domestic Product
Net
Exports of
Goods
and
Services
1.3 Outline for This Text
5
include a fundamental set of mathematical conditions for input–output models,
known as the Hawkins–Simon conditions.
• Chapter 3 extends the basic input–output framework to analysis of regions and the
relationships between regions. First, “single-region” models are presented and the
various assumptions employed in formulating regional models versus national
models are explored. Next, the structure of an interregional input–output (IRIO)
model, designed to expand the basic input–output framework to capture transactions
between industrial sectors in regions, is presented. An important simplification of the
IRIO model designed to deal with the most common of data limitations in constructing such models is known as the multiregional input–output (MRIO) model. The
basic MRIO formulation is presented and the implications of the simplifying
assumptions explored along with the balanced regional model which captures the
distinction between industrial production for regional versus national markets.
Finally, the chapter summarizes the fast-growing range of applications of MRIO to
multinational and global economic models and issues. Several appendices to this
chapter provide additional development of mathematical tools helpful for conceptualizing and implementing regional models, as well as additional mathematical
refinements for MRIO and balanced regional models. Additional appendices explore
more detailed MRIO examples.
• Chapter 4 deals with the construction of input–output tables from standardized
conventions of national economic accounts, such as the widely-used System of
National Accounts (SNA) promoted by the United Nations, including a basic
introduction to the so-called commodity-by-industry or supply–use input–output
framework developed in additional detail in Chapter 5. A simplified SNA is derived
from fundamental economic concepts of the circular flow of income and expenditure, that, as additional sectoral details are defined for businesses, households,
government, foreign trade, and capital formation, ultimately results in the basic
commodity-by-industry formulation of input–output accounts. The process is illustrated with the US input–output model and some of the key traditional conventions
widely applied for such considerations as secondary production (multiple products
or commodities produced by a business), competitive imports (commodities that are
also produced domestically) versus non-competitive imports (commodities not produced domestically), trade and transportation margins on interindustry transactions,
or the treatment of scrap and secondhand goods. Finally, the chapter concludes with
an examination of issues associated with the level of sectoral and spatial detail in
input–output models, e.g., the potential bias introduced by the level of aggregation of
industries or regions. Appendices to this chapter illustrate the implications of
aggregation bias using IRIO and MRIO models for Japan and the USA.
• Chapter 5 explores variations to the commodity-by-industry input–output framework
introduced in Chapter 4, expanding the basic input–output framework to include
distinguishing between commodities and industries, i.e., the supply of specific
commodities in the economy and the use of those commodities by collections of
businesses defined as industries. The chapter introduces the fundamental
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Introduction and Overview
commodity-by-industry accounting relationships and how they relate to the basic
input–output framework. Alternative assumptions are defined for handling the
common accounting issue of secondary production, and economic interpretations
of those alternative assumptions are presented. The formulations of commoditydriven and industry-driven models are also presented, along with illustrations of
variants on combining alternative assumptions for secondary production. Finally, the
chapter illustrates the problem encountered with commodity-by-industry models,
such as non-square commodity–industry systems, mixed technology options, or the
interpretation of negative elements. Appendices to this chapter provide some alternative derivations of commodity-by-industry transactions matrices, methods for
eliminating negative entries in specific types of commodity-by-industry models
where appearance of such entries is most common, and additional observations on
non-square commodity-by-industry systems.
• Chapter 6 examines key summary analytical measures known as multipliers that can
be derived from input–output models to estimate the effects of exogenous changes
on (1) new outputs of economic sectors, (2) income earned by households resulting
from new outputs, and (3) employment generated from new outputs, or (4) valueadded generated by production. The general structure of multiplier analysis and
special considerations associated with regional, IRIO, and MRIO models are
developed. Extensions to capture the effects of income generation for various
household groups are explored, as well as additional multiplier variants and decomposition into meaningful economic components. Chapter appendices expand on
mathematical formulations of household and income multipliers.
• Chapter 7 presents the so-called supply side input–output model. It is discussed both
as a quantity model (the early interpretation) and as a price model (the more modern
interpretation). Relationships to the standard Leontief quantity and price models are
also explored. In addition, the fast-growing literature on quantification of economic
linkages and analysis of the overall structure of economies using input–output data is
examined. Finally, approaches for identifying key or important coefficients in input–
output models and alternative measures of coefficient importance are presented.
• Chapter 8 introduces and illustrates the basic concepts of structural decomposition analysis
(SDA) within an input–output framework. The concept of decomposition of multipliers,
introduced in Chapter 6, is revisited later in Chapter 11 applied to Social Accounting
Matrices (SAMs) as a way to analyze economic structure. The application of SDA to
MRIO is developed to introduce a spatial context. The chapter explores many applications
and summaries of their results. Appendices (some of which appear in the supplementary
online resources for this text, http://www.cambridge.org/millerandblair) to this chapter
develop extended presentations of additional decomposition results as well as an overview
of early applied studies and some further mathematical results.
• Chapter 9 introduces approaches designed to deal with the major challenge in input–
output analysis that the kinds of information-gathering surveys needed to collect
input–output data for an economy can be expensive and very time consuming,
resulting in tables of input–output coefficients that are outdated before they are
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1.3 Outline for This Text
7
produced. These techniques, known as partial survey and nonsurvey approaches to
input–output table construction, are central to modern applications of input–output
analysis. The chapter begins by reviewing the basic factors contributing to the
stability of input– output data over time, such as changing technology, prices, and
the scale and scope of business enterprises. Several techniques for updating input–
output data are developed, and the economic implications of each described. The
bulk of the chapter is concerned with the widely utilized biproportional scaling (or
RAS) technique and some related “hybrid model” variants.
• Chapter 10 surveys a range of partial survey and nonsurvey estimation approaches
for creating input–output tables at the regional level. Variants of the commonly used
class of estimating procedures using location quotients are reviewed; these presume
a regional estimate of input–output data can be derived using some information
about a target region. The RAS technique developed in Chapter 9 is applied using a
base national table or a table for another region and some available data for the target
region. Techniques for partial survey estimation of commodity flows between
regions are also presented, along with discussions of several real-world multinational
applications, including the China–Japan Transnational Interregional Model, Leontief’s
World Model, and models from the Global MRIO Lab.
• Chapter 11 expands the input–output framework to a broader class of economic
analysis tools known as social accounting matrices (SAM) and other so-called
extended input–output models to capture activities of income distribution in the
economy in a more comprehensive and integrated way, including especially employment and social welfare features of an economy. The basic concepts of SAMs are
explored and derived from the SNA introduced in Chapters 4 and 5, and the
relationships between SAMs and input–output accounts are presented. The concept
of SAM multipliers as well as the decomposition of SAM multipliers into components with specific economic interpretations are introduced and illustrated. Finally,
techniques for balancing SAM accounts for internal accounting consistency are
discussed, and several illustrative applications of the use of SAMs are presented.
• Chapter 12 explores the extension of the input–output framework to more detailed
analysis of energy consumption associated with industrial production, including
some of the complications that can arise when measuring input–output transactions
in physical units of production rather than in monetary terms of the value of
production. The chapter reviews early efforts to develop energy input–output analysis, compares them with contemporary approaches, and examines the strengths and
limitations of the alternatives commonly used today. Special methodological considerations such as adjusting for energy conversion efficiencies are developed along
with several illustrative applications, including estimation of the energy costs of
goods and services, impacts of new energy technologies, and energy taxes. Finally,
the role of structural change of an input–output economy associated with changing
patterns of energy use is illustrated, building on the more general approaches to
structural decomposition analysis using input–output models developed in
Chapter 8. An appendix to this chapter, included in the supplementary online
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8
Introduction and Overview
resources for this text (http://www.cambridge.org/millerandblair), develops the
strengths and limitations of alternative energy input–output formulations in more detail.
• Chapter 13 reviews the extensions of the input–output framework to incorporate
activities of environmental pollution and elimination associated with economic activities as well as the linkages of input–output to models of ecosystems. The chapter
begins with the augmented Leontief model for incorporating pollution generation and
elimination, from which many subsequent approaches have been developed. The
chapter then describes the now widespread application of input–output analysis to
environmental lifecycle assessment and establishing a “pollution footprint” for industrial activity. The special case of analyzing the relationship between global climate
change and industrial activity with a carbon footprint is then explored along with using
input–output to attribute pollution generation to the demands driving consumption
compared with the more traditional attribution of pollution generation to the sectors of
industrial production necessary to meet that demand. The chapter continues with a
“generalized” input–output framework which assumes that pollution generation (as
well as other measurable factors associated with industrial production, such as energy
or material consumption measured in physical units or employment measured in
person-years) simply vary in direct proportion to the level of industrial production.
Applications are presented of the generalized input–output formulation to measuring
impacts of specified changes to industrial activity and to planning problems where the
objective is to seek an optimal mix of industrial production subject to input–output
relationships between industrial sectors and to constraints on factors associated with
industrial production, such as pollution, energy use, and employment. In exploring the
application of the generalized input–output framework to planning problems, basic
concepts of linear and multi objective programming are introduced. Finally, expansion of the input–output framework to include ecologic sectors as a means to trace
more comprehensively economic–ecosystem relationships is presented along with a
variety of illustrative applications.
• Chapter 14 describes so-called mixed input–output models that are driven by a mix
of output and final demand specifications rather than driven either solely by specification by final demand or total output. This chapter also introduces dynamic input–
output models that more explicitly capture the role of capital investment and utilization in the production process.
• Chapter 15 briefly describes some additional extensions to input–output analysis for
which space does not permit a detailed treatment in this text, including measuring
total factor productivity, modeling economic impacts of disasters, the inoperability
input–output model, accounting for alternative technologies, and linkages to econometric or computable general equilibrium models.
• Appendix A is an introductory review of matrix algebra concepts and methods used
throughout this text, including matrix operations such as addition, multiplication,
transposition, inversion, and partitioning.
• Appendix B summarizes a highly aggregated series of the US input–output tables
referenced and used in exercise problems in a number of chapters or in the exercise
problems and solutions associated with each chapter. The data, exercise problems,
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References
9
and solutions, and a number of other data sets referenced throughout this text are
included in supplementary appendices included on the Internet website associated with
this book (http://www.cambridge.org/millerandblair) and summarized in Section 1.4.
• Appendix C provides an historical account of the early development of input–output
analysis, including a “pre-history” of the concepts that led to Leontief’s work as well
as the many methodological developments and applications since.
1.4
Internet Website and Text Locations of Real Datasets
A website associated with this text, http://www.cambridge.org/millerandblair, includes
supplementary appendices in three general areas: (1) additional text (appendices) in
selected areas that were not possible to include in the printed text for a variety of
reasons, (2) exercise problems and solutions aligned with chapters of this text along
with a computational workbook providing expanded discussion of the problems and
solutions, and (3) downloadable datasets of many of the examples and problems
printed in the text, as well as a library of supplementary real but highly aggregated
datasets referenced throughout this text for various regions and nations as well as
illustrative interregional input–output (IRIO) and multiregional input–output (MRIO)
data and social accounting matrices (SAM). As noted, additional details about these
supplemental resources are included in Appendix B.
References
Baumol, William. 2000. “Leontief’s Great Leap Forward,” Economic Systems Research, 12,
141–152.
Leontief, Wassily. 1936. “Quantitative Input–Output Relations in the Economic System of the United
States,” Review of Economics and Statistics, 18, 105–125.
1941. The Structure of American Economy 1919–1939. New York: Oxford University Press.
Miller, Ronald E. and Peter D. Blair. 1985. Input–Output Analysis: Foundations and Extensions.
Englewood Cliffs, NJ: Prentice-Hall.
2009. Input–Output Analysis: Foundations and Extensions (Second Edition). Cambridge:
Cambridge University Press.
Polenske, Karen R. 1999. “Wassily W. Leontief, 1905–1999,” Economic Systems Research, 11,
341–348.
2004. “Leontief’s ‘Magnificent Machine’ and Other Contributions to Applied Economics,” in Erik
Dietzenbacher and Michael L. Lahr (eds.), Wassily Leontief and Input–Output Economics. New
York: Cambridge University Press, pp. 9–29.
Polenske, Karen R. and Jiří V. Skolka (eds.). 1976. Advances in Input–Output Analysis. Proceedings
of the Sixth International Conference on Input–Output Techniques. Vienna, April 22–26, 1974.
Cambridge, MA: Ballinger.
Rose, Adam and William Miernyk. 1989. “Input–Output Analysis: The First Fifty Years,” Economic
Systems Research, 1, 229–271.
Stone, Richard. 1984. “Where Are We Now? A Short Account of Input–Output Studies and Their
Present Trends,” in United Nations Industrial Development Organization (UNIDO),
Proceedings of the Seventh International Conference on Input–Output Techniques. New York:
United Nations, pp. 439–459. [Reprinted in Ira Sohn (ed.). 1986. Readings in Input-Output
Analysis. New York: Oxford University Press, pp. 13–31.]
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2
2.1
Foundations of Input–Output
Analysis
Introduction
In this chapter we begin to explore the fundamental structure of the input–output
model, the assumptions behind it, and some of the simplest kinds of problems to which
it is applied. Later chapters will examine the special features that are associated with
regional models and some of the extensions that are necessary for particular kinds of
problems – for example, in energy or environmental studies or as part of a broader
system of social accounts.
The mathematical structure of an input–output system consists of a set of n linear
equations with n unknowns; therefore, matrix representations can readily be used. In
this chapter we will start with more detailed algebraic statements of the fundamental
relationships and then go on to use matrix notation and manipulations more and more
frequently. Appendix A contains a review of matrix algebra definitions and operations that are essential for input–output models. While solutions to the input–output
equation system, via an inverse matrix, are straightforward mathematically, we will
discover that there are interesting economic interpretations to some of the algebraic
results.
2.2
Notation and Fundamental Relationships
An input–output model is constructed from observed data for a particular economic
area – a nation, a region (however defined), a state, etc. In the beginning, we will
assume (for reasons that will become clear in Chapter 3) that the economic area is a
country. The economic activity in the area must be able to be separated into a
number of segments or producing sectors. These may be industries in the usual
sense (e.g., steel) or they may be much smaller categories (e.g., steel nails, and
spikes) or much larger ones (e.g., manufacturing). The necessary data are the flows
of products from each of the sectors (as a producer/seller) to each of the sectors (as
a purchaser/buyer); these interindustry flows, or transactions (or intersectoral
flows – the terms industry and sector are often used interchangeably in input–
output analysis) are measured for a particular time period (usually a year) and in
10
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2.2 Notation and Fundamental Relationships
11
monetary terms – for example, the dollar value of steel sold to automobile manufacturers last year.1
The exchanges of goods between sectors are, ultimately, sales and purchases of
physical goods – tons of steel bought by automobile manufacturers last year. In
accounting for transactions between and among all sectors, it is possible in principle to
record all exchanges either in physical or in monetary terms. While the physical measure
is perhaps a better reflection of one sector’s use of another sector’s product, there are
substantial measurement problems when sectors actually sell more than one good (a
Lexus ES 350 and a Toyota Corolla are distinctly different products with different prices;
in physical units, however, both are cars). For these and other reasons, then, accounts are
generally kept in monetary terms, even though this introduces problems due to changes
in prices that do not reflect changes in the use of physical inputs. (In Section 2.6 we will
explore the implications of a data set in which transactions are expressed in physical
units – for example, tons of steel sold to the automobile sector last year.)
One essential set of data for an input–output model are monetary values of the
transactions between pairs of sectors (from each sector i to each sector j); these are
often designated as zij . Sector j’s demand for inputs from other sectors during the year
will have been related to the amount of goods produced by sector j over that same
period. For example, the demand from the automobile sector for the output of the steel
sector is very closely related to the output of automobiles, the demand for leather by
the shoe-producing sector depends on the number of shoes being produced, etc.
In addition, in any country there are sales to purchasers who are more external or
exogenous to the industrial sectors that constitute the producers in the economy – for
example, households, government, and foreign trade. The demands of these units – and
hence the magnitudes of their purchases from each of the industrial sectors – are
generally determined by considerations that are relatively unrelated to the amount
being produced. For example, government demand for aircrafts is related to broad
changes in national policy, budget levels, or defense needs; consumer demand for
small cars is related to gasoline availability, and so on. The demand of these external
units, since it tends to be much more for goods to be used as such and not to be used as
an input to an industrial production process, is generally referred to as final demand.
Assume that the economy can be categorized into n sectors. If we denote by xi the
total output (production) of sector i and by fi the total final demand for sector i’s
product, we may write a simple equation accounting for the way in which sector i
distributes its product through sales to other sectors and to final demand:
xi ¼ zi1 þ þ zij þ þ zin þ fi ¼
n
X
zij þ fi
(2.1)
j¼1
1
In Chapters 4 and 5 we will explore more recent distinctions between “commodities” and “industries” and see
how these observations lead to alternative representations of the input–output model.
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Foundations of Input–Output Analysis
The zij terms represent interindustry sales by sector i (also known as intermediate
sales) to all sectors j (including itself, when j ¼ i). Equation (2.1) represents the
distribution of sector i output. There will be an equation like this that identifies sales
of the output of each of the n sectors:
x1 ¼ z11 þ þ z1j þ þ z1n þ f1
..
.
xi ¼ zi1 þ þ zij þ þ zin þ fi
..
.
(2.2)
xn ¼ zn1 þ þ znj þ þ znn þ fn
Let
2
3
x1
6 . 7
x ¼ 4 .. 5,
xn
2
z11
6 ..
Z ¼4 .
zn1
3
2 3
f1
z1n
6 .. 7
..
.. 7
. . 5 and f ¼ 4 . 5
znn
fn
(2.3)
Here and throughout this text we use lower-case bold letters for (column) vectors, as in
f and x (so x0 is the corresponding row vector), and upper case bold letters for matrices,
as in Z. With this notation, the information in (2.2) on the distribution of each sector’s
sales can be compactly summarized in matrix notation as
x ¼ Zi þ f
(2.4)
where i represents a column vector of 1’s (of appropriate dimension – here n). This is
known as a “summation” vector (Appendix A, Section A.8). The important observation is that post-multiplication of a matrix by i creates a column vector whose elements
are the row sums of the matrix. Similarly, i0 is a row vector of 1’s, and premultiplication
of a matrix by i0 creates a row vector whose elements are the column sums of the
matrix. We will use summation vectors often in this and subsequent chapters.
Consider the information in the jth column of z’s on the right-hand side:
2 3
z1j
6 .. 7
6 . 7
6 7
6 zij 7
6 7
6 . 7
4 .. 5
znj
These elements are sales to sector j – j’s purchases of the products of the various producing
sectors in the country; the column thus represents the sources and magnitudes of sector j’s
inputs. Clearly, in engaging in production, a sector also pays for other items – for example,
labor and capital – and uses other inputs as well, such as inventoried items. Together, all of
these primary inputs are termed the value added in sector j. In addition, imported goods
may be purchased as inputs by sector j. All of these inputs (value added and imports) are
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13
2.2 Notation and Fundamental Relationships
often lumped together as purchases from what is called the payments sector, whereas the
z’s on the right-hand side of (2.2) serve to record the purchases from the processing sector,
the interindustry inputs (or intermediate inputs). Since each equation in (2.2) includes the
possibility of purchases by a sector of its own output as an input to production, these
interindustry inputs may include intraindustry transactions as well.
The magnitudes of these interindustry flows can be recorded in a table, with sectors
of origin (producers) listed on the left and the same sectors, now destinations (purchasers), listed across the top. From the column point of view, these show each sector’s
inputs; from the row point of view, the figures are each sector’s outputs; hence the
name input–output table. These figures are the core of the input–output model and of
input–output analysis.
2.2.1 Input–Output Transactions and National Accounts
As was suggested by Table 1.1, an input–output transactions (flow) table, such as that
shown in Table 2.1, constitutes part of a complete set of income and product accounts
for an economy. To emphasize the other elements in a full set of accounts, we consider
a small, two-sector economy. We present an expanded flow table for this extremely
simple economy in Table 2.2. (We examine more of the details of a system of national
accounts in Chapter 4.)
Table 2.1 Input–output table of interindustry flows of goods
Buying Sector
Selling Sector
1
..
.
i
..
.
n
1
...
j
...
n
z
.. 11
.
z
.. i1
.
zn1
...
z
.. 1j
.
z
.. ij
.
znj
...
z1n
..
.
zin
..
.
znn
...
...
...
...
Table 2.2 Expanded flow table for a two-sector economy
Processing Sectors
Processing
Sectors
Payments
Sectors
Total Outlays (x0 )
1
2
Labor
Other Value Added
Imports
1
2
z11
z21
l1
n1
m1
x1
z12
z22
l2
n2
m2
x2
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Final Demand
c1
c2
lC
nC
mC
C
i1
i2
lI
nI
mI
I
g1
g2
lG
nG
mG
G
e1
e2
lE
nE
mE
E
Total
Output (x)
x1
x2
L
N
M
X
14
Foundations of Input–Output Analysis
The component parts of the final demand vector for sectors 1 and 2 represent,
respectively, consumer (household) purchases, purchases for (private) investment
purposes, government (federal, state, and local) purchases, and sales abroad (exports).
These are often grouped into domestic final demand (C þ I þ G) and foreign final
demand (E). Then f1 ¼ c1 þ i1 þ g 1 þ e1 and similarly f2 ¼ c2 þ i2 þ g 2 þ e2.
The component parts of the payments sector are payments by sectors 1 and 2 for
employee compensation (labor services, l1 and l 2 ) and for all other value-added items –
for example, government services (paid for in taxes), capital (interest payments), land
(rental payments), entrepreneurship (profit), and so on. Denote these other value-added
payments by n1 and n2 ; then total value-added payments are v1 ¼ l 1 þ n1 and
v2 ¼ l2 þ n2 for the two sectors.
Finally, assume that some (or perhaps all) sectors use imported goods in producing
their outputs. One approach is to record these import amounts in an imports row in the
payments sector as m1 and m2 .2 Total expenditures in the payments sector by sectors
1 and 2 are l1 þ n1 þ m1 ¼ v1 þ m1 and l 2 þ n2 þ m2 ¼ v2 þ m2 , respectively.
However, it is often the case that the exports part of the final demand column is
expressed as net exports so that the sum of all final demands is equal to traditional
definitions of gross domestic product, i.e., net of imports. In that case a distinction is
often made between imports of goods that are also domestically produced (competitive
imports) and those for which there is no domestic source (non-competitive imports),
and all the competitive imports in the imports row will have been netted out of the
appropriate elements in a gross exports column. Under these circumstances it is
possible for one or more elements in the net export column to be negative, if the value
of imports of those goods exceeds the value of exports. (For example, if an economy
exported €300 million of agricultural products last year but imported €350 million, the
net exports figure for the agricultural sector would be €-50 million.) Also, if the federal
government sells more of a stockpiled item (e.g., wheat) than it buys, a negative entry
in the government column of the final demand part of the table could result. If the
negative number is large enough, it could swamp the other (positive) final demand
purchases of that good, leaving a negative total final demand figure.
The elements in the intersection of the value-added rows and the final demand
columns represent payments by final consumers for labor services (for example, l C
includes household payments for, say, domestic help; l G represents payments to
government workers) and for other value added (for example, nC includes tax payments by households). In the imports row and final demand columns are, for example,
mG , which represents government purchases of imported items, and mE , which represents imported items that are re-exported.
Summing down the total output column, total gross output throughout the economy,
X, is found as
2
The treatment of imports in input–output accounts is much more complicated than this, but for the present we
prefer to concentrate on the overall structure of a transactions table. We return to imports in Section 2.3.4, and in
more detail in Chapter 4.
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2.2 Notation and Fundamental Relationships
15
X ¼ x1 þ x2 þ L þ N þ M
This same value can be found by summing across the total outlays row; namely
X ¼ x1 þ x2 þ C þ I þ G þ E
These are simply two alternative ways of summing all the elements in the table.
In national income and product accounting, it is the value of total final product that
is of interest – goods available for consumption, export, and so on. Equating the two
expressions for X and subtracting x1 and x2 from both sides leaves
LþM þN ¼CþI þGþE
or
L þ N ¼ C þ I þ G þ ðE M Þ
The left-hand side represents gross national income – the total factor payments in the
economy – and the right-hand side represents the total spent on consumption and
investment goods, total government purchases, and the total value of net exports from
the economy. The sum of each side, gross income and total final demand, is the
nation’s gross domestic product (GDP).3 Again, national accounts are examined in
more detail in Chapter 4.
In most developed economies, consumption is the largest individual component of
final demand. For example, in the USA in 2018 the percentages of total final demand
were as follows: personal consumption expenditure (PCE), 60 percent; gross private
domestic investment (including producers’ durable equipment, plant construction,
residential construction, and net inventory change), 15 percent; government purchases
(federal, state and local), 15 percent; foreign exports, 9 percent. [However, in the USA
during the 1942–1945 period (World War II), PCE was between 40 and 48 percent and
for much of the 1950s and 1960s it was under 60 percent.]
2.2.2 Production Functions and the Input–Output Model
In input–output work, a fundamental assumption is that the interindustry flows from i
to j – recall that these are for a given period, say a year – depend entirely on the total
output of sector j for that same time period. Clearly, no one would argue against the
idea that the more cars produced in a year, the more steel will be needed during that
year by automobile producers. Where argument does arise is over the exact nature of
this relationship. In input–output analysis it is as follows: Given zij and xj – for
example, input of aluminum (i) bought by aircraft producers (j) last year and total
3
GDP is the total national income or total final demand for domestic economic activity. Another common
measure for national output and income is the gross national product (GNP), which includes net income receipts
from other countries. For example, if a Japanese-based multinational enterprise produces cars in the USA, the
value of this production would be counted toward the US GDP. If the Japanese multinational sends profits back
to shareholders in Japan, then these profits would be a reduction in the US GNP.
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Foundations of Input–Output Analysis
aircraft production last year – form the ratio of aluminum input to aircraft output, zij =xj
[the units are ($/$)], and denote it by aij :
aij ¼
zij value of aluminum bought by aircraft producers last year
¼
xj
value of aircraft production last year
(2.5)
This ratio is called a technical coefficient; the terms input–output coefficient and
direct input coefficient are also often used. For example, if z14 ¼ $300 and
x4 ¼ $15,000 (sector 4 used $300 of goods from sector 1 in producing $15,000 of
sector 4 output), a14 ¼ z14 =x4 ¼ $300=$15,000 ¼ 0:02. Since a14 is actually
$0:02=$1, the 0.02 is interpreted as the “dollars’ worth of inputs from sector 1 per
dollar’s worth of output of sector 4.”
From (2.5), aij xj ¼ zij . This is trivial algebra, but it presents the operational form in
which the technical coefficients are used. In input–output analysis, once a set of
observations has given us the result a14 ¼ 0:02, this technical coefficient is assumed
to be unchanging in the sense that if one asked how much sector 4 would buy from
sector 1 if sector 4 were to produce a total output (x4 ) of $45,000, the input–output
answer would be z14 ¼ a14 x4 ¼ ð0:02Þð$45,000Þ ¼ $900 when output of sector 4 is
tripled, the input from sector 1 is tripled. The aij are viewed as measuring fixed
relationships between a sector’s output and its inputs. Economies of scale in production are thus ignored; production in a Leontief system operates under what is known as
constant returns to scale.
In addition, input–output analysis requires that a sector use inputs in fixed
proportions. Suppose, to continue the previous example, that sector 4 also buys inputs
from sector 2, and that, for the period of observation, z24 ¼ $750. Therefore
a24 ¼ z24 =x4 ¼ $750=$15,000. For x4 ¼ $15,000, inputs from sector 1 and from
sector 2 were used in the proportion p12 ¼ z14 =z24 ¼ $300=$750 ¼ 0:4. If x4 were
$45,000, z24 would be (0.05)($45,000) ¼ $2,250; since z14 ¼ $900 for x4 ¼ $45,000,
the proportion between inputs from sector 1 and from sector 2 is $900=$2,250 ¼ 0:4,
as before. This reflects the fact that
p12 ¼ z14 =z24 ¼ a14 x4 =a24 x4 ¼ a14 =a24 ¼ 0:02=0:05 ¼ 0:4
This proportion is the ratio of the technical coefficients, and, since the coefficients are
fixed, the input proportion is fixed.
For the reader with some background in basic microeconomics, we can identify the
form of production function inherent in the input–output system and compare it with
that in the general neoclassical microeconomic approach. Production functions relate
the amounts of inputs used by a sector to the maximum amount of output that could be
produced by that sector with those inputs. An illustration is
xj ¼ f z1j , z2j ; . . . , znj , vj , mj
Using the definition of the technical coefficients in (2.5), we can see that in the
Leontief model this becomes
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2.2 Notation and Fundamental Relationships
xj ¼
17
z1j z2j
znj
¼
¼ ¼
a1j a2j
anj
(This ignores, for the moment, the contributions of vj and mj .)
A problem with this extremely simple formulation is that it is meaningless if a
particular input i is not used in production of j, since then aij ¼ 0 and hence zij =aij is
infinitely large. Thus, the more usual specification of the kind of production function
that is embodied in the input–output model is
z1j z2j
znj
; ;;
xj ¼ min
a1j a2j
anj
where min (x, y, z) denotes the smallest of the numbers x, y, and z. In the input–output
model, for those aij coefficients that are not zero, these ratios will all be the same, and equal
to xj – from the fundamental definition of aij in (2.5). For those aij coefficients that are
zero, the ratio zij =aij will be infinitely large and hence will be overlooked in the process of
searching for the smallest among the ratios. This specification of the production function
in the input–output model reflects the assumption of constant returns to scale; multiplication of z1j , z2j , . . . , znj by any constant will multiply xj by the same constant. (Tripling all
inputs will triple output; cutting inputs in half will halve output, etc.)
For the reader who is acquainted with the economist’s production function geometry,
we show four alternative representations of production functions in input space for a
two-sector economy in Figure 2.1. A linear production function, depicted in Figure 2.1
(a), assumes that output is a simple linear function of inputs, which means that the inputs
are infinitely substitutable for each other for any level of output. The figure shows a set of
isoquants (constant output lines) depicting higher and higher levels of output.
A classical production function, depicted in Figure 2.1(b), also shows a set of
isoquants (now constant output curves) depicting higher and higher levels of output.
For a given value of z1j in Figure 2.1(b), increasing z2j leads to increases in xj –
intersections with higher-value isoquants. In this case input substitution is also possible
but not linearly, as indicated by the isoquants showing alternative input combinations
that generate the same level of output. For example, moving rightward along a particular
isoquant in Figure 2.1(b) can be accomplished by reducing the amount of input 2 and
increasing the amount of input 1, or leftward by reducing z1j and increasing z2j .
The shape of the isoquants in Figure 2.1(b) reflects two specific classical assumptions
about how inputs are combined to produce outputs. The negative slopes of the isoquants
represent the fact that as the amount of one input is decreased, the amount of the other
input must be increased in order to maintain the level of production indicated by a
specific isoquant. The fact that the curves bulge toward the origin (mathematically, their
convexity) reflects the economist’s law of diminishing marginal productivity.4 The
4
From basic microeconomics concepts, recall that the slope of an isoquant (assuming that these are smooth
functions) at any point is the ratio of the marginal productivities of inputs 1 and 2. These marginal productivities, in turn, are the partial derivatives of the production function (also assumed smooth) with respect to each of
=∂x1
the inputs – thus the slope is ∂f
∂f =∂x2 . As we move rightward along an isoquant, the amount of input 2 used
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18
Foundations of Input–Output Analysis
(a)
(b)
(c)
(d)
Figure 2.1 Production functions in input space. (a) Linear production function. (b) Classical
production function. (c) Leontief production function. (d) Activity analysis production function
“expansion path” representing input combinations that are used for various levels of
output is a curve from the origin through the points of tangency between isocost
(constant cost) lines – dashed in Figure 2.1(b) – and the isoquants.
In the Leontief model, the isoquant “curves” of constant output appear as in
Figure 2.1(c). Once the observed proportion of inputs 1 and 2 is known, as
p12 ¼ z1j =z2j , then additional amounts of either input 1 or input 2 alone are useless
from the point of view of increasing the output of j. Only when availabilities of both
input 1 and input 2 are increased can xj increase; and only if the amounts of increase
of 1 and 2 are in the proportion p12 will all the available amounts of both be used up.
decreases and the amount of input 1 used increases. By diminishing marginal productivity, then, ∂f =∂x1
decreases and ∂f =∂x2 increases; hence the slope decreases, as is true for the isoquants in Figure 2.1(b).
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2.2 Notation and Fundamental Relationships
Of course, the “true” geometric representation should be in n-dimensional input
space, with a separate axis for each of the n possible inputs, but the principles are
the same when only two inputs are considered. From the Leontief production
function, if z1j , z2j , . . . , zðn1Þj were all doubled but znj were only increased by
50 percent (multiplied by 1.5), then the minimum of the new ratios would be
znj =anj and the new output of sector j would be 50 percent larger. There would be
excess and unused amounts of inputs from sectors 1, 2, . . ., (n 1). But since inputs
are not free goods, sector j will not buy more from any sector than is needed for its
production, and thus the input combinations chosen by sector j will lie along the ray,
as represented in Figure 2.1(c). In short, Leontief production functions require inputs
in fixed proportions where a fixed amount of each input is required to produce one
unit of output.
Figure 2.1(d) shows an activity analysis production function, which is a generalization of the Leontief production function and is a piece-wise linear approximation of the
classical production function. Each isoquant is represented by a connected set of line
segments. Each segment is a linear production function applicable over a limited range
of combinations of inputs to produce a given level of output.
Once the notion of a set of fixed technical coefficients is accepted, (2.2) can be
rewritten, replacing each zij on the right by aij xj :
x1 ¼ a11 x1 þ þ a1i xi þ þ a1n xn þ f1
..
.
xi ¼ ai1 x1 þ þ aii xi þ þ ain xn þ fi
..
.
(2.6)
xn ¼ an1 x1 þ þ ani xi þ þ ann xn þ fn
These equations serve to make explicit the dependence of interindustry flows on the
total outputs of each sector. They also bring us closer to the form needed in input–
output analysis, in which the following kind of question is asked: If the demands of the
exogenous sectors were forecast to be some specific amounts next year, how much
output from each of the sectors would be necessary to supply these final demands?
From the point of view of this equation, the f1 , . . . , fn are known numbers, the aij are
known coefficients, and the x1 , . . . , xn are to be found. Therefore, bringing all x terms
to the left,
x1 a11 x1 a1i xi a1n xn ¼ f1
..
.
xi ai1 x1 aii xi ain xn ¼ fi
..
.
xn an1 x1 ani xi ann xn ¼ fn
and, grouping the x1 together in the first equation, the x2 in the second, and so on,
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Foundations of Input–Output Analysis
ð1 a11 Þx1 a1i xi a1n xn ¼ f1
..
.
ai1 x1 þ ð1 aii Þxi ain xn ¼ fi
..
.
(2.7)
an1 x1 ani xi þ ð1 ann Þxn ¼ fn
These relationships can be represented compactly in matrix form. In matrix algebra
notation, a “hat” over a vector denotes a diagonal matrix with the elements of the
2
3
x1 0
6 . ..
. 7
vector along the main diagonal, so, for example, x^ ¼ 4 ..
. .. 5. From the basic
0 xn
2
3
1=x1 0
6
..
.. 7.
definition of an inverse, ðx^Þðx^Þ1 ¼ I, it follows that x^1 ¼ 4 ...
.
. 5
0
1=xn
^ creates a matrix
Also, postmultiplication of a matrix, M, by a diagonal matrix, d,
^ (Appendix A, Section
in which each element in column j of M is multiplied by d j in d
A.7). Therefore, the n n matrix of technical coefficients can be represented as
A ¼ Zx^1
(2.8)
Using the definitions in (2.3) and (2.8), the matrix expression for (2.6) is
x ¼ Ax þ f
(2.9)
2
3
ð1 a11 Þ
a12
a1n
6 a21
ð1 a22 Þ a2n 7
6
7
Let I be an n n identity matrix, then ðI AÞ ¼6
7
..
..
..
..
4
5
.
.
.
.
an1
an2
ð1 ann Þ
and the complete n n system shown in (2.7) is just5
ðI AÞx ¼ f
(2.10)
For a given set of f ’s, this is a set of n linear equations in the n unknowns,
x1 , x2 , . . . , xn , and hence it may or may not be possible to find a unique solution. In
fact, whether or not there is a unique solution depends on whether or not (I A) is
singular; that is, whether or not ðI AÞ1 exists. The matrix A is known as the
technical (or direct input) coefficients matrix. From the basic definition of an inverse
6 0,
for a square matrix (Appendix A), ðI AÞ1 ¼ ð1=jI AjÞ½adjðI AÞ. If jI Aj ¼
5
This is parallel to the form Ax ¼ b that is usually used to denote a set of linear equations. The difference is
purely notational; since it is standard in input–output analysis to define the technical coefficients matrix as A,
then the matrix of coefficients in the input–output equation system becomes (I A). Similarly, convention is
responsible for denoting the right-hand sides of the input–output equations by f (for final demand) instead of b.
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21
2.3 An Illustration of Input–Output Calculations
then ðI AÞ1 can be found, and using standard matrix algebra results for linear
equations the unique solution to (2.10) is given by
x ¼ ðI AÞ1 f ¼ Lf
(2.11)
where ðI AÞ1 ¼ L ¼ ½l ij is known as the Leontief inverse or the total requirements
matrix.
In more detail, the equations summarized in (2.11) are
x1 ¼ l 11 f1 þ þ l 1j fj þ þ l 1n fn
..
.
xi ¼ li1 f1 þ þ l ij fj þ þ l in fn
..
.
xn ¼ ln1 f1 þ þ l nj fj þ þ lnn fn
(2.12)
This makes clear the dependence of each of the gross outputs on the values of each of
the final demands. Readers familiar with differential calculus and partial derivatives
will recognize that ∂xi =∂fj ¼ lij .
2.3
An Illustration of Input–Output Calculations
Numerical Example: Hypothetical Figures – Approach I
Impacts on Industry Outputs We now turn to a small numerical example, as
presented in Table 2.3. For the moment, the final demand elements and the value-added
elements have not been disaggregated into their component parts.
The corresponding table of input–output coefficients (Table 2.4) is found by dividing
each flow in a particular column of the producing sectors in Table 2.3 by the total output
(row sum) of that sector. Thus, a11 ¼ 150=1000 ¼ 0:15; a21 ¼ 200=1000 ¼ 0:2;
a12 ¼ 500=2000 ¼ 0:25; and a22 ¼ 100=2000 ¼ 0:05. In particular,
150 500 1=1000
0
A ¼ Zx^1 ¼
200 100
0
1=2000
2.3.1
Table 2.3 Flows zij for the hypothetical example
To Processing Sectors
From Processing Sectors
Payments Sector
Total Outlays ðxi Þ
1
2
1
2
Final Demand ðfi Þ
Total Output ðxi Þ
150
200
650
1,000
500
100
1,400
2,000
350
1,700
1,100
3,150
1,000
2,000
3,150
6,150
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22
Foundations of Input–Output Analysis
Table 2.4 Technical coefficients (the A matrix) for the hypothetical example
Sector 1 (Agriculture)
Sector 2 (Manufacturing)
Sector 1 (Agriculture)
Sector 2 (Manufacturing)
.15
.20
.25
.05
The A matrix is shown in Table 2.4. To add specificity for the remainder of this
example, we assume sector 1 represents “Agriculture” and sector 2 “Manufacturing.”
The principal way in which input–output coefficients are used for analysis is as
follows. We assume that the numbers in Table 2.4 represent the structure of production
in the economy; the columns are, in effect, the production recipes for each of the sectors,
in terms of inputs from all the sectors. To produce one dollar’s worth of manufactured
goods, for example, 25 cents’ worth of agricultural products and 5 cents’ worth of
manufacturing products are needed as intermediate ingredients. These are, of course,
only the inputs needed from other producing sectors; there will be inputs of a more “nonindustrial” nature as well, such as labor, from the payments sectors. For an analysis of
interrelationships among productive sectors, these are not of major importance.
We can now ask the question: If final demand for agriculture output were to increase
to $600 next year and that for manufacturing output were to decrease to $1,500 – for
example, because of changes in government spending, consumers’ tastes, and so on –
how much total output from the two sectors would be necessary in order to meet this
new f1
600
. In the year
¼
new demand? We denote this new demand as f new ¼
1,500
f new
2
1,000
350
, precisely because, in
, we saw that x ¼
of observation, when f ¼
2,000
1,700
producing to satisfy final demands, each sector must also produce to satisfy the
demands for inputs into the processes of production themselves. Now we are asking,
new x
new
new
new
for f 1 ¼ 600 and f 2 ¼ 1,500, what are the elements of x ¼ 1new ? To satisfy
x2
new
can
be
no
less
than
$600
and
x
no
less
than
$1,500.
These would
the demands, xnew
1
2
be the necessary outputs – the “direct effects” – if neither product were used in
production and all output were directly available for final demand. But since both
products serve as inputs, in a manner that is reflected in the technical coefficients of
Table 2.4, it seems clear that, in the end, more than $600 worth of agriculture goods
and more than $1,500 worth of manufacturing output will have to have been produced
in order to meet the new final demands. That is, there will be “indirect effects” as well.
Both of these effects are captured in the input–output model.
In the 2 2 case, jI Aj ¼ ð1 a11 Þð1 a22 Þ a12 a21 (Appendix A) and
ð1 a22 Þ
a12
adjðI AÞ ¼
a21
ð1 a11 Þ
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2.3 An Illustration of Input–Output Calculations
23
Table 2.5 Flows zij for the hypothetical example associated with xnew
To Processing Sectors
From Processing Sectors
1
2
Payments Sector
Total Outlays ðxi Þ
1
2
Final Demand ðf i Þ
Total Output ðxi Þ
187.13
249.50
810.89
1,247.52
460.40
92.08
1,289.11
1,841.58
600
1,500
1,100
3,200
1,247.52
1,841.58
3,200.00
6,289.10
:15 :25
:85 :25
so ðI AÞ ¼
; hence jI Aj ¼
:20 :05
:20 :95
0:7575 ¼
6 0 and we know that L ¼ ðI AÞ1 can be found. Here we have
1:2541 :3300
L¼
:2640 1:1221
For this example, A ¼
Assuming that technology (as represented in A), does not change, the needed total
outputs caused by f new are then found as in (2.11):
1:2541 :3300
600
1,247:52
¼
(2.13)
xnew ¼ Lf new ¼
:2640 1:1221 1,500
1,841:58
¼ $1,247:52 and xnew
¼ $1,841:58 are one measure of the
These values xnew
1
2
impact on the economy of the new final demands.6
With this result for xnew, it is straightforward to examine the changes in all elements
in the interindustry flows table (as in Table 2.3) caused by f new. From the definition of
coefficients in (2.8), Z ¼ Ax^. With a constant A matrix and a new vector of total
600
new 187:13 460:40
new
new
new
,
; along with f
outputs, x , we find Z ¼ Ax^
¼
1,500
249:50 92:08
we have the results shown in Table 2.5.
The elements in the Payments Sector are found as the difference between new total
outputs (total outlays) and new total interindustry inputs for each sector. (For the
example we assume no change in payments sector transactions with final demand.)
Notice that sector 1’s purchases are larger (reflecting an increase in final demand for
that sector) and sector 2’s purchases are smaller (reflecting smaller demand for that
sector).
The input–output model allows us to deal equally easily with changes in demands
and outputs instead of levels. Here and throughout, we use superscripts “0” to represent
6
Here xnew
and xnew
are shown to two decimals for comparison with results from an alternative approach in
1
2
Section 2.3.2. These xnew values reflect computer calculations carried out with more than four significant digits
and, hence, often will (as here) differ (to the right of the decimal point) from what the reader will produce with a
hand calculator using the four-digit elements shown for A. In any actual analysis, such detail might be
questionable because of the much less accurate data from which the technical coefficients are derived (compare
the figures in Table 2.3).
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24
Foundations of Input–Output Analysis
the initial (base year) situation and “1” for values of variables after the change in
demands [instead of “new” as we did in (2.13)]. Assuming that technology is
unchanged means A0 ¼ A1 ¼ A and L0 ¼ L1 ¼ L, so x0 ¼ Lf 0 and x1 ¼ Lf 1 ; letting
Δx ¼ x1 x0 and Δf ¼ f 1 f 0
Δx ¼ Lf 1 Lf 0 ¼ LΔf
250
247:5
In this example, Δf ¼
, giving Δx ¼
and so
200
158:4
(2.14)
1,247:5
247:5
1,000
¼
þ
x ¼ x þ Δx ¼
1,841:6
158:4
2,000
1
0
This corresponds to the result in (2.13), except for rounding.
Other Impacts In many cases, the dollar value of each sector’s gross output
may not ultimately be the most important measure of the economic impact following a
change in exogenous demands. Gross output requirements could be translated into
employment effects (in either dollars of earned income or physical terms – for
example, person-years), or effects on value-added, or energy consumption (of a
particular type, e.g., petroleum), or pollution emissions (again, of a particular type,
e.g., CO2 ), and so forth. In each instance, we need a set of appropriate coefficients with
which to convert outputs into associated effects. For illustration we consider employment in monetary terms. Let the value of employment in the two sectors be denoted as7
e0 ¼ ½e1 e2 A vector of employment coefficients contains the base-year employment in each sector
divided by that sector’s base-year gross output, x01 and x02 ,
e0c ¼ e0 x^1 ¼ e1 =x01 e1 =x02 ¼ ½ ec1 ec2 Then the vector of total labor income generated in each sector by the new exogenous
final demand can be found as ε ¼ ^e 0c x1 ¼ ^e 0c Lf 1 , namely
"
#" # "
#
ec1 0
x11
ec1 x11
ε¼
¼
0 ec2
x12
ec2 x12
To continue with the numerical example, suppose that ec1 ¼ 0:30 and ec2 ¼ 0:25
give the dollars’ worth of labor inputs per dollar’s worth of output of the two sectors.
(We will examine the role of labor inputs and household consumption in an input–
output model in some detail in Section 2.5.) Then
7
Later in this chapter (and still later, in Chapter 6 on multipliers) we will need to alter this notation to be able to
accommodate additional possibilities.
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2.3 An Illustration of Input–Output Calculations
ε ¼ ^e 0c x1 ¼
:30 0
0 :25
25
374:26
1,247:52
¼
460:40
1:841:58
These are the values of labor inputs purchased by the two sectors.
If, additionally, we have an occupation-by-industry matrix, P, where pij is the
proportion of sector j employment that is in occupation i, then ~ε ¼ P^ε gives a matrix
of employment by sector by occupation type. For example, with k occupation types
and two sectors,
2
3
p11 p12
6
.. 7
P ¼ 4 ...
. 5
pk1
and
pk2
2
p11 ec1 x11
..
~ε ¼ P^ε ¼ 6
4
.
pk1 ec1 x11
3
p12 ec2 x12
7
..
5
.
pk2 ec2 x12
Column sums would give total labor use by sector; row sums give total employment of
a particular occupational category across all sectors. (The vector Pε shows employment by occupational category, aggregated across all sectors.)
Suppose that our economy has three occupational groups: (1) engineers, (2) bankers,
and (3) farmers, and
2
3
0 :8
P ¼ 4 :6 :2 5
:4 0
(For example, this says that 40 percent of the agricultural labor force are farmers;
80 percent of the manufacturing labor force are made up of engineers, etc.) Then
2
3
2
3
0
368:32
0 :8 374:26
0
~ε ¼ P^ε ¼ 4 :6 :2 5
¼ 4 224:56 92:08 5
0
460:40
:4 0
149:70
0
Column sums of ~ε are 374.26 and 460.40, as expected (the elements of ε). Row
sums give the economy-wide (across both sectors) employment of engineers, farmers,
and bankers, respectively. If sectoral disaggregation is not necessary, then
2
3
2
3
0 :8 368:32
374:26
Pε ¼ 4 :6 :2 5
¼ 4 316:64 5
460:40
:4 0
149:70
gives employment by occupational type, across sectors.
Awide variety of such conversion coefficients vectors (as in ^e 0c ) or matrices (as in P) is
possible. For example, in arid regions, water-use coefficients, w0c ¼ ½ wc1 wc2 , could
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26
Foundations of Input–Output Analysis
be used in w0c x0 to assess the water consumption associated with new outputs generated
by new final demands. We explore these kinds of alternative impacts again in Chapter 6
on input–output multipliers, and in Chapters 12 and 13, some of the energy and environmental repercussions of final demand impacts are discussed in detail.
2.3.2 Numerical Example: Hypothetical Figures – Approach II
Consider the same economy, whose 2 2 technical coefficients matrix is given in
600
1
. We can examine the
Table 2.4 and for which the projected f vector is
1,500
question of outputs necessary to satisfy this final demand in a more intuitive way that is
less mechanical than finding elements in an inverse matrix.
1. Initially, it is clear that agriculture needs to produce $600 and manufacturing
$1,500. If the sectors are going to meet the new final demands, they could not get
away with producing less than these amounts.
2. However, to produce $600, agriculture needs, as inputs to that productive process,
(0.15)($600) ¼ $90 from itself and (0.20)($600) ¼ $120 from manufacturing. These
figures come from the coefficients in column 1 of the A matrix – the production
recipe for agriculture. Similarly, to produce its $1,500, manufacturing will have to
buy (0.25)($1,500) ¼ $375 from agriculture and (0.05)($1,500) ¼ $75 from itself.
Thus, agriculture must, in fact, produce the $600 noted in (1), plus another
$(90 þ 375) ¼ $465 more, to satisfy the needs for inputs that it has itself and also
that come from manufacturing. Similarly, manufacturing will have to produce an
additional $(120 þ 75) ¼ $195 to satisfy its own need plus that of agriculture for
inputs to produce the “original” $600 and $1,500.
3. In item (2), we found the interindustry needs that resulted from production of $600
in agriculture and $1,500 in manufacturing. These were $465 and $195, respectively. But now we realize that this “extra” production will also generate interindustry needs – in order to engage in the production of $465, agriculture will need (0.15)
($465) ¼ $69.75 from itself and (0.20)($465) ¼ $93 from manufacturing. Similarly,
manufacturing will now additionally need (0.25)($195) ¼ $48.75 from agriculture
and (0.05)($195) ¼ $9.75 from itself. The total new demands for the two sectors are,
thus, $(69.75 þ 48.75) ¼ $118.50 and $(93 þ 9.75) ¼ $102.75.
4. At this point we realize that it is necessary to treat the additional $118.50 for agriculture
and $102.75 for manufacturing in the same fashion as the $465 and $195 in item (3).
Hence, we find additional required outputs of $43.46 and $28.84 from the two sectors.
5. Continuing in this way, we find that eventually the numbers become so small that
they can be ignored (for example, less than $0.005).
Looking at the total impact of a particular set of final demands this way is described
as looking at the “round-by-round” effects. The initial demands generate a need for
inputs from the productive sectors; this is the “first round” of effects, as found in item
(2). But these outputs themselves generate a need for additional inputs – “second
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2.3 An Illustration of Input–Output Calculations
27
round” effects – as found in item (3); and so forth. For the present example, these
figures have been collected in Table 2.6.
For agriculture, the sum of these round-by-round effects, $647.53, plus the original
demand of $600, is $1,247.53; for manufacturing, the total is $341.57 þ $1,500 ¼
$1,841.57. These total outputs (except for small rounding errors) are the same as those
found by using the Leontief inverse, where x11 ¼ $1;247:52 and x12 ¼ $1;841:58. (It
was for this comparison that the two-decimal accuracy was kept in the Leontief-inverse
approach to this example.)
In this second view of the numerical example we have developed something of a
feeling for the way in which external (final) demands are transmitted through the
productive sectors of the economic system. In fact, we see that the elements of
ðI AÞ1 are really very useful and important numbers. Each captures, in a single
number, an entire series of direct and indirect effects. (The equivalence between
Approaches I and II is examined for the general case in Appendix SA2.1.)
2.3.3
Numerical Example: Mathematical Observations
1:2541 :3300
The inverse in this small example, L ¼
, illustrates a general
:2640 1:1221
feature of Leontief inverses for input–output models of any size – the diagonal
elements are larger than 1. This is entirely consistent with the economic logic of the
round-by-round approach. From (2.13)
x11 ¼ ð1:2541Þð600Þ þ ð0:3300Þð1,500Þ
Looking at the first product on the right, the new final demand of $600 for agriculture
output is multiplied by 1.2541. This can be thought of as (1 þ 0.2541)(600). The
(1)(600) reflects the fact that the $600 new agriculture demand must be met by
producing $600 more agriculture output. The additional (0.2541)(600) captures the
additional agriculture output required because this output is also used as an input to
production activity in both agriculture and also manufacturing. Similarly, from (2.13),
x12 ¼ ð0:2640Þð600Þ þ ð1:1221Þð1,500Þ
and the same logic explains why the coefficient (1.1221) relating manufacturing output
to new final demand for manufacturing goods, $1,500, must be greater than 1.
We examine why both of the diagonal elements in L will be greater than 1 in the
two-sector case. (A more complicated derivation can be used for the general n-sector
input–output model, and it is also apparent from the power series discussion in Section
2.4.) For this 2 2 example, as we saw in Section 2.3.1,
1
l12
l
¼
½adjðI AÞ
L ¼ 11
l 21 l22
jI Aj
1
ð1 a22 Þ
a12
¼
a21
ð1 a11 Þ
ð1 a11 Þð1 a22 Þ a12 a21
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28
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Table 2.6 Round-by-round impacts (in dollars) of f 11 ¼ $600 and f 12 ¼ $1,500
Round
0
1
2
3
4
5
6
7
8 þ 9 þ 10 þ 11
Lf 1
Sec. 1
Sec. 2
600
1,500
465.00
195.00
118.50
102.75
43.46
28.84
13.73
10.13
4.60
3.25
1.50
1.08
0.50
0.35
0.24
0.17
1,247.52
1,841.58
Cumulative Total
Sec. 1
Sec. 2
1,065.00
1,695.00
1,183.50
1,797.75
1,226.96
1,826.59
1,240.64
1,836.72
1,245.29
1,839.97
1,247.52
1,841.58
Percent of Total Effect Captured
Sec. 1
85.40
Sec. 2
92.00
94.90
97.60
98.40
99.20
99.50
99.70
99.80
99.90
1,247.52
1,841.58
2.3 An Illustration of Input–Output Calculations
29
So, for example,
l11 ¼
ð1 a22 Þ
1
¼
a12 a21
a12 a21
ð1 a22 Þ ð1 a11 Þ 1 a11 þ
ð1 a22 Þ
ð1 a22 Þ
Assuming that ð1 a22 Þ > 0, l 11 > 1 if the denominator on the right-hand side is less
than 1, which it will be when a11 > 0 and/or a12 a21 > 0 since ð1 a22 Þ > 0.
Similar reasoning shows that l 22 ¼ ð1 a11 Þ=jI Aj > 1 under similar reasonable
conditions on the aij .
Whether or not the off-diagonal elements are larger than 1 depends entirely on the
sizes of a12 and a21 , relative to |I A|. In most actual input–output tables, with a rather
detailed breakdown of sectors, the off-diagonal elements in L will be less than 1, as in
(2.13). However, for example, if a21 in Table 2.4 had been 0.70 instead of 0.20, so that
the coefficients matrix had been
:15 :25
A¼
:70 :05
then
1:5020 :3953
L¼
1:1067 1:3439
Notice that a coefficient as large as a21 ¼ 0:7 which says that there is 70 cents’
worth of sector 2 output in a dollar’s worth of sector 1 output – is not likely to be seen
6 jÞ, and
often in real tables. The sizes of the between-sector technical coefficients, aij ði ¼
of the off-diagonal elements in L, are related to the level of sectoral detail (that is, the
number of sectors) in the model. We will return to this topic in Chapter 4, when we
consider the effects of aggregating (combining) sectors in an input–output model. (In
Appendix 2.2 we examine the conditions under which a Leontief inverse matrix will
always contain only non-negative elements, as logic suggests should always be the case.)
2.3.4 Numerical Example: The US 2003 Data
We present a highly aggregated, seven-sector version of the 2003 US input–output
coefficients matrix and its associated Leontief inverse in Tables 2.7 and 2.8. (Appendix
B describes a series of such tables over time for the US economy at 7- and 17-sector
levels of aggregation, located on the supplementary resource web site to this text as
Appendix SD.1.) It is important to note that Tables 2.7 and 2.8 represent domestically
produced inputs for the US economy; this requires explanation.
Imports are generally divided into two categories: “competitive” and “non-competitive” imports (or “competing” and “non-competing”).
• Competitive imports are goods that have a domestic counterpart (that is, are also
produced in the USA). For example, grapes from Chile that are used to make grape
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30
Foundations of Input–Output Analysis
Table 2.7 The 2003 US domestic direct requirements matrix, A
Sector
1
2
3
4
5
6
7
Agriculture
Mining
Construction
Manufacturing
Trade, Transportation, & Utilities
Services
Other
1
2
3
4
5
6
7
.2008
.0010
.0034
.1247
.0855
.0897
.0093
.0000
.0658
.0002
.0684
.0529
.1668
.0129
.0011
.0035
.0012
.1801
.0914
.1332
.0095
.0338
.0219
.0021
.2319
.0952
.1255
.0197
.0001
.0151
.0035
.0339
.0645
.1647
.0190
.0018
.0001
.0071
.0414
.0315
.2712
.0184
.0009
.0026
.0214
.0726
.0528
.1873
.0528
5
6
7
Table 2.8 The 2003 US domestic total requirements matrix, L ¼ ðI A Þ1
Sector
1
2
3
4
5
6
7
1
2
3
4
Agriculture
1.2616 .0058 .0131 .0576 .0037 .0069 .0072
Mining
.0093 1.0748 .0122 .0343 .0193 .0033 .0073
Construction
.0075 .0034 1.0047 .0064 .0065 .0111 .0250
Manufacturing
.2292 .1192 .2615 1.3419 .0692 .0856 .1261
Trade, Transportation, & Utilities .1493 .0850 .1371 .1563 1.0887 .0598 .0853
Services
.2383 .2931 .2700 .2918 .2712 1.4116 .3138
Other
.0243 .0239 .0231 .0367 .0280 .0297 1.0338
jelly in the USA, where domestically grown grapes are also used in grape
jelly recipes.
• Non-competitive imports have no domestic counterpart. For example, coffee beans
from Brazil used by US coffee roasting firms (coffee beans are not grown in
the USA).
Some national tables (the USA is one example) show competitive imports within the
transactions table, so that sales of grapes to jelly producers include both domestic and
foreign sources. This correctly reflects the total amount of grapes needed by domestic
producers. However, it causes problems when input–output models are used for impact
analysis. Briefly put, this is because an analyst is usually interested in the economic
consequences on the domestic (or regional or local) economy of an exogenous demand
change. With Chilean grapes in a transactions matrix, and hence in the associated A
and L matrices, some of the demand repercussions measured by the model would in
fact be felt by Chilean grape growers. For this reason, we present here US data based
on a domestic transactions matrix (ZD ) in which the transactions matrix (Z) has been
purged of competitive imports. In matrix terms, ZD ¼ Z M, where M is a matrix of
competitive imports. This “scrubbing” of the matrix is not always easy to do if the data
are lumped together in a published Z table (as is the case in the USA), but it is very
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2.3 An Illustration of Input–Output Calculations
31
important when the question is one of impacts of final demand changes on the
domestic economy (and this is usually the question of interest).8
Spending on non-competitive imports usually appears in a row in the payments
sector (a single value indicating a sector’s payments for all non-competitive imports).
We return to these issues in Chapter 4.
The effects on US output of various final-demand vectors can be easily quantified
using L in Table 2.8. For example, suppose that there were increased foreign
demand (the export component of the final-demand vector) for agricultural and
manufactured items of $1.2 million and $6.8 million, respectively. Here (in millions
of dollars)
2
3
1:2
6 0 7
7
6
6 0 7
7
6
7
Δf ¼ 6
6 6:8 7
6 0 7
7
6
4 0 5
0
and, using (2.14), we find from L in Table 2.8 that (in millions of dollars)
2
3
1:9114
6 :2444 7
6
7
6 :0526 7
6
7
7
Δx ¼ 6
6 9:1249 7
6 1:2421 7
6
7
4 2:2709 5
:2788
As might be expected, the greatest effect, $9.125 million, is felt in the manufacturing sector. The next-greatest effect, $2.271 million, is felt in services. Also,
agriculture output would increase by $1.911 million and trade, transportation, and
utilities would increase by $1.242 million. Effects on the remaining three sectors
are less than $1 million. The total new output effect throughout the country,
obtained by summing the elements in Δx, is $15.125 million; this is generated by
a total new exogenous demand of $8 million. This again illustrates the multiplicative effect in an economy of an exogenous stimulus via an increase in one or more
components of final demand. These multiplier effects will be discussed in further
detail in Chapter 6.
8
By contrast, if one is interested in the structure of production (“production recipes”) and if or how they have
changed over time (structural analysis), it may be more useful to have competitive imports included in the Z
matrix and hence reflected in A and L, since such imports are certainly part of those recipes. These issues are
explored in Chapter 4.
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32
Foundations of Input–Output Analysis
2.4
The Power Series Approximation of ðI AÞ1
2.4.1 A Note on Computer Speeds and Capacities
In preparing input–output tables for many real-world applications of the model, in
which one wants to maintain a reasonable distinction between sectors (e.g., so that
sectors producing aluminum storm windows and women’s apparel are not lumped
together as a single sector labeled “manufacturing”), tables with hundreds of sectors
are not unusual. Early in the history of input–output studies, computer speed and
capacity posed real problems for implementation of input–output models – inversion
of large matrices was simply not possible. It was often necessary to aggregate the data
into a smaller number of sectors. We will say more about sectoral aggregation later
(especially in Section 4.9) since clearly industrial (sectoral) detail is lost in the process.
Other approaches were also applied to help reduce the computational burden. For
example, important matrix algebra results provide a way to carry out matrix inversion
calculations sequentially on a series of smaller submatrices of (I A) using a partitioned
matrix approach or to generate an approximation to ðI AÞ1 , requiring no inverses at
all. Moreover, both of these results have useful economic interpretations as well. The
approximation procedure with no inverses is examined in Section 2.4.2.
Since the early days of input–output analysis, exponential increases in computing
capacity and reductions in the cost of computing have removed many of the practical
obstacles to manipulating and inverting even very large matrices, so the computing
shortcuts just described, along with many others, are no longer necessary.9 There is no
simple way to characterize the combination of features contributing to the historical
evolution of computing power. But as a simplistic comparison we note that, when the
first edition of this text appeared in 1985, a typical microprocessor (those used in early
personal computers) could execute on the order of 1.25 million instructions per second
(MIPS). This was vastly more computing power than was available in the 1950s and
1960s when input–output was emerging as a widely applied tool, and it was more
capable than a mainframe computer in the mid-1970s, when the commonly used IBM
370 was often referred to as a “1 MIPS machine.” When the second edition of this text
appeared in 2009, a typical microprocessor could execute 175,000 MIPS and, at the
writing of the current edition, a typical microprocessor includes multiple processor
cores that enable multiple “threads” of computations to be executed simultaneously,
executing over 300,000 MIPS. And this does not include the equally exponentially
increasing capacity of computer storage and other features of modern computers. As a
result, in the life of this textbook (since the first edition), computing capacity has all but
vanished as a significant constraint on applying input–output analysis.
9
In 1939 it reportedly took 56 hours to invert a 42-sector table (on Harvard’s Mark II computer; see Leontief,
1951a, p. 20). In 1947, 48 hours were needed to invert a 38-sector input–output matrix. However, by 1953 the
same operation took only 45 minutes (Morgenstern, 1954, p. 496; also, see Lahr and Stevens, 2002, p. 478). By
1969, a 100-sector matrix could be inverted in between 10 and 36 seconds, depending on the computer used
(Polenske, 1980, p. 15). Today, inversion of matrices with thousands of sectors takes only seconds on even
desktop computers.
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2.4 The Power Series Approximation of (IA)1
33
2.4.2 The Power Series Approximation
Despite these computer advances, we examine a useful matrix algebra result generally
applicable to ðI AÞ matrices that makes possible an approximation to ðI AÞ1
requiring no inverses at all. This approach has been used in many applications and has
the added advantage of having a useful economic interpretation.
By definition, we know that A is a non-negative matrix with aij 0 for all i and j.
(This characteristic is often written as A 0, where it is understood that not all
aij ¼ 0.)10 The sum of the elements in the jth column of A indicates the dollars’ worth
of inputs from other sectors that are used in making a dollar’s worth of output of sector
j. In an open model, given the economically reasonable assumption that each sector
uses some inputs from the payments sector (labor,
P other value added, etc.), then each of
these column sums will be less than one ð ni¼1 aij < 1 for all jÞ. (We will see, in
Section 2.6, that this column sum condition need not apply to tables based on physical,
not monetary, measures of transactions and outputs.) For
Pn input–output coefficients
matrices with these two characteristics aij 0 and
i¼1 aij < 1 for all j – it is
possible to approximate the gross output x associated with any final demand f without
finding ðI AÞ1 .
Consider the matrix product
ðI AÞ I þ A þ A2 þ A3 þ þ An
where, for square matrices, A2 denotes AA, A3 ¼ AAA ¼ AA2 , and so on. This
premultiplication of the series in parentheses by (I A) can be accomplished by first
multiplying all terms in the right-hand parentheses by I and then multiplying all terms
by (A). This leaves only ðI AÞnþ1 ; all other terms cancel – for A2 there is a A2 ,
for A3 there is a A3 , and so on. Thus
ðI AÞ I þ A þ A2 þ A3 þ þ An ¼ I Anþ1
(2.15)
If it were true that for large n (more formally, as n ! ∞), the elements in Anþ1 all
become zero, or close to zero (i.e., Anþ1 ! 0), then the right-hand side of (2.15) would
be simply I, and the matrix series that postmultiplies (I A) in (2.15) would therefore
constitute the inverse to (I A), from the fundamental defining property of an inverse.
For any matrix, M, if we sum the absolute values of the elements in each column,
the largest sum is called the norm of M – denoted N(M) or ||M||.11 For example, for the
coefficients matrix A given in Table 2.4, N(A) ¼ 0.35, the sum of the elements in the
first column. (The sum of the elements in column 2 is 0.30.) For a pair of matrices, A
and B, that are conformable for the multiplication AB, there is a theorem stating that
10
A more exact characterization of vectors and matrices is often needed for more advanced matrix algebra results.
See Appendix A, Section A.9, where A > 0 is used for the case when A > 0 and A 6¼ 0.
11
A norm is just a measure of the general size of the elements in a matrix. (A measure of the size of the matrix
itself is given by the dimensions of the matrix.) For example, a non-negative m n matrix that has all elements
smaller than 0.1 will have a smaller norm than one that has all elements larger than 10. There are many possible
definitions of the norm of a matrix. The one used here (maximum column sum of absolute values) is one of
the simplest.
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34
Foundations of Input–Output Analysis
the product of the norms of A and B is no smaller than the norm of the product
AB N ðAÞN ðBÞ N ðABÞ. By replacing B with A, it follows that N ðAÞN ðAÞ N A2 or ½N ðAÞ2 N A2 and finally, continuing similarly,
½N ðAÞn N ðAn Þ
(2.16)
As was noted at the beginning of this section, all column sums of a “reasonable”
value-based A matrix will be less than one, so we know that N(A) < 1. Moreover, since
aij 0, we also know that aij N ðAÞ; no element in a non-negative matrix can be larger
than the largest column sum. Thus: (1) since N(A) < 1, ½N ðAÞn ! 0 as n ! ∞; (2) from
(2.16), this means that N ðAn Þ ! 0 also as n ! ∞; and (3) finally, then, all elements in
An must approach zero, since no single element in a non-negative matrix can be larger
than the norm of that matrix. This is the result that we are interested in. The right-hand
side of (2.15) becomes simply I as n gets large and so
(2.17)
L ¼ ðI AÞ1 ¼ I þ A þ A2 þ A3 þ (This is analogous to the series result in ordinary algebra that 1=ð1 aÞ ¼
1 þ a þ a2 þ a3 þ , for jaj < 1.) Notice that the terms on the right-hand side of
(2.17) are all positive. Even if some aij are zero, the increasing number of products of
A virtually guarantees that no zeros will be in evidence at the end of the summation.12
This means that L will contain only positive elements. (Appendix 2.2 looks into the
issue of positivity of L in more detail.)
Then x ¼ ðI AÞ1 f can be found as
x ¼ I þ A þ A2 þ A 3 þ f
(2.18)
Removing parentheses, this is
x ¼ f þ Af þ A2 f þ A3 f þ ¼ f þ Af þ AðAf Þ þ A A2 f þ (2.19)
Each term after the first can be found as the preceding term premultiplied by A. In
many applications it has been found that after about A7 or A8, the terms multiplying f
become insignificantly different from zero. Even with modern-day computer capacity
and speed, there still may be times when the approximation in (2.18) or (2.19) may
prove useful (for example, since matrix multiplications are much more straightforward
than inversion, especially of a large matrix).13
Returning to the original A matrix and the f vector of the example in Section 2.3
:15 :25
(and dropping the “0” superscripts for simplicity), where A ¼
and
:20 :05
600
, we have
f¼
1,500
12
13
As mentioned, the elements in any particular Ak do approach zero – which is the whole point.
Alternatively, some analysts have used the power series approximation as a framework for introducing
“dynamic” concepts into input–output models. We explore these ideas briefly in Section 14.4.7.
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2.5 Open Models and Closed Models
"
If ¼
"
Af ¼
"
Af¼
2
"
A3 f ¼
"
A4 f ¼
"
A5 f ¼
"
A6 f ¼
"
A7 f ¼
35
#
600
1,500
:15
:25
:20
:05
:0725
:0400
:0209
:0165
:0073
:0050
:0024
:0017
:0008
:0006
:0003
:0002
#"
600
# "
465
#
¼
1,500
195
#"
# "
#
:0500
600
118:50
¼
:0525 1,500
102:75
#"
# "
#
:0206
600
43:44
¼
:0126 1,500
28:80
#"
# "
#
:0063
600
13:83
¼
:0048 1,500
10:20
#"
# "
#
:0021
600
4:59
¼
:0015 1,500
3:27
#"
# "
#
:0007
600
1:53
¼
:0005 1,500
1:11
#"
# "
#
:0002
600
:48
¼
:0002 1,500
:42
We see that the individual terms in the power series approximation (except for
rounding errors) simply represent the magnitudes of the round-by-round effects, as
recorded in Table 2.6. (The reader should reconsider the algebra of the round-by-round
calculations to be convinced that, in fact, they were equivalent to premultiplication of f
by a series of powers of the A matrix.) Thus, it is possible that one may capture “most”
of the effects associated with a given final demand by using the first few terms in the
power series. As illustrated in Table 2.6, for our small example more than 98 percent of
the total effects in both sectors was captured in three rounds.14 And as we will see later
in this book, additional extensions of the input–output framework have been built on
the power series approximation (e.g., structural path analysis, Section 8.5).
2.5
Open Models and Closed Models
The model that we have dealt with thus far, x ¼ ðI AÞ1 f, depends on the existence
of an exogenous sector, disconnected from the technologically interrelated productive
sectors, since it is here that the important final demands for outputs originate. The basic
14
In two early (1950s) regional input–output applications, six rounds (Isard and Kuenne, 1953) and eight rounds
(Miller, 1957) were used, in addition to “. . . rough extrapolations to cover the infinite number of succeeding
rounds” (Isard and Kuenne, 1953, p. 298).
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36
Foundations of Input–Output Analysis
kinds of transactions that constitute the activity of this sector, as we have seen, are
consumption purchases by households, sales to government, gross private domestic
investment, and shipments in foreign trade (either gross exports or net exports –
exports from a sector less the value of imports of the same goods). In the case of
households, especially, this “exogenous” categorization is something of a strain on
basic economic theory. Households (consumers) earn incomes (at least in part) in
payment for their labor inputs to production processes, and, as consumers, they spend
their income in rather well patterned ways. And in particular, a change in the amount
of labor needed for production in one or more sectors – say an increase in labor inputs
due to increased output – will lead to a change (here an increase) in the amounts
available to households as a group for purchases of consumption goods. Although
households tend to buy goods for “final” consumption, the amount of their purchases is
related to their income, which depends on the outputs of each of the sectors. Also, as
we have seen, consumption expenditures are usually the largest single element of final
demand – often more than two-thirds of the total final-demand figure.15
Thus, one could move the household sector from the final-demand column and
labor-input row and place it inside the technically interrelated table, making it one of
the endogenous sectors. This is known as closing the model with respect to households. Input–output models can be “closed” with respect to other exogenous sectors as
well (for example, government sales and purchases); however, closure with respect to
households is more usual. It requires a row and a column of transactions for the new
household sector – the former showing the distribution of its output (labor services)
across the various sectors and the latter showing the structure of its purchases (consumption) distributed among the sectors. It is customary to add the household row
and column at the bottom and to the right of the transactions and coefficients tables.
Dollar flows to consumers, representing wages and salaries received by households
from the n sectors in payment for their labor services, would fill an (n þ 1)st row [znþ1,1 , . . . , znþ1, n ]. Dollar flows from consumers, representing the values of household
2
3
z1, nþ1
6
7
purchases of the goods of the n sectors, would fill an (n þ 1)st column: 4 ... 5.
zn, nþ1
Finally, the element in the (n þ 1)st row and the (n þ 1)st column, znþ1, nþ1 , would
include, for example, household purchases of labor services. Thus, Table 2.1 would
have one new row, at the bottom, and one new column, at the right, as indicated in
Table 2.9.
The ith equation, as shown in (2.1), would now be modified to
xi ¼ zi1 þ þ zij þ þ zin þ zi, nþ1 þ fi∗
15
(2.20)
There are usually many other sources of spendable income for a household – for example, from dividends and
interest payments, government, and/or private pensions for older retirees, etc. These are very real issues for
model closure (and they become more important with regional models) but we bypass them at this point to
concentrate on the basics of the procedure.
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37
2.5 Open Models and Closed Models
Table 2.9 Input–output table of interindustry flows with households endogenous
Buying Sector
Selling Sector
..1
.
i.
..
n
Households (Labor)
1
j
n
Households
(Consumers)
..z11
.
z. i1
..
zn1
znþ1, 1
..z1j
.
..zij
.
znj
znþ1, j
..z1n
.
..zin
.
znn
znþ1, n
..z1, nþ1
.
..zi, nþ1
.
zn, nþ1
znþ1, nþ1
where fi∗ is understood to represent the remaining final demand for sector i output –
exclusive of that from households, which is now captured in zi, nþ1 . In addition to this
kind of modification on each of the equations in (2.2), there would be one new
equation for the total “output” of the household sector, defined to be the total value
of its sale of labor services to the various sectors – total earnings. Thus
xnþ1 ¼ znþ1,1 þ þ znþ1, j þ þ znþ1, n þ znþ1, nþ1 þ f ∗
nþ1
(2.21)
The last term on the right in (2.21) would include, for example, payments to
government employees.
Household input coefficients are found in the same manner as any other element in
an input–output coefficients table: The value of sector j purchases of labor (for a given
period), znþ1, j , divided by the value of total output of sector j (for the same period),
xj , gives the value of labor services used per dollar’s worth of j’s output;
anþ1, j ¼ znþ1, j =xj . For the elements of the household purchases (consumption) column,
the value of sector i sales to households (for a given period), zi, nþ1 , is divided by the
total output (income earned) of the household sector, xnþ1 . Thus, household “consumption coefficients” are ai, nþ1 ¼ zi, nþ1 =xnþ1 . A drawback to this approach is that
now household behavior is “frozen” in the model in the same way as producer
behavior (constant coefficients).
The ith equation in the fundamental set given in (2.6) becomes
xi ¼ ai1 x1 þ þ ain xn þ ai, nþ1 xnþ1 þ fi∗
(2.22)
and the added equation which relates household output to output of all of the sectors is
xnþ1 ¼ anþ1,1 x1 þ þ anþ1, n xn þ anþ1, nþ1 xnþ1 þ f ∗
nþ1
(2.23)
Similarly, parallel to the equations in (2.7), we now have, rewriting (2.22) for the ith
equation,
ai1 x1 þ ð1 aii Þxi ain xn ai, nþ1 xnþ1 ¼ fi∗
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38
Foundations of Input–Output Analysis
And, for the household equation, rewriting (2.23),
anþ1,1 x1 anþ1, n xn þ ð1 anþ1, nþ1 Þxnþ1 ¼ f ∗
nþ1
Let the row vector of labor input coefficients, anþ1, j ¼ znþ1, j =xj , be denoted by
h0r ¼ ½anþ1,1 ; . . . , anþ1, n , and let the column vector of household consumption
2
3
a1, nþ1
6
7
coefficients, ai, nþ1 ¼ zi, nþ1 =xnþ1 , be hc ¼ 4 ... 5 and let h ¼ anþ1, nþ1 .16 Denote
an, nþ1
the (n þ 1) (n þ 1) technical coefficients matrix with households included.
by A
Using partitioning to separate the old A matrix from the new sector,
A hc
A¼
hr h
Let x denote the (n þ 1)-element column vector of gross outputs
2
3
x1
6 .. 7 x
6
7
x ¼ 6 . 7 ¼
xnþ1
4 xn 5
xnþ1
Also, let f ∗ be the n-element vector of remaining final demands for output of the
original n sectors and f the (n þ 1)-element vector of final demands, including that for
the output of households
2 ∗ 3
f1
6 .. 7 ∗ f
7
f ¼ 6
6 .∗ 7 ¼ ∗
f nþ1
4 f 5
n
f∗
nþ1
Then the new system of n þ 1 equations, with households endogenous, can be
represented as
Þx ¼ f
ðI A
(2.24)
or
IA
hc
h0r ð1 hÞ
x
xnþ1
¼
f∗
f∗
nþ1
(2.25)
That is, we have the set of n equations
ðI AÞx hc xnþ1 ¼ f ∗
16
In the initial numerical illustration in Section 2.3.1, for simplicity we used e0c for the vector of employment
coefficients. These are seen to be the elements in h0r , which is the notation frequently used in closed models.
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2.5 Open Models and Closed Models
39
Table 2.10 Flows (zij ) for hypothetical example, with households endogenous
To
From
1
2
Household
Consumption
(C )
Other Final
Demand
( f *)
Total
Output
(x)
1
2
Labor Services (L)
Other Domestic
Payments (N)
Imports (M)
Total Outlays (x0 )
150
200
300
325
500
100
500
800
50
400
50
300
300
1,300
150
250
1,000
2,000
1,000
1,675
25
1,000
100
2,000
200
1,000
150
2,150
475
6,150
[a matrix rearrangement of (2.22)] and the added one for households
h0r x þ ð1 hÞxnþ1 ¼ f ∗
nþ1
[a matrix rearrangement of (2.23)]. Together these determine the values of outputs for
the n original sectors x1 , . . . , xn and the value of household services used (wages
paid) to produce those outputs xnþ1 . If the (n þ 1) (n þ 1) coefficients matrix is
non-singular, the unique solution can be found using an inverse matrix in the usual
way:
1 ∗ IA
hc
f
x
¼
(2.26)
h0r ð1 hÞ
f∗
xnþ1
nþ1
or
Þ1f ¼ L
f
x ¼ ðI A
Consider again the information given in Table 2.3. Suppose that the household
consumption part of final demand and the household labor input part of the
payments sector are as shown in Table 2.10. Of the $650 bought by sector 1 from
the payments sectors (Table 2.3), $300 was for labor services; of the $1,400 bought
by sector 2, $500 was for labor inputs. Also, of the $1,100 which represented
purchases of final-demand sectors from the payments sectors, $50 was paid out by
households for labor services (e.g., domestic help); government purchases of labor
were $150. The $300 would record household payments to government (taxes), and
so forth.
The total output of the household sector, as in (2.23), is (here n þ 1 ¼ 3), x3 ¼ z31 þ
z32 þ z33 þ f ∗
3 ¼ 300 þ 500 þ 50 þ 150 ¼ 1,000. The household input coefficients,
anþ1 , j ¼ znþ1, j =xj , are: a31 ¼ 300=1,000 ¼ 0:3, a32 ¼ 500=2,000 ¼ 0:25 and a33 ¼
50=1,000 ¼ 0:05; h0r ¼ ½ 0:3 0:25 and h ¼ 0.05. Similarly, household consumption
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40
Foundations of Input–Output Analysis
coefficients, ai, nþ1 ¼ zi, nþ1 =xnþ1 are a13 ¼ 50=1,000 ¼ 0:05 and a23 ¼ 400=1,000 ¼
:05 17
. Therefore,
0:4; thus hc ¼
:4
2
3
2
3
:25
:05 7
:25
:05 7
6 :85
6 :15
6
7
6
7
A ¼ 6 :2
:95
:4 7
:05
:4 7, ðI AÞ ¼ 6 :2
4
5
4
5
:3
:25
:95
:3
:25
:05
|
|
|
|
|
|
|
|
| - - - - - - - - - - - - - -
| - - - - - - - - - - - |
|
|
|
and
2
1
Þ
ðI A
¼
¼L
11
L
L21
6 1:3651
12
L
6
:5273
22 ¼ 6
L
4
:5698
:4253
1:3481
|
|
|
|
3
:2509 7
7
:5954 7
5
1:2885
(2.27)
- - - - - - - - - - - - - -| - - - - - -
:4890
|
|
Consider again the numerical example in Section 2.3 (again we ignore the “0” and “1”
superscripts for simplicity). There we assumed a change in the final-demand vector such
that f1 went from 350 to 600 and f2 from 1,700 to 1,500. Referring now to Table 2.10,
simply for illustration, suppose that this entire final-demand change was concentrated in
the Other Final Demand sector. In fact, let it represent a change in the demands of the
federal government [part of the Other Final Demand column (fi∗ ) in Table 2.10]. These
new demands of $600 and $1,500 represent increases in both cases, from the current
levels of $300 and $1,300 for all non-household final-demand categories.
The most straightforward comparison is now to use the 3 3 Leontief inverse
2
3
600
1
Þ in (2.27) in conjunction with f ¼ 4 1,500 5 to find the impact of these
ðI A
0
changes in the final demands for the outputs of sectors 1 and 2 on the two original
sectors, plus the added impact due to closure of the model with respect to households.
We have
2 3
2
32
3 2
3
1:3651 :4253 :2509
600
1:456:94
x1
4 x2 5 ¼ x ¼ 4 :5273 1:3481 :5954 54 1,500 5 ¼ 4 2,338:51 5
:5698 :4890 1:2885
0
1,075:48
x3
In the earlier example of Section 2.3, with households exogenous to the model, the
new outputs were x1 ¼ $1,247:46 and x2 ¼ $1,841:55. The new (larger) values –
$1,456.94 and $2,338.51, respectively – reflect the fact that additional outputs are
necessary to satisfy the anticipated increase in consumer spending, as reflected in the
17
In this illustration we are ignoring the real-world possibility that total labor earnings do not equal total
household consumption (recall footnote 14). This is an even more important issue in regional models, and
we will encounter it again below and in subsequent chapters.
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2.5 Open Models and Closed Models
41
household consumption coefficients column, expected because of the increased household earnings due to increased outputs from sectors 1 and 2 and hence increased wage
payments. Using the labor input coefficients, a31 ¼ 0:3 and a32 ¼ 0:25, the necessary
household inputs for the original gross outputs (when households were exogenous)
would be
a31 x1 þ a32 x2 ¼ ð0:3Þð1, 247:46Þ þ ð0:25Þð1, 841:55Þ ¼ 834:63
As would be expected, outputs are increased for all three sectors, due to the
introduction of the formerly exogenous household sector into the model. The
example serves to illustrate an expected outcome – namely that when the added
impact of more household consumption spending due to increased wage income is
explicitly taken into consideration in the model, the outputs of the original sectors in
the interindustry model are larger than is the case when consumer spending
is ignored.
In this section we have introduced the basic considerations involved in moving
households from final demand into the model as an endogenous sector – closing the
model with respect to households. Similar kinds of data and algebraic extensions
would be needed if other exogenous sectors – for example, federal, or state and/or
local government activities – were to be made endogenous in the model. However,
because the value of consumption tends to be the largest component of final demand
and because of the relatively direct linkage between earned income and consumption
and between consumption and output, the household sector is the one final-demand
sector that is most often moved inside the model.
In practice, however, the issue is far more subtle, and the procedure is more
complicated than might be suggested by the discussion in this section. All of the
previous reservations about the aij apply here as well, if not with greater force. For
each additional dollar of received earnings, households are assumed to spend 5 cents
on the output of sector 1, 40 cents on the output of sector 2, and so on. Those
coefficients, which reflect average behavior during the observation period when
household income was $1,000 (a13 ¼ 50=1;000 and a23 ¼ 400=1;000), are assumed
to hold for the additional, or marginal, amounts of household earnings associated with
the new outputs of sectors 1 and 2. One approach, particularly at the regional level, is
to divide consumers into two groups: established residents, for whom the new income
associated with new production would represent an addition to current earnings, and
new residents (in-migrants), who may have moved in search of employment and for
whom the new income represents total earnings. For the former group, a set of
marginal consumption coefficients might be appropriate, while for the latter group
average consumption coefficients would be relevant.
In addition, spending patterns of consumers, especially out of additions to (or
reductions in) disposable income, will depend on the income category in which a
particular consumer is located. An addition of $100 to the spendable income of a
worker earning $40,000 per year is likely to be spent differently than an additional
$100 in the hands of an engineer with an annual income of $250,000, and both
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42
Foundations of Input–Output Analysis
will no doubt differ from the way in which the $100 would be spent by a
previously unemployed person. In effect, this is simply noting that inputs to the
household sector (consumption) per dollar of output (household income) will not
be independent of the level of that output. Yet such independence is assumed in
the way that the direct input coefficients are used in an input–output model; each
sector’s production function (column of direct input coefficients) is assumed to
represent inputs per dollar’s worth of output, regardless of the amount (level) of
that output.
Another approach, then, is to disaggregate “the” household sector into several
sectors, distinguished by total income. For example, $0–$20,000, $20,001–$40,000,
$40,001–$60,000; and so on. Consumption coefficients, by sector, could then be
derived for each income class. We will return to this issue in a regional context in
Chapter 3 and in Chapter 11 when examining social accounting matrices. A very
thorough discussion of an approach for incorporating a disaggregated household sector
into the endogenous part of an input–output model, using a good deal of matrix
algebra, can be found in Miyazawa (1976). We explore that model in more detail in
Chapter 6.
Further disaggregations of the household sector have been proposed and incorporated in input–output analysis. These frameworks fall into the category of what are
known as “extended” input–output models. [For a concise overview of the issues
involved see Batey, Madden, and Weeks (1987) or Batey and Weeks (1989).] The idea
is to separate income payments to and consumption patterns of different household
groups – for example, established versus new residents (noted at the beginning of this
section) and currently employed versus unemployed. And, as noted in this section,
there is the additional issue of identifying other possible sources of consumer incomes
that are not directly related to the economic sectors in the model.
One could imagine a process of moving, one by one, each of the remaining sectors
from the final-demand vector into the interindustry coefficients matrix, constructing
rows of input coefficients and columns of purchase coefficients until there were no
exogenous sectors at all. This is termed a completely closed model. However, the
economic logic behind fixed coefficients in the case, say, of a government sector is less
easy to accept than for the productive sectors, and completely closed models are
seldom implemented in practice.18
2.6
The Price Model
2.6.1 Overview
Leontief originally developed the input–output model in physical units (bushels of
wheat, yards of cloth, man-years of labor, etc.).19 In particular, he assumed that direct
18
The original work done by Leontief, however, was in the framework of a completely closed model of the USA
for 1919. See Leontief (1951b).
19
See, for example, Leontief (1951a, 1951b, 1986) and Leontief et al. (1953).
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2.6 The Price Model
43
input coefficients, A, are based on physical quantities of inputs divided by physical
quantities of output. These data were then converted to a table of (base year) transactions in value terms by using (base year) unit prices – for a bushel of wheat, a yard of
cloth, and a man-year of labor. He writes (Leontief, 1986, pp. 22–23):
All figures [in the value transactions table] . . . can also be interpreted as representing physical
quantities of the goods or services to which they refer. This only requires that the physical unit in
which the entries . . . are measured be redefined as being equal to that amount of output of that
particular sector that can be purchased for $1 at [base year] prices . . . In practice the structural
matrices are usually computed from input–output tables described in value terms . . . In any case, the
input coefficients [A] – for analytical purposes . . . must be interpreted as ratios of two quantities
measured in physical units [emphasis added].
As already noted, input–output data are usually assembled, and input–output studies
are generally carried out in monetary (value) units.
However, particularly with the emergence of energy and environmental concerns,
mixed-units models have been developed, where economic transactions are recorded in
monetary terms and ecological and/or energy transactions are recorded in physical
terms (tons, BTUs, joules, etc.).20 Another line of inquiry has led to input–output
tables in common physical units (e.g., all transactions and outputs measured in tons).
Stahmer (2000) gives an overview of early work in this area, including tables for
Germany for 1990 in both monetary and physical units – sometimes designated MIOTs
and PIOTs, respectively. (There are problems in trying to measure outputs of services
in physical units.)21 We explore a small illustration in Section 2.6.8, using an aggregation of the German data.
2.6.2 Physical versus Monetary Transactions
We return to the illustration in Section 2.3. Suppose the physical unit measures for
outputs are bushels for sector 1 (agriculture) and tons for sector 2 (manufacturing), and
that transactions measured in these physical units are shown in Table 2.11, where we
now use d i for physical amounts delivered to final demand and qi for physical amounts
of total output.
If we know the per-unit prices of the two products, the information in Table 2.11 can
be converted to monetary units. For example, if the price per bushel is $2.00 and the
price per ton is $5.00, then the monetary transactions table is exactly as shown in
Table 2.11 Transactions in physical units
1
2
20
21
1
2
di
qi
Physical Units of Measure
75
40
250
20
175
340
500
400
bushels
tons
These issues are explored further in Chapters 12 and 13.
Stahmer (2000) also introduces the notion of data measured in time units, leading to TIOTs.
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Foundations of Input–Output Analysis
Table 2.12 Transactions in monetary units (see Table 2.3)
1
2
1
2
fi
xi
$ Price per Physical Unit
150
200
500
100
350
1,700
1,000
2,000
2
5
Table 2.13 Transactions in revised physical units
1
2
1
2
di
qi
Revised Physical Units of
Measure
150
200
500
100
350
1,700
1,000
2,000
1/2 bushels
1/5 tons
Table 2.3. Now, redefine the physical units of measurement for each sector to be the
amount that can be bought for $1.00; that is, so that the per-unit price for each sector’s
output is $1.00. This simply means that we measure the physical output of sector 1 in
half bushel units and the physical output of sector 2 in fifths of a ton. Then, in these
revised units, the information in Table 2.12 can be reinterpreted as recording transactions in physical units, as in Table 2.13 – for example, 500 half-bushels of sector
1 output were bought by sector 2 (for $500), 2,000 fifths of a ton of sector 2 output
were delivered to final demand (for $2,000), etc.
In practice, sectors produce more than one good, and the assumption of one price for
a sector’s output is unrealistic. And in any case, monetary tables are assembled on the
basis of recorded values of transactions; price and quantity are generally not
recorded separately.
2.6.3 The Price Model Based on Monetary Data
Monetary transactions are arranged as usual, where for notational simplicity we
assume that all value added (v) is represented by labor (Table 2.14). As we saw in
Section 2.2.1, when all inputs are accounted for in the processing and payments
sectors, then the jth column sum (total outlays) is equal to the jth row sum (total
output). Thus, summing down the jth column in Table 2.14,
xj ¼
n
X
zij þ vj
(2.28)
x0 ¼ i0 Z þ v0
(2.29)
i¼1
or
where, as earlier, v0 ¼ ½v1 ; . . . , vn , total value-added expenditures by each sector.
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2.6 The Price Model
45
Table 2.14 Transactions in monetary terms
Sectors
Sectors
1
j
n
Final Demand
Total Output
1
2.
..
n
Labor
z11
z. 21
..
zn1
v1
z1j
z. 2j
..
znj
vj
z1n
z. 2n
..
znn
vn
f1
..f 2
.
fn
f nþ1
x1
..x2
.
xn
xnþ1
Substituting Z ¼ Ax^, x0 ¼ i0 Ax^ þ v0 , and postmultiplying by x^1,
x0 x^1 ¼ i0 Ax^x^1 þ v0 x^1
or
i0 ¼ i0 A þ v0c
(2.30)
where v0c ¼ v0 x^1 ¼ ½v1 =x1 ; . . . , vn =xn . The right-hand side of (2.30) is the cost of
inputs per unit of output. Output prices are set equal to total cost of production (in the
general case, this will include an allocation for profit and other primary inputs in v0 and
hence in v0c ), so each price is equal to 1 [the left-hand side of (2.30)]. This illustrates the
unique measurement units in the base year table – amounts that can be purchased for
~ 0 ¼ ½p~1 ; . . . , p~n , then the
$1.00. If we denote these base year index prices by p~j, so p
input–output price model is:
~ 0 A þ v0c
~0 ¼ p
p
(2.31)
~ 0 ¼ v0c ðI AÞ1 ¼ v0c L
p
(2.32)
~ 0 ðI AÞ ¼ v0c and
which leads to p
Frequently the model is transposed and expressed in terms of column vectors rather
than row vectors. In that case,
~ ¼ ðI A0 Þ1 vc ¼ L0 vc
p
(2.33)
1
[The interested reader can show that, given ðI AÞ1 ¼ L, then ðI A0 Þ ¼ L0 .]
~ , are determined by the exogenous values (costs) of
From (2.32), index prices, p
primary inputs. For a two-sector model,
p~1 ¼ l 11 vc1 þ l 21 vc2
p~2 ¼ l 12 vc1 þ l 22 vc2
The logic is that changes in labor input prices (or, more generally, primary input price
changes) lead to changes in sectoral unit costs (and therefore output prices, not output
quantities) via the fixed production recipes in A, and hence in L and L0 . For example,
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Foundations of Input–Output Analysis
Table 2.15 The Leontief quantity and price models
Leontief Quantity Model
(Demand-pull) [Prices fixed; quantities
change]
Exogenous Variables
Endogenous Variables
Leontief Price Model
(Cost-push) [Quantities fixed; prices
change]
Exogenous Variables
Endogenous Variables
f 1 ¼ f 1i
or
Δf ¼ ½Δf i x1 ¼ L0 f 1
or
Δx ¼ L0 ðΔf Þ
h
i
1
v1c ¼ x^ 0 v1 ¼ v1j =x0j
or
h
i
1
Δvc ¼ x^ 0 ðΔvÞ ¼ Δvj =x0j
0
p~ 1 ¼ L0 v1c
or
0
Δ~
p ¼ L0 ðΔvc Þ
cost increases are passed along completely as intermediate input price increases to all
purchasers, who in turn pass on these increases by raising their output prices accordingly, etc. As opposed to the demand-pull input–output quantity model, described
throughout this chapter until this section, the price model in (2.32) or (2.33) is more
completely known as the cost-push input–output price model (Oosterhaven, 1996;
Dietzenbacher, 1997). In it, quantities are fixed and prices change. Table 2.15 summarizes the two (dual) models where, again, superscripts “0” and “1” indicate values
before and after accounting for the exogenous change. Examples in Section 2.6.4
illustrate the workings of this model.
2.6.4
Numerical Examples Using the Price Model Based on Monetary Data
Example 2.1: Base Year Prices Table 2.16 contains data from Table 2.10 to
construct an added row to reflect labor as the only primary input. The corresponding
direct inputs matrix is
2
3
:15 :25 :11
¼ 4 :20 :05 :54 5
A
(2.34)
:65 :70 :35
Using A for the 2 2 submatrix of sector 1 and sector 2 coefficients,
0 0
1:254 :264
1
L ¼ ðI A 0 Þ ¼
:330 1:122
From the base year data, v0c ¼
(2.34). Thus, in (2.33),
(2.35)
a31
:65
in
from the bottom row of A
¼
a32
:70
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2.6 The Price Model
47
Table 2.16 Transactions for hypothetical example with one primary input
1
2
3 (Labor)
1
2
fi
xi
150
200
650
500
100
1,400
350
1,700
1,100
1,000
2,000
3,150
0 0 0 0
1:254 :264
:65
1:00
¼
p~ ¼ L vc ¼
:330 1:122 :70
1:00
(2.36)
This reproduces the base year index prices, as expected.
Example 2.2: Changed Base Year Prices The value-based cost-push price
model is generally used to measure the impact on prices throughout the economy of
new primary-input costs (or a change in those costs) in one or more sectors. Again,
suppose that these costs consist entirely of wage payments and that wages in sector 1
increase by 30 percent (from 0.65 to 0.845), while those in sector 2 remain unchanged.
The vector of new labor costs is
:845
1
vc ¼
:700
and, from (2.33),
0
1:254 :264
:845
1:245
~ 1 ¼ L0 v1c ¼
p
¼
(2.37)
:330 1:122 :700
1:064
Relative to the original index prices p~01 ¼ 1:00 and p~02 ¼ 1:00 , sector 1’s price has
gone up to 1.245 (a 24.5 percent increase), and sector 2’s price has increased by
6.4 percent.
As with the demand-driven input–output model, described throughout this chapter
until the price model was introduced in Section 2.6.3, this exercise can just as well be
carried out in the “Δ” form of the model, namely
0
(2.38)
Δ~
p ¼ L0 Δvc
:195
, where (0.195) ¼ (0.30)(0.65), and using (2.38),
In this case, Δvc ¼
0
0 0
:245
1:254 :264
:195
¼
Δ~
p ¼ L Δvc ¼
:064
:330 1:122
0
(2.39)
The results in either (2.37) or (2.39) convey the same information – the economywide effect of the 30 percent wage increase in sector 1 is that the price of sector
1 output goes up by 24.5 percent and that of sector 2 increases by 6.4 percent. In this
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48
Foundations of Input–Output Analysis
cost-push input–output price model, we find relative price impacts – the absolute
values of those prices, even in the base year, are not explicit in the model.
Notice that if labor costs are only a part of the value-added component for sector 1,
then a 30 percent increase in wages in sector j will generate a less than 30 percent
increase in vcj for example, if wages comprise 40 percent of sector j’s value-added
payments, and no other value-added costs increase, a 30 percent wage increase
translates into a 12 percent increase in vcj . The effects of primary input price decreases
can also be quantified in the same way by the models in (2.32) [or (2.33)] or (2.38).
2.6.5 Applications
An early example of the use of this kind of input–output price model is provided by
Melvin (1979), where the price effects of changes in corporate income taxes are
estimated for both the USA and Canada, using an 82-sector US table for 1965 and a
110-sector Canadian table for 1966. Another illustration is provided by Duchin and
Lange (1995), who use the price model framework to assess price effects of alternative
technologies in the US economy. Based on US 1963 and 1977 data, they use 1977 technology with 1963 factor prices to assess the price effects of the change in technology
over that period. Similarly, using projections to 2000, they examine the price effects of
technology change over 1977 to 2000. (They also change technology in the A matrix one
column at a time; and this is done in the context of a dynamic price model. We explore
dynamic input–output models in Chapter 14.) A few additional examples include Lee,
Blakeslee, and Butcher (1977) at a regional level, Polenske (1978) for a multiregional
example, Marangoni (1995) for Italy, Valadkhani and Mitchell (2002) for Australia,
Bazzazan and Batey (2003) for extensions and an application to a 43-sector data set for
Iran in 1994, and Dietzenbacher and Velázquez (2007), who include an analysis of costpush effects of changes in water prices.
2.6.6 The Price Model Based on Physical Data
In this section we examine the implications of an input–output model based on a set of
data in physical units, as was shown in Table 2.11. Here, in Table 2.17, we let sij
Table 2.17 Flows in physical units
Sectors
Sectors
1
2
n
Final Demand
Total Output
1
2
..
.
n
Labor
s11
s. 21
..
sn1
snþ1, 1
s12
s. 22
..
sn2
snþ1, 2
s1n
s. 2n
..
snn
snþ1, n
d1
d. 2
..
dn
d nþ1
q1
..q2
.
qn
qnþ1
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2.6 The Price Model
49
represent the physical quantity of i goods shipped to j; [e.g., bushels of agricultural
products (i) sold to manufacturers (j)], d i is deliveries to final demand (e.g., in bushels
for agricultural demand), and qi is total sector i production (e.g., total bushels produced
by agriculture). Again, for simplicity, let the exogenous payments (value added) sector
consist exclusively of labor inputs (measured in person-days).
Reading across any row in Table 2.17 we have the basic accounting relationships in
physical units:
qi ¼ si1 þ þ sij þ þ sin þ d i ¼
n
X
sij þ d i
(2.40)
j¼1
[Compare with (2.1) in value terms.] Using obvious matrix definitions, this is
q ¼ Si þ d
(2.41)
This is the physical-units parallel to (2.4).
Direct input coefficients in physical terms are defined as
cij ¼
sij
or C ¼ S^
q 1
qj
(2.42)
For the example of agricultural input into manufacturing (Table 2.11), this would be
250/400 = 0.625 (bushels per ton). Then, in a series of steps that parallel the earlier
development of the value-based model in Section 2.2, substitution into (2.41) gives
q ¼ C^
q i þ d ¼ Cq þ d
from which
q ¼ ðI CÞ1 d
(2.43)
This is the physical-units model parallel to (2.11).
Introduction of Prices Suppose also that we know the per-unit price for each
sector’s output, pi , and the labor cost per person-hour, pnþ1 . Then, as Leontief observes
in the quotation above, we can easily convert the basic data to the value units from
earlier in this chapter:
xi ¼ pi qi
(2.44)
zij ¼ pi sij
(2.45)
fi ¼ pi d i
(2.46)
Multiplying (2.40) on both sides by pi gives
xi ¼ pi qi ¼
n
X
pi sij þ pi d i ¼
j¼1
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n
X
j¼1
zij þ fi
(2.47)
50
Foundations of Input–Output Analysis
or x ¼ Zi þ f. As expected, these produce the original accounting relationships in (2.1)
and (2.3) in value terms.
In Section 2.6.1, the representation of total outputs in termsPof column sums of
Table 2.14 was given in monetary terms in (2.28), namely xj ¼ ni¼1 zij þ vj. Column
sums are not meaningful in Table 2.17 since elements in each row are measured in
different units. The objective now is to introduce the results from (2.44) and (2.45)
into (2.28). Assume, for now, that the wage rate is pnþ1 (dollars per person-hour)
across all sectors. Then znþ1, j ¼ pnþ1 snþ1, j ¼ vj ; this represents sector j’s total expenditure on labor – the price, pnþ1 , times total person-hours of labor, snþ1, j . Then (2.28)
becomes
pi qj ¼
n
X
pi sij þ pnþ1 snþ1, j
(2.48)
j¼1
Dividing by qj (which we assume is not zero),
pj ¼
n
X
pi sij =qj þ pnþ1 snþ1, j =qj ¼
i¼1
n
X
pi cij þ pnþ1 cnþ1, j
(2.49)
i¼1
In matrix form, this is
p0 ¼ p0 C þ υ0c
(2.50)
where p0 ¼ ½p1 ; . . . , pn , C is defined in (2.42) and υ0c ¼ pnþ1 ½cnþ1,1 ; . . . , cnþ1, n . So υ0c
represents the labor cost (price) per unit of physical output – for example, labor costs
per ton of output [$/ton ¼ ($/person-hour) (person-hours/ton)].
Labor costs were assumed to be uniform across all sectors; thus we have only pnþ1
and not pnþ1, j . This can easily be extended to encompass differing labor costs (perhaps
reflecting labor of differing skills) among sectors. Equation (2.50) defines the unit price
for each sector’s output as equal to the total costs (interindustry plus primary inputs) of
producing a unit of that output. (In general, there will be more than one component to
primary input costs for each sector, but the principles remain the same.)
From (2.50),
p0 ¼ υ0c ðI CÞ1
(2.51)
As before, we can transpose both sides of (2.50) and (2.51) to have the prices in a
column vector instead of a row vector,
1
p0 ¼ C0 p þ υc and p ¼ ðI C0 Þ υc
(2.52)
This is the Leontief price model based on physical units. These structures are completely parallel to those in (2.32) and (2.33) for the monetary-based index-price model.
For the n ¼ 2 case, we have
p1 ¼ p1 c11 þ p2 c21 þ υc1
p2 ¼ p1 c12 þ p2 c22 þ υc2
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(2.53)
2.6 The Price Model
51
and
p1
p2
ð1 c11 Þ
¼
c12
Relationship between A and C
aij ¼
c21
ð1 c22 Þ
1 υc1
υc2
(2.54)
Direct input coefficients in value terms are
zij
or A ¼ Zx^1
xj
Therefore [from (2.44) and (2.45)]
p sij
p
aij ¼ i ¼ cij i
pj qj
pj
!
(2.55)
In matrix terms22
1 1 ^ Sðp
^q
^ Þ1 ¼ p
^ C^
^ p
^ C^
^
A¼p
q q
¼p
p 1
(2.56)
Either the value-based coefficients, A, or the physical coefficients, C, are assumed
fixed in applications of the input–output model. However, assuming fixed cij (in effect,
a fixed “engineering” production function) has been seen by many as less restrictive
than fixed aij (a fixed “economic” production function), because in the latter case both
a physical coefficient, cij , and a price ratio, pi =pj , are assumed unchanging.23
2.6.7
Numerical Examples Using the Price Model based on Physical Data
Example 2.3: Base Year Prices Consider again the two-sector economy
(agriculture and manufacturing) in Table 2.11 closed with an added row showing labor
inputs and final demand (consumption). From that table we can find the physical
technical coefficients [as in (2.42)]
2
3
:15
:625
:556
6
7
7
¼6
(2.57)
C
6 :08
:05
1:079 7
4
5
:13
:35
:349
|
|
|
|
- - - - - - - - - -| - - - - |
|
for the (closed) technical coefficients matrix that includes households; C will
We use C
represent the 2 2 matrix in the upper-left corner – technical coefficients connecting
are
the two producing sectors in the economy. Note that c23 > 1; column sums in C
meaningless, since each row is measured in different units.
22
23
When two matrices, M and N, satisfy the relationship M ¼ ^v N^v 1 , they are said to be similar.
Economists have held differing opinions on this question of the plausibility of the assumption of stability for
physical versus value-based coefficients. For early examples, see Klein (1953), who suggests that aij ’s may be
more stable than cij ’s, and Moses (1974), who argues the opposite.
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Foundations of Input–Output Analysis
The relationships in (2.54) are
2 ¼ ð2Þð:15Þ þ ð5Þð:08Þ þ ð10Þð:13Þ ¼ :30 þ :40 þ 1:30
5 ¼ ð2Þð:625Þ þ ð5Þð:05Þ þ ð10Þð:35Þ ¼ 1:25 þ :25 þ 3:50
(2.58)
If we use the base-period value-added-per-unit-of-output figures,
υ0c1 ¼ p3c31 ¼ ð10Þð0:13Þ ¼ 1:30 and υ0c2 ¼ p3c32 ¼ ð10Þð0:35Þ ¼ 3:50
along with
0 1
ðI C Þ
1:243 :106
¼
:825 1:122
in p ¼ ðI C0 Þ1 υc ,
(from the 2 2 upper-left submatrix of C)
"
#"
# "
#
" 0#
1:254 :106
1:3
2:00
p1
0 1 0
¼
¼ ðI C Þ υc ¼
:825 1:122 3:5
5:00
p02
(2.59)
(2.60)
This generates the base year prices, as expected.
Example 2.4: Changed Base Year Prices Continuing with this physical
coefficients model, suppose that the wage costs in sector 1 increase from $10.00 to
$13.00 (a 30 percent increase) while those in sector 2 remain unchanged
1
1:69
ð13Þ ð:13Þ
1
0
1
. Then
¼
p31 ¼ $13:00 and p32 ¼ p32 ¼ 10:00 , so υc ¼
3:50
ð10Þ ð:35Þ
"
#"
# "
#
" 0#
1:254
:106
1:69
2:49
p1
1
¼
(2.61)
¼ ðI C0 Þ υ0c ¼
:825 1:122 3:50
5:32
p02
Specifically, p11 ¼ $2:49 (an increase of 24.5 percent over p01 ¼ $2:00) and p12 ¼ $5:32
(a 6.4 percent increase over p02 ¼ $5:00). This illustrates the operation of the cost-push
input–output price model based on physical input coefficients. It generates the new
prices directly (from which percentage changes can easily be found). In Section 2.6.4
we found these percentage increases directly from the index-price model in (2.37).
2.6.8 The Quantity Model Based on Physical Data
Data in physical units can also form the core of an input–output quantity model, as in
q ¼ ðI CÞ1 d in (2.43) – before the introduction of prices. Using the data from the
two numerical examples immediately above,
1:254 :825
:150 :625
1
C¼
and ðI CÞ ¼
:106 1:122
:080 :050
Base year outputs are correctly generated by
500
1:254 :825
175
1 0
0
¼
q ¼ ðI CÞ d )
400
:106 1:122 340
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2.6 The Price Model
53
Table 2.18 Transactions in physical terms (Germany, 1990) (millions of tons)
Primary
Secondary
Tertiary
Primary
Secondary
Tertiary
Final Demand
Total Output
2,248
27
5
1,442
1,045
69
336
206
51
84
708
36
4,110
1,986
161
and, for example, doubling demand doubles outputs,
1:254 :825
350
1,000
q1 ¼ ðI CÞ1 d1 )
¼
:106 1:122 680
800
This is completely parallel to the demand-driven model in monetary terms, except
that units of measurement are consistent only across each row. This means that the new
demands (350 bushels and 680 tons) lead to production of 1,000 bushels and 800 tons.
Notice the units in ðI CÞ1 . For example, in the first column, 1.254 represents direct
and indirect bushels of output per bushel of final demand, and 0.106 is direct and
indirect output of tons per bushel of final demand.
An early real-world illustration of this kind of model based on physical units appears
in Stahmer (2000)24. This consists of a 12-sector input–output data set in physical terms
for Germany in 1990 (an aggregation of a 91-sector model), where all transactions and
outputs are measured in a common physical unit – tons. Hubacek and Giljum (2003)
generate a three-sector aggregation of these data for the illustrations in their study (see
also Giljum and Hubacek, 2004). In particular, transactions are shown in Table 2.18.
As in the illustration in (2.57), the associated direct inputs matrix, here C, has
coefficients that are larger than 1:
2
3
:7261
2:0870 7
6 :5470
6
7
C ¼ 6 :0066
:5262
1:2795 7
4
5
:0012
:0347
:3168
|
|
|
|
|
- - - - - - - - - - - - - - - - - - |
|
As we saw with (2.43), this does not pose any problems for the usual input–output
calculations; here the Leontief inverse is easily found to be
2
3
2:3185
4:7204
15:9220
6
7
6
7
ðI CÞ1 ¼ 6 :0502
2:5486
4:9262 7
4
5
:0067
:1380
1:7425
|
|
|
|
|
- - - - - - - - - - - - - - - - - - - - - |
|
24
Also available as: “The Magic Triangle of Input–Output Tables,” paper presented to the 13th International
Input–Output Association Conference on Input–Output Techniques, Macerata, Italy, August, 2000. An updated
version is in Stahmer (2010). Hoekstra and van den Bergh (2006) discuss and compare six early PIOTs – for
The Netherlands, Germany, Denmark, Italy, Finland, and also the EU.
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54
Foundations of Input–Output Analysis
Some elements are large; these are associated with the large elements in C, but they are
not inappropriate in the context of this PIOT. The reader can easily check the validity
of this inverse from the base-case data, namely
2
32
3
2
3
2:3185 4:7204 15:9220
84
4,110
6
76
7
x ¼ 4 1,986 5 ¼ ðI CÞ1 f ¼ 4 :0502 2:5486 4:9262 54 708 5
161
:0067
:1380 1:7425
36
Despite the unusual elements in C, the power series approximation to the Leontief
inverse – I þ C þ C2 þ C3 þ – works just fine, although slowly; it requires
37 terms to come within four-digit accuracy. Here are some of the terms:
2
3
2
3
:0077 :0971 :3551
:0002 :0030 :0111
6
7
6
7
C10 ¼ 4 :0011 :0167 :0616 5, C20 ¼ 4 :0000 :0005 :0018 5,
:0001
:0018
:0067
:0000 :0001 :0002
2
3
0 :0001 :0003
6
7
C30 ¼ 4 0
0
:0001 5, C37 ¼ 0
0
and
Iþ
37
X
k¼1
!
Ck
0
0
2
3
2:3185 4:7204 15:9220
¼ 4 :0502 2:5486 4:9262 5 ¼ ðI CÞ1
:0067 :1380 1:7425
The interested reader with access to combinatorial algebra software on a computer
2
3
:4530 :7261 2:0870
might check that for this illustration, with ðI CÞ ¼ 4 :0066 :4738 1:2795 5,
:0012 :0347
:6832
the Hawkins–Simon conditions are satisfied, meaning that all seven principal minors of
ðI CÞ are positive (Appendix 2.2).
Interest in physical input–output tables has increased dramatically with the recognition of needs for quantitative assessment of the energy and environmental impacts of
consumption. A number of studies for individual countries have emerged and considerable interest is now centered on tracking these responsibilities across national boundaries. This will be examined in some detail in Chapters 12 and 13.
2.6.9 A Basic National Income Identity
From (2.43), q ¼ ðI CÞ1 d; from (2.51), p0 ¼ υ0c ðI CÞ1 , and postmultiplying this
by d,
p0 d ¼ υ0c ðI CÞ1 d ¼ υ0c q
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2.7 Summary
55
The total value of spending (exogenous final demand, p0 d) equals the total value of
earnings (payments to exogenous primary inputs, υ0c q), or national income spent equals
national income received.
2.7
Summary
We have introduced the basic structure of the input–output model in this chapter. After
investigating the special features of sectoral production functions that are assumed in the
Leontief system, we examined its mathematical features. Importantly, the model is
expressed in a set of linear equations, and we have tried to indicate the connection
between the purely algebraic solution to the input–output equations, using the Leontief
inverse matrix, and the logical, economic content of the round-by-round view of
production interrelationships in an economy. Both the algebraic details as well as the
economic assumptions needed to close the model with respect to households were
discussed. Some of the special problems associated with the concept of household
consumption coefficients have been addressed in applications, especially at the regional
level. We also introduced the Leontief price model, a logical (and mathematical)
companion to the quantity model, and we explored alternatives to both models when
the underlying data are measured in physical rather than monetary terms. Table 2.19
summarizes the alternatives. (Information in the monetary row is in Table 2.15.)
We turn to regional input–output models in Chapter 3. It is important to add the
regional dimension; many if not most important policy questions are not purely national
in scope. Rather, analysts (even at the national level) are interested in differential
regional effects of, say, a change in national government policy regarding exports. It is
important to know not only the total magnitudes of the new outputs, by sector, that come
about because of stimulation of exports, but also to know something of their geographical incidence – is a particularly depressed area helped by such export stimulation, or
does the increased output occur largely in areas that are economically more healthy?
Extensions of the basic model to deal with issues of this sort will occupy us in Chapter 3.
Table 2.19 Alternative input–output price and quantity models
Measurement Units
Quantity Model
1
Price Model
Monetary
x ¼ ðI AÞ f
(2.11)
p~ 0 ¼ vc0 ðI AÞ1
(2.32)
or
~ ¼ ðI A0 Þ1 vc
p
(2.33)
Physical
q ¼ ðI CÞ1 d
(2.43)
p0 ¼ vc0 ðI CÞ1
(2.51)
or
1
p ¼ ðI C0 Þ vc
(2.52)
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56
Foundations of Input–Output Analysis
The framework for regional extensions also applies to the increasingly important area of
multinational applications, where “regions” are replaced by “nations.”
There are two appendices to this chapter. Online Appendix SA2.1 (summarized
below as Appendix 2.1) addresses the relationship between two approaches used for
computing the impact of new final demands: (1) the Leontief inverse and (2) round-byround calculation of total impacts. Appendix 2.2 (below) explores the Hawkins–Simon
conditions which ensure a “viable” input–output economy – one in which any vector
of non-negative final demand induces a vector of non-negative industrial productions
or, equivalently, that the elements of the Leontief inverse must all be positive.
Appendix 2.1
The Relationship between Approaches I and II
Supplemental Appendix SA2.1, located on the internet web site associated with this text
(http://www.cambridge.org/millerandblair), examines the conceptual and mathematical
connections between two alternative approaches to computing the Leontief inverse
which are applied to the two-sector numerical example developed in Section 2.3.
Appendix 2.2
The Hawkins–Simon Conditions
No matter how many terms we use in the series approximation to ðI AÞ1 in (2.17),
it is clear that each of the terms contains only non-negative elements, since all aij 0.
As noted in Section 2.4, not only is A 0, but A2 0, . . ., An 0; therefore
I þ A þ A2 þ is a matrix of non-negative terms. If the elements of f are all
non-negative, then the associated x will contain non-negative elements also. This is
what one would expect; when faced with a set of non-negative final demands it would
be meaningless in an economy to find that one or more of the necessary gross outputs
were negative.25 For a Leontief system with A 0 and N(A) < 1 [so that the results in
(2.17) hold], we know that negative outputs will never be required from any sector to
satisfy non-negative final demands.
One could also explore conditions under which f 0 would always generate x 0
by examining the general definition ðI AÞ1 ¼ jI 1 Aj ½adjðI AÞ (Appendix A). For
the simplest, two-sector case,
2
ð1 a22 Þ
6
jI Aj
ðI AÞ1 ¼ 6
4 a21
jI Aj
25
3
a12
jI A j 7
7
ð1 a11 Þ 5
jI A j
In some models, as we have seen, negative values could have meaning. When both x’s and f’s are defined as
“changes in,” namely Δx and Δf, then a result like Δx3 ¼ 400 is interpreted as a decrease of $400 in sector
3’s output.
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Appendix 2.2
57
and all of the elements in ðI AÞ1 must be non-negative – the numerators must all be
non-negative and the denominator must be positive (the denominator must not be zero,
either). Or, all numerators could be non-positive and the denominator negative.
We have already noted that aij > 0 and that N(A) < 1 and (also by their definition)
all aij < 1.26 Thus all numerators in ðI AÞ1 are non-negative. Therefore, if
jI Aj > 0, all elements in the 2 2 Leontief inverse will be non-negative.
Hawkins and Simon (1949) investigated the issue of non-negative solutions to more
general equation systems. For a system in which A 0 (as in the input–output case)
but in which no restriction is placed on the column sums of A, they found for the 2 2
case that necessary and sufficient conditions to assure x 0 are27
ðaÞ ð1 a11 Þ > 0 and ð1 a22 Þ > 0
ðbÞ jI Aj > 0
(A2.2.1)
These conditions have a straightforward geometrical interpretation for the 2 2 case.
We examine the solution-space representation. The fundamental relations
ðaÞ ð1 a11 Þx1 a12 x2 ¼ f1
ðbÞ a21 x1 þ ð1 a22 Þx2 ¼ f2
(A2.2.2)
define a pair of linear equations in x1 x2 space. By setting one variable at a time equal to
zero in each equation, it is easy to find the intercepts of each line on each axis. These
are shown in Figure A2.2.1a, for arbitrary (but positive) f1 and f2 . (Assume that both
a12 and a21 are strictly positive, i.e., that each sector sells some inputs to the other. In a
highly aggregated model this is virtually certain to be the case.)
As long as ð1 a11 Þ > 0 and ð1 a22 > 0 the first Hawkins–Simon condition in
the 2 2 case – for f1 > 0 and f2 > 0, the intercept of (A2.2.2)(a) on the x1 -axis will be
to the right of the origin and the intercept of (A2.2.2)(b) on the x2 -axis will be above
the origin. Therefore, for non-negative total outputs, it is required that these two
equations intersect in the first quadrant; this means that the slope of equation (a) must
be greater than the slope of equation (b). These slopes are:
f1
ð1 a11 Þ
a12
¼
For equation ðaÞ ¼
f1
a12
ð1 a11 Þ
f2
a21
ð1 a22 Þ
For equation ðbÞ ¼
¼
f2
ð1 a22 Þ
a21
26
27
As we saw in Section 2.6, this need not be the case in input–output tables denominated in physical rather than
monetary terms – for example, liters of input per kilogram of output. See also Chapters 12 and 13.
The matrix algebra requirement for a unique solution to (I A)x ¼ f is that |I A| 6¼ 0. Now we are further
restricting this determinant to only positive values.
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58
Foundations of Input–Output Analysis
(a)
Figure A2.2.1a Solution space representation of (A2.2.2); a12 > 0 and a21 > 0
(b)
Figure A2.2.1b Solution space representation of (A2.2.2); a21 ¼ 0
and thus the slope requirement is ð1 a11 Þ=a12 > a21 =ð1 a22 Þ. Multiplying both
sides of the inequality by ð1 a22 Þ and by a12 – both of which are assumed to be strictly
positive – does not alter the direction of the inequality, giving ð1 a11 Þð1 a22 Þ >
a12 a21 or ð1 a11 Þð1 a22 Þ a12 a21 > 0, which is just |I – A| > 0, the second
Hawkins–Simon condition in the 2 2 case.
The effects of less interdependence in the two-sector economy are illustrated in
Figures A2.2.1b and A2.2.1c. If a21 ¼ 0, meaning that z21 ¼ 0 (sector 1 uses no inputs
from sector 2), then the slope of the line labeled (b) is zero. It is a horizontal line
intersecting the x2 -axis at the height f2 =ð1 a22 Þ. This is to be expected; the gross
output necessary from sector 2 depends only on final demand for the output of sector 2,
f2 , and the amount of intraindustry input that sector 2 buys from itself, a22
(Figure A2.2.1b). Similarly, if a12 ¼ 0 – sector 2 buys no inputs from sector 1 – line
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Appendix 2.2
59
(c)
Figure A2.2.1c Solution space representation of (A2.2.2); a12 ¼ 0
(a) in the figure will have an infinite slope; it will be vertical through the point
f1 =ð1 a11 Þ on the x1 -axis (Figure A2.2.1c).
The geometry of the 2 2 case does not generalize easily, at least for n > 3. For this,
we need some matrix terminology. The minor of an element aij in an n n square
matrix, A, is defined as the determinant of the ðn 1Þ ðn 1Þ matrix remaining
when row i and column j are removed from A (Appendix A). Another kind of minor
that is associated with a matrix (not with a particular element in a matrix) is a principal
minor. If none, or one, or more than one row and the same columns are removed from
A, the determinant of the remaining square matrix is a principal minor of A. Using the
concept of principal minors, the Hawkins–Simon conditions for the 2 2 case in
(A2.2.1) can be expressed compactly as the requirement that all principal minors of
(I A) be strictly positive – (a) in (A2.2.1) results from removing row and column 1 or
row and column 2, (b) in (A2.2.1) results from removing no rows and columns. (It is
impossible to remove more than n 1 rows and columns; if all n are gone, there is no
matrix left.)
For a 3 3 matrix A, removal of row and column 1, or row and column 2, or row
and column 3 leaves, in each case, a square 2 2 matrix. The determinants of those
three matrices are all principal minors of A (sometimes called second-order principal
minors, because they are determinants of 2 2 matrices). Moreover, removal of rows
and columns 1 and 2, or rows and columns 1 and 3, or rows and columns 2 and 3
leaves, in each case, a square 1 1 matrix (the determinant of a 1 1 matrix is defined
simply as the value of the element itself ); these are the three first-order principal
minors of A. By extension, the third-order principal minor in this case is just the
determinant of the entire 3 3 matrix, when no rows and columns are removed. Thus,
there are seven principal minors in a 3 3 matrix.
This principal minor rule can be generalized; namely, regardless of the size of n, the
parallel to (A2.2.1) is that all principal minors of (I A) – first-order, second-order, . . .,
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60
Foundations of Input–Output Analysis
nth-order – should be positive. The interested reader might try writing out the seven
principal minors of a 3 3 (I A) matrix. In the 4 4 case there are 15 principal
minors. (For the reader familiar with the mathematics of combinations, this number is
found as C 40 þ C 41 þ C 42 þ C 43 ¼ 1 þ 4 þ 6 þ 4 ¼ 15.) This gives some idea of the
way in which the complexity of these rules increases with the number of sectors in
the input–output model, and extension and application of the results in (A2.2.1) to
conditions for an n n system, with n even modestly large, would be cumbersome and
tedious, even though the definition of principal minors of a matrix presents a simple
way of expressing the rule for the general case. These conditions are totally impractical
to check for large, real-world input–output systems. [For example, for a 10-sector
model, the number of principal minors is 1,023 (!).]
However, there is a large amount of published work on alternative sets of conditions
on A and f that serve to identify when non-negative final demands will generate
non-negative outputs. Dietzenbacher (2005) provides an extremely simple sufficient
condition. If the original data are Z0 > 0 and f 0 0 (with at least one f 0i > 0), then
1
L0 ¼ I A0
> 0 and x1 ¼ L0 f 1 0 for any f 1 0. These requirements on Z0 and
f 0 are easily checked by inspection, bypassing the need for the Hawkins–Simon
principal minors. In fact, as noted in Dietzenbacher (2005), the positivity condition,
Z0 > 0, can be relaxed to the requirement of non-negativity, Z0 0, using an assumption that allows Z0 to contain many zeros.28 This allows for the more realistic case,
especially in highly disaggregated models, of zero-valued intermediate flows between
some sectors. An additional benefit is that derivation of these results does not depend
on a0ij < 1. When tables are based on transactions measured in physical terms it is
entirely possible that some coefficients will be larger than 1 and hence that N(A) > 1 –
as we saw in Section 2.6.
References
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in Extended Input–Output Models: A Comparative Theoretical and Empirical Analysis,”
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Batey, Peter W. J. and Melvyn J. Weeks. 1989. “The Effects of Household Disaggregation in
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(eds.), Frontiers of Input–Output Analysis. New York: Oxford University Press, pp. 119–133.
Bazzazan, Fatemeh and Peter W. J. Batey. 2003. “The Development and Empirical Testing of
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Dietzenbacher, Erik. 1997. “In Vindication of the Ghosh Model: A Reinterpretation as a Price
Model,” Journal of Regional Science, 37, 629–651.
28
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eigenvalues. The interested reader is referred to the thorough discussion of these and other mathematical issues
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3
3.1
Input–Output Models at the
Regional Level
Introduction
Originally, applications of the input–output model were carried out at national levels –
for example, to assess the impact on the individual sectors of the US economy of a
change from war to peacetime production as the end of World War II approached. Over
time, interest in economic analysis at the regional level – whether for a group of states
(as in a federal reserve district), an individual state, a county, or a metropolitan area –
has led to modifications of the input–output model which attempt to reflect the
peculiarities of a regional (subnational) problem. There are at least two basic features
of a regional economy that influence the characteristics of a regional input–
output study.
First, although the data in a national input–output coefficients table are obviously
some kind of averages of data from individual producers located in specific regions,
the structure of production in a particular region may be identical to or it may differ
markedly from that recorded in the national input–output table. Soft drinks of a
particular brand that are bottled in Boston probably incorporate basically the same
ingredients in the same proportions as are present in that brand of soft drink produced
in Kansas City or Atlanta or in any other bottling plant in the USA. On the other hand,
electricity produced in eastern Washington by water power (Coulee Dam) represents
quite a different mix of inputs from electricity that is produced from coal in
Pennsylvania or by means of nuclear power or “wind farms” elsewhere. For these
reasons, the early methodology for regional input–output applications – which used
national input coefficients with some minor modifications – has given way to coefficients tables that are tailored to a particular region on the basis of data specific to
that region.
Secondly, it is generally true that the smaller the economic area, the more dependent
that area’s economy is on trade with outside areas – transactions that cross the region’s
borders – both for sales of regional outputs and purchases of inputs needed for
production. That is, one of the elements that contributed to the exogenous finaldemand sector in the model described in Chapter 2 – exports – now will generally
be relatively much more important, and a higher proportion of inputs will be imported
from producers located outside of the region. To exaggerate, a one-world economy
63
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64
Input–Output Models at the Regional Level
would have no “foreign trade,” since all sales and purchases would be internal to the
worldwide “region,” whereas an urban area depends very much on imports and exports
(imports of components to aircraft production and exports of airliners from the greater
Seattle urban area).
In this chapter we will explore some of the attempts that have been made to
incorporate these features of a regional economy into an input–output framework.
Such regional input–output models may deal with a single region or with two or more
regions and their interconnections. The several-region case is termed interregional
input–output analysis (in one version) or multiregional input–output analysis (in
another version). We will explore each of these kinds of regionalized input–output
models, as well as what is known as the balanced regional model. We will also
examine more recent developments of multinational models – where the “regions”
are now “nations.” In these cases, the connections between nations take the form of
international trade flows, and data on such transactions are sometimes more readily
available than data on flows between regions – although other issues arise (differing
currencies, tariffs, etc.).
At the regional level, some of the earliest single-region applications are found in
Moore and Petersen (1955), Isard and Kuenne (1953), Miller (1957), and Hirsch
(1959). A very thorough discussion and documentation of the details involved in
producing a regional input–output table during the early period in the development
of this area of application is provided by Isard and Langford (1971) – in this case the
region was the Philadelphia Standard Metropolitan Statistical Area – and in Miernyk
et al. (1967) for Boulder, Colorado, and Miernyk et al. (1970) for West Virginia.
Overviews of early regional input–output models can be found in Polenske (1980,
chapter 3) and in Miernyk (1982). A large amount of continuing work in this area
appears in such journals as Economic Systems Research, Journal of Regional Science,
International Regional Science Review, and Papers in Regional Science.1 In addition,
numerous regional input–output tables and studies using these tables have been
published by the appropriate sub-national agencies (state and local governments or
their counterparts outside the USA) for whom the analysis was done, or by universities
where the work was done.
In Section 3.6 we indicate some examples of how the geographic scale of connectedregion models has evolved in both the micro- and macroscopic directions from these
earliest applications – down to models of as small an area as an inner-city neighborhood and up to what are now world (global) models. Examples of regional applications
will also be discussed in Chapter 6 on multipliers and in Chapter 10 on estimating
regional data. Much of the material on regional and interregional input–output models
in this chapter and subsequently throughout this text is also covered (in less detail) in
Miller (1998), Oosterhaven and Hewings (2014), and Oosterhaven, Polenske, and
Hewings (2019).
1
There is also a growing number of environmental applications using multiregional or multinational models (see
Chapter 13).
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3.2 Single-Region Models
3.2
65
Single-Region Models
3.2.1 National Coefficients
Generally, regional input–output studies attempt to quantify the impacts on the producing sectors located in a particular region that are caused by new final demands for
products made in the region. Early regional studies (Isard and Kuenne, 1953; Miller,
1957) used a national table of technical coefficients in conjunction with an adjustment
procedure that was designed to capture some of the characteristics of the regional
economies, since specific coefficients tables for the particular regions did not exist.2
We use a superscript r to designate “region r” in the same way that subscript i
denoted “sector i” in the discussion in Chapter 2. Thus, just as xi was used to denote
the gross output of sector i, let xr ¼ ½xri denote the vector of gross outputs of sectors
r
in region r. Similarly, f r ¼ ½f i represents the vector of exogenous demands for
goods made in region r. For example, if r denotes Washington State, one element of
f r could be an order from a foreign airline for commercial aircraft from Boeing
in Washington.
The problem in these early regional studies was that only a national technical
coefficients matrix, A, was available, but what was needed, essentially, was a matrix
showing inputs from firms in the region to production in that region. Denote this
rr
unknown matrix by Arr ¼ arr
ij , where aij is the amount of input from sector i in r per
dollar’s worth of output of sector j in r. (This anticipates notation later for many-region
models, where we will need two superscripts to identify origin and destination regions,
just as i and j are origin and destination sectors.) Assume, in the absence of evidence to
the contrary, that local producers use the same production recipes as are shown in
the national coefficients table, meaning that the technology of production in each
sector in region r is the same as in the nation as a whole. Nonetheless, in order to
translate regional final demands into outputs of regional firms ðxr Þ, the national
coefficients matrix must be modified to produce Arr (locally produced goods in local
production).
Early studies carried out this modification through the use of estimated regional
supply percentages, one for each sector in the regional economy, designed to show the
percentage of the total required outputs from each sector that could be expected to
originate within the region. One straightforward way to estimate these percentages,
using data that may often be obtainable at the regional level, requires knowledge of (1)
total regional output of each sector i, xri , (2) exports of the product of each sector i from
region r, eri , and (3) imports of good i into region r, mri . Then, one can form an
expression for the proportion of the total amount of good i available in region r that
was produced in r (the regional supply proportion of good i). We denote this by pri,
where
2
The “regions” were the Greater New York–Philadelphia urban-industrial region (consisting of 2 counties in
Connecticut, 11 in New York, 19 in New Jersey, and 5 in Pennsylvania) in the first case, and the states of
Washington, Oregon, and Idaho in the second.
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66
Input–Output Models at the Regional Level
pri ¼
(xri eri )
r
ðxi eri þ mri Þ
The numerator is the locally produced amount of i that is available to purchasers in r;
the denominator is the total amount of i available in r, either produced locally or
imported. (Thus pri 100 is an estimate of the regional supply percentage for sector i
in region r – the percentage of good i available in r that was produced there.)
Assuming that we can estimate such proportions for each sector in the economy, each
element in the ith row of the national coefficients matrix could be multiplied by pri to
generate a row of locally produced direct input coefficients of good i to each local
producer. If we arrange these proportions in an n-element column vector, pr , then our
^ r A. For a two-sector model, this is
working estimate of the regional matrix will be Arr ¼ p
r
r
p1 a11 pr1 a12
p1 0
a11 a12
r
rr
^ A¼
A ¼p
¼ r
p2 a21 pr2 a22
0 pr2 a21 a22
^ r AÞ1 f r . This uniform modification of the
For any f r we could then find xr ¼ ðI p
elements in a row of A is a strong assumption. It means, for example, that if the
aircraft, kitchen equipment, and pleasure boat sectors in Washington all use aluminum
(sector i) as an input, all three sectors buy the same percentage, pri , of their total
aluminum needs from firms located within the state. :15 :25
In the two-sector example in Chapter 2 we had A ¼
. Assume that this
:20 :05
is a national table, and that we want to create Arr from it, and that there is no evidence
that the basic structure of production in the region differs from the national average
structure reflected in A. The unique features of the region, however, are captured in the
regional supply percentages. Using regional output, export, and import data, suppose we
estimate that 80 percent of sector 1 goods will come from firms in that sector within the
region, but only 60 percent of sector 2 goods can be expected to be supplied by regional
:8
. Suppose that the projected (new) final demand in the
firms in sector 2, so pr ¼
:6
600
(this is the final demand vector that was used for some of the
region is f r ¼
1,500
numerical examples in Chapter 2). Then
:8 0
:8 0
:15 :25
:12 :20
r
r
rr
^ ¼
^ A¼
p
,A ¼ p
¼
,
0 :6
0 :6 :20 :05
:12 :03
rr 1
ðI A Þ
1:169 :241
¼
:145 1:061
and using this regional inverse directly,
1:169 :241
600
1,062:90
r
rr 1 r
x ¼ ðI A Þ f ¼
¼
:145 1:061 1,500
1,678:50
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(3.1)
3.2 Single-Region Models
67
This tells us that the total output that will need to be produced in the region by sectors
1 and 2 is $1,062.90 and $1,678.50, respectively.
Subsequently to this early work in regional input–output analyses, attempts have
been made to model the characteristics of a regional economy more precisely. Many
“regionalization” procedures have been proposed, tested, and modified over the years
in both multiregional and (more recently) multinational settings. We examine some
additional early work briefly in Section 3.2.2 and then return to the issue again in
Chapter 10.
3.2.2 Regional Coefficients
We noted in Section 3.1 that electricity produced in Washington will most likely have
a different production recipe (column of technical coefficients) from electricity
made in Pennsylvania. These regionally produced electricities are really two different
products – “hydroelectric power” and “coal-fired electrical power.” As another
example, consider the aircraft sector. In a national table, this would include the
manufacture of a mix of commercial, business, and personal aircraft. One input to this
sector would be the huge jet engines used on Boeing commercial airliners in Seattle.
On the other hand, the aircraft sector in a regional table for the state of Florida might
reflect the manufacture of small general aviation aircraft by Piper in Vero Beach, for
which the jumbo jet engines are not an input at all; in a Washington table, however, jet
engines are an extremely important input.
Sectors in even very disaggregated national input–output tables will be made up of a
variety of products – as in the aircraft sector example. And firms within that sector,
located in various regions of the country, will generally produce only a small number
of those products – Boeing in Washington does not produce small general aviation
aircraft; Piper in Florida does not produce jet airliners that can carry upwards of 300
passengers. This illustrates the so-called product-mix problem in input–output; firms
classified in the same sector actually produce different sets of products. The most
straightforward way to avoid this problem is to survey firms in the region and construct
what is called a survey-based regional input–output table. In conducting such a survey,
one can pose essentially two variants of the basic question. In asking firms in sector j in
a particular region about their use of various inputs, the question can be:
1. How much sector i product did you buy last year in making your output? (For
example, how much aluminum did aircraft manufacturers in Washington State buy
last year?), or
2. How much sector i product did you buy last year from firms located in the region?
(For example, how much aluminum used by aircraft producers in Washington was
purchased from producers in Washington?)3
3
If it is also possible to determine how much came from firms located outside the state then one has the
beginnings of an interregional or multiregional model. These are discussed in Sections 3.3 and 3.4.
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68
Input–Output Models at the Regional Level
In the former case a truly regional technical coefficients table would be produced; this
would better reflect production practices in the region than does the national table – it
would eliminate the input of large jet engines into the manufacture of private aircraft in
Florida, for example. But it would not address the question of how much of each
required input came from within the region and how much was imported. On the other
hand, a set of coefficients based on inputs supplied from firms within the region for
outputs of firms in the region would reflect regional production technology. These are
termed regional input coefficients. They are to be distinguished from regional technical
coefficients since they do not always accurately describe the technology of regional
firms, but rather only the way in which local firms use local inputs. (Intraregional input
coefficients would be an even more precise, although cumbersome, description.4)
Rather than adapt a national coefficients table through application of regional supply
proportions, some regional analysts have tried to derive true regional input coefficient
tables through surveys of regional establishments using variants of question 2. A long
series of tables for Washington State illustrates this kind of survey-based modeling
effort, specifically for the state for 1963, 1967, 1972, 1982, 1987, 2002, 2007, and
2012. (There is also a Washington table for 1997 produced mainly by a nonsurvey
estimating technique; nonsurvey approaches are explored in some detail in Chapters 9
and 10.)5 In chronological order, the Washington tables appeared in Bourque and
Weeks (1969), Beyers et al. (1970), Bourque and Conway (1977), Bourque (1987),
Chase, Bourque, and Conway (1993), Beyers, Lefberg, and Lin (2011), Beyers,
Baldwin, and Lin (2012), and Roberts and Beyers (2020). Many comparative studies
have been based on these data sets.
To examine this kind of extension, we need more complicated notation. We continue
to use a superscript r for the region in question. Then let zrr
ij denote the dollar flow of
goods from sector i in region r to sector j in region r.6 Just as the order of subscripts is
“from–to” with respect to sectors, the order of superscripts indicates “from–to” with
respect to geographic locations. If we had a complete set of data on zrr
ij for all n sectors
in the regional economy, and also data on gross outputs (xrj ) of each sector in the
region, a set of regional input coefficients could be derived as
arr
ij ¼
zrr
ij
xrj
(3.2)
r
¼ [xrj ]; then the regional input coefficients matrix is
Let Zrr ¼ [zrr
ij ] and x
ðnnÞ
ðn1Þ
1
Arr ¼ Zrr ðx^r Þ
(3.3)
Tiebout (1969, p. 335) used “direct intraregional interindustry coefficient,” which is completely precise but also
rather cumbersome.
5
The 2002, 2007, and 2012 Washington tables are available at https://ofm.wa.gov/washington-data-research/
economy-and-labor-force/washington-input-output-model/. This website also includes an entire historical series
of the Washington transactions tables for each of the years noted in the text.
6
We need double superscripts because later we will also measure interindustry flows between regions – as in zsr
ij .
4
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3.2 Single-Region Models
69
^ r A.)
(This is what was approximated in the early regional studies described above by p
Then the impacts on regional production of a final-demand change in region r would
be found as
xr ¼ ðI Arr Þ1 f r
(3.4)
3.2.3 Closing a Regional Model with Respect to Households
The Washington State models noted in Section 3.2.2 were closed with respect to
households in the manner described in Chapter 2 – by adding a household
consumption column and a labor input row. One extension to the process of endogenizing households in an input–output model is to add more than one row and
column to the direct input coefficients matrix, in order to distinguish various kinds of
consumers. This approach is frequently implemented at a regional level, although it
can apply equally well to national models. As usual, the impacts of projected
increases in final demand will be increased sectoral outputs, and therefore increased
payments for labor services. The basic idea is that a distinction should be made
between consumption habits of various kinds of consumers – for example, at a subnational level, those of established residents of the region, who may experience
an increase in their incomes (for example, due to productivity increases) and
the consumption patterns of new residents, who may move into the region in
anticipation of employment (new income). This distinction apparently originated with
Tiebout (1969), where they are designated intensive and extensive income growth,
respectively.
The reason for the distinction is that current residents may spend each dollar of
new income according to a set of marginal consumption coefficients, while new
residents may distribute their purchases according to a set of average consumption
coefficients.
The presumption should be clear: as new residents move in to fill jobs at the same wage rate as
established residents, average consumption propensities are relevant. Insofar as regional income rises
because of increased per capita incomes, marginal consumption propensities apply. (Tiebout, 1969,
p. 336)
If sales, by sector, could be broken down into those to new residents and those to
existing residents, and if labor payments, by sector, could be similarly disaggregated,
then marginal and average household consumption coefficients could be derived.
Similarly, knowing each sector’s outputs, “old” and “new” labor inputs per dollar’s
worth of output could be found. These would form two additional rows and columns
with which to close the model.
In practice, such data are not so conveniently available. Tiebout (1969) describes the
derivation of extensive and intensive coefficients in a regional model for the state of
Washington. Miernyk et al. (1967) investigate essentially the same issue for their
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70
Input–Output Models at the Regional Level
pioneering Boulder, Colorado, input–output study.7 In addition, an attempt was made
in the Boulder study to disaggregate the income increases to existing residents by
income class, with lower marginal consumption propensities in higher income classes
(see Miernyk et al., 1967, esp. chapter V).
Instead of disaggregating households into “old” and “new” residents, Blackwell
(1978) proposes a tripartite division into intensive and extensive (current residents and
new residents, respectively) and also redistributive, which is that portion of any new
income that goes to previously unemployed local residents. The distinction between
currently employed and currently unemployed workers is also explored in some detail
by Madden and Batey (1983, and elsewhere).8 The considerable work of Madden and
Batey and their colleagues on “extended” input–output models is representative of a
large body of research linking population and economic models. It is summarized in
Batey and Madden (1999), which also contains references to a great deal of earlier
work by them and by others. Miyazawa (1976) also investigates extensions to multiple
categories of consumption spending and income recipients. We further explore various
model closures (including the Miyawaza formulation) in Chapter 6 when we investigate input–output multipliers.
Additional issues also arise when closing a model to households at the regional
level. For example, consumer shopping by mail-order (and now more frequently
online) increases the possibilities for consumer spending that flows out of the region.
Such transboundary income flows also occur frequently when large metropolitan areas
are located close to regional boundaries (e.g., New Jersey residents shopping in New
York City). Conversely, spending in a city by visiting tourists represents a monetary inflow that is unrelated to income earned by in-region residents. (See Rose and Stevens,
1991, for a comprehensive discussion of these transboundary income and expenditure
flow problems.)9
3.3
Many-Region Models: The Interregional Approach
Single-region models of the sort described in Section 3.2 represent one approach to
modeling a regional economy in input–output terms. What they fail to do, however, is
to recognize in an operational way the interconnections between regions. The one
region of interest (region r) was essentially “disconnected” from the rest of the country
within which it is located, in the sense that its production recipes are reflected in an
intraregional matrix, Arr . For a country made up of several regions, a number of
7
Tiebout’s contribution in formulating this distinction between extensive and intensive consumption propensities in a region is noted by Miernyk et al. (1967, p. 104, n. 9). A draft of Tiebout’s paper was completed by
1967 and was published posthumously in 1969, following his death in January, 1968.
8
Other early examples of “extended” models with households endogenized (by no means an exhaustive list)
include Schinnar (1976), Beyers (1980), Gordon and Ledent (1981), Ledent and Gordon (1981), and Joun and
Conway (1983). These combined models are sometimes referred to as demo-economic – or also as ecodemographic. The demo-economic components reflect inputs from various labor (household) groups, and the
eco-demographic components capture activity such as consumption by various household types.
9
We return to this issue in Section 6.3 on multipliers in regional models.
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3.3 Many-Region Models: The Interregional Approach
71
important questions have several-region implications. Next year’s national defense
budget might include a large order for a certain type of aircraft built in California, the
overhaul of one or more ships in Virginia, and modernization and upgrading of an
army base in New Jersey. Each of these activities can be expected to have ramifications
not only within the region (state, in this example) where the activity takes place, but
also in other states. The total economic effect is therefore likely to be larger than the
sum of the regional effects in California, Virginia, and New Jersey. Firms outside
California will produce goods that will be imported to California for aircraft production; those firms, in turn, may import goods from other states for their production.
Materials for ship overhaul will come to Virginia from suppliers outside that state.
Electronic parts for the base upgrading in New Jersey may be imported from elsewhere
and the electronics firms, in turn, will need both local (wherever they are located) and
imported inputs, and so on.
A fundamental problem in many-region input–output modeling is therefore the
estimation of the transactions between regions. One approach, the interregional model,
requires a complete (ideal) set of both intra- and interregional data. For the two-region
ss ss
case, this means knowing xr ¼ ½xri , xs ¼ ½xsi , Zrr ¼ zrr
ij , and Z ¼ zij along with
Zrs ¼ zrs
recording transactions from sector i in region r to sector j in region
ij – sr
s – and Z ¼ zsr
ij – in which flows from s to r are captured. It is the last two matrices
that cause the most trouble. In practice, it is never the case that one has such detailed
information, and the requirements grow quickly with the number of regions – a threeregion model has six interregional matrices, a four-region model has 12, and so on.
Alternative forms of many-region input–output models were created and elaborated
by members of the Harvard Economic Research Project (HERP) under Leontief’s
direction, from its inception through the 1960s.10 Taken chronologically, the interregional input–output model (IRIO) structure was first described by Isard (1951) and
elaborated in Isard et al. (1960). (This is often labeled the “Isard model”.) Leontief
et al. (1953) sketched the framework of an intranational input–output model (often
referred to as a “balanced regional model”; Section 3.5). This was later applied to
assess the sectoral and regional impact of a cut in US arms spending in Leontief et al.
(1965). The multiregional input–output model (MRIO) was (almost simultaneously)
described in Chenery (1953) (a two-region model for Italy) and in Moses (1955) (a
nine-region US model) – thus the label “Chenery–Moses model.” Finally, Leontief and
Strout (1963) proposed a gravity-model approach to estimation of interregional flows
in a connected-region input–output model.11 In this section we explore the interregional input–output (IRIO) model.
10
HERP was started at Harvard by Leontief in 1948 and continued until 1972. Thorough accounts of this
formative work can be found in Polenske (1995, 2004).
11
Isard et al. (1960, esp. chapter 11) described gravity models and explored their potential for estimating
interregional interactions (including commodity flows) in detail. We explore the gravity approach and others
in Section 10.6.1, on estimating interregional flows.
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Input–Output Models at the Regional Level
Table 3.1 Interindustry, interregional flows of goods
Purchasing Sector
Region r
Region s
Selling Sector
1
2
3
1
2
Region r
zrr
11
zrr
21
zrr
31
zsr
11
zsr
21
zrr
12
zrr
22
zrr
32
zsr
12
zsr
22
zrr
13
zrr
23
zrr
33
zsr
13
zsr
23
zrs
11
zrs
21
zrs
31
zss
11
zss
21
zrs
12
1
2
3
Region s
1
2
zrs
22
zrs
32
zss
12
zss
22
3.3.1 Basic Structure of Two-Region Interregional Input–Output Models
For purposes of illustration, we consider a two-region economy (for example, in Italy,
northern Italy and southern Italy; or, in the USA, New England and the rest of the
USA). Using r and s, as before, for the two regions, let there be three producing sectors
(1, 2, 3) in region r and two (1, 2) in region s. Suppose that one has information for
sr
region r on both intraregional flows, zrr
ij , and interregional flows, zij . There will be nine
of the former and six of the latter. Suppose, further, that the same kind of information is
available (perhaps through a survey) on the use of inputs by firms located in region s,
ss
zrs
ij and zij . This complete table of intraregional and interregional data can be represented as
rr
Z
Zrs
Z¼
Zsr Zss
Table 3.1 indicates the full set of data.12
In the regional models of Section 3.2, we utilized intraregional information only – as
in (3.2), (3.3), and (3.4). We now want to incorporate much more explicitly the
interregional linkages, as represented by information in Zrs and Zsr .
These off-diagonal matrices need not be square. Here Zrs has dimensions 3 2 and
sr
Z is a 2 3 matrix. The on-diagonal matrices are always square; for this example, Zrr
and Zss are 3 3 and 2 2, respectively. While the elements in Zrs represent “exports”
from region r and simultaneously “imports” to region s, it is usual in regional input–
output work to refer to these as interregional trade (or simply trade) flows and to use
the terms export and import when dealing with foreign trade that crosses national, not
just regional, boundaries.
By surveying firms in both regions on their purchases of locally produced inputs and
inputs from the other region, one would accumulate the data shown in the various
12
To be more consistent with already-familiar subscript notation, one could denote the regions by 1 and 2,
21
respectively. Then an element such as zsr
13 would be denoted z13 . However, for purposes of exposition it seems
clearer to use lowercase letters to designate regions; for example, so as to avoid having z’s with four different
numbers attached to them.
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3.3 Many-Region Models: The Interregional Approach
73
columns of Table 3.1. On the other hand, the data in Table 3.1 could also be gathered
by asking firms in each region how much they sold to each sector in their region and
how much they sold to sectors in the other region. This would generate the figures
shown in the various rows of Table 3.1.13
Consider again the basic equation for the distribution of sector i’s product, as given
in (2.1):
xi ¼ zi1 þ zi2 þ þ zij þ þ zin þ fi
One of the components recorded in the final-demand term was exports of sector i
goods. In the two-region interregional input–output model, that part of f i that represents sales of sector i’s product to the productive sectors in the other region (but not to
consumers in the other region) is removed from the final-demand category and
specified explicitly. For our two-region example, the output of sector 1 in region r
would be expressed as
rr
rr
rs
zrs
xr1 ¼ zrr
11 þ z12 þ z13 þ
11 þ z12
|fflfflfflffl
ffl{zfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
Sector 1 intraregional,
Sector 1 interregional,
interindustry sales
interindustry sales
þ
f r1
|{z}
(3.5)
Sector 1 intraregional
sales to final demand
There will be similar equations for xr2 and xr3 , and also for xs1 and xs2 . The regional input
coefficients for region r were given in (3.2). There will also be a set for region s,
ass
ij ¼
zss
ij
xsj
(3.6)
Interregional trade coefficients are found in the same manner, where the denominators
are gross outputs of sectors in the receiving region. Here these are
ars
ij ¼
zrs
zsr
ij
ij
sr
and
a
¼
ij
xsj
xrj
(3.7)
Using these regional input and trade coefficients, (3.5) can be re-expressed as
r
r
rr r
rr r
rs s
rs s
xr1 ¼ arr
11 x1 þ a12 x2 þ a13 x3 þ a11 x1 þ a12 x2 þ f 1
(3.8)
Again, there will be similar expressions for xr2 , xr3 , xs1 , and xs2. [Compare (2.6), where
there was no regional dimension – no superscripts r and s – and where there were n
sectors.] Following the same development as in Chapter 2, by moving all terms
involving xr or xs to the left, (3.8) becomes
r
r
rr r
rr r
rs s
rs s
1 arr
(3.9)
11 x1 a12 x2 a13 x3 a11 x1 a12 x2 ¼ f1
There are similar equations with f2r , f3r , f1s , and f2s on the right-hand sides.
13
Usually, one has some (not complete) information on purchases and also some (not complete) information on
sales. The problem then is to produce a table from possibly inconsistent data. This reconciliation problem is
discussed in Section 10.9.
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74
Input–Output Models at the Regional Level
For the present example, Arr [(3.3)] is
2 rr
a11
6 rr
rr
A ¼ 4 a21
arr
12
arr
22
7
arr
23 5
arr
31
arr
32
arr
33
arr
13
3
1
Also, for this example, Ass ¼ Zss ðx^s Þ , and the two trade coefficients matrices are
1
1
Ars ¼ Zrs ðx^s Þ and Asr ¼ Zsr ðx^r Þ . Using these four matrices, the five equations of
which (3.9) is the first can be represented compactly as
ðI Arr Þxr Ars xs ¼ f r
Asr xr þ ðI Ass Þxs ¼ f s
(3.10)
where f r is the three-element vector of final demands for region r goods, and f s is the
two-element vector of final demands for region s goods.
We define the complete coefficients matrix for a two-region interregional model as
consisting of the four submatrices
rr
Ars
A
A¼
Asr Ass
For the current example, this will be a 5 5 matrix. Similarly, let
2
3
r
r
I
0
x
f
ð33Þ ð32Þ 5
x¼ s , f ¼ s , I¼4
0
I
x
f
ð23Þ
ð22Þ
Then (3.10) can be expressed as
ðI AÞx ¼ f
(3.11)
as in (2.10). To highlight the structure of (3.11), it can be expressed less compactly as
rr
r r
I 0
A
Ars
x
f
¼ s
(3.12)
0 I
Asr Ass
xs
f
Note that, in using an interregional model of this kind for analysis, not only is
stability of the (intra)regional input coefficients necessary (the elements of Arr and
Ass ), but also interregional input coefficients in Ars and Asr are assumed unvarying
over time. Thus, both the structure of production in each region and interregional trade
patterns are “frozen” in the model. For a given level of final demands in either or
both regions, the necessary gross outputs in both regions can be found in the usual
input–output fashion as x ¼ ðI AÞ1 f. As is clear from (3.12), this complete (I A)
matrix will be larger than that for the single-region model – if both regions are
divided into n sectors, the single-region matrix would be of size n n and the full
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3.3 Many-Region Models: The Interregional Approach
75
two-region interregional model would be 2n 2n, which means four times as many
(possible) elements of information are needed (many of which may be zero, of course).
However, aside from these dimensionality effects, the analysis proceeds along
similar lines.
The advantage is that the model captures the magnitude of effects on each sector in
each region; interregional linkages are made specific by sector in the supplying
region and by sector in the receiving region. The accompanying disadvantages are
primarily the greatly increased data needs and the necessary assumptions of constancy of interregional trading relationships. If it is not always easy to accept the idea
of constant input coefficients in general, in the national input–output model, it may
be even more difficult to believe that imports of good i per dollar’s worth of sector j
output in a specific region remain constant, no matter how much sector j’s output
changes.
3.3.2 Interregional Feedbacks in the Two-Region Model
Consider an increase in the demand by a foreign airline for commercial aircraft
produced in Washington State (region r). Certain subassemblies and parts will be
purchased from sectors outside the region (for example, jet engines from Connecticut,
region s). This stimulus of new output in Connecticut because of new output in
Washington is often called an interregional spillover. The increased demand for aircraft
will increase the demand for engines and consequently for all of the direct and indirect
inputs to the manufacture of jet engines, one of which might be extruded aluminum
components made in Washington. This idea is illustrated in Figure 3.1.
The downward arrow connecting Washington output to Connecticut output represents an interregional spillover effect; the upward arrow from Connecticut to
Washington is also an interregional spillover – the first originates in Washington
ðr ! sÞ, the second originates in Connecticut ðs ! rÞ. The loop (two arrows)
connecting Washington output to itself, via Connecticut output, represents an interregional feedback effect ðr ! rÞ; in other words, Washington needs more inputs from
Connecticut and therefore Connecticut needs more inputs from everywhere, including
Figure 3.1 Increases in Washington final demands affecting Washington outputs
via Connecticut
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76
Input–Output Models at the Regional Level
Washington. The interregional model in its two-matrix-equation form [in (3.10)]
allows one to isolate exactly the magnitude of such interregional feedbacks.
Suppose, in (3.10), that we read xr , xs , f r , and f s as “changes in” – that is, Δxr , Δxs ,
Δf r , and Δf s . Given a vector of changes in final demands in the two regions, we can
find the consequent changes in gross outputs in both regions. Assume, for simplicity,
that Δf s ¼ 0; we are assessing the impacts in both regions of a change in final demands
in region r only. Under these conditions, solving the second equation in (3.10) for xs
gives
xs ¼ ðI Ass Þ1 Asr xr
and putting this into the first equation, we have
ðI Arr Þxr Ars ðI Ass Þ1 Asr xr ¼ f r
(3.13)
Note that a single-region model (for region r), as in (3.4), would be ðI Arr Þxr ¼ f r .
The “extra” (second) term, subtracted on the left in (3.13),
Ars ðI Ass Þ1 Asr xr
(3.14)
represents exactly the added demands made on the output of region r because of
interregional trade linkages; it is an interregional feedback term. Consider the
various parts, starting at the right: (a) Asr xr captures the magnitude of flows from s
to r because of increased output in r (the value of engines that are shipped from
Connecticut to Washington for installation in the new airplanes), (b) ðI Ass Þ1 Asr xr
then translates these flows into total direct and indirect needs in s to produce the
required shipments from s (Connecticut production in all sectors needed to supply
the engines for shipment to Washington), and (c) Ars ðI Ass Þ1 Asr xr indicates the
magnitude of the additional sales from r to s that will be necessary to sustain
the total s-based production found in (b) [new outputs from Washington sectors
to satisfy Connecticut demand for inputs to Connecticut production quantified
in (b)].14
Thus the strength and importance of interregional linkages depend not only on the
elements of the interregional input coefficients matrices – Ars and Asr , in this example –
but also on the full set of regional input coefficients in the other region, as represented
by ðI Ass Þ1. It is precisely these kinds of spatial linkages that distinguish complete
interregional models from single-region models. Since the feedback term is subtracted
from ðI Arr Þxr in (3.13), a given value of f r will generate a larger xr than in a
single-region analysis in order that the required shipments to region s can be met, as
well as the usual intraregional shipments, Arr xr . In terms of outputs, the single- and
14
The arrows in Figure 3.1 indicate the directions of transmission of demands to producers. The output responses
to those demands travel in the opposite direction along the arrows.
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3.3 Many-Region Models: The Interregional Approach
77
Table 3.2 Flow data for a hypothetical two-region interregional case
Purchasing Sector
Region r
Selling Sector
Region r
1
2
3
1
2
Region s
Region s
1
2
3
1
2
150
200
300
75
50
500
100
500
100
25
50
400
50
60
25
25
200
60
200
150
75
100
40
250
100
two-region models will generate xr ¼ ðI Arr Þ1 f r and xr ¼ ðI Arr Ars Lss Asr Þ1 f r ,
respectively.
3.3.3 Numerical Example: Hypothetical Two-Region Interregional Case
To illustrate for the two-region case, suppose that the figures in Table 3.2 represent the
data in Table 3.1. Also, let
2
3
200
6
7
2
3
r 6 1,000 7
200
6
7
515
f
f r ¼ 4 1,000 5 and f s ¼
50 7
, so that f ¼ s ¼ 6
6
7
f
450
6
7
50
515
4
5
-
-
-
-
450
Thus
2
1,000
3
6
7
3
r 6 2,000 7
1,000
6
7
1,200
x
xr ¼ 4 2,000 5, xs ¼
1,000 7
, and x ¼ s ¼ 6
6
7
x
800
6
7
1,000
4 1,200 5
2
-
-
-
-
800
and Arr is found to be
Similarly,
:1667
Ass ¼
:1250
2
3
:150 :250 :050
Arr ¼ 4 :200 :050 :400 5
:300 :250 :050
2
3
:0208 :0938
:3125
:0750 :0500 :0600
rs
sr
4
5
, A ¼ :1667 :1250 , A ¼
:1250
:0500 :0125 :0250
:0500 :0500
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78
Input–Output Models at the Regional Level
so
2
Arr
A¼
Asr
and define
L11
L¼
L21
6 :1500
6
6
6 :2000
rs
A
6
¼ 6 :3000
Ass
6
6
6 :0750
4
:0500
:2500
:0500
:0500
:4000
:2500
:0500
:0500
:0600
|
:0208
|
|
:1667
|
|
|
:0500
|
3
:0938 7
7
:1250 7
7
7
:0500 7
7
7
:3125 7
5
:1250
- - - - - - - - - - - - - - - - - - - | - - - - - - - - - - - -
:0125
:0250
2
:1667
|
|
|
:1250
|
3
|
6 1:4234
6
6
6 :6346
L12
6
¼ 6 :6383
L22
6
6
6 :2672
4
:1468
:4652
:2909
|
|
:6707
|
:5369
1:3363
|
:2000
:1973
:0908
:0926
1:4237
|
|
:1917
:4092
:2501
:3041 7
7
:4558 7
7
7
:3108 7
7
7
:5473 7
5
1:2538
|
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
|
1:3406
|
|
|
:2155
We use L11 , L12 and so on because later it will be necessary to refer to these individual
submatrices in L, and they are to be distinguished from Lrr ¼ ðI Arr Þ1 and
Lss ¼ ðI Ass Þ1 , which are often used to denote Leontief inverses associated with
regional direct input coefficients matrices.
Impacts on the sectors in both regions of various new final-demand vectors in either
or both regions can now be found. For example, with new demand of 100 for the
output of sector 1 in region r, ðf new Þ0 ¼ ½ 100 0 0 0 0 , and, using L,
3
2
142:34
7
6
6 63:46 7
r new 7
6
ð
Þ
x
new
xnew ¼
¼6
63:83 7
s new ¼ Lf
7
6
ðx Þ
7
6
4 26:72 5
14:68
|
|
- - - - -
The new outputs in region s of sectors 1 (26.72) and 2 (14.68) that result from the new
demand in region r reflect interregional spillovers – economic stimulus in a region
other than the one in which the exogenous change occurs (in this case spillovers from
region r to region s).
It is to be emphasized that the final demands in the interregional input–output model
are for outputs produced in a particular region. That is, f r1 ¼ 100 means that there is a
final demand of 100 for sector 1 goods that are produced in region r. If sector 1 were
aircraft production and region r were Washington, new orders from a foreign airline for
Boeing commercial airliners would be represented in the value for f r1.
Using these hypothetical data, we can illustrate the differences between the results
from a single-region model for region r alone and the results from this two-region
interregional model. From the information on Arr alone we find
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3.3 Many-Region Models: The Interregional Approach
2
1:3651
Lrr ¼ ðI Arr Þ1 ¼ 4 :5273
:5698
79
3
:4253 :2509
1:3481 :5954 5
:4890 1:2885
2
3
100
Using this single-region model with ðf r Þnew ¼ 4 0 5 linkages, as in (3.4), we have
0
2
32
3 2
3
1:3651 :4253 :2509
100
136:51
xrS ¼ Lrr f r ¼ 4 :5273 1:3481 :5954 54 0 5 ¼ 4 52:73 5
:5698
:4890 1:2885
0
56:99
We use a subscript S to make clear that these are outputs in the single-region model,
and we drop the superscript “new.” With the complete two-region model we had, for
region r,
2
3
142:34
xrT ¼ 4 63:46 5
63:83
Here, xrT reminds us that these are outputs in the two-region interregional model. The
difference in results for region r is seen to be
3
3 2
2
3 2
5:83
136:51
142:34
xrT xrS ¼ 4 63:46 5 4 52:73 5 ¼ 4 10:73 5
6:84
56:99
63:83
Each region r output is larger in the interregional model because the interregional
feedbacks are captured in that model. One measure of the “error” that would be
involved in ignoring these feedbacks – in using a single-region model as opposed to
an interregional model – would be given by the percentage of total output in region r
that one fails to capture when using a single-region model only. Total output over all
sectors in region r in the two-region model is i0 xrT ¼ 269:63. Total output estimated
in the single-region model is i0 xrS ¼ 246:23. By this measure, the underestimate
that occurs in using the single-region model is i0 xrT i0 xrS ¼ 23:40, or
ð23:40=269:63Þ 100 ¼ 8:7 percent of the total true (two-region model) output.
Formally, this overall percentage error measure is found as
OPE ¼ i0 xrT i0 xrS =i0 xrT 100 ¼ i0 xrT xrS =i0 xrT 100
It thus becomes an interesting empirical question to try to assess the importance of
interregional feedbacks in real-world regional input–output models. If it turned out that
the error caused by ignoring interregional linkages when assessing the impact of new
region r final demands on region r outputs was quite small, then one might argue that
(at least for such questions) the apparatus of an interregional model would be unnecessary. The answer will depend, in part, upon the relative strengths of the interregional
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80
Input–Output Models at the Regional Level
linkages; in the two-region model this means on the magnitudes of the elements in Ars
and Asr . Precisely this question has been investigated; however, the results are inconclusive. The conclusion from an early set of experiments was that interregional
feedback effects were likely to be very small (an OPE of less than one half of one
percent) (see Miller, 1966, 1969). Other studies have tended to confirm the relative
smallness of interregional feedback effects by comparing output multipliers from
single- and many-region input–output models (Chapter 6). There has been work on
derivation of upper limits on the percentage error that could be expected in certain
interregional input–output models when the interregional feedbacks are ignored (in
particular, Gillen and Guccione, 1980; Miller, 1986; Guccione et al., 1988).
The error caused by ignoring interregional feedbacks is strongly influenced by the
level of self-sufficiency in region r – whether or not region r is relatively dependent on
inputs from region s. This is because higher dependence is reflected in larger coefficients in Asr which, again as in (3.14), generate a larger feedback term. Self-sufficiency
is also a function of the geographic size of the region. In a two-region model with
Nebraska (region r) and the rest of the USA (region s), the average element in Asr will
be larger than in a two-region model in which region r is the USA west of the
Mississippi and region s is the USA east of the Mississippi. However, in the
Nebraska (r)/rest-of-the-USA (s) example, the elements in Ars (reflecting rest-of-theUSA dependence on inputs from Nebraska) will be generally much smaller than in
the USA West (r)/USA East (s) example. Thus, it is not easy to generalize on how the
geographical size of the respective regions ultimately influences the size of the
interregional feedbacks.
In any case, a single-region model, by definition, cannot capture effects outside of
that region (spillovers) in regional/sectoral detail, and there are many kinds of economic impact questions that have important ramifications in more than one region of a
national economy. In these cases, some kind of connected-region model is essential.
The interregional input–output framework provides one such approach. Feedbacks and
spillovers in input–output models will be examined again in Chapter 8, when we
discuss multiplier decompositions.
Some analysts (for example, Oosterhaven, 1981) suggest that measurement
of feedback effects should be based not on total impacts (direct and indirect),
but rather should be found as percentages of indirect impacts only – without the
first term
0 inr the0 rpower
0 r series or with f netted out from gross outputs in
OPE ¼ i xT i xS =i xT 100. This means
0 r
i xT i0 f i0 xrS i0 f = i0 xrT i0 f 100
OPEn ¼
¼ i0 xrT i0 xrS = i0 xrT i0 f 100
This “net” measure is larger than OPE (except in the trivial case when f ¼ 0);
i0 x r
i0 x r
namely OPE n ¼ ðOPE Þ i0 xr T i0 f . In our numerical example, i0 xr T i0 f ¼ 1:59 and
T
T
i0 xrT i0 f
n
n
OPE ¼ 13:8. Alternatively, 100 ðOPE=OPE Þ ¼ 100 i0 xr
indicates the
T
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3.3 Many-Region Models: The Interregional Approach
81
percentage of the net measure that is captured by the original measure. In the example,
this is 63 percent.
3.3.4 Interregional Models with More than Two Regions
The fundamental structure of models with more than two regions is identical to the
two-region case in Section 3.3.1, although the numbers of matrices and their sizes
increase. The objective is to capture explicitly the various economic connections
between and among the several regions in a multiregional economy. For example, in
a three-region model (regions 1, 2, and 3), the complete coefficients matrix would be
2 11
3
A12 A13
A
(3.15)
A ¼ 4 A21 A22 A23 5
A31 A32 A33
and the parallel to (3.10) is
I A11 x1 A12 x2 A13 x3 ¼ f 1
(3.16)
A12 x1 þ I A22 x2 A23 x3 ¼ f 2
A31 x1 A32 x2 þ I A33 x3 ¼ f 3
2 13
2 13
2
3
I 0 0
f
x
With x ¼ 4 x2 5, f ¼ 4 f 2 5 and I ¼ 4 0 I 0 5, the complete three-region interre0 0 I
gional
x3
f3
input–output model is still represented as (I – A)x ¼ f. The underlying logic is the same
as that for the two-region model, and the equations in (3.16) can be built up in the same
way as were those in (3.10). Also, the magnitudes of the interregional feedback effects
can be made specific.
The extension to a p-region model is straightforward. The parallel to (3.16) is
I A11 x1 A12 x2 A1p xp ¼ f 1
..
(3.17)
.
p1 1
p2 2
pp p
p
A x A x þ ðI A Þx ¼ f
(The interested reader can construct the parallel expressions for A, I, f, and x.)
The data requirements increase quickly with the number of regions. Assuming that
all regions are divided into n sectors (not a necessary requirement at all – each region
could have a different number of sectors), a complete two-region interregional model
requires data for four coefficients matrices of size n n, a three-region model contains
nine n n matrices, a four-region model has 16 such matrices, and a p-region model
has p2 such n n matrices. However, interregional models with a relatively small
number of regions may be useful, since one region can always be defined as the “rest of
the country” or the “rest of the world.” A three-region model might concentrate on a
particular county, region 2 could be the “rest of the state,” and region 3 the “rest of the
nation” (outside the state).
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Input–Output Models at the Regional Level
In large and more complex multiregional (and multinational) models (Section 3.6.3)
the feedback and spillover effects can be much more complex and larger. (See
Oosterhaven, 2019, chapter 2, for a critical overview of several feedback/spillover
studies using large multinational models.)
3.3.5 Implementation of the IRIO Model
Clearly, the interregional input–output model requires a large amount of detailed data.
For this reason, there have been few real-world applications. Perhaps the most ambitious attempts at implementation are contained in the impressive series of Japanese
survey-based interregional tables, with nine regions and (ultimately) 25 sectors, beginning with 1960 and updated every five years. [See Ministry of International Trade and
Industry (MITI), various years; this was reorganized as the Ministry of Economy,
Trade and Industry (METI) in 2001.] This very rich data source has generated a
number of Japanese comparative regional studies (see, for example, Akita, 1994,
1999; Akita and Kataoka, 2002).
3.4
Many-Region Models: The Multiregional Approach
While a complete interregional model of the sort described in Section 3.3 is generally
impossible to implement for very many regions and/or sectors because of the enormous
amounts of data that it requires, the approach has inspired modifications and simplifications in the direction of a more operational framework. One attempt in this direction
uses the “Chenery-Moses” approach (noted in Section 3.3) for consistent estimation of
the intra- and interregional transactions required in the IRIO model. It has come to be
known as a multiregional input–output model. It contains counterparts to the regional
input coefficients matrices – as in Arr – and the interregional input (trade) coefficients
matrices – as in Ars . In both cases the attempt has been to specify a model in which the
data are more easily obtained.
Polenske examined and implemented three versions of the MRIO model – the
Chenery-Moses version (also known as a “column-coefficient” model), an alternative
row-coefficient version, and one using the gravity model approach of Leontief and
Strout (1963).15 Problems with the latter two approaches ultimately precluded their
use, and the column-coefficient model was chosen as the structure on which to develop
the US MRIO model (Polenske, 1970a, 1970b, 1980, 1995, section 2, 2004, section 8;
Bon, 1984).
3.4.1 The Regional Tables
The multiregional input–output model uses a regional technical coefficients matrix, Ar ,
in place of the regional input coefficients matrix, Arr . These regional technical
15
Leontief and Strout (1963) “devised the multiregional input–output (MRIO) accounts” (Polenske and Hewings,
2004, p. 274).
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3.4 Many-Region Models: The Multiregional Approach
83
coefficients, arij , can be produced from responses to the question “How much sector i
product did you buy last year in making your output?” [Question (1) in Section 3.2],
where they were contrasted with the regional input coefficients, arr
ij . Information
regarding the region of origin of a given input is ignored; one only needs information
on the dollars’ worth of input from sector i used by sector j in region r. These
transactions are usually denoted by zrij, where the dot indicates that all possible
geographical
locations for sector i are lumped together.16 These coefficients are defined
zrij
r
as aij ¼ xr and Ar ¼ [arij ].
j
In practice, when actual regional data on technology are not available, estimates of
regional technical coefficients matrices are sometimes made using what is known as
the product-mix approach. The basic assumption is that input requirements per unit of
output are constant from region to region at a very fine level of industrial classification,
but that an important distinguishing characteristic of production at the regional level is
the composition of sector outputs, when one is dealing with more aggregate sectors. To
return to our earlier illustration of the product-mix problem, when two-engine commercial jets are made in Washington (or anywhere else), they use, among other things,
two jet engines as inputs; when single-engine propeller-driven private aircraft are made
in Florida or in any other state, they use a one propeller engine as one of the inputs to
production. But the important fact to capture is that the output of the sector designated
“aircraft” in a Washington table is composed of a vastly different mix of products
(commercial jets) than the “aircraft” sector in Florida (private/corporate airplanes).
To illustrate, assume that sector 2 is food and kindred products, and that it contains
only three subsectors, which can be designated by their outputs: tomato soup (sector
2.1), chocolate bars (sector 2.2), and guava jelly (sector 2.3). Assume that the national
technical coefficients from sector 8, paper and allied products, to each of these subsectors are: 0.005, 0.009, and 0.003. (These represent various aspects of packaging –
labels, wrappers, etc.) Suppose that we want to derive coefficients for inputs from
sector 8 to sector 2, a82 , for New Jersey (region J) and for Florida (region F). The data
that we would need are shown in Table 3.3, where N designates national data. The
food and kindred products sector was composed of only tomato soup ($700,000) and
chocolate bars ($300,000) output (no guava jelly) in New Jersey; in Florida it was
made up of tomato soup ($80,000) and guava jelly ($420,000) – no chocolate bars.
Purchases of paper and allied products as inputs to New Jersey food and kindred
products production over the period covered by the output figures in Table 3.3 are then
assumed to be the sum of
aN8,2:1 xJ2:1 ¼ ð0:005Þð700,000Þ ¼ 3,500
aN8,2:2 xJ2:2 ¼ ð0:009Þð300,000Þ ¼ 2,700
aN8,2:3 xJ2:3 ¼ ð0:003Þð0Þ ¼ 0
16
Sometimes a small o is used, primarily because it is easier to read.
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84
Input–Output Models at the Regional Level
Table 3.3 Data needed for conversion of national to regional coefficients
via the product-mix approach
National Data
To Sector 2: Food and Kindred Products
Subsectors
2.1
2.2
(Tomato Soup)
(Chocolate Bars)
From Sector 8: Paper and Allied Products
aN8, 2:1 ¼ :005
aN8, 2:2 ¼ :009
2.3
(Guava Jelly)
aN8, 2:3 ¼ :003
Regional Data
Outputs (in 1,000 dollars) by subsector of sector 2
(New Jersey)
(Florida)
xJ2:1 ¼ 700
xF2:1 ¼ 80
xJ2:2 ¼ 300
xF2:2 ¼ 0
xJ2:3 ¼ 0
xF2:3 ¼ 420
Total Outputs (Sector 2)
xJ2 ¼ 1, 000
xF2 ¼ 500
for a total of $6,200 in necessary inputs from sector 8 to production in sector 2 in New
Jersey. Since xJ2 ¼ xJ2:1 þ xJ2:2 þ xJ2:3 ¼ 1,000,000,
aJ82 ¼ 6,200=1,000,000 ¼ 0:0062
Similarly, for Florida,
aN8,2:1 xF2:1 ¼ ð0:005Þð80, 000Þ ¼ 400
aN8,2:2 xF2:2 ¼ ð0:009Þð0Þ ¼ 0
aN8,2:1 xF2:3 ¼ ð0:003Þð420, 000Þ ¼ 1,260
The total Florida inputs from sector 8 would be estimated as $1,660. Since
xF2 ¼ 500,000, we have
aF2 ¼ 1,600=500,000 ¼ 0:0033
Formally,
J J J x2:1
x2:2
x
N
N
þ a8,2:2 J þ a8,2:3 2:3
J
J
x2
x2
xJ2
x2
F F F aN8,2:1 xF2:1 þ aN8,2:2 xF2:2 þ aN8,2:3 xF2:3
x2:1
x2:2
x2:3
N
N
N
¼
a
þ
a
þ
a
aF82 ¼
8,2:1
8,2:2
8,2:3
F
F
F
x2
x2
x2
xF2
aJ82 ¼
aN8,2:1 xJ2:1 þ aN8,2:2 xJ2:2 þ aN8,2:3 xJ2:3
¼ aN8,2:1
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3.4 Many-Region Models: The Multiregional Approach
85
The regional coefficients derived in this way are weighted averages of the national
detailed coefficients, where the weights are the proportions of subsector outputs to total
output of the sector (e.g., xJ2:1 =xJ2 ) in each state.
3.4.2 The Interregional Tables
The interconnections among regions in the multiregional input–output model are
captured in an entirely different way from the interregional input–output framework.
Trade flows in the multiregional model are estimated by sector, again to take advantage
of the kinds of data likely to be available. For sector i, let zrs
i denote the dollar flow of
good i from region r to region s, irrespective of the sector of destination in the
receiving region.17 These flows will include shipments to the producing sectors in
region s as well as to final demand in s. Thus, there is, for each sector, a shipments
matrix of the sort shown in Table 3.4.
Note that each of the column sums in this table represents the total shipments of
good j into that region from all of the regions in the model; this total, for column s, is
denoted in the table for good i by T si :
ps
2s
rs
T si ¼ z1s
i þ zi þ þ zi þ þ zi
(3.18)
If each element in column s is divided by this total, we have coefficients denoting the
proportion of all of good i used in s that comes from each region r (r ¼ 1, . . ., p). These
proportions are denoted crs
i :
zrs
i
¼
crs
i
T si
Table 3.4 Interregional shipments of commodity i
Receiving Region
Shipping Region
1
2
s
p
1
2
..
.
r
..
.
p
Total
z11
i
z21
i
z12
i
z22
i
...
z1s
i
z2s
i
z1p
i
z2p
i
zrs
i
..
.
zps
i
T si
zrp
i
zpp
i
T pi
17
..
.
zr1
i
..
.
zp1
i
T 1i
..
.
zr2
i
..
.
zp2
i
T 2i
rs
rs
To be consistent with the notation zrij or zor
ij , this should properly be zi or zio . However, when the blank space is
in the second subscript position, it is easier to distinguish than when it is in the first superscript position, and so
we avoid the double subscript option.
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86
Input–Output Models at the Regional Level
For later use, these coefficients are rearranged as follows. For each possible origindestination pair of regions, denote by crs the n-element column vector
2 rs 3
c1
6
. 7
7
crs ¼ 6
4 .. 5
crs
n
These elements show, for region s, the proportion of the total amount of each good
used in s that comes from region r. Finally, construct ^c rs ,
2 rs
3
c1 0 . . . 0
6 0 crs
7
2
6
7
rs
6
7
^c ¼ 6 .
(3.19)
7
.
4 .
5
0
0
. . . crs
n
for r, s ¼ 1, . . ., p. Note that there will be intraregional matrices in this set. For
example, there will be a matrix ^c ss , namely
2 ss
3
c1 0 . . . 0
6 0 css
7
2
6
7
ss
6
7
^c ¼ 6 .
(3.20)
7
4 ..
5
0
0
. . . css
n
s
ss
whose elements, css
i ¼ zi =T i , indicate the proportion of good i used in region s that
came from within region s.
3.4.3 The Multiregional Model
In this section we emphasize the structural parallels between the multiregional model
and the interregional model.18 Consider a small two-sector, two-region example, where
r
s
a11 ar12
a11 as12
r
s
A ¼ r
, A ¼ s
a21 ar22
a21 as22
^c rs ¼
rs
c1
0
ss
c1
ss
^
,
c
¼
rs
c2
0
0
0
css
2
Then the multiregional input–output model uses the matrix
18
In online Appendix SA3.1, the basic relationships in the multiregional model are derived from standard
economic and input–output theory.
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3.4 Many-Region Models: The Multiregional Approach
^c A ¼
rs
s
" rs s
c1 a11
s
crs
1 a12
s
crs
2 a21
s
crs
2 a22
87
#
as an estimate of Ars in the interregional input–output model. Similarly,
" ss s
#
s
c1 a11 css
1 a12
ss s
^c A ¼ ss s
s
c2 a21 css
2 a22
in the multiregional model replaces Ass in the interregional model. Therefore, the
multiregional input–output model embodies the same assumption as was used in the
earlier regional models with estimated supply percentages. Looking at the top rows of
the ^c rs As and ^c ss As matrices, note that both sectors 1 and 2 in region s are assumed to
have the same proportion of their total use of commodity 1 supplied from region r,
ss
namely crs
1 , and the same proportion supplied from within region s c1 .
Suppose that sector 1 in both regions r and s is electricity production and sector 2 in
region s is automobile production, then if crs
1 ¼ 0:6, this means that 60 percent of all
electricity used in making electricity in region s comes from region r and 60 percent of
all electricity used in automobile manufacture in region s also comes from region r.
And similarly, since in this two-region model it would be true that css
1 ¼ 0:4, 40 percent
of the electricity used in both electricity production and automobile production in s
comes from within that region.
Since the interregional shipments recorded in Table 3.4 include sales to both
producing sectors and final-demand users in the receiving region, the final demands
in region s are met in part by firms within the region ð^c ss f s Þ and in part by purchases
from firms in region r ð^c rs f s Þ. To continue the illustration with crs
1 ¼ 0:6, where sector
1 is electricity production, 60 percent of the final demand for electricity in region s will
also be satisfied by producers in region r.
The multiregional input–output counterpart to (3.10) for the interregional model is
therefore
ðI ^c rr Ar Þxr ^c rs As xs ¼ ^c rr f r þ ^c rs f s
^c sr Ar xr þ ðI ^c ss As Þxs ¼ ^c sr f r þ ^c ss f s
Let
A¼
Ar
0
rr
^c
0
,
C
¼
^c sr
As
(3.21)
r
r
^c rs
x
f
,
x
¼
,
and
f
¼
^c ss
xs
fs
so that (3.21) can be represented as
ðI CAÞx ¼ Cf
(3.22)
x ¼ ðI CAÞ1 Cf
(3.23)
and the solution will be given by
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Input–Output Models at the Regional Level
The extension to more than two regions is straightforward. Equations for the threeregion model would be
I ^c 11 A1 x1 ^c 12 A2 x2 ^c 13 A3 x3 ¼ ^c 11 f 1 þ ^c 12 f 2 þ ^c 13 f 3
^c 21 A1 x1 þ I ^c 22 A2 x2 ^c 23 A3 x3 ¼ ^c 21 f 1 þ ^c 22 f 2 þ ^c 23 f 3
^c 31 A1 x1 ^c 32 A2 x2 þ I ^c 33 A3 x3 ¼ ^c 31 f 1 þ ^c 32 f 2 þ ^c 33 f 3
[Compare (3.16), for the three-region interregional model.] By appropriate extension
of matrices A, C, x, and f to incorporate three regions, the fundamental model is still
ðI CAÞx ¼ Cf, as in (3.22), with solution x ¼ ðI CAÞ1 Cf, as in (3.23).
Finally, when there are p regions, let
2
3
2 11
3
2 3
2 3
^c
A1 0 0
x1
f1
^c 1p
6 0 A2 0 7
6 ^c 21 ^c 2p 7
6 x2 7
6 f2 7
6
6
6 7
6 7
7
7
A ¼ 6 ..
7, C ¼ 6 ..
7, x ¼ 6 .. 7, and f ¼ 6 .. 7
..
4 .
5
4
5
4
5
4 . 5
.
.
.
p1
pp
p
p
^c
x
fp
0 0 A
^c
Then ðI CAÞx ¼ Cf and x ¼ ðI CAÞ1 Cf still represents the system and its
solution; only the dimensions of the matrices have changed.
3.4.4 Numerical Example: Hypothetical Two-Region Multiregional Case
Assume that we have the flow data in Table 3.5, representing total inputs purchased by
producing sectors in each region, regardless of whether these are locally produced or
imported from the other region. These are the Zr ¼ [zrij ] and Zs ¼ [zsij ] data.
2
3
2
3
1,000
1,200
Suppose, further, that xr ¼ 4 2,000 5 and xs ¼ 4 800 5, so that the regional technical
1,000
1,500
r
s
coefficients matrices, Ar ¼ aij and As ¼ aij , are
2
3
2
3
:225 :300 :110
:188 :406 :083
Ar ¼ 4 :250 :063 :425 5, As ¼ 4 :292 :250 :180 5
:325 :350 :150
:300 :300 :133
Table 3.5 Flow data for a hypothetical two-region multiregional case
Purchasing Sector
Region r
Region s
Selling Sector
1
2
3
1
2
3
1
2
3
225
250
325
600
125
700
110
425
150
225
350
360
325
200
240
125
270
200
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89
3.4 Many-Region Models: The Multiregional Approach
Table 3.6 Interregional commodity shipments for the hypothetical two-region multiregional case
Commodity 1
r
s
T
Commodity 2
Commodity 3
r
s
r
s
r
s
800
310
T r1 ¼ 1,110
200
890
T s1 ¼ 1,090
1,300
300
T r2 ¼ 1,600
700
500
T s2 ¼ 1,200
900
325
T r3 ¼ 1,225
100
1,175
T s3 ¼ 1,275
For the trade proportions, we need measures of the total amount of each good, i, that
is available in each region – T ri and T si , in (3.18). Table 3.6 provides an example of
these data. (Note that the row sums for each sector in each region must be the total
output for that sector in that region, as recorded in the appropriate x vector.) The
s
rs
proportions crs
i ¼ zi =T i are easily found. Here
2
3
2
3
2
3
2
3
:721
:183
:279
:817
crr ¼ 4 :812 5, crs ¼ 4 :583 5, csr ¼ 4 :188 5, and css ¼ 4 :417 5
:735
:078
:265
:922
Thus, the building blocks in this example for the two-region multiregional input–
output model are
2
3
:300
:110
0
0
0 7
6 :225
6
7
6 :250
:063
:425
0
0
0 7
6
7
7
r
6
6
:325
:350
:150
0
0
0 7
A
0
6
7
A¼
¼
6
7
0 As
6 0
0
0
:188
:406
:083 7
6
7
6
7
0
0
0
:292
:250
:180
6
7
4
5
0
0
0
:300
:300
:133
|
|
|
|
|
|
|
|
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
|
|
|
|
|
|
and
2
rr
^c
C ¼ sr
^c
6 :721
6
6 0
6
6
rs
6 0
^c
6
¼
ss
6
^c
6 :279
6
6
6 0
4
0
0
0
|
|
|
:812
0
0
:735
|
|
3
:183
0
0
:583
0
0
:817
0
0
:417
0
0
|
|
0 7
7
0 7
7
7
:078 7
7
7
0 7
7
7
0 7
5
:922
- - - - - - - - - - - - - - - - -| - - - - - - - - - - - - - - - -
0
0
|
|
|
:188
0
0
:265
Therefore
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|
|
|
|
90
Input–Output Models at the Regional Level
2
6 1:127
6
6 :628
6
6
6 :512
ðI CAÞ1 C ¼ 6
6
6 :625
6
6
6 :238
4
:472
:477
:300
1:317
:606
:526
1:101
:369
:250
:385
:205
:445
:589
|
|
|
|
|
|
|
:478
:418
:552
1:115
:335
:470
1:224
:456
:278
:650
:594
:529
3
:153 7
7
:323 7
7
7
:247 7
7
7
:216 7
7
7
:167 7
5
1:232
- - - - - - - - - - - - - - - - - - -| - - - - - - - - - - - - - - - - - - |
|
|
|
|
|
|
(3.24)
and, for example, the impacts of new final demands of 100 for sector 1 outputs by
consumers in each region – that is, with f 0 ¼ ½ 100 0 0 100 0 0 – are found,
as in (3.23),
2
3
160:50
6 118:00 7
6
7
6
7
6
7
84:70
1
6
7
x ¼ ðI CAÞ Cf ¼ 6
7
6 184:90 7
6
7
4 51:60 5
|
|
- - - - - -
2
3
2
3
106:60
160:50
184:90
So, xr ¼ 4 118:00 5 and xs ¼ 4 51:60 5.
84:70
106:60
0
Similarly, if f ¼ ½ 100 0 0 0 0 0 , which represents new final demands of
100 for sector 1 output by consumers in region r only, we find
2
3
112:70
6 62:80 7
6
7
6
7
6 51:20 7
6
7
x¼6
7
6 62:50 7
6
7
4 23:80 5
47:20
|
|
- - - - -
2
3
62:50
Exactly as in an interregional model, xs ¼ 4 23:80 5 reflects interregional spillovers in
47:20
the multiregional system, in this case from region r (the location of the final demand
change) to region s.
It is important to bear in mind, from the general statement of the multiregional input–
output model in (3.22) or (3.23), that both intermediate demands, Ax, and final demand,
f, are premultiplied by the matrix C; this distributes these demands to supplying sectors
across regions. Thus f r and f s represent demands by (shipments to) the final-demand
sectors in regions r and s, respectively, not final demands for the products of regions r
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3.4 Many-Region Models: The Multiregional Approach
91
and s (as in the interregional input–output model). The operation Cf converts these
demands into a set of shipments by each region to contribute toward satisfaction of the
final demands. In the two-region model here, f r is satisfied in part by shipments from
sectors in region r, ^c rr f r and in part by shipments from sectors in region s, ^c sr f r . An
example of a typical element in f r might be new energy demands by a state government
resulting from a new state office building in region r in that state. Depending upon the
particular region, some or all of that energy demand will be met from within region r, the
rest from outside the region. This is reflected in the appropriate elements in ^c rr and ^c sr .
Thus, if one wants to assess the impacts of new region-specific final demands (such
as from a foreign airline for Boeing airliners, as in the interregional example in Section
3.3) it is necessary to replace Cf by, say, f ∗ , which represents the new final demands
already distributed appropriately to the region or regions of interest, and then to find
x ¼ ðI CAÞ1 f ∗
(3.25)
This is to be contrasted with (3.23). Continuing with the data for this example,
2
3
:471
:359
:258
:345
:135 7
6 1:463
6
7
6 :668
7
1:483
:720
:526
:600
:290
6
7
6
7
6
7
:604
:572
1:445
:274
:327
:145
1
6
7 (3.26)
ðI CAÞ ¼ 6
7
6 :314
:298
:263
1:428
:676
:212 7
6
7
6
7
:167
:221
:292
1:326
:162 7
6 :216
4
5
:409
:376
:329
:636
:734
1:308
|
|
|
|
|
|
|
| - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
|
|
|
|
|
|
If ðf ∗ Þ1 ¼ 100 represents the value of new foreign airline orders for aircraft produced
in region r, we would find, using (3.25)
2
3
146:30
6
7
6 66:80 7
6
7
6 60:40 7
6
7
x¼6
7
6 31:40 7
6
7
6
7
4 21:60 5
r
- - - - -
40:90
3.4.5 The US MRIO Models
The first large-scale implementation of the MRIO framework was initiated at the
Harvard Economic Research Project (HERP) and was further developed by
Professor Karen Polenske and her associates at MIT. In its most detailed form, this
is a model for 1963, with 51 regions (the 50 states and Washington, DC) and 79 sectors
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92
Input–Output Models at the Regional Level
in each region. A thorough description of the model and its construction is provided
in Polenske (1980). There was a second estimation and implementation of the
MRIO framework for the 1977 US economy involving researchers at MIT and also
Jack Faucett Associates, Inc., an economics consulting firm (see Jack Faucett
Associates, Inc., 1981–1983). Since then there have been some additional attempts
at creating multiregional input–output models for the USA. Because of widespread
use, this system is viewed as an alternative to the IRIO model; it could also be
seen as an approach to estimating the intra- and interregional elements of an IRIO
framework.19
Most implementations of interregional/multiregional input–output structures in
recent decades have been generated through a combination of techniques and estimating procedures, all designed to estimate the numbers (especially the interregional
transactions/coefficients) needed for the MRIO framework. These are generally known
as “hybrid” techniques; they are a blend of some survey information, expert opinion,
and mechanical approaches. Some of these are explored in more detail in Chapter 10.
3.4.6 Numerical Example: The Chinese Multiregional Model for 2012
In 2003 the Institute of Developing Economies (Tokyo), in conjunction with the
Japanese External Trade Organization, published an ambitious set of multiregional
input–output data for China in 2000, with 30 sectors and eight regions, which was
updated to 2012. Okamoto and Ihara (2005) and Mi et al. (2018) provide detailed
discussions of table construction and a number of comparative regional economic
analyses that use the Chinese multiregional input–output framework.
Tables 3.7–3.9 contain data for a highly aggregated version of the Chinese 2012
tables, with three sectors and three regions (this is for illustration purposes only).20 The
transactions are denominated in billions of yuan (CNY) [also known as renminbi,
meaning “people’s currency” (RMB)].21 We can easily trace the effects of hypothesized changes in final demands throughout the sectors and regions of the Chinese
economy in this three-region illustration. For example, assume that there is an increase
of ¥100 million in export demand for manufactured goods from the East, we would use
E 0
¼ ½ 0 100 0 0 0 0 0 0 0 Δf
|
|
|
|
in conjunction with the total requirements matrix in Table 3.9 to assess the impacts of
this final demand change throughout the economy. We can examine similar implications of the same amount of increased
export
0 demand for manufactured goods in each
of the other regions, using in turn Δf C ¼ ½ 0 0 0 0 100 0 0 0 0 for
19
|
|
|
|
An early comparison of the MRIO and IRIO models is provided in Hartwick (1971).
These data are from the international consortium, China Emission Accounts and Datasets (CDEDs), an
overview of which is available in Mi et al. (2017 or 2018). Details of the regional and sectoral aggregations
can be found in Appendix 3.2.
21
The symbol usually seen is ¥, although sometimes with just one horizontal stroke. With two lines it is the same
as the symbol for the Japanese yen.
20
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Table 3.7 Chinese interregional and intraregional transactions, 2012 (in ¥ billions)
East
2012
Central
West
1
2
3
1
2
3
1
2
3
Total
Output
East
1
2
3
Agric & Mining
Manufacturing
Services & Utils
88
127
62
354
2,807
610
72
710
954
3
8
4
7
111
41
2
32
28
2
11
4
5
73
10
2
48
21
665
5,878
3,776
Central
1
2
3
Agric & Mining
Manufacturing
Services & Utils
6
5
1
21
115
10
27
43
20
58
70
42
169
594
165
43
209
240
2
6
1
3
26
2
2
23
5
405
1,551
1,071
West
1
2
3
Agric & Mining
Manufacturing
Services & Utils
Total Output
7
4
2
665
43
67
10
5,878
28
34
25
3,776
4
5
3
405
8
21
9
1,551
6
11
8
1,071
70
58
43
444
147
341
121
1,063
43
226
251
1,163
444
1,063
1,163
16,016
93
94
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Table 3.8 Direct input coefficients for the Chinese multiregional economy, 2012
East
2012
Central
West
1
2
3
1
2
3
1
2
3
East
1
2
3
Agric & Mining
Manufacturing
Services & Utils
.13214
.19123
.09284
.06029
.47750
.10379
.01908
.18810
.25278
.00667
.02091
.00999
.00442
.07134
.02651
.00163
.02966
.02644
.00484
.02393
.00825
.00509
.06889
.00908
.00181
.04086
.01793
Central
1
2
3
Agric & Mining
Manufacturing
Services & Utils
.00910
.00712
.00153
.00357
.01954
.00169
.00726
.01143
.00537
.14257
.17397
.10266
.10872
.38279
.10647
.04014
.19504
.22447
.00489
.01265
.00254
.00278
.02402
.00188
.00170
.01975
.00432
West
1
2
3
Agric & Mining
Manufacturing
Services & Utils
.00988
.00615
.00249
.00730
.01136
.00172
.00733
.00890
.00654
.00952
.01234
.00632
.00514
.01382
.00559
.00581
.01024
.00779
.15773
.12975
.09748
.13792
.32115
.11396
.03725
.19436
.21590
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Table 3.9 Leontief inverse matrix for the Chinese multiregional economy, 2012
East
2012
Central
West
1
2
3
1
2
3
1
2
3
East
1
2
3
Agric & Mining
Manufacturing
Services & Utils
1.19521
.53447
.22644
.15494
2.11099
.31853
.07136
.56038
1.42975
.02550
.16442
.06721
.03916
.34605
.13190
.02290
.20382
.10026
.02092
.14750
.04831
.03537
.30560
.08097
.02339
.21701
.07621
Central
1
2
3
Agric & Mining
Manufacturing
Services & Utils
.02404
.04890
.01525
.02645
.09304
.02376
.02453
.06232
.02339
1.23296
.42835
.22328
.24438
1.79622
.28117
.12780
.48235
1.37415
.01816
.06010
.01729
.02401
.10356
.02505
.01835
.08374
.02398
West
1
2
3
Agric & Mining
Manufacturing
Services & Utils
.02761
.03425
.01596
.03408
.05624
.02130
.02703
.04282
.02384
.03013
.05199
.02751
.03308
.06910
.03370
.02655
.05153
.03026
1.24807
.30104
.20045
.28120
1.61418
.27220
.13249
.42055
1.35511
95
96
Input–Output Models at the Regional Level
0
export demands in the South and Δf W ¼ ½ 0 0 0 0 0 0 0 100 0 for
export demands in the Rest of China.
Premultiplying each of these vectors, in turn, by the total requirements matrix in
Table 3.9 produces the results shown in Table 3.10. The new export demand generates
differing own-region economic effects, depending on the region in which the manufacturing sector experiences the new export demand. When the demand is for manufactured goods made in the East, the total output of all sectors in that region increases
by ¥258 million. If the demand is for Central manufactured goods, the total value of
new outputs in that region is ¥232 million, and, when the new demand is for
manufactured goods from the West, output of all sectors there increases by ¥217
million. Interregional spillovers to each of the other regions are indicated by the other
entries in the output columns of Table 3.10. Adding spillovers to own-region impacts,
we see that total national effects of the ¥100 million stimulus for manufacturing are
¥283, ¥298, and ¥274 million, respectively, when the stimulus is in the East, the
Central region, and the West of China, respectively.
Many other observations can be made with the aid of results like those in Table 3.10.
For example, in terms of interregional spillovers, it is clear that the largest external
effect occurs when the demand is in the Central region since total output generated in
the other regions is ¥66 million (¥52 million in the East and ¥14 million in the West).
Spillovers from new demand in the other two regions are ¥25 million (14 þ 11) from
new demand in the East and ¥57 million (42 þ 15) from new demand in the West. In
this highly aggregated example from China, it is clear that the East dominates it terms
of within-region effects (¥258 million) and the Central region dominates with respect
to generation of national output (¥298 million) and also in terms of interregional
spillover effects (¥66 million).
3.5
|
|
|
|
The Balanced Regional Model
3.5.1 Structure of the Balanced Regional Model
A model that has a different sort of “regional” character was proposed in Leontief et al.
(1953, chapter 4) and has been implemented in specific applications, including an
analysis of the effects in the US economy, on both sectors and regions, of a diversion of
production away from military goods and to non-military consumer goods (Leontief
et al., 1965). This has been called a balanced regional model (or intranational model).
The basic mathematical structure of this model is identical to that of the interregional
input–output model, but the interpretation of each of the components of the model is
rather different. The entire analytical structure is based on the observation that in any
national economy there are goods with different kinds of market areas. There are some
goods for which production and consumption are equal (“balance”) only at the national
level. These are goods that have essentially a national (or, indeed, international) market
area – sectors such as automobiles, aircraft (total airliner production in Washington
6¼ total demand for aircraft in Washington), furniture, and agriculture. On the other
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Table 3.10 Region- and sector-specific effects (in ¥ millions) of a ¥100 million increase in final demand for manufacturing goods, China, 2012
2012
East
Central
West
Demand in East
1
2
3
Agric & Mining
Manufacturing
Services & Utils
1
2
3
1
2
3
Total Output
15
211
32
258
Agric & Mining
Manufacturing
Services & Utils
3
9
2
14
Agric & Mining
Manufacturing
Services & Utils
3
6
2
11
100
283
Demand in Central
100
Total Output
4
35
13
52
24
180
28
232
3
7
3
14
Demand in West
298
100
Total Output
4
31
8
42
2
10
3
15
28
161
27
217
274
97
98
Input–Output Models at the Regional Level
hand, there are other sectors for which production and consumption tend to balance at a
lower geographical level; they serve a regional or local rather than a national market.
Examples might be electricity, real estate, warehousing, and personal and repair
services (the number of shoeshines produced in an urban area equals the demand for
shoeshines in that area). Clearly there is in reality an entire spectrum of possibilities,
from sectors that serve extremely small local markets (shoe repair) to large national
and international markets (aircraft). To illustrate the model structure with a simple
example, we suppose that all sectors can be assigned to either a national (N) or a
regional (R) category. (One possible criterion for classification of sectors would be the
percentage of interregional as opposed to intraregional shipments of the products of
that sector.)
Then, from a table of national input coefficients, one can rearrange the sectors so
that, for example, all the regional sectors are listed first, and all the national sectors
follow. Let sectors 1, 2, . . ., r represent the regionally balanced sectors and let sectors
r þ 1, . . ., n represent nationally balanced sectors. Then, the rearranged table of
national input coefficients will be
RR
ARN
A
(3.27)
A¼
ANR ANN
Let xR and f R (r-element column vectors) represent total output and final demand for
the regional sectors, and let xN and f N , which are (n r)-element column vectors,
represent output and final demand for the national sectors. Define
R
R
f
x
x ¼ N and f ¼ N
x
f
Then, in exactly the same spirit as the two-region interregional input–output model,
we have (I A)x ¼ f. Here this is
I ARR xR ARN xN ¼ f R
(3.28)
ANR xN þ I ANN xN ¼ f N
It is important to notice that the R and N superscripts do not refer here to specific
geographic locations of sectors, as in the interregional model. Rather, they serve to
partition the sectors into two types – those whose market areas are national and those
N
whose market areas are regional.22 For example, a typical element aRN
ij xj of the vector
ARN xN in (3.28) records inputs from sector i (in the regionally balanced set of sectors)
to sector j (in the nationally balanced set of sectors). This will become clearer in the
next numerical example.
22
Partitioning of this sort can be done for a wide variety of purposes. For example, if one is particularly interested
in energy-producing sectors, one might want to divide all sectors into two groups – those that produce energy
and those that do not produce energy. Partitioned matrices will be employed frequently in the remainder of this
book. Important results on inverses of partitioned matrices are presented in Appendix A.
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3.5 The Balanced Regional Model
99
More compactly, in partitioned matrix form,
R R ARN
I ARR
xN ¼ f N
x
f
ANR
I ANN
and so
xR
xN
¼
I ARR
ANR
ARN
I ANN
R f
fN
(3.29)
Using regular solution procedures, we find the total outputs of each sector in each of
the two categories, due to an exogenous change in final demand for the outputs of one
or more national sectors and/or one or more regional sectors. For example, in the armsreduction study, there was assumed to be a 20 percent across-the-board decrease in
government demand for the output of military-related goods, some of which were
produced by national sectors (e.g., aircraft) and some of which were produced by
regional sectors (e.g., warehousing), and an assumed across-the-board increase in nonmilitary final demands. Hence, elements in both f R and f N experienced change.
Thus far, there is nothing explicitly spatial in the model. The categorization of either
nationally balanced or regionally balanced sectors deals only with the size of the
market areas involved. For regional sectors, we need to have the new final demands,
f RðsÞ , the final demand for
f R , distributed across regions. That is, we need
P toRðshave
Þ
R
¼ f . In addition, we need, for
regionally balanced goods in region s, where
sf
each region, s, an estimate of the proportion of the output of each nationally balanced
sector that is produced in region s, namely
2 s 3
prþ1
6 .. 7
s
p ¼4 . 5
psn
^ s xN indicates that part of the output of new national goods, xN , that must
The vector p
be produced by sectors r þ 1 through n in region s. SincePthe elements of ps are the
s
proportions
P s of total national output that occur in region s, s pi ¼ 1 for i ¼ r þ 1, . . .,
^ ¼ I.
n, or s p
Total output in region s is an n-element vector
RðsÞ x
ðsÞ
(3.30)
x ¼ N ðsÞ
x
where xRðsÞ contains the outputs of the r regionally balanced goods that are made in
^ s xN Þ indicates production of nationally balanced goods that
region s, and xN ðsÞ ¼ ðp
occurs in region s.
The xRðsÞ term involves two components: (1) production in region s to meet regionspecific final demand for regionally balanced goods, f RðsÞ (e.g., production in Michigan
to satisfy interindustry needs and new final demand in Michigan for electricity
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100
Input–Output Models at the Regional Level
produced in that state) and (2) production in region s to turn out that region’s share of
nationally balanced goods, xN ðsÞ (e.g., Michigan electricity used as an input to
Michigan production of automobiles to satisfy part of the nationwide demand for
automobiles). That is,
1
1
xRðsÞ ¼ I ARR f RðsÞ þ I ARR ARN xN ðsÞ
(3.31)
1
1
^ s xN
¼ I ARR f RðsÞ þ I ARR ARN p
Remember, from (3.27), that all the coefficients in the A matrices reflect national
technology; the “R” and “N” serve to partition this national technology into two types
of sectors. Production in each particular
region is assumed to utilize this same
RR
matrix and its inverse. In Appendix 3.3,
technology, as reflected in the I A
these results are derived directly from observations on the inverse of the partitioned
matrix in (3.29).
For the allocation of region R’s share of production of nationally balanced goods,
found in (3.29), we have
s N
^x
(3.32)
xN ðsÞ ¼ p
In this way, then, the balanced regional model allocates the impacts of new f R and f N
demand to the various sectors in each region.
3.5.2 Numerical Example
An example will illustrate more exactly how this works. Let
2
:15
:05
6 :10
6
RR
6 :03
:10
:02
ARN
A
¼6
A¼
NR
NN
6
A
A
6 :12
:03
:20
4
:10
:02
:25
|
|
|
|
3
:03 7
7
:10 7
7
7
:10 7
5
:15
|
- - - - - - - - - - - - - - - - |
|
|
|
(3.33)
and
2
100
3
7
R 6
6 100 7
f
6
7
f ¼ N ¼6
7
f
4 200 5
- - -
200
Then x is found as in (3.29)
2
xR
x¼ N
x
168:30
3
6
7
6 163:40 7
6
7
¼6
7
4 325:70 5
- - - - -
354:70
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(3.34)
3.6 The Spatial Scale of Regional and Interregional Models
101
These figures represent total outputs, throughout the nation, of the four sectors.
Assume that there are three regions in the country and that the region-specific
distribution of final demands f R is
40
50
10
Rð1Þ
Rð2Þ
Rð3Þ
,f
,f
¼
¼
¼
f
30
30
40
1
:6
:2
:2
2
3
and that p ¼
,p ¼
, and p ¼
. We find I ARR
from the data
:3
:4
:3
in A;
1:117 :186
RR 1
IA
¼
:037 1:117
1
Using (3.31),
Rð1Þ
x
67:47
72:73
28:05
Rð2Þ
Rð3Þ
¼
, x
¼
, x
¼
51:75
52:97
58:65
(3.35)
[Note, as must be the case in a consistent model, that xR , in (3.34), is indeed
^ 1, p
^ 2 , and p
^ 3 , the distribution of nationally balxR ¼ xRð1Þ þ xRð2Þ þ xRð3Þ .] Using p
anced goods across regions is found as
195:40
65:14
65:14
^ 1 xN ¼
^ 2 xN ¼
^ 3 xN ¼
xN ð1Þ ¼ p
, xN ð2Þ ¼ p
, xN ð3Þ ¼ p
106:40
141:90
106:40
(3.36)
where xN must equal xN ¼ xN ð1Þ þ xN ð2Þ þ xN ð3Þ , because of the way in which the p
are defined.
Putting the results in (3.35) and (3.36) together, as in (3.30), we have
2
3
2
3
2
3
67:47
72:73
28:05
6
7
6
7
6
7
6 51:75 7 ð2Þ 6 52:97 7 ð3Þ 6 58:65 7
ð1Þ
6
6
6
7
7
7
(3.37)
x ¼6
7, x ¼ 6 64:14 7, x ¼ 6 65:14 7
4 195:40 5
4
5
4
5
- - - - -
- - - - -
- - - - -
106:40
141:90
106:40
The entire outputs in (3.34) have been allocated across the three regions. As noted,
production in each region is assumed to utilize the same technology, as reflected in
I ARR . But the model does recognize that production, whether of goods with a
national market area or with a subnational market area, occurs in geographically
specific locations, and the information in the distribution of the f N elements and in
the ps vectors reflects this spatial distribution of production.
3.6
The Spatial Scale of Regional and Interregional Models
As illustrated with the US and Chinese examples in Sections 3.4.5 and 3.4.6, multiregional models often employ a partitioning of a country into an exhaustive set of
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102
Input–Output Models at the Regional Level
regions. However, other connected-spaces models have also been developed, both at
smaller (sub-urban areas) and larger (multinational) scales. We cite some examples
here (drawn from many), starting at the neighborhood, city, and community levels, and
then moving on to multinational models.
3.6.1 Cities or Smaller Areas
• Cole (1987) describes a model for the city of Buffalo, New York, and Cole (1999)
looks at an inner-city neighborhood in Buffalo.
• Robison and Miller (1988, 1991) consider small Idaho timber economies (logging/
sawmills) – in the latter reference consisting of six communities (five containing
sawmills; combined population around 20,000). They term these “community”
input–output models. In Robison (1997) the model is for a rural two-county region
in central Idaho (total population less than 12,000) which was disaggregated into
seven community-centered sub-county regions.
• Hewings, Okuyama, and Sonis (2001) present a four-region metropolitan area
model. Three of the regions are sub-divisions of the City of Chicago, and the fourth
is composed of the remaining counties making up the Chicago metropolitan area (six
counties in all).
• Yamada (2015) describes construction of a 14-subregion model for the Nagoya
(Japan) metropolitan area.
3.6.2 States or Other National Subdivisions
• West (1990) contains a summary of Australian input–output models in single-region
and connected-region frameworks.
• Boomsma and Oosterhaven (1992) describe a variety of two-region Dutch models
made up of one region of interest and the rest of The Netherlands as the second
region.
• Jackson et al. (2006) and Schwarm, Jackson, and Okuyama (2006) suggest a new
approach to generating data for the 51-state US model.
• Richardson, Gordon, and Moore (2007, and numerous other citations) create a 51state US MRIO model.
3.6.3 Multicountry (or Multinational) Areas
The definitive framework that foresees expansion from a “multiregional” to a “multinational” input–output perspective is found in Leontief’s world model.23 This huge
23
This is outlined in Leontief (1974; his Nobel Memorial Lecture). The evolution of the model is presented in
Fontela (2004); see also Duchin (2004) or Leontief, Carter, and Petri (1977) for more background.
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3.6 The Spatial Scale of Regional and Interregional Models
103
and ambitious project from the early 1970s was sponsored by the United Nations as
part of its search for “. . . possible alternative policies to promote development while at
the same time preserving and improving the environment” (United Nations, 1973,
p. 2.).
The final version consisted of 15 regions (four advanced industrial countries, four
centrally planned economies) and two groups of developing countries (three resourcerich and four resource-poor), each with 48 sectors, including eight exhaustible
resources as well as eight types of major pollutants and five types of abatement
activities (since the motivation was one of environmental impacts). Data were assembled for the base year of 1970, and projections were made from that time to 1980,
1990, and 2000. National input–output tables formed the basis of the intraregional data
sets; various accounting practices and sectoring schemes created many consistency
issues.
Starting in the early 2000s, with the continuing advancement of computer technology (speed and capacity), work has accelerated on developments of truly allencompassing world (or global) input–output (WIO or GIO) models, which are
sometimes called, even more completely, global multiregional input–output
(GMRIO) models. These are in the spirit of the Leontief vision, but are now able
to be much more disaggregated and sophisticated. Two prominent examples are: (1)
the World Input–Output Database (WIOD), which is a multinational research effort
coordinated at the University of Groningen involving 11 international partners that
produced a model covering 43 countries and 56 industry sectors (Dietzenbacher
et al., 2013) and (2) the Eora Global Supply Chain Database, which was created
and is maintained primarily by researchers at the School of Physics, University of
Sydney, Australia. This is a large data base derived largely from United Nations
covering 190 countries at an industry detail of 20–500 sectors, altogether comprising
a 15,909 sector MRIO table (Lenzen et al., 2013). More recently the European
Commission agency, Eurostat, initiated a new effort labeled Full International and
Global Accounts for Research in Input–Output Analysis, known as FIGARO,
intended to annually compile intercountry tables for EU member nations and, in
collaboration with the OECD, expanded to global coverage (Remond-Tiedrez and
Rueda-Cantuche, 2019).
We explore these GMRIO models in more detail in Chapter 10 and their
application to international trade and global environmental issues in Chapters 12
and 13. These models employ more modern sectoral classifications and terminology, which are also discussed in the following chapters. In developing these large
and complex models it was necessary to use a variety of estimating procedures,
particularly for the trade data (Section 3.4.2), some of which are covered in
Chapters 9 and 10. As a consequence of this broadening of scope of how MRIO
models are constructed, the term multiregional input–output model has come to
encompass a wider range of methods than those that underpin the original Chenery–
Moses MRIO model.
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104
3.7
Input–Output Models at the Regional Level
Summary
In this chapter we have explored some of the most important modifications needed for
the basic input–output model (Chapter 2) when analysis is to be carried out at a
regional or multiregional level. We have seen that the input–output framework can
be used either to study one single region in isolation, or it can be employed in studying
one or more regions whose economic connections are made explicit in the model.
While the representations of these connected regional models appear quite complicated, the models are logical extensions of the basic input–output structure that are
designed to (1) reflect possibly differing production practices for the same sectors in
different regions and (2) capture the trade relationships between sectors in different
regions.
In more recent decades, work has been carried out with multinational (world or
global) input–output models. These have come about as a result of the increasing
economic interdependence of nations – as exemplified, for example, in the European
Union. They are also massively dependent on large-capacity, high-speed computers.
We will explore some of these models in Chapter 10, because they involve various
kinds of “hybrid” approaches to estimation of the necessary data. In this kind of
framework, impacts of alternative development policies in less-developed countries
can be studied for global impacts. These models have been particularly useful in
assessing international transmission of environmental impacts, for example enabling
analysts to identify both industrial and spatial sources of pollution emissions through
the various stages of an entire multinational supply chain (discussed in more detail in
Chapters 12 and 13.)
This chapter has three appendices. The first Online Appendix SA3.1 (summarized
below as Appendix 3.1) explores the basic relationships of the multiregional
input–output model (MRIO). Appendix 3.2 (below) defines the regional and industry
aggregation for the Chinese multiregional model used in Section 3.4.4. Appendix 3.3
(below) develops the balanced regional model discussed in Section 3.5 and the
procedure for inversion of a partitioned ðI AÞ matrix used in that development.
Appendix 3.1
Basic Relationships in the Multiregional Input–Output Model
Supplemental Appendix SA3.1, located on the internet web site associated with this
text (http://www.cambridge.org/millerandblair), further explores the fundamental relationships defining the Multiregional Input–Output (MRIO) Model which form the
basis for (3.22) and (3.23) in the main text of this chapter.
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Appendix 3.2
105
Appendix 3.2 Sectoral and Regional Aggregation in the 2012 Chinese
Multiregional Model
Figure A3.2.1 Regional aggregation in the 2012 Chinese multiregional model
Table A3.2.1 Regional classifications in the 2012 Chinese multiregional model
3-Region Aggregation
Provinces and Municipalities
East
Beijing, Tianjin, Hebei, Liaoning, Jilin, Heilongjiang, Shanghai, Jiangsu,
Zhejiang, Fujian, Shandong, Guangdong, Hainan
Shanxi, Anhui, Jiangxi, Henan, Hubei, Hunan
Inner Mongolia, Guangxi, Chongqing, Sichuan, Guizhou, Yunnan,
Shaanxi, Gansu, Qinghai, Ningxia, Xinjiang
Central
West
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106
Input–Output Models at the Regional Level
Table A3.2.2 Sectoral aggregation in the 2012 Chinese multiregional model
3-Sector Aggregation
Industry Sectors
Natural Resources
Agriculture
Mining & Processing
Light Industry
Energy Industry
Heavy Industry & Chemical Industry
Construction
Transportation & Telecommunications
Services
Commercial Services
Other
Manufacturing &
Construction
Services & Other Sectors
Appendix 3.3 The Balanced Regional Model and the Inverse of a Partitioned
(I A) Matrix
We use the results from Appendix A on the inverse of a partitioned matrix. For the
balanced regional model, let
I ARR
ARN
S T
E F
1
and
ð
I
A
Þ
ðI AÞ ¼
¼
¼
U V
G H
ANR
I ANN
Then, from (3.29)
xR ¼ Sf R þ Tf N
xN ¼ Uf R þ Vf N
(A3.3.1)
This generates total output throughout the nation of both regionally balanced goods
ðxR Þ and nationally balanced goods ðxN Þ.
In this case, using the partitioned inverse results, defining (A3.3.1), we have
1 1
I þ ARN U T ¼ I ARR ARN V
S ¼ I ARR
h
i1
1
1
V ¼ I ANN ANR I ARR ARN
(A3.3.2)
U ¼ VANR I ARR
Substituting for S and T in (A3.3.2), from (A3.3.1),
1
1
xR ¼ I ARR f R þ I ARR ARN Uf R þ Vf N
(A3.3.3)
But xN , as in (A3.3.2), is just the Uf R þ Vf N term on the right-hand side of
(A3.3.3), so
1
1
(A3.3.4)
xR ¼ I ARR f R þ I ARR ARN xN
To distribute both xR and xN production to individual regions, we need the regional
distribution of final demands for regional goods, f RðsÞ , for each region s, and we need
the regional distribution of production of each of the nationally balanced goods, ps , for
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References
107
each region. Then, to add the spatial dimension, for a specific region s, f R becomes
^ s xN . Therefore
f RðsÞ and xN becomes xN ðsÞ , which is p
1
1
^ s xN
(A3.3.5)
xRðsÞ ¼ I ARR f RðsÞ þ I ARR ARN p
This is (3.31) in the text.
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4
4.1
Organization of Basic Data
for Input–Output Models
Introduction
A formidable challenge in using input–output analysis in practice is assembling the
detailed basic data needed to construct input–output tables suited to different needs.
That is, does the table sufficiently characterize the economic area of interest – local,
regional, multiregional, national, multinational, or even global? Does it include the
industries or other economic sectors of interest in a sufficiently detailed way to be
meaningful? Does it adequately represent economic activity for the time-period of
interest? These data are compiled by a variety of means for the specific geographic area
and time-period of interest (or both) and often adapted from other data sources.
In many cases, the data needed for construction of input–output tables are part of a
larger collection of data assembled for a wide variety of socio-economic reasons. This
is the case particularly in national economies where well-developed systems and
standards for collecting economic data exist for such purposes as analyzing economic
impacts of government policy that affects the economy, accountability for distribution
of government revenues, or simply assessing the ongoing health of the economy. These
basic data are often derived from social accounting data assembled in the form of a
system of national (or regional) economic information,1 which is often routinely
collected by means of a periodic census or some other survey.
4.2
Observations on Ad Hoc Survey-Based Input–Output Tables
Sometimes data collected for an input–output table result from an ad hoc survey
designed specifically for that purpose. This is especially common at the regional level,
but these instances have seldom led to data collection for this purpose becoming a
routine part of the region’s annual collection of socio-economic data for the region.
1
For example, the US Department of Commerce’s Bureau of Economic Analysis (BEA) routinely compiles the
US National Economic Accounts. BEA periodically derives the national input–output tables from these
accounts. In 1968 the United Nations published a standardized system of national accounts (SNA) which is
consistent with the discussion presented here; this system is widely applied in the literature – originally in
United Nations (1968), and more recently in successive United Nations publications (1993, 1999, 2004, 2009,
2018a, 2018b). Viet (1994) surveys the common practices adopted by many nations compared with the SNA.
Chapter 5 explores many of these considerations.
112
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4.3 Common Methods for Generating Input-Output Tables
113
The methods, conventions, and standards adopted in carrying out ad hoc surveys
designed specifically for constructing input–output tables vary widely depending upon
circumstances. Among the classic historical efforts in North America is Leontief’s
original work for the US national input–output tables (summarized in Appendix C).
Others include regional efforts such as the State of Utah (Moore and Peterson, 1955), the
St. Louis Metropolitan Area (Hirsch, 1959), the state of Kansas (Emerson, 1969), the
Philadelphia Regional Input–Output Study (Isard and Langford, 1971), Nova Scotia
(Schaffer, 1978), and the Washington State series of input–output tables (e.g., Bourque
and Conway, 1977, and Chase, Bourque, and Conway, 1993). Beyond the US experience, of course, there are many cases in many regions and nations around the world, e.g.,
O’Connor and Henry (1975), Bulmer-Thomas (1982), and Tukker and Dietzenbacher
(2013). The researchers referenced here, along with many others, have published
chronicles of their experiences, including elaboration of alternative mechanisms for
dealing with data shortages, methods for reconciling inconsistent information, and many
other best practices in construction of these so-called survey-based input–output tables.
4.3
Observations on Common Methods for Generating Input–Output Tables
In planning for virtually any input–output analysis effort, the most desirable situation is
a statistically robust data source that includes precisely: (1) the geographic area under
consideration; (2) the time-period of interest; (3) the level of sectoral detail of interest;
and (4) the necessary detail characterizing economic interactions with other geographic
regions. Such a situation is the most desirable, but the least likely. The construction of
full survey-based input–output tables is a major undertaking that is both complex and
expensive, often prohibitively so.
Far more common is the situation in which researchers must adapt previously
constructed input–output tables to reflect current circumstances or to make assumptions about the similarity of the geographic area of interest to that of an area where an
input–output table already exists. Techniques for adapting existing tables either over
time or across space are often referred to as nonsurvey methods. Sometimes surveys of
selected industry sectors or other institutions may exist that are only part of what would
constitute a complete survey of the economy of interest, in which case many methods
have been developed to incorporate the selected new information into a strategy for
constructing a new table. Such techniques are often referred to as partial survey
methods. These nonsurvey and partial survey alternatives to full survey-based construction of input–output tables have been an active area of research for over 30 years.
Chapter 9 explores the basic methods commonly used and Chapter 10 explores
commonly applied extensions of such methods.
As noted in Section 4.1, in many cases where use of input–output models is part of a
government’s analysis of economic structure and performance, the collection of data
needed for construction of input–output tables is part of a larger and more routine
collection of national economic statistics assembled for a wide variety of socio-economic
planning and policy analysis reasons. Since the 1950s, many nations have increasingly
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114
Organization of Basic Data for Input–Output Models
adopted common conventions for organizing national economic data. We examine many
of these conventions in the balance of this chapter.
As we develop the most common conventions for national economic accounting that
are particularly relevant to input–output analysis, we can also set the stage for some
key enhancements to the input–output framework to deal with complications arising
from collection of data. For example, for practical reasons, researchers must often
collect the data for input–output tables (and many other purposes) directly from
business firms or establishments, and subsequently aggregate these data to characterize
a specific industrial sector. However, it is quite common for a firm to produce multiple
products or services that are associated with differently defined industrial sectors.
Hence, it is important to have standardized ways for our accounting conventions to
accommodate such situations. The current version of the System of National Accounts
(SNA) incorporates many of the conventions used in national economic accounting. In
Chapter 5, we deal with many of the important ways to use these conventions to
address the complications of secondary production and other features of the economic
accounts, but, for now, we develop the basic concepts of the SNA.
4.4
A System of National Economic Accounts
In an appendix to a landmark 1947 United Nations (UN) report entitled Measurement of
National Income and the Construction of Social Accounts, British economist Richard Stone
set forth the basic framework for the standardized system of national economic accounts
that is most commonly used around the world today (United Nations, 1947, and Stone,
1947). A subsequent 1950 UN report, A System of National Accounts and Supporting
Tables, formalized the SNA and, finally, in 1968, Stone led the team that produced an
integrated input–output framework and system of national accounts (United Nations,
1968). That work, coupled with Stone’s related original and subsequent contributions
(e.g., Stone, 1961), was recognized in his award of the 1984 Nobel Prize in Economic
Sciences “for having made fundamental contributions to the development of the system of
national accounts and hence greatly improved the basis for empirical economic analysis.”2
In Stone’s work on national income and production accounting, he describes input–
output transactions tables as a “bridge between statistics that can actually be collected
about the productive process and the requirements of applied economic analysis”
(Stone, 1961). Fashioning an ability to build this bridge in an organized way, however,
was a key development in making input–output the practical tool it has become. In the
following, we describe the relationships between input–output tables and national (or
regional) economic accounts and, in the process, show how to derive input–output
tables from such accounts. We focus on the SNA introduced at the beginning of this
section, which continues to be developed to refine the methodology and broadly
standardize the definitions and accounting rules that are used widely today.
2
From the citation announcing the award of The Bank of Sweden 1984 Prize in Economic Sciences in Memory of
Alfred Nobel, awarded December 8, 1984, as reported in Sveriges Riksbank (1984).
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4.4 A System of National Economic Accounts
115
Published updates to the SNA standards and conventions have appeared in the series of
UN publications, United Nations (1993, 1999, 2004, 2009, 2018a, 2018b).
Box 4.1 summarizes the historical evolution of the SNA accounts. By 2016 all
OECD nations had adopted the revised 2008 standards and the SNA remains the de
facto international national accounting standard. The UN reported (Matteo, 2018) that,
as of 2018, 93 percent of member states had “conceptual compliance” with the SNA
compared with 70 percent in 2009, although implementation of those standards has
been much slower.3 In general, the 2008 revision of the SNA accounts includes a
number of new features that provide for:
1. a more consistent treatment of intellectual property as assets along with treatment of
capital services;
2. a more comprehensive treatment of financial services;
3. incorporation of improved standards accounting for goods and services in global
value chains;
4. a more standardized treatment of public sector transactions such as those associated
with public–private partnerships and loan guarantees; and
5. a new approach to measuring activity within households on an informal basis – the
so-called informal economy.
A key concept in the development of the SNA throughout its history has been
adoption of an accounting structure in which not only economic production could be
subdivided so as to display the commodity flows between industries (the traditional
basis for Leontief’s model), but also that such information could be reconciled with all
the relevant information tracing the flows of income and wealth ultimately associated
with those flows. In the SNA, we accomplish this by means of a series of “balance
sheets” for the key sectors of the economy. For the purposes of this text, and in
particular in this chapter, we focus on the use of the SNA to facilitate construction
of input–output models, but it is important to recognize that the SNA provides the basis
for providing a national “balance sheet” and for describing and analyzing economic
change for many forms of economic decision-making.4
The data included in the SNA also enables expansion of the basic input–output
framework to systematically handle such issues as secondary production in the economy,
as noted in Section 4.3. We introduce these so-called commodity-by-industry concepts in
this chapter, but in Chapter 5 we develop them in much more detail and chronicle their
implications for input–output models. In addition, the SNA provides the basis for
broader social accounting modeling, building once again on the Leontief model. We
chronicle these extensions, called Social Accounting Matrices (SAM), in Chapter 11.
3
The 2008 SNA was prepared under the auspices of the Inter-Secretariat Working Group on National Accounts
(ISWGNA), which consists of five organizations: the Statistical Office of the European Communities (Eurostat),
the International, Monetary Fund (IMF), the Organisation for Economic Cooperation and Development
(OECD), the United Nations Statistics Division and regional commissions of the United Nations, Secretariat
and the World Bank. These five organizations jointly published the 2008 SNA. See Van de Ven (2015).
4
United Nations (1968, chapter 1, p. 12).
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Organization of Basic Data for Input–Output Models
Box 4.1 Historic Versions of the System of National Accounts
The principal objective of the System of National Accounts (SNA) is to provide “a
comprehensive conceptual and accounting framework for compiling and reporting macroeconomic statistics for analyzing and evaluating the performance of an economy.” The
framework, introduced in 1947 with the Report of the Sub-Committee on National Income
Statistics of the League of Nations Committee of Statistical Experts, with efforts led by
Richard Stone, which called for and specified a set of standards for the first SNA. Adopted
as a common standard by many nations and periodically updated, the SNA incorporates
changing economic circumstances, methodological improvements, and technical capabilities for compiling the relevant data.
1953. The United Nations Statistical Commission (UNSC) published the first SNA. It
established a set of six standard accounts and 12 standard tables presenting detail and
alternative classifications of the flows in the economy. The concepts and definitions of the
accounts were widely applicable for most countries, including developing countries.
1960. The first revision reflected comments on country experience in implementing the
1953 SNA.
1964. The second revision improved the consistency of the SNA accounts with the
International Monetary Fund’s Balance of Payments Manual.
1968. A new release of the SNA accounts expanded its scope substantially and significantly. For the purposes of this text, this included:
(1) adding specifically the input–output accounts and balance sheets;
(2) adopting conventions for providing estimates at constant prices; and
(3) bringing the SNA and the International Monetary Fund’s (IMF) Material Product
System (MPS) closer together
These revisions established new standards for concepts, definitions, classifications, and conventions used in compilation of balance of payments and international investment position statistics for IMF member countries that regularly
report balance of payments data to the IMF.
1993. This SNA represents a major advance in national accounting and embodies the
result of harmonizing the SNA and other international statistical standards more
completely than in previous versions.
2008. This major update of the 1993 SNA addresses issues reflected changes in the
world economic environment, advances in methodological research, and the practical
needs of users in assembling data implementing the SNA. With this 2008 update, the
UN, the European Commission (EU), the Organization for Economic Co-operation
and Development (OECD), the IMF, and The World Bank all formally endorsed and
adopted the SNA as the international statistical standard for national accounts.
Among the most important methodological changes with the 2008 update were
additional standards for compiling national records, incorporating research and
development and weapons systems into the treatment of capital, recording of pension
entitlements, classification of corporate offices (e.g., of multinational corporations),
treatment of holding companies, and financial services.
Source: United Nations, 2018. Department of Economic and Social Affairs, Statistics Division,
“Historic Versions of the System of National Accounts” (https://unstats.un.org/unsd/
nationalaccount, accessed April 9, 2018).
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4.4 A System of National Economic Accounts
117
4.4.1 The Circular Flow of Income and Consumer Expenditure
As noted earlier, the principal goal of the SNA is to provide “a framework within
which the statistical information needed to analyze the economic process in all of its
many aspects could be organized and related.”5 Conceptually, this takes us back to the
fundamental roots of much of economic thought – the notion of a circular flow of
economic resources in an economy introduced by Cantillon and Quésnay in the
eighteenth century. Appendix C chronicles these conceptual beginnings.
We start with the simplest of economies, in which there are only businesses that
produce goods and services and household consumers that purchase them. The
consumers also manage and work for the businesses and, hence, receive income
from them, the value of which we presume is exactly equal to the total value of their
purchases (see Figure 4.1). This is the fundamental tenet of the circular flow of
income and expenditures in that the total value of production can be measured either
by the value of all goods and services delivered to households or by the payments for
the factors of production delivered by consumer households, as depicted in
Figure 4.1.
From now on in this chapter we depict only the flow of money associated with
transactions, e.g., in Figure 4.1, the income received by consumers in return for
their labor services and the expenditures made by consumers in return for goods and
services delivered by businesses. We begin with a simple example, depicted in
Figure 4.2.
In the figure, consumers receive $575 million dollars as income for the labor
services that they provide to businesses (Q) and, in turn, they use that income to
purchase the same value of goods and services from businesses (C). That is, the total of
expenditures equals the total of all income, so in this simplest of economies we capture
the circular flow in the equally simple (4.1):
Q¼C
Figure 4.1 The circular flow of income and expenditures
5
United Nations (1968, chapter 1, p. 13).
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(4.1)
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Organization of Basic Data for Input–Output Models
Figure 4.2 Circular flow example: Point of departure
4.4.2 Savings and Investment
As our first refinement of this most simple of economies, recall that businesses,
producing the goods and services that they deliver to final consumers, also consume
inputs other than labor services, such as raw materials and capital equipment. So, once
again, as introduced in Chapter 2, we refer to deliveries to final markets as final
demand, deliveries of goods and services to other businesses as intermediate output,
and purchases of goods and services by businesses that are not resold as intermediate
goods to other firms or consumers, i.e., they become long-term depreciable assets, as
capital investments.
For the time being, we will leave industrial production and use of intermediate
output associated with that production – the major focus of input–output analysis –
within the block labeled “Businesses” in our flow diagram. Figure 4.3 depicts
the addition of capital expenditures as a portion of total expenditures in the
economy.
In general, consumers spend some portion of their income but also save or
invest some portion for longer-term financial gain. In Figure 4.3, the portion of
consumer income not spent on final goods is referred to as personal savings (S)
and, in our simple economy, we add a capital market, to hold personal savings
on behalf of consumers and lend it to businesses to purchase capital goods as
investments.
Note that, in the example (Figure 4.3), the total income generated by businesses
remains $575 million, reducing the total of current consumer expenditures to $525
million, with a $50 million residual not used for current consumption reserved as
savings (S) in the capital markets. This, in turn, provides that same amount as investment resources (I) to businesses acquiring capital goods.
As we proceed, sequentially unbundling activities to reflect increasingly more
realistic complexity in the economy, we will find we need to identify a way to include
the valuation of not only the transactions or flows in the economy, but also we
seek to reconcile those transactions with the valuation of the assets and liabilities
or stocks in the economy. We accomplish this by means of a national “balance sheet.”
For the moment, however, we will only account for transactions or flows that relate to
accumulations of stocks, not the valuation of the stocks themselves.
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4.4 A System of National Economic Accounts
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Figure 4.3 Introduction of savings and investment into the circular flow of income and expenditures
Figure 4.4 Introduction of depreciation into the circular flow of income and expenditures
Returning to the example, we are generally interested in measuring the key
economic flows in our economy, income, and expenditures, over a standardized
period, e.g., a year. Expenditures on capital, which we can loosely define as the use
of goods and services that extend beyond the standardized period, accumulate in the
economy as a stock, only a portion of which is “consumed” during the current
period. This is the reason why, in national income accounting parlance, we often
refer to this depreciation of a capital investment as a capital consumption
allowance. Figure 4.4 includes a capital consumption allowance, labeled
depreciation (D), to reflect the depreciation of the accumulated stock of capital in
the current period. Investment flows to businesses (I) are then drawn from the stock
of capital.
We can begin to keep track of the transactions systematically in our simple economy
by maintaining some traditional accounting balance sheets (sometimes called “T”
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Organization of Basic Data for Input–Output Models
Table 4.1 Basic national accounts: Example economy
Debits
Production (Domestic Product Account)
Income (Q)
Credits
575
Total
575
Consumption (Income and Outlay Account)
Purchases of consumption goods (C)
525
Savings (S)
35
Total
560
Accumulation (Capital Transactions Account)
Purchase of capital goods (I)
50
Depreciation (D)
15
Total
35
Sales of consumption goods (C)
Sales of capital goods (I)
Total
525
50
575
Income (Q)
Depreciation (D)
Total
575
15
560
Savings (S)
35
Total
35
accounts since the two columns are usually separated and labeled with lines
resembling a T; hence the name, “T” account). There is one such account for each
major type of economic activity: (1) production by businesses, often called the
Domestic Product Account, (2) consumption by consumers, often called the Income
and Outlays Account, and (3) capital accumulation in the capital markets, often called
the Capital Transactions Account. For our example so far, these accounts are shown in
Table 4.1.6
We can also summarize these double entry bookkeeping accounts by the following
simple balance equations:
Q¼CþI
(4.2)
CþS ¼QþD
(4.3)
I þD¼S
(4.4)
4.4.3 Adding Overseas Transactions: Imports, Exports, and Other Transactions
Our next refinement recognizes that if some of the businesses in our simple economy
are located overseas, it will be important for a variety of reasons to account for them
separately. We can do this by defining and adding the various transactions with
overseas businesses as imports (M) and exports (X) of goods and services, consumer
6
For more detailed expositions on the relationship between macroeconomics, national economic accounts, and
input–output analysis, see Gordon (1978), Sommers (1985), and especially United Nations (1968, 1993, 1999,
2004).
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4.4 A System of National Economic Accounts
121
Figure 4.5 Addition of the rest of world account
expenditures overseas (called overseas transfers, O), the net of lending and borrowing
of capital overseas (net overseas lending, L) and consumer income received from
overseas (H). For our example, Figure 4.5 depicts these additional transactions along
with the addition of a new category of economic activity and corresponding balance
sheet called “Rest of World.”
We refer to the corresponding balance sheet as the Balance of Payments Account,
which is included in Table 4.2 along with the other revised accounts.
We can again summarize these expanded double entry bookkeeping accounts by the
following set of balance equations:
QþM ¼CþI þX
(4.5)
CþSþO¼QþDþH
(4.6)
I þDþL¼S
(4.7)
X þH ¼M þOþL
(4.8)
Conceptually, a useful way to visualize these balance equations is to draw a circle
around each block in Figure 4.5, then calculate the sum of all the transactions going
into the block, which will equal the sum of all the transactions leaving the block
or account.
4.4.4 The Government Sector
Finally, the major role of government in most economies suggests it should also be
included explicitly as a major activity in our portfolio of national economic accounts,
which we label the Government Account. This involves highlighting government
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Organization of Basic Data for Input–Output Models
Table 4.2 Basic national accounts including rest of world
Debits
Credits
Production (Domestic Product Account)
Consumer Income Payments (Q)
Purchases of Imports (M)
550
50
Total
600
Consumption (Income and Outlay Account)
Purchases of consumption goods (C)
500
Net Transfers Overseas (O)
10
Savings (S)
35
Total
545
Accumulation (Capital Transactions Account)
Purchase of capital goods (I)
75
Depreciation (D)
19
Net Lending Overseas (L)
21
Total
35
Rest of World (Balance of Payments Account)
Purchases of Exports (X)
25
Net Overseas Income (H)
14
Total
39
Sales of consumption goods (C)
Sales of capital goods (I)
Sales of Exports (X)
Total
500
75
25
600
Income (Q)
Depreciation (D)
Net Overseas Income (H)
Total
550
19
14
545
Savings (S)
35
Total
35
Sales of Imports (M)
Net Transfers Overseas (O)
Net Borrowing Overseas (L)
Total
50
10
21
39
transactions, including taxes paid by consumers (T), government purchases of goods and
services (G), and government deficit spending (B). For our example, these additions are
reflected in Figure 4.6. Table 4.3 shows the corresponding modified national accounts.
We, once again, summarize these double entry bookkeeping accounts by the
following balance equations:
QþM ¼CþI þX þG
(4.9)
CþSþOþT ¼QþDþH
(4.10)
I þDþLþB¼S
(4.11)
X þH ¼M þOþL
(4.12)
G¼T þB
(4.13)
4.4.5 The Consolidated Balance Statement for National Accounts
For convenience, we can summarize all the double entry bookkeeping accounts (and
corresponding balance equations) we have accumulated so far much more succinctly in
a single Balance Statement, provided in Table 4.4.
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4.4 A System of National Economic Accounts
123
Table 4.3 Basic national accounts including the government sector
Debits
Production (Domestic Product Account)
Consumer Income Payments (Q)
Purchases of Imports (M)
Credits
550
50
Total
600
Consumption (Income and Outlay Account)
Purchases of Consumption Goods (C)
475
Net Transfers Overseas (O)
10
Taxes (T)
20
Savings (S)
40
Total
545
Accumulation (Capital Transactions Account)
Purchase of Capital Goods (I)
75
Depreciation (D)
19
Government Deficit Spending (B)
5
Net Lending Overseas (L)
21
Total
40
Rest of World (Balance of Payments Account)
Purchases of Exports (X)
25
Net Overseas Income (H)
14
Total
Government (Government Account)
Government Purchases (G)
39
Total
25
25
Figure 4.6 Addition of the government account
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Sales of Consumption Goods (C)
Sales of Capital Goods (I)
Government Purchases (G)
Sales of Exports (X)
Total
475
75
25
25
600
Income (Q)
Depreciation (D)
Net Overseas Income (H)
550
19
14
Total
545
Savings (S)
40
Total
40
Sales of Imports (M)
Net Transfers Overseas (O)
Net Borrowing Overseas (L)
Total
Taxes (T)
Government Deficit Spending (B)
Total
50
10
21
39
20
5
25
124
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Table 4.4 Balance statement for the basic national accounts
Debits
Prod
Cons
Capital
Accum
Credits
Govt
Rest of
World
475
75
25
50
550
19
14
10
40
21
25
20
545
40
Economic
Transaction
C
I
Consumption Goods
Capital
Goods
Exports
Imports
Income
Depreciation
Overseas Income
Transfers Abroad
Savings
Net Lending Abroad
Govt
Expenditure
Taxes
Govt. Deficit
Spending
Totals
X
M
Q
D
H
O
S
L
G
T
B
5
600
Var
Name
25
39
Prod
Cons
Capital
Accum
Govt
Rest of
World
475
75
25
50
550
19
14
10
40
21
25
20
5
600
545
40
25
39
4.4 A System of National Economic Accounts
125
Table 4.5 Basic national accounts balance statement in matrix form
Prod
Production
Consumption
Capital Accum
Rest of World
Government
Total
Cons
Cap
RoW
Govt
Total
475
75
19
25
14
25
21
5
40
600
545
40
39
25
39
25
550
50
600
40
10
20
545
Subsequently, and even more compactly, we can represent the transactions (each of
which appears twice in our current tables) by entries in a matrix with the nature of the
transaction (source and destination) to be inferred from the transaction’s position in the
matrix (Table 4.5). Through this example, the reader may anticipate that we are
gradually working our way to the more familiar input–output table format. This will
be apparent ultimately by subdividing the Production and Consumption transactions to
show activities and output in specific industries and the use of specific products. For
example, the Production–Consumption transaction, instead of being a single number
(475 in the example), will be represented by a matrix (often called the Use matrix),
with rows indicating specific products or commodities and columns indicating
specific industries.
At this point, we have characterized the major economic activities in our simple
economy by representing them in a matrix that captures the information in the
following basic series of principal national economic accounts:
1. Production of Goods and Services or the Domestic Product Account
2. Consumption of Goods and Services or the Income and Outlay Account
3. Accumulation of Capital or the Capital Transactions Account
4. Imports and Exports or the Balance of Payments Account
5. Government or the Government Account
So far, with the partial exception of transactions associated with capital accumulation,
we have accounted for only transactions or flows in the economy. We have largely
ignored accounting for the total value of accumulated assets and liabilities or stocks in
the economy. Conceptually we can accomplish this by incorporating the balance
statement into an accounting balance sheet with a valuation of opening net assets in
the economy, i.e., the depreciated value of tangible assets held plus the excess of any
financial claims held as assets over financial claims issued as liabilities, which is
defined as the “net worth” of the economy.
As always, we measure all these activities in terms of the value of the transaction,
but, so far, we have not specified the prices used in valuing these transactions or how
we can accommodate price changes. Conceptually, we seek to incorporate all the
economic activities we have described so far as well as market changes in prices of
goods and services. As we learned in Chapter 2, in input–output analysis, we will most
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Organization of Basic Data for Input–Output Models
often assume fixed prices in any given analysis, but we need to be able to account for
year-to-year revaluations of assets and liabilities.
As we devise a “system” of national accounts to track the evolution of the economy
from one period to the next, for any given time period, we will trace the transformation
of an “opening” balance sheet into a “closing” balance sheet. This relationship between
opening and closing balances is evident in two equivalent ways. First, the sum of net
assets at the end of the period is equal to the sum of net assets at the beginning of the
period, net domestic and foreign investment during the period, and revaluations needed
to adjust the value of assets previously acquired or liabilities previously issued to the
prices in place at the closing date. Second, the net worth at the end of the period is
equal to the net worth at the beginning of the period plus new savings accumulated
during the period and revaluations resulting from price changes.
4.4.6 Expressing Net Worth
Now we can add the notion of Initial and Closing asset values, which, in moving from
one to the other, are transformed by interaction of consumption, production, capital
accumulation, and net exports, as well as asset and liability revaluations resulting from
price changes of goods and services. With two basic observations, it follows that the
opening and closing balance sheets relate to one another by the effect of price
revaluations and the net effect of capital transactions. The two observations are the
following: (1) total saving in the economy is equal to net capital investment and (2) net
worth can only be revalued by applying to it the revaluation determined from the price
changes of tangible assets and financial claims. Figure 4.7 depicts the relationship
between opening and closing balance sheets for our example.
Opening 700
Liabilities
840 Closing
40
(T)
20
(Q) 550
Government
Production
(B) 5
25 (Businesses)
(G)
Consumption
(H) 14
(O)
10
(C)
475
40
(S)
100
= S = 40
Liabilities
(M)
50
Rest of World
-19
(D)
Revaluations
25
(X)
(L)
-21
(I)
75
Accumulation
(Capital Markets)
Opening
Assets
40
= I + D + L + B = 75 - 19 - 21 + 5
700
Figure 4.7 Net worth
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100
Closing
840 Assets
127
4.4 A System of National Economic Accounts
Note that this expanded schematic reflects two changes. First is a reformatting of
Figure 4.6, specifically highlighting (4.11), which is the balance equation for allocation
of savings, i.e., consumer income not spent on current consumption is otherwise
invested either directly as private investments or indirectly as net taxes. We define
the difference between gross payments for taxes and that portion of taxes returned to
consumers as welfare transfers. Second, savings during the period become additions to
the stock of accumulated capital so the opening value of assets is increased by the
amount of savings, or equivalently the level of investment diminished by depreciation,
net lending abroad, and government deficit spending, which is
S ¼I þDþLþB
(4.14)
For our example, the accumulated saving is S ¼ 40 and, if revaluations due to
price changes amount to 100, then the opening asset value (net worth) of 700 is
increased at the end of the period by sum of accumulated savings and revaluations,
or 700 þ 40 þ 100 ¼ 840. We can express this much more succinctly by expanding the
basic matrix presented in Table 4.5 to include opening net worth or net opening assets
(NOA) and closing net worth or net closing assets (NCA), which includes revaluations,
as shown in Table 4.6.
We can recap the balance equations, now including the net worth balances, by
defining R as the total of revaluations due to price changes, W1 as the opening net
worth, and W2 as the closing net worth to yield the following:
QþM ¼CþI þGþX
(4.15)
CþSþOþT ¼QþDþH
(4.16)
I þDþLþB¼S
(4.17)
Table 4.6 Matrix of national accounts including net-worth calculations
NOA
Prod
Net Opening Assets
(NOA)
Production
Rest of World (RoW)
Government
Cap
RoW
Govt
75
25
25
19
14
Reval
NCA
100
840
700
475
Consumption
Capital Accumulation
Cons
550
700
40
50
10
20
Revaluations (Reval)
Net Closing Assets
(NCA)
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21
5
100
840
128
4.5
Organization of Basic Data for Input–Output Models
X þH ¼M þOþL
(4.18)
G¼T þB
(4.19)
W2 ¼ W1 þ S þ R
(4.20)
National Income and Product Accounting Conventions
So far in this chapter, we have developed the system of national income and product
accounts sufficiently that we can review some of the traditional assumptions and
conventions often used in compiling these accounts. For the most part these conventions are completely consistent with the basic input–output framework developed in
Chapter 2, but some of the peculiarities of nomenclature are different and worth noting.
Perhaps the most important general characteristic is that the overall system is closed,
i.e., the system taken together accounts for all activities in the economy such that the
value of total production is equal to total consumption (production and consumption
are balanced). This will be of more interest in Chapter 11, when we expand input–
output table to a Social Accounting Matrix (SAM), formulated specifically to represent
all components of the closed system of national accounts.
The following are some basic principles included in most common systems of
national accounts:7
• Double Entry Bookkeeping. Government agencies generally maintain national
economic accounts in the form of a traditional double entry bookkeeping accounting
system, tabulating total economic output on the debit side and the total resulting
income flows on the credit side. That is, by adopting the financial accounting
convention of a “T” account described earlier we can characterize a business
establishment’s production account (BEPA) in two columns, with debits to the
account (expenses and profits) recorded in the left-hand column and credits (sales
and other revenue) recorded in the right-hand column. If net income before taxes is
viewed as a payment to capital, then total income, recorded as employee compensation and earnings from real and financial property, is equal to total cost
including depreciation.
• Output Equals Demand. Total output of the economic system is exactly equal to
total demand or, equivalently, gross national product is the same as gross national
expenditure. Perhaps the key concept is that inventories are essentially a reservoir
that appears as either a slack or a surplus variable for output. If demand in final
markets (final demand) falls short of output, inventories would rise – an inventory
accumulation. If demand exceeds output, then the result is an inventory drawdown –
a negative demand or inventory liquidation. The result is that output always equals
7
United Nations (1968, 1993, 1999), Gordon (1978), Sommers (1985).
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4.6 Assembling the Input–Output Accounts: The US Case
129
demand with inventories providing the accounting convention for handling surpluses
or shortages during any given time-period.
• Requiring Total Expenditure to Equal Total Income. We have presumed in our
running example that total output in the economy is equivalent to both total
expenditures as well as total income generated. This means if consumers spend less
than their total generated income, then the unspent balance goes into savings, as we
derived from the original concept of a circular flow. Conversely, in current terms,
businesses typically spend more than their generated income, which comes in the
form of net investment. In general, the total of savings in the economy is equal to
total investment, as in the example.
• Avoiding Double Counting of Output of Goods and Services. We measure
economic output as the value of output delivered to final demand. If copper produced
by a copper company is counted both when it is delivered to a circuit board maker
and in the final sale of the circuit board that incorporates the original copper, then the
copper is counted twice. Rather, we think of the value of the copper as embedded in
the value of the circuit board and we measure the value added at each stage of
production, the sum of which is total economic output.
• Valuing Output at Market Value. Generally, we assume that all industry output is
valued at the prevailing producers’ market price or the sales price. Some output does
not pass though a market, however, e.g., food produced and consumed on a family
farm. In such cases, we generally estimate the market value and define it imputed
income or expenditure.
• Measuring Gross Domestic Product. The value of economic output includes the
value of tangible goods as well as new construction and services. It is called the
Gross Domestic Product (GDP) since contributions to GDP are compiled as gross
measures, i.e., recorded prior to accounting for depreciation (or capital consumption
allowances). Note that the GDP measures the total amount of goods and services
produced within a country’s geographic borders. A related measure, the Gross
National Product (GNP), measures the total amount of goods and services that a
country’s citizens produce regardless of where they produce them. For example,
GNP for the USA includes such items as corporate profits that multinational firms
earn in overseas markets whereas, in GDP terms, such profits contribute to
GDP overseas.
• Excluding Revaluations. As noted in Section 4.4.2, during any given time period,
the national accounts do not include capital gains and losses – the creation or
destruction of value not attributable to real economic output or income generated.
That is, revaluations resulting from price changes provide the principal connection
between time-periods, but valuation during any given period excludes them.
4.6
Assembling the Input–Output Accounts: The US Case
Our final task in getting us from the basic concept of the circular flow of income and
expenditures and tracing income and expense through the national economic accounts
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back to where we started in Chapter 1, namely the input–output transactions table, is to
expand the national economic accounts to include industry and commodity levels of
detail. As noted earlier, in practice, researchers often compile the data needed to
provide this detail on interindustry transactions through a census or survey of all
economic activity of establishments or firms involved in the economy. For example,
in the USA, the so-called benchmark Input–Output Accounts (IOAs) are prepared
every five years for years ending with 2 and 7 (e.g., 1992, 1997, 2002, 2007, and 2012)
based on the quinquennial US economic census and other data sources.8 Various
government agencies, such as the Department of Labor through its surveys of prices
and the Department of Commerce’s Census Bureau through its surveys of retailers and
manufacturers, collect the basic data, but the Bureau of Economic Analysis (BEA), an
agency of the Commerce Department, prepares the IOAs (see BEA, 2008).
Historically, a primary goal of the US economic census has been to estimate the nation’s
GDP (and GNP), as reported in the detailed series of Gross Domestic Product by Industry
(GDPI) Accounts (see, for example, Yuskavage and Pho, 2004). The GDPI accounts are
central to the assembling of the formal national economic accounts recording the nation’s
overall generation of income and production, the National Income and Product Accounts
(NIPA). The level of industry production and intermediate consumption detail collected
during the census also provides a convenient basis for construction of the interindustry
IOAs as well, but the sequence of preparation of these data is in reverse order from the
discussion so far in this chapter. That is, since the 1950s, in order to provide as much
consistency as possible between various economic accounting systems used by the BEA,
the preparation of these systems is coordinated among data collection processes. BEA
generally assembles the IOAs are first and then uses them in the process of constructing
other national economic accounts, such as the GDPI and NIPA (see Jaszi, 1986).
BEA publishes the US IOAs in the form of two tables defined in terms of production
and consumption of defined goods and services or commodities and groups of economic
establishments or industries that may produce more than one commodity, but organizing
both commodities and industries according to a standardized classification scheme. BEA
adopted the North American Industrial Classification System (NAICS) in 1997 as a
replacement for the former Standard Industrial Classification (SIC) and other systems for
organizing US economic data. The NAICS underwent subsequent revisions in 2002,
2007, and 2012 along with a major revision in 2017 with the principal goal “to modify or
create industries to reflect new, emerging, or changing activities and technologies.”9
8
A great deal of the data used in constructing the US benchmark IOAs comes from the quinquennial US
economic census, but additional data is often utilized from other sources, particularly in some economic sectors
such as natural resources and mining, financial activities, and services; see Webb (1995), Lawson et al. (2002)
and Moyer et al. (2004a, 2004b).
9
The United States adopted NAICS in 1997 to reflect more accurately the changing structure of North American
economies since the early 1980s, including in particular the large growth in services-producing industries. For
example, the NAICS defines 575 services-producing industries as opposed to 407 in the SIC. About 250 of the
358 new industries defined in the NAICS-based IO industry classification system are services-producing
industries (see Horowitz and McCulla, 2001; McCulla and Moylan, 2003). The major revision to the NAICS
is provided in OMB (2017).
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4.7 Additional Considerations
131
The first key table of the IOAs is the so-called Use table, which provides information on the consumption or use of commodities by industries or by final demand
sectors, such as households, government, capital investment, or exports. A column of
the Use table is an industry or final-demand sector, and the rows indicate the use of
commodities by that industry or final-demand sector. Also included in the Use table are
rows corresponding to components of value added by industry, such as employee
compensation, business taxes, and other value added. Table 4.7 shows the basic
organization of the Use table. Note that in this table the sectoral designations for
industries and commodities are the same, i.e., the Use matrix is square with the labels
of industries and commodities identical. This simply means that we organize accounting for industries and commodities in the economy by the same sector definitions. This
is a standard convention adopted for the US input–output tables and many others for a
variety of reasons, but it need not be, as we will find later in this chapter and in much
more detail in Chapter 5. The numbers of industries and commodities and their
definitions can be different, in which case the Use table may be non-square.
The second key table of the IOAs is the so-called Supply table. Early development of
the commodity-by-industry conventions referred to the Make table, which is the
transpose of the Supply table. The Supply table provides information on industry
production of commodities. A row of the Supply table corresponds to a commodity,
and columns indicate the production of that commodity by different industries. If there
were a one-to-one correspondence between industries and commodities, i.e., each
industry produces one and only one commodity, then the Supply table would be square
and contain non-zero elements only along its main diagonal. Table 4.8 provides an
illustration of the basic organization of the Supply table. Later in this chapter (Section
4.7.2), as an illustration, and in much more detail in Chapter 5, we examine alternative
ways of combining the information in Supply and Use tables to fashion the interindustry transactions matrix used in input–output models.
Finally, the income categories comprising value-added inputs to industries usually
include wages and salaries paid to employees, rental and proprietors’ income, profits,
taxes, interest, adjustments to inventories and non-competitive imports and, in
Table 4.7 (the Use table), all of these transactions are aggregated into the row labeled
Total Value Added, which specifies the total of value-added inputs for each industry.
For an extensive discussion of the supply and use framework for national economic
accounts, see Beutel (2017).
4.7
Additional Considerations
An important challenge in assembling IOAs, beyond simply coping with the scale and
expense of a comprehensive census or survey, has to do with timeliness for the various
uses of these data. If an industry’s technology is changing rapidly, such as the
computer industry over the decades of the 1980s and 1990s, lengthy time lags in
availability of the IOAs can lead to very misleading results if used in economic
modeling. In such situations, nonsurvey tools coupled with partial surveys of the type
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Table 4.7 The commodity-by-industry use table
Industries
Nat
Res
Commodities
Natural Resources
Construction
Manufacturing
Transportation
Utilities
Information
Financial Services
Other Services
Value Added
Employee Compensation
Indirect Business Taxes
Other Value Added
Total Industry Output
Const
Manuf
Transp
Final Demand
Utils
Fin
Srvcs
Oth
Srvcs
Pers
Cons
Invest
Net
Exports
Interindustry Transactions
Gross Domestic Product
Govt
Total Commodity
Output
4.7 Additional Considerations
133
Table 4.8 The industry-by-commodity supply table
Industries
Nat
Res
Commodities
Natural Resources
Construction
Manufacturing
Transportation
Utilities
Information
Financial Services
Other Services
Total Industry Output
Const
Manuf
Transp
Utils
Fin
Srvcs
Oth
Srvcs
Total
Commodity
Output
Commodity Production by Industry
discussed in detail in Chapters 9 and 10 are often used. Specifically, in the USA, data
for the years intervening the availability of the US benchmark IOAs are provided by
the Annual Input–Output Accounts (AIOAs) where more frequent but more highly
aggregated annual surveys are used to update earlier benchmark input–output
accounts. The basic conventions for compiling the AIOAs are reported in Okubo,
Lawson, and Planting (2000), Kuhbach and Planting (2001), and Planting and
Kuhbach (2001). Other conventions for compilation of the US benchmark and annual
accounts and their relationships are chronicled in Horowitz and Planting (2009).
Conceptually, in constructing an input–output model, the most important components of a system of national economic accounts are: (1) the national income and
product accounts (NIPAs), where we started, and (2) the interindustry or input–output
accounts (IOAs). The former, as we have found so far in this chapter, present the
aggregated productive output of the national10 economy, that is, the GDP both in terms
of final products or final demands and in terms of income categories or value added
inputs to industries.11 The IOAs present interindustry flows of goods and services
which, with a number of adjustments we describe in Section 4.7.2, ultimately become
the interindustry transactions matrix.
The NIPAs and IOAs comprise perhaps the bulk of the basic data of the national
economic accounts (at least those relevant to input–output analysis). As noted in
Section 4.2, they are collected by establishment or individual business unit, often as
10
11
For the most part, the following discussion will apply to regional as well as to national accounts.
Recall from Chapter 2 that the sum of all final demands equals GNP, which also equals the sum of all income
types or “charges against” GNP.
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Organization of Basic Data for Input–Output Models
part of a census. For the present, we will invoke three additional simplifying assumptions to facilitate our discussion of reconciling the IOAs with the NIPAs:
• Inventory Adjustments. We ignore, for the time being, the complications of
inventory adjustments, that is, we assume no changes in inventories. For example,
we presume all purchases of automobiles produced occur during the current year and
are not held over until the next year. We saw earlier that inventory adjustments are
essential to provide the balance between total consumption and total output.
• Secondary Products. By distinguishing between commodities and industries, we
assume an industry can produce multiple products (called secondary production). In
the basic framework we allowed each industry to produce one and only one distinct
commodity (or service), e.g., automobile manufacturers make only automobiles, not
additional automobile parts which may be classified as a different industry category.
The implications of this distinction will become much clearer in Chapter 5 and ways
of accommodating this distinction in modeling become straightforward by using the
Supply matrix to allocate secondary production in an organized way.
• Capital Formation. We ignore, for the present, transactions of capital goods
between industries. We, instead, assign them to final demand (gross private domestic
capital formation). For example, new car assembly equipment purchased by automobile manufacturers is a capital good acquisition, recorded as a final demand for
capital by the manufacturer and not as an interindustry transaction.
These assumptions are all somewhat limiting and can be relaxed with additional
modifications that we address in later chapters. The first of these simplifications will
require relatively minor adjustments later in this chapter, but the last two will require
major changes in the way we construct input–output models. While there are different
ways to deal with the problem of secondary production, ultimately, we will resort to
alternative commodity-by-industry model formulations that we introduce later in this
chapter and develop in more detail in Chapter 5. Modeling of capital formation is the
principal concern of dynamic input–output models and is a subject of Chapter 14.
The basic IOAs we have focused on in this chapter, by themselves, are far from
adequate for constructing a useful input–output model. To make the derived table a
useful analytical tool, we must deal with the simplifications that were made earlier. We
now discuss key conventions and modifications to the basic input–output accounting
framework that are designed to deal with several of these simplifying assumptions
(others are addressed in subsequent chapters).
4.7.1 Secondary Production: Method of Reallocation
As noted, in the construction of IOAs information is compiled by establishment or
individual business unit; we assigned an establishment to a defined “industry” category
according to the output of the establishment which comprises the primary source of
revenues (primary product). Many business units, however, may produce substantial
amounts of products that do not belong to the primary product industry classification;
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4.7 Additional Considerations
135
such products are termed secondary products. For example, many automobile manufacturers may produce automobile parts in addition to fully assembled automobiles, or
petroleum refiners may produce petrochemicals as a by-product to producing gasoline
or other petroleum products.
Early input–output studies, such as the pre-1972 US national tables, treated secondary
products in the following manner. First, selected secondary products were “reallocated,” that
is, the level of secondary production and its constituent inputs were assigned to the sector
defining that product as a primary output. Such treatment was used only for industries where
secondary production comprised a significant fraction of total output. All other secondary
products in the economy were treated as if they had been sold by the producing sectors to the
sectors for which those products were classified as primary. To accomplish this, a table of
transfers was constructed recording these imaginary sales. However, adding the matrix of
transfers to the basic transactions matrix double counts the value of secondary products and
consequently inaccurately inflates total output. This was done in order to ensure that the
secondary products were distributed correctly to consumers, at the expense of inflating the
total outputs of some industries that are secondary producers.
Example 4.1: Reallocation of Secondary Production Reallocation (sometimes referred to as redefinition) of secondary production, as just mentioned, involves
factoring out the amount of secondary product produced as well as the inputs used in
that production and reassigning both to the industry for which the product is classified
as primary. However, this requires that a firm allocate its inputs between the production
of primary and secondary products; in effect, it is necessary to break the firm into two
independent subfirms – one a producer of the primary product and the other a producer
of the secondary product. Most firms do not record data in a form that permits this
accounting easily, so a less desirable treatment of secondary production is often
employed in input–output studies. For example, the US Department of Commerce
(prior to preparation of the 1972 national input–output table) reassigned the output of
secondary production to the sector designated as the primary producer, but did not
reassign the corresponding inputs. This amounted to a double counting of the inputs
required for secondary production, which we see in the following example (US
Department of Commerce, 1969, 1974, 1980; Vaccara, Shapiro, and Simon, 1970).
Consider a three-industry economy; the matrix of interindustry transactions and
vector of total outputs are given in Table 4.9.
Table 4.9 Input–output transactions: Example 4.1 (millions of dollars)
Industry
Industry 1
Industry 2
Industry 3
1
2
3
Total Outputs
266
267
340
378
110
340
230
224
468
1,000
1,500
1,200
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Organization of Basic Data for Input–Output Models
Suppose that firms in both industries 1 and 3 are secondary producers of product 2,
that is, industry 1 produces $100 million worth of product 2 in addition to $900 million
worth of product 1, and industry 3 produces $10 million worth of product 2 as well as
$1,190 million worth of product 3. As mentioned in the introduction to Section 4.7, the
proper distribution of output is often desirable in input–output studies. A convention
often employed is to treat the secondary product as if it were sold to the industry for
which the product is classified as primary.
In this example, if the secondary production for all producers was “transferred”
and
to the correct primary producer, then the revised transactions
matrix, Z,
2
3
266 378 230
¼ 4 367 110 234 5 and
corresponding total outputs vector, x, would be Z
2
3
340 340 468
1,000
¼ 4 1,610 5. Note that we accomplish this simply by adding the amount of secondary
Z
1,200
production, termed a transfer, to the industry for which the secondary product is
classified as primary. That is, the $100 million worth of product 2 produced by industry
1 is added to the original z21 transaction as if it were sold by industry 2 to the secondary
producer, industry 1. Similarly, the $10 million worth of product 2 produced by industry
3 is added to the original z23 transaction. Finally, total output of industry 2 is increased by
the sum of all secondary production of product 2. However, the total outputs of industries
1 and 3 are not decreased, since the inputs required in secondary production were not
reallocated; this inflates total output, since secondary production is counted twice.
In many earlier input–output studies (prior to the 1980s and especially prior to
widespread adoption of the SNA framework) such transferring of secondary production
was used except where secondary production comprised a large portion of total output of
an industry, in which case both secondary products and inputs were reallocated.
4.7.2 Secondary Production: Commodity-by-Industry Accounting
A more realistic classification scheme that accounts for industrial production by commodity type rather than industry category eliminates the somewhat clumsy and biased
accounting of reallocating secondary production. More recent studies, including the US
National Tables compiled for years since 1972, redefine all secondary production by
establishing a set of “commodity-by-industry” accounts. In Chapter 5 we examine the
commodity-by-industry accounting framework in detail. For the present, we can introduce the basic features of accounting for secondary production in a commodity-byindustry framework as we expand the way we represent production and consumption in
the national accounting system to include the input–output accounts.
Example 4.2: Commodity-by-Industry Accounts The following illustrates the
use of commodity-by-industry accounts in fashioning an input–output model with a
two-industry and two-commodity example. Consider the consolidated commodity-byindustry accounts in Table 4.10.
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4.7 Additional Considerations
137
Table 4.10 Example 4.2, use and supply accounts (millions of dollars)
Industries
Use Table
Commodities
A
B
Value Added
Total
Supply Table
Commodities
Total
A
B
A
B
Final Demand
Total Output
12
10
68
90
8
7
95
110
80
83
163
100
100
Industries
A
B
90
10
0
100
90
110
Total Output
100
100
In this example, industries are defined as A and B corresponding to the primary
products of the establishments included in the definition of these two industries; that is,
industry A’s primary product is commodity A and industry B’s primary product is
commodity B. Industry A produces only commodity A (all establishments included in
defining industry A produce only commodity A, i.e., there is no secondary product).
Establishments assigned to industry B, however, while primarily producing commodity B, also produce, as a secondary product, some amount of commodity A. In
this simple economy, industry A produces $90 million worth of commodity A and
industry B produces $100 million worth of commodity B and $10 million worth of
90 0
commodity A. We define the Supply matrix for this example as V ¼
.
0 100
While the Supply matrix is a complete picture of the economy, it does not provide
information about the interindustry activity in an economy, such as deliveries of
commodities to other industries or to final demand. The interindustry activity is defined
12 8
by the Use matrix, which for the example (Table 4.10) is U ¼
: Also from the
10 7
80
table we can define the vector of commodity final demands, e ¼
, the vector of
83
100
total commodity outputs, q ¼
, the vector of total value-added inputs,
100
90
v ¼ ½ 68 95 , and the vector of total industry outputs, x ¼
.
100
4.7.3 Reconciling with the National Accounts
Recall the matrix version of the summary of the national accounts for our example,
shown in Table 4.5. We can expand our representation of consumption in the economy,
which is currently reflected by a single number indicating the net value of total
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Organization of Basic Data for Input–Output Models
consumption (545 in our example). The expansion will capture the role of individual
industries and specific products (goods and services). For the example, let us consider
an economy with three industries (Natural Resources, Manufacturing, and Services)
which produce five products (agricultural products, energy, manufactured goods,
financial services, and other services).
We presume that industries consume products (commodities) as interindustry transactions in the course of delivering their own product(s) to other industries as well as
final customers. For our example, in Table 4.10, we allocate total consumption
transactions of commodities between interindustry transactions and sales to final
customers as well as the government, the total of which is C ¼ 550 (total domestic
consumption) from Table 4.5. We also allocate capital accumulation and net foreign
income, which total D ¼ 19 and H ¼ 14, respectively, from Table 4.5. The total of all
outputs then is 545. There are, of course, many possible interindustry transactions
tables such that the relevant interindustry totals for C, D, and X are consistent with the
totals provided in Table 4.5. One such transactions table is shown as Table 4.11.
The interindustry transactions portion of Table 4.11, i.e., including only the transactions among the industry sectors, the reader should recognize as the Use matrix, as
defined in Section 4.7.2, since it depicts the commodities used by each industry in
producing its output. This is, of course, as before, analogous to the interindustry
transactions matrix in the input–output analysis framework, except that in the basic
framework both rows and columns refer to specific industries, rather than commodities
and industries, respectively, for rows and columns.
While one could easily express the transactions matrix in industry-by-industry
terms, the SNA adopts a commodity level of detail in order to provide a more
transparent picture of industrial production functions. There are other benefits as well
Table 4.11 Production account allocated to individual products and sectors
Industry
Consumption
Commodity
Production
Nat
Res
Agriculture
Energy
Manufacturing
Fin Services
Other Services
Totals
25
13
10
10
8
Manuf Srvcs
10
7
20
10
30
15
9
7
25
20
Final Demand
Total
Intermed
Output
50
29
37
45
58
219
Total Final Consumption
Total Final Demand
Total Domestic
Consumption
Pers
Cons
Exp
Govt
Exp
Net
Overseas
Cap Acc Inc
Total
Output
62
23
‒5
3
133
35
25
‒7
5
87
65
9
‒4
2
109
55
25
‒1
3
127
17
15
‒2
1
89
234
97
D ¼ ‒19
H ¼ 14
545
331 ¼ 234 + 97
326 ¼ 234 + 97 ‒ 19 + 14
C ¼ 550 ¼ 219 + 234 + 97 ¼ 219 + 331
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4.7 Additional Considerations
139
Table 4.12 Commodity by industry supply matrix: Running example
Industries
Natural
Resources
Manufacturing
Services
Total
Commodity
Output
Agriculture
88
45
0
133
Energy
68
10
9
87
Manufactured Products
0
98
1
109
Financial Services
0
10
17
127
Other Services
0
33
6
89
156
196
193
545
Commmodities
Total Industry Output
that we will see in the following and in more detail in Chapter 5. In national accounting
parlance, the commodity-by-industry interindustry transactions tables are also often
referred to as Supply and Use tables. Note that, as before, in the Use matrix rows
designate commodities and columns designate industries. In matrix terms, the accounting identities are q ¼ Ui þ e and x0 ¼ i0 U þ v. These relationships for our running
example are illustrated in Table 4.11.
The sources of production or supply in the economy are depicted as before in the
Supply matrix. Recall that the Supply matrix is constructed in dimensions of industries
by commodities, whereas before the row entries indicate the production of commodities by an industry. Hence, the column sums of the Supply matrix form the vector of
total industry production in the economy, while the row sums form the vector of total
commodity production in the economy. For our earlier running national accounts
example, there are many possible Supply matrices, the only strict requirement being
that the columns equal total industry output and the row sums equal total commodity
output. One such matrix is given in Table 4.12.
It is now possible to show for our running example a complete consolidated set of
commodity-by-industry accounts that illustrates the relationships between Use and
Supply matrices and the measures of total industry value added, commodity final
demand, and total industry and commodity output. This is shown in Table 4.13.
Note that in the conosolidated table the Supply matrix is shown as its transpose (earlier
defined as the Make matrix). Chapter 5 explores construction of input–output models
from these accounts.
4.7.4 Producers’ and Consumers’ Prices
Most input–output studies value the entries in input–output accounts (and subsequently the transactions matrix) in producers’ prices, that is, the prices at which the
seller completes the transaction (sometimes called free-on-board or FOB prices,
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Organization of Basic Data for Input–Output Models
Table 4.13 Consolidated commodity-by-industry input–output accounts: Running example
Commodities
Industries
Fin
Othr
Agric Energy Manuf Srvcs Srvcs
Nat
Final
Total
Res Manuf Services Demand Output
Commodities
Agric
25
Energy
Manuf
10
15
83
13
7
9
58
87
10
20
7
72
109
Fin Services
10
10
25
82
127
Other
Services
Industries
Nat
Resources
Manufacuting
8
30
20
31
89
88
68
0
0
0
156
45
10
98
10
33
196
Services
0
9
11
117
56
193
133
87
109
127
89
Value Added
Total Output
90
156
119
196
133
117
193
discussed more in Section 4.7.5). The purchaser incurs the producer’s price plus trade
and transportation margins (and often excise taxes). The convention in many input–
output studies is to assign the margins on all interindustry transactions in a column to
the industry responsible for the margin. That is, all wholesale and retail trade margins
on all inputs to an industry are summed and recorded as the “trade” entry in that
column. Similarly, all transportation margins on inputs are summed and recorded as the
input entry for “transportation.” Hence, the trade and transportation sectors are not
really treated as producing and consuming sectors in the economy, but only as “passthrough sectors.”
These conventions simply mean that the input–output table does not actually trace
flows through the trade and transportation sectors, since this would depict an economy
where industries and final customers would make most of their purchases from and
sales to these two industries alone. Instead, transactions are depicted as flowing
directly from producer to consumer, bypassing trade and transportation. This is done
to show the links between producers, consumers, and final customers.
Since trade and transportation margins for all transactions into an industry are
accumulated as single values for each industry, they in effect become service inputs
to that industry. Hence, the sum of all inputs measured in producers’ prices plus the
value of all transportation and trade margins valued as service inputs (hence, valued in
de facto producers’ prices) is then the value of all inputs in consumers’ prices (the
column sums of the transactions matrix).
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4.7 Additional Considerations
141
Example 4.3: Trade and Transportation Margins Suppose we have a foursector input–output economy with two manufacturing sectors, A and B, and two
service sectors, trade and transportation (for simplicity we return to the industryby-industry accounting framework for the moment). The service sectors act as both
interindustry sectors as well as a repository for all markups or margins. The
interindustry transactions paid in millions of dollars including both trade and
transportation margins - that is, in purchasers’ or consumers’ prices – are given
final demands including margins by f and total outputs including margins by
by Z,
x defined by
2
3
2
3
2
3
36 46 83 24
475
664
6
7
6
7
6
7
¼ 6 76 78 94 35 7, f ¼ 6 263 7, and x ¼ 6 546 7
Z
4 8 7 8 4 5
4 120 5
4 147 5
3
1
5
1
150
160
Suppose that the trade and transportation margins in millions of dollars are as given in
Table 4.14. The sum of the margins in this table is the difference between the
purchasers’ prices and producers’ prices.12 For example, the transaction z11 ¼ $36
million incurs a trade markup of $9 million and a transport markup of $7 million,
leaving a so-called direct allocation of $20 million. Likewise, the transaction z43 ¼ 5
is transport markup on trade services, for example, transport costs associated with
Table 4.14 Example trade and transportation margins: Example 2
A
B
Industry A
Industry B
Transportation
Total Margins
9
5
3
17
Industry A
Industry B
Trade
Total Margins
7
6
8
21
12
Trade
Transp.
Final Demand
Trade Margins
10
11
8
7
1
5
19
23
6
4
1
11
50
20
20
90
Transportation Margins
4
9
8
7
7
8
19
24
5
6
4
15
75
13
50
138
The SNA treatment of prices is slightly more detailed in that first so-called basic prices are defined as the
“amount receivable by the producer from the purchaser for a unit of a good or service produced as output minus
any tax payable, and plus any subsidy receivable, by the producer as a consequence of its production or sale. It
excludes any transport charges invoiced separately by the producer” (UN, 2009, p. 101). Hence: “The
producer’s price is the amount receivable by the producer from the purchaser for a unit of a good or service
produced as output minus any value added tax (VAT), or similar deductible tax, invoiced to the purchaser”
(Ibid.), also excluding any transport charges invoiced separately by the producer. In short, producers’ prices are
basic prices plus any taxes (excluding any VAT) and less any subsidies and purchasers’ prices are producers’
prices plus the sum of any VAT (not deductible by the purchaser), invoiced transportation charges, as well as
wholesale and retail margins.
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Organization of Basic Data for Input–Output Models
transactions between wholesale and retail trade. The direct allocation for this transaction is zero. Similarly, the transaction z34 ¼ 4 is the trade markup on transportation
services, for example, the markup imposed by a principal carrier that subcontracts
transport services from a secondary carrier.
If we factor out (subtract) the margins from all the interindustry transactions in
purchasers’ prices, the result is the direct allocations matrix:
2
3
2
3
2
3
20 32 63 13
350
478
6 65 62 80 25 7
6 230 7
6 462 7
7
6
7
6
7
Zd ¼ 6
4 0 0 0 0 5, f d ¼ 4 100 5, and xd ¼ 4 100 5
0 0 0 0
100
100
Note that we have also factored the margins out of final demand and total output and
termed these vectors f d and xd , respectively. The vector of column sums of the trade
margins, labeled “total margins” in the table, represents the sums of all trade margins on
inputs to industries; for example, the first element of this vector, $17 million, is the sum
of all trade margins on inputs to industry A. If we add this vector to the trade row of the
direct allocations matrix, we are, in effect, distributing the trade margins as a service of
the trade sector. Similarly, if we assign the “total transportation margins” to the transportation row of the direct allocations matrix, we account for transportation margins as a
service of the transportation industry. In this way, we do not trace the flows of goods and
services through the trade and transportation sectors, but, instead, treat them as service
inputs to producing sectors and record the flows directly from producer to consumer. The
result is an interindustry transactions matrix in producers’ prices:
2
3
2
3
2
3
20 32 63 13
350
478
6 65 62 80 25 7
6 230 7
6 462 7
7
6
7
6
7
Z¼6
4 17 19 23 11 5, f ¼ 4 190 5, and x ¼ 4 260 5
21 19 24 15
238
317
Methods of valuation in current use in the literature are discussed in more detail in
Bulmer-Thomas (1982).
4.7.5 Accounting for Imports and Exports
Imports and exports of goods and services play a key role in the functioning of most
economies and in many economies the scale of imports relative to domestic production
in some sectors comprise a significant enough share of total production to warrant
analysis for a variety of reasons, such as assessing its impact on the balance of
payments – the difference in total value between payments into and out of a country
(see Section 4.4.3). The scale and scope of this analysis varies considerably among
nations and organizations (see United Nations, 1973, appendix II; 2018b) according to
availability of data, the characteristics of key sectors, and other factors.
In this section we summarize a collection of the most widely used approaches to
treating imports, as chronicled in United Nations (2018b), and subsequently by others
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4.7 Additional Considerations
143
such as Eurostat, e.g., as described in Kronenberg (2012, 2019) and Többen and
Kronenberg (2015), who supplement the collection with an additional approach
utilized in the European System of Accounts (ESA 95, European Commission [EC],
1996) and successive installments (EC, 2008, 2013). First, however, it is important to
settle how imports are valued in an economy. In an input–output framework, imports
are usually divided into two basic groups: (1) imports of commodities that are also
domestically produced (competitive imports) and (2) imports of commodities that are
not domestically produced (non-competitive imports).13 The distinction is that competitive imports can be represented in a technical coefficients matrix, while noncompetitive imports cannot, unless sectors are added to represent them.
Valuation of Imports The key consideration is to ensure that competitive
imports are valued in comparable terms to their domestically produced counterparts.
That is, recall that domestically produced goods are valued at what are commonly
referred to as “basic values,” i.e., excluding taxes and other marginal additions (e.g.,
transportation and distribution). So, the valuation for imports which corresponds most
closely conceptually to that used for domestic products is the value of import goods at
the border of the importing country. This value is often termed the domestic border or
port value. The most-commonly-applied conventions for determing port value are
known as cost, insurance, and freight (CIF) and free on board (FOB), the latter of
which was referred to briefly in Section 4.7.4 in drawing the distinction between
producers’ and consumers’ prices. Both CIF and FOB (as well as some other conventions) delineate where liability of sellers and buyers officially begins and ends in the
process of importing commodities, as well as the respective responsibilities of buyers
to sellers and vice versa as they relate to economic transactions.
The CIF value is the basic value plus expenses paid by a seller to cover the costs,
insurance, and freight of a buyer’s order while it is in transit. The port value can then be
determined as the CIF value of imports plus protective import duties (tariffs), i.e., the
sum of three components: (a) the value of goods when leaving the exporting country,
(b) freight charges to the domestic port of entry, and (c) insurance charges. Import tariffs
for competitive imports are margins that are important to include since they raise the cost
of an import relative to its corresponding domestically produced counterpart. Other
margins, such as distributive or transportation margins involved in transferring imported
goods from their port of entry to purchasers are customarily treated in the same way as
similar margins on the transfer of domestic goods from producer to purchaser.
The primary difference between CIF and FOB valuation is which party is responsible for the expenses up to the point of loading the product onto the transport vessel.
Under CIF terms, the seller is responsible for trade and transportation margins such as
13
In the literature the terms comparable and non-comparable imports are used interchangeably with competitive
and non-competitive imports, respectively. Note that the treatment of competitive imports outlined here has
been adopted only in more recent input–output studies, such as the 1972 US National Input–Output Table and
subsequent benchmark tables; earlier studies treated competitive imports in the same manner as secondary
products; see Ritz (1979).
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Organization of Basic Data for Input–Output Models
purchasing export licenses for the product, covering the cost and contracts of moving
or carrying the goods, providing insurance to protect the value of the order, providing
inspections of products, and insuring the cost of any damage or destruction to the
goods. With FOB valuation the buyer assumes more responsibility for these margins.
With any of the import conventions, we can represent competitive imports by adding
them as transactions to the matrix of domestic transactions (as in the case of transfers
of secondary products), essentially as if they are produced domestically. However, the
“domestic port value” (in effect, value in producers’ prices) of all imports of a
particular commodity is then often included as a negative entry in final demand. The
purpose of this adjustment is to assure that the total output of an industry, computed as
the row sum of the interindustry transactions to other industries and final-demand
allocation, is total domestic production, net of imports.
The variation in conventions for valuing imports presents a significant dilemma that
leads with no surprise to wide variations in accounting for imports. Using CIF (and
FOB) valuation of imports to determine the domestic border or port values for imports
means that the sum of the CIF values of imports will overstate the value of imports in
computing the balance of payments if any of the freight and insurance activities are
carried out by domestic producers. To compensate or correct, this amount can be
entered in the import of commodities as negative values for transport and services, and
in the input–output transactions table as negative inputs into the transport and services
industries in the determining the total value of those cells. Alternatively, the value of
these freight and insurance services provided by domestic producers could be included
as an export of those commodities. This approach avoids the possible appearance of a
negative entry for transport when the import row is disaggregated, but it involves
double counting of these services (both as part of the total value of imports and as part
of domestic output), which needs correction in some way.
In the following we describe five commonly applied approaches to treating imports,
as chronicled in United Nations (1973, 2018b) and Kronenberg (2012). They can all be
derived from a common starting point, i.e., mathematically, we begin with the familiar
production and consumption identities:
x ¼ Zi þ f
(4.21)
x0 ¼ i0 Z þ v
(4.22)
We denote the vector of total outputs as x, the matrix of interindustry transactions as Z,
the vector of total final demand as f, and the vector of total value added as w. For use
later we define the components of final demand with f ¼ y þ e , where e denotes
product exports and y denotes the remaining components of final demand (personal
consumption, capital investment, inventory change, and government expenditures).14
14
For simplicity, the illustrations of approaches to imports in this section will be for the case of industries and
commodities defined by the same sectors, but the same observations apply for commodity by industry
formulations, e.g., as in Kronenberg (2012).
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4.7 Additional Considerations
145
For the present purposes and if imports are incorporated in interindustry transactions, as is often the case (e.g., for the US input–output tables), we can write Z as the
sum of domestic transactions, Zd , and imports, Zm , i.e., Z ¼ Zd þ Zm , and account for
total imports as m ¼ i0 Zm , the total value of all imports to each interindustry sector, or
as n ¼ Zm i, the total value of interindustry production that is imported. However, in
this case, if the expectation is that total output refers to total domestic production (as is
also often the case), we must make a correction. We can subtract the total value of
imports from the consumption and production identities so that they become
x ¼ Zd þ Zm i þ f n and x0 ¼ i0 Zd þ Zm þ v m. For convenience we define
v ¼ v m, total value added less imports, and f ¼ f n, total final demand less
imports, and we can rewrite the production and consumption identities as
x ¼ Zd þ Zm i þ f and x0 ¼ i0 Zd þ Zm þ v
(4.23)
From these definitions we can now illustrate the various approaches commonly used to
represent imports with a running example, the the point of departure of which is shown
in Example 4.4.
Example 4.4: Competitive and Non-competitive Imports Table 4.15 shows a
domestic transactions matrix, Z; final demand vector, f; and total outputs vector, x; for
a two-sector (industries A and B) input–output economy in millions of dollars.
10 20
70
For these basic data, using the coventions just defined, Z ¼
,f¼
,
30
40
30
100
, from which we can compute v ¼ x0 i0 Z ¼ ½ 60 40 . We presume
and x ¼
100
industry A exports $30 million of its product and Industry B exports $10 million of its
product. In addition, we presume that industry A consumes $10 million worth of B that
is imported rather than domestically produced, in addition to the $30 million worth of
B that is domestically produced. Also, both industries A and B consume another
product, C, that is only produced overseas (a non-competitive import) – $5 million
and $4 million worth for outputs of Industries A and B, respectively. The way imports
are represented will define the alternative approaches below: (A) Classifying Imports
by Commodity, (B) Classifying Imports by Purchaser, (C) Explicit Representation of
Non-competitive Imports, (D) Maintaining Separate Tables of Imported and Domestic
Products, and (E) Total Supply. As the point of departure, however, we represent all
Table 4.15 Example 4.4, basic information
Millions of Dollars
A
B
Final Demand
Total Output
Industry A
Industry B
10
30
20
40
70
30
100
100
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146
Organization of Basic Data for Input–Output Models
imports as part of the total value added to interindustry production, v, which is
exogenous to the matrix of interindustry transactions.
Approach A: Classifying Imports by Commodity In this approach, all imports
are assigned a commodity classification and, when applicable, combined with production by domestic industries. These totals of commodity supplies, combining domestically produced and imported sources, are allocated to various purchasers. The principal
advantage of this approach is that no information is required on the origin (domestic or
foreign) of commodities purchased by each sector. As a result, such tables are easier to
assemble than others that involve more detailed treatment of imports.
Mathematically, this means using the equations in (4.23). For the current example,
10 20
0 0
10 20
0 0 1
d
m
m
þ
¼
. Also, since n ¼ Z i ¼
¼
Z¼Z þZ ¼
30 40
10 0
40 40
10 0 1
70
0
70
0
¼
. Similarly, since m ¼ i0 Zm ¼
, we can compute f ¼ f n ¼
30
10
20
10
0 0
½1 1
¼ ½ 10 0 , we can write v ¼ v i0 Zm ¼ ½ 60 40 ½ 10 0 ¼ ½ 50 40 .
10 0
Finally, as illustrated in Example 4.4, we can separate final demand into its
components, defining e asexports and y as all other value added, so f ¼ y þ e.
30
40
For the example, e ¼
so y ¼ f e ¼
.
10
20
The consumption identity equation now says the value of total domestic input is the
sum of the value of all interindustry inputs (including imported sources) and all valueadded inputs less the value of imports, i.e., x0 ¼ i0 Z þ v. The production identity
equation now says total domestic output used in the economy is the sum of all
interindustry outputs (including those imported) and all final uses net of imports, i.e.,
exports minus imports (net exports), i.e., x ¼ Zi þ y þ ðe nÞ. In the second equation, the terms e and n are commonly combined as e n and recorded as a
combined vector of net exports. The relationships for Approach A applied to the
example are summarized in Table 4.16.
The main disadvantage of Approach A is that the impact of a certain level of final
demand on domestic production versus that of imports cannot be assessed accurately.
Instead, we are forced, in the absence of other information, to assume that each sector
Table 4.16 Example 4.4a, approach A, classifying imports by commodity
Z
A
B
y
e
n
x
A
B
v
x0
10
40
50
100
20
40
40
100
40
20
30
10
0
‒10
100
100
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4.7 Additional Considerations
147
of demand requires imports in the same proportion to domestic supplies. Later we will
refer to this very commonly applied assumption as import similarity. Despite being
widely applied, in many situations this is often an unrealistic assumption. That is,
purchases by any consuming sector may generate a demand for imports either through
the direct purchase of imported goods or required as direct or indirect inputs into the
domestic supplying industry. These demands for imports most likely vary considerably
among sectors, and using average import ratios (import similarity) for each commodity
will consequently likely lead to inaccuracies.
Note also that, in this approach, Zm includes only competitive imports, while noncompetitive imports remain classified as value added since doing otherwise would
require adding new sectors to the interindustry transactions matrix for products that are
not produced domestically. Adding new import-only sectors is certainly possible and
will be employed in subsequent approaches. Ultimately the logical extension is
formation of IRIO or MRIO models as defined in Chapter 3 to trace the origin and
destination of imports and exports more comprehensively. A common compromise
alternative, short of such models, is to include a single sector of non-competitive
imports in the interindustry transactions matrix that includes all non-competitive
imports, which will become apparent with subsequent approaches.
Approach B: Classifying Imports by Purchaser In this approach, competitive
imports are included as a supplemental value-added row, m. Mathematically, this
means that Z ¼ Zd (including only domestic transactions), so that x0 ¼ i0 Z þ w ¼
i0 Z þ m þ v where v ¼ v m (again, all value added other than competitive imports)
and x ¼ Zi þ y þ e. Note that in this approach u ¼ Zi and v ¼ i0 Z are not inflated by
a double counting of imports, so, if, as is customary, the expectation is that x represents
total domestic production, it is not necessary to subtract n from final demand for x to
represent domestic total output. The relationships for the example employing
Approach B are summarized in Table 4.17.
The main disadvantage of this commonly used approach is that it excludes competitive imports from the interindustry transactions table entirely, and hence conceals the
role such imports play in interindustry activity, which, in a vibrant economy with
substantial international trade, reduces the capacity for analyzing the degree to which
imports compete with domestically produced goods. If an import is a substitute for a
Table 4.17 Example 4.4b, approach B, classifying imports by purchaser (use)
Z
A
B
y
e
x
A
B
m
v
x0
10
30
10
50
100
20
40
0
40
100
40
20
30
10
100
100
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148
Organization of Basic Data for Input–Output Models
domestic product then an increase in the purchases of the domestic product may
unpredictably occur at the expense of the import (or vice versa), causing unwarranted
changes in the domestic input coefficients.
In classifying various approaches to imports, the United Nations (1973) observes
that, in estimating the totals of imports by commodity (required in this approach for the
last row of the table), with commonly available data sources it is likely that this
approach will require considering the split between imports and domestic production
in the input cells in each column in any event and, if such information is available,
other approaches, such as Approach D, may be more suitable.
Approach C: Explicit Representation of Non-competitive Imports In this
approach, non-competitive imports are explicitly included and not aggregated in value
added. We accomplish this by introducing a new sector to incorporate non-competitive
imports since by definition no sectors for such products or services exist in the
domestic economy. Competitive imports remain represented, as in Approach B. The
relationships for the example employing Approach C are summarized in Table 4.18.
The new sector for the non-competitive imports is the added Industry C and competitive importss are aggregated into m (as in Approach B) and netted out of v, leaving the
sum of all other value-added inputs, v.
As the world economy becomes more interconnected or globalized, the distinction
between competitive and non-competitive imports becomes more important in input–
output analysis because, if non-competitive commodities are needed, they must be
imported, but they can still stimulate domestic production in other sectors quite
significantly. If this distinction is not made the analysis will proceed as if any
commodity would be obtained in given proportions from domestic production and
imports. The United Nations (1973) observes that it will often be the case that noncompetitive imports will be restricted to a small number of commodities but where the
value of imports of those commodities is high, e.g., the imports of oil for many nations.
Classifying commodities as competitive or non-competitive may also depend on the
level of sectoral aggregation, with more non-competitive imports appearing with more
disaggregation of commodities.
Table 4.18 Example 4.4c, approach C, explicit representation of
non-competitive imports
Z
A
B
C
y
e
n
x
A
B
C
m
v
x0
10
30
5
10
45
100
20
40
4
0
36
100
0
0
0
0
0
0
40
20
0
30
10
0
0
0
9
100
100
0
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4.7 Additional Considerations
149
Approach D: Maintaining Separate Tables of Imported and Domestic
Products If data are available to specify all imported goods by commodity, then we
can presume that both Zd and Zm are available. Hence, in this approach, the
competitive imports to each sector, which were included as aggregated into a separate
value-added row, m, in Approaches B and C, are now included as a separate imports
matrix, disaggregated into the commodity row where they appear. Similarly, there will
be twin vectors for each final demand category as well. We continue to represent noncompetitive imports explicitly as added sectors, as in Approach C. The relationships
for the example are summarized in Table 4.19.
The statistical requirements of this approach are obviously much more substantial
than the other approaches but, if such data are available, the approach allows for much
more flexibility in the treatment of imports and permits a very clear analysis of the
impact of demand on domestic and foreign supplies. Generally, however, available
data often do not include the origin (domestic or foreign) of purchased commodities,
especially if purchased through wholesale dealers.
Approach E: Total Supply This approach is a slight modification of Approach
A or D if both Zd and Zm are specified. It is not included in the original UN
classification (United Nations, 1973) but added by Kronenberg (2012), and it is
included here because it is commonly applied in Europe. The principal distinction is
that a supplemental vector, s, designated total supply, is the sum of total domestic
production and the value of total imports, i.e., s ¼ x þ n. Since there is an added sector
for non-competitive imports, Industry C, the vector n includes both competitive and
Table 4.19 Example 4.4d, approach D, separate imported and domestic product tables
Zd
A
B
C
yd
ed
xd
A
B
C
vd
xd0
10
30
0
60
100
20
40
0
40
100
0
0
0
0
0
40
20
0
30
10
0
100
100
0
Zm
A
B
C
vm
xm0
A
0
10
5
0
15
B
0
0
4
0
4
C
0
0
0
0
0
ym
0
0
0
em
0
0
0
xm
0
10
9
Z
A
B
C
v
x0
A
10
40
5
45
100
B
20
40
4
36
100
C
0
0
0
0
0
y
40
20
0
e
30
10
0
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n
0
10
9
x
100
100
0
150
Organization of Basic Data for Input–Output Models
non-competitive imports, which can be interpreted as a representation of a sector’s
balance of payments, i.e., if the economy produces more than it consumes of a sector’s
output then the sector is a net exporter and, conversely, is a net importer if the economy
consumes more than it produces of a sector’s output.
Recall that in Approach A the matrix Z includes both domestic transactions and
competitive imports, but imports were subtracted from final uses so that x would
include total output from only domestic production. In Approach E, the accounting
identity is related to total available supply of a product rather than solely its domestic
production, i.e., including both domestic production and imports. The corresponding
accounting identities become
s0 ¼ x0 þ m
s0 ¼ i0 Z þ v þ m ¼ i0 Z þ v
(4.24)
s¼xþn
s ¼ Zi þ y þ ðe nÞ þ n
s ¼ Zi þ y þ e ¼ Zi þ f
(4.25)
This is a commonly applied if not standardized approach adopted by Eurostat in
compiling many European input–output tables for the European Commission (EC,
2013). The relationships for the example employing Approach E are summarized in
Table 4.20.
The Choice among Approaches to Representing Imports Selecting from
among these approaches usually involves tradeoffs among specific requirements of a
study and available data. The simplest is Approach A, since it does not distinguish
purchase by origin – domestic or imported. However, as noted in the development of
Approach A, it cannot provide accurate estimates of the import requirements, distinguished from domestic production, for a given vector of final demand since it assumes an
average import content for all buyers. Approach D (or E with both Zd and Zm
specified), by contrast, is the most comprehensive and provides a clear and accurate
Table 4.20 Example 4.4e, approach E, total supply
Z
A
B
C
y
e
s
A
B
C
v
m
s0
10
40
5
45
0
100
20
40
4
36
10
110
0
0
0
0
9
9
40
20
0
30
10
0
100
110
9
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4.7 Additional Considerations
151
analysis of import requirements but also requires the most information. Approach B
requires somewhat less information than Approach D but provides a much less comprehensive picture. Approach C is a compromise that, with relatively little additional effort
compared with Approach A, identifies a substantial proportion of imports and treats
them separately from domestically produced commodities. For a more detailed
comparison of these approaches, see United Nations (1973) or Kronenberg (2012).
4.7.6 Removing Competitive Imports from Total Transactions Tables
In Tables 2.6 and 2.7, we presented direct and total requirements tables for the US
2003 input–output tables where the underlying transactions were “scrubbed” of
competitive imports so that impacts on the domestic economy could be analyzed.
The US input–output tables are routinely published with competitive imports
included as part of interindustry transactions, with negative entries for imports added
to final demand so that the sum of intermediate production and final demand equals
total domestic production and the sum of all final demands equals total gross
domestic product (Approach A). In this section we consider several commonly
applied approximation techniques for removing imports from tables prepared in such
a manner.
First, as above, we assume that interindustry transactions can be divided into
domestic transactions and competitive imports, i.e., Z ¼ Zd þ Zm where Zd is the
matrix of domestic transactions and Zm is the matrix of imports. Hence, as above, the
vector comprising the row sums of Zm is the vector of the use of total imports given by
n ¼ Zm i. As noted earlier, for the US tables, n is included as a negative final demand in
order for x to represent total domestic output, so we define g as the vector of final
demand other than imports such that f ¼ g þ ðnÞ. We can also define m ¼ i0 Zm , the
vector of column sums of Zm , each element of which is the value of all imports to the
industry designated by each column. In many situations we may know mbut not Zm .
If we know Zm then, since x ¼ Zi þ f, we can write x ¼ Zd þ Zm i þ ðg nÞ
and, separating terms, we find x ¼ Zd i þ Zm i þ ðg nÞ. Since n ¼ Zm i, the terms n
and Zm i cancel each other out so it follows that x ¼ Zd i þ g. Also, since total value
added is computed as the residual of intermediate production and the value of total
it follows
that x0 ¼ i0 Z þ v. Finally,
output, i.e., v ¼ x0 i0 Z, then, rearranging terms,
again separating terms, we have x0 ¼ i0 Zd þ Zm þ w or x0 ¼ i0 Zd þ i0 Zm þ v,
which, since m ¼ i0 Zm , we can write as x0 ¼ i0 Zd þ m þ v. We define the new total
value added vector as v ¼ m þ v, which adds total imports to other components of
value added, then x0 ¼ i0 Zd þ v, i.e., the total value of imports has moved from the
interindustry inputs, Z, to value added.
As noted in the development of Approach B for handling imports in Section 4.7.5,
in practice, we often face the situation where we know m but not Zm . That is, we may
know the total value of steel imports, but not the value of steel imports to each
industry individually. The following are the two approximation procedures
for estimating the matrices of domestic transactions and interindustry imports.
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152
Organization of Basic Data for Input–Output Models
They both rely on an assumption commonly referred to as import similarity where for
each product the mix of imports and domestically produced goods is assumed to be
the same for all consuming sectors, but the mix may be different for each product.
For example, the mix of imported and domestically produced agricultural goods is
the same for all consumers of agricultural goods, and the amount of imported steel as
a fraction of total steel used in production is the same for both automobile production
and shipbuilding. This assumption may not be very realistic in many developed
economies, but it is often necessary due to the limits of available data, as in the
construction of the MRIO model introduced in Chapter 3. The limitations of this
approach are discussed in NRC (2006).
Approximation Method I If we do not know Zm , we can approximate it by
allocating n proportionately to the distribution of intermediate output by the following.
^ 1 Z , where
First, create the matrix of intermediate output proportions as B ¼ u
~m ¼ n
^ B as an approximation
u ¼ Zi is the vector of intermediate outputs. We define Z
m
m
~ m i. Hence, we can define our
of Z , for which we can guarantee that n ¼ Z i ¼ Z
~ m and, from before, g ¼ f þ n,
~d ¼ Z Z
approximation of domestic transactions, Z
~ d þ g.
so that x ¼ Z
Approximation Method II In Method I, n ¼ Zm i , which assumes implicitly
that there are no imports consumed directly by final demand. That often is not the case,
of course, but if we make the additional simplifying assumption that for each industry
the fraction of a given input supplied by imports is the same for each industry and that
same fraction also applies to consumer and government expenditures, then that same
fraction of total output is attributable to imports. That is, assume that for each industry
that fraction is given by ri so that ni ¼ ri xi . We multiply through the equation
Pn
Pn
zij þ fi by ri to yield ri xi ¼
xi ¼
j¼1 r i zij þ ri fi or, by recalling that
Pn j¼1
ui ¼ j¼1 zij , we can write ni ¼ ri xi ¼ ri ui þ ri fi or ri ¼ uinþfi . We can use ri to define
i
an estimate of the domestic transactions matrix by ~z dij ¼ zij ri zij ¼ ð1 ri Þ zij. In
~ m ¼ ^r Z as the estimate of the
~ d ¼ Z ^r Z so we can define Z
matrix terms, this is Z
~ m i as the vector of total
~¼Z
matrix of interindustry imports. We can now define n
interindustry imports. If we define hi ¼ ri fi as the estimate of the vector of
imports consumed directly by final demand, which in matrix terms is h ¼ ^r f , then
d
~ m i þ ðg nÞ. As before,
~ þZ
~ , so we can use x ¼ Zi þ f to write x ¼ Z
n¼hþn
~ di þ Z
~ m i þ ðg h n
~ Þ or
separating terms and substituting, we have x ¼ Z
d
m
m
~
~
~
~ Þ. Also as before, the terms n
~ and Z i cancel each
x ¼ Z i þ Z i þ ðg ^r f n
~ d i þ g ^r f. In Method I, we defined g as the
other out so that what remains is x ¼ Z
vector of final demand other than imports, which includes only interindustry imports,
so if we now define g as the vector of final demand other than both interindustry
imports and imports consumed directly by final demand, i.e., g reduced by imports
~ d i þ g.
consumed directly in final demand, h, then g ¼ g h and we can write x ¼ Z
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4.7 Additional Considerations
153
While it may not be obvious, it is useful to observe that method I is equivalent to
method II if we create the matrix of intermediate output proportions as B ¼ x^1 Z
^ 1 Z, as in Method I. Hence, the approximated matrix of imports is
instead of B ¼ u
m
~ ¼n
Method II, by either calculation, of course, is often a somewhat more
^ B.
Z
realistic approximation for most economies. We illustrate the two methods in
Example 4.5.
Example 4.5: Import Scrubbing We define an input–output economy with
2
3
2
3
350 0
0
1,000
Z ¼ 4 50 250 150 5, x ¼ 4 500 5
200 150 550
1,000
From Z and x, we compute the corresponding values of
2
3
2
3
650
350
f ¼ x Zi ¼ 4 50 5, u ¼ Zi ¼ 4 450 5, v ¼ x0 i0 Z ¼ ½ 400 100 300 100
900
and, consequently,
2
3
2
3
:35 0 0
1:5385
0
0
2:5
:8333 5:
A ¼ Zx^1 ¼ 4 :05 :5 :15 5 L ¼ ðI AÞ1 ¼ 4 :4487
:2 :3 :55
:9829 1:6667 2:7778
We presume this is a “US-type” table where the transactions matrix includes competitive imports, so Z ¼ Zd þ Zm and f ¼ g n , and we define (arbitrarily for this
2
3
2
3
100 0
0
250 0
0
example) Zm ¼ 4 25 50 30 5, which means Zd ¼ Z Zm ¼ 4 25 200 120 5,
25 50 100
175 100 45
2
3
2
3
100
750
m
4
5
4
n ¼ Z i ¼ 105 , g ¼ f þ n ¼ 155 5 , and the balance equation, x ¼ Zd i þ g, holds:
175
275
3
2
3
2
32 3 2
750
1
1,000
250 0
0
x ¼ 4 500 5 ¼ Zd i þ g ¼ 4 25 200 120 5 4 1 5 þ 4 155 5. The new total value275
1
1,000
175 100 450
0 d
0
added vector becomes v ¼ m þ v ¼ x i Z ¼ ½55 200 430. This inflates the original
vector of total valued added, v ¼ ½ 400 100 300 by the total value of all imports to
each industry, m ¼ ½ 150 100 130 .
Where we know m but not Zm , in the following, we use the two approximation
methods outlined above for estimating the separate matrices of domestic transactions
and interindustry imports.
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154
Organization of Basic Data for Input–Output Models
^ 1 Z ¼
For
Approximation
Method
I,
we
first
generate
B¼u
3
1
0
0
4 :1111 :5556 :3333 5 in order to allocate n across interindustry output by
:2222 :1667 :6111
2
3
100
0
0
~ mi ,
~m ¼ n
^ B ¼ 4 11:667 58:333
35 5. Since in this technique, n ¼ Zm i ¼ Z
Z
38:889 29:167 106:94
we can compute
2
3
250
0
0
~d ¼ Z Z
~ m ¼ 4 38:333 191:667
Z
115 5
161:111 120:833 443:056
2
~ d i þ g, holds:
Note that in this case the balance equation, x ¼ Z
2
32 3 2
3
2
3
250
0
0
1
750
1,000
~ d i þ g ¼ 4 38:333 191:667
115 5 4 1 5 þ 4 155 5
x ¼ 4 500 5 ¼ Z
161:111 120:833 443:056
1
275
1,000
~ d ¼ ½ 550:556 187:5 441:944 ,
The new total value added vector is v ¼ x0 i0 Z
which is the original total value added vector, v ¼ ½ 400 100 300 , inflated by the
~ ¼ ½ 150:556 87:5 141:944 , but, as
total value of all imports to each industry, m
developed so far, this assumes that no imports are consumed directly in final demand.
For Approximation Method II we begin by calculating the scaling factors
2
3
:1
ni
, which for the example are in the vector r ¼ 4 :21 5 and, hence,
ri ¼
ui þ fi
:175
2
3
:1 0
0
~ d ¼ Z ^r Z ¼
^r ¼ 4 0 :21
0 5.
We
can
then
compute
Z
0
0 :175
2
3
2
3
35
0
0
315
0
0
4 39:5 197:5 118:5 5, Z
~ m ¼ ^r Z ¼ 4 10:5 52:5 31:5 5, and h ¼ ^r f ¼
35 26:25 96:25
165 123:75 453:75
3
2
2
3
685
65
4 10:5 5 so that g ¼ g h ¼ 4 144:5 5. We can show that the balance equation
257:5
17:5
2
3
2
32 3
1,000
315
0
0
1
~di þ ~ d i þ g ¼ 4 39:5 197:5 118:5 5 4 1 5þ
x¼Z
g still holds: x ¼ 4 500 5 ¼ Z
1,000
165 123:75 453:75
1
2
3
685
4 144:5 5:
257:5
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4.7 Additional Considerations
155
~ d i þ g, now accounts for only domestic transactions,
This balance equation, x ¼ Z
but interindustry imports are reassigned to total value added.
The new total value added vector, v ¼ x0 i0 Zd ¼ ½ 480:5 178:75 427:75 ,
inflates the original vector of total value added, v ¼ ½ 400 100 300 , by all inter~ ¼ ½ 80:5 78:75 127:75 ,
industry imports to each industry, i.e., this time, m
excluding the value of imports consumed directly in final demand. While perhaps
not intuitively obvious, this procedure is equivalent to creating the matrix of intermedi ¼ x^1 Z, instead of B ¼ u
^ 1 Z defined for Method I. For
ate output proportions as B
the example,
2
32
3 2
3
1=1,000
0
0
350 0
0
:35 0 0
¼ x^1 Z ¼4 0
1=500
0 5 4 50 250 150 5¼4 :1 :5 :3 5
B
0
0
1=1,000
200 150 550
:2 :15 :55
Hence,
2
32
3 2
3
100 0
0
:35 0
0
35
0
0
¼ 4 0 105 0 5 4 :1 :5 :3 5¼ 4 10:5 52:5 31:5 5
~m ¼ n
^B
Z
0
0 275
:2 :15 :55
35 26:25 96:25
and, as before,
2
3
315
0
0
~ m ¼ 4 39:5 197:5 118:5 5
~d ¼ Z Z
Z
165 123:75 453:75
The application of the alternative estimation procedures for the example is summarized
in Table 4.21.
Implications of the Estimating Assumptions Method II is used commonly for
removing imports to create D if M is not known directly. Examples of its application to
are OTA (1988), NRC (2006), Eldridge and Harper (2010), Guo and Planting (2013),
and Howse (2017). Recall that the key assumption for both Methods I and II is of
import similarity, i.e., for each industry product the mix of imports and domestically
produced goods is the same across all consuming sectors for that product, but is or may
be different for each product.
NRC (2006) uses this assumption to analyze the US content of imports and the
foreign content of US exports as one possible way of gauging the implications of
the globalization of industry for the overall health of the US economy.
Dietzenbacher, Albino, and Kühtz (2005) propose another alternative to both
Methods I and II.
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156
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Table 4.21 Approximation methods for “scrubbing” interindustry transactions of competitive imports: Example 4.5
Original Data
Original US Type Table
Approximation Method I
Industry
Total
Final
Demand
Total
Output
Estimated Table
Industry
Total
250
345
725
750
155
275
1,000
500
1,000
250
0
0 250
38.33 191.67
115 345
161.11 120.83 443.06 725
Approximation Method II
Final
Demand
Total
Output
Estimated Table
Industry
Total
Final
Demand
Total
Output
750
155
275
1,000
500
1,000
315
39.50
165
0
0
315
197.50 118.50 355.50
123.75 453.75 742.50
685
144.50
257.50
1,000
500
1,000
519.5
321.25 572.25
480.5
178.75 427.75
1,000
35
10.50
35
80.50
500 1,000
0
0
35
52.50 31.50
94.50
26.25 96.25 157.50
78.75 127.75
287
Domestic
Transactions
250
25
175
0
200
100
0
120
450
Interindustry
Total
Value Added
450
300
570
550
200
430
1,000
100
25
25
150
500
0
50
50
100
1,000
0
30
100
130
Direct
Requirements
.2500
.0250
.1750
.0000
.4000
.2000
Total Direct
Value Added
.4500
.5500
.6000
.4000
Total
Requirements
1.3333 .0000 .0000
.1514 1.7974 .3922
.4793 .6536 1.9608
1.3333 .0000 .0000
.1684 1.7644 .3643
.4588 .7656 1.9536
1.4599 .0000 .0000
.1994 1.8139 .3935
.5313 .8218 2.0089
Total
1.9641 2.4510 2.3529
1.9606 2.5300 2.3179
2.1905 2.6357 2.4024
Total Output
Interindustry
Imports
Total
449.44 312.50 558.06
1,180
550.56 187.50 441.94
2,500
1,180
1,000
100
11.67
38.89
150.56
500
0
58.33
29.17
87.50
1,000
0
35
106.94
141.94
.0000
.1200
.4500
.2500
.0383
.1611
.0000
.3833
.2417
.0000
.1150
.4431
.3150
.0395
.1650
.0000
.3950
.2475
.0000
.1185
.4538
.5700
.4300
.4494
.5506
.6250
.3750
.5581
.4419
.5195
.4805
.6425
.3575
.5723
.4278
100
105
175
380
2,500
100
105
175
380
1,087
2,500
4.7 Additional Considerations
157
4.7.7 Adjustments for Inventory Change
Inventories in an input–output model are not quite equivalent to the conventional
definition of that term. In input–output models, inventory change is usually taken to
mean the change in inventories of an industry’s primary product, regardless of which
industry or industries hold the inventories. For example, coal inventories held by electric
power plants are classified as coal inventory. The traditional definition is usually restricted
to the inventory actually held by the industry producing the product. This modified
definition is adopted in input–output to ensure that the row total of the transactions and
final demands is equal to total current output of the industry. If we ignore inventory
depletion or addition, then the row totals are total consumption, not output.
4.7.8 Adjustments for Scrap
Input–output accounts typically deal with scrap as a production by-product. That is, it
is assumed that no industry produces scrap on demand, so scrap is the result of
production to meet other demands. This is typically accomplished by calculating the
ratio of non-scrap output to industry output for each industry and then applying these
ratios to the market shares matrix in order to account for total industry output. In
Chapter 5 we develop a variety of methods for handling secondary production and byproducts, but one commonly applied technique to adjust for scrap is the following.
First, we recall that g ¼ i0 V (the column sums of the Supply matrix) and assume
some portion of total commodity production, g, is scrap, h, so we can write
g ¼ i0 V þ h. If, as is commonly assumed, scrap production is related to total production by a constant ratio, we can write hi ¼ ci g i where ci is the ratio of the value of
scrap produced in industry i to the industry’s total output. In matrix terms this is
expressed as h ¼ ^c g. We can rewrite g ¼ i0 V þ h as g h ¼ i0 V and substitute
h ¼ ^c g to yield g ^c g ¼ i0 V and, hence ðI ^c Þg ¼ i0 V. Finally, we multiply through
both sides by ðI ^c Þ1 to yield g ¼ ðI ^c Þ1 i0 V or, equivalently, g ¼ ðI ^c Þ1 V0 i,
the Supply matrix adjusted for scrap.
where we can define the bracketed quantity as V,
4.7.9 Special Considerations for Regional and Multiregional Models
As noted in Section 4.7.2, commodity-by-industry accounting is increasingly becoming a de facto standard for input–output accounts where secondary production is
significant and suitable data sources are available. This is especially true in regional
and multiregional situations, but additional considerations are important as well. For
example, Jackson and Schwarm (2011) emphasize the importance of the commodityby-industry model configuration in regional circumstances. Alternatives for model
configuration, which derive principally from the necessary assumptions about secondary production, are developed in detail in Chapter 5. They examine the implications of
alternative choices which depend mostly on practical considerations, such as availability of the necessary data and the degree to which “suitable mechanisms can be
identified for allocating national industry imports to subnational regions” (p. 195).
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158
Organization of Basic Data for Input–Output Models
4.7.10 Summary
The basic interindustry transactions matrix or supply and use tables are only the very
beginning of a process to compile integrated input–output accounts. Decisions about
how to handle secondary production of products and services are especially important,
but other factors are important as well, such as accounting for imports and exports,
inventory change, and adjustments for scrap. We explore many of these issues in
subsequent chapters. For overviews of key issues, see Rueda-Cantuche (2017) and
Beutel (2017).
4.8
Valuation and Double Deflation
In comparing input–output data for different years, it is often important to distinguish
changes attributable to prices from other sources of difference. This essentially
involves converting tables originally valued at nominal prices for the year in which
the data for the table were collected (current prices) to corresponding tables valued at
constant prices for some established base time period, usually a base year. A common
method for accomplishing this is called double deflation, which refers to a two-step
process (hence the “double”).
The first step is to first “deflate” intermediate inputs, final demands, and total outputs
valued at current prices in the accounting period, which means multiplying all intermediate inputs, final demand, and total output by commodity price indices computed
as current prices for each commodity normalized by the corresponding commodity’s
base year price. That is, the price index is simply a ratio of the price of a commodity in
the designated year to the corresponding price in the base year. All the outputs of an
industry are adjusted by this index, i.e., we adjust both the deliveries to other industries
and those to final demand by the same price index for that industry’s output.
The second step is to derive a value-added price index by using the fundamental
identity that the value of total outputs must always be equal to the value of total inputs.
That is, compute the price index necessary to balance this identity.
We illustrate the process of double deflation by recalling that interindustry transactions in value terms refer to a physical transaction and a corresponding price, i.e.,
zij ¼ pi sij
(4.26)
where zij is the dollar transaction of industry i’s output consumed by industry j, pi is
the price per physical unit of industry i’s output; and sij is the physical units transaction
of industry i’s output consumed by industry j. We can rearrange terms in (4.21) to
z
sij ¼ pij and if we define a superscript to denote the accounting period, we can write
i
z1
z2
zt
zn
i
i
i
i
sij ¼ pij1 ¼ pij2 ¼ . . . ¼ pijt ¼ . . . ¼ pijn for 1, 2, . . ., n accounting periods (usually years). If
we choose any arbitrary year to be the reference or base year (b), we can write
b
b
zb
zt
p
p
sij ¼ pijb ¼ pijt or zbij ¼ pit ztij : The term pit is the price index for industry i in year t
i
i
i
i
relative to base or reference year b. Similarly, we define final demand and total output
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159
4.8 Valuation and Double Deflation
of industry i’s output as d i and qi corresponding to those quantities in value terms, fi
and xi . Hence, fi ¼ pi d i and qi ¼ pi xi and, once again introducing superscripts to
f1
f2
ft
fn
i
i
i
denote accounting periods, we can write d i ¼ pi1 ¼ pi2 ¼ . . . ¼ pit ¼ . . . ¼ pin and
i
xt
xn
x1
x2
qi ¼ pi1 ¼ pi2 ¼ . . . ¼ pit ¼ . . . ¼ pin :
i
i
i
i
di ¼
f bi
pbi
¼
f ti
pti
and qi ¼
xbi
pbi
By introducing a base year, b, we can write
b
b
p
p
¼ : Rearranging terms, f bi ¼ pit f ti and xbi ¼ pit xti .
xti
pti
i
i
This and the earlier expression for transactions allows us to “deflate” transactions,
final demand, and total output from year t to base year b or if we do this for multiple
years we can express all years’ values for these quantities in year b’s prices, or,
b
p
so-called constant prices. In sum, if we define π ti ¼ pit as the price index or deflator
i
for industry i, then we can write zbij ¼ π ti ztij , f bi ¼ π ti f ti and xbi ¼ π ti xti : In matrix terms
we define the vector of price indices as πt ¼ ½π t1 π t2 . . . π tn so that we can write
^ t ÞZt , f b ¼ ðπ
^ t Þ f t and xb ¼ ðπ
^ t Þxt :
Zb ¼ ðπ
Since we have deflated Z, f, and x all by the same price index, both the basic identities,
Zt i þ f t ¼ xt and Zb i þ f b ¼ xb , continue to hold. That is, for each industry we have
simply multiplied through the distribution of all output to intermediate consumers and
to final demand by the same price. However, we need to ensure that total outputs are
0
0
equal to total inputs, as well, i.e., the fundamental identity i0 Zb þ ðvb Þ ¼ ðxb Þ must
0
b
hold as well, where ðv Þ is the as yet undetermined deflated vector of value added
inputs – undetermined since we have not yet specified a price index for value added.
We only have a deflator for interindustry inputs. Here we should observe that if an
industry sector experiences price changes for all of its intermediate inputs (including
the price of its own output), then the value added is the only term left that can change if
the value of total inputs is to remain equal to the value of total outputs. Hence, in order
to maintain this identity we can compute the new value added as the residual, i.e.,
0
0
ðvb Þ ¼ ðxb Þ i0 Zb ; wecan
define the deflator for value added for each industry
vbi
t
simply as the ratio ri ¼ vt , which we can express as vbi ¼ rti vti or, in matrix terms,
i
1
^ t and, rearranging terms, we have ^r t ¼ v^ b ðv^ t Þ .
^ b ¼ rb v
v
Example 4.6: Double Deflation We define a three-sector economy for year
2 with industry prices for two different years, years 1 and 2, in Table 4.22.
Table 4.22 Double deflation: Example 6
Industry Transactions
Price
Sector
1
2
3
Final Demand
Total Output
Year 1
Year 2
1
2
3
Value Added
10
5
22
88
20
25
3
34
30
12
7
87
65
40
104
209
125
82
136
2
2
3
7
6
5
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160
Organization of Basic Data for Input–Output Models
To express the transactions, final demand, and total output in year 1 (which we
define as the base year) prices, first we compute the vector of price indices, πt , as
3
2
3 2
:286
2=7
^ t Zt , f b ¼ π
^ t f t , and
πt ¼ 4 2=6 5¼ 4 :333 5. Hence, we can compute Zb ¼ π
:600
3=5
t t
b
^ x as
x ¼π
2
32
3 2
3
:286
0
0
10 20 30
2:9 5:7 8:6
t
:333 0 5 4 5 25 12 5 ¼ 4 1:7 8:3 4:0 5
^ Zt ¼ 4 0
Zb ¼ π
0
0
600
22 3 7
13:2 1:8 4:2
2
32
3 2
3
:286
0
0
65
18:6
^tf t ¼ 4 0
:333
0 5 4 40 5 ¼ 4 13:3 5
fb ¼ π
0
0
:600
104
62:4
2
32
3 2
3
:286
0
0
125
35:7
^ t xt ¼ 4 0
:333
0 5 4 82 5 ¼ 4 27:3 5
xb ¼ π
0
0
:600
136
81:6
From the original data, ðvt Þ0 ¼ ½ 88 34 87 , and we can compute the necessary
value added to ensure that total inputs remain equal to total outputs as
b 0 b 0
v ¼ x i0 Zb ¼ ½ 18:0 11:5 64:8 Hence, we can find the value-added deflator as
2
32
3 2
3
18 0
0
1=88 0
0
:204 0
0
1
^r t ¼ ^
v t Þ ¼ 4 0 11:5 0 5 4 0 1=34 0 5 ¼ 4 0 :338 0 5
v b ð^
0
0 64:5
0
0 1=87
0
0 :745
The method of double deflation, while widely used (as in United Nations, 1993), has
many disadvantages for deflating input–output tables, perhaps not the least of which is
that all elements in a row of the transactions matrix are deflated by the same index. In
many economies interindustry prices may vary considerably and, hence, deflating by
the same index can be misleading or even wrong. Even without variation in interindustry prices for a commodity, a single price index for that commodity may only be
plausible at very high levels of sectoral disaggregation where products are more
distinct. Such an assumption can be very misleading at higher levels of aggregation
where multiple products are represented. These and other problems with double
deflation are discussed in Dietzenbacher and Hoen (1999a, 1999b), Hoen (2002),
Reih (2008), and Daniel, Flynn, and Graham (2017). A perhaps preferable alternative
to double deflation is biproportional scaling (also known as the RAS technique), which
will be explored in detail in Chapter 9.
Some researchers, such as Dietzenbacher and Temurshoev (2012), question the
relative usefulness of price deflation for many types of input–output impact analysis
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4.9 The Aggregation Problem
161
problems. In that work, they investigated the relative importance in input–output
impact analysis of expressing basic data in terms of current or constant prices. For
example, in estimating gross outputs in constant prices and employment as required for
an exogenously specified vector of final demands in current prices, they compared
three methods that express data: (1) in current prices plus gross output deflators; (2) in
constant prices plus final demand deflators; and (3) in both current and constant prices
to derive cell-specific price indices. They found that all three methods essentially
provide very similar predictions, for economy-wide gross output and employment
impacts in particular. The level of aggregation also complicates interpretation of
methods of price deflation, as discussed in Dietzenbacher and Hoen (1999a, 1999b).
Other issues and proposed solutions for dealing with price deflation in input–output
analysis are analyzed in Statistics Canada (2001), Rampa (2008), Reich (2008),
Daniel, Flynn, and Graham (2017), and Rueda-Cantuche et al. (2018).
4.9
The Aggregation Problem: Level of Detail in Input–Output Tables
Input–output tables are used at widely differing aggregations of sectoral detail, of
geography and of time. As a general matter, aggregation reveals tradeoffs in stability
and homogeneity. For example, the more disaggregated a table representing an economy, the more heterogenous the production mix is among the firms defining that
industry in the table. That is, the number of industrial sectors defined in an input–
output table (often referred to as the level of sectoral aggregation) is usually decided in
the context of the problem considered. For example, it may be important to determine
whether to distinguish between fully assembled automobiles and automobile parts
produced separately by a specific automobile manufacturer; for many problems, a
more aggregated sector labeled “automobiles and parts” may be sufficient. Conversely,
the more aggregated a table, the more stable the characterization of the economic
sector since a more disaggregated table reflects the diversity in production characteristics among highly similar firms in more detail.15
The geographic scope of an input–output table presents similar tradeoffs. A larger
geographic scope likely includes variations in the production characteristics across the
region of interest. Avelino (2017) argues further that any bias introduced by aggregation in space and time depends upon the aggregation level in the other dimension. For
example, he illustrates:
. . . in more aggregated industries it is more likely to observe the production structure varying within
the year, especially if seasonality affects the aggregated sectors distinctly. In this case, the intra-year
oscillation in its interindustrial flows and total production may not follow the same proportion
derived from the annual input–output table. (Avelino, 2017, p. 314)
15
Avelino (2017, p. 313) describes this characteristic as “the more aggregated the matrix is, the more stable the
calculated technical coefficients are, since the lower cross-elasticities derived generate lower substitution
effects, supporting the constant input–output ratios assumption.”
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Organization of Basic Data for Input–Output Models
While, on balance, temporal aggregation annually in compilation of input–output
tables is widely accepted, it can at times lead to misleading results, since it averages
across any seasonality in the production structure within the year. As Avelino observes
“. . . the variance of annual coefficients is a weighted sum of the variance of intra-year
coefficients and their temporal shares, which filters most of the within-year heterogeneity, as shown by Sevaldson (1970) for the case of industrial aggregation” (Avelino,
2017, p. 314).
Similarly, in multiple-region models – that is, interregional or multiregional models,
as defined in Chapter 3 – the number of regions considered in the model (the level of
spatial aggregation) should be selected at a level to match the problem being considered (see Park and Richardson, 2015). For example, if we are interested in the
impacts of increased coal development on regions in the USA, how should states be
grouped into regions (assuming the basic data are state-specific) to construct an
applicable model? An additional and often important consideration is what information, if any, is lost in performing either a sectoral or spatial aggregation?
Finally, and perhaps often overwhelmingly, other factors such as computational
expense or availability of data may also be overriding considerations in decisions
about the level of spatial, sectoral, or temporal aggregation.
4.9.1 Investigating Aggregation Bias
Since the early 1950s, considerable attention has been given in the literature to
establishing criteria for and measuring the effects of aggregation of sectors in input–
output models. Representative earlier examples include Hatanaka (1952), Balderston
and Whitin (1954), McManus (1956), Malinvaud (1956), Theil (1957), Ara (1959),
Morimoto (1970), Kymn (1990), Cabrer, Contreras, and Miravete (1991), and Olsen
(1993). Many of these efforts were aimed at compensating for limited computing
capabilities at the time. Today the issues center more around bias introduced by
sectoral or, in the case of multiregional or interregional models, spatial aggregation
or the definition of regions in input–output models. The questions of the level of
aggregation (number of sectors or regions) is likely to be even more important at the
regional level, where good data are often unavailable or difficult and prohibitively
expensive to obtain (see Doeksen and Little, 1968; Williamson, 1970; Hewings, 1972;
Stevens and Lahr, 1993; Wenz et al., 2015). Miller and Blair (1981) and Blair and
Miller (1983) examine the subject of spatial aggregation in more detail for interregional and multiregional input–output models, respectively.
More recently there has been renewed attention to temporal stability of technical
coefficients, such as in Temurshoev, Webb, and Yamono (2011) and Torres-González
and Yang (2019). Various econometric methods have been applied for deriving or
updating technical coefficients for intervals of time less than a year, such as over the
years in Romanoff and Levine (1981), ten Raa (1986), Aulin-Ahmavaara (1990),
Israilevich et al. (1997), and more recently in Ryaboshlyk (2006), Donaghy, BaltaOzkan, and Hewings (2007), Kratena et al. (2013), and Avelino (2017). The stability of
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4.9 The Aggregation Problem
163
input–output coefficients was a fundamental concern in Leontief’s original work (see
Leontief, 1955), in which he devised a series of tests for the original 1919 and 1929
tables, which proved inconclusive. However, upon completion of the 1939 table in
1944, and compared with the earlier 1919 and 1929 tables, he concluded that the
comparison across two decades demonstrated a sufficient degree of stability across
most coefficients for that level of sectoral aggregation. We revisit the subject of
stability of input–output coefficients in Chapter 9.
In Section 4.9.2 we examine the basic effects of aggregation on input–output models
by investigating several measures of the bias or error introduced by aggregation.
4.9.2 The Aggregation Matrix
Before examining the effects of aggregation, let us develop a systematic way of
accomplishing aggregation of sectors in an input–output table. First, define a matrix
S, the aggregation matrix, to be a k n matrix of ones and zeros, where k is the
number of sectors in the to-be-created aggregated version of the input–output table and
n is the number of sectors in the existing unaggregated version of the table. The
locations of ones in row i of S indicate which sectors of the unaggregated table will be
grouped together as sector i in the aggregated table.
For example, let n ¼ 4 and k ¼ 3; suppose that sectors 2 and 3 of the disaggregated
table are to be combined. Then the aggregation matrix that accomplishes this is
2
3
1 0 0 0
S ¼ 4 0 1 1 0 5. Let Z denote the unaggregated 4 4 transactions matrix and
0 0 0 1
∗
f be the corresponding aggregated 3 3 transactions matrix. Similarly, f and f ∗ are
the unaggregated and aggregated vectors of final demand, respectively. Recall that our
aim is to aggregate sectors 2 and 3 of the unaggregated model; for f this can easily be
accomplished by premultiplying by S:
2 3
2
3
2
3 f1
f1
1 0 0 0 6 7
f2 7 4
5
(4.27)
f ∗ ¼ Sf ¼ 4 0 1 1 0 5 6
4 f 3 5 ¼ f2 þ f3
f4
0 0 0 1
f4
For Z, this can be accomplished by
2
1 0
Z∗ ¼ SZS0 ¼ 4 0 1
0 0
2
0
1
0
z11
Z∗ ¼ 4 z21 þ z31
z41
2
3 z11
0 6
z21
0 56
4 z31
1
z41
z12
z22
z32
z42
z13
z23
z33
z43
32
1
z14
60
z24 7
76
z34 5 4 0
z44
0
z12 þ z13
z22 þ z23 þ z32 þ z33
z42 þ z43
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0
1
1
0
3
z14
z24 þ z34 5
z44
3
0
07
7
05
1
(4.28)
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Organization of Basic Data for Input–Output Models
The new corresponding vector of total outputs x* can be computed as
x∗ ¼ Z∗ i þ f ∗
(4.29)
where, as before, i is a column vector of ones.
We can also use the aggregation matrix to reorder sectors. For example, the matrix S
used in (4.27) is the one that introduces the least sector labeling rearrangement into the
aggregated matrix; that is, the original first sector remains sector 1 and the original
“last” sector, 4, becomes the “last” sector, 3, in the aggregated model. Alternatively,
2
3
0 1 1 0
S ¼ 4 1 0 0 0 5 groups original sectors 2 and 3 together and labels them sector
0 0 0 1
1 in the aggregated matrix, labels the original sector 1 as sector 2 in the aggregated
matrix, and the original sector 4 as sector 3 in the aggregated matrix.
If we are given a new set of final demands, f, for which we wish to compute the
corresponding total output needed to support that final demand, we can compute the
Leontief inverse matrices for both unaggregated and aggregated versions of the model:
1
1
.
ðI AÞ1 and ðI A∗ Þ where A ¼ Zx^1 and A∗ ¼ Z∗ x^∗
As in the case of the initial set of final demands, for a new vector of final demands,
~f , the aggregated vector of new final demands is ~f ∗ ¼ S~f . Hence, impact analysis
1
yields ~
x ¼ ðI AÞ1~f and x~∗ ¼ ðI A∗ Þ ~f ∗ . Note that, except under very special
circumstances which we describe later, x~∗ 6¼ S~x ; the difference between x~∗ and S~x is
one indication of the bias introduced by aggregating the input–output table from four
to three sectors.
Example 4.7: Sectoral Aggregation We begin with a four-sector input–
output model defined by
2
3
2
3
2
3
26:5 75:0 46:0 53:0
659:5
860
6 34:0 5:0 68:0 68:0 7
6 1,835:0 7
6 2,010 7
7
6
7
6
7
Z¼6
4 41:5 38:0 52:0 83:0 5, f ¼ 4 2,515:5 5 and x ¼ 4 2,730 5
33:5 6:0 53:0 67:0
1,560:5
1,720
Let us consider two alternative sectoral aggregations of this model, given respectively
2
3
2
3
1 0 0 0
0 1 0 0
by the aggregation matrices S1 ¼ 4 0 1 0 0 5 and S2 ¼ 4 0 0 1 0 5: S1
0 0 1 1
1 0 0 1
combines sectors 3 and 4 of the four-sector model into sector 3 of a three-sector
model, leaving sectors 1 and 2 unaggregated. S2 combines sectors 1 and 4 of the foursector model into sector 3 of a three-sector model and assigns sectors 2 and 3 of the
four-sector model to sectors 1 and 2, respectively, in a three-sector model.
From (4.22), (4.23), and (4.24) we can compute the corresponding aggregated
values of f, Z, and x for the two alternative aggregation schemes. For the S1 aggregation scheme, we have,
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4.9 The Aggregation Problem
2
3
2
659:5
26:5 75:0
4 1,835:0 5, Z∗ ¼ S1 ZS0 ¼ 4 34:0 5:0
f∗
¼
S
f
¼
1
1
1
1
4,076:0
75:0 44:0
2
3
860
∗
∗
∗
4
x1 ¼ Z1 i þ f 1 ¼ 2,010 5
4,450
Similarly, for the S2 aggregation scheme, we have
2
3
2
1,835:0
5:0
4 2,515:5 5, Z∗ ¼ S2 ZS0 ¼ 4 38:0
¼
S
f
¼
f∗
2
2
2
2
2,220:0
81:0
2
3
2,010
∗
∗
4 2,730 5
x∗
2 ¼ Z2 i þ f 2 ¼
2,580
68:0
52:0
99:0
165
3
99:0
136:0 5, and
255:0
3
102:0
124:5 5, and
180:0
We can now compute the technical coefficients matrix and Leontief inverse for each of
the aggregation schemes. For S1, the result is
2
3
2
3
:031 :037 :037
1:036 :039 :026
1
∗ ∗ 1
^1
¼ 4 :04 :003 :003 5, and I A∗
¼ 4 :044 1:005 :034 5
A∗
1 ¼ Z1 x
1
:087 :022 :057
:097 :041 1:064
For S2, we have
2
3
2
3
:002 :025 :04
1:005 :027 :044
1
1
∗
^∗
¼ 4 :019 :019 :048 5 and I A∗
¼ 4 :022 1:022 :054 5
A∗
2
2 ¼ Z2 x
2
:04 :036 :07
:044 :041 1:079
Suppose we are given a new final demand, ~f , which is presented to the economy as
2 3
10
6 10 7
~f ¼ 6 7. For the two alternative aggregations, the corresponding final-demand
4 10 5
10
2 3
2 3
10
10
∗
~
~
~
4
5
4
vectors are ~f ∗
¼
S
¼
S
f
¼
10
and
f
f
¼
10 5: The corresponding total
1
2
1
2
20
2
3 20
2
3
11:26
11:2
∗ 1 ~∗
∗ 1 ~∗
f 1 ¼ 4 11:16 5 and x~∗
f 2 ¼ 4 11:51 5:
output vectors are x~∗
1 ¼ I A1
2 ¼ I A2
22:52
22:43
If we use the unaggregated model in impact analysis, the total output vector is
2
3
11:3
6 11:2 7
7
~
x ¼ ðI AÞ1~f ¼ 6
x 1 from the original unaggregated matrix
4 11:51 5, where A ¼ Z^
11:13
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166
Organization of Basic Data for Input–Output Models
of transactions, Z, and vector of total outputs,
the vector
2 x. If we
3 aggregate 2
3 ~x by the
11:3
11:2
two aggregation schemes, we obtain S1 ~x ¼ 4 11:2 5 and S2 ~x ¼ 4 11:51 5:
22:64
22:43
∗
~
~
~
Note that while x~∗
x
are
quite
different,
x
x
are
identical.
That is, no
and
S
and
S
1
2
1
2
error is introduced in the second aggregation scheme S2 . We will see more formally
later why this is true, but, for the time being, we2can examine the original unaggregated
3
:031 :037 :017 :031
6 :04 :003 :025 :04 7
7
matrix of technical coefficients, A ¼ Zx^1 ¼ 6
4 :048 :019 :019 :048 5: Note that
:039 :003 :019 :039
the first and last columns of A are identical, that is, the two industries have identical
production characteristics. In the S2 aggregation scheme, these two industries are
aggregated into one; this was not the case in the S1 aggregation scheme. This should
not be surprising, however, since two industries with the same production function are,
by definition, the same industry and, hence, there should be no bias introduced
by aggregation.
4.9.3 Measures of Aggregation Bias
We can define total aggregation bias (e.g., Morimoto, 1970) as the difference between
the vector of total outputs in the aggregated system and the vector obtained by
aggregating the total outputs in the original unaggregated system. As in Example
4.7, for some new vector of final demands, f, the total output vector in the unaggregated model is x ¼ ðI AÞ1 f . The total output vector in the aggregated model is
1
x∗ ¼ ðI A∗ Þ f ∗ , and the total aggregation bias is defined as
τ ¼ x∗ Sx
(4.30)
1
That is, τ ¼ ðI A∗ Þ f ∗ SðI AÞ1 f, or τ ¼ ðI A∗ Þ1 S SðI AÞ1 f.
Using the power series results,
τ ¼ I þ A∗ þ A∗2 þ . . . S S I þ A þ A2 þ . . . f
(4.31)
¼ ðA∗ S SAÞ þ A∗2 S SA2 þ . . . f
We define the first term of this series as the “first-order” aggregation bias (Theil, 1957);
that is,
φ ¼ ðA∗ S SAÞf
(4.32)
There are two key theorems that capture the basic implications of aggregation bias in
intput–output models, especially the circumstances under which it will vanish. One has
to do with the nature of the A and A* matrices, that is, with the structural characteristics
of the economy; the other has to do with the nature of the final-demand vectors, f and f *,
being studied. We summarize these theorems here and provide additional detail and
examples in online Appendix SA4.1.
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4.9 The Aggregation Problem
167
Theorem 4.1 The total aggregation bias vanishes (i.e., τ ¼ 0) for any φ if and only if
A∗ S ¼ SA. This follows from the expression for τ in (4.26) since, if A∗ S ¼ SA, then
ðA∗ Þ S SA2 ¼ A∗ A∗ S SAA ¼ A∗ ðSAÞ ðA∗ SÞA ¼ 0
2
and similarly, for higher-order terms in the series. This theorem suggests that if two (or more) sectors
have identical interindustry structures (i.e., equal columns in the A matrix, as we found in the
example), then aggregation of these sectors will result in zero total aggregation bias.
Theorem 4.2 If some sectors are not aggregated and the new final demands occur only
in unaggregated sectors, the first-order aggregation bias will vanish.
For a general three-sector economy, the unaggregated and aggregated technical coefficients
matrices, A and A∗ , respectively, are
2
z11 =x1
6
A ¼ 4 z21 =x1
z31 =x1
z12 =x2
z22 =x2
z32 =x2
z13 =x3
3
z11 =x1
7
z23 =x3 5 and A∗ ¼
ðz21 þ z31 Þ=x1
z33 =x3
ðz12 þ z13 Þ=ðx2 þ x3 Þ
ðz22 þ z23 þ z32 þ z33 Þ=ðx2 þ x3 Þ
The unaggregated sector is sector 1 (in both the aggregated and unaggregated models). Consider
2 3
f1
final-demand vectors for which only the unaggregated elements are non-zero: f ¼ 4 0 5
0
f
and f ∗ ¼ Sf ¼ 1 . This theorem asserts that the first-order aggregation bias, φ ¼ ðA∗ S SAÞf,
0
is zero for final demands such as those given as f and f ∗ , as defined in the introduction to this
section. Thus, if one is studying the effect of new final demand only for sector 1’s output in an nsector model, any combination of sectors 2 through n into fewer sectors will generate no first-order
aggregation bias. Additional general theorems on sectoral aggregation bias based on statistical
properties are discussed in Gibbons, Wolsky, and Tolley (1982).
4.9.4 Spatial Aggregation Bias
Theorems 4.1 and 4.2 are stated in terms of sectoral aggregation, but they also
have implications for spatial aggregation in interregional models. In general, the
conditions of Theorem 4.1 are almost certain not to be met as one combines
regions in an interregional input–output model, but the conditions of Theorem
4.2 will be met in many cases. Aggregation bias in interregional and multiregional
input–output models, or spatial aggregation, is a very straightforward extension of
sectoral aggregation. Most simply it can be thought of as aggregating regions each
with the same level of sectoral detail to a reduced number of the regions considered, although including regions at different sectoral aggregations would, of
course, also be possible. Additional discussion and examples are provided in
online Appendix SA4.1.
As a general matter it appears that spatial aggregation in both IRIO and MRIO
models produce only modest aggregation bias. Hence, for questions pertaining to one
or more specific regions, it appears that an MRIO (or IRIO) model in which those
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168
Organization of Basic Data for Input–Output Models
regions are distinct, while the rest of the economy is spatially aggregated into the
“remaining” region, can often prove to be entirely adequate. The subjects of spatial
aggregation applied to IRIO and MRIO models are discussed in more detail in Miller
and Blair (1981) and Blair and Miller (1983), respectively. Examples of spatial
aggregation for the three-region Japanese interregional and the US multiregional
input–output models are included for the interested reader in online Appendix SA4.1.
In recent years with considerably expanded computational capacity increasingly
available, very large MRIO models, including global models, have been widely
applied (discussed in more detail in Chapters 3 and 13). Lenzen (2019) and others
have considered methods for optimizing aggregation of such very large models and
concluded that while clustering sectors with similar characteristics was frequently
identified as the method associated with the lowest general error level, especially
where considerable aggregation is necessary, for large MRIO systems such methods
remain computationally challenging. Lenzen found Structural Path Analysis (SPA),
discussed in Chapter 8, to be the most intuitively appealing and provides “a straightforward approach to realize groupings that account for the specific purpose of a
specific study” (Lenzen, 2019, p. 19). Other researchers over the years have explored
a wide range of approaches for minimizing error in aggregation of input–output tables,
such as Fisher (1969), Neudecker (1970), Blin and Cohen (1977), Roy, Batton, and
Lesse (1982), Cabrer, Contreras, and Miravete (1991), Oksanen and Williams (1992),
Olsen (1993, 2001), Andrew, Peters, and Lennox (2009) and, as noted, Lenzen (2019).
4.10
Harmonization of Input–Output Data
As the various adjustments to data in fashioning an input–output table from multiple
sources of data just described illustrate, much of the time and effort in assembling
input–output tables is invested in ensuring that the data are adjusted in a transparent
way to yield an internally consistent and current table. The problem is compounded
significantly in assembling multiregional or multinational tables since, most often,
different nations have different standards and traditions in collecting and maintaining
the relevant data. Even assembling current tables utilizing earlier industry and
product classification systems along with new ones can be especially challenging.
This very common situation prompted the development of various organized procedures for reconcling a classification systems and providing “bridges” among them
such as the RACE conversion method (Rueda Cantuche, Amores, and RemondTiedrez, 2020).
The process of adjusting data from different sources, time periods, levels of aggregation, systems of valuation, treatments of imports and exports, interregional trade, and
many other features has come to be known as harmonization and is central to contemporary input–output studies. Walmsley et al. (2018) provide a summary of the most
common issues. Guo, Webb, and Yamano (2009), Inomata (2016), and Stanger (2018)
provide additional details for specific cases. Murray and Lenzen (2013) focus on
multiregional input–output tables. Chapters 9 and 10 discuss many of the most common
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References
169
tools used for harmonization. The emerging Erostat project, Full International and
Global Accounts for Research in Input–Output Analysis, known as FIGARO
(Remond-Tiedrez and Rueda-Cantuche, 2019), applies many of these techniques in the
course of producing its collection of intercountry supply, use, and input–output tables.
4.11
Summary
In this chapter we have explored some of the most important practical issues associated
with applying input–output analysis, namely construction of the basic input–output
tables as part of a system of national accounting conventions and data collection. The
chapter focuses primarily on a System of National Accounts (SNA), including the
integral Input–Output Accounts (IOAs), derived from basic concepts of the circular
flow of income and expenditure.
In order to define interindustry production and consumption within the SNA, the
framework includes conventions for distinguishing between commodities and industries, i.e., production and consumption of defined goods and services or commodities
and industries or groups of economic establishments that produce those commodities,
with an individual industry perhaps producing more than one commodity.
The commodity-by-industry framework lays the foundation for more detailed examination of commodity-by-industry models in Chapter 5, alternatives to full surveybased construction of input–output tables in Chapters 7 and 8, and extensions to the
basic input–output framework in later chapters, such as the SNA as the basis for
broader social accounting in Chapter 11.
Finally, this chapter examines some of the key considerations in defining the level of
sectoral detail in input–output models, especially measures of bias introduced by
sectoral, regional, and temporal aggregation.
Online Appendix SA4.1 (summarized below as Appendix 4.1) provides additional
discussion and examples of aggregation bias introduced in Section 4.9.
Appendix 4.1
Supplemental Discussion of Aggregation Bias
Supplemental Appendix SA4.1, located on the internet web site associated with this
text (http://www.cambridge.org/millerandblair), develops two key theorems providing
the basis for measures of first-order and total aggregation bias in input–output models.
Also explored are these measures applied to interregional (IRIO) and multiregional
(MRIO) models, known as spatial aggregation bias. These measures are applied, as
examples, to historical Japanese IRIO and US MRIO models.
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5
5.1
The Commodity-by-Industry
Approach in Input–Output
Models
Introduction
In this chapter we explore a key variation in the underlying data sets from which an
input–output model is constructed. As noted in Chapter 4, an important challenge in
assembling input–output tables is to account in an organized way for industrial firms
that produce more than one product, which, of course, is quite common in modern
economies. By explicitly distinguishing between industries and products or commodities in compiling interindustry data, we can allocate the production of both principal
and secondary products or by-products to the industrial sectors primarily associated
with those products. As introduced in Chapter 4, this accounting involves a
“commodity–industry” format with two key tables: (1) a Use table that defines all
the commodity inputs to defined industries in the economy, and (2) a Supply table that
defines the usually multiple sources of supply among industrial producing sectors of
the commodities consumed in the economy.
The need for accounting for secondary production in an organized way was a major
reason for the introduction of the commodity–industry accounting system, but data
organized in this way are also more easily integrated with a broader system of national
accounts (SNA) for a country, as we saw in Chapter 4. These commodity–industry
accounts lead to input–output models that have more complicated mathematical
structures than those developed in Chapters 2 and 3, and these commodity–industry
models are the concern of this chapter.
The Eurostat manual (Eurostat/European Commission, 2008) provides an excellent
and comprehensive discussion of the commodity–industry framework. There, as in
many other publications, the term “product” is used instead of “commodity.” We will
use “commodity” in this text because that is the predominant terminology associated
with the early derivations and discussions of this system, in the 1960s and 1970s, and it
continues to be used by many analysts.
As chronicled in Chapter 4, the commodity-by-industry accounting framework
originated largely in the work of Richard Stone and his associates (Stone, 1961;
Cambridge University, 1963). It was proposed in 1968 by the United Nations as a
standard for data gathering in countries throughout the world (United Nations, 1968),
and subsequently has become a feature of data collection and input–output statistics
176
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5.1 Introduction
177
virtually everywhere (United Nations et al., 1993). Viet (1994) reviews input–output
data collection and assembly practices in 53 countries during the 1970s and 1980s.
Specific examples include Canada, where this framework has been used at both a
national and a regional scale since the early 1960s (Statistics Canada, 1981) and the
US, where national data have been collected and presented in commodity-by-industry
form starting with the 1972 tables.1 It has also become the template for countries in the
European Community (Eurostat, 1996, describing the European System of Accounts,
ESA, 1995) – Denmark (annual tables since 1966), the Netherlands, and Norway
provide examples. The commodity–industry approach indeed provides a framework
in which secondary products, by-products, etc. can be much more explicitly accounted
for; however, it also introduces new problems (including the possibility of negative
coefficients or transactions), as we will see in Section 5.5.3.
The underlying observation is that industries use commodities to make commodities.
It is commodities that are the inputs to industrial processes and that are used to satisfy
final demands. An industry can be thought of as defined by its primary product
(commodity), but some, if not many, industries also produce additional commodities
as secondary products. There are several kinds of non-characteristic or secondary
products, joint products, by-products, subsidiary products; we will investigate some
of these distinctions later. In order to highlight the differentiation between commodities
and industries, assume that the commodity index, i, runs from 1 to m and the industry
index, j, runs from 1 to n. If every commodity produced in an economy is primary to
some industry in that economy, then the number of commodities and the number of
industries will be the same, m ¼ n. Initially we will investigate this case in some detail.
In many economies there may be benefits to situations when m 6¼ n, but mathematical
complications can arise, which we explore in Section 5.6.2
5.1.1 The Use Matrix
In ordinary input–output models with n sectors, an element of the n n transactions
matrix Z ¼ ½zij represents the value of purchases of industry i output by industry j. In
addition, there is an n-element vector of total industry outputs, x ¼ ½xj , where
xj ¼ zj1 þ þ zjn þ fj
(5.1)
and fj is industry j’s sales to final demand. In matrix form, this is
1
For extensive discussions of data collection and modeling efforts and conventions in a number of countries
using some version of commodity–industry accounts, see Franz and Rainer (1989) and Viet (1994). Since
their original compilation, the pre-1972 US input–output tables have since been “backcast” and reformulated
in the commodity-by-industry framework for all tables since 1946; these tables are available at www.bea.gov
and highly aggregated versions are included with this text as described in Appendix B and in online
Appendix SD1.
2
If m > n, aggregation of commodity accounts could proceed until m ¼ n; similarly, if m < n, industry accounts
could be aggregated. This is often done in practice (again, see the papers in Franz and Rainer, 1989). But, of
course, aggregation covers up information from the originally more detailed data sets.
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178
The Commodity-by-Industry Approach in Input–Output Models
x ¼ Zi þ f
(5.2)
and direct input (technical) coefficients, A ¼ ½aij , are defined as
A ¼ Zx^1
(5.3)
Throughout this chapter, the adjectives “ordinary” and “original” will be used with
“input–output” to refer to the model that is derived from (5.1)–(5.3), as in Chapter 2
and in (5.11) and (5.12). These are the relationships as Leontief first articulated them
and that are reflected, in the case of the US, in pre-1972 originally-published input–
output data.3
In the commodity-by-industry approach, the interindustry transactions matrix, Z, is
replaced, initially, by the Use matrix, UðciÞ ¼ ½uij , where uij is the value of purchases of
commodity i by industry j.4 Thus the “industries use commodities” part of “industries use
commodities to make commodities” is quantified in U, which is sometimes also called the
absorption or input matrix. In conjunction with total industry output, x, the parallel to
ordinary technical coefficients, aij , would appear to be bij ¼ uij =xj or, in matrix terms,
B ¼ Ux^1
(5.4)
in which column j represents the value of inputs of each commodity per dollar’s worth of
industry j’s output.5 The dimensions of B are therefore commodities-by-industries.
However, we will see that among the other matrices that emerge in this system, some will
have the dimensions “commodity-by-commodity”; others will be of “industry-by-commodity” or “industry-by-industry” structure. For this reason, in what remains we will use
the general term “commodity–industry” to characterize this accounting framework generally and all the variations of models that are derived from it. If we also have information on
commodity sales to final demand, this can be arranged as in the example shown in Table 5.1.
5.1.2 The Make Matrix
As might be expected, the matrix showing how industries make commodities is termed
the Make matrix, usually denoted V (it is also called the output matrix).6 Table 5.2
provides an example.
An element of V, vij , shows the value of the output of commodity j that is produced
by industry i; hence, the dimensions of V are industries-by-commodities. In this
3
As noted in the introduction to this chapter, more recent compilations of the US input–output tables include
reformulation of the pre-1972 tables to the commodity–industry format. Aggregated versions of these tables are
described in Appendix B and the data included as spreadsheets in online Appendix SD1.
4
Normally, the parentheses below a matrix indicate its dimensions – number of rows and number of columns. In
this section we will sometimes use expressions like (c i) to help us remember which dimension enumerates
commodities (in this case, rows) and which enumerates industries (in this case, columns). Thus, we will write
that U has “commodity-by-industry dimensions.”
5
The notation B ¼ ½bij is also used for the coefficients matrix in a supply-side model (Chapter 7) and for capital
coefficients in a dynamic input–output model (Chapter 14). Its use in (5.4) in the commodity-by-industry literature is
fairly widespread, and in general the context of any discussion should make clear which meaning is intended.
6
Use of V for the Make matrix and v0 for the row vector of value added elements is also standard in the input–
output literature and, again, should not lead to confusion when read in appropriate context.
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5.2 The Basic Accounting Relationships
179
Table 5.1 The use matrix (U) and other data for a two-commodity, two-industry
hypothetical example (in dollars)
Industries
Commodities
1
2
Value Added (w0 )
Total Industry Outputs (x0 )
Final Demand for
Commodities
Total Commodity
Output
1
2
(e)
(q)
12
10
68
90
8
7
95
100
80
83
100
100
Table 5.2 The make matrix (V) and other data for a two-commodity, two-industry
hypothetical example (in dollars)
Commodities
Industries
Total Commodity
Outputs (q0 )
1
2
Total Industry Output
1
2
(x)
90
10
100
0
100
100
100
100
example, industry 1 produces only its primary product, commodity 1, but the output of
industry 2 consists of $100 worth of its primary product, commodity 2, and also $10
worth of commodity 1, which is a secondary product in industry 2. In an economy in
which there is no secondary production, the Make matrix will be diagonal and, as we
will see in the balance of this chapter, all of the commodity–industry results reduce to
the original Leontief industry-based approach.
As the development and application of commodity-by-industry approaches have
evolved over the last two decades it has become common to present the Make matrix
as its transpose, V0 , referred to as the Supply matrix, S, i.e., S ¼ V0 . There is some
appeal to using S instead of V, at least in representing the data, since the U and S tables
can be shown together more compactly, e.g., Table 5.3 shows both ways of presenting
all of the data for the running example in a commodity–industry framework. We will
principally use V in the following development of the commodity-by-industry
approach, but will use S at various points throughout the text when it is convenient.
5.2
The Basic Accounting Relationships
In the ordinary input–output model, the basic accounting relationship for total (industry) output is given in (5.1) and (5.2). The commodity–industry framework accounts
for both total industry output (x) and total commodity output (q) separately. From the
data in the Make matrix, total output of any industry is found by summing over all
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180
The Commodity-by-Industry Approach in Input–Output Models
Table 5.3 The complete set of commodity–industry data
Using the Use and Make Matrix
Commodities
1
Commodities
Industries
2
1
2
1
2
Final Demand
Total Output
12
10
8
7
80
83
e
100
100
q
U
Industries
1
2
90
10
0
100
90
110
x
V
Value Added
68
Total Inputs
100
q0
100
95
v0
90
x0
163
110
Using the Use and Supply Matrix
Industries
Commodities
1
2
1
2
90
0
10
100
Final Demand
100
100
q
S
Commodities
1
2
12
10
8
7
80
83
e
95
163
U
Value Added
68
Total Inputs
90
v0
x0
Total Output
100
100
q
110
commodities produced by that industry. These totals are the row sums of V (or the
column sums of S ¼ V0 ),
xj ¼ vj1 þ þ vjm or xj ¼ s1j þ þ smj
(5.5)
x ¼ Vi or x0 ¼ i0 S
(5.6)
or, in matrix terms,
Similarly, total output of any commodity can be found by summing over all
industries that produce that commodity. These totals are the column sums of V (or
the row sums of S)
qi ¼ v1i þ . . . þ vni or q1 ¼ si1 þ . . . þ sin
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(5.7)
5.3 Technology and Total Requirement Matrices
181
or, in matrix terms,
q0 ¼ i 0 V
or
q ¼ V0 i ¼ Si
(5.8)
Alternatively, and as illustrated for the example in Table 5.1,
qj ¼ uj1 þ þ ujn þ ej
(5.9)
q ¼ Ui þ e
(5.10)
or
The original input–output model combines (5.2) and (5.3). Substituting Z ¼ Ax^
from (5.3) into (5.2), and recalling that x^i ¼ x, gives
x ¼ Ax þ f
(5.11)
Rearranging terms, the operational form of the model becomes
x ¼ ðI AÞ1 f ¼ Lf
(5.12)
where L is the familiar Leontief inverse, L ¼ ðI AÞ1 . The driving force for use of
the model is the exogenous vector of final demand for industry outputs. In conjunction
with the L (total requirements) matrix, industry outputs necessary to sustain the final
demand are determined.
The commodity–industry approach uses (5.10) and (5.4) in the same way as (5.2)
and (5.3), respectively. From (5.4), U ¼ Bx^, and substituting into (5.10) gives
q ¼ Bx þ e
(5.13)
as a parallel to (5.11) in the ordinary input–output model. The problem is that,
unlike (5.11), one cannot generate a total requirements matrix, as in (5.12), directly
from (5.13), because (5.13) contains commodity output (q) on the left-hand side
and industry output (x) on the right-hand side. Approaches for generating the total
requirements matrix under a variety of assumptions is the focus of much of the
following.
5.3 Technology and Total Requirement Matrices in the
Commodity–Industry Approach
One solution to the problem of generating a total requirements matrix analogous to
(5.12) for (5.13) is to find an expression transforming industry outputs, x, to commodity outputs, q – or, alternatively, to transform commodity outputs (and commodity final
demand, e) into industry terms. The data needed for such transformations are found in
the Make matrix, whose row sums are industry outputs and whose column sums are
commodity outputs. Two alternative algebraic options of using the information in the
Make matrix are described in Sections 5.3.1 and 5.3.2, each of which have quite
different economic interpretations.
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5.3.1 Industry Source of Commodity Outputs
The first option is to define d ij ¼ vij =qj (each element in column j of V is divided by the
jth column sum, qj ), so that d ij denotes the fraction of total commodity j output that
was produced by industry i. Forming a matrix of these commodity output proportions
D ¼ d ij , we have
ðicÞ
D ¼ V^
q 1
For the numerical example,
:9
D¼
:1
0
1
(5.14)
From column 1, for example, we see that 90 percent of the total amount of commodity
1 made in the economy was produced by industry 1 and 10 percent was produced by
industry 2. D is often called the market shares matrix. By definition, each of the
column sums of D is unity, defining the entire “market.”
5.3.2 Commodity Composition of Industry Outputs
The second option is to define cij ¼ vij =xi (each element in row i of V is divided by the
ith row sum,xi ), so that cij denotes the fraction of total industry i output that is in the
form of commodity j. For later purposes it will turn out to be convenient to have these
industry output proportions arranged in a matrix with commodities-by-industries
dimensions (remember that V has industry-by-commodity dimensions). Define V0 as
the supply matrix, with commodity-by-industry dimensions; then the matrix of these
industry output proportions is found as7
C ¼ V0 x^1
For the numerical example,
C¼
1
0
:0909
:9091
(5.15)
The second column, for example, says that 90.9 percent of the value of industry 2’s
output consisted of commodity 2 and 9.1 percent was accounted for by commodity 1.
C is sometimes called the product mix matrix or the commodity mix matrix. By
definition, each column sum in C is unity.
5.3.3 Generating Total Requirements Matrices
The results in (5.14) and (5.15) – in conjunction with (5.6) and (5.8) – provide two
alternative linear transformations between commodity and industry outputs. Using
(5.14),
7
C is another letter that serves more than one purpose in the input–output literature. Recall from Chapter 3 that it
is also used for the matrix of regional trade proportions.
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5.3 Technology and Total Requirement Matrices
183
D ¼ V^
q 1 ) D^
q ¼ V ) D^
q i ¼ Vi
and from (5.6)
Dq ¼ x
(5.16)
q ¼ D1 x
(5.17)
This also means
if D is square and non-singular.8
A compact statement of the relationships in (5.13) and (5.16) is provided as
follows:9 from (5.16), x – Dq ¼ 0; from (5.13), –Bx þ q ¼ e. These are two matrix
equations in x and q; in partitioned matrix form they can be represented as
0
I
D x
¼
e
B
I
q
Alternatively, using (5.15),
C ¼ V0 x^1 ) Cx^ ¼ V0 ) Cx^i ¼ Cx ¼ ðV0 Þi
and from (5.8)
Cx ¼ q
(5.18)
x ¼ C1 q
(5.19)
so
again provided that C is square and non-singular.
A compact statement of the results in (5.13) and (5.18) is: from (5.18), Cx – q ¼ 0,
and from (5.13), again, –Bx þ q ¼ e. This pair of relationships in x and q can be
represented in partitioned matrix form as
C I x
0
¼
B I
q
e
Using D One solution to the dilemma posed by the presence of both x and
q in (5.13) is provided by (5.16). Substitute Dq for x in (5.13),
q ¼ BðDqÞ þ e ¼ ðBDÞq þ e
8
9
For the moment we will assume that D (and C) are non-singular. In Section 5.5.3 we will explore how
important (and how likely) these assumptions are.
This parallels the representation in Jack Faucett Associates, Inc. (1981–1983, Vol. 5, pp. 11-4 and 11-5), which
was developed in the context of the US multiregional input–output model for 1977.
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The Commodity-by-Industry Approach in Input–Output Models
from which
q ¼ ðI BDÞ1 e
(5.20)
The inverse on the right-hand side, which is called a commodity-by-commodity total
requirements matrix, connects commodity final demand to commodity output. It thus
plays the role of ðI AÞ1 in the ordinary input–output model, (5.12). It is to be noted
that the “parallel” to the A matrix (direct input requirements) in the ordinary model
appears now to be BD [and not simply B alone, as seemed initially the case when B
was defined in (5.4)].
Using (5.20), and since Dq ¼ x,
x ¼ [DðI BDÞ1 ]e
(5.21)
The bracketed matrix on the right connects commodity final demand to industry
output. It is an industry-by-commodity total requirements matrix.
There are alternative possible expressions for total requirements matrices. For
example, premultiplying both sides of (5.13) by D gives, since Dq ¼ x,
x ¼ DBx þ De
and
x ¼ [ðI DBÞ1 D]e
(5.22)
so the bracketed expression on the right-hand side is also an industry-by-commodity
total requirements matrix.10
Using C A second transformation of (5.13) is easily accomplished using
(5.19) – as long as C1 exists. Substitute C1 q for x in (5.13),
q ¼ B C1 q þ e ¼ BC1 q þ e
from which
1
q ¼ I BC1 e
(5.23)
It is apparent that the inverse on the right-hand side is also a commodity-by-commodity
total requirements matrix, connecting commodity final demand to commodity output,
and it differs from the expression in (5.20) which has the same name. Thus another
“parallel” to the A matrix in the ordinary Leontief inverse is BC1 .
Using (5.23), and since C1 q ¼ x,
1
x ¼ [C1 ðI BC1 Þ ]e
10
(5.24)
This reflects a general matrix algebra result (for non-singular D). For example, starting at (5.22) ðI DBÞ1 D ¼
1
1 1
1 1
1 1
D ðI DBÞ
¼ D B
¼ D BDD1
¼ ðI BDÞD1
¼ DðI BDÞ1 . This is the total
requirements matrix in (5.21).
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5.3 Technology and Total Requirement Matrices
185
Here we have an industry-by-commodity total requirements matrix (in brackets) on the
right, and this differs from the expression in (5.21) with the same name.11
To introduce some order into this apparent profusion of alternatives to the Leontief
inverse in the ordinary input–output model, it is instructive to go behind the matrix
algebra and investigate the basic assumptions that underpin these results, as in (5.20)
and (5.21) as compared with (5.23) and (5.24). Recall that transformations of the data
in the Make matrix gave us the industry output proportions, in C, and the commodity
output proportions, in D. In the remainder of this section, we follow the classification
approach that has traditionally been used since the outset of the commodity–industry
discussion, for example as in the system of national accounts (SNA) described in
United Nations (1968); in Section 5.5.3 we will present an alternative and more
recent view.
5.3.4 “Industry-Based” Technology
The commodity-by-commodity total requirements matrix in (5.20) was derived from
q ¼ ðBDÞq þ e
The matrix BD plays the role of a technical coefficients matrix, showing commodity
inputs per dollar’s worth of commodity output. For our example,
12 8 1=90
0
:1333 :0727
B¼
¼
10 7
0
1=110
:1111 :0636
and so
:1333 :0727
BD ¼
:1111 :0636
:9
:1
:1273 :0727
0
¼
:1064 :0636
1
Using B1 and B2 for the two columns in B, this product can be shown as
BD ¼ ½B1 ð0:9Þ þ B2 ð0:1Þ B1 ð0Þ þ B2 ð1Þ
The columns in BD are seen to be convex combinations of the columns in B, where the
weights come from the elements in each column of D. (This simply means that
BD ¼ α1 Β1 þ α2 B2 , where, α1, α2 0 and α1 þ α2 ¼ 1.) Thus, BD embodies the
assumption that commodity inputs to commodity j production are weighted averages
of commodity inputs to each industry that produces commodity j (from the B matrix),
and the weights are the proportions of each industry’s contribution to total commodity j
output (from the D matrix). A given commodity can have differing input structures if it
is produced by more than one industry. In this example, the first column of BD reflects
the fact that 90 percent of the total amount of commodity 1 that is available in the
economy is produced by industry 1 (using the production recipe embodied in Β1 ) and
11
1
Using the same algebra as in footnote 9, this can be expressed as x ¼ [ I C1 B ]C1 e.
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The Commodity-by-Industry Approach in Input–Output Models
10 percent of total commodity 1 output is produced in industry 2 (using that industry’s
production recipe, as embodied in B2 ).
All commodities produced by an industry are assumed to have the same input
structure, as given by that industry’s column in the B matrix. This is shown in the
example by the fact that B2 , the recipe for industry 2 production, appears in both of the
columns of BD. That part (10 percent) of commodity 1 that is produced in industry
2 and that part of commodity 2 (100 percent) that is produced in industry 2 are both
made according to the industry 2 production technology, given in B2 .12 For this reason,
BD is said to embody industry-based technology (or simply industry technology
hereafter), and since its dimensions are commodities-by-commodities, it is sometimes
denoted AI :13
ðccÞ
AI ¼ BD
ðccÞ
The inverse ðI BDÞ1 in (5.20) is therefore referred to, more completely, as the
commodity-by-commodity total requirements matrix under industry technology.
A matrix of technical coefficients that is more parallel to A in the original input–
output model (connecting industry inputs per unit of industry output) arose in the
derivation of (5.22), where
x ¼ ðDBÞx þ De
and it is clear that DB shows inputs from industries per dollar’s worth of industry
production. Its dimensions are industries-by-industries, and it is thus seen to be
comparable to the technological coefficients matrices, A, in the original industry-byindustry input–output models; we will denote it by AI .
ðiiÞ
Carrying out the pre-multiplication of B by D in the small numerical example shows
exactly how the commodity inputs (in B) are distributed back to the industries where
they are made:
:1200 :0655
:9 0 :1333 :0727
¼
AI ¼ DB ¼
:1244 :0709
:1 1 :1111 :0636
ðiiÞ
Using D1 and D2 for the two columns in D,
DB ¼ ½ D1 ð0:1333Þ þ D2 ð0:1111Þ D1 ð0:0727Þ þ D2 ð0:0636Þ Consider, for example, the second column in DB. It disaggregates b12 ¼ 0:0727
(commodity 1 input per dollar’s worth of industry 2 output) and b22 ¼ 0:0636 (commodity 2 input per dollar’s worth of industry 2 output) into two components (vectors).
The first,
12
It has been argued that this may be an appropriate assumption for commodities that are by-products of an
industry’s production process.
13
This notation may seem cumbersome, but in view of the various alternative direct requirements matrices that
will emerge in this and subsequent sections, it is essential to identify precisely both the dimensions and the
technology assumptions that underpin these matrices.
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5.3 Technology and Total Requirement Matrices
187
:0655
:9
ð:0727Þ ¼
:0073
:1
shows the industry 1 (90 percent) and industry 2 (10 percent) contributions to the total
0.0727 needed of commodity 1. Similarly, industry 1 and 2 proportions of the 0.0636
of commodity 2 used by industry 2 are 0 and 1 (the elements of D2 ), and so the vector
showing industry origins of commodity 2 input to industry 2 is
0
0
ð0:0636Þ ¼
1
:0636
The sum of these two vectors indicates the inputs from industry 1 and 2, respectively,
per dollar’s worth of industry 2 output. This is the second column in AI ; there is a
ðiiÞ
similar interpretation for the first column.
5.3.5 “Commodity-Based” Technology
With the industry technology assumption, industry input structures (in the columns of B)
are the basic data, and commodity input structures are found as weighted averages of
these columns. An alternative point of view would suggest that a given commodity
should have the same input structure in all of the industries that produce it.14 In this
case, commodity inputs to industry j production (the elements of the jth column of B)
are viewed as weighted averages of commodity inputs to commodity production for
each of the commodities that industry j makes, and the weights are the proportions of
each commodity in industry j’s total output. This is known as the commodity-based
technology, or simply commodity technology, assumption.
From our small example, we found
1 :0909
C¼
0 :9091
with dimensions commodities-by-industries. We know B, with dimensions commodities-by-industries also. The (presently unknown) commodity-by-commodity technological coefficients matrix can be denoted AC . The commodity technology
ðccÞ
assumption is that B ¼ ðAC Þ C. For the small example, letting ðAC Þ ¼ ½ AC1
ðccÞ
B ¼ ½ AC1
AC2 1
0
:0909
:9091
ðccÞ
AC2 ,
or
½ B1
14
B2 ¼ ½ ðAC1 Þð1Þ þ ðAC2 Þð0Þ
ðAC1 Þð0:0909Þ þ ðAC2 Þð0:9091Þ This may be an appropriate assumption for subsidiary products that are produced by an industry in a separate
facility, employing a similar technology to that used by the industry to which the commodity is primary.
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The Commodity-by-Industry Approach in Input–Output Models
From B ¼ ðAC Þ C, the (unknown) matrix of commodity inputs per dollar’s worth of
ðccÞ
commodity production is found as AC ¼ BC1 (again, provided C is square and
ðccÞ
non-singular). Here
AC ¼ BC
ðccÞ
1
:1333 :0727
¼
:1111 :0636
1
0
:1333 :0667
0:1
¼
:1111 :0589
1:1
1
Hence, for example, ðI BC1 Þ in (5.23) is properly described as the commodityby-commodity total requirements matrix under commodity technology.
The matrix of direct commodity inputs per dollar’s worth of commodity output
under the industry technology assumption was
:1273 :0727
:1333 :0727 :9 0
¼
AI ¼ BD ¼
:1064 :0636
:1111 :0636 :1 1
ðccÞ
Clearly, the two technology assumptions can and generally will lead to different direct
commodity input matrices, as in this example. The “size” of this difference and,
perhaps more importantly, the resulting differences in the corresponding total require1
ments matrices – ðI BDÞ1 and ðI BC1 Þ – is a topic of continuing research and
empirical examination with real-world data sets.
From (5.13) and (5.19),
(5.25)
x ¼ C1 Bx þ C1 e
so under the assumption of commodity technology it is clear that the matrix C1 B
plays the role of DB in an industry technology model and A in the ordinary input–
output models; namely, it records industry inputs per dollar of industry outputs. Let
AC ¼ C1 B; from the numerical example
ðiiÞ
AC ¼ C1 B ¼
ðiiÞ
1
0
0:1
1:1
:1222
:1333 :0727
¼
:1222
:1111 :0636
:0664
:0700
As expected, this differs from AI ¼ DB, calculated in Section 5.3.4. (In this particular
ðiiÞ
numerical example, the difference is not great, but in general there is no reason to
expect that DB ¼ C1 B.)
We explore the economic content of the operation C1 B. Let C1 B ¼ T (instead of
AC , to simplify notation). For the general two-industry case,
ðiiÞ
t 11 t 12
T¼
t 21 t 22
Then
B ¼ CT ¼
1 :0909
0 :9091
t 11
t 21
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t 12
t 22
189
5.3 Technology and Total Requirement Matrices
and, for example,
b
B2 ¼ 12
b22
1
:0909
¼
ðt Þ þ
ðt Þ
0 12
:9091 22
The vector multiplying t 12 disaggregates industry 1 input into commodity inputs –
commodity 1 (100 percent) and commodity 2 (0 percent) – reflecting the commodity
composition of industry 1 output (column 1 of C). Similarly, the vector multiplying t 22
distinguishes industry 2 input as composed of commodity 1 (9.1 percent) and commodity 2 (90.9 percent), from column 2 of C. Adding these together gives B2 , showing
commodity 1 and 2 inputs per dollar of industry 2 output. A similar analysis of the
composition of B1 can be carried out.
5.3.6 Direct Requirements (Technical Coefficients) Matrices Derived from Basic Data
In the ordinary input–output model, the direct requirements matrix is derived directly
from interindustry flows, Z, and industry outputs, x, as in A ¼ Zx^1 in (5.3). In (5.4),
we saw how commodity-to-industry flows, U, and industry outputs, x, were used to
calculate direct requirements in terms of commodity inputs per dollar’s worth of
industry output B ¼ Ux^1 . In the commodity technology models, the matrix that
relates commodity inputs per dollar’s worth of commodity output is AC ¼ BC1 .
ðccÞ
Since C ¼ V0 x^1 , AC is found directly from the basic data in Tables 5.2 and 5.3 as
ðccÞ
1 1 1
AC ¼ BC ¼ Ux^1 V0 x^1
¼ Ux^ [x^ðV0 Þ1 ] ¼ UðV0 Þ1
(5.26)
ðccÞ
and the matrix that relates industry inputs per dollar’s worth of industry output is
1 1 AC ¼ C1 B ¼ [x^ðV0 Þ ] Ux^
ðiiÞ
In contrast to (5.26), further simplifications are not possible.
In the industry technology models, AI ¼ BD relates commodity inputs to each
ðccÞ
dollar’s worth of commodity output. Since D ¼ V^
q 1 ,
1 1 V^
q
AI ¼ BD ¼ Ux^
ðccÞ
(5.27)
Finally, industry inputs per dollar’s worth of industry output under the industry
technology assumption are found from basic data as
1 1 q
Ux^
AI ¼ DB ¼ V^
ðccÞ
If one wants to compare direct requirements matrices for, say, the US economy both
before and after 1972, it is AI ¼ DB and AC ¼ C1 B that are comparable to A, since
ðiiÞ
ðiiÞ
they have industry-by-industry dimensions of the earlier tables. Notice that these
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The Commodity-by-Industry Approach in Input–Output Models
four definitions of direct requirements matrices will all be equal in the case of no
^,
secondary production in any industry. This means that V is diagonal, V ¼ x^ ¼ q
Vi ¼ x ¼ i0 V ¼ q, and so in all four cases, above, A ¼ UV1 ¼ UðV0 Þ1 .
5.3.7
Total Requirements Matrices
Approach I: Starting with Technical Coefficients Results thus far for total
requirements matrices [from (5.20)–(5.24)] are collected together in Table 5.4. Since in
each case the exogenous force driving the model is final demand for commodities,
these are called commodity-demand driven models. We continue, for now, to assume
that C is non-singular. An alternative presentation of the four cases is explored briefly
in Section 5.5.3.
These commodity-by-commodity results are derived from (5.13) through transformations that generate either
q ¼ AI q þ e or
ðccÞ
q ¼ AC q þ e
ðccÞ
and then
q ¼ ðI AI Þ1 e
or
ðccÞ
q ¼ ðI AC Þ1 e
ðccÞ
These total requirements matrices have exactly the same structure as the Leontief
inverse in the original input–output model – namely, the inverse of a matrix containing
technical coefficients subtracted from an identity matrix.
It is also possible to derive total requirements matrices for industry-demand
driven models, replacing e by an equivalent expression involving f in appropriate
equations [from among (5.20)–(5.25)]. In commodity–industry models, one of the
basic premises is that commodities are the products of industries, and therefore it is
commodities that are used to satisfy final demand. Hence the notion of “industry
final demand,” f (the exogenous driving force in ordinary input–output models), is
not very meaningful in commodity–industry models. However, for analyses of
structural change in an economy it is necessary to have consistent data sets for
two or more years. For example, for comparisons of US input–output tables across
time (in which some of the data are pre-1972), it is useful to have an industry-byindustry format, since this was inherent in the original input–output models, and
their inverses, as in (5.12).
Table 5.4 Total requirements matrices, commodity-demand driven models
Industry Technology
Commodity-by-Commodity
Industry-by-Commodity
ðI BDÞ1
h
i
DðI BDÞ1
Commodity Technology
1
I BC1
h
1 i
C1 I BC1
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5.3 Technology and Total Requirement Matrices
191
Table 5.5 Total requirements matrices, industry-demand driven models
Industry Technology
Industry-by-Industry
Commodity-by-Industry
1
ðI DBÞ
D1 ðI BDÞ1
Commodity Technology
1
I C1 B
1
CðI C1 BÞ
For industry technology models, in which Dq ¼ x, the assumption can be made that
the same commodity-to-industry transformation is valid for final demands, that is,
De ¼ f. Similarly, for commodity technology models, in which Cx ¼ q, the same
industry-to-commodity transformation can be used for final demands, i.e., Cf ¼ e. For
example, from (5.22), since De ¼ f, x ¼ ðI DBÞ1 f and, using q ¼ D1 x, we have
q ¼ D1 ðI DBÞ1 f. The latter is the only industry technology result that requires
D1 . Parallel results can be derived for commodity technology models. These are
collected together in Table 5.5. (As in Table 5.4, it is clear that the transformation from
industry technology to commodity technology involves replacement of D by C1
throughout.)
It is worth re-emphasizing that most real-world applications of the commodity–
industry input–output model assume that final demand for commodities is the exogenous driving force, so the results in Table 5.4 are of primary interest. In Table 5.5 the
industry-by-industry case (first row) is useful principally for studies in which
commodity–industry tables are compared with earlier data in the original input–output
1
format, as in (5.11) and (5.12). Thus, for example, both ðI DBÞ1 and I C1 B
are candidates if one is making comparisons with total requirements matrices for the
original pre-1972 US economy. The commodity-by-industry results (second row) are
included in Table 5.5 primarily for completeness – they are of little practical use.
Approach II: Avoiding C1 in Commodity Technology Cases The only case in
which D appears in a total requirements matrix in an industry technology model is in
the relatively unimportant commodity-by-industry format. On the other hand, in the
commodity technology model, C1 is everywhere, and this presents a problem if C is
singular. (It also presents a problem if, as is common, it contains negative elements, as
we will see in the next section.) However, there is an alternative derivation that
circumvents the singularity issue, although it does not create parallels to a technical
coefficients matrix, as in Approach I. Starting again with (5.13) and (5.18),
1
Cx ¼ Bx þ e ) ðC BÞx ¼ e ) x ¼ ðC BÞ1 e
(5.28)
1
Thus, ðC BÞ also serves as an industry-by-commodity total requirements matrix.15
Also, premultiplying both sides of (5.28) by C, and since Cx ¼ q,
15
1
Simple matrix algebra converts ðC BÞ1 to [C1 I BC1 ] (or vice versa), but only if C–1 exists. The
1
point is that the total requirements matrix – ðC BÞ in (5.28) or (5.29) – does not depend on an inverse
for C.
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The Commodity-by-Industry Approach in Input–Output Models
Table 5.6 Rewritten forms of total requirements matrices
Industry Technology
Commodity-Demand Driven Models
1
Commodity-by-Commodity
D1 D1 B
1
1
Industry-by-Commodity
D B
Industry-Demand Driven Models
1
1
Industry-by-Industry
D B D1
1
Commodity-by-Industry
D1 D1 B D1
Commodity Technology
CðC BÞ1
ðC BÞ1
ðC BÞ1 C
CðC BÞ1 C
q ¼ CðC BÞ1 e
(5.29)
This is an alternativeto the commodity-by-commodity total requirements matrix in
1
Table 5.4, I BC1 , that does not require a non-singular C.
Substituting Cf for e on the right-hand sides of (5.28) and (5.29) generates total
requirements matrices with dimensions industry-by-industry and commodity-by-industry, comparable to the results in Table 5.5:
x ¼ ðC BÞ1 Cf
(5.30)
q ¼ CðC BÞ1 Cf
(5.31)
and
The important point is that all four of these results for total requirements matrices
under commodity technology – in (5.28) through (5.31) – do not require that C be nonsingular. (There is a numerical illustration in Section 5.3.8.)
These results are collected together in Table 5.6, along with their counterparts for
industry technology.16 These latter are included primarily for completeness; they are of
little practical interest since they all require D1 , the very inverse that was avoided in
three out of the four industry technology results in Tables 5.4 and 5.5.
Looking down either column in Table 5.6, it is clear that there is an inverse matrix
that is common toall of thetotal requirements matrices in that column. For industry
1
technology this is D1 B , for commodity technology it is ðC BÞ1 . These are
the complete total requirements matrices for the industry-by-commodity case. In the
first column of the table (industry
technology),
it is clear that the other total require
1
through pre- or postmultiplication (or both)
ments matrices differ from D1 B
by D1. As we have seen, under the industry technology assumption, premultiplication
of a matrix (or vector) from industries
by D1 serves to convert the row dimension
1
to commodities. Thus, D1 D1 B
changes the industry-by-commodity total
16
The derivations are similar to those for the commodity technology model cases and are left as an exercise for
the interested reader.
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5.3 Technology and Total Requirement Matrices
193
requirements matrix to commodity-by-commodity form. This is the first matrix in
Table 5.6.
Postmultiplication of a total requirements matrix by D1 is equivalent to premultiplication of a final demand vector by D1 ; in an industry-demand driven model, we
saw that the conversion of final demand to commodity terms is provided by e ¼ D1 f.
This explains the last two matrices in the first column of Table 5.6. A similar
relationship holds for the matrices in the second column (commodity technology).
Recall that, under commodity technology, premultiplication by C transforms the rows
from industry terms to commodity terms, i.e., e ¼ Cf.
Is Singularity Likely to be a Problem in Real-World Models?
numerical example in Table 5.3 we had
100
90
90 0
, and q ¼
, x¼
V¼
100
110
10 100
From this, we found
0
C ¼ V x^
1
¼
90
10
1=90
0
1
In the original
:0909
¼
100
10 1=100
0 :9091
90 0
1=100
0
:9 0
1
¼
D ¼ V^
q ¼
10 100
0
1=100
:1 1
0
and both C and D are non-singular.
For C to be singular, we must have |C| ¼ 0, which means |V0 | ¼ 0 (or |V| ¼ 0).
Similarly, for D to be singular, the requirement is |D| ¼ 0; this also means |V| ¼ 0. As
an example, suppose that the second column of V is the same as the first,
90 90
180
~ ¼V
~
~
~ 0 ð^~x Þ1 ¼
so V ¼
with an associated ~x ¼ Vi ¼
and C
10 10
20
:5 :5 17
~ is singular (as is the associated D
~ ¼ V^
~ q 1 , which the reader
. Clearly, C
:5 :5
can easily check), and so the total requirements matrices under commodity technology,
as expressed in Tables 5.4 and 5.5, cannot be found.
Since industry output has changed from the original example, so has B, and we now
have
1
:0667
:4
:4333
:1
~
~
~ ¼ U x^~
¼
and C B ¼
B
:0556 :35
:4444 :15
17
In a 2 2 matrix both rows and columns must be proportional for singularity. Here, for simplicity, we use the
~ unchanged at
only possible illustration that leaves commodity output (column sums of V and of V)
q0 ¼ ½ 100 100 .
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194
The Commodity-by-Industry Approach in Input–Output Models
~ B
~ is non-singular, we can find
Since C
1
7:2973
4:8649
~
~
¼
CB
21:6219 21:0811
which appears to serve as the industry-by-commodity total requirements matrix under
commodity technology, as expressed in Table 5.6.
This can be easily checked. For example, using the unchanged commodity final
demand (Table 5.3), we find that
1
180
7:2973
4:8649 80
~
~
¼ ~x
¼
CB e¼
20
21:6216 21:0811
83
exactly as would be expected (industry output required for commodity final demand).
Similarly,
1
100
:5 :5
7:2973
4:8649 80
~
~
~
¼q
C CB e¼
¼
100
:5 :5 21:6216 21:0811
83
~ C
~ B
~ 1 is the commodity-by-commodity total requirements matrix under
and C
commodity technology
(Table
1 5.6).
~
~
is that it contains negative elements. These are implausThe trouble with C B
ible; for example, an increase in final demand for commodity 1 leads to a decrease in
industry 2 output. We will explore the issue of negative elements in total requirements
matrices in more detail in Section 5.5. Here we
simply
1illustrate the problems that they
~
~
in the standard way – namely
create. For example, suppose we were to use C B
to assess the impact on industry outputs ðΔ~x Þ of some change in final demand for
316
100
,
, Δ~x ¼
commodities (Δe). As the reader can easily check, for Δe ¼
370
85
which is difficult if not impossible to interpret meaningfully. Remember that originally
180
80
180
new
, so an increase demands to e ¼
and ~
x¼
e¼
generates
20
83
168
496
x~new ¼
. As we will see in Section 5.5, negative elements are problematic
350
with commodity technology models even when C is non-singular.
In any event, how likely is it that C (or D – or V) will be singular in any real-world
90 90
~ ¼
model? Not very. In this small illustration, the implication of V
10 10 is that
ðicÞ
industry 1 produces 90 percent of the output of commodity 1 and also 90 percent of the
output of commodity 2. But if industries are named on the basis of their primary
product, there will only be one primary product per industry, and industry 1 could not
produce 90 percent of the output of commodity 2 which is, by definition, primary to
industry 2. In fact, each industry should produce more than one-half of the output of its
primary commodity, if the commodity is truly “primary” to that industry.
There are matrix algebra results that are very pertinent here. A matrix M is said to
have a dominant diagonal if
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5.3 Technology and Total Requirement Matrices
195
n X
mjj >
mjj for j ¼ 1, . . . , n; i 6¼ j
i¼1
In words, and for a matrix (like V) with non-negative elements (so the absolute value
bars are not needed), in each column the element on the main diagonal is larger than
the sum of all the other elements in that column.18 The important point is that it can be
shown that an n n matrix with a dominant diagonal is always non-singular. In the
case of a V matrix, a dominant diagonal means that more than one-half of the output of
each commodity (each column sum in V) would be made by the corresponding
industry (row) to which that commodity is primary. And, as just noted, since industries
are named for their primary commodity, diagonal dominance of V is to be expected.
This means that singularity of C (or D) may not be a problem in most real-world
commodity–industry input–output models.19
5.3.8 Commodity-by-Industry Configurations for Multiregional Models
Building on Jackson (1998), who addresses the subject of regionalizing national
commodity-by-industry accounts (discussed in more detail in Chapter 10), Jackson
and Schwarm (2011) explore the special circumstances encountered with commodityby-industry model configurations for regional, interregional (IRIO), and multiregional
(MRIO) models. For example, in a basic two-region configuration, the use and make
V1 0
U1 0
and V ¼
. Recalling that q ¼ i0 V (total
matrices would be U ¼
0 U2
0 V2
commodity output) and x ¼ Vi (total industry output), in an IRIO configuration, the
U11 U12
use table would become UIRIO ¼
, specifying the regional commodity
U21 U22
origin and industry destination of commodity use. In this case, D ¼ V^
q 1 and
BIRIO ¼ UIRIO x^1 . Alternatively, the make matrix could be specified as
V11 V12
IRIO
V
, specifying regional industry origin of production and the
¼
V21 V22
^ 1 and
regional destination of the commodity delivered. In this case, DIRIO ¼ VIRIO q
B ¼ Ux^1 . However, it is important to note that both BIRIO and DIRIO cannot be used
simultaneously since each rests on different assumptions about the distribution of
inputs across regions. Jackson and Schwarm explore additional MRIO model configurations and discuss the implications of selection among configurations, emphasizing
that the choices usually hinge on data availability.
18
There are several alternative definitions of dominant diagonal matrices, but these are not necessary for us at this
point. See, for example, Takayama (1985, chapter 4) or Lancaster (1968, chapter R7). Both of these include
discussions of related concepts, including Frobenius theorems and the notion of indecomposable matrices;
these topics are also beyond our needs here.
19
The reader might think about whether diagonal dominance will be more or less likely as the number of
commodities/industries increases. The simplicity of the two-commodity, two-industry case may be misleading.
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196
5.4
The Commodity-by-Industry Approach in Input–Output Models
Numerical Examples of Alternative Direct and Total Requirements Matrices
From the numerical example in Table 5.3, we had
:1333 :0727
1 :0909
:9 0
B¼
C¼
D¼
:1111 :0636
0 :9091
:1 1
1
C
1
¼
0
:1
1:1111 0
1
D ¼
1:1
:1111 1
(Notice the negative element in C1 .) We collect together the associated direct and
total input requirements matrices in this section.
5.4.1
Direct Requirements Matrices
:1273 :0727
:1200 :0655
AI ¼ DB ¼
AI ¼ BD ¼
:1064 :0636
:1244 :0709
ðccÞ
ðiiÞ
:1333 :0667
:1222 :0664
1
1
AC ¼ BC ¼
AC ¼ C B ¼
:1111 :0589
:1222 :0700
ðccÞ
ðiiÞ
5.4.2
Total Requirements Matrices
Commodity-Demand-Driven Models
Industry Technology
Commodity Technology
Commodity-by-Commodity
1
1:1568 :0898
1:1644 :0825
I BC1
ðI BDÞ1 ¼
¼
:1314 1:0782
:1375
1:0723
Industry-by-Commodity
1:0411
:0809
1:1507 :0247
1
1
1 1
DðI BDÞ ¼
C I BC
¼
:2471 1:0871
:1512 1:1795
Industry-Demand-Driven Models
Industry Technology
1
ðI DBÞ
Commodity Technology
Industry-by-Industry
1
1:1478 :0809
1:1507 :0821
1
¼
IC B
¼
:1537 1:0871
:1512 1:0861
Commodity-by-Industry
1
1:2753
:0898
1:1644 :1808
1
1
1
D ðI DBÞ ¼
C IC B
¼
:0262 1:0782
:1375 :9873
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197
5.5 Negative Elements in the Commodity–Industry Framework
Notice that a negative element appears in one of these total requirements matrices. This
reflects the negative element in C1 . (In fact, C1 appears in the other three commodity technology total requirements matrices also, but the influence of the negative
element is mitigated in the products BC1 and C1 B.) We will look into negative
elements in commodity–industry models in some detail in Section 5.5. Lenzen and
Rueda-Cantuche (2012) explore the implications of these alternative configurations
applied to their use in impact analysis.
5.5
Negative Elements in the Commodity–Industry Framework
In the original input–output model one does not expect to find negative elements, either
in an interindustry transactions matrix (Z) or in a total outputs vector (x). This means
that there will not be any negative elements in the technical coefficients matrix (A) or
in the Leontief inverse.20 However, the commodity–industry format, designed to
improve on the original Leontief framework in accounting for secondary products,
introduces a new problem of its own – the possibility of negatives.
5.5.1
Commodity Technology
Direct Requirements Matrices
0 1
UðV Þ
Consider the structure of AC ¼ BC1 ¼
ðccÞ
for the general 2 2 case:
1
u11 u12 v11 v21
u11 u12
v22
¼ ð1=jVjÞ
AC ¼
u
u
v
v
u
u
v
21
22
12
22
21
22
12
ðccÞ
u v u12 v12 u11 v21 þ u12 v11
¼ ð1=jVjÞ 11 22
u21 v22 u22 v12 u21 v21 þ u22 v11
v21
v11
If V is a dominant diagonal matrix, as it is expected to be, then
jVj ¼ v11 v22 v12 v21 > 0, and the signs of the elements in AC will depend on the
ðccÞ
relative sizes of the uij and vij .
As an illustration, suppose that U is as shown in Table 5.3 and that all elements in V
remain the same except for v21. For what values of v21 would at least one element in
AC be negative? (Notice that v21 only appears in the second column of AC .) This
ðccÞ
ðccÞ
means, at what value of v21 would v21 become larger than either ðu12 v11 =u11 Þ or
ðu22 v11 =u21 Þ? In this case, we have v21 > 60 or v21 > 63, respectively, so the point
at which ð aC Þ12 becomes negative is when v21 > 60 and ð aC Þ22 becomes negative
ðccÞ
ðccÞ
when v21 > 63. For example, as the reader can easily check, if v21 ¼ 60,
20
Extensions of the original framework could accommodate negative elements, as in a pollution-generation
model in which a negative zij might indicate the amount of pollutant i generated in conjunction with production
activity in industry j. The associated aij would also be negative (amount of pollutant i released per unit of
industry j output).
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The Commodity-by-Industry Approach in Input–Output Models
1
12 8 90 60
AC ¼ UðV Þ ¼
10 7
0 100
ðccÞ
:1333
0
12 8 :0111 :0067
¼
¼
:1111 :0033
10 7
0
:01
0 1
:1333 :0013
while
for
v21 ¼ 61,
A ¼
and
for
v21 ¼ 64,
ðccCÞ
:1111 :0022
:1333 :0053
AC ¼
. So AC exhibits a kind of unsatisfactory instability; there
:1111 :0011
ðccÞ
ðccÞ
is no obvious reason why v21 ¼ 60 is any more economically plausible than v21 ¼ 61,
yet this one-unit variation means the difference between a reasonable direct requirements matrix and a much less reasonable one. For example, the implication of
:1333 :0013
AC ¼
is that production of commodity 2 releases rather than
:1111 :0022
ðccÞ
consumes an amount of commodity 1, even though, as shown in U, industry 2
consumes positive amounts of that commodity as a production input. And
:1333 :0053
AC ¼
, when v21 ¼ 64, is even more implausible.
:1111 :0011
ðccÞ
1
From AC ¼ UðV0 Þ and the basic definition of the inverse of a 2 2 matrix –
ðccÞ
1
in this case ðV0 Þ ¼ ð1=jV, Þ½adjðV0 Þ – we recognize that negative elements in AC
ðccÞ
mean that at least one of the off-diagonal elements in V0 will be negative.21 As the
1
examples so far in this section illustrate, a negative element in ðV0 Þ may or may not
translate into one or more negative elements in AC – for v21 ¼ 60 it does not but for
ðccÞ
v21 ¼ 61 and larger it does.
It is worth carefully examining the operations involved in AC for this small 2 2
ðccÞ
u11 v22 u12 v12 u11 v21 þ u12 v11
case where AC ¼ ð1=jVjÞ
. To simplify the exposu21 v22 u22 v12 u21 v21 þ u22 v11
ðccÞ
ition, suppose v12 ¼ 0; that is, industry 1 produces commodity 1 only, whereas industry
2 produces some of both commodities. In this case, as the reader can check, AC
ðccÞ
becomes
½u12 =v22 ðu11 =v11 Þðv21 =v22 Þ
u =v
AC ¼ 11 11
u21 =v11 ½u22 =v22 ðu21 =v11 Þðv21 =v22 Þ
ðccÞ
Consider the element that measures commodity 1 input per unit of commodity 2
output – ð aC Þ12 ¼ u12 =v22 ðu11 =v11 Þðv21 =v22 Þ. First of all, u12 =v22 normalizes the
ðccÞ
input of commodity 1 to industry 2, u12 , as if all output of industry 2 were in the form
of commodity 2. But some u12 went to industry 2 for production there of commodity 1.
21
Of course if V is diagonal, there will be no off-diagonal elements in ðV0 Þ1 . A diagonal V means that all
production in the economy is primary, none secondary, and there is no need for the entire commodity–industry
apparatus (see Section 5.6). These observations strictly hold only for a two-commodity/two-industry example.
A more general argument is needed for the case of m commodities and n industries, where m ¼ n > 2 and where
both jV0 j and [adj (V0 )] have more complicated structures.
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5.5 Negative Elements in the Commodity–Industry Framework
199
Under the commodity technology assumption, the recipe for commodity 1 production
is the same in both industries, and from the first column in AC we know that ðu11 =v11 Þ
ðccÞ
represents commodity 1 input per unit of commodity 1 output, wherever produced.
From the second row of V, we know that industry 2 made v21 units of commodity
1 while also producing v22 units of commodity 2 – so ðv21 =v22 Þ represents commodity
1 production in industry 2 per unit of commodity 2 production there. Hence the per
unit recipe for commodity 1 times the number of units ðu11 =v11 Þðv21 =v22 Þ must be
netted out of u12 =v22 to account for the fact that industry 2 used u12 to make both
commodity 2 and commodity 1. And what we want in ð aC Þ12 is just that part of
ðccÞ
commodity 1 input that was used for commodity 2 production. A similar argument
holds for ð aC Þ22 .22 From column 2 of AC it is clear that if the negative term in
ðccÞ
ðccÞ
either element exceeds the positive term, a negative coefficient will result.
Transactions Matrices In the original input–output model, the underlying
interindustry transactions matrix is retrieved from A and x as Z ¼ Ax^ [for example,
from (5.3)]. Similarly, an intercommodity transactions matrix (commodity inputs to
support commodity outputs) in a commodity technology model can be derived; denote
it by ZC . In Section 5.3 we saw q ¼ BC1 q þ e, in which BC1 serves as a direct
ðccÞ
^ ¼ BC1 q
^ . The implication of a
inputs matrix – AC ¼ BC1 . Then ZC ¼ AC q
ðccÞ
ðccÞ
ðccÞ
negative element in AC is that the underlying transaction is negative, and this is
ðccÞ
generally viewed as implausible. Since AC is postmultiplied by a (positive) diagonal
ðccÞ
matrix, any negative element in AC will immediately translate into a negative element
ðccÞ
in the corresponding location in ZC .
ðccÞ
90 0
, we found
For the modified example in which v21 ¼ 64, so that V ¼
64 100
:1333 :0053
AC ¼
. In this case, the associated vector of commodity outputs is
:1111 :0011
ðccÞ
20:53 :53
90
^¼
q¼
, and so the resulting transactions matrix is ZC ¼ AC q
17:11 :11
164
ðccÞ ðccÞ
with negative flows exactly where expected.
Using the definitions of B, C, and D in terms of U, V, q, and x, along with matrix
algebra facts on transposes and inverses of products and of diagonal matrices, it is easy
1
to show also that ZC ¼ UðD0 Þ .23 In this form, the original commodity-to-industry
ðccÞ
transactions matrix, U, is converted to a commodity-to-commodity transactions matrix
via postmultiplication by a “conversion” matrix. We examine the notion of generating
22
23
If both v12 6¼ 0 and v21 6¼ 0, the economic logic behind the more complicated expressions that will make up
AC is much more difficult to sort out. And for cases larger than 2 2 it is a lot worse.
ðccÞ
The steps from BC1 to UðD0 Þ1 are purely algebraic. Readers who are interested in this kind of matrix algebra
should work through the details.
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The Commodity-by-Industry Approach in Input–Output Models
ZC or Z via modifications of U under commodity technology ( ZC or ZC ) or
ðiiÞ
ðciÞ
ðccÞ
ðccÞ
ðiiÞ
industry technology ( ZI or ZI ) assumptions in Appendix 5.1. In Appendix 5.2,
ðccÞ
ðiiÞ
building on a result in Appendix 5.1, we explore an approach to eliminating negative
elements if they should appear in ZC (as in Almon, 2000).
ðccÞ
From (5.25) in Section 5.3, we saw that C1 B plays the role of an industry-toindustry direct inputs matrix in the commodity technology model, and therefore the
underlying transactions matrix is ZC ¼ AC x^ ¼ C1 Bx^ ¼ C1 U. Again, any
ðiiÞ
ðiiÞ
(implausible) negative element in AC must reflect a corresponding (equally implausðiiÞ
ible) negative transaction. Since
AC ¼ C
ðiiÞ
1
1
B ¼ x^ðV0 Þ Ux^1
1
the influence on the direct requirements matrix of negative elements in ðV0 Þ is a little
less straightforward than in the case of AC . However, using just a bit more algebra,
ðccÞ
one can find for this example with v12 ¼ 0 that the possible negative elements will be
located in the top row of AC , and that they occur if (1) v21 > u11 v22 =u21 or (2)
ðiiÞ
v21 > u12 v22 =u22 . These work out to be (1) v21 > 120 and (2) v21 > 114:3, respectively. In terms of our example, either of these larger values for v21 is highly improbable
because each of them exceeds v11 ¼ 90 – and we expect diagonal dominance in V.24
In particular, using the v21 ¼ 64 case, introduced in the last paragraph, and the
154
associated new industry output vector x ¼
(column sums of V), we find
100
:0622 :0215
AC ¼
, with no negative elements.
:1822 :0700
ðiiÞ
Total Requirements Matrices Negative elements also appear in total requirements matrices. With AC , as calculated in the last section, when v21 ¼ 60, one of the
ðccÞ
total requirements matrices in the commodity-demand driven model (Table 5.4) contains a negative element:
1:1538
0
1:0767 :6020
1 1
1
1 1
and C I BC
I BC
¼
¼
:1286 1:0033
:2058 1:6054
However, when v21 ¼ 61, there are negative elements in both matrices
1:1536 :0015
1:0753 :6128
1 1
1
1 1
and C I BC
¼
¼
I BC
:1285 1:0021
:2068 1:6133
The same is true for v21 ¼ 64. 1
The first of these matrices, I BC1 , connects commodity final demands to
commodity outputs, so the negative element in the v21 ¼ 61 case means that an
24
These results are of course completely dependent on the specific values in U and V in our small
numerical example.
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5.5 Negative Elements in the Commodity–Industry Framework
201
increase in final demand for commodity 2 generates
a decrease in the output of
1
1 1
commodity 1. The second matrix, C I BC
, connects commodity final
demands to industry outputs, and so increases in final demand for commodity 2 create
a decrease in the output of industry 1. By contrast, as the reader can easily verify, both
total requirements matrices in the industry-demand driven model (Table 5.5)
1
1
and C I C1 B are non-negative under any of the assumptions
I C1 B
about v21 . As mentioned, however, the commodity-demand driven model is generally
the one of interest, since demands for commodities are usually preferred as the
exogenous stimuli in models built on commodity–industry data sets.
5.5.2
Industry Technology
Direct Requirements Matrices In contrast to the situation under commodity
technology, the direct requirements matrix under industry technology –
AI ¼ BD ¼ Ux^1 V^
q 1 , as in (5.27) – can never contain negative elements (as long
ðccÞ
as there are none in U and V) since inversion of the diagonal matrices of (positive)
industry and commodity outputs will never generate negative elements. For example,
under the assumption that v21 ¼ 64 – when both elements in the second column of
AC turn out to be negative – there are no negative elements in AI :
ðccÞ
AI ¼ Ux^
ðccÞ
1
V^
q
1
ðccÞ
¼ BD
1=90
0
90 0
1=154
0
0
1=164 64 100
0
1=100
:0982 :0488
:1333 :0485 :5844 0
¼
¼
:0827 :0427
:1111 :0424 :4156 1
¼
12 8
10 7
Similarly, AI ¼ DB ¼ V^
q 1 Ux^1 , and negative elements will never be present.
ði1Þ
Again, for the example with v21 ¼ 64,
:0779 :0283
AI ¼ DB ¼
:1665 :0626
ðiiÞ
Under industry technology we will never have to deal with the problem of possible
negative transactions – either in ZI or ZI .
ðccÞ
ðiiÞ
Total Requirements Matrices The fact that direct requirements matrices will
be non-negative under industry technology assures that all but one of the associated
total requirements matrices will also be non-negative. Continuing with the data from
the example with v21 ¼ 64, these matrices are
1:1139 :0564
:6510 :0330
1
1
DðI BDÞ ¼
ðI BDÞ ¼
:0960 1:0726
:5590 1:0726
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The Commodity-by-Industry Approach in Input–Output Models
ðI DBÞ1 ¼
1:0905 :0330
:1937 1:0726
D1 ðI BDÞ1 ¼
1:8659 :0564
:5817 1:0492
And, despite the positivity of the direct requirements matrices, we find that
D1 ðI DBÞ1 contains a negative element, reflecting the influence of the negative
element in D1 . It is worth noting, however, that D1 ðI DBÞ1 is the only total
requirements matrix under industry technology that might ever contain negative elements
and, as noted in Section 5.3.7, this is the least interesting or important of these matrices.
5.5.3
Making a Model Choice
Which Model to Choose? There is a large literature discussing the merits
and drawbacks of various models in a commodity–industry framework. For example,
ten Raa, Chakraborty, and Small (1984) rule out an industry technology model in favor
of commodity technology. Later, ten Raa (1988) rejects the commodity technology
model [and hence also the mixed technology model (Section 5.7)], leaving “frustration.” Then Kop Jansen and ten Raa (1990) examine the alternative technical coefficients matrices that are created from Make and Use matrices – A(U, V) – under the
1
assumptions of commodity technology AðU; VÞ ¼ UðV0 Þ , as in (5.26)], industry
1
1
technology ½AðU; VÞ ¼ UhV0 ii VhV0 ii , as in (5.27), and recalling that Vi ¼ x and
V0 i ¼ q] and various mixed technologies. They evaluate these against the backdrop of
a set of four “desirable properties”: Material Balance [x ¼ Ax þ y becomes
AðU; VÞV0 i ¼ Ui], Financial Balance [revenues ¼ costs for each sector, expressed as
i0 AðU; VÞV0 ¼ i0 U], Price Invariance [new relative prices, p > 0, imply new values in
^ U and V^
the Use and Make matrices, p
p , and the resulting coefficients matrix should be
1
^ U, V^
^ AðU; VÞ^
Aðp
pÞ ¼ p
p ], and Scale Invariance [multiplying all inputs and outputs
of each sector, i, by a constant, si , should leave the coefficients unchanged, so, for
s > 0, AðU^s , ^s VÞ ¼ AðU; VÞ]. Only the commodity technology model satisfies all four
criteria (among seven different models examined).
In contrast, the Eurostat Manual (Eurostat/European Commission, 2008), which
sets out recommended standards for data collection for member countries of the
European Union, supports an industry–technology model, for example, as in
1 1 1 q
q
[in (5.27) ] or AI ¼ DB ¼ V^
Ux^ . The
AI ¼ BD ¼ Ux^1 V^
ðccÞ
ðiiÞ
Manual recommends a different classification scheme that reflects observations like
those made by Konijn and Steenge (1995, pp. 34–35):
It can relatively easily be understood that these technology assumptions [commodity technology and
industry technology] are not used to construct industry-by-industry tables. In constructing industryby-industry tables, assumptions are made on the origins and destinations of products [commodities]
and not on the technology of production. Hence, we find the traditional presentation of methods . . . to
be incorrect.
This leads to an alternative presentation, adapted from Eurostat/European Commission
(2008, figure 11.3, p. 310) and summarized for reference in Table 5.7.
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5.5 Negative Elements in the Commodity–Industry Framework
203
Table 5.7 Alternative classifications, total requirements matrices, commodity-demand driven models
Commodity
Technology
Commodity-byCommodity
(I BC1 )
Model A
Industry
Technology
1
Fixed Industry
Sales Structure
Fixed Commodity
Sales Structure
ðI BDÞ1
Model B
Industry-byIndustry
1
(I C1 B) C1
Model C
ðI DBÞ1 D
Model D
Model A: Each commodity is produced in its own specific way, irrespective of the industry where it is
produced. Negative elements may occur.
Model B: Each industry has its own specific way of production, irrespective of its product mix. No
negative elements.
Model C: Each industry has its own specific sales structure, irrespective of its product mix. Negative
elements may occur.
Model D: Each product has its own specific sales structure, irrespective of the industry where it is
produced. No negative elements.
Eurostat then makes the case for Model D (p. 310):
Industry-by-industry tables which are based on the fixed product sales structure (Model D) do not
involve any technology assumptions (A and B), and do not require the application of sometimes
arbitrary methods to adjust for negatives (A and C).
The Eurostat Manual contains a wealth of numerical examples illustrating the
consequences of alternative assumptions. It emphasizes the issues surrounding the
compilation of symmetric input–output tables (SIOTs) – meaning tables with dimensions commodity-by-commodity or industry-by-industry – from the data in Supply and
Use tables (SUTs). The interested reader is referred to the Manual or to Thage (2002,
2005) or Thage and ten Raa (2006) for details. There are also useful numerical
examples in the Manual and in Thage (2005). The issue is primarily with the negatives
that can appear in the commodity technology model, which we turn to next.
Dealing with Negative Values Researchers working with real-world input–
output data repeatedly find that the commodity technology model generates negative direct
input coefficients and transactions (frequently relatively small). For example: “There are
numerous examples of the [commodity technology] method leading to negative coefficients
which are clearly nonsensical from an economic point of view” (United Nations et al., 1993,
section 15.147; quoted in Almon, 2000, p. 28). Table 5.8 provides a few examples.
These kinds of “nonsensical” results have generated two reactions. One is to abandon the
commodity technology model entirely – for example, de Mesnard (2004)25 or Eurostat/
European Commission (2008), along with the references by Thage, noted at the beginning
of this section. Others find that rejection of the commodity–technology model is much too
25
de Mesnard (2004) argues against any version of a model in which C1 (with its negative elements) appears.
He does so by viewing various models in terms of economic circuits (directed impulses). The interested reader
is referred to the article for details.
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204
The Commodity-by-Industry Approach in Input–Output Models
Table 5.8 Examples of negative elements in real-world commodity–technology direct requirements
matrices [ AC ¼ BC1 ]
ðccÞ
Negative as
Percentage
of Total
Reference
Dimensions
Country, Year
ten Raa, Chakraborty,
and Small (1984)
ten Raa and van der
Ploeg (1989)
Rainer (1989)
43 43
Canada, 1977
Number of Negative
Elements
10a in ðI ACB Þ1
39 39
UK, 1975
22b
175 175
Austria, 1976
Steenge (1989)
Steenge (1990)
79 79
14 14
USA, 1977
USA, 1977
Negative elements accounted for 1.4
percent of total value of intermediate
flows in c c model
1.9
116b
3.6
7b
a
b
0.5
1.4
Criterion for selection: > |0.03|.
Criterion for selection: |0.001|.
harsh a judgment (for example, Rainer and Richter, 1992). The other reaction is to propose
“adjustments” to commodity–technology models to avoid negative elements and to deal
with such elements when they occur in practice. Examples of additional literature dealing
with both of these reactions, and others, include (in approximately chronological order):
van Rijckeghem (1967), Stahmer (1985), Rainer (1989), Rainer and Richter (1989),
Steenge (1989, 1990), ten Raa and van der Ploeg (1989), Londero (1990, 1999, 2001),
ten Raa (1995, 2005), Mattey and ten Raa (1997), Almon (2000), and ten Raa and RuedaCantuche (2003, 2007). Clearly this continues to be very much an open question.
When the negative elements are relatively small, they have sometimes simply been
changed to zeros [e.g., in work at the Cambridge (UK) Growth Project, under Stone’s
direction, cited in Armstrong (1975)]. One problem with this approach is deciding
what constitutes a “relatively small” element. Alternatively, in some studies a negative
element has been replaced by a small positive element, with “compensating adjustments in other entries in the matrix so that the overall row and column accounting
constraints were still met” (Armstrong, 1975, p. 80). The “compensating adjustments”
will be somewhat ad hoc, and different researchers might make differing sets of
adjustments. In Appendix 5.2, as an illustration of one approach to dealing with
negatives in commodity-based technology, we investigate a procedure that has been
used successfully in many INFORUM studies for decades (Almon, 2000).
5.6
Non-square Commodity–Industry Systems
If the number of commodities in the input–output accounts is not the same as the number
of industries, then various matrices in the commodity–industry modeling system will be
“rectangular” rather than square.26 In this section we explore some of the implications of
this m 6¼ n possibility. In principle, this can mean either m > n or m < n.
26
Strictly speaking, a square is a rectangle all of whose sides are the same length. In general, however, in the
commodity-industry literature, “rectangular” is used to indicate a “non-square” system in which the number of
commodities does not equal the number of industries.
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205
5.6 Nonsquare Commodity–Industry Systems
Table 5.9 A three-commodity, two-industry example
Commodities
1
Commodities
2
Industries
3
1
2
3
1
2
Final Demand
Total Output
18
20
2
12
16
6
70
40
7
e
100
76
15
q
U
Industries
1
2
90
10
10
66
V
8
7
Value Added
108
83
x
68
Total Output
100
76
q0
15
108
v0
x0
49
117
83
We consider these in turn. The case of more commodities than industries (m > n) is
seen often in real-world input–output accounts. Less usual is the case of fewer
commodities than industries (m < n), although it can sometimes be found in disaggregated versions of data sets – that is, at the data-collection phase, for instance, when
“dummy” industries are used to account for such products as scrap, used/second-hand
goods, or import duties. Since it is general practice to aggregate to m n levels before
implementing an input–output model, we will consider only the m > n case in this
section. In Appendix SA5.3, brief attention is given to the much less important
m < n situation.
As an illustration, let m ¼ 3 (commodities) and n ¼ 2 (industries). The dimensions of
the matrices that are the building blocks of the commodity–industry model are
U , V , x , and q . This leads to
ð32Þ ð23Þ ð21Þ
ð31Þ
^ 1
B ¼ U x^1 , C ¼ V0 x^1 , and D ¼ V q
ð32Þ
ð32Þ ð22Þ
ð32Þ
ð32Þ ð22Þ
ð23Þ
ð23Þ ð33Þ
Later in this section we will use the following three-commodity, two-industry data for
illustration (Table 5.9). In this case, one can easily find that
2
3
2
3
:1667 :1446
:8333 :1205
:9 :1316 :5333
4
5
4
5
B ¼ :1852 :1928 , C ¼ :0926 :7952 and D ¼
:1 :8684 :4667
:0185 :1723
:0741 :0843
5.6.1 Commodity Technology
Under commodity technology, there is an immediate problem when trying to convert x
into a function of q on the right-hand side in (5.13). Recall that the sequence goes from
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The Commodity-by-Industry Approach in Input–Output Models
Cx ¼ q [in (5.18)] to x ¼ C1 q [in (5.19)], in order to convert q ¼ Bx þ e [in (5.13)]
into q ¼ BC1 q þ e, from which we had AC ¼ BC1 . Also, again using (5.13), but
ðccÞ
now premultiplying both sides by C1, we had x ¼ C1 q ¼ C1 Bx þ C1 e, and so
AC ¼ C1 B. Clearly, these operations are in trouble without a well-defined C1 . We
ðiiÞ
explore in online Appendix SA5.3 (for the reader interested in mathematical detail)
why it is that the mathematical notions of inverses for rectangular matrices are
of no help here. The conclusion is that commodity technology models cannot
generate a plausible direct requirements matrix in the case of more commodities than
industries. Consequently, total requirements matrices also cannot be found. The usual
solution is to aggregate commodities (and perhaps also industries) until some level at
which m = n.
Notice that Approach II in Section 5.3.7 (avoiding direct requirements matrices
completely) is of no help here. In particular, since B and C are both m n (here 3 2)
matrices, the inverse that is required in (5.28) – ðC BÞ1 – is just as problematic as
C1 alone.
5.6.2
Industry Technology
Direct Requirements Matrices A rectangular format presents no problems
under industry technology. Substitution of (5.16) – Dq ¼ x – into (5.13) –q ¼ Bx þ e –
is straightforward and requires no inverse; q ¼ BDq þ e. Using B and D from
2
3
:1645 :1475 :1564
Table 5.9, AI ¼ B D ¼ 4 :1859 :1918 :1887 5 [as in (5.27)] has the correct
ð32Þ ð23Þ
ðccÞ
:0239 :0652 :0436
:1842 :1940
3 3 dimensions (commodity-by-commodity) and AI ¼ D B ¼
:1861 :2156
ð23Þð32Þ
ðiiÞ
is, appropriately, a 2 2 matrix relating industry inputs to industry outputs.
Total Requirements Matrices As to total requirements matrices, we know
from Tables 5.4 and 5.5 and the discussion in Section 5.5 that the only problem under
industry technology occurs with the commodity-by-industry matrix in Table 5.5, where
D1 plays a role – in D1 ðI DBÞ1 . The problem is exactly the same as in the case
when D is singular. It occurs when trying to move from (5.16), Dq ¼ x, to (5.17),
q ¼ D1 x, only now it is because D is rectangular, not because of singularity. We
explore this problem briefly also in online Appendix SA5.3.
The remaining total requirements matrices, from Tables 5.4 and 5.5, are unhampered by
a rectangular format. The reader can check that for the data in Table 5.9 these are
2
3
1:2599 :2505 :2554
ðI BDÞ1 ¼ 4 :3020 1:3173 :3093 5
:0521 :0961 1:0731
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5.7 Mixed Technology in the Commodity-Industry Framework
DðI BDÞ1 ¼
1
ðI DBÞ
5.7
1:2014 :4500 :8429
:4126 1:2139 :7949
1:2992
¼
:3083
:3214
1:3511
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Mixed Technology in the Commodity–Industry Framework
At even a very detailed level of industry and commodity disaggregation, there are
certain to be many industries that produce more than one commodity. For example,
Danish annual tables (1966–1998) are based on supply and use tables with about 2,750
commodities and 130 industries, and annual tables for the Netherlands start from data
on 800 commodities and 250 industries (Thage, 2002, pp. 3 and 13).
In the USA, correspondences between input–output commodities/industries at the
six-digit level and the four-digit 1987 US Standard Industrial Classification Code
(SIC)27 indicate under SIC 2211, “Broadwoven fabric mills, cotton,” almost four
pages (single spaced) of some 147 individual commodities, among them “Sheets and
sheetings, cotton-mitse (manufactured in the same establishment),” along with such
diverse items as diaper fabrics, mosquito netting, and typewriter ribbon cloth.28 The
output of SIC 2211 (along with four other six-digit SIC industries) is classified under
I-O industry 16.0100, “Broadwoven fabric mills and fabric finishing plants.” On the
other hand, under SIC 2392 “House furnishing, except curtains and draperies,” there
are 43 commodities, including “Sheets, fabric-mfpm (manufactured from purchased
materials),” along with such products as boat cushions, dust cloths, and shoe bags. The
output of SIC 2392 is counted under I-O industry 19.0200, “House furnishings, except
curtains and draperies.”
Of course, as data are aggregated into fewer commodity and industry classifications,
all of these diverse products get lumped together, making for even more heterogeneity
in an industry’s output – for example, SIC 2392 is combined with SIC 2391,
2393–2397, and 2399 into three-digit SIC industry 239, “Miscellaneous fabricated
textile products,” which, in turn, is part of two-digit SIC industry 23 “Apparel and
other finished products made from fabrics and similar materials.”
There is an extensive and often contradictory literature on alternative definitions for
classifying, among others, secondary products, subsidiary products, joint products,
and by-products. Generally, the “principal” (or “primary”) output of a multiproduct
27
This has since been redefined several times since the North American Industrial Classification System (NAICS)
was adopted in 1997 (replacing the Standard Industrial Classification [SIC] system, but the principles remain
the same). Correspondences between the current 2017 NAICS and earlier installments are shown at www
.census.gov/eos/www/naics.
28
It is very informative to investigate the contents of various US SIC “industries.” This is easily done at the
website identified in the previous footnote and also at www.osha.gov/oshastats/sicser.html.
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The Commodity-by-Industry Approach in Input–Output Models
production process is the one that accounts for the maximum value of production
(sometimes maximum value added is used); remaining outputs (if any) are classified as
“secondary.” Primary and secondary together are sometimes called “joint products,”
and a “by-product” is then sometimes defined as a joint product that is of “distinctly
lesser importance . . .” than the other joint product(s) (United Nations, 1966, section
2.60, as quoted in Londero, 2001, p. 39).29 These distinctions can play a role in helping
to establish whether a commodity or an industry technology model is more appropriate
for a particular secondary product.
A variation on these classifications identifies several kinds of secondary products in
the following way (for example, Bulmer-Thomas, 1982, chapter 9).
1. A product whose output level is independent of the level of primary production in
an establishment where it is produced and which is:
a. produced according to the technology used in the industry that produces it as a
primary product. Examples: computer hard drives made by a computer manufacturer (e.g., IBM) according to the production “recipe” used by other computer
hard drive manufacturers and sold to other computer manufacturers; cotton
sheets that are made in a cotton mill (as noted in the examples introducing this
section). Such secondary products would be logical candidates for the commodity technology assumption.
b. produced according to the technology used in the industry where it is produced
as a secondary product. Example: computer services developed by an aircraft
establishment (e.g., Boeing, for aircraft design) then marketed as a secondary
product. This would appropriately be treated under an industry technology
assumption.
2. A product whose output level is not independent of that of the primary product in an
establishment and for which:
a. there is another industry that makes it as a primary product. This is classified as a
“by-product.” Example: ethylene generated during petroleum refining (but also
produced in “natural gas” plants). This kind of secondary product appears not to
conform well to either technology assumption.
b. there is not another industry making it as a primary product. This is classified as
a joint product. Examples: wool produced in conjunction with sheep ranching;
hides from cattle raising; radioactive waste generated in producing electrical
power at a nuclear plant; ash, generated by coal burning electric power plants,
that is used as a hardening agent in some types of road surfaces (e.g., airport
runways). Here it is at least clear that a commodity technology model is
not appropriate.
In any event, the key issue is how to “assign” each of the various secondary products to
a particular production technology in a commodity–industry input–output system. In
29
This reference also includes some discussion of the confusing and contradictory language employed in various
publications – including several from the United Nations.
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5.7 Mixed Technology in the Commodity-Industry Framework
209
practice, there are about as many approaches as there are real-world accounts and
models.30
In years prior to the adoption of commodity–industry accounting systems, input–
output modelers recognized this obvious fact of secondary production in many industries (“sectors” in pre-commodity–industry days) and dealt with it in a number of ways.
Suppose that industry j produces not only commodity j (its primary product) but also
some of commodity k (which is primary to industry k), as a secondary product. One
general approach was to subtract from the value of industry j’s total output the amount
that represents the value of commodity k production and add this amount to the value
of output of industry k. Do the same for inputs; subtract from industry j’s total input
vector those inputs that were used for commodity k production, and then add these to
industry k’s inputs column. As described in Chapter 4, this is known as redefinition (or,
for reasons that we will see in a minute, specific redefinition). Usually, this industry
redefinition is easier said than done, especially as regards inputs. An alternative
approach created transfers of secondary production – in input–output parlance this
meant that industry j’s output of commodity k was treated as if sold by industry j to
industry k and added to industry k’s total output.31
The commodity–industry accounting approach introduces a wider range of options.
It generates what has been called a mechanical redefinition of secondary production –
through use of the commodity technology assumption (for example, as in
AC ¼ BC1 ) or the industry technology assumption (as in AI ¼ BD). And these
ðccÞ
ðccÞ
two technology approaches can be used in a variety of ways. For example, in the US
models, as originally published starting with 1972, secondary products that are “obviously” [however defined; for example, as in (1)(a)] produced under commodity
technology are specifically redefined to their primary industries and for the remainder
the mechanical redefinition of an industry technology model is employed.32
Another refinement that is available with commodity–industry accounts is to use
both commodity technology and industry technology mechanical redefinitions in the
same model, thereby bypassing an exclusively commodity technology model or an
exclusively industry technology model. This is accomplished in what are known as
“mixed-technology” or “hybrid” models.
The essential idea is to divide the Make matrix, V, into two components, so that
V ¼ V1 þ V2
30
(5.32)
For a comprehensive account of the US procedures in the early commodity-industry years, see Ritz (1980).
Rainer and Richter (1992) examine aspects of the Austrian experience. For an overview in a number of other
countries, see Franz and Rainer (1989), and for a comprehensive classification of approaches see ten Raa and
Rueda-Cantuche (2003, esp. table 2).
31
Fukui and Seneta (1985) examine and classify four “conventional” methods for dealing with joint products as
of the mid-1980s.
32
In speaking of the adoption of the commodity–industry approach for the US, starting with the 1972 table, Ritz
writes: “The use of the mechanical redefinition for all secondary products other than those which have been
specifically redefined is a substantial improvement over the transfer treatment used in earlier I-O studies” (Ritz,
1980, p. 51).
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The Commodity-by-Industry Approach in Input–Output Models
where V1 records the making of commodities that are best identified with one of the
two technology assumptions and V2 records the making of commodities associated
with the other technology assumption.33 We can define vectors of total industry outputs
under the two technology assumptions as
x1 ¼ V1 i
and x2 ¼ V2 i
(5.33)
(where, as usual, i is a column vector of l’s). Total industry output then is just
x ¼ x1 þ x2 ¼ ðV1 þ V2 Þi
Similarly, total commodity outputs are identified as
q1 ¼ V01 i and q2 ¼ V02 i
so that
(5.34)
q ¼ q1 þ q2 ¼ V01 þ V02 i
5.7.1 Commodity Technology in V1
We illustrate the general idea by attaching commodity technology to production
recorded in V1 and industry technology to V2 . (This is an arbitrary assignment; we
could equally well decide to reflect industry technology in V1 and commodity technology in V2 . This is explored in the next section.) Using V1 , define
C1 ¼ V01 ðx^1 Þ1
(5.35)
This creates a commodity-by-industry matrix whose i, jth element records the proportion of industry j’s (commodity technology) output that is in the form of commodity
0 i.
This is identical in spirit to the definition of C in (5.15). From (5.35), C1 x^1 i ¼ V1 i
and using (5.34), C1 x1 ¼ q1 , or
x1 ¼ C1
1 q1
(5.36)
assuming that C1 is non-singular. This provides a transformation between x1 and q1 .
Note that, without the subscripts, this is completely parallel to the connection between
x and q given by C in the pure commodity technology case, in (5.18) and (5.19).
To account for the industry technology character of the outputs recorded in V2 ,
define the industry-by-commodity matrix D2 as
^ 1
D2 ¼ V2 q
(5.37)
Notice that the “normalization” of V2 is done using total commodity outputs, not just
those recorded in V2 . This is similar to the definition of D, in (5.14), in the pure industry
33
Detailed derivations, examples, variations, and discussions can be found in, among others, United Nations
(1968), Aidenoff (1970), Gigantes (1970), Cressy (1976), Armstrong (1975), and ten Raa, Chakraborty, and
Small (1984).
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5.7 Mixed Technology in the Commodity-Industry Framework
211
technology case.34 The i, jth element in D2 identifies that fraction of all commodity j
production that is made by industry i under an industry technology assumption. Since
x2 ¼ V2 i [as in (5.33)], a line of argument similar to that used at the beginning of this
section for C1 leads to a transformation between x2 and q (not q2 ); namely
x2 ¼ D2 q
(5.38)
If we put the two pieces of x together, from (5.36) and (5.38), we have
x ¼ x1 þ x2 ¼ C1
1 q1 þ D 2 q
Our interest is in the transformation between all of x and all of q; that means that we
need to replace q1 on the right of C1
1 q1 with some function of q.
To do this, weexamine
the components of q in detail. As above, q ¼ q1 þ q2 . We
^ 1 , so that q2 ¼ ðD2 q
^ Þ0 i. From matrix algebra
know that q2 ¼ V02 i and D2 ¼ V2 q
rules on transposes of products of appropriately dimensioned matrices –
ðMNÞ0 ¼ N0 M0 – and on the product of two diagonal matrices (order of multiplication
makes no difference), it follows that35
^ Þ0 i ¼ D02 i q ¼ hi0 D2 iq
q2 ¼ ðD2 q
Since q1 ¼ q q2 ¼ q hi0 D2 iq, this allows x to be expressed as
1
0
0
x ¼ C1
1 ðq hi D2 iqÞ þ D2 q ¼ C1 ðI hi D2 iÞ þ D2 q
If we define
0
R ¼ C1
1 ðI hi D2 iÞ þ D2
(5.39)
we see the parallel with earlier results. Here the transformation between industry
outputs and commodity outputs is given by x ¼ Rq. Previously, under pure industry
technology, it was x ¼ Dq, and under pure commodity technology, it was x ¼ C1 q.
Notice that R contains elements that are reminiscent of both previous transformations;
namely C1
1 and also D2 . Total requirements matrices under this particular mixed
technology assumption are found as the exact parallels to those under pure commodity
technology or pure industry technology, with C1 or with D replaced by R.
5.7.2 Industry Technology in V1
In order to invoke the industry technology assumption for V1, we define
^ 1
D1 ¼ V1 q
1
(5.40)
This definition is not “parallel” to that for C1 in the sense that the divisors here are not elements of q2 but rather
of q. The reason for this will become clear as the algebra is worked out. Other definitions for D2 could be (and
have been) used, with different algebraic consequences.
35
^ 0 D0 2 i (transpose of a product) ¼ q
^ D02 i (transpose of a diagonal matrix)
^ Þ0 i ¼ q
The specific steps are: ðD2 q
0
^ D2 i i (the vector that is created from row sums of a matrix is the same as the vector that is created from
¼q
^ i (order of multiplication of two diagonal
row sums of the diagonal matrix formed from that vector) ¼ D02 i q
matrices makes no difference) ¼ D02 i q.
34
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The Commodity-by-Industry Approach in Input–Output Models
[compare (5.14) and now also (5.37)]. This identifies an industry-by-commodity
matrix whose i, jth element records the fraction of total commodity j produced under
the industry technology assumption that is made by industry i using that technology.
Along with the definition x1 ¼ V1 i [as in (5.33)], we see that
x1 ¼ D1 q1
(5.41)
which provides the transformation between x1 and q1 . [Without the subscripts, this is
completely parallel to the definition of D in the pure industry technology case, shown
in (5.16).]
To account for the commodity technology character of the outputs in (5.15), define a
commodity-by-industry matrix, C2 , as
C2 ¼ V02 x^1
(5.42)
[Compare (5.15) and now also (5.35).] Here the “normalization” of V2 has been carried
out using total industry outputs, not just those attributable to production in V2 .36 The i,
jth element in C2 represents the fraction of all industry j output that takes the form of
commodity i production under commodity technology. An argument similar to that
introducing D1 at the beginning
of this section shows that this definition of C2 , along
with the fact that q2 ¼ V02 i, leads to a transformation between q2 and x (not x2 );
namely
q2 ¼ C2 x
(5.43)
From (5.33) and the definition of C2 , x2 ¼ ðC2 x^Þ0 i and, following an argument parallel
to that in footnote 35, we find
x2 ¼ C02 i x ¼ hi0 C2 ix
(5.44)
Using q1 ¼ q q2 ¼ q C2 x allows the components of x to be expressed as
x ¼ x1 þ x2 ¼ D1 ðq C2 xÞ þ h i0 C2 ix
and rearrangement, putting x alone on the left, gives
h
i
1
x ¼ ðI þ D1 C2 hi0 C2 iÞ D1 q
We can define
h
i
1
T ¼ ðI þ D1 C2 hi0 C2 iÞ D1
(5.45)
and we see the parallel, again, with earlier results. Now the transformation between
industry outputs and commodity outputs is provided by x ¼ Tq. Thus, T plays the role
36
As in the case of C1 and D2 , this definition of C2 is not “parallel” to that for D1 , since the divisors are not
elements of x2 but rather of x. Again, as noted in footnote 34, other definitions could be used, with differing
algebraic results.
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5.7 Mixed Technology in the Commodity-Industry Framework
213
of R from the previous mixed technology case, and of D and C1 in the industry and
commodity technology cases, respectively, in Section 5.6. Again, too, there are elements
of both industry technology, here in D1 , and commodity technology, here in C2 , embedded
in the transformation matrix, T. Total requirements matrices under this different mixed
technology assumption are found by replacing C1 , D, or R by T in each case.
5.7.3 Numerical Examples with Mixed Technology Assumptions
We continue with the same set of hypothetical data derived from Table 5.3. In
particular
90 0
:1333 :0727
V¼
B¼
10 100
:1111 :0636
Suppose that we decompose V as follows:37
90 0
0 0
V1 ¼
and V2 ¼
3 100
7 0
Example 5.1: Commodity Technology in V1 Here we assume that V1 reflects
commodity technology and V2 embodies industry technology. The necessary pieces of
information are
90
0
93
7
, x2 ¼
, q1 ¼
, q2 ¼
x1 ¼
103
7
100
0
so that
1 90 3
90 3
1 :0291
¼
0 100
0 103
0 :9709
1 0 0 100
0
0 0
D2 ¼
¼
7 0
0 100
:07 0
C1 ¼
and, from (5.39),
:93 :03
R¼
:07 1:03
Notice that, for this numerical illustration, i0 C1 ¼ i0 and i0 R ¼ i0 . [Exercise Problem
5.6 on this text’s supplemental resources website, online Appendix SP1, asks the
reader to show that this is always the case, using the definitions in (5.35) and
(5.39).] From this information, we can find the four total requirements matrices as
37
With such small (2 2) examples, there is relatively little flexibility in the ways that the elements of V can be
split between V1 and V2 , especially since it is likely that all of the production represented in v11 and v22 would
be assigned to either V1 or V2. (Of course, at such a high level of aggregation as we have in 2 2 examples, it
is impossible to imagine that either commodity is in fact just one product.)
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The Commodity-by-Industry Approach in Input–Output Models
Commodity-by-Commodity
1:1591 :0876
1
ðI BRÞ ¼
:1332 1:0764
Commodity-by-Industry
1:2372 :1211
1
1
R ðI BRÞ ¼
:0644 1:0469
Industry-by-Commodity
1:0739 :0492
RðI BRÞ1 ¼
:2184 1:1148
Industry-by-Industry
1:1487 :0812
ðI RBÞ1 ¼
:1530 1:0868
If we compare the matrices in Section 5.4.2, we see that these results lie somewhere
between those for the “pure” industry technology model and those for the “pure”
commodity technology model.
Example 5.2: Industry Technology in V1 Now we attribute production in V1
to the industry technology assumption. In this case, we have
1 90 0
93 0
:9677 0
¼
D1 ¼
3 100
0 100
:0323 1
1 0 7 90 0
0 :0636
C2 ¼
¼
0 0
0 110
0
0
from which
:9656 :0656
T¼
:0344 1:0656
These results illustrate that i0 D1 ¼ i0 and i0 T ¼ i0 . [Again, Exercise Problem 5.6 asks
for a general proof of this, using in this case (5.40) and (5.45).] The four total
requirements matrices are found to be
Commodity-by-Commodity
1:1617 :0850
1
ðI BTÞ ¼
:1354 1:0743
Commodity-by-Industry
1:1976 :1535
1
1
T ðI TBÞ ¼
:1041 1:0146
Industry-by-Commodity
1:1129 :0116
1
TðI BTÞ ¼
:1842 1:1477
Industry-by-Industry
1:1496 :0816
1
ðI TBÞ ¼
:1521 1:0864
Here, also, the total requirements matrices lie between those for the two “pure”
technology model illustrations in Section 5.4.2.
5.7.4 Additional Mixed Technology Variants
Yet another variant is what is known as the by-product technology model. Here all
secondary production is categorized as by-products, and these are treated as negative
inputs. Since uij is total input of commodity i for production by industry j, the net input
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215
5.7 Mixed Technology in the Commodity-Industry Framework
of commodity i into industry j becomes uij vji (for i 6¼ j). Using a hat on a matrix M
(square or otherwise) to denote a diagonal matrix whose elements are the mii elements
in M, and using an “upside down” hat to denote the matrix M but with its mii elements
^ þ M),
̌ the net use matrix is seen to be
replaced by zeros (so that M ¼ M
0 v21
u11 u12
0
̌
U V ¼
u21 u22
v12 0
and the technology matrix relating commodity inputs to commodity outputs, under the
by-product technology assumption, can be represented as
1
^
(5.46)
AB ¼ U V̌ 0 V
For our numerical example,
12 8
0
AB ¼
10 7
0
10
0
90 0
0 100
1
:1333 :0200
¼
:1111 :0700
and we see that, as in the commodity technology models, negative elements are possible.
The associated total requirements matrix (with a not-surprising negative element) is
1:1507 :0247
1
ðI A B Þ ¼
:1375 1:0723
Other variants are also possible. For example, ten Raa, Chakraborty, and Small (1984)
suggest a combination of a mixed technology assumption with the by-product technology assumption. Letting V1 contain primary and ordinary secondary products (defined
as those for which the commodity technology will be invoked), ten Raa, Chakraborty,
and Small propose the by-product technology assumption rather than the industry
technology assumption for V2. In this case, the direct inputs matrix, call it ACB , is
1
ACB ¼ U V02 V01
(5.47)
For the numerical example, this works out to be
1 12 8
0 7
90 3
:1333 :0060
ACB ¼
¼
10 7
0 0
:1111 :0667
0 100
and
1
ðI ACB Þ
1:1548 :0074
¼
:1375 1:0723
If the commodity technology model is invoked for all secondary products, then
V1 ¼ V, V2 ¼ 0 and so ACB ¼ AB ¼ AC , as in (5.26). On the other hand,
if all secondary products are by-products for which industry technology is used, then
^ V2 ¼ V,
̌ and so ACB ¼ AB in (5.46). The distinction between which secondary
V1 ¼ V,
products are ordinary or by-products is an empirical one; ten Raa, Chakraborty, and Small
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216
The Commodity-by-Industry Approach in Input–Output Models
investigate the question using regression analysis to examine whether primary and
secondary output of an industry are proportional. If they are, the secondary output is
classified as a by-product; if they are not, it is classified as an ordinary secondary product.
Negative numbers in the direct inputs matrix will arise in the by-product technology
model when an industry produces more of a particular commodity (as a by-product)
than it uses in production. One brute force approach changes these negatives to zeros.
0
This is equivalent to setting negative elements in the “net transactions” matrix, U V̌ ,
in (5.46), to zero. The logic is: if a particular element uij vji is non-negative, then the
by-product approach correctly records the net use of commodity i by industry j; if
uij vji < 0, then as a first approximation one could assume that all of j’s needs for i
goods are met from j’s own production and so j’s net use of commodity i is zero.
Following this approach for our example,
:1333
0
1:1538
0
and ðI AB1 Þ1 ¼
AB1 ¼
:1111 :0700
:1379 1:0753
An even more radical approach would be to just ignore all secondary production, after
netting it out of an industry’s primary production. In this case, V̌ becomes a null matrix
(primary production is the only production that is accounted for), so, from (5.46),
1
:1333
:0800
1:1667
:1004
1
^ ¼
AB2 ¼ UV
and ðI AB2 Þ ¼
:1111 :0700
:1394 1:0873
The most radical approach of all would be to force the assumption of no secondary
production at all by collecting all of the elements in each row of V into the on-diagonal
element. This then simply reduces to the original Leontief system. In this example, V
90 0
^ ¼V
. Using V in (5.46) (note that V
would be replaced by V ¼ hVii ¼
0 110
¼ 0)
and V
1 12 8 90 0
:1333 :0727
1
AL ¼ UV ¼
¼
(5.48)
10 7
:1111 :0636
0 110
and
1
ðI A L Þ
1:1655 :0905
¼
:1383 1:0787
^ , and all four direct requirements matrices in Section 5.3.6 –
In this case, V ¼ x^ ¼ q
AC , AC , AI , and AI – will be the same [and equal to AL in (5.48)], and the
ðccÞ
ðiiÞ ðccÞ
ðiiÞ
commodity–industry accounting has been completely swept under the rug.
5.8
Summary
In this chapter we have explored the issues that are introduced by a commodity-byindustry accounting approach. This framework was introduced primarily as an attempt
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5.8 Summary
217
to accommodate the real-world fact that sectors (industries) generally produce more
than one product, thereby violating the “one industry/one product” assumption of the
original input–output model (as in Chapter 2). This leads to the possibility of “rectangular” input–output systems, where the number of commodities (products) need not
be the same as the number of industries. In turn, this rectangularity leads to computational problems insofar as inverse matrices are concerned. However, the commodityby-industry approach is also completely valid when the number of commodities and
industries is the same; that is, it still allows for the more realistic representation of
economies in which some industries produce more than one product. The now wellestablished commodity-by-industry approach continues to be developed in other ways
as well. For example, Rueda-Cantuche (2017) surveys and develops new approaches to
construction of commodity-by-industry models and accompanying econometric tests
of available firm-level data for evaluating the relative validity of industry-based or
commodity-based technology assumptions for commodity-by-commodity models.
These developments are also included in the newest edition of the UN Handbook on
Supply, Use, and Input–Output Tables (United Nations, 2018, annexes A, B, and C).
We conclude with Table 5.10, extracted from a larger table in the Eurostat Manual
(Eurostat/Economic Commission, 2008) that records the actual amount of secondary
production in some 24 countries of the EU annually from 1995 to 2003 (not all countries
are reported for all years). These figures may help to put into perspective how significant
secondary activity is (or is not) in these countries. The Manual concludes (p. 309):
In most European countries the reported level of secondary products of industries as well as the
production of products in secondary industries is relatively low. [Thus] the difference between
product-by-product input–output tables and industry-by-industry input–output tables is relatively
small. Both transformations can be regarded as valid options for impact analysis.
There are three appendices to this chapter. Appendix 5.1 (below) develops alternative derivations of transactions matrices from commodity-by-industry accounts.
Appendix 5.2 (below) describes and illustrates Almon’s iterative procedure for eliminative negative numbers, commonly found in commodity technology models. Finally,
online Appendix SA5.3 (summarized below as Appendix 5.3) explores the properties
and methods for deriving left and right inverses in non-square input–output systems.
Table 5.10 Percentage share of secondary product output in total
industry output (European Union countries, 60-sector level)
Highest
Lowest
EU Average
1995
1999
2003
19.0
(Czech Republic)
1.8
(France)
6.1
16.7
(Belgium)
1.7
(Greece)
6.2
12.0
(Hungary)
3.1
(Luxembourg)
7.4
Source: Selected from Eurostat/European Commission (2008, table
11.8, p. 308)
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218
The Commodity-by-Industry Approach in Input–Output Models
Appendix 5.1
Alternative Approaches to the Derivation of Transactions Matrices
The generation of commodity-by-commodity or industry-by-industry transactions
matrices from the commodity-by-industry transaction data in the Use matrix, U, can
be visualized as a process of “adjusting” U to convert it to the proper commodityby-commodity or industry-by-industry dimensions. Since the dimensions of U are
commodities-by-industries, the adjustment must be one that either (1) replaces commodity rows by industry rows, or (2) replaces industry columns by commodity
columns. In what follows, we assume that the appropriate technical coefficients matrix
is known and we examine the consequent structure of the corresponding transactions
matrix.
A5.1.1
Industry Technology
Commodity-by-Commodity Requirements Here, AI ¼ BD; therefore (paralðccÞ
lel to A ¼ Zx^1 ) Ax ¼ Z as introduced in Chapter 2)
1 ^ ¼ Ux^1 V
q ¼ Ux^1 V^
(A5.1.1)
ZI ¼ ðBDÞ^
q q
ðccÞ
Since C ¼ V0 x^1 , then C0 ¼ x^1 V, and hence, from (A5.1.1),
ZI ¼ U C0
ðccÞ
(A5.1.2)
ðciÞ ðicÞ
^ Þ in
The use of D in defining A ¼ BD and the consequent definition of ZI as BDðq
ðccÞ
(A5.1.1) identifies the industry technology assumption. The matrix C is associated
with the commodity technology assumption; its appearance ðC0 Þ in the representation
of ZI in (A5.1.2) is a matter of algebraic convenience; it is simply a result of the
ðccÞ
algebraic definition of C and the algebraic rearrangement of (A5.1.1) that this definition makes possible.
Postmultiplication of U by C0, as in (A5.1.2), serves to rearrange the “destinations”
(columns) of commodity sales (rows) to commodity categories of purchasers rather
than to industries as purchasers. Since, by definition, columns in C all sum to one, row
sums in C0 are also unity. Thus, ZI i ¼ UC0 i ¼ U i; row sums in ZI and U are
ðccÞ
ðciÞ
ðccÞ
ðciÞ
the same. This is as it should be; the redistribution accomplished by ZI ¼ UC0 does
ðccÞ
not change the total intermediate sales of any commodity, only the names that are
given to the purchasers.
Ritz (1980, p. 41) has observed that (A5.1.2) can be re-expressed as
ZI ¼ UðI þ C0 IÞ ¼ U þ UðC0 IÞ
ðccÞ
(A5.1.3)
and that the operation in the UðC0 IÞ term “incorporates the ‘mechanical redefinitions’ required to shift inputs and create a commodity-by-commodity use matrix (or,
in other words, to make the industry classification scheme conform precisely to the
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Appendix 5.1
219
commodity classification scheme).” [Since C0 i ¼ i and Ii ¼ i, (C0 – I)i ¼ 0 and in this
form it remains true that ZI i ¼ Ui.] The numerical example that follows illustrates
ðccÞ
the logic of this adjustment technique.
Using the data from the Table 5.3 example, ZI in (A5.1.3) is found as
ðccÞ
12 8
12 8
1
0
1 0
ZI ¼
þ
10 7
10 7
:0909 :9091
0 1
ðccÞ
12 8
12 8
0
0
¼
þ
10 7
10 7 :0909 :0909
12 8
:7272 :7272
þ
¼
10 7
:6363 :6363
12:7272 7:2728
¼
10:6363 6:3637
(A5.1.4)
Consider, for illustration, the upper-right element; z12 ¼ 8 þ ð12Þð10Þ þ ð8Þð0:0909Þ ¼
7:2728. The original 8 is modified to reflect the fact that commodity 2 comprises only 90.9
percent of industry 2’s total output. Thus 100 – 90.91 ¼ 9.09 percent of the input of
commodity 1 to industry 2 must have been used for the production of commodity 1 and
(8) (0.0909) ¼ 0.7272 is therefore netted out of the original u12 (commodity-by-industry)
transaction of 8. On the other hand, 12 units of commodity 1 are used as inputs to industry
1 production ðu11 Þ. We know from the C matrix that industry 1 produces no commodity 2,
so none of this transaction [(12) (0) ¼ 0] should be added in to produce the estimate of
z12 . The logic behind other elements in Z in (A5.1.4) is similar.
Industry-by-Industry Requirements
Here, A ¼ DB; therefore, since U ¼ Bx^,
ZI ¼ ðDBÞx^ ¼ D U
ðicÞ ðciÞ
ðiiÞ
(A5.1.5)
Premultiplication of U by D serves to rearrange the “origins” (rows) of industry
purchases (columns) to industry categories of sellers rather than commodity categories.
By definition, columns in D all sum to one, i0 ZI ¼ i0 DU ¼ i0 U . Column sums in ZI
ðiiÞ
ðciÞ
ðiiÞ
and U are the same, which is also as it should be; the redistribution accomplished
ðciÞ
by ZI ¼ DU should not change the total intermediate purchases by any industry, only
ðiiÞ
the names given to the sellers (rows).
Alternatively, since (I þ D – I) ¼ D, ZI in (A5.1.5) can be expressed as
ðiiÞ
ZI ¼ ðI þ D IÞU ¼ U þ ðD IÞU
ðiiÞ
(A5.1.6)
and in this case the (D – I)U term provides the adjustment elements to convert U to an
industry-by-industry Use matrix. Since i0 D ¼ i0 and i0 I ¼ i0 , i0 (D – I) ¼ 0, and column
sums in ZI and U remain equal.
ðiiÞ
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220
The Commodity-by-Industry Approach in Input–Output Models
From the numerical example in the text,
12 8
:9 0
1 0
12
ZI ¼
þ
10 7
:1 1
0 1
10
ðiiÞ
12
12 8
:1 1 12 8
¼
þ
¼
10 7
:1 0 10 7
10
10:8 7:2
¼
11:2 7:8
8
7
8
7
þ
1:2
:8
1:2
:8
(A5.1.7)
Again, consider the upper-right element; z12 ¼ 8 þ ð0:1Þð9Þ þ ð0Þð7Þ ¼ 7:2. The
original 8, which is commodity 1 inputs to industry 2, now needs to be converted to
industry 1 inputs to industry 2. From the D matrix, we know that 90 percent of
commodity 1 is produced by industry 1. Therefore 100 – 90 ¼ 10 percent of the
original 8 must be netted out, since it represents industry 2’s production of commodity
1. On the other hand, 7 units of commodity 2 are also used by industry 2 ðu22 Þ. But we
know from the D matrix that none of this comes from industry 1, so the second
(potential) adjustment to the original 8 is (0)(7) ¼ 0. Other elements in ZI in (A5.1.7)
ðiiÞ
can be interpreted similarly.
A5.1.2
Commodity Technology
Commodity-by-Commodity Requirements Here A ¼ BC1 ; therefore
^ ¼ Ux^1 [x^ðV0 Þ1 ]q
(A5.1.8)
ZC ¼ BC1 q
^ ¼ UðV0 Þ1 q
^
ðccÞ
^ 1 V0 and ðD0 Þ1 ¼ ðV0 Þ1 q
^ ; therefore
Since D ¼ V^
q 1 , D0 ¼ q
1
ZC ¼ U ðD0 Þ
ðccÞ
(A5.1.9)
ðciÞ ðicÞ
This requires that V and hence V0 be non-singular, so that C and D0 are non-singular
also. The fact that this is a commodity-by-commodity transactions matrix under the
commodity technology assumption is emphasized by the fact that the matrix C (more
^ in (A5.1.8). The
precisely, C1 ) is used in defining A ¼ BC1 and ZC ¼ BC1 q
ðccÞ
matrix D (industry technology assumption) appears only because of the algebraic
rearrangement which the fact that D ¼ V^
q 1 makes possible. This is parallel to the
way in which the C matrix crept into the commodity-by-commodity requirements
expressions under the industry technology assumption, in (A5.1.1) and (A5.1.2).
As in (A5.1.2), postmultiplication of U serves to rearrange the destinations
(columns) of the commodity sales (rows) – again, the relabeling is from industries to
commodities as purchasers. We know that column sums of D are all unity, so D0 i ¼ i
from which it follows that ðD0 Þ1 i ¼ i. Therefore, from (A5.1.9), ZC i ¼ U i – row
ðccÞ
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ðciÞ
Appendix 5.1
221
sums of ZC and U are the same, as is to be expected and as we also saw was true for
ðccÞ
ðciÞ
ZI and U , in (A5.1.2), under the industry technology assumption.
ðciÞ
ðccÞ
Alternatively, using the same kind of algebraic reasoning as previously, (A5.1.9) can
be written as
1
ZC ¼ U[I þ (D0 )
ðccÞ
1
I] ¼ U þ U[(D0 )
I]
(A5.1.10)
and in this case the matrix ðD0 Þ1 I supplies the adjustment terms to convert the
original Use matrix to commodity-by-commodity terms, but now under the assumption
of a commodity-based technology [as opposed to the adjustment in (A5.1.3), again to
commodity-by-commodity
terms,
but under an industry technology assumption]. It is
easily shown that ðD0 Þ1 I i ¼ 0, so again row sums are not altered by the relabeling of the columns of U.
From the numerical example,
12 8
12 8
1:1111 :1111
1 0
þ
ZC ¼
10 7
10 7
0
1
0 1
ðccÞ
12 8
12 8 1:1111 :1111
¼
þ
10 7
10 7
0
0
12 8
1:3332 1:3332
13:3332 6:6668
¼
þ
¼
(A5.1.11)
10 7
1:1111 1:1111
11:1111 5:8889
Note that this differs from ZI in (A5.1.4), which was derived under the assumption of
ðccÞ
an industry technology. While the numerical example in (A5.1.11) illustrates the
parallels with the adjustments in (A5.1.4) and (A5.1.7), interpretation of the elements
in ðD0 Þ1 I is not as straightforward as in the case of (C0 – I) in (A5.1.4) or (D – I) in
(A5.1.7). The operations in this case are more easily understood by going back to
(A5.1.8) or (A5.1.9) and noting that
1
or U ¼ ZC D0
(A5.1.12)
U ¼ ZC ðV0 Þ1 q
^
ðccÞ
ðccÞ
That is, if the commodity-by-commodity Use matrix, ZC , were known, then postðccÞ
multiplication by D0 would serve to rearrange the column labels (purchasers) from
commodity groups to industry groups. From the numerical example,
:9 :1
0
D ¼
0 1
ðciÞ
and so
ðz11 Þð0:9Þ þ ðz12 Þð0Þ ðz11 Þð0:1Þ þ ðz12 Þð1Þ
U ¼ ZC D ¼
ðz21 Þð0:9Þ þ ðz22 Þð0Þ ðz21 Þð0:1Þ þ ðz22 Þð1Þ
ðciÞ
ðccÞ ðciÞ
0
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(A5.1.13)
222
The Commodity-by-Industry Approach in Input–Output Models
Consider element u12 in this matrix (total input of commodity 1 to industry 2). The second
term, ðz12 Þ(1), reflects the fact that 100 percent of the output of commodity 2 is made by
industry 2, and so all of the sales of commodity 1 to commodity 2 production ðz12 Þ can be
thought of equally well as purchases by industry 2. In addition, however, since 10 percent
of the output of commodity 1 is produced by industry 2, 10 percent of the sales of
commodity 1 for commodity 1 production ðz11 Þ would be purchased by industry 2.
Thus, u12 ¼ ðz11 Þð0:1Þ þ ðz12 Þð1Þ. Other elements can be interpreted similarly.
Industry-by-Industry Requirements Here A ¼ C1 B and
ZC ¼ C1 B x^ ¼ C1 U
ðicÞ ðciÞ
ðiiÞ
(A5.1.14)
Since i0 C ¼ i0 , i0 C ¼ i0 also, column sums of ZC and U are the same. The premultiplication
ðiiÞ
ðciÞ
of U by C1 serves to relabel the rows from commodities to industries as sellers. Following
the earlier examples, ZC in (A5.1.14) can be re-expressed as
ðiiÞ
ZC ¼ I þ C1 I U ¼ U þ C1 I U
(A5.1.15)
ðiiÞ
and now it is clear that the matrix C1 I represents the adjustment mechanism for
converting U to an industry-by-industry basis, under the commodity-based technology
assumption. All column sums in the adjustment (second) term in (A5.1.15) are zero, so
the row relabeling leaves column sums of ZC and U equal.
ðciÞ
ðiiÞ
Using the numerical example
12 8
1 :1
1 0
12 8
ZC ¼
þ
10 7
0 1:1
0 1
10 7
ðiiÞ
12 8
0 :1 12 8
12 8
1 :7
þ
¼
þ
¼
1
:7
10 7
0 :1
10 7
10 7
11 7:3
(A5.1.16)
¼
11 7:7
This differs (although in this case only slightly) from the ZI matrix in (A5.1.7),
ðiiÞ
derived under the industry technology assumption.
As with the commodity-by-commodity requirements case immediately above, the
logic of this adjustment is easily seen by rewriting (A5.1.14) as
C ZC ¼ U
ðciÞ ðiiÞ
ðciÞ
(A5.1.17)
If the industry-by-industry Use matrix, ZC , were known, premultiplication by C
ðiiÞ
would rearrange row labels (sellers) from industries to commodities. Here
1 :0909
C¼
0 :9091
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ðciÞ
223
Appendix 5.2
and therefore
ð1Þðz11 Þ þ ð0:0909Þðz21 Þ ð1Þðz12 Þ þ ð0:0909Þðz22 Þ
U ¼ C ZC ¼
ð0Þðz11 Þ þ ð0:9091Þðz21 Þ ð1Þðz12 Þ þ ð0:9091Þðz22 Þ
ðciÞ
ðciÞ ðiiÞ
(A5.1.18)
Again, we consider u12, which represents total inputs of commodity 1 into industry 2
production. From the first row of C, we know that 100 percent of industry l’s output
consists of commodity 1 and that 9.09 percent of industry 2’s output is commodity 1.
Thus, all of the industry-to-industry transactions z12 can be viewed as commodity
1 sales to industry 2, while the commodity 1 composition of the industry-to-industry
transaction z22 is represented by ð0:0909Þz22. Thus, u12 ¼ ð1Þz12 þ ð0:0909Þz22 . Other
elements in (A5.1.18) have similar interpretations.
Appendix 5.2
Elimination of Negatives in Commodity Technology Models
A5.2.1 The Problem
A major practical problem with the commodity technology assumption is the very
real possibility of negative entries appearing in the direct requirements matrix and hence
also in the associated transactions matrix. We explored this in Section 5.5. Table A5.2.1
summarizes some of the results from the 2 2 numerical illustrations in that section.
These examples might be viewed as overly simplistic (too small), or exaggerated,
since 40 percent or more of commodity 1 is produced in industry 2 (v21 is large relative to
v11 ). Here are some larger examples that also generate negative elements in direct inputs
matrices (not shown) and therefore also in the associated transactions matrices (shown).
3 3 Example
2
3
2
3
2
3
4 2 4
20 2 1
5:340 :667 5:327
U ¼ 4 2 5 2 5 V ¼ 4 5 25 7 5 ZC ¼ 4 2:225 4:556 2:219 5
6 1 3
3 2 15 ðccÞ
8:364 1:778 3:414
Table A5.2.1 Summary of two commodity/two industry results
U
12
10
12
10
12
10
8
7
8
7
8
7
V
90
60
0
100
90
61
0
100
90
64
0
100
ZC
ðccÞ
AC
ðccÞ
:1333
0
:1111 :0033
:1333 :0013
:1111 :0022
:1333 :0053
:1111 :0011
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20
0
16:67 :33
20:13 :13
16:78 :22
20:53 :53
17:11 :11
224
The Commodity-by-Industry Approach in Input–Output Models
4 4 Example
2
1
6 3
U¼6
4 10
15
4
1
6
3
5
2
1
4
3
2
6
30 4
6 8 20
57
7V ¼ 6
4 5 1
75
2
5 5
3
10 10
5 8 7
7
50 2 5
5 60
2
3
2:327 4:856
6:882
6:589
6 3:181
:443
2:143
6:119 7
7
ZC ¼ 6
4
13:961
4:656 1:079 6:462 5
ðccÞ
23:949 1:648 2:202 :503
5 5 Example (from Almon, 2000)
2
3
2
0 0 0 0 0
70
6 0 0 0 0 07
6 30
6
7
6
7
6
U¼6
6 3 37 0 0 0 7 V ¼ 6 0
4 15 5 0 0 0 5
4 0
28 72 30 5
0
2
0
0
0
0
6 0
0
0
6
1:67
41:67
0
ZC ¼ 6
6
ðccÞ
4 21:67 1:67 0
30
70
30
3
20
0
0
0
180 0
0
0 7
7
0 100 0
0 7
7
0
0 20 0 5
0
0
0 535
3
0 0
0 07
7
0 07
7
0 05
5 0
A5.2.2 Approaches to Elimination of Negative Elements
Whether or not the reader finds these examples compelling, input–output practitioners
repeatedly find that the commodity technology model generates negative elements in
real-world applications (Section 5.5.1). Clearly, if one wants to use a commodity
technology input–output model, this is an issue that needs to be addressed. In Section
5.5.1 we noted a large amount of literature that addresses this issue in a variety of ways.
Here we examine an approach to negatives that is reported in Almon (2000). He and
his associates have used it repeatedly and successfully to transform an observed Use
matrix, U , into a non-negative commodity technology based commodity-by-comðciÞ
modity transactions matrix, ZC .38 The building blocks are the usual commodity–
ðccÞ
38
The procedure has been in use for decades at the University of Maryland’s INFORUM project. It was
mentioned in print at least as early as Almon (1970) and also in Almon et al. (1974). The associated nonnegative direct input coefficients matrix can easily be derived from the non-negative transactions.
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Appendix 5.2
225
industry accounts – a Use matrix and a Make matrix, V, from which D ¼ V^
q 1 , as
usual. Initially, we will explore the approach for a 3 3 model.39
Given
2
3
2
3
u11 u12 u13
d 11 d 12 d 13
U ¼ 4 u21 u22 u23 5 and D ¼ 4 d 21 d 22 d 23 5
ðicÞ
ðciÞ
u31 u32 u33
d 31 d 32 d 33
we want to find an associated
2
z11
ZC ¼ Z ¼ 4 z21
ðccÞ
z31
z12
z22
z32
3
z13
z23 5
z33
that contains no negative elements.
We have simplified the representation of the elements in ZC (showing them only as
ðccÞ
zij ) to keep the notation as uncluttered as possible. Equation (A5.1.9) from Appendix
5.1 is our point of departure – Z ¼ UðD0 Þ1 or
ZD0 ¼ U
(A5.2.1)
Rewriting as U ZD0 ¼ 0 and adding Z to both sides, we have
Z ¼ U þ ZðI D0 Þ
(A5.2.2)
This suggests an iterative approach that could be used to construct an estimate of Z.
Let the next [(k þ l)st] estimate, Zðkþ1Þ , depend on the current (kth) estimate, Zðk Þ , and
on the elements in U in the following way:
Zðkþ1Þ ¼ U þ Zðk Þ ðI D0 Þ
(A5.2.3)
Given the characteristics of D, and hence of ðI D0 Þ, it is possible to show that this
kind of sequential procedure will in fact converge.40 The process begins (k ¼ 0) by
assuming
Zð 0 Þ ¼ U
(A5.2.4)
We know that this will turn out to be wrong, since U has dimensions commodity-byindustry and our transactions matrix Z must have commodity-by-commodity dimensions. But notice that because both U and our final Z have commodities in the row
dimension, the transformation from the former to the latter must preserve row sums.
Next, from (A5.2.3)
Zð1Þ ¼ U þ Zð0Þ ðI D0 Þ
39
As will become clear, a two-commodity and industry illustration is too small to properly illustrate the
intricacies of the technique.
40
Details are beyond what we need here. See Almon (2000) for further discussion.
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The Commodity-by-Industry Approach in Input–Output Models
and so on:
ð1 Þ
0
Zð2Þ ¼ U þ Z ðI D Þ
..
..
.
.
ZðnÞ ¼ U þ Zðn1Þ ðI D0 Þ
(A5.2.5)
In the iterative steps in (A.5.2.5) we are assured that total commodity outputs for
interindustry use (row sums of U) are preserved because row sums of ðI D0 Þ are
zero – ðI D0 Þi ¼ i i ¼ 0 (since column sums of D are 1). So, with each step,
Zðkþ1Þ i ¼ Ui.
Consider the process one row at a time. Let i U ¼ ½ ui1 ui2 ui3 (the ith row of U,
which is known) and i Z ¼ ½ zi1 zi2 zi3 (the ith row of Z, which we want to find).
Then (A5.2.2) can be expressed as
0
i Z ¼ i U þ i ZðI D Þ
for i ¼ 1, 2, 3. More explicitly,
½ zi1
zi2
zi3 ¼ ½ ui1
ui2
(A5.2.6)
2
ui3 þ ½ zi1
zi2
1 d 11
zi3 4 d 21
d 13
d 21
1 d 22
d 23
3
d 31
d 32 5
1 d 33
(A5.2.7)
The iterative process in (A5.2.3) can be carried out on each row. That is,
iZ
ðkþ1Þ
¼ i U þ i Zðk Þ ðI D0 Þ
(A5.2.8)
with
ð0Þ
¼ iU
(A5.2.9)
¼ i U þ i Zð0Þ ðI D0 Þ
(A5.2.10)
iZ
and then (A5.2.8), starting with k ¼ 0
iZ
ð1Þ
and so on, as in (A5.2.5).
Specifically, for the 3 3 illustration, here are the three linear equations in (A5.2.8)
written out explicitly for the i ¼ 1 case [elements of the first row of Zðkþ1Þ ]:
z11
ðkþ1Þ
¼ u11 þ ð1 d 11 Þz11 d 12 z12 d 13 z13
z12
ðkþ1Þ
¼ u12 d 21 z11 þ ð1 d 22 Þz12 d 23 z13
ðkþ1Þ
¼ u13 d 31 z11 d 32 z12 þ ð1 d 33 Þz13
z13
ðk Þ
ðk Þ
ðk Þ
ðk Þ
ðk Þ
ðk Þ
ðk Þ
ðk Þ
ðk Þ
Notice that these equations are a 3 3 illustration of the relationships in the transposed
version of (A5.2.8), namely
0 0
ðkþ1Þ 0 0 ðk Þ
0
¼ i U þ i Z ðI D0 Þ ¼ i U þ ðI DÞ i Zðk Þ
iZ
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Appendix 5.2
227
To add further specificity, we use the D matrix from the 3 3 numerical example
introduced in Section A5.2.1:
2
32
31 2
3
20 2 1
28 0 0
:7143 :0690 :0435
^ Þ1 ¼ 4 5 25 7 54 0 29 0 5 ¼ 4 :1786 :8621 :3043 5
D ¼ Vðq
3 2 15
0 0 23
:1071 :0690 :6522
and so
z11
ðkþ1Þ
¼ 4 þ 0:2857z11 0:0690z12 0:0435z13
z12
ðkþ1Þ
¼ 2 0:1786z11 þ 0:1379z12 0:3043z13
ðkþ1Þ
¼ 4 0:1071z11 0:0690z12 þ 0:3478z13
z13
ðk Þ
ðk Þ
ðk Þ
ðk Þ
ðk Þ
ðk Þ
ðk Þ
ðk Þ
ðk Þ
(A5.2.11)
In general it is not easy to come up with a convincing set of names for commodities and
industries in a small (say 3 3) numerical example – “Agriculture,” “Manufacturing,”
and “Services” are too aggregate to make much sense as “commodities.” And at a finer
level of detail – e.g., “Cheese,” “Ice Cream,” and “Other Foodstuffs” – it is usually
difficult to create a reasonable numerical illustration unless many of the elements in the
Use matrix are zero; for example, ice cream is an unlikely input to cheese
manufacturing.41 At the same time, it is helpful in sorting out what is going on in
(A5.2.11) to have some specificity, so (with apologies to Almon, 2000) we opt for
1 ¼ cheese, 2 ¼ ice cream, and 3 ¼ other foodstuffs, without going into a careful
analysis of the plausibility of each and every element in U.
Consider how the (k þ l)st estimate of the input of cheese (i ¼ 1) into production of
the commodity other foodstuffs (j = 3) is built up on the basis of the current estimate
ðkþ1Þ
(iteration k), i.e., z13 , the third of the three equations in (A5.2.11). We start on the
right with u13 – the original observed input of cheese into the other foodstuffs industry
(4 units). Since the other foodstuffs industry made secondary products – cheese
ðv31 ¼ 3Þ and ice cream ðv32 ¼ 2Þ – we need to net out the cheese that was used by
other foodstuffs for both of those non-primary products. (Remember that we are
building a commodity-to-commodity transactions table.) First we deal with the cheese
ðk Þ
produced secondarily by other foodstuffs. z11 is the “current” estimate of commodity
1 input into commodity 1 production (cheese into cheese). But 10.71 percent of cheese
ðk Þ
production occurs in other foodstuffs. So (0.1071)z11 accounts for the cheese used by
other foodstuffs to make a secondary product, cheese, and it is netted out of the u13
ðk Þ
ðk Þ
transaction: (–0.1071)z11 . Similarly, z12 is the current estimate of commodity 1 input
into commodity 2 production (cheese into ice cream). But 6.9 percent of the ice cream
ðk Þ
produced is as a secondary product for other foodstuffs, so (–0.0690)z12 nets out from
u13 other foodstuffs’ use of cheese for another of its secondary products.
ðk Þ
Finally, we have (0.3478)z13 . This reflects the fact that 34.78 percent of the
commodity other foodstuffs is not made by the other foodstuffs industry, but rather
41
With larger examples, it is easier, as we will see in Section A5.2.3 with the 5 5 example.
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The Commodity-by-Industry Approach in Input–Output Models
by the cheese industry (4.35 percent; d 13 ¼ 0:0435) and by the ice cream industry
(30.43 percent; d 23 ¼ 0:3043). So we need to add to u13 these amounts of cheese used
ðk Þ
elsewhere to make other foodstuffs as a secondary product: (0.0435)z13 for what is
ðk Þ
made by cheese and (0.3043)z13 for what is made by ice cream.
By the property of D matrices that column sums are 1, we recognize that the amount
ðkþ1
ðk Þ
ðk Þ
ðk Þ
added in z13 ð0:3478Þz13 – is exactly right, namely (0.0435)z13 þ ð0:3043Þz13 .
The absolute amount of cheese to other foodstuffs that is netted out of the first two
equations in (A5.2.11) is exactly what is added back into that transaction in the third
equation. And the same is true for cheese-to-cheese and cheese-to-ice cream adjustments in these equations. The amounts netted out in two of the equations are added
back in the other equation – transactions are simply “rearranged,” and nothing is “lost.”
This illustrates the two issues to be addressed in converting a use table from a
commodity-by-industry format to a commodity-by-commodity format. For each uij we
need to:
1. Remove from each uij the sales of commodity i to industry j that were used as inputs
in production of j’s non-primary (secondary) products. For example, for u13 this
ðkþ1Þ
ðk Þ
means transformations across the z13 row, except for the z13 term, in (A5.2.11).
2. Add to each uij the sales of commodity i to producers other than j who made j as a
secondary product. For u13 this means transformations to the first and second terms
ðk Þ
in the z13 column in (A5.2.11), and this modification is captured in the third term of
this column.
The iterative routine for row-by-row creation of ZC allows us to see when and how
negatives first emerge, and at that time corrective interventions, like the Almon
“purification” procedure, can begin. (This corrective procedure requires that U and V
be square matrices.) It is instructive to look in detail at the first step in the (A5.2.5)
sequence – Zð1Þ ¼ U þ UðI D0 Þ.
For this 3 3 example we have
2
32
3
4 2 4
:2857 :1786 :1071
UðI D0 Þ ¼ 4 2 5 2 54 :0690 :1379 :0690 5
6 1 3
:0435 :3043 :3478
2
:8308 1:6558
¼ 4 :1394 :2763
1:5147 1:8466
3
:8248
:1364 5
:3318
2
3
4:8308
:3442 4:8248
This leads to Zð1Þ ¼ 4 2:1394 4:7237 2:1364 5, and already on this first step a
7:5147 :8466 3:3318
ð1Þ
negative flow appears – z32 ¼ 0:8466: In a small example like this, we could easily
see the trouble coming, since element (3,2) in UðI D0 Þ shows that 1.8466 units will
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Appendix 5.2
229
be taken from u32 , which has only 1 unit to give up. But in larger real-world
applications this kind of visual approach is not possible. The Almon procedure is a
way of identifying approaching negatives and dealing with them as they arise. It
examines each iteration one row at a time.42
For illustration in this example we look in detail at the third row of Zð1Þ , as in
(A5.2.10):
2
3
:2857 :1786 :1071
ð1 Þ
¼ 3 U þ 3 UðI D0 Þ ¼ ½ 6 1 3 þ ½ 6 1 3 4 :0690 :1379 :0690 5
3Z
:0435 :3043 :3478
¼ ½6
1 3 þ ½ 1:5147 1:8466 :3318 ¼ ½ 7:5147 :8466 3:3318 Alternatively, when written out in detail, the individual calculations that are aggregated
in the matrix multiplication operation are made clear. These are simply the first step for
the equations in (A5.2.10) for i ¼ 3:
ð1Þ
z31 ¼ 6 þ 6ð0:2857Þ þ 1ð0:0690Þ þ 3ð0:0435Þ ¼ 6 þ 1:7142 0:0690 0:1305 ¼ 7:5147
ð1Þ
z32 ¼ 1 þ 6ð0:1786Þ þ 1ð0:1379Þ þ 3ð0:3043Þ ¼ 1 1:0716 þ 0:1379 0:9129 ¼ 0:8466
ð1Þ
z33 ¼ 3 þ 6ð0:1071Þ þ 1ð0:0690Þ þ 3ð0:3478Þ ¼ 3 0:6426 0:0690 þ 1:0434 ¼ 3:3318
(A5.2.12)
Note that the elements in column j of ðI D0 Þ show up as the elements in row j of these
ð1Þ
equations, because of the transposition as in (A5.2.8), and also that z32 has become
negative after this first step.
ð1Þ
Viewed this way, the problem that generates a negative z32 is immediately apparent.
Continuing with the 1 ¼ cheese, 2 ¼ ice cream, 3 ¼ other foodstuffs story, we see that
other foodstuffs (commodity 3) shipped 1 unit to the ice cream industry (u32 ¼ 1). But
the ice cream industry also made secondary products – cheese (v12 ¼ 2) and other
foodstuffs (v32 ¼ 2) – and we need to remove the other foodstuffs inputs to both of
these products that are non-primary to producing the commodity ice cream. In this case
the removal suggests trouble, because we need to take away 6 0:1786 ¼ 1:0716
units to account for cheese production and 3 0:3043 ¼ 0:9129 units to account for
foodstuffs production – a total of 1.9845 units to be subtracted. In this case, the original
allocation is 1 unit, and an additional 1 0.1379 units are being added at this point
because of the foodstuffs used in making ice cream in the cheese and foodstuffs
industries. This means 1.9845 units are to be subtracted from 1.1379 units, leaving
42
We thank Fred Pallada, former economist of the Central Planning Bureau, The Hague (Netherlands), for his
initial inquiries in 2015 regarding our coverage of this material in Appendix 5.2 of the second edition of this
book, for his efficient programming of the technique, for our many email discussions over the ensuing months,
and for alerting us to the Vollebregt and van Dalen (2001) paper.
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The Commodity-by-Industry Approach in Input–Output Models
us with the deficit of 0.8466. A small algebraic rearrangement helps with visualization
of the problem.
^ , so
A rewrite of the basic iterative equation, replacing i U with i0 i U
0
ð1Þ
^ ðI D0 Þ, generates useful results. For row 3 we now have,
¼ iU þ i iU
iZ
ð1Þ
^ ðI D0 Þ
¼ 3 U þ i0 3 U
3Z
2
32
3
6 0 0
:2857 :1786 :1071
¼ ½ 6 1 3 þ ½ 1 1 1 4 0 1 0 54 :0690 :1379 :0690 5
0 0 3
:0435 :3043 :3478
and, in particular,
3
1:7142 1:0716 :6426
0
4 :0690
^
:1379
:0690 5
3 U ðI D Þ ¼
:1305 :9129 1:0434
2
(A5.2.13)
This disaggregates
product 3 UðI D0 Þ in such a way that the elements in
the matrix
0
^
column j of 3 U ðI D Þ show the magnitudes of each of the changes that convert u3j to
ð1Þ
its corresponding z3j at this step – exactly the elements in (A5.2.12) and in the text in
the preceding paragraph
–and the 0j-th column sum gives the total adjustment for u3j. Here,
^ ðI D Þ ¼ ½ 1:5147 1:8466 :3318 , leading to Zð1Þ ¼
as we saw earlier, i0 3 U
3
ð1Þ
½ 7:5147 :8466 3:3318 . Column 2 contains the elements that produce z32 < 0:
To concentrate on the potential difficulty with the negative off-diagonal elements in
0
^
3 U ðI D Þ, we create a variant of this matrix, replacing each on-diagonal element
by the absolute value of the sum of the negative elements in that column; here for row
3 at iteration 1 this is
2
3
:1995 1:0716 :6426
ð1 Þ
¼ 4 :0690
1:9845 :0690 5
3M
:1305
:9129
:7116
ð1Þ
The Almon procedure then compares each on-diagonal element 3 mjj with its
ð1Þ
corresponding u3j . When u3j < 3 mjj , the recommendation is to scale back the elemð1Þ
ents in column j. This is the case here with u32 ¼ 1 and 3 m22 ¼ 1:9845 (shown in
ð1Þ
ð1Þ
bold), and, as we saw, this results in z32 < 0: So, from 3 M an “adjustment” matrix,
~ ð1Þ
3 M , is created:
ð1Þ
1. In those columns j where u3j < 3 mjj
scaling factor
ð1 Þ
ð1Þ
sj ¼ u3j =mjj .
the current elements are multiplied by a
ð1Þ
In this example where u32 < 3 m22 , we have
ð1 Þ
s2 ¼ ð1=1:9845Þ ¼ 0:5039, and each element in column 2 will be reduced to
50.39 percent of its current value. The total amount removed from u32 ¼ 1 will
be precisely 1 unit; and
ð1Þ
2. In those columns k where u3k 3 mkk there is no problem, so for those elements is
ð1 Þ
sk ¼ 1 (no adjustment is needed).
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Appendix 5.2
231
Notice that the scaling in (1) does not take into account the fact that there is also an
ð1Þ
addition to z3j along with the decreases. This appears as element (2, 2) in
(A5.2.13), namely 0.1379.
2 ð1 Þ 3
s1
7
~ ð1Þ ¼ 3 Mð1Þ^s ð1Þ where s ¼ 6
Put compactly, 3 M
4 sð21Þ 5. In this case we have
ð1 Þ
s3
2
3
2
3
1
:1995 :5400 :6426
~ ð1Þ ¼ 4 :0690 1:0000 :0690 5. This matrix is
sð1Þ ¼ 4 :5039 5 and 3 M
1
:1305 :4600
:7116
0
^
subtracted from 3 U ðI D Þ to give a new adjustment matrix
2
3
1:5147 :5316
0
ð1Þ
^ ðI D 0 Þ M
~ ð1 Þ ¼ 4 0
¼ 3U
:8621
0 5
3Δ
3
0
:4529 :3318
Elements in row j of 3 Δð1Þ represent adjustments to u3j that come about from
reallocation of shipments of j to production of each of the commodities 1, 2, and 3,
respectively. So adding across row j gives the total adjustment for u3j. In this case, our
new and final estimate of 3 Zð1Þ is
0
ð1 Þ
ð1Þ
Z
¼
U
þ
Δ
i
¼ ½ 6 1 3 þ ½ :9831 :8621 :1211 3
3
3
¼ ½ 6:9831 :1379 2:8789 ð1Þ
Notice that indeed we now have z32 ¼ 0:1379; the original u32 ¼ 1 has been wiped out
because of the ice cream industry’s non-primary production of cheese and other foodstuffs, but at the same time 0.1379 units of ice cream are added because of the non-primary
production of the commodity ice cream in the cheese and other foodstuffs industries.
The interested reader can derive results for 1 Zð1Þ and 2 Zð1Þ in the same way (no other
negatives will be encountered at this point), and putting it all together gives
3
2 ð1Þ 3 2
4:8310 :3442 4:8248
1Z
Zð1Þ ¼ 4 2 Zð1Þ 5 ¼ 4 2:1396 4:7238 2:1365 5
ð1Þ
6:9831 :1379 2:8789
3Z
These first-step estimates are shown in Table A5.2.2.43
A5.2.3
Results of the Iterative Procedure
3 3 Example Here is the commodity technology-based commodity-by commodity transactions matrix ZA that results from using the iterative procedure on
ðccÞ
the 3 3 illustration in Section A5.2.1:
43
These results are based on a D matrix with elements shown with four-decimal accuracy. They may sometimes
differ a bit from those in Table A5.2.2 that were generated through matrix multiplications carried out with and
rounded down from basic data with more significant digits.
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The Commodity-by-Industry Approach in Input–Output Models
Table A5.2.2 Steps in the iterative procedure for the 3 3 example
ðk Þ
k
ðk Þ
ðk Þ
zi1
zi2
zi3
0
1
2
3
4
5
6
4
4.8310
5.0242
5.0693
5.0817
5.0856
5.0868
2
0.3442
0.0475
0.0065
0.0009
0.0001
0.0000
4
4.8248
4.9283
4.9241
4.9174
4.9143
4.9132
ðk Þ
2 Z (Row 2)
0
1
2
3
4
5
6
7
2
2.1396
2.1927
2.2128
2.2204
2.2233
2.2244
2.2248
5
4.7238
4.6192
4.5796
4.5647
4.5590
4.5569
4.5560
2
2.1365
2.1881
2.2076
2.2149
2.2177
2.2188
2.2191
6
6.9834
7.2009
7.2525
7.2688
7.2688
7.2697
1
0.1379
0.0190
0.0026
0.0001
0.0001
0.0000
3
2.8787
2.7801
2.7449
2.7312
2.7312
2.7303
1Z
3Z
0
1
2
3
4
5
6
ðk Þ
ðk Þ
(Row 1)
(Row 3)
2
3
5:087
0
4:913
ZA ¼ 4 2:225 4:556 2:219 5
ðccÞ
7:270
0
2:730
Table A5.2.2 indicates a number of the steps in the procedure. Iterations stop for any
particular row when some criterion is met – for example, when the differences between
ðk Þ
successive values for all zij are less than some prespecified level. We show the results
to four decimal places in order to give a feeling for how things develop. (In practice,
each row went through additional iterations, until there were changes in the sixth
decimal place only, but that level of detail is unnecessary for this simple illustration.)
As we would expect, for rows in ZC in which there are no negative elements, the
ðccÞ
iterative approach generates exactly the same vector (here row 2) in ZA . As the reader
ðccÞ
can verify, in moving from one iteration to the next, the amount(s) that are subtracted
from the original u’s in a given row are exactly balanced (except perhaps for rounding)
by the amount(s) that are added to other u’s in that same row (preservation of row sums
https://doi.org/10.1017/9781108676212.006 Published online by Cambridge University Press
Appendix 5.2
233
from the original U matrix). For example, for row 1, moving from k ¼ 0 to k ¼ 1,
(4.8310 – 4) þ (4.8248 – 4) ¼ 1.6558 ¼ (2 – 0.3442).
4 4 Example
The original transactions matrix
2
3
2:327 4:856
6:882
6:589
6 3:181
:443
2:143
6:119 7
7
ZC ¼ 6
4
13:961
4:656 1:079 6:462 5
ðccÞ
23:949 1:648 2:202 :503
is transformed into
2
3
0
4:031 5:966 6:003
6 2:983
0
2:118 5:899 7
7
ZA ¼ 6
4
12:859
4:730
0
6:412 5
ðccÞ
21:646
0
2:354
0
5 5 Example This example is from Almon (2000), where the unsatisfactory Z matrix is shown (with its negative elements) but the modified Z that results
from the iterative procedure (without negatives) is not. We saw in Section A5.2.1, in
the 5 5 example, that those U and V matrices generated
2
3
0
0
0 0 0
6
0
0
0 0 07
6
7
1:67
41:67
0 0 07
ZC ¼ 6
6
7
ðccÞ
4 21:67 1:67 0 0 0 5
30
70
30 5
0
containing “the infamous negative flows” (Almon, 2000, p. 31). After using the
iterative procedure, we have
2
3
0 0 0 0 0
6 0 0 0 0 07
6
7
0 40 0 0 0 7
ZA ¼ 6
6
7
ðccÞ
4 20 0 0 0 0 5
30 70 30 5 0
In the relatively plausible story that goes along with this example, the commodities
are: (1) cheese, (2) ice cream, (3) chocolate, (4) rennet,44 and (5) other. Notice that in
ZA , in particular, there is (thankfully) no chocolate used to make cheese zA31 ¼ 0 , nor
ðccÞ
is there rennet going into (what would turn out to be curdled) ice cream zA42 ¼ 0 .
44
Rennet is a milk curdling agent, used in making cheese, but certainly not in ice cream.
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234
The Commodity-by-Industry Approach in Input–Output Models
As a final illustration, here is a 5 5 Use matrix that differs very little from the
original one that we used:
2
3
0 0 0 0 0
6 0 0 0 0 07
6
7
7
~
U¼6
6 4 36 0 0 0 7
4 14 6 0 0 0 5
28 72 30 5 0
There is only a one-unit difference in four elements from their original counterparts in
U; u~31 and u~42 are one unit larger, u~32 and u~41 are one unit smaller. In this case, in
conjunction with the original V,
2
3
0 0 0 0 0
6 0 0 0 0 07
6
7
~
0 40 0 0 0 7
ZC ¼ 6
6
7
ðccÞ
4 20 0 0 0 0 5
30 70 30 5 0
This is exactly the same as the ZA ðccÞ that was derived from the original 5 5 U and
~ were
V, which makes sense. The one-unit variations that differentiated U from U
essentially ignored when the iterative procedure created a no-nonsense commodity-tocommodity transactions matrix from U.
This example contains one further (possible) surprise. If we elect to use the industry
technology assumption instead of commodity technology, then we find that the
transactions matrix is
2
3
0
0
0 0 0
6 0
0
0 0 07
6
7
6
q ¼ 6 8:25 31:75 0 0 0 7
ZI ¼ BD^
7
ðccÞ
4 11:75 8:25 0 0 0 5
32:06 67:94 30 5 0
This is (appropriately and appealingly) characterized as “massive nonsense,” and to
call this a commodity-to-commodity table “. . . would be little short of scandalous”
(Almon, 2000, p. 31). Why? Because now we find that chocolate is being used as an
input to make cheese, and rennet is part of the recipe for ice cream. Bad cooking; and a
nice illustration that neither technology assumption is flawless.
Appendix 5.3
Left and Right Inverses in Non-square Input–Output Systems
Supplemental Appendix SA5.3, located on the internet web site associated with this
text (http://www.cambridge.org/millerandblair), develops the most common basic formulations of so-called generalized inverses for non-square input–output systems.
These formulations are illustrated with commodity-by-industry models using both
industry-based and commodity-based technology assumptions.
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References
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6
6.1
Multipliers in the Input–Output
Model
Introduction
One of the major uses of the information in an input–output model is to assess the
effect on an economy of changes in elements that are exogenous to the model of that
economy. For example,
Leontief input–output economics derive their significance largely from the fact that output multipliers
measuring the combined effects of the direct and indirect repercussions of a change in final demand
were readily calculated. (Steenge, 1990, p. 377)
In Chapters 2 and 3 we presented several numerical illustrations of the ways in which
assumed changes in final-demand elements (e.g., federal government spending, household consumption, exports) were translated, via the appropriate Leontief inverse, to
corresponding output changes in the industrial sectors of the economy. When the
exogenous changes occur because of the actions of only one “impacting agent” (or a
small number of such agents) and when the changes are expected to occur in the short
run (e.g., next year), this is usually called impact analysis. Examples are a change in
federal government defense spending or in consumer demand for recreation vehicles.
On the other hand, when longer-term and broader changes are examined, then we are
dealing with projections and forecasting. If we project the levels of final demand for outputs
of all sectors in an economy five years hence, and estimate, using the Leontief inverse, the
outputs from all sectors that will be needed to satisfy this demand, this is an exercise in
forecasting. As the period of projection gets longer, the accuracy of such an exercise tends
to decrease, both because our ability to forecast the new final demands accurately (the
elements of f) will diminish and also because the coefficients matrix – the elements of A and
hence of L – may have become outdated. (The issue of temporal stability of input–output
coefficients is examined in Chapter 7.) If the model is built from commodity–industry
accounts, then it is the matrices B, C, and/or D that may become out of date.
In either impact analysis or forecasting, the general form of the model is x ¼ Lf or
Δx ¼ LΔf, and the usefulness of the result, x (or Δx), will depend on the “correctness”
of both the Leontief inverse and the final-demand vector. Our primary concern in this
section is with the elements aij , and hence with L ¼ ðI AÞ1 . The f (or Δf) incorporates the assumed or projected behavior of one or more final-demand elements, and
accuracy in the estimation of these elements is also of paramount importance to
238
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6.2 General Structure of Multiplier Analysis
239
generating an accurate result. When the question is one of impact, then the finaldemand value or values are usually completely specified – for example, what is the
impact, by sector, of a new order for $2.5 million worth of sector j output by the federal
government? Then Δf contains 2.5 (million) in the jth row and zeros elsewhere.
Alternatively, to find x for some future year requires a projection of both A and f to
that year. We will investigate some of the approaches for changing A over time in
Chapter 7. The projection of f is a problem that is often approached via econometric
models. The input–output forecasts of 1985 industrial outputs (and employment) for
the US economy in Almon et al. (1974, chapters 8 and 9) depend on detailed and
painstaking projections of each of the components of final demand – personal consumption expenditures, investment in capital equipment, construction, inventories,
imports and exports, and government expenditures (1974, chapters 2–7, respectively).
In some but by no means all “joined” input–output and econometric models, the
econometric model provides a forecast of the final demands, which then “drive” the
input–output model. (There is a growing literature on this issue of the interactions
between input–output models and econometric models, particularly at the regional
level. Some of this is explored briefly in Chapter 15.)
A number of summary measures, derived from the elements of L, are often
employed in impact analysis; these are input–output multipliers. We examine multipliers in this chapter.
6.2
General Structure of Multiplier Analysis
Several of the most frequently used types of multipliers are those that estimate the
effects of exogenous changes on outputs of the sectors in the economy or on income
earned by households or employment (jobs) created in each sector because of the new
outputs. Additionally, many other generalized multipliers are possible, such as value
added that is created in each sector in the economy because of the new outputs or any
one of many possible negative impacts; e.g., pollution generated or resources used in
producing the new outputs. We examine income multipliers in some detail in this
section; extensions to other effects follow in Section 6.2.3.
The notion of multipliers rests upon the difference between the initial effect of an
exogenous change and the total effects of that change. The total effects can be defined
either as the direct and indirect effects (found from an input–output model that is open
with respect to households) or as direct, indirect, and induced effects (found from a
model that is closed with respect to households).1 The multipliers that incorporate
1
In some discussions of multipliers in an input–output model, what we have called the initial effect is termed the
direct effect. For later exposition – for example, in looking at shortcut methods for finding multipliers – when
the power series approximation
ðI AÞ1 ¼ I þ A þ A2 þ A3 þ will be used, it seems to us preferable to associate “initial” with the I term, “direct” with A, and “indirect” with
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240
Multipliers in the Input–Output Model
direct and indirect effects are also known as simple multipliers. When direct, indirect,
and induced effects are captured, they are often called total multipliers.
6.2.1 Output Multipliers
An output multiplier for sector j is defined as the total value of production in all sectors
of the economy that is necessary in order to satisfy a dollar’s worth of final demand for
sector j’s output.
Simple Output Multipliers For the simple output multiplier, this total production is obtained from a model with households exogenous. The initial output effect
on the economy is defined to be just the initial dollar’s worth of sector j output needed
to satisfy the additional final demand. Then, formally, the output multiplier is the ratio
of the direct and indirect effect to the initial effect alone.
We continue with the small example in Chapter 2, Section 2.3, where
:15 :25
A¼
:20 :05
and
1:254 :330
L¼
:264 1:122
(In the remainder of this book we will sometimes keep three figures to the right of the
decimal point and sometimes four, depending on the purposes of the numerical
1
indicates an additional dollar’s worth of final
illustration.) Note that Δf ð1Þ ¼
0
0
indicates, similarly, an
demand for the output of sector 1 only, and Δf ð2Þ ¼
1
additional dollar’s worth of final demand for the output of sector 2 only.
Consider Δf ð1Þ; the implications for sectors 1 and 2 are found as LΔf ð1Þ. Denote this
by Δxð1Þ, so
1:254 :330
1
1:254
Δxð1Þ ¼
¼
(6.1)
:264 1:122
0
:264
l 11
.
l 21
The additional outputs of $1.254 from sector 1 and $0.264 from sector 2 are required
for a dollar of new final demand for the output of sector 1 only. The $1.254 from
sector 1 represents $1.00 to satisfy the original new dollar of final demand plus
an additional $0.254 for intra- and interindustry use. The $0.264 from sector 2 is for
intra- and interindustry use only. The sector 1 output multiplier, mðoÞ1 , is defined as
the sum of the elements in the Δx(1) column, namely $1.518, divided by $1;
This is, of course, just the first column of L https://doi.org/10.1017/9781108676212.007 Published online by Cambridge University Press
6.2 General Structure of Multiplier Analysis
241
mðoÞ1 ¼ $1:518=$1 ¼ 1:518, a dimensionless number. The $1 in the denominator is the
initial effect on sector 1 output of the new dollar’s worth of final demand for sector 1’s
product; the dollar’s worth of final demand becomes an additional dollar’s worth of
sector 1 output as the first term in the series assessment of total direct and indirect effects
on sector 1 production. Formally, using i0 ¼ ½1 1 as usual to generate column sums
mðoÞ1 ¼ i0 Δxð1Þ ¼
n
X
l i1
(6.2)
i¼1
where n ¼ 2 in this example.
Similarly,
l
:330
0
1:254 :330
¼ 12
¼
Δxð2Þ ¼
1:122
1
:264 1:122
l22
and
mðoÞ2 ¼ i0 Δxð2Þ ¼
n
X
l i2
(6.3)
i¼1
Here mðoÞ2 ¼ 1:452. In general, the simple output multiplier for sector j is
mðoÞj ¼
n
X
l ij
(6.4)
i¼1
Thus, for example, if a government agency were trying to determine the differential
effects of spending an additional dollar (or $100, or $1,000,000, or whatever amount)
on the output of a sector, comparison of output multipliers would show where this
spending would have the greatest impact in terms of total dollar value of output
generated throughout the economy. Note that when maximum total output effects are
the exclusive goal of government spending, it would always be rational to spend all the
money in the sector with the largest output multiplier. Even with anticipated expenditures of $1,000,000, there would be no reason, on the basis of output multipliers alone,
to divide that spending between the sectors.
Of course, there might well be other reasons – taking into account strategic factors,
equity, capacity constraints for sectoral production, and so on – for using some of the
new final-demand dollars on the output of the other sector (or sectors, when n > 2).
Note also that multipliers of this sort may overstate the effect on the economy in
question if some sectors are operating at or near capacity and hence some of the needed
new inputs would have to be imported to the economy and/or outputs from some
sectors would be shifted from exports and kept in the economy for use as inputs.
Phenomena such as these will assume even more importance in regional models.
We see that L is a matrix of sector-to-sector multipliers, l ij , relating final demand in
sector j to output in sector i. Output multipliers (column sums of L) represent sectorto-economy multipliers, relating final demand in sector j to economy-wide output.
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242
Multipliers in the Input–Output Model
For an n-sector model, denote the row vector of these multipliers by
mðoÞ ¼ mðoÞ1 ; . . . , mðoÞn .2 With i0 ¼ ½1; . . . , 1, we have
ð1nÞ
Sector-demandto-sector-output
multipliers
z}|{
mðoÞ ¼ i
L
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
Sector-demandto-economy-wide-
(6.5)
output multipliers
We will see that many additional input–output multiplier variations build on this
representation. All that is required is to alter the elements in the multiplier matrix so
that, instead of (Δfj ¼ 1) ! ðΔxi Þ, they represent (Δfj ¼ 1) ! (some function of Δxi ),
such as employment or energy use or pollution emissions.
Total Output Multipliers If we consider the input coefficients matrix closed
with respect to households (as described in Section 2.5) we capture in the model the
additional induced effects of household income generation through payments for labor
services and the associated consumer expenditures on goods produced by the various
sectors.3 Continuing with the example from Section 2.5, the augmented coefficient
matrix, with an added household row and column, was
2
3
:15
:25
:05
6
7
7
¼6
A
6 :20
:05
:40 7
4
5
:30
:25
:05
|
|
|
|
- - - - - - - - -| - - |
|
and the Leontief inverse, with elements lij , was
2
:425
6 1:365
1
6
Þ ¼ 6 :527
¼ ðI A
L
1:348
4
:570
:489
|
|
|
|
3
:251 7 11
L
7
:595 7 ¼ L21
5
1:289
|
- - - - - - - - - - - - - - - - - |
|
12
L
22
L
(6.6)
as in (2.27) but rounded here to three decimals. We have added the partitioned matrix
representation because it will be useful in much of what follows in this chapter. Clearly,
Strictly speaking, one expects a row vector to include a “prime” in its designation, as with x and x0 in earlier
chapters. However, here and throughout this discussion of multipliers we simply define various rows of
multipliers without the prime to save on notational complexity.
3
As noted in Chapter 2, there are many other sources of personal income in addition to wages. We return to this
issue in Section 6.3.
2
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6.2 General Structure of Multiplier Analysis
243
¼ l ij also relate final-demand changes to sector outputs, only now
the elements in L
these are in a model with households endogenous, and hence the effects tend to be larger.
To assess the impact of a new dollar’s worth of final demand for sector 1 output, we
2 3
1
would now form the three-element vector Δf ð1Þ ¼ 4 0 5 (meaning no exogenous
0
change in demand for sector 2 output or for labor services), and find exactly the first
namely
column of L,
2
3
1:365
f ð1Þ ¼ 4 :527 5
Δxð1Þ ¼ LΔ
:570
[Compare (6.1).] Adding these elements gives a parallel to (6.2),
ðoÞ1 ¼ i0 Δxð1Þ ¼
m
nþ1
X
l i1 ¼ 2:462
(6.7)
i¼1
with n ¼ 2, as before but now with i0 ¼ ½1, 1, 1. (In what follows we assume that i or i0
always has appropriate dimensions for the multiplication in which it is involved.)
(n ¼ 2 for our example)
Sums of the first n elements in each of the columns of L
represent the total output multiplier effects over the original n sectors only – the
11 . When interest is centered
truncated output multipliers. They can be found as i0 L
on the total output multipliers for the original n sectors (for example, to be compared
with the simple output multipliers for these same n sectors), these truncated output
½oðtÞj ;
multipliers are of interest. Denote these truncated total output multipliers by m
½oðtÞ1 ¼ 1:892.
here m
The total output multiplier for sector 2 is
ðoÞ2 ¼
m
nþ1
X
l i2 ¼ 2:262
(6.8)
i¼1
½oðtÞ2 ¼ 1:773. In general, for sector j, the total output multiplier is given by
and m
ðoÞj ¼
m
nþ1
X
l ij
i¼1
½oðt Þj ¼
and the truncated total output multiplier is m
(6.9)
Pn i¼1 l ij . In compact matrix terms,
and m
11
ðoÞ ¼ i0 L
½oðtÞ ¼ i0 L
m
(6.10)
Example 6.1 : The US Input–Output Model for 2003 We again use the sevensector 2003 US model. The Leontief inverse was shown as Table 2.8 in Chapter 2 and
is not repeated here. The simple output multipliers are easily found to be
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244
Multipliers in the Input–Output Model
mðoÞ ¼ ½1.9195 1.6051 1.7218 1.9250 1.4868 1.6081 1.5985
In this case, the largest multipliers are associated with manufacturing (4) and agriculture (1). This is hardly surprising, considering the seven-sector level of aggregation.
Output Multipliers in Commodity–Industry Models With commodity-byindustry models output multipliers can be found as column sums of the relevant total
requirements matrices (open or closed with respect to households). In Table 6.1 we
collect the results for total requirements matrices from Tables 5.4 and 5.5 in
Chapter 5.4
For example, for the commodity-by-commodity total requirements matrix under
industry technology, the row vector of these output multipliers is i0 ðI BDÞ1 . Notice
that since i0 D ¼ i0 (column sums of D are all 1), the same output multipliers
will be found for the industry-by-commodity total requirements matrix:
i0 DðI BDÞ1 ¼ i0 ðI BDÞ1 . The same will be true for any other pair of matrices
(vertically) in the table. This is because (1) i0 C ¼ i0 (C is constructed so that is true), (2)
i0 D1 ¼ i0 (this is easy to show, given i0 D ¼ i0 ) and (3), similarly, i0 C1 ¼ i0 . This result
is what we would expect – summing down the columns in a total requirements matrix
(over all rows) should give the same result, irrespective of the row labels (“commodities” or “industries”).
The results in the tables below are for the total requirements matrices in the
numerical examples from Chapter 5. They illustrate the identical results for pairs of
matrices.
Table 6.1 Total requirements matrices in commodity–industry models
Commodity-Demand Driven Models
Commodity-by-Commodity
Industry-by-Commodity
Industry-Demand Driven Models
Industry-by-Industry
Commodity-by-Industry
4
Industry Technology
Commodity Technology
ðI BDÞ1
DðI BDÞ1
1
I BC1
[C1 I BC1 1 ]
ðI DBÞ1
D1 ðI DBÞ1
1
I C1 B
[C I C1 B 1 ]
Lenzen and Rueda-Cantuche (2012) suggest using the data in supply and use tables (SUTs) directly to generate
multiplier impacts both for industries and for products, and they emphasize the importance of this flexibility
when incorporating external data that is either product- or industry-specific. For example, final demand is
product-specific, while pollution-generation is industry-specific.
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6.2 General Structure of Multiplier Analysis
245
Commodity-Demand-Driven Models
Industry Technology
ðI BDÞ
1
Commodity Technology
Commodity-by-Commodity
1
1:1568 :0898
1:1644 :0825
I BC1
¼
¼
:1314 1:0782
:1375 1:0723
Output Multipliers ½ 1:2882 1:1680 1:0411
¼
:2471
1
DðI BDÞ
½ 1:3019 1:1548 Industry-by-Commodity
1
:0809
1:1507 :0247
C1 I BC1
¼
1:0871
:1512 1:1795
Output Multipliers ½ 1:2882 1:1680 ½ 1:3019 1:1548 Industry-Demand-Driven Models
Industry Technology
Commodity Technology
Industry-by-Industry
1
1:1478 :0809
1:1507 :0821
1
IC B
¼
ðI DBÞ ¼
:1537 1:0871
:1512 1:0861
Output Multipliers ½ 1:3015 1:1680 ½ 1:3019 1:1682 1
1
1
D ðI DBÞ
Commodity-by-Industry
1
1:2753 :0898
1:1644
C I C1 B
¼
¼
:0262 1:0782
:1375
Output Multipliers ½ 1:3015 1:1680 :1808
:9873
½ 1:3019 1:1682 5
6.2.2 Income/Employment Multipliers
Generally an analyst is more likely to be interested in economic impacts of new final
demand as measured by jobs created, increased household earnings, value added
generated, etc., and not simply gross output by sector. In this section we explore
impacts on household earnings; the approach is exactly the same whether we measure
this impact in terms of income (monetary), jobs (physical), value added or “bads,” such
as energy consumed or pollution generated. In what follows, we illustrate using
income, but this applies equally well to other impacts, as we will see.
Income Multipliers One straightforward approach is simply to convert the
elements in L into dollars’ worth of employment using labor-input coefficients – either
monetary (wages earned per unit of output, as in ½anþ1, 1 ; . . . , anþ1, n ) or physical
(person-years, or some such measure, per unit of output). We begin with transactions
information; let h0 (for households) denote the row vector of these data. In the
monetary case, this is h0 ¼ ½znþ1, 1 ; . . . , znþ1, n , in physical terms it would be some
5
This is not equal to 0.1808 þ 0.9873 only because of rounding in the total requirements matrix.
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Multipliers in the Input–Output Model
measure of numbers of employees in each sector. Then h0c ¼ h0 x^1 is the row of
associated household input coefficients. For example, in the two-sector case this is6
1=x1
0
0 1
¼ ½ h1 =x1 h2 =x2 ¼ ½ hc1 hc2 ¼ h0c
h x^ ¼ ½ h1 h2 0
1=x2
Again, in monetary terms these are the elements in ½anþ1, 1 ; . . . , anþ1, n , used in the
example in Section 6.2.1 when the model was closed with respect to households. In that
income received per dollar’s worth of sector j output.
case anþ1, j ¼ znþ1, j =xj is household
1
l
Associated with Δf ¼
, we found output effects in the first column of L 11 ,
l 21
0
as in (6.1). The conversion of this first column to income terms is accomplished by
weighting
the first element by anþ1, 1 and the second element by anþ1, 2, giving
anþ1, 1 l 11
. In general, then, using mðhÞj for the simple household income multiplier
anþ1, 2 l 21
for sector j,
n
X
anþ1, i l ij
(6.11)
mðhÞj ¼
i¼1
Again, “simple” refers to the fact that these multipliers are found using elements in the
L matrix, with households exogenous.
In matrix terms, the row vector of these simple household income multipliers is
mðhÞ ¼ h0c L ¼ h0 x^1 L
(6.12)
Recall from (6.5) that the vector of output multipliers was compactly represented as
mðoÞ ¼ i0 L. Now, in (6.12), the summation row, i0 , is replaced by the row of labor
input coefficients, h0c . We can deconstruct this in the same way as in (6.5):
Sectordemand
tosector income
multipliers
½MðhÞ
zfflfflffl}|fflfflffl{
^ 0x^1 L
mðhÞ ¼ h0c L ¼ h0 x^1 L ¼ i0 h
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
(6.13)
Sectordemand
toeconomywide
income multipliers
½mðhÞ
^ 0 x^1 L converts the inverse matrix of final demand-to-output multiIn particular,7 h
pliers in L into a matrix of final demand-to-sector income multipliers, MðhÞ. Then
This labor input coefficients row was denoted h0R in Chapter 2 in initial discussions of model closure with
respect to households. The prime is standard for row vectors, and the “R” was to distinguish its elements from
those in hC which denoted a column of household consumption coefficients.
7
The “prime” (transpose) notation could be dropped from h since it doesn’t matter whether it is a row vector or a
column vector when it is converted into a diagonal matrix. We retain it to emphasize that this is the row vector of
labor payments from the original transactions matrix.
6
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6.2 General Structure of Multiplier Analysis
247
0
^ x^1 L ¼ h0 x^1 L ¼ h0 L generates a vector of final demand-to-economy-wide
i0 h
c
income multipliers, mðhÞ (the column sums of the converted inverse). In this generic
format, the simple output multipliers in (6.5) can be thought of as
mðoÞ ¼ i0 L ¼ x0 x^1 L ¼ i0 x^0 x^1 L
Continuing the same example, we had anþ1, 1 ¼ 0:3 and anþ1, 2 ¼ 0:25. Thus
mðhÞ1 ¼ ð0:3Þð1:254Þ þ ð0:25Þð0:264Þ ¼ 0:376 þ 0:066 ¼ 0:442
and
mðhÞ2 ¼ ð0:3Þð0:33Þ þ ð0:25Þð1:122Þ ¼ 0:099 þ 0:281 ¼ 0:380
1,000
0
,
In matrix terms, (6.12), from h ¼ ½ 300 500 and x ¼
2,000
1:254 :330
:376 :099
^ 0 x^1 L ¼ :3 0
MðhÞ ¼ h
¼
0 :25
:264 1:122
:066 :281
and
mðhÞ ¼ ½ :442 :380 In this illustration, mðhÞ1 ¼ 0:442 indicates that an additional dollar of final demand
for the sector 1 output would generate $0.442 of new household income, when all direct
and indirect effects are converted into dollar estimates of income. If earnings in individual
sectors are of interest, we see that $0.376 would be earned by employees in sector 1 and
$0.066 would be earned by sector 2 employees. And similarly, mðhÞ2 ¼ 0:380 is disaggregated into earnings of $0.099 in sector 1 and $0.281 in sector 2. From this example,
using this measure of effectiveness, dollars of final demand – for example, new government purchases – generate more dollars of new household income when they are spent on
the output of sector 1 than when they are spent on the output of sector 2.
in (6.6) are weighted similarly, total (direct plus indirect plus
If the elements in L
induced) income effects or household income multipliers are obtained. As before,
the parallel to mðhÞ in (6.11) is
using an overbar to denote a multiplier derived from L,
j
ðhÞj ¼
m
nþ1
X
anþ1, ilij
(6.14)
i¼1
For the closed model, the n-element vector of total income multipliers for the n
sectors is
0
L
11
0 L11
ðhÞ ¼ m
ðhÞn ¼ hc anþ1:nþ1
ðhÞ1 ; , m
(6.15)
m
21 ¼ hc L
21
L
½1ðnþ1Þ
½ðnþ1Þn
ðhÞj > mðhÞj for two reasons: (1) Even
[Compare (6.12).] This makes clear that m
11 are
though the weights in h0c are the same for both models, the inverse elements in L
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Multipliers in the Input–Output Model
ðhÞj includes the additional term
consistently larger than those in L, and (2) Each m
anþ1, nþ1 lnþ1, j . [In the case of truncated multipliers, only (1) is relevant.]
In our numerical example in Section 2.5, h0 c ¼ ½ 0:3 0:25 00:05 and
2
3
1:365
:425
:251
6
7
7
¼ L11 L12 ¼ 6
L
6 :527
1:348
:595 7 [this is (2.27), rounded to three
L21 L22
4
5
:570
:489
1:289
|
|
|
|
| - - - - - - - - - - - - - - - - |
|
ðhÞ ¼ ½ 0:570 0:489 . Decomposing the mathematics,
figures after the decimal], so m
we have
ðhÞ1 ¼ ð0:3Þð1:365Þ þ ð0:25Þð0:527Þ þ ð0:05Þð0:570Þ ¼ 0:570
m
and
ðhÞ2 ¼ ð0:3Þð0:425Þ þ ð0:25Þð1:348Þ þ ð0:05Þð0:489Þ ¼ 0:489
m
Notice that these total income multipliers for sectors 1 and 2 are equal to
lnþ1, 1 and l nþ1, 2 , the elements of L
21 . Recall the interpretation of any element lij ; it
measures the total (direct, indirect, and induced) effect on sector i output of a dollar’s
worth of new demand for sector j output. Thus, lnþ1, j is the total effect on the output of
the household sector (the total value of labor services needed) when there is a dollar’s
worth of new final demand for goods of sector j. This is precisely what we mean by the
total household income effect or total household income multiplier;
ðhÞ ¼ L21
m
(6.16)
(In Appendix 6.1, the relationship between the total household income multipliers and
is shown exactly, using matrix algebra results on the
the bottom-row elements of L
inverse of a partitioned matrix.) Again, if we are only interested in household incomegenerating effects originating in the n original sectors, we would calculate a truncated
11
½hðtÞ1 , by summing down the columns of L
total household income multiplier, m
0
½hðtÞ ¼ hc L11 . For the example, m
½hðt Þ1 ¼ 0:541 and m
½hðtÞ2 ¼ 0:465.
only; m
In this and all subsequent discussions in this chapter, all results hold if A and L are
understood to be direct and total requirements matrices in a commodity-industry
model – as for example with AI ¼ BD and LI ¼ ðI BDÞ1 . We illustrated the case
ðccÞ
ðccÞ
of output multipliers for various commodity–industry models in Section 6.2.1.
Type I and Type II Income Multipliers With income multipliers, one has
some choice regarding what should logically be termed the initial effect of new final
demand. With output multipliers it was fairly clear that the initial effect of a new
dollar’s worth of final demand for sector j output is that sector j production must
increase by one dollar (and eventually, of course, by more than that dollar). With
income effects, the same dollar’s worth of new demand for sector j becomes, initially,
the same dollar’s worth of new output by sector j; this is what we considered the initial
effect in developing the household income multipliers. However, the initial dollar’s
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6.2 General Structure of Multiplier Analysis
249
worth of new output from sector j means an initial additional income payment of anþ1, j
to workers in sector j. Hence anþ1, j could be viewed as the initial income effect of the
new demand for sector j output.
Thus, there is another kind of simple income multiplier, often called the type
I income multiplier, for any sector j. This has the direct and indirect income effect,
or the simple household income multiplier (6.4) as a numerator, and uses as a
denominator not the initial dollar’s worth of output but rather its initial (direct) labor
income effect, anþ1, j . Let mðhÞIj represent this type I income multiplier for sector j, so
n
P
anþ1, i l ij
mðhÞIj ¼ i¼1
anþ1, j
¼
mðhÞj
(6.17)
anþ1, j
In matrix terms,
1
1
¼ h0c L h^0c
mðhÞI ¼ mðhÞ h^0c
To illustrate for the two-sector case, this is
1=hc1
0
mðhÞ ¼ ½ mðhÞ1 mðhÞ2 ¼ ½ mðhÞ1 =hc1
0
1=hc2
(6.18)
mðhÞ2 =hc2 For our numerical example,
mðhÞI1 ¼ 0:442=0:3 ¼ 1:473
mðhÞI2 ¼ 0:380=0:25 ¼ 1:520
The idea, as usual, is to relate an overall effect to an initial effect, but now the initial
effect is measured in units that are specific to the multiplier – e.g., total income/initial
income, total value added/initial value added (in monetary units), total pollution/initial
pollution (pounds), or total energy use/initial energy use (BTUs), etc.
When expressed inPthis way, these are often known as normalized multipliers. To
summarize, mðhÞj ¼ ni¼1 anþ1, i lij indicates the economy-wide household income per
Pn
anþ1, i l ij
unit of final demand for product j while mðhÞIj ¼ i¼1
gives the economy-wide
anþ1, j
household income per unit of direct income in sector j (that was generated by that unit
of final demand).
Again, if the coefficients matrix is closed with respect to households, income effects
similar to these type I multipliers can be calculated; these have been called type II
income multipliers:8
8
The designations “type I” and “type II” seem to have originated with Moore (1955). Calculation of these
measures (in a regional setting) was pioneered by Moore and Petersen (1955) for Utah and later by Hirsch
(1959) for St. Louis. As noted, “normalized” is now often used rather than “type I.”
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Multipliers in the Input–Output Model
nþ1
P
mðhÞIIj ¼ i¼1
anþ1, il ij
anþ1, j
¼
ðhÞj
m
anþ1, j
(6.19)
A row vector of type II income multipliers for the original n sectors can be defined
21 ¼ [lnþ1, 1 , lnþ1, 2 ; , l nþ1, n ], namely
using L
21 h^0c 1
(6.20)
mðhÞII ¼ L
Again, for the numerical example,
mðhÞII1 ¼
0:570
¼ 1:900
0:3
mðhÞII2 ¼
0:489
¼ 1:956
0:25
The parallel between this measure and the type I effect in (6.17) is the same as that
ðhÞj and mðhÞj . The
between the total and simple household income multipliers – m
ðhÞj from
numerator for mðhÞIj is mðhÞj from (6.11); the numerator for mðhÞIIj is m
ðhÞj, we can alternatively define
(6.14). Thus, for exactly the same reasons as for m
mðhÞIIj as
mðhÞIIj ¼ l nþ1, j =anþ1, j
(6.21)
These multipliers show by how much the initial income effects (0.3 and 0.25) are
blown up when direct, indirect, and induced effects (due to household spending
Truncated type
because of increased household income) are taken into account, via L.
11 only. In
II income multipliers would be found, as usual, by considering columns in L
II
II
this example they are m½hðt Þ1 ¼ 1:803 and m½hðt Þ2 ¼ 1:860.
It is generally conceded that Type I multipliers probably underestimate economic
impacts (since household activity is absent) and Type II multipliers probably give an
overestimate (because of the rigid assumptions about labor incomes and attendant
consumer spending). For example, Oosterhaven, Piek, and Stelder (1986, p. 69)
suggest
These two multipliers [Type I and Type II] may be considered as upper and lower bounds on the true
indirect effect of an increase in final demand; a realistic estimate generally lies roughly halfway
between the Type I and Type II multipliers.
Relationship between Simple and Total Income Multipliers or between Type I
and Type II Income Multipliers To the extent that the results of an input–output
analysis with households exogenous tend to underestimate total effects, total or type
II multipliers may be more useful than simple or type I multipliers in estimating
potential impacts. Or some in-between figure might be more realistic, as noted in the
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6.2 General Structure of Multiplier Analysis
251
quote, above, but deciding exactly where between these two limits may be problematic. However, if one is primarily interested in ranking or ordering the sectors – which
sector has the largest multiplier, which has the next largest, and so on – then type
I multipliers are just as useful as type II (and usually easier to obtain), because the ratio
of type II to type I income multipliers can be shown to be a constant across all sectors.
ðhÞj =anþ1, j and mðhÞIj ¼ mðhÞj =anþ1, j , mðhÞIIj =mðhÞIj ¼ m
ðhÞj =mðhÞj .
Since mðhÞIIj ¼ m
What is now claimed is that mðhÞIIj =mðhÞIj ¼ k (a constant) for all j. Moreover, k can be
This represents a computational advantage. To
easily found without any need for L.
show that this ratio is a constant requires that we apply some facts on the inverse of
the partitioned matrix L. This is done in Appendix 6.2, for the interested reader. In
ðhÞ1 ¼ 0:570, mðhÞ2 ¼ 0:380,
our illustrative example we found mðhÞ1 ¼ 0:442, m
I
II
ðhÞ2 ¼ 0:489, mðhÞ1 ¼ 1:473, mðhÞ1 ¼ 1:900, mðhÞI2 ¼ 1:520, and mðhÞII2 ¼ 1:956.
m
ðhÞ1 =mðhÞ1 ¼ 0:570=0:442 ¼ 1:29, mðhÞII1 =mðhÞI1 ¼
Therefore (to two decimals), m
ðhÞ2 =mðhÞ2 and
1:90=1:47 ¼ 1:29, and the same values can be found for m
II
I
mðhÞ2 =mðhÞ2 , so k ¼ 1.29 for this example.
Which Multiplier to Use? As a practical matter, the choice between multi ðhÞj ] or by mðhÞIj [and mðhÞIIj ] depends on the
plier effects as measured by mðhÞj [and m
nature of the exogenous change whose impact is being studied. If that change is, for
example, an increase in federal government spending on output of the aircraft sector,
then the most useful figures may be those that convert the total dollar value of new
government spending into total new income earned by households in the economy –
ðhÞj . Using mðhÞ1 ¼ 0:442 and mðhÞ2 ¼ 0:380
the income multipliers mðhÞj and m
from the example, we would estimate that a tariff policy that would increase foreign
demand for sector 1 goods by $100,000 would ultimately lead to an increase of (0.442)
($100,000) ¼ $44,200 in new income earned, while a policy that increased export
demand for sector 2 goods by $100,000 would generate (0.380)($100,000) ¼ $38,000
in new household income earned. If we also attempt to capture the consumer spending
ðhÞ1 and
that is associated with income earned, in a closed model, we would use m
ðhÞ2 and find (0.570)($100,000) ¼ $57,000 and (0.489)($100,000) ¼ $48,900,
m
respectively. In either case, we find that stimulation of export demand for sector
ðhÞj =mðhÞj ¼ k (here
1 output generates the larger effect, as expected, because m
1.29), so the largest simple multiplier will be the largest total multiplier.
The impacts of decreases can be assessed just as easily. Suppose that management
teams in two different industries, i and j, were considering moving a large assembly
plant out of the country because of lower labor costs abroad. If these plants had annual
payrolls of $pi and $pj , respectively, then a measure of the total household income lost
throughout the national economy because of the contemplated relocations would be
given by mðhÞIi pi and mðhÞIj pj or by mðhÞIIi pi and mðhÞIIj pj , if one wants to include
induced households consumption effects. For example, using mðhÞI1 ¼ 1:473 and
mðhÞI2 ¼ 1:520 from our example, if a plant in industry 1 with an annual payroll of
$100,000 were to move out of the country, we would estimate a total income loss of
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252
Multipliers in the Input–Output Model
(1.47)($100,000) ¼ $147,300 throughout the economy. Similarly, if a plant in industry
2, with an annual payroll of $250,000, were to move out of the economy, we could
estimate the total loss to household income throughout the economy because of this
out-movement to be (1.520)($250,000) ¼ $380,000. Again, if we capture consumer
spending using a closed model, our estimates, using mðhÞII1 ¼ 1:900 and
mðhÞII2 ¼ 1:956, would be a (1.900)($100,000) ¼ $190,000 income loss from the outmovement of the plant in industry 1 and a (1.956)($250,000) ¼ $489,000 income
decrease from loss of the plant in industry 2.
Even More Income Multipliers As noted in Section 3.2.3, in an important
early study of Boulder, Colorado, Miernyk et al. (1967) implement a model that
distinguishes between consumption propensities of new residents in a region and those
of established residents. In addition, current residents were divided into income classes
(four in this study), and separate regional consumption functions were estimated for
each income class. The results of this approach have been termed type III income
multipliers, and they are smaller, sector by sector, than the type II income multipliers.
This is to be expected, since marginal consumption coefficients, associated with
current residents’ consumption habits, were smaller than average consumption coefficients, associated with new residents’ consumption habits and which are the exclusive
basis of the type II multipliers.9
Although the ratio of type III to type II income multipliers is not constant across
sectors, the range was only 0.87–0.91, with an average of 0.88. Since the (constant)
ratio of type II to type I income multipliers in this study was 1.34, this means that the
ratio of type III to type I income multipliers averaged 1.18. If a similar narrow range of
ratios of type III to type II income multipliers were found in other regional studies in
which households were similarly disaggregated, it would be possible to approximate
type III income multipliers across all sectors by appropriate “inflation” of the type
I multiplier. In the Boulder study, the inflating factor would be 1.18.
Further, Madden and Batey (1983 and elsewhere) derive a type IV income
multiplier. Like the type III multipliers, these are (generally) larger than type I but
smaller than type II income multipliers. The distinction here is between the spending
patterns of currently employed local residents and the spending patterns of currently
unemployed local residents.10 The models giving rise to these four kinds of multipliers
9
In the Boulder study, the average (aggregate) household consumption coefficient, for the products of all
31 sectors of the local economy, is 0.40. (This is i0 hC , using the household column in the Boulder study.) The
marginal (aggregate) household consumption coefficients for the products of the same 31 sectors, are 0.31,
0.21, 0.16, and 0.02 for the four income classes; their average is 0.1730. (Calculated from tables IV-2 and V-4a,
respectively, in Miernyk et al., 1967.) The type III multipliers in the Boulder study were found not from the
Leontief inverse of a model that had been closed with respect to households in this disaggregated way but
rather in an iterative, round-by-round fashion.
10
Conway (1977) proposed applying the terms “type A” and “type B” multipliers to the numerators of “type I”
and “type II” multipliers. The motivation is to facilitate studies of changes in multiplier values over time. When
the multiplier is a ratio in which both numerator and denominator elements change over time, a change in a
multiplier value can reflect changes in either the numerator or in the denominator, or in both.
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6.2 General Structure of Multiplier Analysis
253
Table 6.2 Model closures with respect to households
Measured Effects
Model
Direct + Indirect
Induced*
Model Closure
1
2
Direct + Indirect
Direct + Indirect
None
Intensive
3
Direct + Indirect
Intensive + Extensive
4
Direct + Indirect
Intensive + Extensive +
Redistributive
None
Single household row and
column
Two household rows and
columns
Three household rows
and columns
Income
Multiplier
Type I
Type II
Type III
Type IV
* Intensive effects are associated with indigenous workers and marginal consumption coefficients.
Extensive effects are associated with in-migrants and average consumption coefficients.
Redistributive effects are associated with unemployed residents and their consumption propensities
based on benefit payments.
are discussed and summarized in Batey and Weeks (1989). Table 6.2 provides an
overview.
6.2.3 Additional (Generalized) Multipliers
It is often argued that value added is a better measure of a sector’s contribution to an
economy since it captures the value that is added by the sector in engaging in
production – the difference between a sector’s total output and the cost of its intermediate inputs. The principles behind value-added multipliers are identical, and the results
in (6.11)–(6.20) again remain valid. The only new information required is a set of
sectoral value-added coefficients, calculated as v0c ¼ v0 x^1 , where v0 is a vector of
value added amounts in each sector. Parallel to (6.13), the matrix of sector specific
value added multipliers is MðvÞ ¼ ^v 0 x^1 L, and the associated vector of economy-wide
value added multipliers is mðvÞ ¼ v0c L ¼ v0 x^1 L.
The same kind of multiplier calculation can relate an exogenous final demand
change to a negative effect caused by production activities rather than a positive one.
For example, knowing the amount of energy needed by each sector [in appropriate
units, e.g., British thermal units (BTUs)] to produce its output, g0 ¼ ½g 1 ; , gn , we
can generate energy-use coefficients as g0c ¼ g0 x^1 , and the associated energy use
matrix and multipliers are Mðg Þ ¼ ^g 0 x^1 L and mðg Þ ¼ g0c L ¼ g0 x^1 L. Or, in terms
of other burdens on an economy, pollution coefficients can be found from information
on pollution of a specific kind (pounds of sulfur dioxide, for example) generated by
each sector during production, d0 ¼ ½d 1 ; , d n . The associated pollution-generation
coefficients would then be dc ¼ d0 x^1 and the associated pollution-generation matrix
and multipliers would be Mðd Þ ¼ d^0 x^1 L and mðd Þ ¼ d0c L ¼ d0 x^1 L. Energy and
environmental extensions and applications are discussed in Chapters 12 and 13.
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254
Multipliers in the Input–Output Model
(Different researchers sometimes use different letters for the same activity – e.g.,
pollution – so the reader must be attentive.)11
6.2.4 Summary
Table 6.3 presents a summary of the results in Sections 6.2.1–6.2.3. Table 6.4 summarizes these multiplier results in a set of generic templates. We use z0 to denote the
row of transaction-like data, for whatever activities are of interest (as in g0 and d0 ); then
z0c is the appropriate row vector of coefficients. When z0 ¼ x0 , z0c ¼ i0 , and we have
traditional output multipliers. [Contrary to subsequent notation, we denoted these as
mðoÞ, for “output,” rather than mðxÞ.] Note that Type I and II output multipliers are
0
meaningless; they are identical to simple and total output multipliers since ^i ¼ I.
When z0 ¼ h0 or z0 ¼ v0 we have household (either income or employment) or valueadded multipliers, respectively. When z0 ¼ g0 we have energy-use multipliers, and so
11 , which is
forth.12 These multipliers are also found in truncated form, as in z0c L
0
0
equivalent to setting zc ¼ ½zc 0 in Table 6.4.
Just as Type I and II income multipliers involve “normalization” through division by
anþ1, j , analysts often choose to normalize a generalized multiplier calculation for
sector j by the initial effect, zcj . Thus, what are termed Type I and II multipliers in
Table 6.4 can be expressed more generally as normalized multipliers in an open
(households exogenous) or a closed (households endogenous) model as
n
P
mðzÞ
n
1 i¼1
¼ z0c L ^z 0c
¼
n
P
zci lij
zcj
ðzÞ
and m
n
11 ^z 0 1 ¼ i¼1
¼ z0c L
c
zcilij
zcj
where, in the closed-model case the multipliers are usually truncated, as shown here.
6.3
Multipliers in Regional Models
In Section 6.2 we presented the basic concepts of various input–output multipliers. All
of these multipliers, which quantify impacts on the economy under study, rely on the
fact that the A matrix (as well as the associated coefficients for income, employment,
value added, etc.) must represent interindustry relationships within that economy. In
particular, if sector i is agriculture and sector j is food processing, aij must represent the
value of inputs of agricultural products produced within the economy (not imported)
per dollar’s worth of output of the food-processing sector in the same economy.
11
Rueda-Cantuche (2011) presents a SUBE (Supply-Use Based Econometric) approach in which multipliers can
be estimated from rectangular supply and use tables without a Leontief inverse. This includes an application to
measurement of European CO2 emissions.
12
Lenzen (2003) presents a comprehensive discussion of “generalized” multipliers, reflecting various kinds of
environmental/ecosystem degradation (land disturbance, harmful emissions) and resource depletion (uses of
energy, land, water, etc.). We will return to this in Section 7.2 on linkages.
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Table 6.3 Input–output multipliers
Output Effects
Income Effectsa
Exogenous Change
Δfj ¼ 1
Δfj ¼ 1
Initial Effect (N) (sector j)
Δxj ¼ 1
Total Effect (T) in open model
(Direct + Indirect)
Simple Multiplier (T/N) (open model)
n
X
l ij
i¼1
Simple income multiplier
mðhÞj ¼
Type I income multiplier
n
X
anþ1, i lij =Δfj
mðhÞIj ¼
i¼1
i¼1
i¼1
[(6.4)]
[(6.11)]
¼ mðhÞj =anþ1, j
[(6.17)]
nþ1
X
l ij
n
X
anþ1, i lij =anþ1, j
i¼1
n
X
¼
anþ1, i lij
i¼1
Total Multiplier ðT =N Þ (closed model)b
anþ1, i l ij
i¼1
Simple output
multiplier
n
X
lij =Δfj
mðoÞj ¼
nþ1
X
Δ in sector j payments to labor ¼ anþ1, j
n
X
i¼1
n
X
¼
l ij
in closed model (Direct
Total Effect (T)
+ Indirect + Induced)
Δxj ¼ 1
anþ1, il ij
i¼1
Total output multiplier
nþ1
X
lij =Δfj
ðoÞj ¼
m
i¼1
nþ1
X
lij
¼
Total income multiplier
nþ1
X
ðhÞj ¼
m
anþ1, il ij =Δfj
i¼1
nþ1
X
¼
anþ1, il ijj ¼ l nþ1, j
i¼1
i¼1
[(6.9)]
[(6.14)]
Type II income multiplier
nþ1
X
anþ1, il ij =anþ1, j
mðhÞIIj ¼
i¼1
ðhÞj =anþ1, j ¼ lnþ1, j =anþ1, j
¼m
[(6.19), (6.21)]
For income effects, anþ1, j ¼ znþ1, j =xj , where znþ1, j ¼ sector j’s payments to households (labor). For employment effects, replace znþ1, j with sector j’s
employment measured in physical units. For value-added effects, replace znþ1, j with sector j’s value-added payments.
b
For truncated total multiplier effects, sum over i ¼ 1, . . . , n rather than i = 1, . . ., n þ 1.
a
255
256
Multipliers in the Input–Output Model
Table 6.4 General multiplier formulas
Multiplier
Matrix Definition
Simple
mðzÞ ¼ z0c L
L
ðzÞ ¼ z0c 11
m
L21
where
z0c ¼ z 0c znþ1, nþ1 =xnþ1
11
½zðtÞ ¼ z0c L
m
1
I
0
mðzÞ ¼ zc L ^z 0c
21 ^z 0c 1
mðzÞII ¼ L
Total
Truncated
Type I
Type II
6.3.1 Regional Multipliers
Very often an analyst is interested in impacts at a regional level. For example, the federal
government may be trying to decide where to award a new military contract and have as
one of its concerns the stimulation of economic development in one or more lessdeveloped regions. A state government may wish to allocate funds for labor skill training
in one or more industries among several counties with currently above-average levels of
unemployment, and so on. In a single-region input–output model, as in Section 3.2, the
^ r A matrix represented one way of trying to capture regional interrelationships
Ar ¼ p
among sectors, and the various kinds of multipliers discussed above would acquire a
spatial dimension by using the elements of Ar and its associated
Leontief inverse.
:15 :25
For example, in Section 3.2 a national table, A ¼
, was modified because
:20 :05
of the assumption that in region r the basic technology of production in sectors 1 and 2 was
essentially the same as that reflected in the two columns of A, but the proportions of inputs
required from sectors 1 and 2 that could be expected to come from within the region were
:8
r
r
r
p1 ¼ 0:8 and p2 ¼ 0:6, so p ¼
, and
:6
:12 :20
1:169 :241
r
r
r
r 1
^ A¼
A ¼p
and L ¼ ðI A Þ ¼
:12 :03
:145 1:061
Hence the regional simple output multipliers, as in (6.4), are mðoÞr1 ¼ 1:314 and
mðoÞr2 ¼ 1:302. Recall from Section 6.2.1 that the output multipliers in the original A
matrix were mðoÞ1 ¼ 1:518 and mðoÞ2 ¼ 1:452. The difference, of course, is due to
the fact that the elements of A have been reduced, using the regional percentages in
pr , to reflect the need for imports to supply some of the necessary production.
Similarly, external output multipliers (not regional – denoted ~r ) are mðoÞ1~r ¼
1:518 1:314 ¼ 0:204 for sector 1 and mðoÞ2~r ¼ 1:452 1:302 ¼ 0:150 for sector
2. The interpretation of these is similar to that for other output multipliers: for each
dollar’s worth of final demand in the region for sector 1 output, 20.4 cents’ worth of
inputs will be needed from firms in all sectors outside of the region. And for each
dollar’s worth of final demand in the region for sector 2 output, this figure is 15 cents.
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6.3 Multipliers in Regional Models
257
Household closure in a regional model presents additional complications. In addition to the fact that some household receive income from non-wage sources such as
savings and non-wage payments (e.g., pensions, private, and/or government), welfare
payments, rental, or dividend income, regional households may also receive transboundary income flows (e.g., wages from employment in an adjacent region).13 If we
have estimates of household inputs, household consumption, and income earned in the
region, the model can be closed with respect to households, allowing calculation of
regional total output multipliers. If we assume that the household input coefficients in
the region are the same as those for the nation as a whole and that these represent labor
supplied by workers living in the region, then ar31 ¼ 0:30, ar32 ¼ 0:25, and ar33 ¼ 0:05.
Also, if we assume that sectors 1 and 2 supply 80 percent and 60 percent, respectively,
of consumer needs (the same percentages as they supply of the needs for production),
then ar13 ¼ ð0:8Þð0:05Þ ¼ 0:04 and ar23 ¼ ð0:6Þð0:40Þ ¼ 0:24 so
2
3
2
3
:20
:04 7
:282
:123 7
6 :12
6 1:217
r 1
r
6
7
6
7
r
¼ 6 :263
A ¼ 6 :12
:03
:24 7and L ¼ I A
1:164
:305 7
4
5
4
5
:30
:25
:05
:453
:395
1:172
|
|
|
|
|
|
|
- - - - - - - - -| - - - -
|
| - - - - - - - - - - - - - - - - -
|
|
|
|
ðoÞr1 ¼ 1:933 and
Therefore, the regional total output multipliers, as in (6.9), are m
ðoÞr2 ¼ 1:841.
m
With information on regional labor inputs (in monetary terms) and household consumption coefficients, various income multipliers could be found for the region. Valueadded multipliers could also be found in exactly parallel ways. No new principles are
involved in assessing multiplier effects with a single-region table instead of a national
table. However, with many-region input–output models, a wider variety of multipliers is
possible. We examine these in the interregional and multiregional cases in turn.
6.3.2 Interregional Input–Output Multipliers
With interregional and multiregional input–output models output, various multiplier
effects can be calculated (a) for a single region (region r), (b) for each of the other
regions, (c) for the “rest of the economy” (aggregated over all regions outside of r),
and (d) for the total, many-region (national) economy.
We illustrate the possibilities using a set of hypothetical data for a two-region model.
Consider the following coefficients matrices for an interregional model with (the same)
three sectors in each region
13
This problem is fully explored in Emonts-Holley, Ross, and Swales (2021), where six alternative approaches to
closure are discussed, several of which require more information than is usually available in input–output
accounts. Wage payments accounted for around 60 percent of total household income in Scotland in 2009
(Emonts-Holley, Ross, and Swales, 2014).
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258
Multipliers in the Input–Output Model
2
A¼
Arr
Asr
6 :150
6
6 :200
6
6
rs
6 :300
A
6
ss ¼ 6
A
6 :075
6
6
6 :050
4
:025
:250
:050
:050
:400
:250
:050
:050
:060
:013
:025
|
|
|
|
|
|
:100
:100
|
|
|
|
|
|
2
L¼
L11
L21
:094
:167
:125
:050
:050
:167
:313
:125
:125
:250
:250
3
:017 7
7
:133 7
7
7
:000 7
7
7
:067 7
7
7
:047 7
5
:133
|
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
and
:021
6 1:462
6
6 :721
6
6
6 :678
L12
¼6
6
L22
6 :318
6
6
6 :177
4
:346
:506
:332
1:514
:761
|
|
|
|
|
|
:578
1:378
:253
:251
|
(6.22)
3
:259
:382
:558
:629
:318
:390
1:428
:649
:268
1:315
:598
:695
:147 7
7
:324 7
7
7
:147 7
7
7
:190 7
7
7
:114 7
5
1:300
| - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
:123
:124
:365
:365
|
|
|
|
|
|
|
(6.23)
Recall from Chapter 3 that we use subscript numbers for elements (submatrices) of a
partitioned interregional L matrix because Lrr and Lss are used for ðI Arr Þ1 and
ðI Ass Þ1 , respectively.
Intraregional Effects For exogenous changes in final demands for
region r goods (the first three elements in a six-element f vector), the elements
in the 3 3 submatrix L11 represent impacts on the outputs of sectors in region r.
Here
2
3
1:462 :506 :332
(6.24)
L11 ¼ 4 :721 1:514 :761 5
:678
:578 1:378
Simple intraregional output multipliers for region r are found as the column sums
of L11 ;
mðoÞrr ¼ i0 ½L11 ¼ ½ 2:861 2:598 2:471 (6.25)
Similarly, for region s,
mðoÞss ¼ i0 ½L22 ¼ ½ 2:294 2:659 1:604 (6.26)
rr If we had household input coefficients in monetary terms for regions r anþ1, j and
ss s anþ1, j , we could find simple intraregional household income multipliers and type
I income multipliers. Note that finding total intraregional output multipliers, household
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6.3 Multipliers in Regional Models
259
income multipliers, or type II income multipliers requires that we have labor input
coefficients (in monetary terms) and household consumption coefficients for four
different matrices. Initially, the input coefficients matrix for region r Arr in
(6.22) – must be closed with respect to households. This then adds a row to Ars and
a column to Asr . The former represents inputs of labor from region r to sector 1, 2, and
3 production in region s (for example, commuters). The latter represents purchases of
outputs of sectors 1, 2, and 3 in region s by consumers located in region r (imports of
consumer goods). For complete consistency, in order to capture income-generating
effects throughout the entire (here, two-region) system, the input coefficients matrix for
region s Ass in (6.22) – should also be closed with respect to households. This then
additionally requires a new row in Asr and a new column in Ars . These new coefficients represent inputs of labor from region s to production in r and purchases by
and L
matrices, for our
consumers in s of goods made in r, respectively. Thus the A
numerical example, would grow from 6 6 to 8 8.
matrix, total intraregional output multipliers, household income multiGiven this L
pliers, and type II income multipliers for region r would be found using the elements
Similarly, using intraregional
from the upper left submatrix – now 4 4 – in L.
physical labor input coefficients or value-added coefficients for both regions, total
intraregional employment or value-added multipliers and type II multipliers could
be found.
Interregional Effects The essence of an interregional (or multiregional)
input–output model is that it includes impacts in one region that are caused by changes
in another region; these are often termed the interregional spillover effects. In our
example, these are reflected in the L12 and L21 matrices; here
2
3
:318 :253 :251
L21 ¼ 4 :177 :123 :124 5
(6.27)
:346 :365 :365
Consider, ðl 21 Þ23 ¼ 0:124; this indicates that for each dollar’s worth of final demand
for the output of sector 3 in region r, 12.4 cents’ worth of output from sector 2 in region
s is required as input.
Thus, in an interregional input–output model, we can calculate simple interregional
multipliers, mðoÞsr
j – the total value of output from all sectors in region s used to satisfy
a dollar’s worth of final demand for sector j in region r. Here,
mðoÞsr ¼ i0 ½L21 ¼ ½ :841 :741 :740 (6.28)
These are output impacts that are transmitted across regional boundaries – here from r
(where the exogenous change occurs) to s (where production occurs). As the reader can
perhaps imagine by now, we have the same set of possibilities for measuring various
interregional income effects, interregional employment effects, and total interregional
effects using the same kinds of calculations as for intraregional effects, now using L21
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260
Multipliers in the Input–Output Model
21 if the regions were closed with respect to households). Interregional effects
(and L
whose origins are in new final demand in region s would be calculated using the
12 ). Here
elements of L12 (or L
mðoÞrs ¼ i0 ½L12 ¼ ½ 1:135 1:401 :618 (6.29)
National Effects Assuming, once again, that there are exogenous increases
in final demands for region r goods and hence in outputs of region r sectors, we can
denote as national effects the sums of columns in both L11 and L21 . (These could
logically also be termed total effects, but we have used total, as contrasted with simple,
for effects that are calculated from a matrix that has households endogenous.)
Arranged as row vectors,
"
#
L
11
¼ ½ 3:702 3:339 3:211 mðoÞr ¼ i0
L21
(6.30)
"
#
L
12
mðoÞs ¼ i0
¼ ½ 3:429 4:060 2:222 L22
For the two-region interregional system, let mðoÞ ¼ ½mðoÞr mðoÞs . Here
mðoÞ ¼ i0 L ¼ ½ 3:702 3:339 3:211
|
|
3:429 4:060 2:222 (6.31)
A policy implication from these figures is that a dollar’s worth of government spending
on the output of sector 2 in region s would have the greatest impact throughout the
two-region economy, as measured by total output (direct plus indirect) required from
all sectors in both regions. Similarly, if the government is interested in acquiring goods
from sector 1 or sector 3, the greatest national (both regions) economic impact will
occur if the purchases are made from firms in region r.
Again, using information on labor inputs or value added in each region, simple and
type I income, employment and value-added effects could be calculated at the national
(all regions) level. Similarly, for a system in which all regions have been closed with
respect to households, total national output, income, employment and value-added
effects and type II multipliers can be found.
Sectoral Effects As a final kind of multiplier, we can find the impact on
sector i throughout the entire country, because of a dollar’s worth of final demand for
sector j in either region. (Since this crosses regional boundaries, it is also a kind of
“national” effect.) Denote this simple output multiplier as mðoÞrij and mðoÞsij . For our
example,
mðoÞr13 ¼ ðl 11 Þ13 þ ðl 21 Þ13 ¼ 0:332 þ 0:251 ¼ 0:583
mðoÞs21 ¼ ðl 22 Þ21 þ ðl 12 Þ21 ¼ 0:268 þ 0:558 ¼ 0:826
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6.3 Multipliers in Regional Models
261
and so on. With additional region-specific information (labor input or value-added
we
coefficients) we could find various simple or type I effects; with elements from L,
would find total multipliers and type II effects. (These kinds of sectoral effects are only
meaningful when each region contains the same sectors.)
More than Two Regions With models of more than two regions, there
are no new principles involved, although the possibilities increase. For example,
with three regions, one can trace interregional effects now in six different ways:
(1) exogenous changes in region 1 affecting outputs in region 2 and/or region 3,
(2) exogenous changes in region 2 affecting outputs in region 1 and/or region 3, and
(3) exogenous changes in region 3 affecting outputs in region 1 and/or region 2.
6.3.3 Multiregional Input–Output Multipliers
All of the multipliers found in the interregional input–output model have their counterparts in the multiregional model. This is to be expected, since the multiregional
model is an attempt to capture all of the connections in the interregional model using a
simpler set of data. Each of the components in the interregional case – for example, Arr
and Ars – has its counterpart estimate – ^c rr Ar and ^c rs As – in the multiregional case.
A thorough exploration of multipliers in the multiregional input–output model can be
found in DiPasquale and Polenske (1980).
The final form of the multiregional model was
x ¼ ðI CAÞ1 Cf
(6.32)
0
is a block diagonal matrix whose submatrices represent regional
As
rr
^c rs
^c
technical (not regional input) coefficients and C ¼ sr
, where the components
^c
^c ss
of the submatrices in C represent flows between regions in the form of proportions of a
commodity in a region that come from within the region and from each of the
other regions.
The important point to be recalled is that in the interregional model the exogenous
sectors represent final demands, wherever located, for goods made by producers in a
particular region. In the multiregional model, the f’s represent demands exercised by
exogenous sectors located in a given region for goods, wherever produced. For a tworegion multiregional model, it is the ^c rr and ^c sr matrices that spatially distribute the
final demand in region r between producers in r and producers in s.
For example, assume that there are two sectors in each of the two regions and that
we want to assess the impact throughout the two-region system of an increase of $100
100
r
and
in final demand for good 1 by households located in region r, so f ¼
0
0
fs ¼
;
0
Ar
Here A ¼
0
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262
Multipliers in the Input–Output Model
2
3
100
7
r 6
6 0 7
f
6
7
f ¼ s ¼6
7
f
4 0 5
0
- - -
Let
:7 0
:2
rs
^c ¼
, ^c ¼
0 :4
0
0
:3
sr
, ^c ¼
:3
0
rr
Then
0
:8 0
ss
, ^c ¼
:6
0 :7
2
rr
^c
C ¼ sr
^c
^c
^c ss
rs
3
6 :7
6
60
¼6
6
6 :3
4
0
0
|
:4
|
0
|
:6
|
:2
|
0
|
07
7
:3 7
7
7
07
5
:7
- - - - - - -| - - - - - -
:8
|
0
|
and the Cf term that postmultiplies ðI CAÞ1 in (6.32) is
2
32
3 2 3
100
:7
0
:2
0
6
76
7 6 70 7
6
76
7 6 7
60
6 0 7 6 0 7
:4
0
:3 7
6
7
6
7¼6 7
Cf ¼ 6
76
7 6 7
6 :3
0
:8
0 7 6 0 7 6 30 7
4
54
5 4 5
|
|
|
|
- - - - - - -| - - - - - - -
-
-
-
-
|
|
0
:6
|
|
0
:7
0
0
The impact of the new $100 is not felt exclusively in region r, rather only $70 (70
percent) is presented as new demand for good 1 made in region r, and $30 (30 percent)
turns out to be new demand for good 1 in region s.
The C matrix distributes the final demands in the multiregional model across supplying
regions in accordance with the percentages embodied in the components of C.
Premultiplication of Cf by ðI CAÞ1 then converts these distributed final demands into
necessary outputs from each sector in each region in the usual way. Thus, the matrix from
which the various multipliers are derived in the multiregional model is ðI CAÞ1 C.
In the numerical illustration in Section 3.4.4, with two regions of three sectors each,
we found
2
3
:447
:300
:478
:418
:153 7
6 1:127
6
7
6 :628
1:317
:606
:552
1:115
:323 7
6
7
6
7
6
7
:512
:526
1:101
:335
:470
:247
1
6
7
ðI CAÞ C ¼ 6
7
6 :625
:369
:250
1:224
:456
:216 7
6
7
6
7
:385
:205
:278
:650
:167 7
6 :238
4
5
:472
:445
:589
:594
:529
1:232
|
|
|
|
|
|
|
- - - - - - - - - - - - - - - - - - - | - - - - - - - - - - - - - - - - - - |
|
|
|
|
|
|
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6.3 Multipliers in Regional Models
263
in (3.24). This matrix plays the same role for multiplier analysis in the multiregional
L11 L12
model that L ¼ ðI AÞ1 ¼
in (6.23) did for the interregional case. We
L21 L22
examine some of these possibilities; the parallels with the interregional case should be
clear, so the illustrations need not be exhaustive. To emphasize the parallel, we define
L11 L12
L ¼ ðI CAÞ1 C ¼
(6.33)
L21 L22
Intraregional Effects Column sums of elements in L11 and L22 are simple
intraregional output multipliers. These multipliers correspond to (6.25) and (6.26); here
mðoÞrr ¼ i0 L11 ¼ ½ 2:267 2:290 2:007 mðoÞss ¼ i0 L22 ¼ ½ 2:096 1:635 1:615 (6.34)
As before, income, employment or value-added multipliers could be found if we had
the requisite additional data. Closing the multiregional model with respect to households, in order to be able to calculate total and type II multipliers, requires the addition
of regional labor input coefficient rows and household consumption coefficient
columns to each of the regional input matrices in A, and it requires estimates of
rs
crr
nþ1, nþ1 , cnþ1, nþ1 , and so on – these are the proportions of household demands for
labor services that are expected to be supplied from within and from outside of each
region. These coefficients would be added to the lower right of each diagonal matrix
A
using overbars to indicate a model in which
Þ1 C,
^c rr , ^c rs , etc. Given ðI C
households have been made endogenous, we could find these various intraregional
multipliers in the usual way, from the upper left and lower right submatrices. Also,
with information on value added in each sector in each region, value-added multipliers
could be found as in the interregional case.
Interregional Effects As in the interregional model, these effects are derived
from L12 and L21 . Here, corresponding to (6.28) and (6.29), we have
mðoÞsr ¼ i0 ½L21 ¼ ½ 1:335 1:199 1:044 mðoÞrs ¼ i0 ½L12 ¼ ½ 1:365 2:003 :723 (6.35)
National Effects Corresponding to (6.30), we have the following simple
output multipliers that reflect production in all sectors in all (here, the two) regions to
support a dollar’s worth of new final demand for a particular good. Here
"
#
L11
r
0
¼ ½ 3:602 3:489 3:051 mðoÞ ¼ i
L21
(6.36)
"
#
L12
s
0
mðoÞ ¼ i
¼ ½ 3:461 3:638 2:338 L22
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264
Multipliers in the Input–Output Model
Thus, a new dollar’s worth of demand from households located in r for good
2 generates a total of $3.49 new output throughout the entire multiregional system.
Arranged in a single row vector, and parallel to (6.31), we have
mðoÞ ¼ i0 L ¼ ½ 3:602 3:489 3:501 3:461 3:638 2:338 (6.37)
and similar kinds of policy implications can be drawn from these figures. For example,
assume that the government could stimulate consumer demand in a particular region
for a particular product (e.g., through tax credits, as for insulation and storm windows
in cold regions). The greatest overall (national) effect, as measured by these simple
national output multipliers, would come from consumer demand in region s for good 2.
Sectoral Effects Finally, as with the interregional model, we can assess the
impact on sector i throughout the economy of one dollar’ s worth of new final demand
in region r for good j. For example, mðoÞr13 ¼ ðℓ 11 Þ13 þ ðℓ 21 Þ13 ¼ 0:300 þ 0:250 ¼
0:550, mðoÞs21 ¼ ðℓ 22 Þ21 þ ðℓ 12 Þ21 ¼ 0:278 þ 0:552 ¼ 0:830, and so on.
Final Demand for Goods Made in a Particular Region If one is using the
version of the multiregional input–output model in which impacts of new regionspecific final demands are being assessed (as in the example of a foreign airline’s new
order for Boeing jetliners made in the state of Washington), where
x ¼ ðI CAÞ1 f ∗
as in (3.25) in Chapter 3, then all of the multiplier calculations outlined earlier in this
section would be found from the elements in ðI CAÞ1 rather than ðI CAÞ1 C.
The ðI CAÞ1 matrix for this numerical example was given in (3.26). The interested
reader may wish to find the various multipliers, as in (6.34)–(6.37).
More than Two Regions As before, with models of more than two regions,
there are no new principles involved, although the possibilities for multiplier calculations increase. For example, with three regions, there are three possible settings in
which to calculate various intraregional multiplier effects and six in which to calculate
interregional effects.
In Section 3.4.6 we introduced a three-sector, three-region aggregation of the
Chinese 2012 multiregional model. The L ¼ ðI CAÞ1 C matrix for that model is
repeated in Table 6.5. (This was Table 3.9 in Chapter 3.) The regional aggregations
used in this table result in very large geographic aggregates, and the relative uniformity
of the simple output multipliers across regions, as indicated in the tables to follow,
reflects this. Simple intra- and interregional output multipliers for this Chinese model
are presented in Table 6.6. In addition, simple national (all-region) multipliers
are shown.
For example, a ¥1 change in final demand in the East for manufacturing (sector 2)
requires ¥0.143 from all sectors in the Central region and ¥0.112 from the West. In
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Table 6.5 Leontief inverse matrix, L, for the Chinese multiregional economy, 2012
East
2012
Central
West
1
2
3
1
2
3
1
2
3
East
1
2
3
Agric & Mining
Manufacturing
Services & Utils
1.19521
.53447
.22644
.15494
2.11099
.31853
.07136
.56038
1.42975
.02550
.16442
.06721
.03916
.34605
.13190
.02290
.20382
.10026
.02092
.14750
.04831
.03537
.30560
.08097
.02339
.21701
.07621
Central
1
2
3
Agric & Mining
Manufacturing
Services & Utils
.02404
.04890
.01525
.02645
.09304
.02376
.02453
.06232
.02339
1.23296
.42835
.22328
.24438
1.79622
.28117
.12780
.48235
1.37415
.01816
.06010
.01729
.02401
.10356
.02505
.01835
.08374
.02398
West
1
2
3
Agric & Mining
Manufacturing
Services & Utils
.02761
.03425
.01596
.03408
.05624
.02130
.02703
.04282
.02384
.03013
.05199
.02751
.03308
.06910
.03370
.02655
.05153
.03026
1.24807
.30104
.20045
.28120
1.61418
.27220
.13249
.42055
1.35511
265
266
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Table 6.6 Simple intra- and interregional output multipliers for the Chinese multiregional input–output system, 2012
Region and Sector Experiencing a One-Unit Change in Final Demand
East
Central
West
2012
1
2
3
1
2
3
1
2
3
East
Central
West
Nation
1.95612
.08820
.07783
2.12214
2.58446
.14325
.11162
2.83933
2.06149
.11024
.09369
2.26541
.25714
1.88459
.10964
2.25136
.51711
2.32176
.13587
2.97475
.32697
1.98430
.10835
2.41963
.21674
.09555
1.74956
2.06185
.42194
.15262
2.16758
2.74214
.31660
.12607
1.90816
2.35083
6.4 Miyazawa Multipliers
267
view of the sectoral breakdown used in this model it is not surprising that manufacturing has the largest simple output multiplier in each region and in the nation as a whole,
or that the services and utilities sector has the second-largest multipliers, with agriculture and mining a rather distant third.
In terms of regional dependencies, we see from Table 6.6 that the Central region is
much more dependent on the East than on the West for the inputs that would be needed
to satisfy one unit of final demand in each of the sectors in the Central region – from
the sums of the three elements in the East row for the Central region, 1.101, vs. the
sums of the three elements in the West row, 0.354. Similar aggregate measures can be
derived for the other regions.
Sector-specific simple output multipliers, mðoÞrij , are shown in Table 6.7. There is a
great deal of uniformity across regions. For example, ¥1 of new demand for agriculture
and mining output by households located in the East, Central, or West regions
generates a national impact in terms of ¥ worth of new output in sector 1 of 1.247,
1.289, or 1.287 in the three regions, respectively. Similarly, ¥1 worth of new final
demand for services and utilities generates a need for inputs of ¥0.666, 0.738, or 0.721
worth of new manufacturing output in the three regions, respectively. The figures are
generally similar in other rows of Table 6.7. Again, this is primarily because of the very
large sizes of the three regions in this model illustration.
Hioki (2005) presents an empirical analysis for the Chinese economy, using the
same Chinese MRIO data but at greater levels of disaggregation. This is an analysis of
the magnitude of interregional spread or “trickle down” effects, especially from eastern
Chinese (coastal) regions to the less developed western (inland) regions. The study
calculated intraregional and interregional simple output multipliers for an eight-region,
17-sector version of the CMRIO model. Illustrative of the kinds of conclusions drawn
in this study is the observation that around 20 percent of the total output in the Central
region is induced by final demands of the coastal regions (p. 170). This suggests that
the government’s strategy, begun during the 1980s, favoring development of the
coastal regions (which it was thought would then lead to spillovers inland) has
“actually started to work to a certain extent” (p. 171).14
6.4
Miyazawa Multipliers
The important work of Miyazawa (1976) on endogenizing households in an input–
output model generates various multiplier matrices.15 A comprehensive overview of
the explicit demographic–economic interactions in the Miyazawa structure and its
applications can be found in the collection of papers in Hewings et al. (1999). In this
section we depart from some of the notation used elsewhere in this book, in order to be
14
15
We examine some of the details of construction of this multiregional model in Section 10.7.
The definitive work is Miyazawa (1976), although there were several articles preceding that monograph. Most
of these were in the Hitotsubashi Journal of Economics in the 1960s and early 1970s and were not widely
known outside of Japan. Later work by Sonis and Hewings (1993, 1995) on extended multiregional Miyazawa
multipliers can also be found in that journal, as well as elsewhere (e.g., Sonis and Hewings, 2000).
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268
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Table 6.7 Sector-specific simple output multipliers for the Chinese multiregional input–output system, 2012
Sector and Region Experiencing a One-Unit Change in Final Demand
Agriculture and Mining (1)
2012
Agric. & Mining
Manufacturing
Serv. & Utils.
Manufacturing (2)
Services & Utilities (3)
East
Central
West
East
Central
West
East
Central
West
1.24687
.61763
.25765
1.28859
.64476
.31800
1.28715
.50864
.26605
.21547
2.26026
.36359
.31661
2.21137
.44676
.34058
2.02334
.37822
.12291
.66553
1.47697
.17725
.73771
1.50466
.17423
.72130
1.45530
6.4 Miyazawa Multipliers
269
consistent with that used by Miyazawa, since virtually all subsequent discussion and
application of the Miyazawa framework has continued to use his notation. Specifically,
this means that we will now define B ¼ ðI AÞ1 (instead of L, since Miyazawa uses
L for another purpose, as we will see in Section 6.4.1).
6.4.1 Disaggregated Household Income Groups
We assume that households can be separated into q distinct income-bracket groups and
that payments by producers to wage earners in each of those groups can be identified.
Let V ¼ ½vgj , where vgj represents income paid to a wage earner in income bracket g
ðqnÞ
(g ¼ 1, . . , q) per dollar’s worth of output of sector j. This is a generalization (to q
rows) of the single row of household input coefficients or labor input coefficients in
Chapter 2, hr ¼ ½anþ1, 1 ; . . . , anþ1, n . Similarly, let C ¼ ½cih , where cih is the amount
ðnqÞ
of sector i’s product consumed per dollar of income of households in income group h
(h ¼ 1, . . . , q); this is a generalization (to q columns) of the single column of
2
3
a1, nþ1
6
7
household consumption coefficients in Chapter 2, hc ¼ 4 ... 5, and yet another
an, nþ1
use for C in input–output discussions. So, the augmented matrix of coefficients is
2
3
A
C
¼ 4 ðnnÞ ðnqÞ 5, and the expanded input–output system is
A
V
0
ðqnÞ
ðqqÞ
A
x
¼
V
y
C
0
∗
x
f
þ
y
g
(6.38)
where y is a vector of total income for each of the income groups, f ∗ is a vector of
ðn1Þ
ðq1Þ
final demands excluding household consumption (now endogenized), and g is a
ðq1Þ
vector of exogenous income (if any) for the income groups.
Assume that g ¼ 0; then the two matrix equations in the system in (6.38) are
ðq1Þ
x ¼ Ax þ Cy þ f ∗ and y ¼ Vx
(6.39)
From (6.38),
x
IA
¼
y
V
C
I
1 ∗ f
0
(6.40)
Using results on inverses of partitioned matrices (Appendix A) it is not difficult to
show that the elements of the partitioned inverse in (6.40) can be expressed as
∗
1
1
x
f
VB
BC
ð
I
VBC
Þ
B½I
þ
C
ð
I
VBC
Þ
(6.41)
¼
y
0
ðI VBCÞ1 VB
ðI VBCÞ1
where, as noted, B ¼ ðI AÞ1 .
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270
Multipliers in the Input–Output Model
This can be simplified if, following Miyazawa, we define VBC ¼ L and
KðI LÞ1 ¼ ðI VBCÞ1 , so that
2
3
B(I þCKVB) BCK " ∗ #
ðnqÞ
ðnnÞ
x
6
7 f
¼4
(6.42)
5
y
KVB
K
0
ðqnÞ
ðqqÞ
Miyazawa defines L as the matrix of “inter-income-group coefficients” and K as the
“interrelational income multiplier” matrix. A typical element of L is lgh ¼ vgi bij cjh ; this
shows the direct increase in the income of group g resulting from expenditure of an
additional unit of income by group h. Reading from right to left, household demand
(expenditure) of cjh by group h for the output of sector j requires bij cjh in output from
sector i and this, in turn, means income payments from sector i in the amount of
vgi bij cjh to households in group g. Similarly, each element in K ¼ ðI LÞ1 indicates
the total increase (direct, indirect, and induced) in the income of one group that results
from expenditure of an additional unit of income by another group. (An illustration of
this approach can be found in the matrix of interrelational income multipliers, K, for
11 income groups in the USA for 1987 that is shown in Rose and Li, 1999.)
From (6.42),
x ¼ BðI þ CKVBÞf ∗
(6.43)
y ¼ KVBf ∗
(6.44)
and
In (6.43), the effect of final demands on outputs is seen to be the product of two distinct
matrices. The first is the Leontief inverse of the open model, B. The second is
(I þ CKVB); this augments the final demand stimulus, If ∗ , by CKVBf ∗, which
endogenizes the total income spending effect. Again, starting at the right, Bf ∗ generates
the initial output (without household spending), VBf ∗ indicates the resultant initial
income payments to each group, KVBf ∗ multiplies that into total income received in
each group – this is exactly what is described by the result in (6.43) – and, finally,
CKVBf ∗ translates that received income into consumption (demand) by each group on
each sector’s output. Miyazawa denotes KVB the “multi-sector income multiplier”
matrix (or the “matrix multiplier of income formation”), indicating the direct, indirect,
and induced incomes for each income group generated by the initial final demand.
6.4.2 Miyazawa’s Derivation
Miyazawa first derives the results on the interrelational multiplier matrix without
reference to partitioned matrices [in Miyazawa (1976, chapter 1, sections II(2)–III
(1)); the partitioned inverse structure appears later in chapter 1, section III(3)]. He
makes extensive use of partitioned matrices later in the book – especially in Part 2 on
internal and external matrix multipliers. This is a direction that has been explored and
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6.4 Miyazawa Multipliers
expanded considerably in much of the work of Sonis, Hewings, and others [summarized in Sonis and Hewings (1999), which also contains an extensive set of references to
their work]. A second direction of research that extends the input–output framework to
incorporate interactions between economic and demographic components is associated
with the many publications of Batey, Madden, and others [summarized in Batey and
Madden (1999), again with many references].
We present Miyazawa’s initial approach here primarily for completeness, and
because the results are often discussed (briefly) in this form in the literature. He begins
with
x ¼ Ax þ CVx þ f ∗
from (6.39). From this,
x ¼ ðI A CVÞ1 f ∗
(6.45)
1
and with B ¼ ðI AÞ , straightforward matrix algebra gives
ðI A CVÞ ¼ B1 CV BB1 ¼ ðI CVBÞB1
Substituting into (6.45),
1
x ¼ [ðI CVBÞB1 ] f ∗
and, from the rule for inverses of products,
x ¼ BðI CVBÞ1 f ∗
(6.46)
In this form, we find the original Leontief inverse, B, postmultiplied by ðI CVBÞ1,
which Miyazawa termed the “subjoined inverse matrix.”
A further variation is possible and is sometimes used. Starting with (6.46) and, as
earlier, with VBC ¼ L and K ¼ ðI LÞ1 , then
KðI VBCÞ ¼ I
Premultiply both sides by C and postmultiply both sides by VB,
CKðI VBCÞVB ¼ CVB or
CKðVB VBCVBÞ ¼ CVB
Factor out VB to the left and then subtract both sides from I, giving
I CKVBðI CVBÞ ¼ I CVB or
I ¼ CKVBðI CVBÞ þ I CVB
Regrouping terms
I ¼ ðI þ CKVBÞ ðI CVBÞ
and so, from the fundamental definition of an inverse,
ðI CVBÞ1 ¼ ðI þ CKVBÞ
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Multipliers in the Input–Output Model
Putting this result into (6.46) gives
x ¼ BðI þ CKVBÞf ∗
(6.47)
as in (6.43).
Miyazawa suggests that if labor input coefficients, in V, and household consumption
coefficients, in C, are less stable than interindustry coefficients (in A and consequently
in B), there is an advantage to using the format in (6.47) instead of (6.45). Namely, a
revised subjoined inverse, ðI CVBÞ1 , whose order is n, can be found by using K,
whose order is q “. . . [which] in most cases is very much smaller than n . . .”
(Miyazawa, 1976, p. 7). However, inverting large matrices is no longer the concern
that it was in the 1970s.
From (6.47), household income, y ¼ Vx, is seen to be
y ¼ VBðI þ CKVBÞf ∗ ¼ ðI þ VBCKÞVBf ∗ ¼ ðI þ LKÞVBf ∗
But since K ¼ ðI LÞ1 , ðI LÞK ¼ I, LK ¼ K I, so ðI þ LKÞ ¼ K, and
y ¼ KVBf ∗
(6.48)
as in (6.44).
6.4.3 Numerical Example
We expand the numerical example from Chapter 2, assuming a three-sector economy
with households divided into two income groups. Let the augmented coefficients
matrix be.
2
3
:25
:05
:1
:05 7
6 :15
6
7
:05
:4
:2
:1 7
6
6 :2
7
7
¼ A C ¼6
A
6 :3
:25
:05
:01
:1 7
V 0
6
7
6
7
:1
:08
0
0 7
6 :05
4
5
:12
:05
:1
0
0
|
|
|
|
|
|
|
- - - - - - - - - - - - - - |- - - - - - - - |
|
|
|
In particular, laborincome coefficients
for the two household groups are given in the
:05 :1 :08
, and consumption
coefficients
for those same two
two rows of V ¼
2
3
:12 :05 :1
:1 :05
groups are given in the two columns of C ¼ 4 :2
:1 5.
:01 :1
2
3
1:3651 :4253 :2509
Given V, C, and B ¼ ðI AÞ1 ¼ 4 :5273 1:3481 :5954 5, the relevant Miyazawa
:5698 :4890 1:2885
matrices are easily found to be
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273
6.4 Miyazawa Multipliers
VBC ¼
:0574 :0454
:0601 :0480
and K ¼ ðI VBCÞ1 ¼
1:0642 :0507
:0671 1:0536
For example, in this illustration, a direct increase of $1 in income to households in
group 1 leads to a 6.7 cent ðk 21 Þ increase in income payments to households in group 2.
Similarly,
:1898 :2162 :1960
KVB ¼
:2716 :1894 :2106
In this case, for example, an additional unit of final demand for the goods of sector
1 generates 27.16 cents in new income for group 2. Furthermore,
2
3
2
3
1:4445 :4994 :3234
:2476 :1545
BðI CVBÞ1 ¼ 4 :6496 1:4609 :7062 5 and BCK ¼ 4 :3642 :2492 5
:6577 :5644 1:3648
:4923 :2258
(The reader can make appropriate interpretations of the elements in each of these
matrices.)
In this case, the Leontief inverse for the augmented system can easily be found
directly; it is16
2
3
1:4445
:4994
:3234
:2476
:1545
6
7
6
7
6
1:4609
:7062
:3642
:2492 7
6 :6496
7
7
Þ1 ¼ B
¼ B11 B12 ¼ 6
ðI A
6
7
:6577
:5644
1:3648
:1923
:2258
21 B
22
B
6
7
6
7
:2162
:1960
1:0642
:0507 7
6 :1898
4
5
:2716
:1894
:2106
:0671
1:0536
|
|
|
|
|
|
|
- - - - - - - - - - - - - - - - - - - - - -| - - - - - - - - - - - - - |
|
|
|
are exactly as expected, namely K ¼ B
22,
and the correspondences with elements in B
12 and BðI CVBÞ1 ¼ B
11 .
21 , BCK ¼ B
KVB ¼ B
6.4.4 Adding a Spatial Dimension
We saw in Chapter 3 that interregional or multiregional input–output models were
conveniently represented in partitioned matrix form. To incorporate the Miyazawa
structure into an IRIO- or MRIO-style model, assume that we have p regions
ðk; l ¼ 1; . . . ; pÞ with n sectors ði; j ¼ 1; ; nÞ each, and that we have identified q
household income groups ðg; h ¼ 1; . . . ; qÞ in each region. Then the augmented A
matrix would be
16
rather than L
to be consistent with the Miyazawa literature.
Again, we use B
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Multipliers in the Input–Output Model
2
A
ðnpnpÞ
¼6
A
4
V
ðpqnpÞ
3
C
ðnppqÞ 7
0
5
ðpqpqÞ
where
2
A11
6 ðnnÞ
6
..
A ¼ 6 ...
.
4
ðnpnpÞ
Ap1 ðnnÞ
A1p
3
2
ðnnÞ
C11
7 h i
6 ðnqÞ
6
kl
.. 7
..
¼ aij , C ¼ 6 ...
. 7
.
5
4
ðnpqpÞ
App
Cp1 ðnnÞ
ðnqÞ
Cp1
3
ðnqÞ
7 .. 7
¼ cklih ,
. 7
5
Cpp
ðnqÞ
and
2
V11
6 ðqnÞ
6
..
V ¼ 6 ...
.
4
ðpqnpÞ
Vp1 ðqnÞ
V1p
3
ðqnÞ
7 h i
.. 7
¼ vklgj :
. 7
5
Vpp
ðqnÞ
Notice that consumption coefficients require knowledge of the spending habits of
consumers in each income group in each region on goods from each sector in each
region. Similarly, the labor input coefficients require knowledge on payments to
laborers in each income group in each region by each sector in each region.
The elements in the partitioned inverse in (6.42) will have the same dimensions as
namely
A,
2
3
BðI þ CKVBÞ BCK " ∗ #
6
ðnppqÞ 7 f
x
ðnpnpÞ
7
¼6
4
5
y
0
KVB
K
ðpqnpÞ
ðpqpqÞ
Clearly, this is potentially very demanding of data. However, an illustrative application
can be found in Hewings, Okuyama, and Sonis (2001) for a 53-sector, four-region
model (Chicago and three surrounding suburbs), without division into income groups –
that is, n ¼ 53, p ¼ 4, and q ¼ 1. In this case the income formation impacts are across
regions rather than income groups. In particular, K is a 4 4 matrix; as is shown in
Table 6.8.17
Reading down column 1 for illustration, we find that, from an increase of $1 in
income in Region 1, an additional $0.23 is generated in Region 1, $0.11 in Regions
2 and 3, and $0.44 in Region 4. Column sums have an interpretation similar to the
more usual output multipliers; they indicate the new income generated throughout the
four-region system (Chicago metropolitan area) of an additional $1 in income in the
region at the top of the column. Row sums are a measure of additional income in each
17
For additional data and details on this application, see Hewings and Parr (2007).
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6.5 Gross and Net Multipliers in Input–Output Models
275
Table 6.8 Interrelational interregional income multipliers
Region of Income Origin
Region of Income Receipt
1
2
3
4
Row Total
1
2
3
4
Column Total
1.23
.11
.11
.44
1.81
.12
1.28
.03
.56
1.99
.16
.13
1.06
.50
1.85
.07
.05
.01
1.77
1.90
1.57
1.57
1.14
3.28
Source: Hewings, Okuyama, and Sonis (2001, table 9)
region at the left as a result of a $1 income increase in each region. (As with row sums
of the usual Leontief inverse, these are generally less useful results than the column
sums.) Often, results in empirically derived interrelational multiplier matrices are
normalized in some way to account, for example, for differences in sizes of the regions
being studied. A complete interregional Miyazawa analysis would require that we
distinguish several income brackets in each region (that is, q > 1) and then create
consumption coefficients and labor input coefficients for each of those brackets (in
each region).
6.5
Gross and Net Multipliers in Input–Output Models
6.5.1 Introduction
Leontief’s earliest formulations (for the USA in 1919, 1929, and 1939) were in terms
of “net” accounts. The fundamental balance equations had no zii or aii terms; in the
empirical tables the on-diagonal elements were zero.
[The interindustry transactions table] would naturally have many empty squares. Those lying along
the main diagonal are necessarily left open because our accounting principle does not allow for
registration of any transaction within the same firm . . . (Leontief, 1951, p. 13)
The output of an industry . . . is defined with exclusion of the products consumed by the same
industry in which they have been produced. Thus a11 ¼ a22 ¼ ¼ aii ¼ ¼ amm ¼ 0
by definition. (Leontief, 1951, p. 189)
The 1947 US input–output tables discussed and published in Evans and Hoffenberg
(1952) include on-diagonal transactions, coefficients, and inverse elements; in that
sense these tables are “gross.” They point out that the inverse figures can be adjusted to
exclude intra-sector transactions, but they do not suggest that as a preferable alternative.18 In Leontief et al. (1953, chapter 2 by Leontief ) the equations in the text are
gross, but the tables and the equations in the Mathematical Note to chapter 2 are net. In
18
In contrast, Georgescu-Roegen (1971) argues that diagonal elements in an input–output model (“internal
flows”) must be suppressed.
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Multipliers in the Input–Output Model
virtually all later publications (for example, Leontief, 1966, chapters 2 and 7) ondiagonal elements are included.19 (For a thoughtful discussion of net and gross input–
output accounts, see Jensen, 1978.) This net/gross distinction led to the concept of
input–output “net” multipliers, which we explore in Section 6.5.2.
6.5.2 Multipliers in the Net Input–Output Model
We consider only square systems. Generating a net model simply means that the
main diagonals of Z and A contain only zeros, and that the gross output vector is
reduced by the amount of each sector’s intraindustry transactions. As usual, denote
^ and
^ the diagonal matrix containing the elements zii . Then let Znet ¼ Z Z,
by Z
^ this latter is a diagonal matrix of sectoral outputs in the net system from
x^net ¼ x^ Z;
which on-diagonal (intrasectoral) transactions have been removed.20 As usual, input
coefficients are found for the net system as
^ x^ Z
^ 1
Anet ¼ Znet ðx^net Þ1 ¼ Z Z
and
^ x^ Z
^ 1
ðI Anet Þ ¼ I Z Z
We now examine an alternative expression for the right-hand side. [This demonstration appears to have originated in Weber, 1998 (in German). It does not seem to be
widely known, at least outside the German-speaking world.] Using the observation that
^ x^ Z)
^ 1 ¼ I, it can be shown that21
(x^ Z)(
^ 1
ðI Anet Þ ¼ ½ðI AÞx^ x^ Z
Taking the inverse of both sides,
1
^ 1 g
Lnet ¼ ðI Anet Þ1 ¼ f½ðI AÞx^(x^ Z)
and using the matrix algebra rule for inverses of products (for appropriately sized
matrices) that ðMNPÞ1 ¼ P1 N1 M1 ,
^ x^1 ðI AÞ1 ¼ x^net x^1 L
Lnet ¼ (x^ Z)
(6.49)
Early input–output tables in the UK (for example, for 1954 and 1963) were presented in “net” form (UK,
Central Statistical Office, 1961, 1970). Fifteen-sector versions of these tables appear in Allen and Lecomber
(1975) and Barker (1975).
20
Alternative notation uses Z instead of Znet , and similarly for Anet and xnet . We avoid that convention because it
becomes cumbersome when the vector xnet needs a hat to indicate the associated diagonal matrix – and a “^”
on top of a “∨” is just too much.
21
This particular expression for the identity matrix may seem unmotivated, but it cleverly allows for a significant
rewriting of the expression for (I – Anet ). For the interested reader, the derivation is:
^ x^ Z
^ 1 Z Z
^ x^ Z
^ 1 ¼ x^ Z
^ Z Z
^ x^ Z
^ 1 ¼ ðx^ ZÞ x^ Z
^ 1 ¼
ðIAnet Þ ¼ x^ Z
^ 1 :
^ 1 ¼ ½ðIAÞ^
IZ^
x 1 x^ x^ Z
x x^ Z
19
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6.5 Gross and Net Multipliers in Input–Output Models
277
from which
ðx^net Þ1 Lnet ¼ x^1 L
(6.50)
^ L, where A
^ ¼ Z^
^ x^1 L ¼ I A
^ x 1 .]22
[Notice from (6.49) that Lnet ¼ (x^ Z)
Consider household income multipliers for the two systems. Given a vector of
total household income by sector, zh ¼ ½znþ1, 1 ; . . . ; znþ1 , n , then h ¼ zh x^1 and
hnet ¼ zh ðx^net Þ1 are the vectors of earnings coefficients in the gross and net systems,
respectively. From (6.50),
zh ðx^net Þ1 hnet ¼ zh x^1 L
or
hnet Lnet ¼ hL
Thus, the income multipliers in the two systems are equal, and therefore for studies in
which these kinds of multiplier results are of interest, it makes no difference which
model is used.
This result is equally valid for most other multipliers – value-added, household
income, pollution-generation, energy use, etc. – associated with productive activity
(Table 6.4). The only exception is for output multipliers – m(o) ¼ i0 L and
mðoÞnet ¼ i0 Lnet ; they will not be equal,23 since from (6.49) Lnet ¼ x^net x^1 L.
However, the transformation from one to the other is straightforward, namely
mðoÞnet ¼ i0 Lnet ¼ i0 x^net x^1 L
or
mðoÞnet ¼ i0 L ¼ i0 x^ðx^net Þ1 Lnet
(Recall that order of multiplication of diagonal matrices makes no difference.)
Numerical Example We use the previous example but now disregard the
fact that sector 3 is households, and simply treat this as a general three-sector
model illustration.
2
3
2
3
2
3
0 500 50
1,000
150 500 50
^ ¼ 4 200 0 400 5. If x ¼ 4 2,000 5,
Let Z ¼ 4 200 100 400 5 so Znet ¼ Z Z
300 500 0
1,000
300 500 50
2
3
2
3
2
3
850
0 :2632 :0526
:15 :25 :05
A ¼ 4 :2 :05 :4 5; xnet ¼ 4 1,900 5, Anet ¼ Znet ðx^net Þ1 ¼ 4 :2353 0 :4211 5.
950
:3529 :2632 0
:3 :25 :05
22
This fact was noted by Evans and Hoffenberg (1952, p. 140) who used a verbal argument and not a matrix
algebra demonstration.
23
Except for the trivial and uninteresting case when x =xnet .
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Multipliers in the Input–Output Model
2
3
2
3
1:3651 :4253 :2509
1:1603 :3615 :2133
Then L¼4 :5273 1:3481 :5954 5 and Lnet ¼ ðIAnet Þ1 ¼ 4 :5010 1:2807 :5656 5.
:5698 :4890 1:2885
:5414 :4646 1:2241
In this case,
mðoÞ ¼ i0 L ¼ ½ 2:4623
2:2624 2:1348 mðoÞnet ¼ i0 Lnet ¼ ½ 2:2026 2:1067
2
2:0030 3
0
0
1:0526
0 5 and so mðoÞ ¼ i0 x^ðx^net Þ1 Lnet ¼
0
1:0526
1:1765
Here x^ðx^net Þ1 ¼ 4 0
0
2
3
1:1603 :3615 :2133
½ 1:1765 1:0526 1:0526 4 :5010 1:2807 :5656 5 ¼ ½ 2:4623 2:2624 2:1348 :5414 :4646 1:2241
as expected.
Finally, let zh ¼ ½100 120 80 (household income payments); then
h ¼ ½ 0:10 0:06 0:08 and hnet ¼ ½ 0:1176 0:0632 0:0842
from which
hL ¼ hnet Lnet ¼ ½ :2137 :1625 :1639 again as expected.
6.5.3
Additional Multiplier Variants
(Indirect Effects)/(Direct Effects) A number of analysts have taken the view
that multipliers should not include the initial stimulus, as they do when the basic definition
is “total effects”/“direct effects.” For example, for output multipliers this means the $1 of
new final-demand for sector j which turns into $1 of new sector j output. The usual
resolution is simply to subtract 1 from each of the elements in m(o). This is equivalent to
replacing L by (L – I) in the formula for m(o), since i0 ðL IÞ ¼ i0 L i0 I ¼ mðoÞ i0 .
(For example, see Oosterhaven, Piek, and Stelder, 1986.)24 Of course this will not change
24
Since ðL IÞ ¼ L I L1 ¼ LA or ðL IÞ ¼ I L1 L ¼ AL these modified multipliers could also be
found as i0 AL or i0 LA (see de Mesnard, 2002, or Dietzenbacher, 2005).
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6.5 Gross and Net Multipliers in Input–Output Models
279
the rankings of the sectors, but it certainly has implications for other kinds of calculations
in which the multipliers are used.
The same adjustment [subtracting 1 or using (L – I)] is appropriate for any Type
I or Type II multiplier (Tables 6.3 and 6.4). As an example, when r ¼ h, the
^ 1 ^ 1 ¼ hLh
Type I multiplier, mðhÞ ¼ hL would be converted to hðL IÞh
^ 1 ¼ mðhÞ i0 .
hlh
“Growth Equalized” Multipliers Policy-makers may wish to know the
impact on a particular sector of a general expansion in final demand in all sectors
(for example, to help identify “bottlenecks”) or of changing patterns of final demand.
One approach involves what have been called “growth-equalized” multipliers. (See,
for example, Gray et al., 1979, and Gowdy, 1991, for these and many additional
multipliers.) The motivation is clear: “. . . size variation among economic sectors
prevents meaningful comparisons of multipliers . . . to add $1 of output to some sectors
represents a much larger rate of growth than it would for other sectors” (Gray et al.,
1979, pp. 68, 72, respectively).
Consider output multipliers; again, the principles are the same for all the other
possible multipliers. The idea begins with the multiplier matrix, MðoÞ ¼ L. Row
sums, MðoÞi ¼ Li, indicate output effects in each sector when final demand for each
sector increases by $1.00. This is generally considered an unlikely scenario; an
obvious variation is to posit an unequal increase in final demand across sectors.
1
1
For example, instead of Li one could use Lhf ði0 f Þ i, where hf ði0 f Þ i is a diagonal
matrix
P showing each sector’s final demand as a proportion of total final demand,
fj = j fj ; that is, a measure of relative sector size (or importance). (Base-year output
P
0 1
proportions, xj = j xj , could also be used.) Element
P (i, j) in the matrix Lhf ði f Þ i
shows the effect on sector i output of a $ fj = j f increase in j’s final demand.
1
Then Lhf ði0 f Þ ii shows the multiplier effect on each sector’s output of a $1 finaldemand increase distributed across sectors according to their proportion of total
final demand.
Another possibility is to use equal percentage, not absolute, demand increases across
sectors. This is the “growth equalization.” For example, elements of the column vector
[M(o)](0.01)f ¼ (0.01)Lf indicate output effects in each sector when final demand
for each sector increases by one percent, and (0.01)i0 Lf ¼ (0.01)[m(o)]f indicates
the economy-wide total output generated. We illustrate with the same three-sector
figures.
For the example,
2
3
2
3
:1714
0
0
300
1
:7429
0 5
f ¼ 4 1,300 5 and hf ði0 f Þ i ¼ 4 0
0
0
:0857
150
In this case,
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Multipliers in the Input–Output Model
2
3
2
3
:2340 :3159 :0215
:5714
1
1
Lhf ði0 f Þ i ¼ 4 :0904 1:0015 :0510 5 and Lhf ði0 f Þ ii ¼ 4 1:1429 5
:0977 :3633 :1104
:5714
Using a one percent increase for the growth equalization illustration,
2
3
4:0953 5:5284
:3764
Lhð0:01Þf i ¼ 4 1:5820 17:5250 :8930 5
1:7095 6:3576 1:9328
and
i0 Lhð0:01Þf i ¼ ½ 7:3868 29:4110 3:2022 Recall that for this example the simple output multipliers were
mðoÞ ¼ i0 L ¼ ½ 2:4623
2:2624 2:1348 and we see that the relative importance of the sectors is altered (now it is
final demand for sector 2 that is the most stimulative; previously – in m(o) – it was
sector 1).
Another Kind of Net Multiplier The multipliers in Tables 6.3 and 6.4 are
designed to be used with (multiplied by) final demand. Oosterhaven and Stelder
(2002a, 2002b) have observed that in the real world, “practitioners” sometimes
(perhaps often) use them incorrectly, to multiply total sectoral output (or value added
or employment). So they propose net multipliers (the terminology could be confusing;
these are not the multipliers in a net model, as in Section 6.5.2). Essentially, they
simply convert a standard multiplier so that it can be used in conjunction with total
outputs. For example, their Type I net output multipliers are i0 L^f c , where
^f c ¼ ^f x^1 ¼ hfj =xj i; in their terms, fj =xj is the fraction of j’s output that may “rightfully
be considered exogenous” (Oosterhaven and Stelder, 2002a, p. 536). Specifically, they
“decompose” the usual output multiplier calculation driven by final demands, i0 Lf, to
one driven by outputs, namely
i0 Lf ¼ mðoÞf ¼ mðoÞ^f i ¼ mðoÞ^f x^1 x^i ¼ mðoÞ^f c x ¼ i0 L^f c x
(6.51)
The net multiplier matrix is thus L^f c and the associated vector of economy-wide
multipliers is i0 L^f c ¼ mðoÞ^f c .
A small (two-sector) case shows the elements in detail:
2
3
2
3
f1
f2
f1
l12
0
6 l 11 x1
7
x2 7
x1
l 11 l 12 6
6
7
0 ^
6
7
¼
½
1
1
i Lf c ¼ ½ 1 1 6
7¼
5
l 21 l 22 4
4
f2
f1
f2 5
0
l 21
l22
x2
x1
x2
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6.6 Multipliers and Elasticities
ðl 11 þ l 21 Þ
f1
x1
ðl 12 þ l 22 Þ
f2
x2
f
¼ mðoÞ1 1
x1
mðoÞ2
f2
x2
Each sector’s standard output multiplier is scaled down to reflect the fraction of its
total output that is represented by its final demand. Other multipliers can be
similarly modified.
This work generated considerable discussion and a lengthy and elaborate exchange
(de Mesnard, 2002, 2007a, 2007b; Dietzenbacher, 2005; Oosterhaven, 2007), with a
variety of interpretations and alternative terminology. In the end, “net contribution” or
“net backward linkage” indicators were suggested as a more appropriate label than
“multiplier.” We will return to an aspect of this in Chapter 7 on linkage measures in
input–output models.
6.6
Multipliers and Elasticities
6.6.1 Output Elasticity
Another approach to compensating for differences in industry size is one step further
from simply considering percentage increases in final demand (as with growth equalized multipliers). The idea is to measure both the stimulus and its effect in percentage
terms – in this case the percentage change in total output (or income or employment,
etc.) due to a percentage change in a given industry’s final demand. [See, for example,
Mattas and Shrestha (1991) or Ciobanu, Mattas, and Psaltopoulos (2004).] These
(percentage change)/(percentage change) measures are “elasticities” in economics
terms.
In particular, consider a one percent change in fj only, so ðΔf Þ0 ¼ 0;...; ð0:01Þ fj , ...,0 .
2 3
l 1j
6 .. 7
Then Δx ¼ LΔf ¼ 4 . 5ð0:01Þ fj . The economy-wide output change is i0 Δx¼
l nj
2 3
l1j
0 6 .. 7
i 4 . 5ð0:01Þfj ¼mðoÞj ð0:01Þ fj . This percentage change in total output (across all indusl nj
tries) that is generated by ð0:01Þ fj has been labeled the output elasticity of industry j
oej and is defined as
oej ¼ 100 ði0 Δx=i0 xÞ ¼ 100 mðoÞj ð0:01Þfj =i0 x ¼ mðoÞj fj =i0 x
(It would be more precise to call this an output-to-final demand elasticity, to distinguish it from other elasticities in Section 6.6.2.)
Modification of any of the other multipliers in Section 6.2.2 – through multiplication
0 by fj =i x – produces exactly parallel results, giving income, employment, etc.,
elasticities to final demand. Note that these are very similar to “growth-equalized”
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P fj = j fj , while here it is
multipliers;
in
that
case,
the
modification
was
produced
by
P f j = j xj .
6.6.2
Output-to-Output Multipliers and Elasticities
Direct Effects Starting with zij ¼aij xj , consider the direct effect of an
exogenous change in industry j’s output Δxj Δxj ! Δzij ¼ aij Δxj . This Δzij represents new i output directly required by j, so Δxi ¼ Δzij , and thus Δxi ¼ aij Δxj or
Δxi =Δxj ¼ aij . Now consider a one percent increase in j’s output Δxj ¼ ð0:01Þxj ; this
means Δxi ¼ ð0:01Þaij xj . So the (i, j)th element of the matrix (0.01)Ax^ measures the
direct effect of j’s one percent increase in output on industry i. Expressed as a
percentage of i’s output, we have 100ðΔxi =xi Þ ¼ 100ð0:01Þaij xj =xi ¼ aij xj =xi . And, in
matrix form, this is the (i, j)th element of the matrix x^1 Ax^, showing the direct
effect on industry i’s output (percentage change) resulting from a one percent change
in industry j’s output. This is a direct output-to-output elasticity. We will meet
the matrix x^1 Ax^ again in Chapter 7, where we explore supply-side input–output
models.
Total Effects Elements of the Leontief inverse matrix translate final demand
changes into total output changes – Δxi ¼ l ij Δfj and lij ¼ Δxi =Δfj . These encompass
direct and indirect effects, and they are at the heart of the multipliers explored in
previous sections in this chapter. Again, it would be slightly cumbersome but completely accurate to call l ij an output-to-final-demand multiplier. Consider lij, the ondiagonal element in the jth column of L; l jj ¼ Δxj =Δfj or Δxj ¼ l jj Δfj. Define l∗
ij as
l ij =ljj ; then
l∗
ij ¼ l ij =l jj ¼ Δxi =Δfj = Δxj =Δfj ¼ Δxi =Δxj
∗
or Δxi ¼ l ∗
ij Δxj. Thus, l ij could be (and has been) viewed as a total output-tooutput multiplier.
∗
The matrix of these multipliers, L∗ ¼ l ij , is created by dividing each element in a
^ 1 (as usual, L
^
column of L by the on-diagonal element for that column – L∗ ¼ LðLÞ
is a diagonal matrix created from the on-diagonal elements in L). Then each of the
elements in column j of L∗ indicates the amount of change in industry i output (the
row label) that would be required if the output of industry j were increased by one
dollar.25
25
This is equivalent to the “total flow” approach of Szyrmer (for example, Szyrmer, 1992). He makes a case for
the unsuitability of the usual output multipliers (from the standard demand-driven input–output model) for a
wide variety of real-world impact studies. Some analysts argue that the initial exogenous one-dollar stimulus
should be removed from the “total effect” calculation. As was seen in Section 6.5.3, this can be accomplished
by replacing L by (L – I). The interested reader should see de Mesnard (2002) and Dietzenbacher (2005) for
details.
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6.7 Summary
283
Suppose, then, that industry j is projected to increase its output to some new amount,
xj . Postmultiplication of L∗ by a vector, x, with xj as its jth element and zeros elsewhere,
will generate a vector of total new outputs, x∗ , necessary from each industry in the
economy because of the exogenously determined output in industry j. That is,
x∗ ¼ L∗ x
(6.52)
We return to this matrix in Chapter 14 in the context of “mixed” input–output models
in which final demands (for some industries) and gross outputs (for the other industries) are specified exogenously.
Moving to elasticity terms, the (i, j)th element of (0.01)Lx^ gives the (total) new
output in industry i caused by a one percent output increase in industry j. So, exactly
parallel to the direct elasticity case, the (i, j)th element of x^1 Lx^ gives the percent
increase in industry i total output due to an initial exogenous one percent increase in
industry j output – the “direct and indirect output elasticity of industry i with respect to
the output in industry j” (Dietzenbacher, 2005, p. 426). We will also meet this matrix,
x^1 Lx^, again in Chapter 7 in the discussion of supply-side input–output models.
6.7
Summary
In this chapter we have introduced the reader to a wide variety of multipliers that are
frequently calculated and used in real-world applications of the input–output framework. While the array may seem bewildering at first glance, it is, in fact, incomplete.
For example, instead of using household input coefficients, as in (6.11), to generate a
household income multiplier, one can weight the elements of a column of L by the
parallel concept of “government input” coefficients, representing a dollar’s worth of
government payments by a sector per a dollar’s worth of that sector’s output. These
would be the elements needed in the added row of an A matrix that was being closed
with respect to government operations, not households. In this way, we would generate
government multipliers. And similarly, other multipliers associated with exogenous
sectors can be calculated – for example, foreign trade multipliers.
The use of the input–output framework for impact analysis, due to changing final
demands, using multipliers, constitutes one of the most frequent uses of the model. In
subsequent chapters we will explore extensions to deal specifically with energy
(Chapter 12) and environmental problems (Chapter 13), and alternative uses of the
model, in which the data are transformed into alternative summary measures of
economic activity such as decomposition of changes over time and linkage analysis,
in which the relative “importance” of sectors is assessed.
We explored the added richness of the Miyazawa formulation of a “closed” model in
which various income–consumption–output impacts can be isolated. And we also
examined some of the many variations on early multiplier formulations – for example,
when the approach is changed from (direct + indirect effects)/(direct effects) to
(indirect effects)/(direct effects) – which essentially means subtracting one from a
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284
Multipliers in the Input–Output Model
traditional multiplier. Finally, we examined the conversion of (multiplier) effects into
elasticity terms, giving percentage changes due to a one percent increase in an
industry’s final demand or output.
There are two appendices to this chapter. Appendix 6.1 demonstrates the equivalence of total household income multipliers and the elements in the bottom row of the
Leontief inverse. Appendix 6.2 shows the mathematical relationship between Type
I and Type II income multipliers introduced in Section 6.2.
Appendix 6.1 The Equivalence of Total Household Income Multipliers and the
Þ1
Elements in the Bottom Row of ðI A
Consider the general representation of our 3 3 model closed with respect to households (sector 3), and its inverse, similarly partitioned.
2
3
ð
Þ
1
a
a
a
11
12
13
6
7 E F
6
7
ðI AÞ ¼ 6 a21
ð1 a22 Þ
a23 7 ¼ G H
4
5
a31
a32
ð1 a33 Þ
|
|
|
|
|
- - - - - - - - - - - - - - - - - - - - - - - - - |
|
2
6 l 11
1
6
Þ ¼ L
¼ 6 l 21
ðI A
4
l 31
l 12
l 22
l 32
|
|
|
|
3
l13
7 S
l23 7
7¼
U
5
l33
- - - - - - - -| - - |
|
T
V
From results on inverses of partitioned matrices in Appendix A, particularly (2) in
(A.4), GS þ HU ¼ 0. Here, since H ¼ 1 a33 , we can write U ¼ a33 U GS, or
l 31 l 32 ¼ a33 l 31 l 32 þ ½ a31 a32 l 11 l 12
l 21 l 22
Written out and rearranged, this is
l 31 ¼ a31l11 þ a32l 21 þ a33l31
l 32 ¼ a31l 12 þ a32l 22 þ a33l 32
The three terms on the right-hand sides are exactly the terms in (6.12) –
Pnþ1
ðhÞj ¼ i¼1
m
anþ1, il ij for j ¼ 1 and j ¼ 2, where the (n þ 1) ¼ 3 and i ¼ 3 terms are
ðhÞ2 ¼ l32 , and this is
ðhÞ1 ¼ l 31 and m
those in the household row (or column). Thus, m
ðhÞj, for a model of any size with households endogenous. This
always true, for any m
ðhÞj ¼ l nþ1, j.
is (6.13), namely m
Appendix 6.2
Relationship between Type I and Type II Income Multipliers
To examine the value of the ratio between type II and type I income multipliers, we
again use results on the inverse of a partitioned matrix. To begin, we note, for any
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Appendix 6.2
285
sector j, that both multipliers – in (6.14) and (6.15) – have the same denominator,
anþ1, j , and thus the ratio of the two multipliers for sector j is
Rj ¼
mðhÞIIj
mðhÞIj
l nþ1, j
¼P
n
anþ1, i lij
(A6.2.1)
i¼1
¼ L11 L12 , the numerator of the ratio in (A6.2.1) is the jth
In matrix terms, with L
21 L
22
L
21 and the denominator is the corresponding element of h0c L. Thus, the
element of L
n-element row vector of these ratios is
21 h0 L 1
(A6.2.2)
R ¼ ½R1 ; . . . , Rn ¼ L
c
The reader should be
P clear that this matrix operation divides each l nþ1, 1 , . . . , lnþ1, n by
the corresponding ni¼1 anþ1, i l ij . (Recall also that the notation hxi is used instead of x^
when the vector being diagonalized is represented by a matrix expression containing
several elements, so that the hat does not fit easily.)
Again using results from Appendix A on the inverse of a partitioned matrix
2
3
ð
Þ
a
a
1
a
11
12
13 7
6
E F
7
Þ ¼ 6
,
[specifically (A.5)], and with ðI A
6 a21 ð1 a22 Þ a23 7 ¼
G H
4
5
a31
a32 ð1 a33 Þ
|
|
|
|
|
- - - - - - - - - - - - - - - - - - - - |
|
we see that the components in (A6.2.2) are
21 ¼ L
21 GE1 ¼ L
22 ðGLÞ and h0 L ¼ GL
L
c
22 ðGLÞ ½hGLi
Thus R ¼ L
ð11Þ ð1nÞ
ðnnÞ
1
22 ½1, . . . , 1 ¼ L
22 i0 ; that is, the ratios are all
¼L
ð11Þ
ð1nÞ
the same and are equal to the element in the lower-right of the closed model inverse.
For the numerical example in Section 6.2.2, we found that the ratio of these
multipliers, which we designated k, was 1.29. Recall the inverse for our small example,
2
3
1:365
:425
:251
6
7
7
¼6
in (6.6), namely L
6 :527
1:348
:595 7, where, in particular (to two deci4
5
:570
:489
1:289
22 ¼ 1.29. (Differences are due to rounding and the detailed precision of the
mals), L
inversion process.)
This constancy of the ratios of the two types of multipliers was apparently first
demonstrated by Sandoval (1967), in an article in which he showed that the ratio is
Þj=jðI AÞj, the ratio of the determinants of the Leontief matrices (not
equal to jðI A
inverses) of the closed and open models. [The reader familiar with determinants
can easily verify this for the numerical example in this chapter – jðI AÞj ¼ 0:7575,
Þj ¼ 0:587875 and (to two decimal places) jðI AÞj=jðI A
Þj ¼ 1:29: In
jðI A
|
|
|
|
|
- - - - - - - - - - - - - - - - - |
|
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286
Multipliers in the Input–Output Model
producing his result, Sandoval did not use results from the inverses of partitioned
matrices but rather from the general definitions of inverses in terms of determinants
and cofactors. [Other discussions of these topics can be found in Bradley and Gander
(1969), Katz (1980), and ten Raa and Chakraborty (1983).]
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7
7.1
Supply-Side Models, Linkages,
and Important Coefficients
Supply-Side Input–Output Models
7.1.1 The Early Interpretation
In 1958, Ghosh presented an alternative input–output model based on the same set
of base-year data that underpin the demand-driven model in earlier chapters, namely
Z, f, and v, from which x follows as x ¼ Zi þ f or as x0 ¼ i0 Z þ v0 . In the demanddriven model, direct input coefficients are defined in A ¼ Z^x 1 , leading to
x ¼ ðI AÞ1 f ¼ Lf. In this case the Leontief inverse relates sectoral gross outputs
to the amount of final product (final demand) – that is, to a unit of product leaving the
interindustry system at the end of the process. The alternative interpretation that Ghosh
suggests relates sectoral gross production to the primary inputs – that is, to a unit of
value entering the interindustry system at the beginning of the process.
This approach is made operational by essentially “rotating” or transposing our
vertical (column) view of the model to a horizontal (row) one. Instead of dividing
each column of Z by the gross output of the sector associated with that column, the
suggestion is to divide each row of Z by the gross output of the sector associated with
that row. We use B to denote the direct-output coefficients matrix that results.1 For a
two-sector example, this means
b11 b12
0
z11 =x1 z12 =x1
1=x1
z11 z12
B¼
¼
¼
¼ ^x 1 Z (7.1)
b21 b22
z21 =x2 z22 =x2
0
1=x2 z21 z22
These bij coefficients represent the distribution of sector i’s outputs across sectors j that
purchase interindustry inputs from i; these are frequently called allocation coefficients,
as opposed to technical coefficients, aij . Using
x 0 ¼ i0 Z þ v 0
where v0 ¼ ½v1 ; , vn and
1
!
Early presentations used A for these coefficients and A# for the traditional demand-side coefficients, which
we have denoted simply by A. This served to make visually explicit the two points of view:
! A# resulting
from uniform division of all elements in each column of Z by the associated column output, and A resulting from
division of all elements in each row of Z by the associated row output.
289
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Supply-Side Models, Linkages, and Important Coefficients
Z ¼ ^x B
(7.2)
x0 ¼ i0 ^x B þ v0 ¼ x0 B þ v0
(7.3)
x0 ¼ v0 ðI BÞ1
(7.4)
G ¼ ðI BÞ1
(7.5)
from (7.1), we have
x ¼ x0 . From this,
since i0 ^
Define
with elements g ij . This has been called the output inverse, in contrast to the usual
Leontief inverse, L ¼ [l ij ] ¼ ðI AÞ1 (the input inverse). Element g ij has been
interpreted as measuring “the total value of production that comes about in sector j
per unit of primary input in sector i” (Augustinovics, 1970, p. 252). Then, (7.4) is
x0 ¼ v 0 G
(7.6)
In terms of changes in v, we would find the associated output changes as
Δx0 ¼ ðΔv0 ÞG
(7.7)
As we have seen with the Leontief price model in Section 2.6, we can equally well
transpose all elements so that the resulting vector of gross outputs is a column rather
than a row. In that case, (7.3) will be
x ¼ B0 x þ v
(7.8)
from which
1
x ¼ ðI B0 Þ v
(7.9)
x ¼ G0 v
(7.10)
Since2 G0 ¼ ðI B0 Þ1 , (7.9) is
This is the version of the model that we will use in what follows. However, many
analysts use the form in (7.6) and (7.7). Again, in terms of changes in v we would have
Δx ¼ G0 ðΔvÞ
(7.11)
The basic assumption of the supply-side approach is that the output distributions in
bij are stable in an economic system, meaning that if output of sector i is, say, doubled,
then the sales from i to each of the sectors that purchase from i will also be doubled.
2
This follows from matrix algebra results that ðA BÞ0 ¼ A0 B0 and ðA0 Þ
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1
0
¼ A1 .
7.1 Supply Side Input–Output Models
291
Instead of fixed input coefficients, fixed output coefficients are assumed in the supplyside model.
For sector j in the n-sector case, from (7.10) we have
xj ¼ v1 g 1j þ þ vi g ij þ þ vn g nj
(7.12)
Recall the typical equation in the solution to the demand-driven model, from (2.12) in
Chapter 2:
xi ¼ l i1 f 1 þ þ l ij fj þ lin f n
The effect on output of sector i, Δxi , of a $1.00 change in final demand for sector j
goods (Δfj ¼ 1), is given by l ij. (Again, for readers
who are familiar with differential
¼
l
.)
Column
sums
of
L
¼
l ij were seen (Chapter 6) to be output
calculus, ∂xi =∂f
ij
Pnj
l
denotes
the
total
new
output
throughout all n sectors of the
multipliers;
i¼1 ij
economy that is associated with a $1.00
increase
in
final demand for sector j. Row
Pn
sums of L can also be interpreted; j¼1 lij shows the total new sector i intermediate
sales to all sectors that would be needed if there were a $1.00 increase in the final
demands for the outputs of each of the n sectors in the economy.
From (7.12), the effect on sector j output, Δxj , of a $1.00 change in the availability
of primary inputs to sector i (Δvi ¼ 1) is given by g ij. (In calculus terms,
∂xj =∂vi ¼ g ij ; note that the order of the subscripts in this partial derivative is the
opposite of that for l ij ). For example, if g ij ¼ 0:67, this has been interpreted to mean
that if there is $1.00 less labor available to sector i as an input to production (due,
say, to a strike), then the amount of reduction in sector j output will be $0.67. The
reduction comes about because, in the input–output framework, a decrease in the
available labor to sector i means a decrease in sector i output and hence in the outputs
of all sectors that depend on sector i’s product as an input to their own production
processes. This represents the same kind of effect, originating in an exogenous supply
change, as is captured in the usual input–output system, which responds to exogenous demand changes.
In this (early) view of the Ghosh model, row and column sums in the output inverse,
G ¼ ðI BÞ1 ¼ [g ij ], were given interpretations that parallel those in the Leontief
Pn
quantity model. Row sums,
j¼1 g ij ¼ g i1 þ þ g in ð¼ ∂x1 =∂vi þ þ ∂xn =∂vi Þ,
were taken to represent the effect on total output throughout all sectors of the economy
that would be associated with a $1.00 change in primary inputs for sector i. This is
the supply-side model’s analog to an output (or demand) multiplier – a column
sum in L. These supplyP
model row sums were termed
input (or supply) multipliers.
n
g
¼
g
þ
þ
g
¼
∂x
=∂v
Also, column sums,
j
1 þ þ ∂xj =∂vn , were
1j
nj
i¼1 ij
interpreted as the total effect on sector j output if there were a $1.00 change in
the supply of primary factors for each of the n sectors in the economy. These
column sums were the supply-side model’s parallel to the row sums of L in the
demand model.
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Numerical Illustration (Hypothetical Data) Let
2
3
2
3
2
3
225 600 110
1,200
265
Z ¼ 4 250 125 425 5, x ¼ 4 2,000 5, f ¼ 4 1,200 5
325 700 150
1,500
325
Then
3
32
225 600 110
1=1,200
0
0
54 250 125 425 5
0
1=2000
0
B¼^
x 1 Z ¼ 4
325 700 150
0
0
1=1,500
2
3
:188 :5
:092
¼ 4 :125 :063 :213 5
:217 :467 :1
and
2
2
3
2
3
1:484 :982 :383
1:484 :316 :521
G ¼ ðI BÞ1 ¼ 4 :316 1:418 :367 5 and G0 ¼ 4 :982 1:418 :971 5
:521 :971 1:394
:383 :367 1:394
Thus, for example, if there were $100 less labor available for sector 1 production
and $300 less for both sector 2 and sector 3 production, we would find, as in (7.11),
2
3 2
32
3 2
3
1:484 :316 :521
100
399:53
Δx1
4 Δx2 5 ¼ 4 :982 1:418 :971 54 300 5 ¼ 4 815:06 5
:383 :367 1:394
300
566:47
Δx3
These figures, Δx1 ¼ 400, Δx2 ¼ 815, and Δx3 ¼ 566, would then be interpreted as the amounts by which the outputs of the three sectors would be reduced,
given the decreases in labor inputs to the sectors.
If Δv1 ¼ 1 and Δv2 ¼ Δv3 ¼ 0,
2
3 2
3
1:484
Δx1
4 Δx2 5 ¼ 4 :982 5
:383
Δx3
These figures represent the total additional outputs possible in each of the three sectors
due to the availability of one more unit of primary inputs to sector 1. If Δv1 ¼ 1 and
Δv2 ¼ Δv3 ¼ 0, these numbers will be negative, representing reduced output in the
sectors. The sum of the elements in row 1 of G (column 1 of G0 ), 2.849, represents the
total potential impact throughout the economy of a $1.00 change in the availability of
primary inputs to sector 1. Again, this is parallel to the concept of the output multiplier
for sector 1 in the ordinary, demand-driven input–output model. It is, in the context of
this supply-side model, an input multiplier for sector 1. Similarly, this kind of input
multiplier for sector 2 is 2.101 and for sector 3 it is 2.886. In this view of the supplyside model, one might use these figures to decide where an additional dollar’s worth of
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7.1 Supply Side Input–Output Models
provision of primary resources (labor, etc.) would be most beneficial to the total
economy, in terms of potential for supporting expanded output. Conversely, these
input multipliers can indicate the potential contracting effects of shortages in primary
inputs to a particular sector. From this point of view, a reduction by $1.00 in the
availability of a scarce resource could lead to a reduction in economy-wide output of
$2,849, $2,101, or $2,886, depending on where the primary input reduction occurs.
Numerical Application (US Data) Giarratani (1978) presents an early application of the Ghosh model. He calculated output coefficients, B, and the associated
output inverse matrix, G, using 78-sector 1967 US data. Supply multipliers ranged
from a high of 4.01 for iron and ferroalloy-ores mining to 1.09 for medical and
educational services and non-profit organizations. With rankings of sectors such as
this, Giarratani suggested that it is possible to determine where primary factor constraints would have the greatest potential for limiting aggregate economic output – for
example, a contemplated labor strike in one or more sectors.
Looking down the jth column of G allows one to identify supply linkages that have
potential for significantly limiting the output of sector j. Among others, Giarratani
considered an energy sector, petroleum refining and related industries (sector 31, the
only secondary energy sector in the 78-sector 1967 US table). Examination of column
31 in the output inverse identifies the following among the largest coefficients: for
sector 8, crude petroleum and natural gas, g8,31 ¼ 0:8605; for sector 27, chemicals and
chemical products, g 27,31 ¼ 0:0513; and for sector 12, maintenance and repair construction, g 12,31 ¼ 0:0504. The suggested interpretation is that interruptions in primary
inputs to these sectors have the largest potential for disruptions in refined petroleum
output.
Other early examples of this kind of empirical analysis using the Ghosh model
include Chen and Rose (1986), on the role of bauxite as a critical input in the
Taiwanese economy, and Davis and Salkin (1984), on the importance of water as an
input in a county in California.
7.1.2 Relationships between A and B and between L and G
Given A ¼ Z^x 1 and B ¼ ^x 1 Z, Z ¼ ð^x ÞB; putting this into the definition of A,
A ¼ ^x B^x 1
(7.13)
(When two matrices, P and Q, are connected by the relation P ¼ MQM1, they are
said to be similar; this is denoted P ~ Q. Thus, we see that A and B are similar
matrices.) Of course, it also follows straightforwardly that
B ¼ ^x 1 A^x
(7.14)
Recall in Section 6.6.2 on elasticities that element (i, j) in the matrix ^x 1 A^x was
shown to capture the direct effect on industry i’s output (percentage change) resulting
from a one percent change in industry j’s output. This was termed a direct output-to-
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output elasticity. Hence, from (7.14), these elasticities are precisely the elements in
B ¼ [bij ].3
Consider (I A). From (7.13), ðI AÞ ¼ I ^x B^x 1 : Since ^x I^x 1 ¼ I,
ðI AÞ ¼ ^x ðI BÞ^x 1
That is, ðI AÞeðI BÞ. Using a basic result on the inverse of a product of matrices
1
ðPQRÞ1 ¼ R1 Q1 P1 we find that, since ðI AÞ1 ¼ [^x ðI BÞ^x 1 ] ,
ðI AÞ1 ¼ ^x ðI BÞ^x 1
(7.15)
L ¼ ^x G^x 1
(7.16)
or
Thus L ~ G. [The interested reader might confirm these similarity relationships in
(7.13) and (7.15) for the small numerical illustration in in Section 7.1.1.] The results in
(7.16) can equally well be written as
G ¼ ^x 1 L^x
(7.17)
Again referring to Section 6.6.2, we saw that element (i, j) in the matrix ^x 1 L^x gives
the percent increase in industry i total output due to an initial exogenous one percent
increase in industry j output – the total output-to-output elasticity of industry i output
with respect to output in industry j. From (7.17) these elasticities are exactly the
elements in G ¼ [gij ].
From these results it is clear that any measures defined for A – such as output
multipliers or backward linkages (Section 8.2.1) – can be found from B, provided that
x is also known. Conversely, input multipliers or forward linkages (Section 7.2) –
defined on B – can be found using A and x.4
7.1.3 Comments on the Early Interpretation
An early application of the Ghosh model is to be found in Augustinovics (1970), where
direct-input coefficients (A) and direct-output coefficients (B) are compared for a
number of countries and over time. However, reservations to this model began to
appear in the early 1980s – for example in Giarratani (1980, 1981). The issue is:
essentially what kind of economic behavior is represented by a system with constant
supply distribution patterns? Ghosh had in mind the context of a planned economy
experiencing severe excess demand, with government-imposed restrictions on supply
3
Using this interpretation, de Mesnard (2001) refers to the aij and bij coefficients as reflecting the absolute and
relative direct influence of sector j on sector i, respectively.
4
^ to
It is easily shown that A and B have the same main diagonal elements; the same is true for L and G. Using M
^ ¼ ^x B^
^
^ x 1 ¼ B
denote the diagonal matrix whose elements are the main diagonal of a square matrix M, A
from (7.13), since order of multiplication of diagonal matrices makes no difference and ^x ^x 1 ¼ I. Exactly the
^
^ ¼ G.
same line of argument shows that L
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7.1 Supply Side Input–Output Models
295
patterns. This is probably not a very general situation in much of the modern world.
However, Giarratani (1981, p. 283) suggested a possibly broader context:
More interesting perhaps is the prospect that this behavior may be the result of voluntary supply
decisions in the same context or, alternatively, given the disruption of some basic commodity. Firms
may well attempt to maintain their existing markets . . . by allocating available product on the basis of
deliveries in more normal times. Casual evidence on the U.S. experience would seem to support
this hypothesis.
It was in this spirit that Giarratani’s application was carried out.
Oosterhaven (1980) raised reservations about the plausibility of the Ghosh model,
and then in the late 1980s a more vigorous exchange took place, particularly in
Oosterhaven (1988, 1989), Gruver (1989), and Rose and Allison (1989). The topic
surfaces again in later comments from Guerra and Sancho (2011) and Oosterhaven
(2012). In essence, the problem is that primary input increases in sector j are transmitted forward in the Ghosh model to output increases in all sectors that buy from j,
without any corresponding increases in primary input use in those sectors. This is
because Δv is viewed as exogenous and (in this example) is fixed at
Δv0 ¼ 0; . . . , 0; Δvj , 0, . . . , 0 : This wreaks havoc with the notion of sectoral production functions where material inputs plus primary inputs are used in fixed proportions.
A similar kind of plausibility issue is raised by Manresa and Sancho (2013, 2020) for
the Leontief model that is closed with respect to households.
7.1.4
Joint Stability
The Issue When the demand-driven input–output model is used in standard
fashion for impact analysis – as in Δx ¼ ðI ΑÞ1 Δf – a crucial assumption is that the
direct-input coefficients matrix, A, remains constant. As a consequence of the connections between A and B, or between L and G, this means that in general B (and
therefore G) cannot remain constant. This came to be known as the “joint stability”
problem.5 A numerical example illustrates the problem nicely. From the data for the
three-sector hypothetical illustration in Section 7.1.1, we also find6
2
3
2
3
:188 :3
:073
1:484 :589 :306
A ¼ 4 :208 :063 :283 5 and L ¼ 4 :527 1:418 :489 5
:271
:35
:1
:651
:729
1:394
It will be useful at this point to use superscripts “0” to represent the base-year data,
i.e., the given A, B, L, and G matrices, as well as the initial output, x, will be denoted
A0 , B0 , and so forth. Vectors and matrices that result from some exogenous change will
2
3
100
be given superscripts “1”. Suppose, for illustration, Δf ¼ 4 40 5; using the demand30
5
6
See, among others, Dietzenbacher (1989), Miller (1989), Rose and Allison (1989), and Chen and Rose (1991).
These matrices, along with B and G in Section 7.1.1, illustrate the relationships shown in footnote 2.
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Supply-Side Models, Linkages, and Important Coefficients
2
3
2
3
181:166
1,381:2
driven model Δx ¼ L0 Δf we find Δxðd Þ ¼ 4 124:057 5 and x1 ðd Þ ¼ 4 2,124:1 5.
136:095
1,636:1
[We use (d) to indicate that these are results from the demand-driven model, in which a
constant A matrix is assumed.] From these results, we find the new transactions matrix
associated with A and the new outputs; namely,
2
3
258:969 637:217 119:980
Z1 ðd Þ ¼ A0 ^x 1 ðd Þ ¼ 4 287:743 132:754 463:560 5
374:066 743:420 163:610
[The reader can easily check that Z1 ðd Þi þ f 1 ¼ x1 ðd Þ.] The direct-output coefficients
matrix associated with these new transactions and new total outputs is found, as in
(7.1), as
2
3
:188 :461 :087
1
B1 ¼ ^x 1 ðd Þ Z1 ðd Þ ¼ 4 :136 :063 :218 5
:229 :454 :1
Recall from above that
2
3
:188
:5 :092
B0 ¼ 4 :125 :063 :213 5
:217 :467 :1
and clearly B1 6¼ B0. [One simple measure of the difference is the average of all of the
P P
(absolute) percentage differences ð1=n2 Þ ni¼1 nj¼1 [jb0ij b1ij j=b0ij ] 100. Here this is
3.58 percent.] The upshot is that, at least in this example (but actually in general), the
assumption of a constant A matrix, used in an impact analysis, carries with it the
requirement that B change as a result of the impact.
An exactly similar problem occurs if one uses the supply-driven model to assess the
impact of a change in primary inputs. For example, from the data for this three-sector
example, v0 ¼ ½400 575 815. Suppose that ðΔvÞ0 ¼ ½ 50 100 20 ; using (7.11)
to assess the output effects of this change in primary inputs, we find ½ΔxðsÞ0 ¼
0
½ 116:221 210:325 83:720 and ½x1 ðsÞ ¼ ½ 1,316:2 2,210:3 1,583:7 . [Now (s)
denotes results from the supply-driven model.] Parallel to the demand-driven example,
there is now a new transactions matrix,
2
3
246:792 658:111 120:654
Z1 ðsÞ ¼ ^x 1 ðsÞ B0 ¼ 4 276:291 138:145 469:694 5
343:139 739:069 158:372
In conjunction with the associated x1 ðsÞ, this Z1 ðsÞ defines the corresponding directinput coefficients matrix, A1 , namely
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297
2
3
:188 :3
:076
1
A1 ¼ Z1 ðsÞ ^x 1 ðsÞ
¼ 4 :210 :063 :3 5
:261 :334 :1
Originally,
2
3
:188 :3
:073
A0 ¼ 4 :208 :063 :283 5
:271 :35
:1
and A1 6¼ A0 . (In this case, the average absolute difference is 2.06 percent.)7
This apparent inconsistency – the fact that the requirement of a constant A (for
demand-driven model impact analysis) implies a non-constant B in the related supplydriven model or that the constant B is needed for supply-driven model impact analysis
carries with it the implication of a non-constant A in the related demand-driven model –
led to several empirical studies on relative joint stability (see, for example, Rose and
Allison, 1989, or Chen and Rose, 1991). In general, the conclusion drawn was that
instability in actual empirical applications was not a major issue.
Conditions under Which Both A and B Will Be Stable Assume that we have
found the new outputs resulting from new final demands using the demand-driven
model x1 ¼ I A0 f 1 , so A1 ¼ A0 . From (7.14),
1
B1 ¼ (^x 1 ) A0 ^x 1
and substituting A0 from (7.13)
1
1
B1 ¼ x^ 1 ^x 0 B0 ^x 0 ^x 1
1
Let ^e ¼ ^
x 1 (^
x 0 ) where ei ¼ x1i =x0i can be thought of as a kind of “growth rate” for
sector i (remember that order of multiplication makes no difference when the matrices
are diagonal); then
B1 ¼ ^e 1 B0^e
A similar story holds if the supply-driven model is used, with B1 ¼ B0 ; namely
A1 ¼ ^e A0^e 1
If each sector’s output changes at the same rate ei ¼ x1i =x0i ¼ λ for all i – then ê ¼ λI
and B1 ¼ ½ð1=λ ÞΙΒ0 ðλΙÞ ¼ B0 . A similar argument shows that A1 ¼ A0 under the
same conditions, after an impact analysis with the supply-driven model.8
^0 ¼ A
^ 1 and B
^0 ¼ B
^ 1 . These relationships are illustrated
For exactly the same reasons as shown in footnote 3, A
by the matrices in this section.
8
For much more detail on these matters see Dietzenbacher (1989, 1997).
7
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Supply-Side Models, Linkages, and Important Coefficients
7.1.5 Reinterpretation as a Price Model
In order to overcome the criticisms and implausibilities in the original view of the
Ghosh model, Dietzenbacher (1997) proposed an alternative interpretation by suggesting that the model be viewed not as a quantity model but as a price model (see also
extensive discussions on alternative interpretations of the Ghosh model in
Oosterhaven, 1996 and de Mesnard, 2009). We illustrate the idea by looking again at
results from the numerical example in Section 7.1.4. Specifically, for
0
0
(v1 ) ¼ (v0 ) þ (Δv)0 ¼ ½ 400 575 815 þ ½ 50 100 20 ¼ ½ 450 675 835 0
we found, using x1 ðsÞ ¼ G0 v1 [(7.10)],
1 0
x ðsÞ ¼ ½ 1,316:2 2,210:3
1,583:7 Suppose that we view the elements in the supply-driven model not as quantities (in
which case elements in Δv are interpreted as changes in the amounts of primary inputs
available to the economy and elements in Δx are interpreted in changes in quantities
produced) but rather as values (in which case elements in Δv reflect changes in the
prices or costs of primary inputs and elements in Δx reflect changes in the values of
outputs). In the demand-driven model of earlier chapters all prices are assumed fixed in
an impact analysis and quantities change as a result of changes in the quantities of final
demands. Now we assume that all quantities are fixed and use the Ghosh model to
assess the repercussions throughout the economy of changes in primary input prices. In
that reinterpretation, we can use the term Ghosh price model, which can reasonably be
looked upon as a cost-push input–output model. Changes in primary input costs are
transmitted throughout the economy as they are passed on (completely) by producers
in the prices of their products that are purchased by other intermediate users, who in
turn increase their prices accordingly, etc.
With this interpretation, we identify the relative price changes easily as the ratios of
elements in x0 to those in x1 ðsÞ, since quantities are fixed and only valuations change.
Define π as the vector of these price ratios,
1 1 ^x ðsÞ
π ¼ ^x 0
(7.18)
where π j ¼ x1j ðsÞ=x0j ¼ p1j q0j =p0j q0j ¼ p1j =p0j (where q0j is a physical measure of the
output of sector j in the base period). For this three-sector example,
2
3 2
3
3 2
1:0968
1,316:2=1,200
x11 ðsÞ=x01
6 1
7
7
6
7 6
07
(7.19)
π¼6
4 x2 ðsÞ=x2 5 ¼ 4 2,210:3=2,000 5 ¼ 4 1:1052 5
x13 ðsÞ=x03
1,583:7=1,500
1:0558
This indicates that (unit) prices of the products of sectors 1, 2, and 3 would rise by
9.68, 10.52, and 5.58 percent, respectively, in response to primary input cost increases
of 12.5 ½¼ ð50/400Þ 100 percent, 17.39 ½¼ ð100/575Þ 100 percent, and 2.45
½¼ ð20/815Þ 100 percent, for the three sectors, respectively.
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7.1 Supply Side Input–Output Models
299
3
3 2
1:484
Δx1
Similarly, when Δν1 ¼ 1 and Δν2 ¼ Δν3 ¼ 0, we found 4 Δx2 5 ¼ 4 :982 5 so
2
3
:383
Δx3
1,201:48
1
4
5
x ðsÞ ¼ 2,000:98 . This can now be interpreted in terms of price ratios for the
1,500:38
three sectors of
2
3 2
3
1,201:48=1,200
1:0012
π ¼ 4 2,000:98=2,000 5 ¼ 4 1:0005 5
1,500:38=1,500
1:0003
2
This says that prices would be expected to increase by 0.12, 0.05, and 0.03 percent in
the three sectors, respectively, in the face of a 0.25 percent ½¼ ð401/400Þ 100
increase in the cost of primary inputs to sector 1 only.
Connection to the Leontief Price Model (Algebra) It is straightforward to
show that the Ghosh price model and the Leontief price model (Section 2.6) generate
exactly the same results. The Ghosh price model finds
1 1 x ðsÞ
π ¼ ^x 0
0
Since ½x1 ðsÞ ¼ G0 ðv1 Þ from (7.10),
1
0
π ¼ (^x 0 ) (G0 ) v1
From G ¼ ^
x 1 L^x in (7.17), G0 ¼ ^x L0 ^x 1 , and so we have
1
0
1
0
1
π ¼ (^x 0 ) [^x 0 ðL0 Þ (^x 0 ) ]v1 ¼ ðL0 Þ (^x 0 ) v1
1
Finally, since primary input coefficients are found as v1ci ¼ v1i =x0i , or v1c ¼ x^ 0 v1,
0 1
0
(7.20)
π ¼ L0 ^x 0 ^x 0 v1c ¼ L0 v1c
In the Leontief price model of Section 2.6, it is also the case that primary input price
changes generate relative price changes [as in (2.33), which is repeated here as (7.21)]:
1
0
~ ¼ I ðA0 Þ0 v1c ¼ L0 v1c
(7.21)
p
~ . The Leontief price (cost-push) model (Section
As (7.20) and (7.21) make clear, π ¼ p
2.6) and the Ghosh price (cost-push) model generate the same results; the former
~ , and the latter in terms of
directly in terms of the vector of relative price changes, p
new outputs, x1 ðsÞ, from which π is found as the ratio of new to old output values.
Connection to the Leontief Price Model (Numerical Illustration) Using data
from the hypothetical example in Section 7.1.1 and 7.1.4, we find the base-year
primary input coefficients as
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2
3 2
3
400=1,200
:3333
v0c ¼ 4 575=2,000 5 ¼ 4 :2875 5
815=1,500
:5433
As expected,
2
32
3 2
3
1:4840 :5266
:6514
:3333
1:0
0
~ 0 ¼ L0 v0c ¼ 4 :5893 1:4179 :7287 54 :2875 5 ¼ 4 1:0 5
p
:3064
:4893 1:3936
:5433
1:0
verifies that all prices are one (“per dollar’s worth of output”) in the base-year Leontief
model.
Now consider the primary input price increases from the example above, namely
1 0 0 0
v ¼ v þ ðΔvÞ0 ¼ ½ 400 575 815 þ ½ 50 100 20 ¼ ½ 450 675 835 In terms of primary input coefficients, we have
2
3 2
3
450=1,200
:3750
v1c ¼ 4 675=2,000 5 ¼ 4 :3375 5
835=1,500
:5566
and using (7.21),
2
32
3 2
3
1:4840 :5266
:6514
:3750
1:0968
0
~ 1 ¼ L0 v1c ¼ 4 :5893 1:4179 :7287 54 :3375 5 ¼ 4 1:1051 5
p
:3064 :4893 1:3936
:5566
1:0558
As expected, these are precisely the same results as we found above for π in (7.19).
Either exercise produces the result that price increases of 9.68, 10.51, and 5.58
percent are to be expected for the output of the three sectors as a result of the primary
input cost increases given in ðΔvÞ0 ¼ ½ 50 100 20 .
A Ghosh Quantity Model Thus far we have seen a Leontief quantity model,
a Leontief price model, and a Ghosh price model. It is logical to expect that a Ghosh
quantity model also exists (Dietzenbacher,
1 1997). From the familiar Leontief quantity
model, x1 ¼ L0 f 1 and L0 ¼ ^x 0 G0 ^x 0 , we have
1
x1 ¼ ^x 0 G0 ^x 0 f 1
Define the new final-demands as proportions (coefficients) of base-period outputs –
1
1
1
f c i ¼ f 1i =x0i and f 1c ¼ ^x 0 f 1 – and premultiply both sides by ^x 0 ,
1
1
1
~x ¼ ^x 0 x1 ¼ ^x 0 ^x 0 G0 ^x 0 f 1 ¼ G0 f 1c
where ~x i ¼ x1i =x0i : In this case changes in final-demand proportions (of gross outputs)
are translated into relative output measures; that is, an index showing new outputs, x1 ,
as proportions of base-period outputs, x0 .
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301
Table 7.1 Overview of the Leontief and Ghosh quantity and price models
Model
Price (Cost-push) [Quantities
fixed; prices change]
Leontief
0 1 1
v1c ¼ ^
x
v
¼ [v1j =x0j ]
0
~ 1 ¼ L0 v1c
p
p~i ¼ x1i ðd Þ=x0i
Ghosh
Coefficient Stability
A1 6¼ A0
B1 ¼ B0
Exogenous Variables
f 1 ¼ f 1i
0 1 1
x
f
f 1c ¼ ^
¼ f 1i =x0i
Endogenous Variables
x1 ðd Þ ¼ L0 f 1
~
x ¼ G0 f 1c
~x i ¼ x1i ðsÞ=x0i
Coefficient Stability
A1 ¼ A0
B1 6¼ B0
Exogenous Variables
Endogenous Variables
Quantity (Demand-pull) [Prices
fixed; quantities change]
v1 ¼ [v1j ]
0
x1 ðsÞ ¼ G0 v1
This is the straightforward algebraic derivation of a Ghosh quantity model. The
reader can explore the logic of the “story” behind it. Table 7.1 gathers together some of
the relevant information about these four models. The quantity and price models –
either Leontief or Ghosh – are often described as “dual” to each other,9 while the
Leontief variant of the quantity model has been described as the “mirror image” of the
Ghosh quantity model, and similarly for the Leontief and Ghosh price models. (Some
of this material appeared earlier in Table 2.15.)
7.2
Linkages and Key Sectors in Input–Output Models
In the framework of an input–output model, production by a particular sector has two
kinds of economic effects on other sectors in the economy. If sector j increases its
output, this means there will be increased demands from sector j (as a purchaser) on the
sectors whose goods are used as inputs to production in j. This is the direction of
causation in the usual demand-side model, and the term backward linkage is used to
indicate this kind of interconnection of a particular sector with those (“upstream”)
sectors from which it purchases inputs. On the other hand, increased output in sector j
also means that additional amounts of product j are available to be used as inputs to
other sectors for their own production – that is, there will be increased supplies from
sector j (as a seller) for the sectors that use good j in their production. This is the
direction of causation in the supply-side model. The term forward linkage is used to
9
There are some rather detailed mathematical discussions on what constitutes a pair of “dual” models. For our
input–output models we simply take the term to mean that one model determines quantities (with prices fixed),
the other determines prices (with quantities fixed), and the fundamental structural relationships (in L0 or in G0 )
are at the heart (although transposed) of each model and its dual.
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Supply-Side Models, Linkages, and Important Coefficients
indicate this kind of interconnection of a particular sector with those (“downstream”)
sectors to which it sells its output.
Starting in the late 1950s, measures have been proposed to quantify such backward
and forward linkages, or economic “connectedness.” Comparisons of the strengths of
backward and forward linkages for the sectors in a single economy provide one
mechanism for identifying “key” or “leading” sectors in that economy (those sectors
that are most connected and, therefore, in some sense, most “important”) and for
grouping sectors into spatial clusters. And, if data are available for more than one time
period, the evolution of these interconnections can be studied. Also, examination of
these measures for similar sectors in different countries provides one method of
making international comparisons of the structure of production.
7.2.1 The Early Measures
If the backward linkage of sector i is larger than that of sector j, one might conclude
that an expansion of sector i’s output would be more beneficial to the economy than
would an equal expansion (in monetary terms) in sector j’s output, in terms of the
productive activity throughout the economy that would or could be generated by it.
Similarly, if the forward linkage of sector r is larger than that of sector s, it could be
said that a given amount of expansion of the output of sector r is more essential to the
economy than an equal-valued expansion in the output of sector s, from the point of
view of the overall productive activity that it could support.
This was the logic behind early discussions and applications of these straightforward
linkage measures. Notable works include Rasmussen (1957),10 Hirschman (1958),
Chenery and Watanabe (1958), Yotopoulos and Nugent (1973), Laumas (1975), and
Jones (1976) – see also the debate among several authors in the May 1976 issue of the
Quarterly Journal of Economics, or the Diamond (1976), Schultz and Schumacher
(1976), and Laumas (1976a) exchange in Kyklos. There have been numerous suggestions for differing definitions and refinements of these linkage and key sector measures
and others of economic connectedness (McGilvray, 1977; Hewings, 1982, among
others). Our purpose here is simply to introduce the reader to some of the most
prevalent of these measures and, in particular, to indicate how they are derived from
information in either the demand-side or the supply-side input–output model.
There have also been numerous suggestions for various ways of combining forward
and backward linkage measures (examples can be found in Hübler, 1979; Meller and
Marfán, 1981; Loviscek, 1982; Cella, 1984; Adamou and Gowdy, 1990; and Clements,
1990). Often overlooked in some of this early work are the differing sizes of sectors;
stimulating a new dollar’s worth of output in some sectors represents a much larger
percentage output change than for other sectors. In some of the multiplier literature this
point was in fact raised and addressed – in, for example, Gray et al. (1979) and Gowdy
10
Hirschman (1958) cites an edition of this book (same title) published by Einar Harcks in Copenhagen in 1956.
This must be a precursor to the 1957 North-Holland edition (also under the Einar Harcks imprint) which is
identified as a “second printing.”
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7.2 Linkages and Key Sectors in Input–Output Models
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(1991); recall the discussion of growth-equalized multipliers in Section 6.5.3.
Normalizations, examined in the next section, are meant to address this issue. This
was later examined, for example, in Oosterhaven (2004, 2007) and in Temurshoev and
Oosterhaven (2014). We explore these newer developments along with the so-called
hypothetical extraction method introduced in Section 7.2.5.
Backward Linkage In its simplest form, a measure of the strength of the
backward linkage of sector j – the amount by which sector j production depends on
interindustry inputs – is given by the
P sum of the elements in the jth column of the direct
input coefficients matrix, namely ni¼1 aij. Since the coefficients in A are measures of
direct effects only, this is called the direct backward linkage:
bðd Þj ¼
n
X
aij
(7.22)
i¼1
In terms of transactions (Z, not A), this is simply
intermediate
Pn inputs
Pof total
P the value
for sector j as a proportion of j’s total output, ni¼1 aij ¼ ni¼1 zij =xj ¼
i¼1 zij =xj .
This definition, in transactions terms, was first proposed by Chenery and Watanabe
(1958). If we define11 bðd Þ ¼ bðd Þ1 ; , bðd Þn , then
bðd Þ ¼ i0 A
(7.23)
To capture both direct and indirect linkages in an economy, column sums of the total
requirements matrix, L ¼ ½lij , were proposed as a total backward linkage measure
(Rasmussen, 1957); these are output multipliers, designated mðoÞj in Section 6.2. For
sector j we have
bðt Þj ¼
n
X
lij
(7.24)
i¼1
The corresponding row vector of these total backward linkage measures for each
sector is
bðt Þ ¼ i0 L ¼ mðoÞ
(7.25)
There is some disagreement in the literature on whether the on-diagonal elements in
A or L should be included or netted out of the summations (see, for example, Harrigan
and McGilvray, 1988). To the extent that these “internal linkages” constitute part of
Hirschman’s (1958, p. 100) “. . . input-provision, derived demand . . . effects,” they are
appropriately included. On the other hand, if one is specifically interested in a sector’s
11
Lower-case b designations for various backward linkage measures should not be confused with elements of the
direct-output coefficients matrix (bij ) in the Ghosh model; the meaning should be clear from the context. Recall
from footnote 2 in Chapter 6 that we define vectors of multipliers as row vectors to avoid a proliferation of
“primes” in the discussion. We continue that convention here, where additional sub- and superscripts will
be needed.
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Supply-Side Models, Linkages, and Important Coefficients
“backward dependence” on or linkage to the rest of the economy, they should be
omitted.
Various normalizations of these measures have been proposed and used in empirical
studies. For example, sector j’s backward linkage could be represented as a proportion
of the average backward linkage for the entire economy
n
P
bðd Þ ¼
j
aij
bðd Þj
i¼1
¼
n
n P
n
P
P
ð1=nÞ
bðd Þk ð1=nÞ
aik
k¼1
i¼1 k¼1
and
ðd Þ ¼
b
i0 A
ni0 A
¼
ði0 AiÞ=n i0 Ai
(7.26)
(where the overbar indicates a normalized measure). Various weighted averages have
also been suggested.12
ðd Þ is unity – ½b
ðd Þið1=nÞ ¼ ni0 0 A i 1 ¼ i00 Ai ¼ 1 – so that
The average value of b
n
i Ai
i Ai
sectors with “above average” (stronger) direct backward linkages have indices that are
greater than one and that those with “below average” (weaker) direct backward
linkages have indices that are less than one. The same logic generates
0
ðtÞ ¼ ni L
b
i0 Li
(7.27)
as a normalized total backward linkage index, also with an average value of unity.
(This is the “Index of the Power of Dispersion” suggested by Rasmussen, 1957.)
Forward Linkage An early measure of direct forward linkages was also
proposed as the row sums in A, Ai, along with an associated total forward linkage
measure, the row sums in L, Li.13 Both of these have been viewed with skepticism,
because they represent a peculiar stimulus – a simultaneous increase of one unit in the
gross outputs of every sector in the case of Ai and an increase of one unit in the final
demands of every sector in the case of Li.
This dissatisfaction led to the suggestion that elements from the Ghosh model would
be more appropriate as forward linkage measures (Beyers, 1976; Jones, 1976). The
row sums Bi were proposed as better measures of direct forward linkage. In terms
of transactions
(Z, not B), this is simply the value of total intermediate sales by sector
P
i – nj¼1 zij – as a proportion of the value of i’s total output – xi ,
Among the first to make an issue of weightings in linkage measures was Laumas (1976b). Others before him
(e.g., Hazari, 1970; Diamond, 1974), however, had used sets of weights other than unit vectors.
13
In normalized form, nLi=i0 Li, this is Rasmussen’s (1957) “Index of Sensitivity of Dispersion.”
12
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7.2 Linkages and Key Sectors in Input–Output Models
305
Table 7.2 Basic linkage measures
bðÞ
f ð Þ
ð Þ
b
f ðÞ
Direct (d)
i0 A
Bi
ni0 A
i0 Ai
nBi
i0 Bi
Total (t)
i0 L
Gi
ni0 L
i0 Li
nGi
i0 Gi
n
X
j¼1
bij ¼
n X
zij =xi ¼
j¼1
n
X
!
zij =xi
j¼1
(This was also first proposed in Chenery and Watanabe, 1958.) In addition, row sums
of the Ghosh inverse, Gi, were suggested as a better measure of total forward linkages.
As with backward linkage measures, inclusion or exclusion of on-diagonal elements is
an issue, and normalizations are usual.
Thus, the parallels to (7.22) and (7.23) for direct and total forward linkages are14
n
X
f ðd Þ i ¼
bij
(7.28)
j¼1
and
f ðt Þ i ¼
n
X
g ij
(7.29)
j¼1
In addition, the same (and other) normalized versions for forward linkages can
be found.
Matrix expressions for these basic results are collected in Table 7.2.
7.2.2 Classifying Backward and Forward Linkage Results
Many studies that look to identify key sectors from their backward and forward linkage
measures usually calculate both (generally in normalized form) and then select those
sectors with a high score on both measures.15 In the normalized form shown in the last
two columns of Table 7.2, this means sectors with both backward and forward linkages
greater than one.16
14
Again we face a notation issue. Lower-case f designations for various forward linkage measures should not be
confused with elements of the final demand vector; in this case, also, the meaning should be clear from
the context.
15
There have been suggestions for “combined” measures to capture “total” linkage. For example, Hübler (1979)
1
proposed column sums from ½I ð0:5ÞðA þ B0 Þ for this purpose. More comprehensive measures of total
linkage come from hypothetical extraction approaches (Section 7.2.5).
16
It is not necessary to create a normalization in such a way that the average value is one. For any set of results
one can easily calculate the mean and then separate the results into those above and those below that mean
value. However, this is particularly easy to do, at least visually, when the mean is 1.
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Table 7.3 Classification of backward and forward linkage results
Direct ½f ðd Þ or Total ½f ðtÞ Forward Linkage
ðd Þ or Total ½b
ð t Þ
Direct ½b
Backward Linkage
Low (<1)
High (>1)
Low (<1)
High (>1)
(I) Generally
independent
(IV) Dependent on
interindustry supply
(II) Dependent on
interindustry demand
(III) Generally
dependent
Often, sectors are distributed over a four-way classification as (1) generally independent
of (not strongly connected to) other sectors (both linkage measures less than 1), (2) generally
dependent on (connected to) other sectors (both linkage measures greater than 1), (3)
dependent on interindustry supply (only backward linkage greater than 1), and (4) dependent on interindustry demand (only forward linkage greater than 1). This can be displayed in a
2 2 table, such as shown in Table 7.3.17 With data for two or more time periods, a table of
this sort for each period will give one indication of the evolution of the economy.18
Applied linkage studies are numerous. Examples include Dietzenbacher (1992) for
the Netherlands; Chow, Lee, and Ong (2006) for Singapore (based on 144-sector
input–output data for 1990, 1995, and 2000 in which the authors chose to calculate
total forward linkages using row sums of L rather than G); and the chapter by
Temursho in ADB (2016),19 which includes, among other measures, backward and
forward linkages in the Kazakhstan economy (for 2005 and 2013).20
7.2.3 Spatial Linkages
Exactly the same kinds of measures can be applied to multi-regional or multi-national
input–output data to assess the types and intensities of spatial interdependence or
connectedness. These address the issue of strength of economic connections among
regions in an economy or countries in a world model and, if data for more than one
period are available, how those connections are changing over time – for example,
increasing regional or national self-sufficiency or increasing interregional or international dependence. These measures can be aggregate – that is, is region r in general
dependent on imports or exports (or both) or relatively self-sufficient? Or they can be
sector/region specific – assessing the import- or export-dependence of sector i in region
r on one sector (or all sectors) in another region (or regions).21 Recalling that total
17
This two-way table arrangement appears to have originated with Chenery and Watanabe (1958).
Further subtleties are possible. Each quadrant can be further subdivided; for example, each quadrant could be
divided into four more categories, those above and those below one standard deviation above the mean.
19
In 2016, Temurshoev adopted Temursho officially as his family name.
20
An alternative approach to key sector identification, presented in García Muñiz, Raya, and Ramos Carvajal
(2008), has been developed and applied in several papers. However, Gurgul and Lach (2018) suggest a logical
flaw in the method. (Also see Gurgul and Lach also for additional references.)
21
In this discussion, reference to “region” can be understood also as “nation” when appropriate.
18
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7.2 Linkages and Key Sectors in Input–Output Models
307
backward linkage is measured by the output multiplier, it is clear that the interregional
multipliers discussed in Section 6.3 get at exactly these kinds of questions. (Early
presentations of the spatial form of linkage measures are in Miller and Blair, 1988 and
rr
rr
Batten and Martellato, 1988.)
L
Ars
Lrs
A
,
L
¼
and
In the two-region (nation) context, we have A ¼
rr
Asr Ass
Lsr Lss
rs
G
G
G¼
. One straightforward set of spatial linkage measures closely paralGsr Gss
lels the sectoral linkage cases summarized in Table 7.2. In this two-region case, the
direct backward linkage of sector j in region r will have both an intraregional and an
interregional component. Specifically,
sr
bðd Þrj ¼ bðd Þrr
j þ bðd Þj ¼
n
X
arr
ij þ
i¼1
n
X
asr
ij
i¼1
One measure of the relative strength of intra- versus interregional (internal versus
external) direct backward dependence is given by the percentages
rr
r
sr
r
100 bðd Þj =bðd Þj and 100 bðd Þj =bðd Þj
or, using an alternative normalization,
sr r
r
bðd Þrr
j =xj and bðd Þj =xj
Parallels can be found for total backward linkages, namely
n
n
X
X
sr
rr
bðtÞrj ¼ bðt Þrr
þ
b
ð
t
Þ
¼
l
þ
lsr
ij
ij
j
j
i¼1
and
i¼1
rr
r
sr
r
100 bðt Þj =bðt Þj and 100 bðt Þj =bðt Þj
sr r
r
bðtÞrr
j =xj and bðt Þj =xj
In compact matrix form, direct and total intra- and interregional backward linkages for
each sector in region r are given by the n elements in the following vectors [the
parallels are (7.23) and (7.25)]
bðd Þrr ¼ i0 Arr and bðd Þsr ¼ i0 Asr
bðtÞrr ¼ i0 Lrr and bðt Þsr ¼ i0 Lsr
and
bðd Þr ¼ bðd Þrr þ bðd Þsr and bðt Þr ¼ bðtÞrr þ bðt Þsr
Ignoring the sectoral detail, one aggregate measure (scalar) of a region’s direct and
total backward linkage to itself and to the other region(s) is found by summing (or
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Table 7.4 Summary of spatial/sectoral linkage measures (two-region example)
Spatial/Sectoral Linkages
Backward
Direct
rr
0
Forward
Total
b ðd Þ ¼ i A ,
bðd Þsr ¼ i0 Asr
Direct
0
rr
bðt Þ ¼ i L ,
bðtÞsr ¼ i0 Lsr
rr
rr
Normalizations include division of each direct
element by bðd Þrj [or each total element by
bðtÞrj ] or by xj ; for example,
ðd Þrr ¼ i0 Arr hbðd Þr i1 or
b
ðd Þrr ¼ i0 Arr ð^x Þ1
b
rr
Total
rr
f ðd Þ ¼ B i,
f ðd Þrs ¼ Brs i
f ðt Þ ¼ Grr i,
f ðt Þrs ¼ Grs i
rr
Normalizations include division of each direct
element by f ðd Þrj [or each total element by
f ðt Þrj ] or by xj ; for example,
f ðd Þrr ¼ hf ðd Þr i1 Brr i or
f ðd Þrr ¼ ð^
x Þ1 Brr i
Spatial Linkages
Backward
Direct
rr
0
Forward
Total
Bðd Þ ¼ i A i,
Bðd Þsr ¼ i0 Asr i
rr
rr
0
Direct
BðtÞ ¼ i L i,
Bðt Þsr ¼ i0 Lsr i
rr
rr
0
Total
F ðd Þ ¼ i B i,
F ðd Þrs ¼ i0 Brs i
rr
F ðt Þ ¼ i0 Grr i,
F ðt Þrs ¼ i0 Grs i
rr
Normalizations include division of each element by n or by i0 x;
ðd Þrr ¼ ð1=nÞi0 Arr i or B
ðd Þrr ¼ ð1=i0 xÞi0 Arr i
for example, B
averaging) over all sectors. For example, using upper-case letters for these aggregate
measures,
ðd Þrr ¼ ð1=nÞi0 Arr i
Bðd Þrr ¼ i0 Arr i or B
and similarly for Bðd Þsr , Bðt Þss , and BðtÞsr . Spatial versions of forward linkages follow
the same kind of pattern. These are summarized in Table 7.4.
Examples of applications for regions in single countries can be found in, among
others, Blair and Miller (1990) and Shao and Miller (1990) for the US economy, Pan
and Liu (2005) and Okamoto (2005) for China.
As noted, with the emergence of multinational and global input–output data sets
(Sections 3.6.3 and 10.8.6) these spatial measures are pertinent to questions of
international economic connections and dependencies and their evolution over time.
Illustrative applications here include Dietzenbacher and van der Linden (1997) for
the countries of the European Community; Wu and Chen (2006) on backward
linkages Taiwan Japan, Korea Japan, China Japan, and also Japan Taiwan,
Japan Korea, Japan China, in 1985, 1990, 1995, and 2000.
Alternative definitions for interregional economic connections are grounded in the
notions of interregional feedbacks and spillovers (Miller and Blair, 1988; see also
Chapter 3). These are closely related to the “hypothetical extraction” method. This
provides a general framework for linkage analysis, and we turn to it in Section 7.2.5.
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7.2 Linkages and Key Sectors in Input–Output Models
309
“Net” Backward and Forward Linkages
Net Backward Linkage Dietzenbacher (2005) proposed another linkage
measure in his interpretation of the Oosterhaven and Stelder net multiplier formulation
(Section 6.5.3). Start with the observation that L^f ¼ [l ij fj ] is a matrix whose (i,j)th
element represents output of i generated by final demand fj . Row sums of L^f are given
by L^f i ¼ Lf ¼ x, whose ith element is simply xi, the output of i generated by all final
demands – the standard interpretation of x. Column sums of L^f are given by i0 L^f ; the
jth element of this row vector is the output needed from all sectors to satisfy final
demand fj – bðt Þj [or mðoÞj ]. The Oosterhaven-Stelder net output multiplier was
defined as i0 L^f c ¼ i0 L^f ^x 1 (a row vector).
Dietzenbacher suggests rewriting this multiplier by replacing ^x by L^f i , giving
1
x 1 ¼ i0 L^f L^f i . In this form, the jth element in this row vector is
i0 L^f c ¼ i0 L^f ^
7.2.4
seen to be a ratio, namely
0
jth column sum of L^f bðtÞj f j
1
¼
i L^f c j ¼ (i0 L^f ^x )j ¼
xj
jth row sum of L^f
In words, the output generated in all industries by final demand for j, f j , divided by the
output generated in j by all final demands, f 1 , . . . , f n . This suggests
a kind of “net”
backward linkage or another key sector measure. In particular, if i0 L^f c j > 1 then the
value of economy-wide output generated by final demand in j is larger than the value
of j’s output generated by all industries’ final demands. So industry j could be said to
be more important for the others than the others are for it.22 By this measure, sector j
might be said to be a key sector.
1
Here is a two-sector illustration of the pieces in this deconstruction, i0 L^f L^f i :
l 11 f 1 l 12 f 2
l11 l12 f 1 0
^
¼
Lf ¼
0 f2
l21 l22
l 21 f 1 l 22 f 2
i0 L^f ¼ ½ l 11 f 1 þ l 21 f 1 l 12 f 2 þ l 22 f 2 ¼ ½ bðt Þ1 f 1 bðtÞ2 f 2 . (Element j in this vector
shows output in all sectors,
bðtÞj fj , caused by j’s final demand.)
l
f
þ
l
f
x
12 2
L^f i ¼ 11 1
¼ 1 (Element i shows output in sector i, xi , caused by all
l21 f 1 þ l 22 f 2
x2
final demands.) 22 f 2 Þ
For example, i0 L^f c 2 ¼ ððll1221 ff 2 þl
¼ bðt Þ2 fx22 . We define this as bðnÞ2 , and in
1 þl 22 f 2 Þ
23
general
bðnÞj ¼ bðt Þj
22
fj
xj
(7.30)
Strictly
0
speaking, this interpretation requires that ljj fj be subtracted from both numerator and denominator in
i L^f c j , to remove j’s influence on itself.
23
We use “n” to indicate net linkage measures.
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Supply-Side Models, Linkages, and Important Coefficients
In matrix form, this is bðnÞ ¼ iL^f ^x 1 .
Net Forward Linkage Temurshoev and Oosterhaven (2014) have proposed
a corresponding net forward linkage measure, as f ðnÞ ¼ ^x 1 v^Gi. Following the same
line of reasoning for deconstructing this expression, replace ^x 1 with its Ghosh model
1
counterpart hi0 v^Gi [from (7.6)] allowing us to write f ðnÞ as hi0 v^Gi v^Gi, whose ith
element is
1
f ðt Þi vi
ith row sum of v^G
x^ v^Gi i ¼
¼
xi
ith column sum of v^G
One suggested interpretation: All sectors’ output values are generated by the primary
input costs of sector i, f ðt Þi vi , divided by the output value of sector i due to all its
primary inputs (Temurshoev and Oosterhaven, 2014, p. 7). If ðv^GÞi > 1, this could be
said to imply that sector i is less dependent on all sectors than all others are on it. So,
f ðnÞi ¼ f ðtÞi
vi
xi
(7.31)
is proposed as a kind of net forward linkage or key sector measure.24
7.2.5
Hypothetical Extraction
Complete Extraction The hypothetical extraction approach was originally
conceived to quantify how much the total output of an n-sector economy would change
(decrease) if a particular sector, say the jth, were completely removed from that
economy. Initially, this was modeled in an input–output context by deleting row and
ðjÞ for the ðn 1Þ ðn 1Þ matrix
column j from the A matrix.25 Early work used A
ðjÞ for the associated Leontief inverse, and f ðjÞ for the correspondwithout sector j, L
ingly reduced final demand vector, so output in the “reduced” economy is found as
ðjÞf ðjÞ . In the full n-sector model, output is x ¼ Lf, so Δj ¼ i0 x i0 xðjÞ is one
xðjÞ ¼ L
aggregate (scalar) measure of the economy’s loss [difference (decrease) in the value of
gross output] if sector j disappears – as such, it is a measure of the “importance” or
total linkage or total impact of sector j. It has sometimes been argued that the first term,
i0 x, should not include the (original) output xj . If xj is omitted, (i0 x xj ) i0 x would
measure j’s importance to the remaining sectors in the economy. In either case,
normalization through division by total gross output (i0 x) and multiplication by
100 produces an estimate of the percentage decrease in total economic activity;
j ¼ 100 i0 x i0 xðjÞ =i0 x . (Online Appendix SA7.1 presents hypothetical extractions
Δ
24
The interested reader can create a two-sector illustration in detail, as was done for the net forward
linkage measure.
25
The original idea seems to have appeared in Paelinck, de Caevel, and Degueldre (1965) (in French) or Strassert
(1968) (in German). The first discussion in English known to us is in Schultz (1976, 1977; the latter paper is a
longer version of the former).
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7.2 Linkages and Key Sectors in Input–Output Models
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in detail in the context of partitioned matrix versions of the Leontief and Ghosh models,
along with citations to much more of the relevant literature.)
In the remainder of this section we adopt alternative notation that is now generally in
use. Denote Aj as the ðn nÞ coefficients matrix in which row and column j have
been nullified (replaced by zeros; sector j’s “removal”), with Lj its associated
Leontief inverse and f j as the final demand vector with fj ¼ 0. Then the output vector
for this reduced economy is given by xj ¼ Lj f j, and with this notation
Δj ¼ i0 ðx xj Þ, since now both output vectors contain n elements.
Calculating hypothetical extractions for an n-sector economy can be tedious –
removing each sector in turn, finding its associated Leontief inverse, and then the
corresponding reduced total outputs, etc., a total of n times. But this work can be
dramatically simplified. Temurshoev [(2010a), based on (2010b)] shows that (using
our notation)
L Lj ¼
1
Lej e0j L ej e0j
ljj
(7.32)
where ej is the jth column of the ðn nÞ identity matrix – all zeros except for a 1 in
location j. [Thus ej e0j is an ðn nÞ matrix of all 0’s except for a 1 on the main diagonal
at location j.] Using this result, Temurshoev (2010a, appendix) shows that the total
linkage measure of sector j’s importance,
Δj ¼ i0 x xj ¼ i0 Lf Lj f j
can be reduced to the incredibly simple expression
Δj ¼
1 0
i Lej e0j x
l jj
(7.33)
Since e0j x ¼ xj and i0 Lej ¼ mðoÞej ¼ mðoÞj ¼ bðt Þj , this becomes
Δj ¼
xj
bðtÞj
ljj
which is a very straightforward transformation of the total backward linkage
measure.26
Because these calculations result from completely removing a sector from the A
matrix (and hence from L as well), they are often referred to as complete hypothetical
extraction backward linkages. For this reason, we change the notation, replacing Δj
with bðcÞj , to be consistent with previous linkage measures, so
bðcÞj ¼
26
xj
bðt Þj
l jj
(7.34)
This result is anticipated in Szyrmer’s work (see footnote 28) and a different sort of proof is given in Gallego
and Lenzen (2005, appendix 1).
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312
Supply-Side Models, Linkages, and Important Coefficients
where the “c” denotes complete hypothetical extraction. This means that in using the
hypothetical extraction approach to calculate this measure of a sector’s importance
only xj =l jj and the original multiplier, bðtÞj [¼ mðoÞj ], are needed for each sector in
turn. This is an enormous advantage over earlier one-by-one extraction procedures.
Notice that this measure is in monetary units (e.g., millions of dollars of output lost
throughout the economy as a result of removing sector j from that economy). Other
linkage measures discussed thus far are dimensionless (e.g., dollars per dollar). So, for
a comparable dimensionless measure, it is usual to normalize the hypothetical extraction output results by the value of sector j’s output, xj . This recognizes the fact that
“large” sectors can be expected to have “large” impacts in absolute terms. The
normalized version of (7.34) – using “n” to denote normalization – is therefore simply
bðcÞnj ¼
1
bðt Þj
ljj
(7.35)
At the sectoral level, the ith element (i 6¼ j) in (7.34), xi xj
i , can be viewed as the
backward dependence of sector j on sector i. Writing out the elements in
i0 ðx xj Þ ¼
xj
l jj
bðtÞj ,
2
3
x1 xj
1
6
7
..
6
7
.
6
7
06
j 7
i 6 xi xi 7 ¼
6
7
..
4
5
.
j
xn xn
2
3
l 1j
6 .. 7
6 . 7
7
xj 0 6
l ij 7
i6
6
l jj 6 . 7
7
4 .. 5
l nj
and hence,
Δxi ðcÞj ¼ xi xj
i ¼
xj
l ij
ljj
ij 27
and in normalized form [Δxi ðcÞj ]n ¼ (xi xj
i )=xj ¼ l jj .
Similarly, a set of complete hypothetical extraction forward linkages can be identified. For sector i this involves replacing row and column i of the direct output
coefficients matrix, B, with zeros, setting the ith element of the primary inputs vector,
v, to zero and then proceeding as above. The end result (Temurshoev and Oosterhaven,
2014, appendix) can be shown to be
l
f ðcÞi ¼
27
xi
f ðt Þi
g ii
(7.36)
When i ¼ j we find Δxj ðcÞj ¼ xj =ljj l jj ¼ xj and [Δxj ðcÞj ]n ¼ 1=ljj ljj ¼ 1; namely, j’s influence on itself is
indeed xj .
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7.2 Linkages and Key Sectors in Input–Output Models
313
This has been termed the complete hypothetical extraction forward linkage for sector
i.28 Normalized versions are created through division by the output of the extracted
sector, xi .
Partial Extraction: Backward Linkage The hypothetical extraction approach
has also been used to develop an additional set of backward and forward linkage
measures (for example, in Dietzenbacher and van der Linden, 1997 or Temurshoev and
Oosterhaven, 2014). For the backward linkage case, assume that sector j buys no
intermediate inputs from any production sector. This is done by replacing (only)
column j in A by a column of zeros.29 Denote this new matrix Aj
c (the
c subscript
j 1
j
¼
I
A
f ¼ Lj
to indicate
that
only
column
j
has
been
zeroed
out).
Then
x
c
c
c f,
0
j
and i x xc has been suggested as another measure of (aggregate) backward
linkage for sector j. Fortunately, this difference can also be found without going
through the convoluted process of one-by-one extraction and recalculation of outputs.
In this case (Temurshoev and Oosterhaven, 2014, appendix)
xj
l jj
bðpÞj ¼ i0 x i0 xj
c ¼
[bðtÞj 1]
(7.37)
This has been labelled the partial hypothetical extraction backward linkage for sector
x
j. Notice, from (7.34), that this is equivalent to bðcÞj ljjj . Normalization leads to
bðpÞj
n
¼
1
l jj
[bðtÞj 1]
Again, if sectoral detail is of interest, the. ith element in xi ðxi Þj
c can be viewed as
the backward dependence of sector j on sector i. Here the derivation of (7.37) makes
clear a subtle variation.30 The relevant vector of differences is
2
3
82 3 2 39
l 1j
x1 ðx1 Þj
>
c
0 >
>
6
7
>
.. 7 6 . 7>
.
6
>
>
6
7
>
..
6
7
>
>
. 7 6 .. 7>
6
7
>
>
6
>
>6 l 7 6 7>
6 x ðx Þj 7
>
>
>
>
ij
6 i
6
7
0
i c 7
<6 7 6 7>
=
6
7
x
.
.. 7 6 . 7
..
7 ¼ j i0 6
i0 6
6
7
. 7>
6
>6 . 7 6
ljj >
. j 7
6
7
7>
>
6 ljj 7 6
1
>
>
6 xj xj c 7
6
7>
>
6
7
>
>
6
7
6
7>
.
>
>
6
7
.
.
>
>
6
7
4
5
..
>
>
.
.
4
5
>
>
4
5
.
.
>
;
:
0
j
l nj
xn ðxn Þc
As noted in Section 7.1.2, gii ¼ l ii , so the reduction term could also be shown as ðxi =l ii Þ.
This is the case considered by Szyrmer (in his dissertation, 1984, and in several publications, such as Szyrmer
and Walker, 1983; Szyrmer, 1992; Szyrmer and Ulanowicz, 1987, and others). The dissertation provides a
proof of (7.36) using notation that differs considerably from current practice, and it also relies on unpublished
material.
30
See Temurshoev and Oosterhaven (2014, appendix) for the derivation and the appearance of ej , the jth column
of an identity matrix.
28
29
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314
Supply-Side Models, Linkages, and Important Coefficients
so Δxi ðpÞj and [Δxi ðpÞj ]n are the same as Δxi ðcÞj and [Δxi ðcÞj ]n for all i 6¼ j.31 These
equivalences are apparent from the partitioned-matrix expressions for these extractions, as shown in online Appendix SA7.1.
The interested reader might notice that these results [in (7.34), (7.35), and (7.37)]
^ 1
can also be expressed in terms of the elements of L∗ ¼ [l∗
ij ] ¼ [l ij =l jj ] ¼ L(L ) ,
which were discussed as output-to-output
multipliers in Section 6.2.2. For example,
P
rearranging (7.34), with bðt Þj ¼ ni¼1 l ij ,
bðcÞj ¼
n
n n
X
X
xj X
lij ¼ xj
l ij =ljj ¼ xj
l∗
ij
l jj i¼1
i¼1
i¼1
This deconstruction also illuminates the interpretation of bðcÞj as a measure of the
importance to the entire economy (in terms of output) of the xj units produced by sector
j and thus of how much would be lost to that economy if sector j with its xj units of
output were to disappear.
Partial Extraction: Forward Linkage A parallel to eliminating column j in A
as a way of identifying backward linkages is the elimination of a sector’s intermediate
sales in the B matrix as a way of identifying forward linkages. That is, replace row i
of the output coefficients
by a row of zeros. Denote this matrix as Bi
r . Then
imatrix
0
1
1
0
0
0
x ¼ v (I B) and xr ¼ v (I Bi
)
indicate
preand
post-extraction
outr
0
puts, and [x0 xi
]i
is
an
aggregate
measure
of
sector
i’s
forward
linkage.
Again,
r
0
is an indication of i’s dependence on sector j as an
each element in x0 xi
r
intermediate output buyer, and normalizations are usual, as in [xi (xi
r )i ]=xi or
)
]=x
.
Also,
as
with
partial
elimination
for
backward
linkages,
the
100 [xi (xi
i
r i
actual calculation need not involve the steps of removing each sector (row) from B in
turn. Rather, again from Temurshoev and Oosterhaven (2014, appendix), it is shown
that a partial hypothetical extraction forward linkage measure for sector i is
0
i¼
f ðpÞi ¼ x0 i xi
r
xi f ðt Þi 1
g ii
(7.38)
As before, l jj can replace gjj . From (7.35), this can be seen to be equivalent to
f ðcÞi xi
l ii
.
In Table 7.5 we summarize the main hypothetical extraction results. The interested
reader can work through the exercise of extending these extraction possibilities to the
spatial context in which a region is hypothetically extracted from its many-region
system in order to assess that region’s backward, forward. and/or total spatial linkages
(importance) to the rest of that system. And again, “region” would be replaced by
“country” in a many-country model.
31
n
For i ¼ j, Δxj ðpÞj ¼ xj xj =ljj ¼ xj l jj 1 =ljj , and [Δxj (pj )] ¼ l jj 1 =l jj .
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7.2 Linkages and Key Sectors in Input–Output Models
315
Table 7.5 Hypothetical extraction linkages
Partial Hypothetical Extraction Linkages
Backward
bðpÞj ¼
xj
l jj
Forward
[bðtÞj 1]
f ðpÞi ¼
xi
g ii
f ðt Þi 1
Normalizations include division
of each element by xj (backward) or xi
P
(forward), or division by nj¼1 xj (and multiplication by 100) to create
percentage
in totaloutput,
decrease
bðpÞ ¼ 100 i0 x i0 xj =i0 x ¼ 100
bðpÞj and
c
j
i0 x
100
f ðpÞ ¼ 0 f ðcÞ
i
i
xi
or values relative to the average,
~b ðpÞ ¼ nbðpÞ =i0 bðpÞ and ~f ðpÞ ¼ nf ðpÞ =f ðpÞ i
j
j
j
i
i
i
Complete Hypothetical Extraction Linkages
Backward
bðcÞj ¼
xj
l jj
Forward
bðt Þj
f ðcÞi ¼
xi
g ii
f ð t Þi
As a percentage of total original
output,
bðcÞ ¼ 100½i0 ðx xj Þ=i0 x ¼ 100
bðcÞj and
j
i0 x
f ðcÞ ¼ 100
f
ð
c
Þ
i
i
x0 i
or as values relative to the average,
~b ðcÞ ¼ nbðcÞ =i0 bðcÞ and ~f ðcÞ ¼ nf ðcÞ =f ðcÞ i
j
j
j
i
i
i
When linkages are being measured in order to make comparisons of the structure of
production between countries, the underlying coefficients matrices, whether A or B,
should be derived from total interindustry transactions data – that is, a particular zij
should include good i used by sector j, whether good i comes from domestic producers
or is imported. This is simply because interest is concentrated on how things are made
in various economies, not on where the inputs come from. On the other hand, if
linkages are being used to define “key” sectors in a particular economy, then the A or B
matrices should include only domestically supplied inputs, since it is the impact on the
domestic economy that is of interest. In studying the economies of less developed
countries, it has been suggested (Bulmer-Thomas, 1982, p. 196) that “linkage analysis
for LDCs is probably the most common use to which their input–output tables have
been put.”
An applied study that uses both backward and forward linkages in a spatial setting is
Dietzenbacher, van der Linden, and Steenge (1993). They concentrate on changes in
spatial backward and forward linkages in the European Community (EC) between
1970 and 1980 for five and seven countries, respectively (missing data caused the
differences in numbers of countries in the two years). A later study that examines both
spatial and sectoral linkages is found in Dietzenbacher and van der Linden (1997). This
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316
Supply-Side Models, Linkages, and Important Coefficients
application is based on 1980 intercountry data for seven countries and 17 sectors of the
European Community. It begins with backward and forward sectoral linkages for each
country. These are then split into domestic and external linkages (the other six
countries). Summations over all sectors give average (backward or forward) linkage
of each country to each other. Then hypothetical extraction is applied to each country –
for example, removal of sector j in Germany leads to an output reduction in all sectors
in Germany and in all sectors in the other six countries. Thus, the initial results for each
extraction indicate each of the individual linkages between (extracted) sector j in
country A and each sector in every country. With seven countries and 17 sectors, each
x vector contains 119 elements. These are translated into percentages (domestic versus
intercountry), giving a measure of each country’s importance in the European
Community economic system. Also sums (and averages) over all sectors in each
country generate linkages between each pair of countries.
7.2.6 Generalized Linkage Measures
In Sections 6.2.2 and 6.2.3 we saw that output multipliers are often not of primary
interest for policy analysis. Income multipliers, reflecting household earnings, were
explored. (The results hold equally well for employment measured in physical terms.)
Additional interest is often in measures of value added, resource use, or environmental
intrusions generated throughout the economy by that output. To that end, multipliers and
linkage measures are easily generalized to capture these impacts (as in Section 6.2.3).32
Recall the deconstruction of household income multipliers [(6.13) in Chapter 6] and
the discussion in Section 6.2.3. Let z0 represent the ð1 nÞ row of “transactions” per
sector – for example, total value added, BTUs of energy used in production, or pounds
of sulfur dioxide emitted. Then z0c ¼ z0 ^x 1 ¼ ½zc1 , zc2 ; , zcn is the corresponding
row of corresponding (z-effect) coefficients for each sector. If z measures value added,
these are value-added coefficients; if z0 ¼ g0 records energy use, or if z0 ¼ d0 measures
pollution generated, these are energy-use or pollution-generation coefficients.
(Remember that for output multipliers, z0 ¼ x0 and z0c ¼ i0 .) In general, then, parallel
to (6.13),
Sector demand
to sectorlevel
zeffect multipliers
½ Mð z Þ zfflfflffl}|fflfflffl{
^z 0 ^x 1 L
mðzÞ ¼ z0c L ¼ z0 ^x 1 L ¼ i0
|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
(7.39)
Sector demand
to economywide
zeffect multipliers
½mðzÞ
32
This point is discussed in some detail in Dietzenbacher and Lahr (2013) along with model variations to deal
with partial extractions, for example due to capacity constraints.
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7.2 Linkages and Key Sectors in Input–Output Models
317
The linkage measures in Table 7.5 can be expressed in these alternative metrics,
P
depending on the policy issue of interest. To do this, replace bðt Þj ¼ mðoÞj ¼ ni¼1 lij
in both locations in the backward linkage column of Table 7.5 with
P
bðt Þzj ¼ mðzÞj ¼ ni¼1 zci l ij , the sum of weighted elements in L, where zci is the z-effect
coefficient for sector i. (The “z” superscript is needed to identify the metric being
used.) So, for example, we have
bðcÞzj ¼
n
xj X
zci l ij ¼
ljj i¼1
xj
bðt Þzj
l jj
Similarly, forward linkages would be calculated using weighted elements in G.
Normalizations of the direct and total backward linkage measures for sector j use zcj ,
the policy variable coefficient per unit of j’s output, as the normalizing measure – e.g.,
economy-wide value added that is created per unit of value added in sector j. These are
parallels to Type I (“normalized”) household income multipliers from Section 6.2.2,
P
mðhÞIj ¼ ni¼1 anþ1, i l ij =anþ1, j [as in (6.17)], where the numerator is the jth column sum
of the income weighted inverse and the denominator is the weight (coefficient) for
sector j. For hypothetical extraction measures, whose results are expressed in monetary
units, the appropriate normalizing agent is a unit of output, xj . Finally, note that net
linkages are normalized (dimensionless) measures by definition, hence there are no
corresponding non-normalized variants. Table 7.6 presents these linkage measures.
(The use of bold face for three of these measures is discussed in Section 7.2.7, below.)
Applications of these measures (especially hypothetical extractions) have grown
enormously in recent years. Temursho (in ADB, 2016) examines total, net, and complete
hypothetical extraction backward and forward linkage measures for Khazakhstan’s
economy in 2005 and 2013 (with 60 and 68 sectors, respectively), using output,
employment, and income metrics. Using a value-added metric, several studies have
examined ways of accounting for a country’s (or a region’s) value-added in exports. For
example, Los, Timmer, and de Vries (2016) start with a simple two-country illustration
rr
rr rs Ars
f
f
A
and
f
¼
(as in Section 3.3.1, but with disaggrein which A ¼
Asr Ass
f sr f ss
gated final demands).33 In this framework they examine the amount of domestic value
added in country r (GDPr ) that can be attributed to its exports. As usual,
rr
1
I Arr Ars
L
Lrs
1
¼ ðI A Þ ¼
L¼
Lsr Lss
Asr I Ass
0
Using vrc to denote the row of value-added coefficients
in r, and setting value-added
0
34
coefficients in s equal to zero, they form v0c ¼ [ vrc
0 ] and find GDPr ¼ v0c Lfi.
33
34
We alter their notation slightly to be consistent with earlier sections of this book.
If value-added coefficients for s were not zeroed out, v0c Lfi would indicate total value added in both countries.
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318
Supply-Side Models, Linkages, and Important Coefficients
Table 7.6 Generalized linkage measures
Backward Linkage Measures
Name
Non-normalized
Pn
i¼1 zci aij
Pn
i¼1 zci l ij
Direct bðd Þzj
Total bðtÞzj
Net bðnÞzj
Normalized
Pn
i¼1 zci aij =zcj
Pn
i¼1 zci l ij =zcj
Does not exist
fj
xj
bðt Þzj
Complete Hypothetical Extraction bðcÞzj
xj
l jj
bðt Þzj
1
ljj
bðtÞzj
Partial Hypothetical Extraction bðpÞzj
xj
l jj
[bðtÞzj 1]
1
ljj
[bðtÞzj 1]
Forward Linkage Measures
Name
Non-normalized
Pn
j¼1 bij zcj
Direct f ðd Þzi
Pn
Total f ðt Þzi
j¼1 g ij zcj
Net f ðnÞzi
Complete Hypothetical Extraction f ðcÞzi
Partial Hypothetical Extraction f ðpÞzi
Normalized
P
( nj¼1 bij zcj )=zci
(
Pn
j¼1 g ij zcj =zci
)
Does not exist
vi
xi
f ðt Þzi
xi
g ii
f ðt Þzi
1
g ii
f ðtÞzi
xi
g ii
f ðtÞzi 1
1
g ii
f ðt Þzi 1
rr
Arr 0
f
0
∗
and f ¼ sr ss (hypothetically extracting
Then, creating A ¼
Asr Ass
f
f
∗
0 ∗ ∗
exports from r to s) they find GDPr ¼ vc L f i; this is total value added in country
r without its exports to s (either intermediate or final demand goods). Thus, they view
the difference, GDPr GDP∗
r , as a measure of the domestic value added that is
embodied in exports from r to s.
Many other studies have used hypothetical extraction approaches in conjunction
with the extensive World Input–Output Database (WIOD). Representative examples
include Los, Timmer, and de Vries (2015), who decompose the value of a final product
into value-added contributions in any country in the world; Los et al. (2017), who take
things down to the regional level in the UK; and Chen et al. (2018) on the effect of
Brexit on regions in the UK and in the rest of the EU. (These latter two incorporate
additional data that is needed to bring analysis possibilities down to the regional level.)
With 40 countries (plus a “Rest of the World” region) and 35 industries in the WIOD,
there are many possibilities for hypothetical extractions, as these studies indicate.
Dietzenbacher, van Burken, and Kondo (2019) formulate a global extraction model
(GEM), an adaption of the hypothetical extraction model (HEM) for a completely
connected world database (WIOD in their case). They point out that in the HEM it is
∗
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7.2 Linkages and Key Sectors in Input–Output Models
319
simply assumed that imports make up for loss of an industry in a region, but in a global
model it is necessary to specify precisely which other country or countries supply those
imports and in what proportions.
Hypothetical extraction approaches to environmental and energy issues have also
proliferated. Specialized journals in which this work appears include Ecological
Economics, Ecological Indicators, Energy Economics, Energy Policy, Journal of
Industrial Ecology, Journal of Cleaner Production, Papers in Regional Science, and
many others. To cite but a very few examples, Duarte, Sánchez-Chóliz, and Bielsa
(2002) on water usage in Spain; Ali (2015), Wang et al. (2013), and Zhao et al. (2015)
on CO2 emissions in Italy, China, and South Africa, respectively; and Zhang et al.
(2017) on the impact of consumption of 13 household groups (urban and rural
households, distinguished, respectively, by eight and five income categories) on carbon
emissions in China.
7.2.7 Which Measures to Use?
It is generally agreed that backward linkage measures should be found from the
Leontief model and forward linkages from the Ghosh model, and also that total are
more appropriate than direct linkages. Partial extraction measures may be appropriate
for certain specific situations, e.g., assessing the effects of a sector’s (rather unlikely)
complete elimination of domestic inputs in favor or imported inputs. Because of the
mathematical relationships between and among these measures, it has been suggested
(e.g., Temurshoev and Oosterhaven, 2014) that the search for “key” sectors might
adequately be conducted by considering the results of only total, net, and complete
hypothetical extraction calculations (indicated in bold in Table 7.6).35
7.2.8 Illustration Using US Data
Results for the US 2003 seven-sector tables (Chapter 2) are collected in Tables
7.7–7.10. The resulting four-way classification table (as in Table 7.3) for the total
linkage measures from Tables 7.7 and 7.8 is shown in panel A of Table 7.11.
We illustrate alternative normalizations in Tables 7.9 and 7.10, all using the output
metric. Normalizations in the first two columns simply create an average value of 1 for
the measures. In the third column, the figures indicate percentage change in (the value
of ) original aggregate output caused by removal of each sector in turn. One way of
plotting results like those in the third column of these tables is by normalizing (again)
by dividing each result in the column by the average of the figures in that column. This
makes 1 the dividing point for the figures (comparable to the other plots). Many other
35
Temurshoev and Oosterhaven (2014) consider in detail the mathematical relationships and similarities among
these linkages and make similar recommendations. For example, it is apparent from inspection that the two
versions of hypothetical extraction measures (complete and partial) differ only by xj =ljj for sector j. See also
Temursho’s chapter in ADB (2016).
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320
Supply-Side Models, Linkages, and Important Coefficients
Table 7.7 Sector j backward linkage results, US 2003 data (using the output metric, z0c ¼ i0 )
bðt Þzj
Sector
1
2
3
4
5
6
7
bðnÞzj
bðcÞzj
Non-normalized
Pn
i¼1 zci l ij
Normalized
Pn
i¼1 zci l ij =zcj
Normalized
1.92
1.61
1.72
1.93
1.49
1.61
1.60
1.92
1.61
1.72
1.93
1.49
1.61
1.60
0.47
0.58
1.50
1.04
0.92
0.89
1.34
fj
xj
bðt Þzj
Non-normalized
($ trillion)
xj
l jj
Normalized
bðt Þzj
bðtÞzj
1
ljj
418
363
1,822
5,581
3,901
10,412
3,501
1.52
1.49
1.71
1.43
1.37
1.14
1.55
Table 7.8 Sector i forward linkage results, US 2003 data (using the output metric, z0c ¼ i0 )
f ðtÞzi
Sector
1
2
3
4
5
6
7
f ðnÞzi
Nonnormalized
Pn
j¼1 g ij zcj
Normalized
Pn
j¼1 g ij zcj =zci
2.46
2.11
1.20
1.76
1.63
1.74
1.27
2.46
2.11
1.20
1.76
1.63
1.74
1.27
f ðcÞzi
Normalized
(by definition)
vi
xi
f ðtÞzi
Non-normalized
($ trillion)
xi
g jj
1.19
1.33
0.69
0.83
1.14
1.10
0.81
Normalized
f ðt Þzi
1
g jj
537
477
1,267
5,109
4,285
11,297
2,778
f ðtÞzi
1.95
1.96
1.19
1.31
1.50
1.24
1.23
Table 7.9 Sector j backward linkage results, US 2003 data (alternative normalizations)
Sector
1
2
3
4
5
6
7
bðt Þzj
bðnÞzj
bðcÞzj
Pn
Pn Pn
n i¼1 zci l ij =
j¼1
i¼1 zci l ij
P
nbðnÞzj = nj¼1 bðnÞzj
P
bðcÞzj = nj¼1 xj
1.13
0.95
1.02
1.14
0.88
0.95
0.94
0.49
0.60
1.56
1.08
0.96
0.92
1.39
2.12
1.84
9.23
28.28
19.77
52.77
17.74
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321
7.3 Identifying Important Coefficients
Table 7.10 Sector i forward linkage results, US 2003 data (alternative normalizations)
f ðt Þzi
Sector
1
2
3
4
5
6
7
Pn
Pn Pn
n i¼1 zci l ij =
j¼1
i¼1 zci l ij
1.42
1.21
0.69
1.01
0.94
1.00
0.73
f ðnÞzi
Pn
nf ðnÞzi =
z
i¼1 f ðnÞi
1.18
1.31
0.68
0.87
1.13
1.08
0.80
f ðcÞzi
Pn
f ðcÞzi =
j¼1 xj
2.72
2.42
6.42
25.90
21.72
57.25
14.08
variants are possible, such as simply plotting the results in a two-dimensional graph
with an origin of 0.
Variations in rankings in standard four-way tables that result from these alternative
normalizations are apparent in the panels of Table 7.11. In panel D we illustrate one
way out of the dilemma when we use a “majority rule” that locates a sector in a
quadrant if it is above 1 by at least two of the three measures.36
7.3
Identifying Important Coefficients
There is a long history and an enormous amount of published work, both theoretical
and empirical, on the impact (transmission, propagation) of errors or changes or
uncertainty in basic input–output data on the model outcomes. This has appeared
under a variety of titles (“probabilistic” or “stochastic” input–output, “error” analysis
and “sensitivity” analysis, and so on). Examples in the “probabilistic” vein go back at
least to Quandt (1958, 1959).37 Approaches that investigate the impacts of discrete
changes in one or more model components go back at least to the early 1950s (Dwyer
and Waugh, 1953; Evans, 1954). It is beyond the scope of this book to explore all of
this literature. (A brief review and a large set of pertinent references can be found in
Lahr, 2001.) Instead, we concentrate on the concept of “important coefficients.”
Early mathematical work on the notion of “important” coefficients (ICs) in an input–
output model explored ways of identifying aij coefficients that have a particularly
strong influence on one or more elements in the model, usually on the associated
Leontief inverse matrix and/or on one or more gross outputs – meaning that Δaij ! a
“large” Δlrs or that Δaij ! a “large” Δxr for one or more r and s. Identification of such
36
There is nothing particularly sophisticated about this rule; it is simply one that takes into account information
contained in each of the measures. See Temursho (in ADB, 2016), who proposes this rule. It is not so useful if
there are only two measures and always has a chance of failing when there is an even number of measures.
37
Also representative of this line of inquiry are Simonovits (1975), Lahiri (1983), West (1986), Jackson and West
(1989), Roland-Hoist (1989), Kop Jansen (1994), Dietzenbacher (1995, 2006), or ten Raa (1995, chapter 14;
2005, chapter 14), and the many additional publications cited in these references.
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322
Supply-Side Models, Linkages, and Important Coefficients
Table 7.11 Classification of hypothetical extraction results, US 2003 data
A. Total Linkages
Forward Linkage f ðt Þzi
Low (<1)
Backward
Linkage
bðt Þzj
Low (<1)
High (>1)
High (>1)
5 (Trade, Transp., Utilities),
7 (Other)
3 (Construction)
2 (Mining), 6 (Services)
1 (Agriculture),
4 (Manufacturing),
B. Net Linkages
Forward Linkage f ðnÞzi
Low (<1)
Backward
Linkage
bðnÞzj
High (>1)
Low (<1)
High (>1)
1 (Agriculture), 2 (Mining), 5 (Trade,
Transp., Utilities), 6 (Services)
3 (Construction), 4
(Manufacturing), 7
(Other)
C. Hypothetical Extraction Linkages
Forward Linkage f ðcÞzi
Low (<column average)
Backward
Linkage
bðcÞzj
Low (<col.
avg.)
High (>col.
avg.)
High (>column average)
1 (Agriculture), 2 (Mining), 3
(Construction), 7 (Other)
4 (Manufacturing), 5 (Trade,
Transp., Utilities), 6 (Services)
D. Majority Rule Classification
Forward Linkage f ðcÞzi
Low (<column
average)
Backward
Linkage
bðcÞzj
Low (<col.
avg.)
High (>col.
avg.)
3 (Construction), 7
(Other)
High (>column average)
2 (Mining), 5 (Trade, Transp., Utilities),
6 (Services)
1 (Agriculture), 4 (Manufacturing)
coefficients can be helpful in deciding where to expend effort in obtaining superior
information for updating or regionalizing a known input–output table using a hybrid
model. And ICs contribute to some studies of key sectors and of what has come to be
known as “fundamental economic structure.” Jackson (1991) suggests that a distinction should be made between coefficient error (e.g., estimation error) and coefficient
change (e.g., technological change).
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7.3 Identifying Important Coefficients
323
Table 7.12 Number of important transactions in the 2000 China MRIO model
Criterion
Number of Cells
Percentage of Total Number of Cells
0
4,715
1,042
84
32
8.19
1.81
0.15
0.06
>(i Zi/n )
>(10) (i0 Zi/n2)
>(100) (i0 Zi/n2)
>(200) (i0 Zi/n2)
2
In what follows, we examine the mathematical underpinnings of these approaches
and then several kinds of studies – primarily influences on inverse elements and on
gross outputs. Reviews of much of this work can be found in Xu and Madden (1991),
Casler and Hadlock (1997), and Tarancón et al. (2008). There are many more published studies than we are able to cite. A large amount of work was done in Germany in
the 1970s and 1980s and published in German, making it somewhat less accessible to a
segment of the English-speaking audience – for example, Schintke (1979, 1984), Maaß
(1980), and numerous references therein.
One very straightforward way to assess “importance” of individual cells in input–
output data is simply to compare each transaction (zij ) with the average transaction
amount ði0 Zi=n2 Þ. This is done in Okamoto (2005) for the 2000 China multiregional
input–output data (CMRIO) made up of eight regions with 30 sectors each – a total of
57,600 potential elements in Z. Table 7.12 shows results for this particular data set
(adapted from Okamoto, 2005, p. 141). A similar approach could also be used on the
data in coefficients matrices (A or B) or total requirements matrices (L or G).
7.3.1 Mathematical Background
These investigations build on early results in Sherman and Morrison (1949, 1950) and
Woodbury (1950) – hereafter SMW – who studied how changes in elements in a (nonsingular) matrix were transmitted to changes in elements in the inverse of that matrix.
(Basic results and additional details are presented in Appendix 7.1.) Given a nonsingular matrix, M, and its inverse, M1 ¼ [μij ], assume that one (or more) elements of
∗
M are changed, i.e., m∗
ij ¼ mij þ Δmij producing M ¼ M þ ΔM. SMW show how
1
the elements of ðM∗ Þ ¼ [μ∗
ij ] can be found by “adjusting” the known elements μij .
This is addressed by Sherman and Morrison (1950) for the case when only one element
is changed, by Sherman and Morrison (1949) for changes in several elements in a
given column or row, and by Woodbury (1950) for changes in elements in several rows
(or columns).38
For the simplest situation, when a single element mij is changed (increased or
decreased) by an amount Δmij , the value of the element in row r and column s of the
new inverse is found to be
38
For much more detail on all of these results, see Miller (2000, appendices 5.2 and 6.1).
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Supply-Side Models, Linkages, and Important Coefficients
μ∗
rs ¼ μrs μri μjs Δmij
1 þ μij Δmij
(7.40)
Building on this result, it is possible to trace the influence of a change (or “error”) in
an element of an A matrix – and hence in (I A) – on the associated Leontief inverse,
L ¼ ðI AÞ1 . In this case, we begin with A∗ ¼ A þ ΔA. Since our interest is in
L∗ ¼ L þ ΔL, the parallel to M∗ ¼ M þ ΔM is
ðI A∗ Þ ¼ ½I ðA þ ΔAÞ ¼ ðI AÞ þ ðΔAÞ
For individual elements in the new inverse, l ∗
rs , the result in (7.40) becomes
l∗
rs ¼ l rs þ
l ri ljs Δaij
1 lij Δaij
(7.41)
[This is (A7.1.4) in Appendix 7.1.] Notice that the signs are reversed from those in
(7.40) because of the way in which ΔA enters the expression for ðI A∗ Þ.
7.3.2 Relative Sizes of Elements in the Leontief Inverse
The following observations on Leontief inverse elements are relevant to the problem of
identifying important coefficients. As we will see, they help to reduce the number of
coefficients that need to be examined when ranking those elements for importance.
Observation 1 From the power series approximation, it is clear that all ondiagonal elements in a Leontief inverse are larger than one. Also, it is virtually always
observed in real-world Leontief inverse matrices that l rs < 1 ðr 6¼ sÞ (off-diagonal
elements are less than one);39 thus l ii > 1 > l rs for all ðr 6¼ sÞ. This will be of use for
the results in (7.42).
Observation 2 In (7.43) and (7.44), it will be of interest to identify the
largest of the ratios lri ljs =l rs for a given i and j. Schnabl (2003, p. 497) reports on results
in Maaß (1980, in German).
Maaß’s calculation showed that the maximum (of these ratios) is attained if r ¼ i and
s ¼ j because then the main diagonal element of the inverse is involved twice and since
the main diagonal element is usually the biggest one in a row or column this gives the
maximum.
Thus, Max l ri l js =lrs ¼ l ii l jj =l ij .
r, s¼1,..., n
39
This is not to say that a counterexample cannot be constructed, but rather that they do not seem to occur in
practice. For example
2
3
2
3
:02 :4 :4
1:7767 1:0779 1:0445
A ¼ 4 :3 :05 :3 5 ) L ¼ 4 :8711 1:6925 :8649 5:
:4 :30 :01
:9818 :9484 1:6942
As an illustration, all the US Leontief inverses in Miller and Blair (1985, appendix B), from 1947 through
1977, at both 23- and seven-sector levels of aggregation, exhibit the properties of Observation 1.
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7.3 Identifying Important Coefficients
325
Finally, Max l ri =xr ¼ l ii =xi . It is not at all obvious that
r¼1,..., n
this should be the case, since the sizes of sectors (as measured by their gross outputs) can
vary greatly in real-world models. Nonetheless, it was observed in some early empirical
observations and is proven to always be the case in Tarancón et al. (2008).40 This is
useful for the results in (7.48).
Observation 3
7.3.3 “Inverse-Important” Coefficients
For the remainder of this section, it will be useful to complicate the notation in order to
be explicit about the element in A that is changed. From (7.41),
ΔlrsðijÞ ¼ l∗
rsðijÞ l rs ¼
lri ljs Δaij
¼ lri l js k 1ðijÞ
1 lji Δaij
(7.42)
∗
where k 1ðijÞ ¼ Δaij = 1 l ji Δaij , a constant for a given i and j, and L∗
ðijÞ ¼ [l rsðijÞ ]
reminds us that the change is in aij . From Observation 1, Δaij will exert the largest
ð1Þ
influence on l ij when r ¼ i and s ¼ j, since then both elements multiplying k ij are
larger than one [when l ri is l ii ð> 1Þ and l js is ljj ð> 1Þ]. Similarly, next-largest influences will be felt in row i or column j of L, since then either l ri ! l ii ð> 1Þ or
ljs ! ljj ð> 1Þ. In virtually all other cases (not row i or column j) both elements of
the product l ri l js are less than one.
From (7.42), the expression for relative changes in Leontief inverse elements is
ΔlrsðijÞ
l ri l js Δaij
l l
¼ ri js k 1ðijÞ
¼ lrs
l rs
lrs 1 l ji Δaij
(7.43)
This is where Observation 2 becomes relevant. Since Max l ri l js =l rs ¼ l ii l jj =lij , it is
r, s¼1, ..., n
clear that, again, Δaij will create the largest relative change on l ij .
In addition, the elements Δl rsðijÞ =lrs in column i and row j of the matrix of
relative changes will all be identical. In column i (when s ¼ i), ΔlriðijÞ =l ri ¼
l ri l ji =l ri k 1ðijÞ ¼ lji k 1ðijÞ , and in row j (when r ¼ j), Δl jsðijÞ =ljs ¼ l ji l js =ljs k 1ðijÞ ¼
lji k 1ðijÞ ¼ Δl riðijÞ =l ri .
Finally, the percentage changes are
Δl rsðijÞ
l ri ljs Δaij
l ri l js 1
1
¼ 100
¼ 100
k ij
prsðijÞ ¼ 100
l rs
1 l ji Δaij l rs
l rs
(7.44)
Again, pijðijÞ will be the largest percentage change caused by Δaij.
It has been suggested (for example, Hewings, 1981) that aij may be viewed as
“inverse-important” if, for a specified “threshold” amount of change in an inverse
element, β, prs > β for one or more r and s, that is, if
40
Sekulić (1968) observed this to be true for the Yugoslav economy in the early 1960s. Similar observations are
made in Schintke (1979 and elsewhere) based on German data. See also results from US data in Table 7.12.
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326
Supply-Side Models, Linkages, and Important Coefficients
prsðijÞ ¼ 100
l ri l js Δaij
1 l ji Δaij
1
>β
lrs
(7.45)
Denote the percentage change in aij by α, so that Δaij ¼ ½α=100aij ; then we have
l ri l js αaij
100
β
(7.46)
100 l ji αaij lrs
for any l rs and a given α and β. For example, let α ¼ 20 and β ¼ 10. This means that aij
will be considered inverse-important if a 20 percent change in its value generates a
10 percent or larger change in one or more elements in the Leontief inverse. The
analyst must specify α and β, on the basis of the particular problem under study.
Given Observations 1 and 2, establishing inverse importance for each aij in an nsector A matrix requires only one application of (7.45) [or (7.46)] – for r ¼ i and
s ¼ j.41 The virtue of the SMW method is that it finds this information about the inverse
by working exclusively with known elements in L and avoiding direct calculation of
the new inverse. This was the whole point of the formulation.
At present, however, finding inverses is not quite the task it was in 1950 when the
SMW approach was developed; at least this is true for matrices that are not “too
large.” Then a straightforward alternative to applying (7.45) or (7.46) is to calculate
directly the L∗
ðijÞ associated with each Δaij and then find the corresponding matrix of
percentage changes, PðijÞ ¼ [prsðijÞ ] ¼ 100f[L∗
ðijÞ L]Lg, where “” indicates element-by-element division.
This line of work, formulating the notion of inverse-important coefficients, was
taken up initially in the early 1980s by Hewings, Jensen, West, and others. Examples
are Jensen and West (1980), Hewings (1981), Hewings and Romanos (1981), and
Hewings (1984). For hybrid (partial-survey) models, the idea is to identify coefficients
(or sectors) for which additional information (survey, expert opinion) would be
particularly useful. But, of course, identifying inverse importance implies that a
relevant matrix of coefficients already exists to supply the elements in results like
(7.45). For updating, there is a base matrix to be updated, and the premise is that
important coefficients at time “t” will also be important at time “t þ 1.” However, there
is no hard evidence to support that argument. In fact, Hewings (1984, p. 325),
cautioned that “. . . only 3 of the cells deemed inverse-important in 1963 [the
Washington State 49-sector model] were similarly identified in 1967.” For regional
models there is often not an “earlier” regional table. In the context of estimating a
coefficients table in a regional context, Boomsma and Oosterhaven (1992, p. 276, n. 3)
observed:
41
If one wants not simply to establish inverse-importance, but also extent [that is, for a given Δaij , how many (and
which) prsðijÞ exceed the β threshold], then [n2 (2n 1)] calculations like those in (7.45) or (7.46) would be
needed – the n2 inverse elements, lrs , for a given aij , less those in row j and column i that are all identical – and
these calculations must be made n times, once for each of the aij . The “field of influence” approach (Section
7.3.6) accomplishes this in one matrix operation.
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7.3 Identifying Important Coefficients
327
Here we have a typical “chicken or egg” problem. Without a regional table one cannot determine the
inverse-important cells and without that information one cannot construct a decent regional table.
Hence, we suggest use of the national table as second best information on inverse-importance.
7.3.4 Numerical Example
We use the two-sector example closed with respect to households from Section 2.5,
namely42
2
3
2
3
:15 :25 :05
1:3651 :4253 :2509
A ¼ 4 :20 :05 :04 5 and L ¼ 4 :5273 1:3481 :5954 5
:30
:25 :05
:5698
:4890 1:2885
2
3
:15 :30 :05
Consider Δa12 ¼ ð0:2Þa12 (that is, α ¼ 20); then A∗ ¼ 4 :20 :05 :40 5, and we can
:30 :25 :05
easily calculate L∗
ð12Þ directly as
2
3
1:4021 :5198 :2926
4 :5416 1:3846 :6115 5
L∗
ð12Þ ¼
:5853 :5285 1:3060
Then
2
3
2:7080 22:2225 16:6345
Pð12Þ ¼ 4 2:7080 2:7080 2:7080 5
2:7080 8:0667 1:3521
As expected, with the change Δa12 , all elements in column 1 and row 2 are identical.
Further, in this illustration ljj > 1 ðj ¼ 1; . . . ; 3Þ, l ij < 1 ði ¼ 1; . . . ; 3, i 6¼ jÞ
(Observation 1) and indeed the largest change caused by Δa12 is in l 12 – here this is
p12ð12Þ ¼ 22:2 percent. If we specify β ¼ 10 as our criterion in (7.46) for inverseimportance – namely when a change of 10 percent or more is experienced by at least
one inverse coefficient – then we see that a12 would be classified as inverse-important
because Δa12 ¼ ð0:2Þa12 causes both l12 and l 13 to be changed by more than 10 percent.
In this small three-sector case, it is relatively easy to modify each element in A, in turn,
by 20 percent, find the associated Leontief inverse and then the associated P matrix.
(Readers are encouraged to do this, at least for several additional aij .) If we continue to
use β ¼ 10 in that series of calculations – for Δa11 ¼ ð0:2Þa11 , , a33 ¼ ð0:2Þa33 – we
will identify a21 , a23 , a31 , and a32 also as important.43 (Higher values of β serve to raise
the bar on eligibility for importance. For example, with β ¼ 20, only a12 and a23 would
be labeled important.)
42
In (Section 2.5 these matrices contained overbars to indicate a model closed with respect to households and to
distinguish them from the earlier open model. At this point the overbars just get in the way of other notation
and will be dropped.
43
In each of the eight cases the largest change caused by that Δaij was in the associated lij , as expected, but four of
those were below the β ¼ 10 threshold.
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328
Supply-Side Models, Linkages, and Important Coefficients
“Importance” could be identified in many other ways, for example, with respect to
0
0
0
0
changes in output multipliers – as in 100f[i0 L∗
ðijÞ i L]i Lg ¼ 100f½i ΔLi Lg. If β
now refers to percentage change in multipliers, only a23 is found to be important with
β ¼ 10; however, at β ¼ 5 the same five coefficients as above are identified.
As noted, the point of the SMW result is that these percentage changes in inverse
coefficients can be found without knowing the new inverse at all. Continuing with
Δa12 ¼ ð0:2Þa12 , consider the percentage change in l13 [p13ð12Þ in Pð12Þ , as shown in
introducing this example]. Using (7.45) with i ¼ 1, j ¼ 2, r ¼ 1, s ¼ 3, and Δa12 ¼ 0:05,
we have
l11 l 23 Δa12 100
ð1:3651Þð0:5954Þð0:05Þ
100
¼
¼ 16:6359
p13ð12Þ ¼
1 l21 Δa12 l 13
1 ð0:5273Þð0:05Þ
ð0:2509Þ
Except for rounding (and the number of significant digits carried in the inversion
programs used to find L and L∗
ð12Þ ), this corresponds to the p13ð12Þ found in Pð12Þ . Any
other value in Pð12Þ could be found in the same way.
The designation of inverse-importance depends crucially on the choice of α and β. In
a study of several of the Washington State 49-sector tables, Hewings (1984) used
α ¼ 30 and β ¼ 20. Out of 49 49 ¼ 2,401 direct input coefficients, between 24 and 42
(1.0–1.7 percent) were judged inverse-important. In a similar study in Sri Lanka (also
Hewings, 1984), 3.5 percent were found important in a 12-sector model (apparently
using the same a and β). There were interesting although not surprising variations in a
two-region Sri Lanka interregional input–output model between intraregional and
interregional coefficients (now with a 24 24 matrix); 3.3 percent of the (possible)
288 intraregional coefficients were important and 0.9 percent of those 288 in the
interregional matrices were important. In a similar study (Hewings and Romanos,
1981) using a 22-sector model for the rural Evros region in Greece, 18 of 484 possible
coefficients (3.7 percent) were important – only here, because of the less-developed
nature of the economy, the critical values used were α ¼ 20 and β ¼ 1. With those same
values, a 22-sector model for the Greek national economy had 38 important coefficients (7.9 percent).
7.3.5 Impacts on Gross Outputs
Early applications of these ideas to the impact of coefficient change on gross
outputs are found in Sekulić (1968) and Jilek (1971).44 In matrix terms,
∗
ΔxðijÞ ¼ x∗
ðijÞ x ¼ LðijÞ f Lf ¼ ΔLðijÞ f. From (7.41), we see that row r of ΔLðijÞ is
44
Δl r1ðijÞ ; , Δl rnðijÞ ¼
l ri Δaij l j1 ; , l jn
1 lji Δaij
Sekulić (1968) and later Jílek (1971) attribute the approach to E. B. Yershof, who contributed a chapter of a
1965 Moscow publication on planning (in Russian). The bases of the approach are in the work of SMW, some
15 years earlier.
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7.3 Identifying Important Coefficients
and therefore
329
2
3
2 3
f1
f
6 . 7
6 .1 7
l ri Δaij l j1 ; , l jn 4 .. 5
ΔxrðijÞ ¼ Δl r1ðijÞ ; . . . , ΔlrnðijÞ 4 .. 5 ¼
1 l ji Δaij
fn
fn
2
3
f1
6 7
But since lj1 ; , l jn 4 ... 5 ¼ xj , this is just
fn
ΔxrðijÞ ¼
lri xj Δaij
¼ l ri k 2ðijÞ
1 l ji Δaij
(7.47)
where k 2ðijÞ ¼ xj Δaij = 1 l ji Δaij . Compared with the expression for ΔlrsðijÞ in (7.42),
ljs has been replaced on the right-hand side by xj. Again, from Observation 1, l ii > l ri
(for r ¼ 1, . . . , n; r 6¼ i), so (7.47) indicates that the largest gross output change from
Δaij will be in sector i (that is, when r ¼ i).45
The relative change in xr is then
ΔxrðijÞ
lri xj Δaij
l ri 2
k
¼
(7.48)
¼
xr
xr ðijÞ
xr 1 l ji Δaij
Here the largest relative change in gross output for a given Δaij will be in sector s, for
which l si =xs ¼ Max ðl ri =xr Þ, and from Observation 3, this will be for sector i. The
r¼1,..., n
interested reader can easily show this to be true for the numerical example. Table 7.13
presents the same calculations for the 2003 US seven-sector data from Chapter 2,
showing that the largest ratios (in bold) are on the main diagonal.
Table 7.13 (l ri =xr ) 106 for the 2003 US seven-sector model
r¼1
r¼2
r¼3
r¼4
r¼5
r¼6
r¼7
45
i¼1
i¼2
i¼3
i¼4
i¼5
i¼6
i¼7
4.5868
.0381
.0071
.0589
.0523
.0261
.0107
.0211
4.4187
.0032
.0306
.0297
.0321
.0105
.0477
.0501
.9449
.0672
.0480
.0295
.0102
.2093
.1408
.0060
.3449
.0547
.0319
.0162
.0136
.0794
.0061
.0178
.3811
.0297
.0124
.0253
.0136
.0105
.0220
.0209
.1544
.0131
.0263
.0301
.0235
.0324
.0298
.0343
.4566
If xi is small relative to other outputs, then a large Δxi may not have much economy-wide importance. There
have been attempts to take this aspect of relative output size into account, but we do not consider this level of
detail. The interested reader might speculate on how this could be done.
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330
Supply-Side Models, Linkages, and Important Coefficients
Table 7.14 Percentage change in x resulting from Δa12 ¼ ð0:2Þa12
i¼1
i¼2
i¼3
2
j¼1
3
4:27
4 :82 5
1:78
2
3
1:73
4 2:74 5
1:99
3
2
1:53
4 1:81 5
7:85
2
j¼2
3
14:02
4 2:71 5
5:85
2
3
:86
4 1:37 5
:99
3
2
2:59
4 3:07 5
13:28
2
j¼3
3
1:37
4 :27 5
:57
2
3
3:54
4 5:61 5
4:07
3
2
:25
4 :30 5
1:31
Finally, multiplication in (7.48) by 100 creates a percentage change,
ΔxrðijÞ
Δaij
l ri xj
¼ 100
100
xr
1 l ji Δaij xr
(7.49)
Table 7.14 contains these percentages for x1, x2 , and x3 from the hypothetical example
as a result of Δa12 ¼ ð0:2Þa12 for all nine direct input coefficients (i, j ¼ 1, 2, 3).
As expected, for any Δaij, the largest changes are found in xi ; in the row for i ¼ 1,
this means Δx1 > Δx2 and Δx1 > Δx3 , and so on in the rows for i ¼ 2 and i ¼ 3. Also,
with α ¼ 20, if the criterion for “importance” is that one or more outputs changes by
β ¼ 10 percent, then a12 and a32 would be labeled most and second-most important.
These percentage changes are indicated in bold in the table. [The interested reader
might speculate on why it is not surprising that coefficients judged important by the
criterion in (7.45) are likely to be tagged as important by the criterion in (7.49).]
Again, the “importance” of any aij could be defined in terms of the impact of relative
or percentage changes in aij on the associated relative or percentage changes in each xr .
Using γr for the (user-specified) threshold on percentage changes in xr ,
100Δaij
lri xj
γr
1 l ji Δaij xr
[Compare (7.45).] Again, with Δaij ¼ ½α=100aij , we have
"
# αaij
lri xj
100αaij
l ri xj
h α i
¼
γr
100 lji αaij xr
1 l ji 100 aij xr
Much of the empirical work in this area is based on a rearrangement of (7.49).
Putting Δaij on the left and converting to relative change in aij , we have
ΔxrðijÞ =xr
Δaij
¼ aij
aij l ji ΔxrðijÞ =xr þ l ri xj =xr
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7.3 Identifying Important Coefficients
331
Table 7.15 Upper threshold on Δaij =aij for γ ¼ 1 percent
i¼1
i¼2
i¼3
j¼1
j¼2
j¼3
4.90
7.46
2.58
1.47
14.92
1.55
14.71
3.73
15.50
Define an allowable error limit, γ, for all sectors r which is just fulfilled by positive
relative deviations Δaij =aij . This is often called a “tolerable limit, TL” and hence
the
name “tolerable limits approach.” As is frequently done, let γ ¼ 100 ΔxrðijÞ =xr ¼ 1
percent; then in percentage terms
100ΔxrðijÞ =xr
Δaij
1
¼ ¼ aij
aij lji 100ΔxrðijÞ =xr þ 100 lri xj =xr
aij l ji þ 100ðlri =xr Þxj
Expressed in this way, we see that the larger the denominator on the right-hand side,
the smaller Δaij =aij. So the upper threshold on Δaij =aij will be determined by
Max lri =xr ,
r¼1,..., n
Δaij
1
aij
aij l ji þ 100 Max ðl ri =xr Þxj
r¼1,..., n
As noted (Observation 3) Max l ri =xr ¼ lii =xi so
r¼1,..., n
Δaij
1
aij
aij lji þ 100ðlii =xi Þxj
(7.50)
establishes an upper limit on the relative change in aij that assures that no gross output will be
changed by more than one percent.46 The smaller Δaij =aij, the more important the coefficient
aij . Table 7.15 shows the right-hand sides of (7.50) for our small numerical example.
From the upper-left element in the table, we learn that a11 could change by as much
as 4.9 percent before any output would be changed by more than one percent.
Similarly, a12 (1.47) is identified as the most important coefficient (smallest value in
Table 7.15), followed by a32 (1.55), a31 (2.58), and so on. While perhaps not of much
interest, we would also conclude that a33 is least important, since it could change by as
much as 15.5 percent before any gross output would be changed by more than one
percent. [Since the result in (7.50) comes directly from the result in (7.49), it should not
be surprising that the importance rankings of the nine coefficients in our numerical
example that are shown in Tables 7.14 and 7.15 are exactly the same.]47
This can be found in Sekulić (1968). Forssell (1989, p. 431) describes it as a measure “developed by Mäenpää
(1981)” but it seems to have been suggested much earlier. The Jugoslav journal in which the Sekulić paper
appeared may not be well known, but the paper was also presented at the Fourth International Conference on
input–output Techniques in Geneva in 1968.
47
Empirical examples identifying important coefficients for a variety of tolerable limits can be found in ArocheReyes (1996, 2002) for Mexico (1970, 1980) in the first case and for Mexico (1971, 1990), Canada (1972,
1990), and the USA (1971, 1990) in the second.
46
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332
Supply-Side Models, Linkages, and Important Coefficients
Table 7.16 Average values in US total requirements matrices
Number of Sectors
^
i0 Li=n
̌ ðn2 nÞ
i0Li=
n¼7
n ¼ 16
n ¼ 61
1.1739
1.1290
1.1113
.0868
.0429
.0133
The denominator on the right in (7.50),
aij l ji þ 100 l ii =xj xj
has been described as a measure of the “degree of importance” of aij (for example,
by Schintke and Stäglin, 1984). In real-world applications it turns out that
100ðlii =xi Þxj , especially for relatively disaggregated input–output models, again
l ji
because of Observation 1 (l ii > 1 > l ij ). In fact, there are usually quite large differences
between the l ii and the l ij . For example, average values of on-diagonal elements (in L)
̌ in Leontief inverses for 2003 US input–output data
and off-diagonal elements (in L)
are shown in Table 7.16.
This suggests that, for any given aij and irrespective of xi and xj ,48 the first term can
be ignored and the measure can be approximated as
100 aij ðl ii =xi Þxj
aij lij þ 100 ðlii =xi Þxj
Using bij ¼ zij =xi ¼ aij xj =xi (the usual “output coefficient” from the Ghosh model) this
has also been expressed as
aij l ij þ 100 ðl ii =xi Þxj
100 bij lii
7.3.6 Fields of Influence
In a number of articles, Sonis and Hewings and their colleagues have developed and
applied the concept of a “field of influence” associated with each coefficient in an A
matrix.49 This is essentially an extension of the Sherman–Morrison approach that
generates in one operation the entire matrix of changes in the Leontief inverse associated
with a given change in a particular aij. Recall that ΔlrsðijÞ is related to Δaij through
Δl rsðijÞ ¼ l ∗
rsðijÞ l rs ¼
l ri l js Δaij
¼ l ri l js k 1ðijÞ
1 l ji Δaij
Of course one could generate counterexamples with very large xi and very small xj so that lji > 100ðl ii =xi Þ=xj .
The point is that this does not seem to happen in real-world applications.
49
The publications are numerous, going back at least to Sonis and Hewings (1989). A fairly compact statement
can be found in Sonis and Hewings (1992) and an application (to the Chicago economy) is presented in
Okuyama et al. (2002).
48
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7.3 Identifying Important Coefficients
333
[This is (7.42).] Finding all the n2 elements in the n n matrix ΔLðijÞ ¼ Δl rsðijÞ would
require ½n2 ð2n 1Þ operations, as we saw in footnote 39. Instead, Sonis and
Hewings propose an efficient alternative.
2 3
l 1i
6 . 7
Let column i and row j of L be denoted L•i ¼ 4 .. 5 and Lj• ¼ l j1 ; . . . , l jn . Then
l ni
the first order (direct) field of influence of the incremental change Δaij is defined by
Sonis and Hewings as the matrix50
2
3
2 3
l 1i l j1 l1i l j2 l1i ljn
l 1i
7
6
6 . 7
6 l 2i l j1 l2i l j2 l2i ljn 7
F½i; j ¼ Li Lj ¼ 4 .. 5 lj1 ; . . . , l jn ¼ 6 .
.
.
..
.. 7
4 ..
5
l ni
l ni l j1 lni l j2 lni ljn
Thus, F½i; j ¼ l ri l js for r, s ¼ 1, . . ., n is the expanded version of the product lri l js on
the right-hand side of (7.42), and the matrix showing the change in each element of L
caused by Δaij is just ΔLðijÞ ¼ F½i; jk 1ðijÞ . Therefore
L∗
Δaij = 1 l ji Δaij F½i; j ¼ L þ F½i; jk 1ðijÞ
ðijÞ ¼ L þ ΔLðijÞ ¼ L þ
Since k 1ðijÞ is a constant for any specific Δaij , corresponding elements of ΔLðijÞ and F½i; j
are proportional and will have the same ordering – for
2 example,
3 largest to smallest.
1:3651
In the numerical example, L•i ¼ L•1 ¼ 4 :5273 5 and Lj ¼ L2 ¼
:5698
½ :5273 1:3481 :5954 so
2
3
:7198 1:8402 :8127
F½1, 2 ¼ L1 L2 ¼ 4 :2781 :7109 :3139 5
:3005 :7682 :3393
Further, Δa12 ¼ 0:05 and k 1ð12Þ ¼ ½ðΔa12 Þ=ð1 l 21 Δa12 Þ ¼ 0:0514 so
2
3
:0370 :0945 :0417
ΔLð12Þ ¼ F½1, 2ð0:0514Þ ¼ 4 :0143 :0365 :0161 5
:0154 :0395 :0174
and it is easily verified that L∗
ð12Þ ¼ L þ ΔLð12Þ .
Sonis and Hewings suggest that inverse-important coefficients can be identified
by comparing their fields of influence.51 The problem is how to reduce the n2
i
to indicate a field of influence in early publications; later (for example, Sonis
j
and Hewings, 1999) this became F[i, j].
51
In other publications they also propose higher-order fields of influence when two or more coefficients change
(with associated mathematical representations that are much more complicated), and they also use some of
these concepts to characterize the fundamental structures of economies and provide alternative kinds of
model decompositions.
50
Sonis and Hewings used F
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334
Supply-Side Models, Linkages, and Important Coefficients
pieces of information in each F½i; j in order to make comparisons across the
Δaij .52 The norms of these matrices offer one possible compact measure; the
trouble is that there are many different definitions of a matrix norm. Among those
that they mention (Sonis and Hewings, 1992, p. 147) are
X f ðlargest individual elementÞ53
kFk ¼ Max
ij
j
kFk ¼
i
X f ðsum of absolute values of all elementsÞ
ij
ij
kFk ¼
X 1=2
f ij
ij
In Chapter 2 we used a largest column sum norm; kFk ¼ Max j
P i
f ij . Further,
[t]he choice of norm ||F|| is the basis of the construction of the rank-size sequence of the elements aij
of the matrix A according to the numerical sizes of the norms ||F[i, j]||. The decision or cutting rule
must be formulated in such a way that only a relatively small number of the elements of the rank-size
sequence will comprise the set of inverse-important coefficients. (Sonis and Hewings, 1992, p. 147)
Returning to our numerical example, we generated fields of influence for each
of the nine coefficients in A using Δaij ¼ ð0:2Þaij – namely, Δa11 ¼ ð0:2Þa11 ,
Δa12 ¼ ð0:2Þa12 , and so on. Tables 7.17 and 7.18 present two summary measures
Table 7.17 Column sums of |F[i, j]| for numerical example
j¼1
aij
i¼1
i¼2
i¼3
3.3612
3.0884
2.9142
1.0471
0.9621
0.9078
j¼2
0.6178
0.5676
0.5356
1.2984
1.193
1.1257
Table 7.18 Sum of all elements in |F[i, j]| (kFk ¼
3.3193
3.0499
2.8779
j¼3
1.4659
1.3469
1.271
1.4031
1.2892
1.2165
1.2042
1.1064
1.044
3.1727
2.9152
2.7508
P ij f ij )
aij
j¼1
j¼2
j¼3
i¼1
i¼2
i¼3
5.0261
4.6181
4.3577
6.0837
5.5898
5.2746
5.7800
5.3108
5.0113
Generally, for comparability, each aij is changed by the same percentage, α, so that Δaij ¼ ðα=100Þaij for all i, j.
In presenting applications identifying important coefficients, Sonis and Hewings (1992) do not specify either
their choice of norm or their coefficient alteration mechanism.
53
There is no need to generate the
entire field of influence matrix if one is then going to summarize the
information by using the max ij f ij norm of that matrix. We know that the largest Δl rsðijÞ is ΔlijðijÞ , and that
f rsðijÞ is proportional to ΔlrsðijÞ so this can be found using the Sherman–Morrison results in (7.42) for Δl ijðijÞ only.
52
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7.3 Identifying Important Coefficients
335
(norms) from these
nine F[i, j] matrices. Table 7.17 contains the column sums, and the
P kFk ¼ Max j i f ij norm is obvious by inspection in each case. Table 7.18 contains
P the kFk ¼ ij f ij norm for the nine coefficients.
7.3.7
Additional Measures of Coefficient Importance
Converting Output to Employment, Income, etc. As noted many times earlier
in this book, gross outputs may not ultimately be the most important measure of
economic impact. Gross output requirements can be translated into employment (for
example, person-years) using employment coefficients (for example, person-hours per
dollar’s worth of each sector’s output). If these coefficients are denoted ec and total
2 3
ε1
6 .. 7
employment in each sector is represented by ε ¼ 4 . 5, then Δε ¼ ^e c Δx converts
εn
changes in outputs to changes in employment. For example, from (7.46),
Δεr ¼ ðec Þr ΔxrðijÞ ¼
ðec Þr l ri xj Δaij
xj Δaij
¼ ðec Þr l ri k 2ðijÞ where k 2ðijÞ ¼
1 lji Δaij
1 l ji Δaij
The largest employment impact of Δaij will thus be in the sector with the largest
ðec Þr l ri , and this is no longer assured to be sector i. Numerous other conversions are
also possible – for example, to changes in income, value added, energy use, environmental impacts, and so forth.54
Elasticity Coefficient Analysis Several authors have suggested a variation of
the measure of relative change that parallels the concept of elasticity in economics (see
Section 6.6), namely the relative change in lrsðijÞ divided by the relative change in aij
Δl rsðijÞ Δl rsðijÞ
Δaij
l
ηlrsðijÞ ¼ rs ¼
¼
Δaij
l rs
aij
aij
Δl rsðijÞ
Δaij
aij
lrs
(7.51)
From (7.43), this is
ηlrsðijÞ ¼
l l a
l l
ri js ij ¼ ri js k 3ðijÞ
l rs
lrs 1 l ji Δaij
(7.52)
where k 3ðijÞ ¼ aij = 1 l ji Δaij . Notice that this differs from the expression for
Δl rsðijÞ =lrs in (7.43) only in that Δaij has been replaced by aij in the numerator. For
any aij, there will be n2 of these elasticities. Then Maaß (1980; cited in Schnabl, 2003)
54
Tarancón et al. (2008) discuss in some detail the identification of important coefficients using alternative
measures of economic welfare.
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336
Supply-Side Models, Linkages, and Important Coefficients
proposed the maximum of these elasticities as another measure of the importance of
aij Max rs (ηlrsðijÞ ). From Observation 2, again, it is clear that
Max rs (ηlrsðijÞ ) ¼
l ii l jj aij
l ij 1 l ji Δaij
So, as noted by Schnabl, this elasticity analysis generates the same results as the
important coefficient analysis above.
Replacement of ΔlrsðijÞ =l rs in the numerator in (7.51) by ΔxrðijÞ =xr will lead to an
expression for the elasticity of gross output with respect to Δaij . And, just as gross
output impacts can be translated into employment, income, value-added, etc. effects,
these variations too can be converted to elasticity measures.
Relative Changes in All Gross Outputs
takes into account changes in all outputs is
A straightforward error measure that
n X
Δxk ðijÞ EðijÞ ¼ i0 ΔxðijÞ ¼
k¼1
Or, to take account of the relative sizes of the sectors, Siebe (1996) suggests
SUMðijÞ ¼
n X
Δxk ðijÞ =xk k¼1
as a measure of importance of each coefficient, Δaij . As with previous measures, this
could be transformed into an aggregate effect on employment, income, valueadded, etc.
Impacts of Changes in More than One Element of the A Matrix Assessing the
importance of each aij relative to all the others is carried out using one or more of the
one-at-a-time approaches. There has also been considerable work on the impacts of
simultaneous changes (errors) in many or all aij coefficients. Indeed Sherman and
Morrison (1949) considered cases with more than one change, but concentrated in a
single row (or column). This was also explored in many publications by Schintke
(1979 and elsewhere) and Schintke and Stäglin (1984 and elsewhere). Since this is
somewhat peripheral to our “important coefficient” interests, we confine some of the
background and results to online Appendix SA7.2.
7.4
Summary
Initially in this chapter we explored the supply-side (Ghosh) model with both its early
and later interpretations, in terms of quantity and price models, respectively. Various
approaches to measuring linkages in an input–output system were the topic of Section
7.2. Early approaches identified backward and forward linkages through appropriate
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Appendix 7.1
337
row and column sums of the Leontief and Ghosh coefficient matrices (A and B) or their
counterpart inverses, L and G. An alternative and more comprehensive view of linkage
measurement grew out of the notion of hypothetical extraction, which can be implemented for backward, forward, or total linkage measures. A detailed classification of
hypothetical extraction possibilities is presented in online Appendix SA7.2. The final
topic considered in this chapter is the problem of how to define (conceptually) and
identify (mathematically) “important” coefficients in an input–output system. Many
approaches have been suggested. A major reason for interest in this topic is that it helps
to identify where one might concentrate resources when trying to improve (for
example, update) an input–output model’s data base.
This chapter has two appendices. Some historical background on methods of
hypothetical extraction, and a formulation systematically examining the implications
of changing elements of the Leontief inverse, often referred to as the ShermanMorrison-Woodbury formulation, are included in Appendix 7.1 (below), and additional mathematical details are explored in online Appendix SA7.2 (summarized
below as Appendix 7.2).
Appendix 7.1
The Sherman–Morrison–Woodbury Formulation
A7.1.1 Introduction
Given a non-singular matrix, M, and its inverse, suppose that one or more elements of
1
M are changed, producing M∗ . The question is: can we find ðM∗ Þ ¼ [μ∗
ij ] by
1
“adjusting” M ¼ [μij ], which is already known? This is addressed by Sherman and
Morrison (1949, 1950) for the case in which only one element is changed and by
Woodbury (1950) for the case in which more than one element is changed. The answer
is “yes,” and the adjustment is relatively simple.55 (Hereafter we will refer to the
“SMW” results.)
Here is an illustration for the case of a change in one element only (Miller, 2000,
pp. 281–286). Given
2
3
2
3
3:5
:5
:5
1 1 1
M ¼ 4 2 0 6 5 and M1 ¼ 4 1:3333 :1667 :3333 5
1:1667 :3333 :1667
3 7 1
consider an M∗ that differs from M only in that 3 has been added to m23 , changing it
2
3
0 0 0
from a 6 to a 9. Let M∗ ¼ M þ ΔM where, in this case, ΔM ¼ 4 0 0 3 5. For later
0 0 0
reference, we can easily find
55
Henderson and Searle (1981) is an important reference on inverses of sums of matrices that seems generally
ignored in the input–output literature. It includes at least six different variations on the SMW results and an
extensive set of references.
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338
The Sherman–Morrison–Woodbury Formulation
2
2:625
1
ðM∗ Þ ¼ 4 1:0417
:5833
3
:25 :375
:0833 :2917 5
:1667 :0833
1
The idea is to find an alternative to the direct computation of ðM∗ Þ , making use only
of M1 and of the size of the change (here Δm23 ¼ 3).56
The heart of the procedure is contained in two matrices (for this example, these are
2 3
0
vectors). Let C ¼ 4 1 5 and R ¼ ½ 0 0 3 ; then ΔM ¼ CR. The trick is to let C be
0
the ith column of an identity matrix (the same size as M), where i identifies the row in
M in which the change occurs, and where R is an appropriately sized null row vector
with the jth element replaced by Δmij. The fundamental result is
1 M C RM1
∗ 1
1
1
1
(A7.1.1)
ðM Þ ¼ M ΔM ¼ M 1 þ RM1 C
This is not as complex as it might appear. The numerator of ΔΜ–1 is the product of a
column vector M–1C and a row vector RM–1, and the denominator is simply a scalar.57
1
The expression for an individual element in ðM∗ Þ follows directly from (A7.1.1).
For a matrix M in which element mij is changed (increased or decreased) by Δmij, the
value of the element in row r and column s of the new inverse, μ∗
rs , is
μ∗
rs ¼ μrs μri μjs Δmij
1 þ μji Δmij
(A7.1.2)
1
The new elements in column i and row j of ðM∗ Þ will be strictly proportional to the
corresponding elements in M1 . For column i, when s ¼ i,
μ∗
ri ¼ μri μri μji Δmij
μri þ μri μji Δmij μri μji Δmij
¼
¼ μri k ij
1 þ μji Δmij
1 þ μji Δmij
where k ij ¼ 1=(1 þ μji Δmij ) is a constant for a given Δmij , and exactly similar algebra
shows that when r ¼ j, u∗
js ¼ ujs k ij .
For the numerical example,
2
3
:5
M1 C ¼ 4 :1667 5, RM1 ¼ ½ 3:5 1 :5 and RM1 C ¼ 1
:3333
so that, from (A7.1.1),
If changes in each of several aij are to be examined, it is helpful to use the notation M∗
ij in order to identify the
specific case under consideration.
57
A similar result can be derived with the roles of R and C interchanged (see Miller, 2000, appendix 5.2).
56
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Appendix 7.1
339
2
3 2
3
1:75
:5 :25
:875
:25 :125
ΔM1 ¼ ð0:5Þ 4 :5833 :1667 :0833 5 ¼ 4 :2917 :0833 :0417 5
1:1667 :3333 :1667
:5833 :1667 :0833
and
1
ðM∗ Þ
¼ M1 ΔM1
2
3 2
3:5
:5
:5
875
4
5
4
¼ 1:3333 :1667 :3333 :2917
1:1667 :3333 :1667
:5833
2
3
2:625
:25 :375
4
¼ 1:0417 :0833 :2917 5
:5833 :1667 :0833
:25
:0833
:1667
3
:125
00417 5
:0833
This is exactly the inverse that was found directly earlier in this Appendix. The reader
1
can easily check the results in (A7.1.2) for any of the elements in ðM∗ Þ .
The (obvious) point is that a change (here an increase of 50 percent) in the value of
just one element in M leads to changes in all elements in M1 . Note that some changes
are increases, as with μ12 (and three other elements), and some are decreases, as with
μ11 (and
four other
elements). Absolute values of the percentage changes can be found
as58 pij ¼ 100(μ∗
ij μij )=μij , or
1
jPj ¼ 100j[ðM∗ Þ
M1 ]M1 j
where “” indicates element-by-element division. Here
2
3
25
50 25
jPj ¼ 4 21:875 50 12:5 5
50
50 50
As expected for this example with a change in m23 , the elements in column 2 and row
1
3 of ðM∗ Þ are proportional to the corresponding elements in M1 , and hence the
percentage changes are all the same.59
A7.1.2 Application to Leontief Inverses
The relevance to input–output models is that one can investigate the influence of
changes (or “errors”) in one or more elements of an A matrix on the associated
Frequently the changes are expressed as (μij μ∗
ij )=μij . This simply reverses signs. If absolute values are used,
it makes no difference.
59
The fact that all these changes are 50 percent (the same as the increase in m23 ) is a coincidence of this example
only. Moreover, some of the changes are 50 percent increases (μ12 and μ31 ) and some are 50 percent decreases
( μ22 ,μ32 and μ33 ).
58
https://doi.org/10.1017/9781108676212.008 Published online by Cambridge University Press
340
The Sherman–Morrison–Woodbury Formulation
Leontief inverse, L ¼ ðI AÞ1 . Here we begin with A∗ ¼ A þ ΔA, but, since our
1
interest is in L∗ ¼ ðI A∗ Þ , the parallel to M∗ ¼ M þ ΔM is
ðI A∗ Þ ¼ ½I ðA þ ΔΑÞ ¼ ðI AÞ þ ðΔΑÞ
and the result in (A7.1.1) becomes
L∗ ¼ L þ
ðLCÞðRLÞ
1 RLC
(A7.1.3)
Notice that negative and positive signs are interchanged, compared to (A7.1.1).
In terms of an individual element in the new inverse, l ∗
rs , the parallel to (A7.1.2) for a
change Δaij is (with notation to remind us of which element in A is changed)
l∗
rsðijÞ ¼ l rs þ
lri l js Δaij
1 l ji Δaij
(A7.1.4)
Again, note the changes in signs, this time compared to (A7.1.2). Define percentage
differences in Leontief inverse elements as Δl rs ¼ (l ∗
rsðijÞ l rs )=l rs ; then
ΔlrsðijÞ
l ri l js Δaij
1
100
¼ 100
(A7.1.5)
lrs
1 l ji Δaij lrs
As before, all elements in row j and in column i of the matrix of absolute percentage
differences will be the same.
Appendix 7.2
Hypothetical Extractions with Partitioned Matrices
Supplemental Appendix SA7.2, located on the internet web site associated with this text
(http://www.cambridge.org/millerandblair), uses the basic operations of partitioned matrices developed in Appendix A to examine hypothetical extraction measures developed in
this chapter. Cases examined include complete extraction of an economic sector, extraction of a sector’s intersectoral relationships, extraction of a sector’s intermediate purchases, extraction of a sector’s intermediate sales, extraction of a sector’s intersectoral
intermediate purchases, and extraction of a sectors intersectoral intermediate sales. Some
comparisons with the Ghosh Model developed in this chapter are also explored.
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8
8.1
Decomposition Approaches
Introduction
When there are two or more sets of input–output data for different years for an
economy, analysts are often interested in trying to disaggregate the total amount of
change in some aspect of that economy into contributions made by its various
components. For example, the total change in gross outputs between two periods
could be associated with changes in technology (as reflected in changes in the
Leontief inverse for the economy over the period), and also changes in final demand
over the same period.1
At the next level, the total change in the Leontief inverse matrix could be disaggregated into a part that is attributable to changes in technology within each sector (as
reflected in changes in the direct input coefficients matrix) and that part associated with
changes in product mix within each sector. Similarly, the change in final demand could
be further broken down into a part that reflects changes in the overall level of final
demand and a part that captures changes in the composition of final demand. And there
are numerous additional options – for example, there is no need to use only two
contributing factors; changes in employment, value added, energy use, etc. may be of
more economic interest than changes in gross outputs; and so on. And in multiregional
or multinational models various trade effects can also be disentangled.
For early general overviews of this literature, see Rose and Casler (1996) or
Dietzenbacher and Los (1997, 1998). Two early empirical examples of this kind of work
can be found in Feldman, McClain, and Palmer (1987) for the USA and Skolka (1989) for
Austria. The literature now abounds with many studies that apply structural decomposition techniques to input–output data. A special issue of Economic Systems Research
contains many articles that explore structural decompositions of energy use and carbon
emissions (Volume 28, Number 2, June, 2016) (see also Chapters 12 and 13).
Originally these decompositions were generally carried out in an additive way, as
discussed in Section 8.2. More recently, a multiplicative version has also been proposed and used (Section 8.3). In addition, several areas of research have focused
1
De Boer and Rodrigues (2020) provide a discussion of decomposition approaches in the context of earlier index
number theory.
347
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348
Decomposition Approaches
specifically on decomposing multiplier matrices into additive or multiplicative components (Section 8.4). Finally, in Section 8.5 we briefly consider structural path
analysis and decomposition, which uses the power series approximation to the
Leontief inverse (Section 2.4).
8.2
Structural Decompositon (Additive)
8.2.1 Initial Decompositions: Changes in Gross Outputs
To get a general idea of the additive structural decomposition analysis (ASDA)
approach, we initially explore gross output changes. Assume that there are two time
periods for which input–output data are available. Using superscripts 0 and 1 for the
two different years (0 earlier than 1), our illustration of structural decomposition in an
input–output model focuses on the differences in the gross output vectors for those two
years. As usual, gross outputs in year t, xt ðt ¼ 0; 1Þ, are found in an input–output
system as
x1 ¼ L1 f 1 and x0 ¼ L0 f 0
(8.1)
1
where f t ¼ the vector of final demands in year t, and Lt ¼ ðI At Þ . Then the
observed change in gross outputs over the period is
Δx ¼ x1 x0 ¼ L1 f 1 L0 f 0
(8.2)
The task is to decompose the total change in outputs into changes in the various
components – in (8.2) that would (at least initially) mean separation into changes in
L ðΔL ¼ L1 L0 Þ and changes in f ðΔf ¼ f 1 f 0 Þ.2 In order to remove the influence
of price changes, we assume that all data are expressed in prices for a common year.
A number of alternative expansions and rearrangements of the terms in (8.2) can be
derived. For example, using only year-1 values for L and only year-0 values for f –
replacing L0 with ðL1 ΔLÞ and f 1 with ðf 0 þ Δf Þ in (8.2) – we have
Δx ¼ L1 f 0 þ Δf L1 þ ΔL f 0 ¼ ðΔLÞf 0 þ L1 ðΔf Þ
(8.3)
This simple algebra produces a straightforward decomposition of the total change in
gross outputs into (1) a part that is attributable to changes in technology, ΔL, in this
case weighted by year-0 final demands ðf 0 Þ, and (2) a part that reflects final-demand
changes, Δf, which are here weighted by year-1 technology ðL1 Þ.
Notice that each term on the right-hand side of (8.3) has a certain amount of intuitive
appeal – for example, ðΔLÞf 0 ¼ L1 f 0 L0 f 0 . The first term quantifies the output that
would be needed to satisfy old (year-0) demand with new (year-1) technology; the
second term is, of course, the output needed to satisfy old demand with old technology.
2
In Section 9.2.1 we explore some of the most frequently used approaches to assessing overall structural change.
One frequently used measure is to compare x1 ¼ L1 f 1 with L0 f 1 , the output that f 1 would have generated with
L0 technology.
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8.2 Structural Decompositon (Additive)
349
So the difference is one reasonable measure of the effect of technology change. And
L1 ðΔf Þ in (8.3) has a similar kind of interpretation.
Alternatively, using only
year-0 values for L and
only year-1 values for f, which
means replacing L1 with L0 þ ΔL and f 0 with f 1 Δf , (8.3) becomes
Δx ¼ L0 þ ΔL f 1 L0 f 1 Δf ¼ ðΔLÞf 1 þ L0 ðΔf Þ
(8.4)
In this case, the technology change contribution is weighted by year-1 final demands
and the final-demand change contribution is weighted by year-0 technology.
These alternatives, in (8.3) and (8.4), are equally valid in the sense that both are
“mathematically correct,” given (8.2) and the definitions ΔL ¼ L1 L0 and
Δf ¼ f 1 f 0 . Yet clearly the measures of the individual contributions from changed
technology and from changed final demands in (8.3) will be different from those in
(8.4), except in the totally uninteresting and implausible case where L1 ¼ L0 and/or
f 1 ¼ f 0 – no change in technology or no change in demand (or no change in either)
over the period. The results in (8.3) and (8.4) can be derived from (8.2) in another way.
For example, adding and subtracting L1 f 0 to (8.2), and rearranging, gives (8.3).
Similarly, adding and subtracting L0 f 1 to (8.2) gives (8.3), after rearrangement.
And there is more. Other expressions emerge if only year-0 or only year-1 values are
used for weights on both change terms. If we use year-0 weights exclusively, so that L1
and f 1 are replaced by ðL0 þ ΔLÞ and ðf 0 þ Δf Þ, then (8.2) becomes
(8.5)
Δx ¼ L0 þ ΔL f 0 þ Δf L0 f 0 ¼ ðΔLÞf 0 þ L0 ðΔf Þ þ ðΔLÞðΔf Þ
In this case, both technology and final-demand changes are weighted by year-0 values,
but an additional (“interaction”) term – ðΔLÞðΔf Þ – has appeared. Unlike the first two
terms in (8.5), this new interaction term does not have an intuitively appealing
interpretation.3
Finally, using only year-1 weights means putting L0 ¼ L1 ΔL and f 0 ¼ f 1 Δf
into (8.2), which becomes
Δx ¼ L1 f 1 L1 ΔL f 1 Δf ¼ ðΔLÞf 1 þ L1 ðΔf Þ ðΔLÞðΔf Þ
(8.6)
again with the same interaction term, only this time it is subtracted rather than added.4
Various researchers have worked with one or more of these four alternatives. For
example, Skolka (1989) presented the first three decompositions;5 Rose and Chen
(1991) work only with the expression in (8.5), although ultimately in an expanded
form. Vaccara and Simon (1968) used the factorizations in (8.3) and (8.4), then
averaged the two measures of final-demand change and the two measures of coefficient
3
Derivation of this result by adding and subtracting like terms in (8.2) is possible but more complicated. In fact, it
requires that L1 f 0 , L0 f 1 , and L0 f 0 all be both added and subtracted and then (considerably) rearranged.
4
This result can be derived by adding and subtracting L1 f 0 , L0 f 1 , and L1 f 1 in (8.2) and (again) extensive
algebraic rearrangement.
5
He also classifies much of the pre-1989 work in this area according to which version of the decomposition
was used.
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350
Decomposition Approaches
change. This is also the approach of Feldman, McClain, and Palmer (1987), Miller and
Shao (1994), and others. Dietzenbacher and Los (1998) examine a wide variety of
possible decompositions and conclude that using an average of results from (8.3) and
(8.4) is often an acceptable approach.6
We can view this as follows. Adding (8.3) and (8.4) gives
2Δx ¼ ðΔLÞf 0 þ L1 ðΔf Þ þ ðΔLÞf 1 þ L0 ðΔf Þ
and so
Δx ¼ ð1=2ÞðΔLÞðf 0 þ f 1 Þ þ ð1=2ÞðL0 þ L1 ÞðΔf Þ
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
Technology change
Final-demand change
(8.7)
[The average in (8.7) is the same as the average of the results in (8.5) and (8.6), as the
reader can easily show.]7
Numerical Example Here is a small numerical illustration of these decompositions. Let
2
3
2 3
2
3
2 3
10 20 25
45
12 15 35
50
Z0 ¼ 4 15 5 30 5, f 0 ¼ 4 30 5, Z1 ¼ 4 24 11 30 5, f 1 ¼ 4 35 5
30 40 5
25
36 50 8
26
From x0 ¼ Z0 i þ f 0 and x1 ¼ Z1 i þ f 1 , L0 and L1 are easily found, as are
2
3
2 3
2 3
:0649 :0941 :0320
5
12
ΔL ¼ 4 :1447 :0607 :0116 5, Δf ¼ 4 5 5 and Δx ¼ 4 20 5
:1448 :0342 :0586
1
20
The alternative decompositions of Δx, for this example, are shown in Table 8.1.8
It should be noted at the outset that input–output structural decomposition studies
generate, by definition, results at the sectoral level. For an n-sector model, each
element in the n-element vector of changes (Δx in the case of gross outputs) will be
decomposed into two or more constituent elements. This means that there is an
Not everyone would agree. Fromm (1968) discusses the index number issues that are involved in finding
averages of measures with weights from different years. In terms of (8.3), the ðΔLÞf 0 term is a kind of Laspeyres
index (original year weights, in f 0 ), and the L1 ðΔf Þ term is a kind of Paasche index (terminal year weights in
L1 ); in (8.4) the Laspeyres and Paasche terms are reversed. He suggests that averaging the two – (8.3) and
(8.4) – gives a “. . . bastard measure of beginning- and end-point quantities and prices” (p. 65).
7
There is some not very illuminating discussion in the literature about terms in (8.3) or (8.4) “absorbing” the
0
1
0
0
interaction term. Starting
with a rearranged (8.5), ðΔLÞf þ ðΔLÞðΔf Þ þ L ðΔf Þ ) ðΔLÞf þ L ðΔf Þ, which
1
incorporates the interaction term ½þðΔLÞðΔf Þ. Equally plausible, however, is viewing
is (8.4), and so ΔLf
(8.5) as ðΔLÞf 0 þ L0 ðΔLÞ þ ðΔLÞðΔf Þ ) ðΔLÞf 0 þ L1 ðΔf Þ which is (8.3), and now it is L1 ðΔf Þ that has
absorbed ½þðΔLÞðΔf Þ. Similar rearrangements of (8.6) will show that ðΔLÞf 0 in (8.3) or L0 ðΔf Þ in (8.4) could
be viewed as absorbing ½ðΔLÞðΔf Þ. Mathematically, the result in (8.7) allocates one-half of the interaction
term to technical change and one-half to final-demand change. See also Casler (2001) for thoughts on the
interaction term.
8
The reader can easily identify the various “absorptions” in the previous footnote in terms of the results in
this table.
6
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351
8.2 Structural Decompositon (Additive)
Table 8.1 Alternative structural decompositions
Output
Change
Technology
Change
Contribution
Final-Demand
Change
Contribution
Interaction
Term
Equation (8.3)
Sector 1
Sector 2
Sector 3
Total
12
20
20
52
.90
8.62
9.01
18.53
11.10
11.38
10.99
33.47
.00
.00
.00
.00
Equation (8.4)
Sector 1
Sector 2
Sector 3
Total
12
20
20
52
.78
9.66
9.96
20.41
11.22
10.34
10.04
31.59
.00
.00
.00
.00
Equation (8.5)
Sector 1
Sector 2
Sector 3
Total
12
20
20
52
.90
8.62
9.01
18.53
11.22
10.34
10.04
31.59
‒.11
1.04
.95
1.88
Equation (8.6)
Sector 1
Sector 2
Sector 3
Total
12
20
20
52
.78
9.66
9.96
20.41
11.10
11.38
10.99
33.47
.11
‒1.04
‒.95
‒1.88
Equation (8.7)
Sector 1
Sector 2
Sector 3
Total
12
20
20
52
.84
9.14
9.49
19.47
11.16
10.86
10.51
32.53
.00
.00
.00
.00
inherent problem in finding appropriate summary measures of results in these studies.
One obvious solution is to use total (economy-wide) figures – in the case of the
decomposition in (8.7), this would be9
i0 ðΔxÞ ¼ i0 ½ð1=2ÞðΔLÞðf 0 þ f 1 Þ þ i0 ½ð1=2ÞðL0 þ L1 ÞðΔf Þ
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Economy-wide technology change effect Economy-wide final-demand change effect
Alternatives include grouping sectors into categories and then finding averages
(simple or weighted) over the smaller numbers of elements in these groupings. For
example: “fastest growing sectors” (say the top x percent), “slowest growing (fastest
declining) sectors” (the bottom x percent), and other sectors [the middle (100 – 2x)
percent], or primary (natural resource related), secondary (manufacturing and processing), and tertiary (support and service oriented) sectors. As will be clear from this small
example and from the empirical studies examined in Section 8.2.5, any such economywide or averaging figures sweep an enormous amount of detail (and, usually, variation)
under the rug.
9
Dividing both sides by n would generate one kind of “average” figure.
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352
Decomposition Approaches
Table 8.2 Sector-specific and economy-wide decomposition results
[equation (8.7)]
Output
Change
Sector 1
12
Sector 2
20
Sector 3
20
Total
52
Change
Percentage
Change
Percentage
Change
Percentage
Change
Percentage
Technology
Change
Contribution
Final Demand
Change Contribution
.84
7
9.14
46
9.49
47
19.47
37
11.16
93
10.86
54
10.51
53
32.53
63
Table 8.2 emphasizes the results from (8.7) including both the absolute and
percentage total output change in each row. (Since these are hypothetical figures for
illustration only, there is no need to be compulsive about detail in the percentages. We
use no places to the right of the decimal.)
Of the economy-wide total output change in this example, 37 percent is seen to be
attributable to technological change and 63 percent results from changes in final
demand. But variation across sectors is large. The technology change contribution to
individual sector output growth varies from 7 to 47 percent and (therefore) the final
demand contribution varies from 53 to 93 percent.
8.2.2 Next-Level Decompositions: Digging Deeper into Δf and ΔL
Of course the story need not and does not end with the decompositions in (8.3)–(8.7).
Changes in final demands, for example, may be the result of a change in the overall
level of final demand or of a change in the relative proportions of expenditure on the
various goods and services in the final-demand vector (the final-demand mix). Or,
indeed, final-demand data may be collected and presented in several vectors, one for
each final-demand category, such as household consumption, exports, government
spending (federal, state, and local), and so on, and the relative importance of these
categories may change.
Similarly, changes in the Leontief inverse result from changes in the economy’s A
matrix – which, in turn, may reflect various aspects of technology change, such as
changes in production recipes (replacing metals with plastics in automobiles), substitutions caused by relative price changes (for domestically produced inputs and also for
imports), reductions in a sector’s materials inputs per unit of output brought about by
economies of scale, and so on – as noted in Section 9.2. We examine some approaches
to account for these “next-level” effects. Before doing that, we need to generalize the
decomposition results.
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8.2 Structural Decompositon (Additive)
353
Additive Decompositions with Products of More than Two Terms The results in
(8.3) can be looked at in the following way, which lends itself to generalization. Let
yt ¼ xt1 xt2 represent the general case in which the product of two variables (scalars,
vectors, matrices or appropriate combinations) defines a dependent variable; the particular example here is xt ¼ Lt f t . Then the decompositions of Δy ¼ x11 x12 x01 x02 in (8.3) and
(8.4) are seen to be of the form Δy ¼ ðΔx1 Þx02 þ x11 ðΔx2 Þ and Δy ¼ ðΔx1 Þx12 þ x01 ðΔx2 Þ,
respectively. Specifically, year-0 weights are to the right of a change term and year-1
weights are to the left in (8.3), and the year-0 and year-l terms are reversed for (8.4).
An approach for the case of more than two terms, as in yt ¼ xt1 xt2 . . . xtn , is to extend
the logic of these two alternatives.10 We begin with the case of n ¼ 3, where
yt ¼ xt1 xt2 xt3 and hence Δy ¼ x11 x12 x13 x01 x02 x03 . Persistent and tedious substitutions from
x11 ¼ x01 þ Δx1 , x12 ¼ x02 þ Δx2 and x13 ¼ x03 þ Δx3 will lead to
Δy ¼ ðΔx1 Þx02 x03 þ x11 ðΔx2 Þx03 þ x11 x12 ðΔx3 Þ
(8.8)
Alternative substitutions and rearrangements will generate
Δy ¼ ðΔx1 Þx12 x13 þ x01 ðΔx2 Þx13 þ x01 x02 ðΔx3 Þ
The usual averaging leads to
Δy ¼ ð1=2ÞðΔx1 Þ x02 x03 þ x12 x13
þ ð1=2Þ x01 ðΔx2 Þx13 þ x11 ðΔx2 Þx03 þ ð1=2Þ x01 x02 þ x11 x12 ðΔx3 Þ
(8.9)
(8.10)
[Notice that the (1=2) terms result from averaging the two expressions for Δy in (8.8)
and (8.9). They are unrelated to the number of elements in each of the terms on the
right-hand sides of Δy.]
There are similar results for n > 3. The pattern is the same in the equations parallel
to (8.8) and (8.9) – year-0 (year-1) weights always appear on the right of the Δx term
and year-1 (year-0) weights always appear on the left. The generalization is straightforward but, again, the algebra is tedious. The parallel to (8.8) is
Δy ¼ ðΔx1 Þ x02 x0n þ x11 ðΔx2 Þ x03 x0n
(8.11)
þ þ x11 x1n2 ðΔxn1 Þx0n þ x11 . . . x1n1 ðΔxn Þ
The parallel to (8.9) has exactly the structure of (8.11) with superscripts “0” and “1”
reversed. We write out the n-variable extension of (8.10), for completeness.
Δy ¼ ð1=2ÞðΔx1 Þ x02 . . . x0n þ x12 . . . x1n
þ ð1=2Þ x01 ðΔx2 Þ x13 . . . x1n þ x11 ðΔx2 Þ x03 . . . x0n
(8.12)
þ þ ð1=2Þ x01 . . . x0n2 ðΔxn1 Þx1n þ x11 . . . x1n2 ðΔxn1 Þx0n
þ ð1=2Þ x01 . . . x0n1 þ x11 . . . x1n1 ðΔxn Þ
10
These are not the only options. See Dietzenbacher and Los (1998) for a very thorough discussion of
alternatives.
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354
Decomposition Approaches
Changes in Final Demand Among the factors that may contribute to
changes in final demands between two periods are: (1) the total amount of all
expenditures for final demands – the final-demand level; (2) the distribution of total
expenditure across final-demand categories – for example, the total value of household
consumption, exports (possibly broken down by countries of destination), government
expenditures (possibly separated into federal, state, and local), and other final
demands, as proportions of total final-demand expenditure; and (3) the product mix
within each particular final-demand category – for example, the proportion of total
household consumption expenditure that goes to computers and computer services.
This is reflected in the coefficients in the bridge matrix (see point c, below).
In an n-sector input–output model, if there are p categories of final demand – instead
of a single final-demand vector, f t – then we have a final-demand matrix,
2 tðn1
3Þ
f 1k
6
7
Ft ¼ f t1 ; . . . ; f tp , where f tk ¼ 4 ... 5, and f tik records the amount of expenditure
ðnpÞ
f tnk
by final-demand category k on the product of sector i in year t. In particular,
a. Ft i ¼ f t , the n-element vector of total final-demand deliveries from each sector in
year t.
b. i0 Ft i ¼ i0 f t ¼ f t , the level (total amount) of final-demand expenditure over all
sectors in year t. 2 t 3
y1
6
7
0
.
0
t
c. Let yt ¼ ði F Þ ¼ 4 .. 5, where ykt ¼ total final-demand expenditure by finalypt
demand category k in year t.
The vector that indicates the distribution of f t across the p final-demand categories is
found as the column sums of Ft divided by f t, or
2 t t3
y1 =f
6 .. 7
t
t
t
t
d ¼ [d k ] ¼ ð1=f Þy ¼ 4 . 5
(8.13)
ðp1Þ
ypt =f t
So d tk represents the proportion of total final-demand expenditure in year t that
originated in category k. Finally, the bridge (product mix) matrix, Bt , is
B
t
1
¼ [btik ] ¼ ðFt Þð^y t Þ
ðnpÞ
(8.14)
So Bt is Ft normalized by its column sums – btik ¼ f ikt =ykt indicates the proportion of
total expenditures by final-demand category k that was spent on the product of sector i
in year t.11
11
This use of B is not to be confused with the output coefficients matrix in the Ghosh model.
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8.2 Structural Decompositon (Additive)
355
With these definitions,
f t ¼ f t Bt dt ¼ Bt yt
(8.15)
Δf ¼ f 1 f 0 ¼ f 1 B1 d1 f 0 B0 d0 ¼ B1 y1 B0 y0
(8.16)
and so
This holds for data with either only one final-demand vector ðp ¼ 1Þ or with several
2 t3
f1
6 .. 7 t
t
t
final-demand categories ðp > 1Þ. In the former case, F ¼ f ¼ 4 . 5, f ¼ yt (a
f tn
t
t t
t t
t
scalar), B is a column vector bi ¼ f i =f ¼ f i =y , and d ¼ 1(also a scalar). In the
latter case, the final-demand matrix, disaggregated by categories, is seen to be
yt .
Ft ¼ Bt ^
Decomposing the final-demand change in (8.16) as in (8.8), (8.9), and (8.10) gives
t
Δf ¼ ðΔf ÞB0 d0 þ f 1 ðΔBÞd0 þ f 1 B1 ðΔdÞ
(8.17)
Δf ¼ ðΔf ÞB1 d1 þ f 0 ðΔBÞd1 þ f 0 B0 ðΔdÞ
(8.18)
and
Δf ¼ ð1=2ÞðΔf ÞðB0 d0 þ B1 d1 Þ þ ð1=2Þ f 0 ðΔBÞd1 þ f 1 ðΔBÞd0 þ ð1=2Þ f 0 B0 þ f 1 B1 ðΔdÞ
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Final-deman level effect
Final-demand mix effect
Final-demand distribution effect
(8.19)
When p ¼ 1, d t ¼ 1, and Δd ¼ 0, and the third terms disappear from (8.17)–(8.19); in
fact, (8.19) is simplified to
(8.20)
Δf ¼ ð1=2ÞðΔf ÞðB0 þ B1 Þ þ ð1=2Þ f 0 þ f 1 ðΔBÞ
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Final-demand level effect
Final-demand mix effect
8.2.3
Numerical Examples
One Category of Final Demand (p ¼ 1) Continuing with the same numerical
illustration,12
2
3
2
3
2
3
:45
:4505
:0005
B0 ¼ 4 :3 5, B1 ¼ 4 :3153 5, ΔB ¼ 4 :0153 5, f 1 ¼ 111, f 0 ¼ 100
:25
:2342
:0158
12
It is necessary to work with more than two decimal places in these calculations, but results will continue to be
rounded to two.
https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
356
Decomposition Approaches
Table 8.3 Sector-specific and economy-wide decomposition results (with two-factor final-demand
decomposition detail)a
Final Demand Change Contribution
Output
Change
Sector 1
12
Sector 2
20
Sector 3
20
Total
52
Technology Change
Contribution
Level
Mix
Total
.84
7
9.14
46
9.49
47
19.47
37
11.05
92
9.35
47
11.45
57
31.85
61
.10
1
1.51
8
‒.93
‒5
.68
1
11.16
93
10.86
54
10.51
53
32.53
63
Change
Percentage
Change
Percentage
Change
Percentage
Change
Percentage
a
In this and later tables, percentages are shown with no decimal places, so there may be (small)
discrepancies between the total effect and the sum of its parts.
Notice that (by definition) the column sums in B0 and B1 must be one and so the
column sum in ΔB must be zero; there must be one or more negative elements in ΔB to
balance one or more positive elements. This means that the final-demand mix effect for
at least one sector – the second term in (8.20) – must be negative. In this numerical
illustration, sector 3 has become relatively less important in total final-demand spending. Putting the Δf results in (8.20) into (8.7) leads to the results shown in Table 8.3.
Two Categories of Final Demand (p ¼ 2) Suppose that data are available on
two categories of final demand – for example, households and all other final demand.
Consistent with the numerical illustration, let
2
3
2
3
20 25
25 25
F0 ¼ f 01 f 02 ¼ 4 10 20 5 and F1 ¼ f 11 f 12 ¼ 4 15 20 5
15 10
18 8
Then
d0 ¼
45=100
:4500
¼
55=100
:5500
and the bridge matrices are
2
3
20 25 1=45
B0 ¼ 4 10 20 5
0
15 10
and d1 ¼
58=111
:5225
¼
53=111
:4775
3
:4444 :4545
0
¼ 4 :2222 :3636 5 and
1=55
:3333 :1818
2
2
3
2
3
25 25 :4310 :4717
1=58
0
B1 ¼ 4 15 20 5
¼ 4 :2586 :3774 5
0 1=53
18 8
:3103 :1509
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8.2 Structural Decompositon (Additive)
357
Table 8.4 Sector-specific and economy-wide decomposition results (with three-factor final-demand
decomposition detail)
Final Demand Change Contribution
Output
Change
Sector 1
12
Sector 2
20
Sector 3
20
Total
52
Change
Percentage
Change
Percentage
Change
Percentage
Change
Percentage
Finally,
:0725
Δd ¼
,
:0725
Technology Change
Contribution
Level
Mix
Dist.
Total
.84
7
9.14
46
9.49
47
19.47
37
11.05
92
9.35
47
11.45
57
31.85
61
.31
3
2.42
12
‒1.65
‒8
1.08
2
‒.21
‒2
‒.91
‒5
.71
4
‒.41
‒1
11.16
93
10.86
54
10.51
53
32.53
63
2
3
:0134
:0172
ΔB ¼ 4 :0364
:0137 5,
:0230 :0309
Δf ¼ 11
Notice that, again by definition, column sum in Δd (as with those in ΔB) must be
zero. This introduces negative elements into both the final-demand mix and distribution effects [the second and third terms in (8.19)]. Inserting the Δf decomposition in
(8.19) into (8.7) generates the results in Table 8.4.
8.2.4
Changes in the Direct Inputs Matrix
Decomposition of ΔL Changes in the Leontief inverse between two time
periods reflect, of course, changes in the underlying direct inputs matrices. One
1
approach to translating ΔA into ΔL proceeds as follows. Given L1 ¼ ðI A1 Þ and
1
L0 ¼ ðI A0 Þ , postmultiply L1 through by ðI A1 Þ
L1 I A1 ¼ I ¼ L1 L1 A1
and premultiply L0 through by I A0
I A0 L0 ¼ I ¼ L0 A0 L0
(8.21)
(8.22)
Rearrange (8.21) and postmultiply by L0
L1 I ¼ L1 A1 ) L1 L0 L0 ¼ L1 A1 L0
(8.23)
Similarly, rearrange (8.22) and premultiply by L1
L0 I ¼ A0 L0 ) L1 L0 L1 ¼ L1 A0 L0
https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
(8.24)
358
Decomposition Approaches
Finally, subtract (8.24) from (8.23)
ΔL ¼ L1 L0 ¼ L1 A1 L0 L1 A0 L0 ¼ L1 ðΔAÞL0
(8.25)
This expression relates the change in the Leontief inverse to the change in A; the
decomposition is a multiplicative one in which ΔA is “doubly weighted” – in this case
by L1 on the left and by L0 on the right. The reader can verify that changing each
premultiplication to a postmultiplication, and vice versa, in deriving (8.21) through
(8.24) will generate the (possibly surprising13) result that, in addition,
ΔL ¼ L1 L0 ¼ L0 A1 L1 L0 A0 L1 ¼ L0 ðΔAÞL1
(8.26)
Since there is only one term on the right in either (8.25) or (8.26) there is no need to
express ΔL as the average of the two expressions; either one will do. Again,
terms will appear if we choose to have only year-0 L0 or only
interaction
1
year-1 L weights. For example, replacing L1 with L0 þ ΔL in (8.25) leads to
ΔL ¼ L0 ðΔAÞL0 þ ðΔLÞðΔAÞL0 . Making the same replacement in (8.26) generates
ΔL ¼ L0 ðΔAÞL0 þ L0 ðΔAÞðΔLÞ. This identifies another instance in which the general
“order makes a difference” rule in matrix algebra is violated; since the second terms
must be equal, we see that ðΔLÞðΔAÞ L0 ¼ L0 ðΔAÞ ðΔLÞ. Also, substituting L1 ΔL
for L0 in both (8.25) and (8.26) will produce ΔL ¼ L1 ðΔAÞL1 L1 ðΔAÞðΔLÞ and
ΔL ¼ L1 ðΔAÞL1 ðΔLÞðΔAÞL1 , respectively. In these two cases, we also find that the
interaction terms are equal –ðΔLÞðΔAÞ L1 ¼ L1 ðΔAÞ ðΔLÞ.
In what follows, we will use the result in (8.25) to convert changes in the Leontief
inverse into changes in the A matrix.14
Decomposition of ΔA There are many ways to create decompositions of ΔA.
Here we illustrate a straightforward disaggregation into column-specific changes only.
Since each column in A reflects a sector’s production recipe, identifying the changes
column-by-column is one way of disentangling the effects of input changes in each of
the sectors in the economy. For expositional simplicity, we denote these as
technology change.
For an n-sector economy,
2 0
3
a11 þ Δa11 a01n þ Δa1n
6
7
..
..
A1 ¼ A0 þ ΔA ¼ 4
5
.
.
a0n1 þ Δan1
13
a0nn þ Δann
The result is surprising in the sense that the order in which matrices appear in matrix multiplication usually
makes a difference in the outcome (in contrast to scalar multiplication).
14
A continuous version of this approach has been noted (for example, Afrasiabi and Casler, 1991, Rose and
Casler, 1996). As in (8.21), with LðI AÞ ¼ L LA ¼ I, using the product rule for differentiation,
ðdL=dtÞ ðdL=dtÞA LðdA=dtÞ ¼ 0 or ðdL=dtÞðI AÞ ¼ LðdA=dtÞ, and postmultiplying by L, ðdL=dtÞ ¼
LðdA=dtÞL.
https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
8.2 Structural Decompositon (Additive)
359
2
3
0 Δa1j 0
6
..
.. 7 represent changes in sector j’s technology –
Let ΔAðjÞ ¼ 4 ...
.
.5
0 Δanj 0
the superscript “ðjÞ” identifies the sector (column) in which coefficients change.15
Then
ΔA ¼ ΔAð1Þ þ þ ΔAðjÞ þ þ ΔAðnÞ ¼
n
X
ΔAðjÞ
|ffl{zffl}
j¼1
(8.27)
Technology change
in sector j
This decomposition of ΔA can be introduced into (8.25), and the resulting expression
for ΔL can then be used in (8.7), which now looks like this:
Δx ¼ ð1=2ÞðΔLÞ f 0 þ f 1 þ ð1=2Þ L0 þ L1 ðΔf Þ
¼ ð1=2ÞL1 ðΔAÞL0 f 0 þ f 1 þ ð1=2Þ L0 þ L1 ðΔf Þ
¼ ð1=2ÞL1 ΔAð1Þ þ þ ΔAðnÞ L0 f 0 þ f 1 þ ð1=2Þ L0 þ L1 ðΔf Þ
(8.28)
¼ ð1=2Þ L1 ΔAð1Þ L0 f 0 þ f 1 þ þ ð1=2Þ L1 ΔAðnÞ L0 f 0 þ f 1
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Effect of technology change in sector 1
Effect of technology change in sector n
þ ð1=2Þ L0 þ L1 ðΔf Þ
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Effect of final-demand change
Numerical Illustration (continued)
For our numerical example,
2
3
2
3
:1000 :2500 :2500
:1071 :1500 :2917
A0 ¼ 4 :1500 :0625 :3000 5 and A1 ¼ 4 :2143 :1100 :2500 5
:3000 :5000 :0500
:3214 :5000 :0667
so
2
:0071
ΔA ¼ 4 :0643
:0214
15
1
:0475
0
3
0:0417
:0500 5
0:0167
The superscript parentheses serve to distinguish A1 , the direct inputs matrix in period 1, from ΔAð1Þ , the matrix
that reflects the technology change in sector 1 only.
https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
360
Decomposition Approaches
and, in particular,
2
:0071 0
ΔAð1Þ ¼ 4 :0643 0
:0214 0
3
2
0
0 :1
0 5 Að2Þ ¼ 4 0 :0475
0
0
0
3
2
0
0
0 5 Að3Þ ¼ 4 0
0
0
0
0
0
3
:0417
:0500 5
:0167
Table 8.5 indicates the additional results from using this technology change
decomposition for our numerical illustration. (Final-demand results repeat those in
Table 8.4.)
8.2.5 Decompositions of Changes in Some Function of x
A number of studies have decomposed not simply gross output change but rather
changes in some variable that depends on output. For example, if we have a set of
sectoral labor inputs, h0 ¼ ½h1 , h2 ; , hn measured either in monetary or physical
units (Chapter 5), then h0c ¼ h0 x^1 is the row of associated labor input coefficients.
Thus, total employment in the economy (a scalar) is h0c Lf and employment by sector (a
^ c Lf. For two different time periods this is ε0 ¼ h
^ 0 x0 ¼ h
^ 0 L0 f 0 and
vector) is ε ¼ h
c
c
^ 1c x1 ¼ h
^ 1c L1 f 1 and changes in employment in each sector is given by
ε1 ¼ h
^ 1 L1 f 1 h
^ 0 L0 f 0
Δε ¼ ε1 ε0 ¼ h
c
c
(8.29)
Decomposition into contributions by the three elements now follows the standard
pattern shown in (8.10). Here this means
^ c L0 f 0 þ L 1 f 1
Δε ¼ ð1=2Þ Δh
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Labor input coefficient change
h 0
i
^ c ðΔLÞf 1 þ h^ 1c ðΔLÞf 0
þ ð1=2Þ h
(8.30)
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
þ ð1=2Þ
Technology change
0 0
^c L þ h
^ 1c L1 ðΔf Þ
h
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Final-demand change
Of course, additional decompositions of ΔL and/or Δf as in Section 8.2.2 are possible.
Exactly the same principles apply for any economic variable that is related to output by
a similar set of coefficients per dollar of sectoral output – pollution generation, energy
consumption, value added, etc.16
16
Dietzenbacher and Los (2000) suggest convincingly that a fundamental assumption of independence of the
variables in an expression being decomposed is most probably violated by expressions like (8.29) where, for
example, changes in energy costs may be accompanied by changes in a sector’s use of certain energy inputs in
favor of cheaper alternatives, leading to a change in the mix of that sector’s inputs. This appears to have been
largely ignored in subsequent studies. Another kind of concern is raised in Nagashima (2018), who demonstrates that in some cases uncertainties in the basic data underlying an input–output model may lead to the
wrong sign on one or more of the decomposition terms.
https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
Table 8.5 Sector-specific and economy-wide decomposition results (with additional technology and final-demand decomposition detail)
Technology Change Contribution
Output Change
Sector 1
12
Sector 2
20
Sector 3
20
Total
52
Change
Percentage
Change
Percentage
Change
Percentage
Change
Percentage
Final Demand Change Contribution
Sector 1
Sector 2
Sector 3
Total
Level
Mix
Distrib.
Total
6.64
55
12.42
62
11.37
57
30.43
59
‒10.25
‒85
1.28
6
‒2.85
‒14
‒11.83
‒23
4.45
37
‒4.56
‒23
.97
5
.86
2
.84
7
9.14
46
9.49
47
19.47
37
11.05
92
9.35
47
11.45
57
31.85
61
.31
3
2.42
12
‒1.65
‒8
1.08
2
‒.21
‒2
‒.91
‒5
.71
4
‒.41
‒1
11.16
93
10.86
54
10.51
53
32.53
63
361
362
Decomposition Approaches
8.2.6 Summary for Δx
For Δx we assemble both the final-demand decomposition (including distribution
across final-demand categories) and the technology change decomposition in the same
expression, primarily for completeness. The expression includes all six of the change
components.
Δx ¼ ð1=2ÞðΔLÞ f 0 þ f 1 þ ð1=2Þ L0 þ L1 ðΔf Þ
¼ ð1=2Þ L1 ΔAð1Þ L0 f 0 þ f 1 þ ð1=2Þ L1 ΔAð2Þ L0 f 0 þ f 1
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Effect of technology change in sector 1
Effect of technology change in sector 2
þ ð1=2Þ L1 ΔAð3Þ L0 f 0 þ f 1 þ ð1=4Þ L0 þ L1 ðΔf Þ B0 d0 þ B1 d1
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Effect of technology change in sector 3
Effect of change in final-demand level
0
þ ð1=4Þ L þ L1 f 0 ðΔBÞd1 þ f 1 ðΔBÞd0 þ ð1=4Þ L0 þ L1 f 0 B0 þ f 1 B1 ðΔdÞ
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Effect of change in final-demand mix
Effect of change in final-demand distribution
(8.31)
8.2.7 ASDA in a Multiregional Input–Output (MRIO) Model
~
The standard form of the MRIO model (Chapter 3) is x ¼ ðI CAÞ1 Cf ¼ LCf,
1
~
where L ¼ ðI CAÞ , A is a technical coefficients matrix indicating intermediate
inputs for each region from both within and outside of the region and C contains input
proportions (both intraregional and interregional shipments). The distinctive feature of
this formulation is that the Leontief-like inverse contains both technical coefficients
and trade proportions.
~
Following (8.10), for x ¼ LCf
we have
0
0 0
~ 1 ðΔCÞf 0
~ ðΔCÞf 1 þ L
~ C f þ C1 f 1 þ ð1=2Þ L
Δx ¼ ð1=2Þ ΔL
(8.32)
0 0
~ C þL
~ 1 C1 ðΔf Þ
þ ð1=2Þ L
~ we follow (8.25)
To disentangle the trade proportions and technical coefficients in L
and then (8.7), namely
~0
~ ¼L
~ 1 ðΔCAÞL
ΔL
and
ΔCA ¼ ð1=2ÞðΔCÞ A0 þ A1 þ ð1=2Þ C0 þ C1 ðΔAÞ
(8.33)
~ 0 in (8.32),
~ ¼L
~ 1 ðΔCAÞL
First, using ΔL
1
0
~ 0 C0 f 0 þ C1 f 1 þ ð1=2Þ L
~ 1 ðΔCÞf 0
~ ðΔCAÞL
~ ðΔCÞf 1 þ L
Δx ¼ ð1=2Þ L
0 0
~ 1 C1 ðΔf Þ
~ C þL
þ ð1=2Þ L
https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
8.2 Structural Decompositon (Additive)
363
and then using (8.33) (and rearranging)
1 0
~ 0 C0 f 0 þ C1 f 1
~ C þ C1 ðΔAÞL
Δx ¼ ð1=4Þ L
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Effect of technology change
1
0 0 0
~ ðΔCÞ A0 þ A1 L
~ C f þ C1 f 1
¼ ð1=4Þ L
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
(8.34)
One effect of trade coefficient change
0
0 0
~ ðΔCÞf 1 þ L
~ 1 ðΔCÞf 0 þ ð1=2Þ L
~ C þ L1 C1 ðΔf Þ
þ ð1=2Þ L
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
A second effect of trade coefficient change
Effect of final-demand change
Notice in particular that the change in trade proportions exerts influence in conjunction
with both the technical coefficients (A0 and A1 ) and also the final demands (f 0 and f 1 ),
since in the MRIO model both A and f are transformed – into CA and Cf.
Embellishments are possible. For example, the final-demand effect might be further
decomposed into level, mix, and/or distribution, as in Section 8.2.2. Furthermore,
some models may feature (or at least propose) separate trade proportions for intermedi~ ∗ Cf f. In that
ate inputs and for final demands, leading to x ¼ ðI Ca AÞ1 Cf f ¼ L
case, ΔCa and ΔCf must be treated separately. This simply leads to more complexity
(more terms) in (8.34). In online Appendix SA8.1, we explore the implications of
~
alternative groupings of the terms in x ¼ LCf
(as has been done in some published
~
~ where y ¼ Cf. (See examples
studies) into either x ¼ Mf, where M ¼ LC, or x ¼ Ly,
in Section 8.2.8.)
8.2.8 Empirical Examples
Analysts are generally interested in structural decompositions because they offer a
means of quantifying the relative importance of various components in understanding
some observed economic change – in early studies this was usually changes in industry
outputs; more recently, changes in labor use, value added, energy use, pollution
emissions, service industry outputs, etc. have also been decomposed. The results of
empirical SDA studies are often used to inform policy decisions – the relative
importance of trade (and hence trade policy) to an economy, the relative importance
of one or more components of final demand (and hence tax or subsidy policy), and so
forth. In this section we mention a few of the pioneering early studies applying
structural decompositions to national input–output data bases and several of the first
with a spatial dimension, using connected-regional (or connected-national) data –
precursors of modern world models. As noted, decompositions generate results at a
sectoral level and summary measures are needed. In Table 8.6, virtually all of the rich
detail in each of the studies cited has been foregone in favor of simple averages in
order to present figures that are comparable across studies.
Studies Using National Models The first study known to us that uses this
approach is Chenery, Shishido, and Watanabe (1962), for Japan over the periods
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364
Decomposition Approaches
1914–1935 and 1935–1954.17 The authors were interested in deviations of later year
output from what it would have been under a regime of proportional growth from an
earlier year. These deviations were decomposed into the effects of (1) changes in
domestic final demand, (2) changes in exports, (3) changes in imports, and (4) changes
in technology (as represented by changes in elements of the A matrix).
A study by Vaccara and Simon (1968), to the best of our knowledge, represents the
first application of this kind of decomposition approach to the US economy. Using
42 industry groups, they measured the amount of output change that was attributable to
final-demand change and the amount due to coefficient change over the 1947–1958
period. As a (very) general conclusion, they found final-demand changes somewhat
more important than changes in technical coefficients.
Bezdek and Wendling (1976) continued this kind of analysis. They factored Δx into
final-demand and coefficient change in a 75-sector model of the US economy for the
1947–1958, 1958–1963, and 1963–1966 periods. In addition, they compared their
decomposition results for 1958–1963 with those reported for Germany (1958–1962) in
Stäglin and Wessels (1972) at a 35-sector level. They found similarity in the industryspecific influences of final-demand change, but not of coefficient change.
The late 1980s and early 1990s saw the beginnings of an explosion of empirical
studies using SDA which continues, more recently emphasizing energy and ecological
contributions. An early example is in Feldman, McClain, and Palmer (1987), which is
frequently cited.18 This study also examined the relative importance in the US economy of changes in final demands and changes in technology – this time over the
1963–1978 period using a very disaggregated 400-sector level of analysis. (The
1978 table was an updated version of the 1972 survey-based national table.)
They use the form x ¼ Ax þ Bf ) x ¼ LBf and then define C ¼ LB so that x ¼ Cf,
where B is the bridge matrix that connects the outputs of some n ¼ 400 sectors to
ðnpÞ
p ¼ 160 categories of final demand.19 Thus their decomposition takes the form
Δx ¼ ðΔCÞf 0 þ C1 ðΔf Þ or Δx ¼ ðΔCÞf 1 þ C0 ðΔf Þ – as in (8.3) and (8.4). They define
structural change broadly – including changes in the structure of production (technical
change, reflected in changes in A) and in the microstructure of expenditure (reflected in
changes in B).20 Overall, the contribution made by coefficient change was 62 percent,
while final-demand change accounted for 38 percent.21
A second frequently cited study from this period is that by Skolka (1989). It
describes the structural decomposition methodology in some detail and applies it to
17
This builds on earlier work by Chenery (for example, Chenery, 1960). A thorough summary of this kind of
analysis in the economic development literature can be found in Syrquin (1988). Illustrative examples include
Fujita and James (1990) – and many other publications by these authors – at the national level and Siegel,
Alwang, and Johnson (1995) for a “growth accounting” study at the regional level.
18
And, less frequently, Feldman and Palmer (1985).
19
This use of C is not to be confused with the trade proportions matrix of the MRIO model.
20
They recognize
that an alternative would be
to group B with f and to use x ¼ LðBfÞ, leading to
Δx ¼ ðΔLÞ B0 f 0 þ L1 ðΔΒf Þ and Δx ¼ ðΔLÞ B1 f 1 þ L0 ðΔBf Þ. See comments on the effect of alternative
groupings on decompositions in Appendix SA8.1.
21
Wolff (1985) used the same mode of analysis to study trends in productivity in the US economy.
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8.2 Structural Decompositon (Additive)
365
a 19-sector data set for Austria (1964–1976). Both net output (value added) change and
employment change were decomposed into an intermediate demand (technology)
component (with separate domestic and imports parts) and a final-demand component
(with separate domestic and exports parts). This and several other early studies are
summarized in online Appendix SA8.2 for the interested reader.
Early Studies with a Spatial Dimension Washington State. Holland and
Cooke (1992) used the structural decomposition framework at a regional (state) level
to study the sources of change in the economy of Washington over 1963–1982, using
the survey-based Washington input–output tables for 1963 and 1982. Reflecting a
concern with the importance of trade for the Washington economy, they separated out
the role of changes in demand (intermediate and final) within the state, within the
rest of the USA (national markets), and outside the USA (international markets).
The US Multiregional Model (Miller and Shao, 1994). Two early implementations of
a multiregional input–output (MRIO) model for the US economy are available for
1963 ðt ¼ 0Þ and 1977 ðt ¼ 1Þ. The 1963 model takes the form
1
x0 ¼ I C0 A0 C0 f 0
and the 1977 model is
1
x1 ¼ I D1 C1 B1 C1 f 1
The C0 and C1 matrices contain the interregional trade proportions for the two years.
However, matrices D1 and B1 reflect technology in the 1977 model (only), which is
based on commodity–industry input–output accounting.22 Similarly, A0 is a matrix of
technical coefficients in the 1963 model (only). Therefore, for simplicity, the superscripts on D, B, and A can be eliminated, giving the following equation for gross
output change over the period:
1
1
(8.35)
Δx ¼ x1 x0 ¼ I DC1 B C1 f 1 I C0 A C0 f 0
The two total requirements
matrices (transforming
final
into outputs) can
1
1demands
1
1
0
0
1
0
~
~
be denoted L ¼ I DC B C and L ¼ I C A C . Then
~ 0f 0
~ 1f 1 L
Δx ¼ L
(8.36)
This parallels (8.2), only now the two total requirements matrices are more compli1
cated than the usual Leontief inverses, Lt ¼ ðI At Þ . In particular, they incorporate
both technology coefficients (D and B in one case, A in the other) and trade proportions (C1 and C0 , respectively). In any event, following (8.7),
22
To be consistent with the 1963 model, in which industry final demands drive industry outputs, the 1977 model
is in industry-by-industry format under the industry-based technology assumption.
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366
Decomposition Approaches
0
0
~ f þ f 1 þ ð1=2Þ L
~ 1 ðΔf Þ
~ þL
Δx ¼ ð1=2Þ ΔL
(8.37)
~ 0.
~ ¼L
~1 L
where now ΔL
In (8.7) the two terms on the right captured the effects of technology change and
final-demand change, respectively. Here, where
~ ¼ I DC1 B 1 C1 I C0 A 1 C0
(8.38)
ΔL
0
~ f þ f 1 term encompasses changes in both technology and trade.
the ð1=2Þ ΔL
~ Technical Coefficients, Trade Structure
Digging Deeper into ΔL:
1
Decomposition 1. Create M ¼ I DC0 B C0 . This represents a kind of hybrid total
requirements matrix that combines 1977 technology (in B and D) with 1963 trade
structure (in C0 ). By subtracting and adding this term in (8.38),
h
i
~ ¼ I DC1 B 1 C1 I DC0 B 1 C0
ΔL
h
1
1 i
þ I DC0 B C0 I C0 A C0
(8.39)
~ made by changing trade
The first term is a measure of the contribution to ΔL
proportions (with constant 1977 technology) and the second measures the effect on
~ of changing technology (with constant 1963 trade proportions). Then (8.39) can be
ΔL
written as
1
~0
~ ¼ L
~ M þ
ML
(8.40)
ΔL
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
Trade change,
Technology change,
1977 technology
1963 trade patterns
1
Decomposition 2. Consider, instead, N ¼ I C1 A C1 . This is a kind of total
requirements matrix that combines 1963 technology (in A) with 1977 trade structure
(in C1 ). Subtracting and adding this term in (8.38) gives
h
i
~ ¼ I DC1 B 1 C1 I C1 A 1 C1
ΔL
h
1
1 i
þ I C1 A C1 I C0 A C0
(8.41)
~ that is due to technology
In this case, the first term is a measure of the influence on ΔL
~
change (with constant 1977 trade proportions) and the second captures the effect on ΔL
of trade proportions change (assuming 1963 technology). Now, (8.41) can be written as
1
~0
~ ¼ L
~ N
þ NL
(8.42)
ΔL
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
Technology change,
Trade change,
1977 trade patterns
1963 technology
Averaging. Averaging the results in (8.40) and (8.42) in the usual way gives
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8.2 Structural Decompositon (Additive)
1
1
~ ¼ ð1=2Þ L
~ þML
~ 0 N þ ð1=2Þ L
~ þNL
~0 M
ΔL
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Technology change effect
367
(8.43)
Trade change effect
and, putting this result into (8.37)
1
1
~ þML
~ 0 N f 0 þ f 1 þ ð1=4Þ L
~ þNL
~0 M f0 þ f1
Δx ¼ ð1=4Þ L
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Technology change effect
Trade change effect
0
~ þL
~ 1 ðΔf Þ
þ ð1=2Þ L
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Final-demand change effect
(8.44)
Digging Deeper into Δf: Level and Mix
The decompositions of Δf given in (8.20) – into level and mix – were also carried out.
The final expression for Δx is
1
1
~ 0 N f 0 þ f 1 þ ð1=4Þ L
~ þNL
~0 M f0 þ f1
~ þML
Δx ¼ ð1=4Þ L
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Technology change effect
Trade change effect
0
0
~ þL
~ 1 ðΔf Þ B0 þ B1 þ ð1=4Þ L
~ þL
~ 1 f 0 þ f 1 ðΔBÞ
þ ð1=4Þ L
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Final-demand level effect
Final-demand mix effect
(8.45)
This was used originally for a 70-sector, 51-region version of the model. The article
presents results for a version aggregated to 10 sectors and nine regions. This means
that in the original study there were 3,570 separate results for each of the decompositions. The figures noted in Table 8.9 are averages over 90 outcomes for each
decomposition. This illustrates again that a structural decomposition analysis for a
reasonably large sized model generates an enormous amount of detail.
A Multicountry Model for the European Community (Oosterhaven and van der
Linden, 1997; see also Oosterhaven and Hoen, 1998). Here the authors are concerned
with changes in value added that are associated with changes in output in a multicountry
input–output setting. The model is a variant of the MRIO model, with 25 sectors, 8
countries, and 4 categories of final demand in each country. Their decomposition
follows the general structure of (8.34), with embellishments. Letting vt and ct represent
column vectors of value added and value added per dollar of output at t, they work with
xt ¼ Lt f t ¼ Lt Bt yt and vt ¼ ^c t xt ¼ ^c t Lt Bt yt
(The bridge matrix, Bt , and yt , which contains final-demand expenditures by finaldemand category k in year t, was examined in Section 8.2.2.)
Then, following (8.12) for n ¼ 4,
Δv ¼ ð1=2ÞðΔ^c Þ L0 B0 y0 þ L1 B1 y1 þ ð1=2Þ ^c 0 ðΔLÞ B1 y1 þ ^c 1 ðΔLÞ B0 y0
þ ð1=2Þ ^c 0 L0 ðΔBÞðy1 Þ þ ^c 1 L1 ðΔBÞy0 þ ð1=2Þ ^c 0 L0 B0 þ ^c 1 L1 B1 ðΔyÞ
(8.46)
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368
Decomposition Approaches
This accounts for the four components that contribute to the change in value added.
The embellishments come from further decompositions of ΔL and ΔΒ.
The European Union. The Dietzenbacher and Hoekstra (2002) study [Table SA8.2.1
(in online Appendix SA8.2)] also has a spatial component because the data used came
from intercountry input–output tables for the European Union (EU). This made it
possible to disaggregate their final-demand component into: household consumption,
other domestic final demands (government consumption, capital stock formation,
inventory stock changes), and exports – to Germany, France, Italy, Belgium,
Denmark, the rest of the EU, and the rest of the world.
Results from some of these early studies are collected together in Table 8.6.
8.3
Structural Decomposition (Multiplicative)
8.3.1 Initial Decompositions: Relative Changes in Total Gross Output
Rather than examining components of the absolute difference in output in two years
ðΔx ¼ x1 x0 Þ, consider instead the relative or percentage difference in aggregate
output over the same period, ði0 x1 Þ=ði0 x0 Þ.23 With gross output, this is not something
that is usually of great interest, but measures closely related to it are. We introduce it
here only to illustrate multiplicative structural decomposition analysis (MSDA) using
data from the previous three-sector ASDA example in Sections 8.2.1 and 8.2.4. As in
those sections, we are given f 0 and f 1 and can find L0 and L1 and, thus, x1 ¼ L1 f 1 and
x0 ¼ L0 f 0 .
A straightforward measure of the relative change in total gross output is, thus
i0 x1 i0 L1 f 1
¼
i0 x0 i0 L0 f 0
This ratio can be rewritten (decomposed) into the product of two relative changes, one
due to technology changes (in L) and a second due to final demand changes (in f):
i0 x1 i0 L1 f 1 i0 L1 f 1
i 0 L0 f 1
¼
¼
0 0 0
i0 x0 i0 L0 f 0 i0 L0 f 1
i|fflffl{zfflffl}
Lf
|fflffl{zfflffl}
Technology
Final-demand
change
(8.47)
change
This decomposition introduces i0 L0 f 1 as the denominator for the first term and the
numerator of the second term, so they cancel out. The relative change over the period is
now shown as the product of a term accounting for technology change over the period,
with constant (year-1) final demands as weights, and a second term measuring final
demand change over the period, weighted by constant (year-0) technology.
23
This type of decomposition of input–output data appears to have originated with Dietzenbacher, Hoen, and Los
(2000). It was further developed in Dietzenbacher, Lahr, and Los (2004), and, for an early study with
11 separate multiplicative factors, see Dietzenbacher, de Groot, and Los (2007). Studies of the relative changes
in the Kobe, Japan economy resulting from the earthquake of 1995 can be found in Okuyama (2014, 2015).
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Table 8.6 Selected early empirical structural decompositions at a regional, interregional, or multiregional level
Details Decomposition Components (percentage of total changea)
Author(s) and Source
Technology/Trade
Holland and Cooke
(1992, table 2)
Washington state; 1963–1982; Δx;
51 sectors
Miller and Shao
(1994, table 4)
US MRIO model; 1963–1977; Δx;
51 regions, 70 sectors; (aggregated
to 9 regions, 10 sectors)
Oosterhaven and van
der Linden (1997)
Intercountry model for EC; 1975–1985;
Δ(value added); 8 countries, 25
sectors
a
Final Demand
5
95
Washington
39
34
Intraregional
coefficients
28
(19, 59)b
Intraregional
coeff.
4
–2
Interregional
coeff.
–2
Rest of US
and world
56
67
Interregionalb
coefficients
6
(–43, 19)
Level
65
(53, 79)
Value-added
coeff.
–3
Level
102c
Mix
2
(–5, 13)
102
Mix
–1
Figures may not add to 100 percent due to rounding.
Figures in parentheses indicate the range of values across the nine regions in the study.
c
This figure is further decomposed into the following percentages: Household consumption, 47; Government consumption, 20; Investment, 13; Exports to
other EC countries, 9; Exports outside the EC, 12.
b
369
370
Decomposition Approaches
As with additive case, this multiplicative decomposition is not unique. Its polar
opposite is found by reversing the weights in each term (that is, the f weights in the first
term change from year 1 to year 0 and the L weights in the second term change from
year 0 to year 1). This gives
i0 x1 i0 L1 f 1 i0 L1 f 0
i 0 L1 f 1
¼
¼
0 1 0
i0 x0 i0 L0 f 0 i0 L0 f 0
i|fflffl{zfflffl}
Lf
|fflffl{zfflffl}
Technology
Final-demand
change
(8.48)
change
This time the new numerator on the left and the new denominator on the right
cancel out.
In additive decompositions, the arithmetic average of each of the elements in the
two polar decompositions was used as a representative measure of the results.
A common solution to the non-uniqueness of multiplicative decompositions like this
is to use the geometric averages of the two polar opposites.24 This leads to the
following result:
i0 x1
¼
i0 x0
i 0 L1 f 1 i 0 L1 f 0
i 0 L0 f 1 i 0 L0 f 0
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Technology
change
0:5
i 0 L0 f 1 i 0 L1 f 1
i 0 L0 f 0 i 0 L1 f 0
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Final-demand
0:5
(8.49)
change
8.3.2 A Note on Arithmetic and Geometric Means (Averages)
P
The simple arithmetic average of n elements, x1 , x2 , , xn , is xa ¼ ð1=nÞ ni¼1P
xi . This
n
is the number which, when added to itself n times,
gives
the
same
result
as
i¼1 xi ;
P
n
a
a
a
a
namely x þ x þ þ x ðn timesÞ ¼ nx ¼ i¼1 xi. For example, with x1 ¼ 2,
P
x2 ¼ 6 and x3 ¼ 18, 3i¼1 xi ¼ 26 and xa ¼ 8:667. The arithmetic average thus serves
as a representative measure of elements in an additive expression.
xg , is the number which
when multiplied by
The geometric mean of n elements,Q
Qn
n
g n
itself n times
gives
the
same
result
as
x
,
so
ð
x
Þ
¼
x
and, taking the nth
i
i
i¼1
i¼1
Qn 1=n
Q3
g
x
.
For
the
same
x
,
x
¼
216
and
x
¼
6.
Geometric averroot, xg ¼
i
i¼1 i
i¼1 i
ages serve as representative measures for elements that are multiplied together (as in
multiplicative decompositions) rather than added. This example may not show it, but
the differences between the two kinds of averages can be substantial – for example,
with x1 ¼ 2 and x2 ¼ 32, xa ¼ 17 and xg ¼ 8:
24
There are, in fact, n! ¼ ðnÞðn 1Þ ð2Þð1Þ different possible decompositions when there are n factors. It is
generally agreed that a geometric average of the two polar opposites is a reasonable solution. (Again, see
Dietzenbacher and Los, 1998; also de Haan, 2001; Dietzenbacher, Lahr, and Los, 2004; de Boer, 2008, 2009.)
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8.3 Structural Decomposition (Multiplicative)
371
8.3.3 Numerical Example (Reexamined)
0 1 1
From the illustration in Section 8.2, ii0 LL0 ff 0 ¼ 332:0051
¼ 1:1857, and using the data from
280
that earlier example in (8.47) and (8.48) gives
1:1857 ¼ 1:0655 1:1128 and 1:1857 ¼ 1:0662 1:1121
|fflfflffl{zfflfflffl}
|fflfflffl{zfflfflffl}
|fflfflffl{zfflfflffl}
|fflfflffl{zfflfflffl}
Technology
Final-demand
Technology
Final-demand
change
change
change
change
The geometric averages are ½ð1:0662Þð1:0655Þ0:5 ¼ 1:0658 for the two technology
change components and ½ð1:1121Þð1:1128Þ0:5 ¼ 1:1125 for the final-demand components, giving the final decomposition
1:1857 ¼ 1:0658 1:1125
|fflfflffl{zfflfflffl}
|fflfflffl{zfflfflffl}
Technology
Final-demand
change
(8.50)
change
This suggests that if only technology were to change, output would grow by 6.58
percent and if only final demands changed, output would grow by 11.25 percent,
whereas if both change, the result would be output growth of 18.6 percent.25 In many
real-world multiplicative decomposition applications, some of the right-hand side
elements will be less than one. So, for example, a result of 0.9255 would be interpreted
as indicating that if only that factor were to change, the result would be a 7.45 percent
decrease in the left-hand side variable.
Note that, unlike the additive decomposition case, we cannot represent the percentage contributions of each of the two right-hand-side factors by dividing by 1.1857 (and
multiplying by 100), since 1:1857 6¼ 1:0658 þ 1:1125. (It may be close here, but it is
not mathematically valid.) As a result, logarithmic transformations have been applied
in an attempt to clarify the interpretation of multiplicative decomposition results. The
natural logarithm of a ratio (a period-1 value divided by a period-0 value) gives
the growth rate of the variable over the period with continuous compounding. From
the logarithmic rule that ln ðx1 x2 xn Þ ¼ ln x1 þ ln x2 þ þ ln xn , it follows that
taking the natural logarithms of both sides of (8.50) will display the results in an
intuitively appealing way, so that the right-hand side components add up to the lefthand side and thus percentage contributions can be found. Using it for this example we
have ln 1:1857 ¼ ln 1:0659 þ ln 1:1125, or 17.03 ¼ 6.38 þ 10.66, suggesting that of
the 17.06 percent continuous growth rate experienced over the period, 6.38 percent
was due to technical changes and 10.66 percent was a result of changes in final
demand – accounting for roughly 37 and 63 percent, respectively, of the total change.26
25
In this particular example, the arithmetic and geometric averages are virtually identical (to four decimal places),
but in many real-world cases the differences can be large.
26
These are the same economy-wide percentages as shown in Table 8.2 for the same two factors in the additive
decomposition case. In fact, economy-wide percentages for additive and multiplicative decompositions with
two explanatory factors will usually be (approximately) equal. Proof of this result is given in online Appendix
SA8.3 for the mathematically inclined reader. With more than two factors and/or longer time periods the
equivalence disappears.
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372
Decomposition Approaches
Note that this continuous growth rate, 17.06 percent, is not quite the same as
i0 L1 f 1
100 ¼ 18.57.27 Logarithmic transformations of this sort first appear in Yang
i0 L0 f 0
and Lahr (2010).28 The logarithmic version of (8.49) is thus
ln
i0 x1
i0 x0
i 0 L1 f 1 i 0 L1 f 0
i 0 L0 f 1 i 0 L1 f 1
¼ ð0:5Þ ln 0 0 1 0 0 0 þ ð0:5Þ ln 0 0 0 0 1 0
iL f
iL f
iL f
iL f
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Technology change
Final-demand change
(8.51)
8.3.4 Multiplicative Decomposition of Changes in Some Function of x
Multiplicative decompositions have been used often in environmental- and energyrelated input–output analyses. (More about this in Chapters 12 and 13.) In a simplified
example (in the spirit of Zhang, Lahr, and Bi, 2016), aggregate gross output, i0 x, is
replaced by a measure of the energy embodied in that output, e0c x (where e0 is the row
vector of total energy inputs by each sector, and e0c ¼ e0 x^1 is the corresponding row
vector of energy-input coefficients).
A Three-Factor Example Using ε for total energy used in producing the
gross outputs in x, ε ¼ e0c x ¼ e0c Lf, and a measure of the rate of change of total
1
embodied energy over two periods is εε0 . Decomposition will now involve three
multiplicative terms, one for each right-hand side factor. One of the polar decompositions is
1 0 1 1 1 0 1 1
0 0 1 1
0 0 0 1
ec L f
ec L f
ec L f
e Lf
ε1
¼ 0 0 0 ¼ 0 1 1 0 0 1 c 0 0 0
(8.52)
0
0
0
0
ε
ec L f
ec L f
ec L f
e0c L f
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
Energy use change
Technology change
Final-demand change
This isolates changes in energy-use coefficients from technology and final demand
effects. Again, the polar opposite can be found,
1 0 1 1 1 0 0 0
1 0 1 0
1 0 1 1
ec L f
ec L f
ec L f
e Lf
ε1
¼ 0 0 0 ¼ 0 0 0 0 0 0 c 0 1 0
(8.53)
0
0
1
ε0
ec L f
ec L f
ec L f
e1c L f
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
Energy use change Technology change Final-demand change
and then geometric means of each of these three effects constitute the final results.
27
28
See Appendix 8.4 for a brief note on various approaches to measuring growth rates.
They cite an unpublished paper by Jojo Jacob: “Structural Change, Liberalization and Growth: The Indonesian
Experience in an Input–Output Perspective,” ECIS, Eindhoven University of Technology, The Netherlands,
January 7, 2003.
https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
8.3 Structural Decomposition (Multiplicative)
ε1
¼
ε0
373
1 0 1 1 1 0 0 0 !0:5
0 0 1 1 1 0 1 0 !0:5
ec L f
ec L f
e Lf
e Lf
c 0 0 1 c 0 0 0
0 1 1 0 0 0
0
0
0
ec L f
ec L f
ec L f
e1c L f
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Energy use change
Technology change
0 0 0 1 1 0 1 1 !0:5
e Lf
e Lf
c 0 0 0 c 0 1 0
0
ec L f
e1c L f
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Final-demand change
(8.54)
A Four-Factor Example One final example illustrates even more complicated (and usual) multiplicative decompositions (Dietzenbacher, Hoen, and Los,
2000). In this case variable of interest is aggregate labor productivity in an
economy, defined as aggregate value added, v ¼ v0c x, divided by aggregate labor
inputs, h ¼ h0c x (where h and v are vectors of total labor inputs and total value added,
respectively, by sector, and v0c ¼ v0 x^1 and h0c ¼ h0 x^1 ). Aggregate labor productivity,
π, is thus
π¼
0
v v0c x v0c Lf
¼
¼
h h0c x h0c Lf
(8.55)
1
For two periods with π 0 ¼ hv0 and π 1 ¼ hv1 , the relative change in aggregate labor
0
1 1
1
ðv1c Þ L1 f 1
v1
h0
v1
¼
.
Here
¼
and
productivity over the period is ππ0 ¼ vv0=h
0
v0
v0
=h0
h1
ðv0c Þ L0 f 0
0
ðh0c Þ L0 f 0
h0
, so the variable of interest is
0
1 ¼
h
ðh1c Þ L1 f 1
1 0 1 1 0 0 0 0
v Lf
h Lf
π1
¼ c 0 0 0 c1 0 1 1
(8.56)
0
π
v0c L f
hc L f
As usual, we want the terms on the right-hand side rewritten so that only one pair of
elements in each ratio is measured at different time periods. For example, the first term,
0
0
0
0
ðv1c Þ L1 f 1
ðv1c Þ L1 f 1 ðv0c Þ L1 f 1 ðv0c Þ L0 f 1
. As in the earlier simpler
0 0 0 can be expanded to
0 1 1 0 0 1 0
0
0
0
0
ðvc Þ L f
ðvc Þ L f ðvc Þ L f ðvc Þ L0 f 0
|fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl}
ð1Þ
ð2Þ
ð3Þ
example, each new denominator cancels the following new numerator, so in fact only
0
ðv1c Þ L1 f 1
remains. But the decomposition provides a disentangling of the individual
0
ðv0c Þ L0 f 0
contributions: first, from relative changes in value added (v0c ) with unchanged L and f
[in (1)], second from changing production structure (L) [in (2)], and finally of changing
final demand (f) [in (3)].
https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
374
Decomposition Approaches
0
ðh0c Þ L0 f 0
A similar decomposition of the second term,
leads to
0
ðh1c Þ L1 f 1
0
0
0
ðh0c Þ L1 f 1 ðh0c Þ L0 f 1 ðh0c Þ L0 f 0
. Putting these two decompositions together and regroup0
0
0
ðh1c Þ L1 f 1 ðh0c Þ L1 f 1 ðh0c Þ L0 f 1
|fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl}
ð4Þ
ð5Þ
ð6Þ
ing gives
1 0 1 1 0 0 1 1
0 0 1 1 0 0 0 1 !
0 0 0 1 0 0 0 0 !
vc L f
hc L f
vc L f
hc L f
v Lf
h Lf
π1
¼ 0 1 1 1 0 1 1 0 0 1 0 0 1 1 c 0 0 0 c0 0 0 1
0
0
0
0
π
vc L f
vc L f
vc L f
h Lf
hc L f
hc L f
|fflfflfflfflffl
ffl{zfflfflfflfflfflffl} |fflfflfflfflfflc ffl{zfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Changes in production structure
Changes in final demand
Changes in
Changes in
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
value added
labor inputs
|fflfflfflfflffl{zfflfflfflfflffl}
|fflfflfflfflffl{zfflfflfflfflffl}
ð1Þ
ð4Þ
ð2Þþð5Þ
ð3Þþð6Þ
(8.57)
As was the case with earlier decompositions, the result in (8.57) is by no means unique.
For example, changes in value added in (1) could have been measured with weights
from period 0 (L0 and f 0 ), changes in L in (2) and (5) could have been weighted by
1 1
1
vc , hc and f 0 , and so forth. The second polar decomposition of ππ0 results from
exactly this kind of reversal of weights, namely
1 0 0 0 0 0 0 0
1 0 0 0 1 0 1 0 !
1 0 1 0 1 0 1 1 !
vc L f
hc L f
vc L f
hc L f
v L f
h Lf
π1
¼ 0 0 0 1 0 0 0 0 1 0 1 0 0 0 c 0 1 1 c1 0 1 0
0
π
v0c L f
v1c L f
v1c L f
h Lf
hc L f
hc L f
|fflfflfflfflffl
ffl{zfflfflfflfflfflffl} |fflfflfflfflfflc ffl{zfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Changes in
value added
Changes in
labor inputs
Changes in production structure
Changes in final demand
(8.58)
Taking the geometric means of each of the four effects in (8.57) and (8.58) gives the
final result:
π1
¼
π0
1 0 1 1 1 0 0 0 !0:5
0 0 1 1 0 0 0 0 !0:5
vc L f
vc L f
h Lf
h Lf
c1 0 1 1 c1 0 0 0
0 1 1 0 0 0
0
0
vc L f
vc L f
hc L f
hc L f
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Value added
Labor inputs
0 0 1 1 1 0 0 0 !0:5
0 0 0 1 1 0 1 0 !0:5
v L f
v Lf
h Lf
h Lf
c 0 0 1 c 0 1 0
c0 0 1 1 c1 0 0 0
v0c L f
v1c L f
hc L f
hc L f
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Production structure
0 0 0 1 1 0 1 0 !0:5
0 0 0 0 1 0 1 1 !0:5
vc L f
vc L f
h Lf
h Lf
0 0 0 0 1 1
c0 0 0 1 c1 0 1 0
0
1
vc L f
vc L f
hc L f
hc L f
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Final demand
https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
(8.59)
8.4 Decomposition of Multipliers
375
We write this out in full primarily to familiarize the reader with the general structure of
this kind of multiplicative decomposition because complex expressions similar to this
can be found in many applied studies, as a prelude to the specific application in that
study. And, as with the earlier multiplicative decompositions, a logarithmic transformation generates a result in which the right-hand side factor contributions add up to the
result on the left-hand side.
There are many applied studies in a multiregional (or multinational) setting in which
inputs and outputs are distributed spatially over regions (or countries), as in Section
8.2.7, or in which final demand is broken down into further categories such as level
effects, mix effects, and/or distribution effects, as in Section 8.2.2. These extensions
and applications result in increasingly more complex decompositions, but the basic
principles remain the same.
Many applications using the multiplicative structural decomposition approach have
appeared since around 2010. For example Xia, Fan, and Yang (2015) assess changes in
Chinese industrial carbon emissions intensities over 2002–2007. These are multiplicatively decomposed into relative changes in technology, final demand structure, and
final demand levels. And Jiang, Dietzenbacher, and Los (2014) use a multiplicative
decomposition approach to assess relative changes over 1997–2002 in Chinese
regional labor productivities as functions of relative changes in regional sectoral
employment shares, output per worker, and value-added coefficients.
8.4
Decomposition of Multipliers
Multiplier decompositions provide another approach to digging deeper into input–
output data structural relationships, this time using data from a single period. Again,
there are both multiplicative and additive versions. We explore these in this section.29
8.4.1 Multiplier Decompositions (Multiplicative)
We start with the fundamental input–output accounting relationship
x ¼ A
ðn1Þ
x þ f
ðnnÞ ðn1Þ
ðn1Þ
(8.60)
from which x ¼ ðI AÞ1 f ¼ Lf. We now introduce some algebra that initially
appears unmotivated but it will soon be clear what is accomplished.30 Given some
~ , adding and subtracting Ax
~ to (8.60) and rearranging produces
A
ðnnÞ
29
30
~ þ Ax
~ þf ) IA
~ x¼ AA
~ xþf
x ¼ Ax Ax
(8.61)
For an overview of these and several others, see Sonis and Hewings (1988) or additional references noted in
Section 15.6.
This decomposition is presented in Pyatt and Round (1979) in the context of social accounting matrices
(SAMs).
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376
Decomposition Approaches
and, solving31 for x,
~ xþ IA
~ 1 f
~ 1 A A
x¼ IA
~ 1 A A
~ ; then this is
Let A∗ ¼ I A
~ 1 f
x ¼ A∗ x þ I A
(8.62)
Next, premultiply both sides of (8.62) by A∗
2
~ 1 f
A∗ x ¼ ðA∗ Þ x þ A∗ I A
and substitute this for A∗ x in the right-hand side of (8.62)
2
~ 1 f þ I A
~ 1 f ¼ ðA∗ Þ2 x þ ðI þ A∗ Þ I A
~ 1 f
x ¼ ðA∗ Þ x þ A∗ I A
(8.63)
Again, solving for x,
2 1
~ 1 f
x ¼ [I ðA∗ Þ ] ðI þ A∗ Þ I A
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl}
M3
M2
(8.64)
M1
In this way the usual Leontief inverse (multiplier) matrix, ðI AÞ1 , has been decomposed into the product of three matrices.
This algebra can be continued. Premultiply both sides of (8.63) by A∗ x,
3
2 ~ 1 f
(8.65)
A∗ x ¼ ðA∗ Þ x þ [A∗ þ ðA∗ Þ ] I A
and, again, substitute for A∗ x in the right-hand side of (8.62)
h
i
3
2 ~ 1 f
x ¼ ðA∗ Þ x þ I þ A∗ þ ðA∗ Þ I A
(8.66)
Solving for x, we now find
3 1
2 ~ 1 f
x ¼ [I ðA∗ Þ ] [I þ A∗ þ ðA∗ Þ ] I A
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl}
M3
M2
(8.67)
M1
[Compare with the results in (8.64).]
In the context of social accounting matrices (Chapter 11), where much of the
fundamental work on multiplier decompositions originated, M1 is said to capture a
“transfer” effect, M2 embodies “open-loop” effects, and M3 contains “closed-loop”
effects. (Pyatt and Round, 1979.) The logic of these labels will be clear in the
interregional context in Section 8.4.2.
31
Here and throughout we assume non-singularity of the matrices whose inverses are shown.
https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
8.4 Decomposition of Multipliers
377
These iterations can continue any number of times. After k steps, the parallel to
(8.66) is
h
i
k
2
k1 ~ 1 f
x ¼ ðA∗ Þ x þ I þ A∗ þ ðA∗ Þ þ þ ðA∗ Þ
IA
(8.68)
and the parallel to (8.67) is
h
i h
i
k 1
2
k1 ~ 1 f
x ¼ I ðA∗ Þ
I þ A∗ þ ðA∗ Þ þ þ ðA∗ Þ
IA
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl}
M2
M3
(8.69)
M1
8.4.2 Multiplier Decompositions in an Interregional Context
For a two-region interregional model (Section 3.3) the input–output accounting relationship x ¼ Ax þ f becomes
r rr
r A
f
Ars xr
x
s ¼
sr
s þ
ss
x
A
A
x
fs
With a view toward decompositions, we can isolate the intraregional and interregional
elements in A; let
rr
rr
Ars
0
A
0 Ars
A
¼
þ
A¼
Asr Ass
Asr 0
0 Ass
rr
A
I Arr
0
0
~
~
Define A ¼
from which I A ¼
. Then, using
0 Ass
0
I Ass
the decomposition in (8.64), for example,
1
ðI Arr Þ1
0
~
M1 ¼ I A
¼
0
ðI Ass Þ1
(from the rule that the inverse for a block-diagonal matrix is made up of the inverses of
the matrices on the main diagonal). Also,
~ 1 A A
~
A∗ ¼ I A
#
"
#"
0 Ars
ðI Arr Þ1
0
¼
0
ðI Ass Þ1
Asr 0
"
#
0
ðI Arr Þ1 Ars
¼
ðI Ass Þ1 Asr
0
and so, again from (8.64),
∗
M2 ¼ I þ A ¼
I
ðI Ass Þ1 Asr
https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
ðI Arr Þ1 Ars
I
378
Decomposition Approaches
Finally, from straightforward matrix multiplication,
ðI Arr Þ1 Ars ðI Ass Þ1 Asr
0
∗ 2
ðA Þ ¼
0
ðI Ass Þ1 Asr ðI Arr Þ1 Ars
and so
h
i
2 1
M3 ¼ I ðA∗ Þ
2h
3
i1
rr 1 rs
ss 1 sr
ð
Þ
A
ð
I
A
Þ
A
0
I
I
A
6
7
¼4
h
i1 5
ss 1 sr
rr 1 rs
0
I ðI A Þ A ðI A Þ A
(again from the rule for the inverse of a block-diagonal matrix).
In terms of intra- and interregional effects, the matrices in M1 are seen to capture
intraregional (Leontief inverse or “transfer”) effects, those in M2 contain interregional
spillover (“open-loop”) effects, and the matrices in M3 record interregional feedback
(“closed-loop”) effects (Round, 1985, 2001; Dietzenbacher, 2002).32 As usual, define
Lrr ¼ ðI Arr Þ1
and
Lss ¼ ðI Ass Þ1
These are the intraregional effects in each region ðM1 Þ. The two spillover matrices in
M2 may be represented as
Srs ¼ Lrr Ars and Sss ¼ Lss Asr
and the two feedback matrices in M3 can be defined as
Frr ¼ ½I Lrr Ars Lss Asr 1 and Fss ¼ ½I Lss Asr Lrr Ars 1
or
Frr ¼ ½I Srs Ssr 1 and Fss ¼ ½I Ssr Srs 1
Therefore, in the two-region interregional context, x ¼ M3 M2 M1 f becomes
r rr
r F
x
0
I Srs Lrr 0
f
¼
0 Fss Ssr I
0 Lss f s
xs
or, carrying out the multiplications,
r rr rr
x
F L
¼
xs
Fss Ssr Lrr
32
Frr Srs Lss
Fss Lss
r f
fs
(8.70)
(8.71)
There have been other definitions of these various effects in the input–output literature, beginning perhaps with
Miller (1966, 1969) but also including, among others, Round (1985, 2001), and Sonis and Hewings (2001).
https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
379
8.4 Decomposition of Multipliers
8.4.3 Multiplier Decompositions (Additive)
An alternative decomposition isolates net effects. Starting with Pyatt and Round’s
multiplicative result in (8.64) [or (8.67), or (8.69)], namely x ¼ Mf, where
M ¼ M3 M2 M1 , Stone (1985) proposed an additive form
M ¼ I þ ðM1 IÞ þ ðM2 IÞM1 þ ðM3 IÞM2 M1
|fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
~1
M
~2
M
~3
M
(This is easily seen to be true by simply carrying out the algebra on the right-hand
side.) Therefore,
x ¼ Mf ¼ If þ ðM1 IÞ f þ ðM2 IÞM1 f þ ðM3 IÞM2 M1 f
|fflfflfflfflffl{zfflfflfflfflffl}
|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
~1
M
~2
M
(8.72)
~3
M
To paraphrase Stone (p. 162) – in the context of an interregional model – we start with
~ 1 f) adds on the net intraregional
a matrix of initial injections, If. The second term (M
~ 2 f) we add in the net interregional spillover effects
effects captured in M1 . Next (in M
~
in M2 . Finally, the fourth term (M 3 f) captures the net interregional feedback effects in
M3 . In the two-region example, these are
rr
0
~ 1 ¼ M1 I ¼ L I
M
0
Lss I
~ 2 ¼ ðM2 IÞM1 ¼
M
~ 3 ¼ ðM3 IÞM2 M1 ¼
M
0
Ssr
Srs
0
Lrr
0
0
Lss
Frr Lrr Lrr
ss sr rr
F S L Ssr Lrr
0
¼
Ssr Lrr
Srs Lss
0
Frr Srs Lss Srs Lss
Fss Lss Lss
While these appear (and are) increasingly complex, they also serve to disentangle the
complex net of intraregional, spillover, and feedback effects.
8.4.4 A Note on Interregional Feedbacks
Interregional feedback effects in a two-region input–output model were explored in
Section 3.3.2. They were defined early on (Miller 1966, 1969) for the specific scenario
of a change in final demand in region r only – so Δf r 6¼ 0 and Δf s ¼ 0. Then a measure
of the interregional feedback effect is found as the difference between the output
change in region r that would be generated by the complete two-region model and
the output change in region r that would be calculated from a single-region model.
These outputs are
ΔxrT ¼ ½ðI Arr Þ Ars Lss Asr 1 Δf r and ΔxrS ¼ ðI Arr Þ1 Δf r
(with subscripts T and S indicating “two-region” and “single-region” models, respect1
ively). Consider the inverse matrix in ΔxrT , [ðI Arr Þ Ars ðI Ass Þ1 Asr ] .
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380
Decomposition Approaches
1. Factoring out ðI Arr Þ gives
1
ðI Arr Þ I ðI Arr Þ1 Ars ðI Ass Þ1 Asr
2. Using the rule that ðMNÞ1 ¼ N1 M1 , we have
1
1
I ðI Arr Þ1 Ars ðI Ass Þ1 Asr ðI Arr Þ
Using Lrr ¼ ðI Arr Þ1 and Lss ¼ ðI Ass Þ1 , we have
ΔxrT ¼ ½I Lrr Ars Lss Asr 1 Lrr Δf r and ΔxrS ¼ Lrr Δf r
Finally, using Frr ¼ ½I Lrr Ars Lss Asr 1 ,
ΔxrT ΔxrS ¼ Frr Lrr Δf r Lrr Δf r ¼ ðFrr Lrr Lrr ÞΔf r ¼ ðFrr IÞLrr Δf
The Frr Lrr term is exactly the upper left element in the multiplier matrix from the
multiplicative decomposition in (8.71), and the ðFrr IÞLrr term (for the difference in
~ 3 from the
gross outputs in the two models) is exactly the upper left element in M
additive decomposition of net effects.
8.4.5 Numerical Illustration
We reconsider the two-region example from Chapter 3, in light of these decomposition
possibilities. In that example we had
2
3
500
50
25
75 7
6 150
6
7
6 200
100
400
200
100 7
rr
6
7
Zrs
Z
6
7
¼
Z¼
6
300
500
50
60
40 7
Zsr Zss
6
7
6
7
100
60
200
250 7
6 75
4
5
50
25
25
150
100
|
|
|
|
|
|
|
- - - - - - - - - - - - - - - - - - - - - - - - |
|
|
|
|
and
2
1,000
3
6
7
2,000 7
r 6
6
7
x
6
7
Z ¼ s ¼ 6 1,000 7
x
6
7
6 1,200 7
4
5
- - - -
800
with associated direct and total requirements matrices of
https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
8.4 Decomposition of Multipliers
2
rr
A
A¼
Asr
and
6 :1500
6
6
6 :2000
rs
A
6
¼ 6 :3000
Ass
6
6
6 :0750
4
:0500
:2500
:0500
:0500
:4000
:2500
:0500
:0500
:0600
:0125
:0250
:4652
:2909
|
|
|
|
|
|
:0208
:1667
:0500
381
3
:0938 7
7
:1250 7
7
7
:0500 7
7
7
:3125 7
5
:1250
|
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
|
|
|
|
:1667
:1250
2
3
6 1:4234
6
6 :6346
6
6
L ¼ 6 :6383
6
6
6 :2672
4
:1468
:6707
1:4237
:5369
1:3363
:2000
:1973
:0908
:0926
|
|
|
|
|
|
|
:1917
:3041 7
7
:4558 7
7
7
:3108 7
7
7
:5473 7
5
1:2538
:4092
:2501
- - - - - - - - - - - - - - - - - - - - - - -| - - - - - - - - - - - - - -
In addition,33
2
1:3651
Lrr ¼ ðI Arr Þ1 ¼ 4 :5273
:5698
and
Lss ¼ ðI Ass Þ1 ¼
|
|
|
|
1:3406
:2155
3
:4253 :2509
1:3481 :5954 5
:4890 1:2885
1:2679
:1811
:4528
1:2075
From these we can generate the additional components needed for these decompositions, namely
2
3
:1119 :1937
:1177 :0691 :0874
rs
rr rs
sr
ss sr
4
5
S ¼ L A ¼ :2654 :2477 and S ¼ L A ¼
:0740 :0242 :0411
:1578 :1790
2
1:0296 :0134
Frr ¼ ½I Srs Ssr 1 ¼ 4 :0535 1:0262
:0343 :0164
and
sr rs 1
F ¼ ½I S S ss
33
3
:0191
:0359 5
1:0228
1:0488 :0599
¼
:0228 1:0297
Remember that Lrr does not designate the 3 3 submatrix in the upper left of L, and similarly Lss is not the
2 2 submatrix in the lower right of L.
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382
Decomposition Approaches
The M matrices for the multiplicative decomposition are easily found to be
2
3
1:3651
:4253
:2509
0
0
6
7
6
7
6 :5273
1:3481
:5954
0
0 7
6
7
6
7
M1 ¼ 6 :5698
:4890
1:2885
0
0 7
6
7
6
7
0
0
0
1:2679
:4528
6
7
4
5
0
0
0
:1811
1:2075
|
|
|
|
|
|
|
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
|
|
|
|
for intraregional transfer effects, as is expected, since only Lrr and Lss appear in this
matrix. Next
2
3
1
0
0
:1119
:1937
6
7
6
7
6 0
1
0
:2654
:2477 7
6
7
6
7
M2 ¼ 6 0
0
1
:1578
:1790 7
6
7
6
7
:0691
:0874
1
0 7
6 :1177
4
5
:0740
:0242
:0411
0
1
|
|
|
|
|
|
|
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
|
|
|
|
contains interregional spillover (“open-loop”) effects only, transmitted from r to s
(upper right) and from s to r (lower left). Finally
2
3
1:0296
:0134
:0191
0
0
6
7
6
7
6 :0535
1:0262
:0359
0
0 7
6
7
6
7
M3 ¼ 6 :0343
:0164
1:0228
0
0 7
6
7
6
7
0
0
1:0488
:0599 7
6 0
4
5
0
0
0
:0228
1:0297
|
|
|
|
|
|
|
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
|
|
|
|
- -
identifies interregional feedback (“closed-loop”) effects.
We first use the multiplicative decomposition to find xnew ¼ M3 M2 M1 f new
for our example (Chapter 3) with ðf new Þ0 ¼ ½100 0 0 0 0. This will generate
2
3
142:34
6
7
6 63:46 7
6
7
6
7
xnew ¼ 6 63:83 7, as we found in that chapter. Now, however, the effects can be
6
7
6 26:72 7
4
5
-
-
-
-
14:68
disentangled. Specifically,
https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
8.4 Decomposition of Multipliers
2
136:51
383
3
6
7
6 52:73 7
6
7
6
7
1. First, M1 f new ¼ 6 56:98 7 indicates the initial impact in region r, the origin of the
6
7
6 0 7
4
5
-
-
-
-
0
final demand change.
2
136:51
3
6
7
6 52:73 7
6
7
6
7
2. Next, M2 M1 f new ¼ 6 56:98 7 adds to (1) the increases in the two sectors of region
6
7
6 24:69 7
4
5
-
-
-
-
13:71
s because of the spillovers from r. Note that outputs in r are unchanged from (1),
since this calculation is concerned with spillovers only. Clearly the difference
between the results in (2) and (1), M2 M1 f new M1 f new , will be the vector of
changes in s only.
2
3
142:34
6
7
6 63:46 7
6
7
6
7
3. Finally, M3 M2 M1 f new ¼ 6 63:83 7 ¼ Lf new then adds in the feedback effects in
6
7
6 26:72 7
4
5
-
-
-
-
14:68
the two regions – in r where the stimulus originated and in s because of the stimulus
from
the3spillovers. In this case, the difference between the results in (3) and (2),
2
5:83
6
7
6 10:73 7
6
7
6
7
6 6:84 7, nets out the feedback effects by themselves. The first three elements,
6
7
6 2:03 7
4
5
-
-
-
-
2 :97 3
5:83
4 10:73 5, are exactly the interregional feedback amounts that we found for region r
6:84
in Chapter 3.
Consider now the components of the additive decomposition
xnew ¼ Mf new ¼ If new þ ðM1 IÞ f new þ ðM2 IÞM1 f new þ ðM3 IÞM2 M1 f new
|fflfflfflfflffl{zfflfflfflfflffl}
|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
~1
M
https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
~2
M
~3
M
384
Decomposition Approaches
These provide the net effects. For this example, these multiplier matrices are
2
3
:3651
:4253
:2509
0
0
6
7
6
7
6 :5273
:3481
:5954
0
0 7
6
7
7
~1 ¼6
M
6 :5698
:4890
:2885
0
0 7
6
7
6
7
0
0
0
:2679
:4528
6
7
4
5
0
0
0
:1811
:2075
|
|
|
|
|
|
|
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
|
|
|
|
2
3
6 0
6
6 0
6
~2 ¼6
M
6 0
6
6
6 :2496
4
:1371
0
0
|
|
|
0
0
0
0
:1859
:1833
|
|
|
:1769
:3814
:2325
:2845 7
7
:4193 7
7
7
:2876 7
7
7
0 7
5
0
|
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
:0841
:0858
:0400
:0400
|
|
|
|
0
0
2
3
6 :0583
6
6 :1072
6
~3 ¼6
M
6 :0684
6
6
6 :0203
4
:0097
:0756
:0753
:0478
:0477
:0141
:0141
:0067
:0067
|
|
|
|
|
|
:0148
:0278
:0176
:0195 7
7
:0365 7
7
7
:0232 7
7
7
:0944 7
5
:0463
|
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
|
|
|
|
:0727
:0343
The pieces of the decomposition in (8.72) are:
2
3
100
6
7
6 0 7
6
7
6
7
1. If new ¼ 6 0 7 is just the initial “shock.”
6
7
6 0 7
4
5
- - -
0
2
36:51
3
6
7
6 52:73 7
6
7
7
~ 1 f new ¼ 6
56:98
2. Then M
6
7 accounts for the indirect effects in r; the sum of (1) and
6
7
6 0 7
4
5
- - - -
0
(2) is just M1 f
new
, by definition.
https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
8.5 Paths
2
0
385
3
6
7
6 0 7
6
7
7
~ 2 f new ¼ 6
3. Next, M
6 0 7 captures the spillovers; this is M2 M1 f new M1 f new , also
6
7
6 24:69 7
4
5
- - - -
13:71
by definition.
2
5:83
3
6
7
6 10:73 7
6
7
7
~ 3 f new ¼ 6
4. Finally, M
6 6:84 7 isolates the contribution from the interregional feedbacks;
6
7
6 2:03 7
4
5
- - - -
:97
by definition this is M3 M2 M1 f new M2 M1 f new .
~ are multiplier matrices,
The matrix components of these decompositions, M and M,
and so various multipliers can be calculated in the same way as was done earlier in this
chapter for L – for example, simple column sums, or weighted sums if employment,
value added, or other economic impacts are of interest.
An empirical example applying these kinds of decompositions can be found in
Zhang and Zhao (2005). They present a detailed set of decompositions of initial,
spillover, and feedback effects derived from the 17-sector version of the 2000
Chinese multiregional (CMRIO) model that has been aggregated spatially into two
mega-regions – Coastal and Non-coastal regions.
An analysis of feedbacks and spillovers in a 40-country, 59-product version of the
WIOD databases for 1995–2009, using information directly from the international
SUTs (without inverses) can be found in Temursho (2018).34
8.5
Paths
Another approach to disentangling data in input–output accounts is structural path
analysis, which goes back at least to Defourny and Thorbecke (1984), a reference that
also includes extensive historical background and earlier work (see also Kahn and
Thorbecke, 1988). More recently, an additional variant, structural path decomposition
has been proposed (Wood and Lenzen, 2009). These path-oriented approaches have
been widely used in many energy and environmental applications, for example in
identifying individual responsibilities for carbon footprints, scarce resource use, and
34
Moran, Wood, and Rodrigues (2018) examine the size of feedback effects for embodied CO2 emissions using
the EXIOBASE3 global MRIO data set for 2011. They find them to be less than two percent of total emissions
(a figure in line with results discussed in Chapter 3 for much simpler interregional models).
https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
386
Decomposition Approaches
other (negative) side-effects from production and consumption. (See Chapters 12 and
13 for examples.)
8.5.1 Structural Path Analysis
As with the multiplier decompositions in Section 8.4, structural path analysis (SPA)
does not depend on data from more than one period. The object is to identify the
various sectoral contributions to an overall effect by viewing the Leontief inverse in its
power series form. Thus, the starting points are the observations that
L ¼ ðI AÞ1 ¼ I þ A þ A2 þ A3 þ and
x ¼ Lf ¼ f þ Af þ A2 f þ A3 f þ As noted, various forms of structural path analysis have found wide application in
environmental and energy studies, and we use these as the setting for a brief
introduction to the procedure. Let εi represent the total input of some factor
(BTU’s of energy, etc.) or unwanted output (e.g., CO2 emissions) accompanying
sector i’s production. Then ε0 ¼ ½ε1 ; . . . , εn is known as an intensity vector, and the
associated factor intensity coefficients (energy use or CO2 emissions per unit of
output) are ε0c ¼ ε0 x^1 .
Incorporation of the Power Series Results Sectoral factor multipliers are
mðεÞ ¼ ε0c L (Section 6.2.3). To illustrate, in a three-sector model,
2
3
l 11 l 12 l 13
mðεÞ ¼ ε0c L ¼ ½ εc1 εc2 εc3 4 l 21 l 22 l 23 5
l 31 l 32 l 33
¼ ½ εc1 l11 þ εc2 l 21 þ εc3 l 31 εc1 l 12 þ εc2 l22 þ εc3 l 32 εc1 l13 þ εc2 l 23 þ εc3 l 33 P
Consider mðεÞ1 ¼ 3i¼1 εci l i1 ¼ εc1 l 11 þ εc2 l21 þ εc3 l 31. This shows the total amount of
CO2 embodied in each unit of sector 1’s output for final demand, and the terms in this
sum, εci l i1 , show each sector’s CO2 contribution to that total.
Elements in the associated factor multiplier matrix MðεÞ ¼ ^ε 0c L ¼
2
3
εc1 l11 εc1 l 12 εc1 l 13
4 εc2 l21 εc2 l 22 εc2 l 23 5 identify all sources of embodied CO2 from each sector
εc3 l31 εc3 l 32 εc3 l 33
generated by one unit of each sector’s output for final demand. These can be viewed
as measures of consumers’ responsibilities for CO2 pollution.
In the structural path analysis approach, further disaggregation of these responsibilities is accomplished using the power series expansion. From this viewpoint,
sectoral multipliers are given by
mðεÞ ¼ ε0c L ¼ ε0c I þ ε0c A þ ε0c A2 þ ε0c A3 þ https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
(8.73)
8.5 Paths
387
Expressing the multiplier for sector i in this disaggregated way, summing down the ith
column in each term on the right-hand side,
n
n X
n
n X
n X
n
X
X
X
εcj aji þ
εcj ajk aki þ
εcj ajk akl ali þ (8.74)
mðεÞi ¼ εci þ
j¼1
k¼1 j¼1
l¼1 k¼1 j¼1
|fflfflfflfflffl{zfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
j!i
j!k!i
j!k!l!i
This disentangles the various paths by which a factor influences a final outcome, in this
case production in sector i. First-order paths from j to i come from ε0c A and are found
P
in nj¼1 εcj aji ; second-order paths from j to i (via k) come from ε0c A2 and are captured
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
j!i
Pn Pn
in k¼1 j¼1 εcj ajk aki ; third-order paths from j to i (via k and l) come from ε0c A3 ,
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
j!k!i
P P P
captured in nl¼1 nk¼1 nj¼1 εcj ajk akl ali ; and so on.35
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
j!k!l!i
h i
ð3Þ
To illustrate how this procedure works, we examine the elements in A3 ¼ aij for
a two-sector model, using (8.74). Here is A3 (the source of third-order paths) written
out explicitly for a two-sector example:
" 3
#
a11 þa11 a12 a21 þa12 a21 a11 þa12 a22 a21 a211 a12 þa11 a12 a22 þa12 a21 a12 þa12 a222
3
A ¼
a21 a211 þa21 a12 a21 þa22 a21 a11 þa222 a21 a21 a11 a12 þa21 a12 a22 þa22 a21 a12 þa322
Consider third-order paths into sector
1 in P
the context of the series representation in
P P
(8.74). These paths are given by 2l¼1 2k¼1 2j¼1 εcj ajk akl al1. We examine the elements
in this nested summation for column 1 beginning with j ¼ 1:
2 X
2
2
X
X
a1k akl al1 ¼ εcj
ða11 a1l al1 þ a12 a2l al1 Þ
εc1
l¼1 k¼1
l¼1
¼ εcj ða11 a11 a11 þ a12 a21 a11 þ a11 a12 a21 þ a12 a22 a21 Þ
Then, for j ¼ 2,
εc2
2 X
2
X
a2k akl al1 ¼ εc2
l¼1 k¼1
2
X
ða21 a1l al1 þ a22 a2l al1 Þ
l¼1
¼ εc2 ða21 a11 a11 þ a22 a21 a11 þ a21 a12 a21 þ a22 a22 a21 Þ
And so
2 X
2 X
2
X
εcj ajk akl al1 ¼ εc1 a311 þ a12 a21 a11 þ a11 a12 a21 þ a12 a22 a21
l¼1 k¼1 j¼1
þ εc2 a21 a211 þ a21 a12 a21 þ a22 a21 a11 þ a222 a21
Which indeed captures all the third-order paths in mðεÞ1 .
35
These are sometimes referred to as “tiers” – that is first-tier paths, second-tier paths, etc.
https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
388
Decomposition Approaches
It is clear that in a model with more sectors and looking at additional higher-order
terms in the power series – ε0c A4 , ε0c A5 , etc. – there will be an enormous number of
paths. In this small example there are eight distinct third-order paths into sector 1 from
all sectors (here two). This illustrates the general result that there will be nω paths of
the ωth order into any given sector in an n-sector model. (Our two-sector example for
third-order paths into sector i indeed identified 23 paths. This is illustrated further in the
numerical illustration, especially in Table 8.9.) For completeness, for any specific i–j
pair in an n-sector model there are nðω1Þ ω-th order paths, as illustrated by any of the
individual elements in A3 . And, finally, the total number of ω-th order paths in the
entire n sector system is nðωþ1Þ ; in A3 we can distinguish a total of 16 (=24 ) third-order
paths. These rules identify the maximum number of possible paths in each case. In
larger models it is likely that some input coefficients, possibly many, will be very small
or zero, virtually eliminating paths in which those coefficients appear.
This means that in decomposing mðεÞi in this way, there could be
1 þ 2 þ 4 þ 8 þ 16 þ paths in a two-sector
Pmodel. In general, through p-th order
paths in an n-sector model, there are up to pω¼0 nω possible paths, a number that
quickly becomes untenable. Consider a more realistic case with, say, a 50-sector
model. If all coefficients are non-zero, the sequence through fourth-order paths will
be 1 þ 50 þ 2,500 þ 6,250,000. Of course, many elements in higher-order paths will
be very small.36 Thus, in applications a number of so-called pruning techniques have
been used to eliminate certain paths (“branches”) to reduce the calculations to reasonable numbers. Also, some studies have arbitrarily set an upper limit on the number of
terms considered in the power series.37
An early application of this kind of analysis to identify environmentally important
paths in the Australian economy can be found in Lenzen (2002), and Lenzen (2007) is
a more general discussion of SPA in ecosystem studies. See also Itoh (2016) for an
analysis via SPA of important interregional spillovers in the Japanese economy.
Numerical Illustration We illustrate with the two-sector model from
Chapter 2 (Table 2.3). Suppose that the production of $1,000 worth of output from
sector 1 was accompanied by the generation of 50 units of pollution (e.g., pounds of
toxic sludge). Similarly, assume that 80 pounds of this pollutant accompanied the
production of the $2,000 output from sector 2. Thus, the pollution generation coefficients are found as
1=1,000
0
¼ ½ :05 :04 ε0c ¼ ε0 x^1 ¼ ½ 50 80 0
1=2,000
36
37
In Section 2.4 we saw the decline in the elements in higher powers of a very simple two-sector model.
Peters and Hertwich (2006) stop at the A7 term in a study with 49 sectors. They also explain their pruning
algorithm in some detail in an appendix, making note of the “tree” structure of the data. See also Lenzen (2007)
for mention and use of tree pruning techniques.
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389
8.5 Paths
600
(again, from
Pollution generated by the new final demand, f new ¼
1,500
Chapter 2), is thus
1:2541 :3300
600
1,247:52
ε0c Lf new ¼ ½ :05 :04 ¼ ½ :05 :04 :2640 1:1221 1,500
1,841:58
¼ 136:039
37:623 24:75
:05 0
1:2541 :3300
600
new
¼
¼
Using ^ε c Lf
6:336 67:326
0 :04
:2640 1:1221 1500
we see that 62.376 units were generated by sector 1 production and 73.663 came from
sector 2 production.38
We use (8.73) and (8.74) for a specific illustration, which means that all terms on the
right-hand sides are post-multiplied by our new final demand vector. That is,
(8.75)
mðεÞf new ¼ ε0c Lf new ¼ ε0c þ ε0c A þ ε0c A2 þ ε0c A3 þ f new
and, for sector i’s contribution to pollution,
2
3
6
7
n
n X
n
n X
n X
n
X
X
X
6
7 new
6εci þ
7f
¼
ε
a
þ
ε
a
a
þ
ε
a
a
a
þ
mðεÞi f new
cj ji
cj jk ki
cj jk kl li
i
6
7 i
4
5
j¼1
k¼1 j¼1
l¼1 k¼1 j¼1
|fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
j!i
j!k!i
j!k!l!i
(8.76)
Examining these right-hand
side
elements
in turn:
:05
0
600
30
¼ ð0:05Þð600Þ ¼ 30
1. ^ε 0c f new ¼
¼
. Explicitly, εc1 f new
1
0 :04 1,500
60
¼ ð0:04Þð1, 500Þ ¼ 60. These direct effects are sometimes referred to as
and εc2 f new
2
zeroth-order paths.
23:25
:05 0
a11 a12
600
new
0
. The paths showing
2. ^ε c Af
¼
¼
7:8
0 :04 a21 a22 1,500
responsibility for pollution by sectors 1 and 2, respectively, are
new
εc1 a11 f new
¼ ð0:05Þð0:15Þð600Þ þ ð0:05Þð0:25Þð1, 500Þ ¼ 23:25
1 þ εc1 a12 f 2
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
4:5
18:75
new
εc2 a21 f new
¼ ð0:04Þð0:2Þð600Þ þ ð0:04Þð0:05Þð1, 500Þ ¼ 7:8
1 þ εc2 a22 f 2
38
|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
4:8
3
Here, as elsewhere throughout this text, figures have been rounded from calculations made with more digits
to the right of the decimal point. They may not match exactly the results from using the printed figures
(e.g., for L).
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390
Decomposition Approaches
Table 8.7 First-order paths
Sector 1
Sector 2
23.25 (17.09)
7.8 (5.73)
Sector 1
a11
4.5 (3.31)
a21
4.8 (3.53)
Sector 2
a12
18.75 (13.78)
a22
3 (2.21)
Table 8.8 Second-order paths
Sector 1
Sector 2
Sector 1
Sector 2
Sector 1
Sector 2
5.925 (4.36)
4.11 (3.02)
a211 þ a12 a21
2.175 (1.60)
a21 a11 þ a22 a21
.96 (0.71)
a211
.675 (0.50)
a21 a11
.72 (0.53)
a12 a21
1.5 (1.10)
a22 a21
.24 (0.18)
a11 a12 þ a12 a22
3.75 (2.76)
a21 a12 þ a222
3.15 (2.31)
a11 a12
2.8125 (2.07)
a21 a12
3 (2.21)
a12 a22
.9375 (0.69)
a222
.15 (0.11)
It will be useful to arrange this information in a table, especially because results get
more complicated for higher-order paths. Table 8.7 presents specifics for each possible
path (here two) for each sector (here two). The figures in parentheses are percentages
of the total pollution created (136.039 units). (In this and subsequent tables, all terms
containing aij ’s are understood to be multiplied by an appropriate εci and f j ; they are
omitted from the tables to reduce clutter and to emphasize the elements from
A, A2 , A3 that comprise each of the paths.)
2
:05 0
600
a11 þ a12 a21 a11 a12 þ a12 a22
. The results
3. ^ε 0c A2 f new ¼
0 :04 a21 a11 þ a22 a21 a21 a12 þ a222
1,500
for second-order paths appear in Table 8.8. There are now four paths for each sector.
This and subsequent tables now have an additional row to break out the individual
h i
ð2 Þ
terms in each element in the higher powers of A – for example, in A2 ¼ aij the
ð2 Þ
element a21 ¼ a21 a11 þ a22 a21 , which requires further decomposition into two paths:
a21 a11 and a22 a21 .
:05 0
3 new
0
4. ^ε c A f
¼
0 :04
3
600
a11 þ a11 a12 a21 þ a12 a21 a11 þ a12 a22 a21 a211 a12 þ a11 a12 a22 þ a12 a21 a12 þ a12 a222
a21 a211 þ a21 a12 a21 þ a22 a21 a11 þ a222 a21 a21 a11 a12 þ a21 a12 a22 þ a22 a21 a12 þ a322 1,500
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8.5 Paths
391
There are now eight paths for each sector. Third-order path results are shown in
Table 8.9.
Overall results of this kind of structural path decomposition are often summarized in
a table similar to Table 8.10. In many real-world applications of this approach the
importance of each particular path is often quite small, and often dozens of paths are
identified, each with a relatively small contribution. For example, Acquaye et al.
(2011) describe a study of greenhouse gases in the biodiesel supply chain in a
178-sector, two-region model (the UK and the Rest of the World). The eight most
important paths account for 24 percent of the total effect, and after that each new path
contributed less than one percent. Other representative studies of this type can be found
in Peters and Hertwich (2006) on the importance of imports for household consumption environmental impacts in the Norwegian economy, or Wood and Lenzen (2009),
which includes a structural path analysis of the Australian wood products industry
in 2005.
Any realistic model will contain more sectors and hence a very large number of
possible paths. Interesting insights can often be found in multinational (world) input–
output models, with multiple countries and regions. Here is just one example showing
h i
ð3Þ
for a three-sector model.
how complexity increases; it comes from A3 ¼ aij
ð3Þ
Consider the paths indicated in a31 ; these are the nine ð32 Þ third-order paths for the
specific ij pair connecting sector 3 to sector 1:
ð3 Þ
a31 ¼ ða31 a11 a11 þ a31 a12 a21 þ a31 a13 a31 Þ þ ða32 a21 a11 þ a32 a22 a21 þ a32 a23 a31 Þ
þ ða33 a31 a11 þ a33 a32 a21 þ a33 a33 a31 Þ
Fourth-order paths will identify 27 (33 ) connections between sector 3 and sector 1, etc.
8.5.2 Structural Path Decomposition
A further decomposition within the context of SPA was suggested by Wood and
Lenzen (2009). If one has information on more than one time period, structural path
decomposition offers a procedure for examining temporal changes in individual paths
in a production chain.
Consider the question of sources of changes in CO2 emissions over time. Total
emissions, by sector, are tε ¼ ^ε 0c Lf. A standard additive structural decomposition
(Section 8.2) gives
Δtε ¼ Δ^ε 0c Lf þ ^ε 0c ðΔLÞf þ ^ε 0c LðΔf Þ
(8.77)
0
This breaks out the individual effects of changes in emission intensities Δ^ε c , changes
in technology ðΔLÞ, and changes in final demands ðΔf Þ, in standard additive
decomposition fashion.
At this point the power-series approximation of L is introduced into each of the
terms in (8.77)
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392
https://doi.org/10.1017/9781108676212.009 Published online by Cambridge University Press
Table 8.9 Third-order paths
Sector 1
Sector 2
2.1731 (1.60)
1.1535 (0.85)
a311 þ a11 a12 a21 þ a12 a21 a11 þ a12 a22 a21
.6263 (0.46)
a21 a211 þ a21 a12 a21 þ a22 a21 a11 þ a222 a21
.3960 (0.29)
Sector 1
Sector 2
Sector 1
Sector 2
a311
.1013 (0.07)
a21 a211
.1080 (0.08)
a11 a12 a21
.2250 (0.17)
a21 a12 a21
.2400 (0.18)
a12 a21 a11
.2250 (0.17)
a22 a21 a11
.0360 (0.03)
a211 a12 þ a11 a12 a22 þ a12 a21 a12 þ a12 a222
1.5469(1.14)
a21 a11 a12 þ a21 a12 a22 þ a22 a21 a12 þ a322
.7575 (0.56)
a12 a22 a21
.0750 (0.06)
a222 a21
.0120 (0.01)
a211 a12
.4219 (0.31)
a21 a11 a12
.4500 (0.33)
a11 a12 a22
.1406 (0.10)
a21 a12 a22
.1500 (0.11)
a12 a21 a12
.9375 (0.69)
a22 a21 a12
.1500 (0.11)
a12 a222
.0469 (0.03)
a322
.0075 (0.00)
8.5 Paths
393
Table 8.10 Rank-ordered paths for the numerical illustration
Rank
Path
Order
1
2
3
4
5
6 (tie)
6 (tie)
7
8
0
0
1
1
1
1
2
2
2
Path
εc2 ! f 2
εc1 ! f 1
εc1 a12 ! f 2
εc2 a21 ! f 1
εc1 a11 ! f 1
εc2 a22 ! f 2
εc2 a21 a12 ! f 2
εc1 a11 a12 ! f 2
εc1 a12 a21 ! f 1
Percent of Total
Pollution Captured
Cumulative
Percent of Total
44.11
22.05
13.78
3.53
3.31
2.21
2.21
2.07
1.10
66.16
79.94
83.47
86.78
88.99
91.20
93.27
94.37
Δtε ¼ Δ^ε 0c I þ A þ A2 þ A3 þ f þ ^ε 0c ΔA þ ΔA2 þ ΔA3 þ f
þ ^ε 0c I þ A þ A2 þ A3 þ ðΔf Þ
This is not quite the end of the story, because ΔA2 and ΔA3 (and changes in any higher
powers of A) must themselves be decomposed in the same additive way, namely
ΔA2 ¼ ΔðAAÞ ¼ ðΔAÞA þ AðΔAÞ and ΔA3 ¼ ΔðAAAÞ
¼ ðΔAÞAA þ AðΔAÞA þ AAðΔAÞ
So finally
Δtε ¼ Δ^ε 0c I þ A þ A2 þ A3 þ f
þ ^ε 0c ðΔA þ ðΔAÞA þ AðΔAÞ þ ðΔAÞAA þ AðΔAÞA þ AAðΔAÞ þ Þf
þ ^ε 0c I þ A þ A2 þ A3 þ ðΔf Þ
(8.78)
At this point the relative contributions of each individual supply chain path to CO2
emissions can be identified (albeit with a lot of calculations).
This kind of structural path decomposition has been explored in a number of applied
studies, including CO2 emissions and their changes in important supply chain components
in the Japanese economy over the period 1990–2000 (Oshita, 2012), sources of change in
important pathways in the Finnish economy (Mattila, 2012), and an examination of how
value chain results differ depending on which MRIO database is used – Eora, EXIOBASE,
GTAP, or WIOD (Owen et al., 2016). The availability of MRIO databases makes it possible
to decompose outcomes within and between countries (or regions). For example, a
consumer in the USA who buys an Audi made in Germany which comes with tires made
in the UK bears at least some responsibility for pollution caused in both Germany
(manufacture of the Audi) and the UK (production of the tires). This kind of boundarycrossing decomposition analysis is now actively pursued by many researchers.39
39
Zhang and Lin (2018) provide an overview of the options for production-based versus consumption-based
accounting for emissions in a Chinese multiregional framework. See also Temursho and Wood (2020) for a
review and application of structural path decomposition in a global multiregional context.
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394
8.6
Decomposition Approaches
Summary
In this chapter we have explored several frameworks for decomposing various input–
output indicators into component parts. When data sets are available for more than one
period, structural decomposition analysis (SDA), in its additive or multiplicative form,
provides an approach to disentangling the sources of change in some aspect of an economy.
In its additive form, this technique relates changes in an absolute indicator; for example,
total output change as a function of changes in demand and technology. In its multiplicative
form, it identifies changes in an intensity or ratio indicator; for example, change in energy
use embodied in total output as a function of relative changes in energy-use coefficients,
technology, and final demand. And further layers of decomposition are also possible.
Multiplier matrices can also be decomposed into meaningful structural blocks. These
are found to have interesting relevance, for example in identifying various kinds of spatial
connections in multiregional and multinational models, such as feedbacks and spillovers.
In addition, by invoking the power series version of the Leontief inverse, structural
path analysis (SPA) can help to identify potentially important supply-chain paths
through which transactions travel, from supplying sector(s) to purchasing sector(s).
This has turned out to be particularly relevant in multiregional and multinational
models as an approach for identifying sources of pollution generation or scarce
resource consumption. Finally, with data for more than one period, structural paths
can be decomposed (SPD) to investigate sources of change in each individual path.
In modern-day large (many-sector) input–output models many of these kinds of
decompositions rapidly generate very large sets of results that are generally difficult to
present and interpret without some kind of aggregation (for example, finding averages);
this, as usual, removes much of the fine detail that the input–output model provides.
There are four appendices to this chapter. Online Appendix SA8.1 (summarized
below as Appendix 8.1) explores alternative additive decompositions introduced in
Section 8.2. Online Appendix SA8.2 (summarized below as Appendix 8.2) summarizes some early additive structural decomposition studies. Online Appendix SA8.3
(summarized below as Appendix 8.3) shows the economy-wide equivalence of additive and multiplicative decompositions SDA effects. Finally, Appendix 8.4 (below)
discusses some implications of growth rates on structural decomposition analysis.
Appendix 8.1
Alternative Additive Decompositions of x ¼ LBf
Supplemental Appendix SA8.1, located on the internet web site associated with this text
(http://www.cambridge.org/millerandblair), explores three alternative variations of
additive structural decomposition introduced in Section 8.2 and illustrated in sections
that follow in that chapter.
Appendix 8.2
Additional Early Additive Structural Decomposition Studies
Supplemental Appendix SA8.2, located on the internet web site associated with this
text (http://www.cambridge.org/millerandblair), summarizes seven early structural
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Appendix 8.4
395
decomposition studies applying the basic concepts of additive structural decomposition developed in this chapter. The studies include applications at various level of
sectoral aggregation from 19 to 477 economic sectors.
Appendix 8.3 The Approximate Economy-wide Equivalence of Additive and
Multiplicative SDA Effects
Supplemental Appendix SA8.3, located on the internet web site associated with this
text (http://www.cambridge.org/millerandblair), explores the approximate economywide equivalence of additive and multiplicative structural decompositions. Several
general theorems are derived and illustrated with World Input–Output Database tables
for the years 1995 through 2011 for 40 countries and 35 economic sectors.
Appendix 8.4
A Note on Growth Rates
If the interval between period 0 and period 1 spans several years, researchers are often
more interested in something like an average annual growth rate. For example, let
xt ¼ 100 and xtþ5 ¼ 150, so the total amount of growth over the period is 50 percent.
One obvious approach is to divide by the number of years in the period. Here this
would produce an “average” growth of 10 percent (10 units) per year.
A measure that reflects an average annual growth rate (g) more precisely is
found from the standard growth rate formula for an n-period interval:
1=n
xtþn ¼ xt ð1 þ g Þn which yields g ¼ ðxtþn =xt Þ 1. For our example, we have
g ¼ ð150=100Þ0:2 1 ¼ 0:084472, or 8.4472 percent per year. (Four-decimals were
used for accuracy in the calculations.) The annual progression over the five years from
xt ¼ 100 looks like this:
xtþ1 ¼ 100 þ 100ð0:084472Þ ¼ 100ð1:084472Þ ¼ 108:4472
xtþ2 ¼ 108:4472 þ 108:4472ð0:084472Þ ¼ 100ð1:084472Þ2 ¼ 117:6079
xtþ3 ¼ 117:6079 þ 117:6079ð0:084472Þ ¼ 100ð1:084472Þ3 ¼ 127:5425
xtþ4 ¼ 127:5425 þ 127:5425ð0:084472Þ ¼ 100ð1:084472Þ4 ¼ 138:3162
xtþ5 ¼ 138:3162 þ 138:3162ð0:084472Þ ¼ 100ð1:084472Þ5 ¼ 150
This illustrates the effects of annual compounding; the gain in year 1 is reflected in the
base from which the gain in year 2 is calculated, and so forth. The end result, 150, is of