Advanced Macroeconomics
A Text for Undergraduates
Jesus Fernandez-Villaverde
Department of Economics
University of Pennsylvania
Dirk Krueger
Department of Economics
University of Pennsylvania
January 2012
ii
Contents
1 Introduction
1.1 The Questions . . . . . . . . . . . . . .
1.2 The Answers . . . . . . . . . . . . . .
1.2.1 Long Run Growth . . . . . . .
1.2.2 Business Cycle Fluctuations . .
1.3 The Approach and the Structure of the
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Measurement and Stylized Facts
2 How We Measure Facts
2.1 National Income and Product Accounts . . . . . . . .
2.1.1 Gross Domestic Product and its Components
2.2 International Data Sources . . . . . . . . . . . . . . .
2.3 Micro Data . . . . . . . . . . . . . . . . . . . . . . .
2.4 Interaction between Theory and Data. . . . . . . . .
2.5 Dichotomy Growth and Cycles: A Good Idea? . . . .
2.6 Appendix: A Primer on Filtering . . . . . . . . . . .
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.1 Decomposing the National Defense Budget . .
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3 Basic Growth and Business Cycle Facts
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II
15
The Basic Model
4 Set-Up of the Basic Model
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Basic Assumptions . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Household Preferences . . . . . . . . . . . . . . . . . . . . . .
4.2.3 The Budget Constraint . . . . . . . . . . . . . . . . . . . . . .
4.2.4 The Household Optimization Problem and the Euler Equation
4.2.5 Consumption Levels and the Permanent Income Hypothesis .
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CONTENTS
4.3
4.4
4.5
4.6
4.2.6 The General Case . . . .
4.2.7 The Steady State . . . .
Firms . . . . . . . . . . . . . .
4.3.1 Basic Assumptions . . .
4.3.2 The Firm Problem . . .
Aggregate Resource Constraint
Competitive Equilibrium . . . .
Characterization of Equilibrium
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5 Social Planner and Competitive Equilibrium
5.1 The Social Planner Problem . . . . . . . . . .
5.2 Characterization of Solution . . . . . . . . . .
5.3 The Welfare Theorems . . . . . . . . . . . . .
5.4 Steady State Analysis . . . . . . . . . . . . . .
5.4.1 Characterization of the Steady State .
5.4.2 Golden Rule and Modi…ed Golden Rule
5.5 Dynamic Analysis . . . . . . . . . . . . . . . .
5.5.1 An Example with Analytical Solution .
5.5.2 How to Analyze the General Model . .
5.6 Appendix A: More Rigorous Math . . . . . . .
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67
Economic Growth
6 Motivation and Facts about Economic Growth
6.1 The World Income Distribution in 2010 . . . . . . . . . . . . . . .
6.2 Changes in Income over Time . . . . . . . . . . . . . . . . . . . .
6.2.1 The Very Long Run . . . . . . . . . . . . . . . . . . . . .
6.2.2 Kaldor’s Growth Facts for Industrialized Countries (Time
for a Given Country) . . . . . . . . . . . . . . . . . . . . .
6.2.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Accounting for Income Growth: Standard Growth Accounting . .
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7 The Neoclassical Growth Model
7.1 Aggregate and Per Capita Variables . . . . . . . . . . . . . . . . . . . . .
7.2 The Social Planner’s Problem . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1 Setup of the Model . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.2 Analysis of the Model: The Lagrangian and its First Order Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Balanced Growth Path Analysis . . . . . . . . . . . . . . . . . . . . . . .
7.3.1 Determination of Growth Rates in the BGP . . . . . . . . . . . .
7.3.2 Wages and Interest Rates in the BGP . . . . . . . . . . . . . . . .
7.3.3 Existence of Balanced Growth Path . . . . . . . . . . . . . . . . .
7.3.4 Full Characterization of the Balanced Growth Path . . . . . . . .
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CONTENTS
7.4 Calibration . . . . . . . . . . . . . . . . . . . . . . .
7.5 Comparative Statics of the BGP . . . . . . . . . . . .
7.6 Transitional Dynamics . . . . . . . . . . . . . . . . .
7.6.1 An Example with Analytical Solution . . . . .
7.6.2 Linearizing the Problem . . . . . . . . . . . .
7.7 Appendix A: Details of the Linearization Procedure .
7.7.1 Preliminary Steps . . . . . . . . . . . . . . . .
7.7.2 Carrying Out the Linear Approximation . . .
7.7.3 Determining the Policy Functions . . . . . . .
7.7.4 Selecting and Characterizing the Stable Root
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8 Neoclassical Growth Model: Confronting the Facts
8.1 Using the Neoclassical Production Function: Growth Accounting
8.2 Balanced Growth Path Predictions . . . . . . . . . . . . . . . .
8.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4 Absolute Convergence . . . . . . . . . . . . . . . . . . . . . . .
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9 Endogenous Growth Theory
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 The Simplest Endogenous Growth Model: The AK Model . . . . . . . .
9.2.1 Social Planner Problem . . . . . . . . . . . . . . . . . . . . . . . .
9.2.2 Empirical Predictions of the Model . . . . . . . . . . . . . . . . .
9.3 An Endogenous Growth Model with Human Capital . . . . . . . . . . . .
9.4 Models with Externalities . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.1 Social Planner Problem . . . . . . . . . . . . . . . . . . . . . . . .
9.4.2 Competitive Equilibrium . . . . . . . . . . . . . . . . . . . . . . .
9.4.3 Comparison and Policy Implications . . . . . . . . . . . . . . . .
9.5 Endogenizing Technological Progress: The Romer Model . . . . . . . . .
9.6 Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.7 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.7.1 The Households . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.7.2 Final Good Sector . . . . . . . . . . . . . . . . . . . . . . . . . .
9.7.3 Intermediate Goods Sector . . . . . . . . . . . . . . . . . . . . . .
9.7.4 Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.7.5 Production Function for Ideas . . . . . . . . . . . . . . . . . . . .
9.7.6 Research Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.7.7 Characterizing the Equilibrium . . . . . . . . . . . . . . . . . . .
9.7.8 Balanced Growth Path Analysis . . . . . . . . . . . . . . . . . . .
9.7.9 Market Equilibrium versus Social Planner . . . . . . . . . . . . .
9.8 A More General Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.9 Productive Public Infrastructure in Endogenous Growth Models (Barro’s
1990 model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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CONTENTS
10 Endogenous Growth Models: Confronting the Facts
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10.1 Balanced Growth Path Predictions . . . . . . . . . . . . . . . . . . . . . 171
IV
Business Cycle Analysis
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11 Business Cycle Facts
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12 The Real Business Cycle Model
12.1 The Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.1 Households Preferences . . . . . . . . . . . . . . . . . . . . . . .
12.1.2 Budget Constraint and First Order Conditions of the Household
12.1.3 Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.4 The Problem of the Firm . . . . . . . . . . . . . . . . . . . . . .
12.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3 Solving for the Equilibrium . . . . . . . . . . . . . . . . . . . . . . . .
12.3.1 An Expression for Equilibrium . . . . . . . . . . . . . . . . . . .
12.3.2 Interpreting the Dynamics of the Model . . . . . . . . . . . . .
12.4 Bringing the Model to the Data . . . . . . . . . . . . . . . . . . . . . .
12.4.1 Measuring the Productivity Shock . . . . . . . . . . . . . . . . .
12.4.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.5 Unconditional Second Moments . . . . . . . . . . . . . . . . . . . . . .
12.6 Impulse Response Functions . . . . . . . . . . . . . . . . . . . . . . . .
12.7 On the Interpretation of Productivity Shocks . . . . . . . . . . . . . . .
12.8 Appendix 1: Steady State of the Model . . . . . . . . . . . . . . . . . .
12.9 Appendix 2: Linearization . . . . . . . . . . . . . . . . . . . . . . . . .
12.10Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.10.1 Social Planner and the Real Business Cycle . . . . . . . . . . .
12.10.2 A Real Business Cycle Model with Stochastic Volatility . . . . .
12.10.3 Changing Capacity Utilization . . . . . . . . . . . . . . . . . . .
12.10.4 Investment-Speci…c Technological Change . . . . . . . . . . . .
12.10.5 Rule-of-Thumb Households . . . . . . . . . . . . . . . . . . . . .
12.10.6 Capitalists and Workers . . . . . . . . . . . . . . . . . . . . . .
12.10.7 Home Production . . . . . . . . . . . . . . . . . . . . . . . . . .
12.10.8 Lotteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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13 Extending the Model I: Fiscal Policy
13.1 Introducing a Government . . . . . . . . . . . . . . .
13.1.1 Taxes and Government Budget . . . . . . . .
13.1.2 Household Preferences . . . . . . . . . . . . .
13.1.3 Budget Constraint and First Order Conditions
13.1.4 The Problem of the Firm . . . . . . . . . . . .
13.1.5 Aggregate Resource Constraint . . . . . . . .
13.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . .
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of the Household
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CONTENTS
vii
13.3 Solving for the Equilibrium . . . . . . . . . . . . . . . . . .
13.3.1 An Expression for Equilibrium . . . . . . . . . . . . .
13.3.2 Interpreting the Dynamics of the Model . . . . . . .
13.3.3 A Permanent Increase in the Tax Rate . . . . . . . .
13.3.4 A Temporay Increase in the Tax Rate . . . . . . . . .
13.4 Government De…cits . . . . . . . . . . . . . . . . . . . . . .
13.4.1 Government Debt . . . . . . . . . . . . . . . . . . . .
13.4.2 Intertemporal Budget Constraint of the Government
13.5 A Ricardian World . . . . . . . . . . . . . . . . . . . . . . .
13.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.6.1 Long Run E¤ects of Fiscal Policy . . . . . . . . . . .
13.6.2 A Dynamic La¤er Curve . . . . . . . . . . . . . . . .
13.6.3 Investment-Speci…c Technological Change . . . . . .
13.6.4 Intertemporal Budget Constraint of Household . . . .
13.7 Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . .
13.7.1 Facts about Money in the Long Run . . . . . . . . .
13.7.2 Facts about Money in the Short Run . . . . . . . . .
13.8 Deep Models of Money . . . . . . . . . . . . . . . . . . . . .
13.9 Money in the Utility Function . . . . . . . . . . . . . . . . .
13.10Money in Equilibrium Theory . . . . . . . . . . . . . . . . .
13.11Price and Wages Rigidities . . . . . . . . . . . . . . . . . . .
13.12A Benchmark New Keynesian Model . . . . . . . . . . . . .
13.13Coordination Failures . . . . . . . . . . . . . . . . . . . . . .
V
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Economic Policy
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14 Stabilization Policy
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14.1 Fiscal Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
14.2 Monetary Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
15 Long Run E¤ects of Policy
15.1 Monetary Policy . . . . . .
15.2 Fiscal Policy . . . . . . . .
15.2.1 Taxation . . . . . .
15.2.2 Social Security . .
15.2.3 Social Insurance . .
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16 An Introduction to Optimal Monetary and Fiscal Policy
VI
Miscellaneous Topics
17 Consumption
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271
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viii
CONTENTS
18 Investment
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19 The Labor Market
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20 Asset Pricing
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21 Open Economy Macroeconomics
285
VII
Epilogue
287
22 The History and Future of Macroeconomics
289
22.1 History of growth theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
List of Figures
4.1 Strictly Concave Utility Function and Risk Premium . . . . . . . . . . .
22
5.1 Dynamics of the Capital Stock . . . . . . . . . . . . . . . . . . . . . . . .
66
6.1
72
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 Dynamics of k^t in the Neoclassical Growth Model . . . . . . . . . . . . . 112
13.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
13.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
ix
x
LIST OF FIGURES
List of Tables
7.1 Parameter Values
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
98
xii
LIST OF TABLES
Preface
Of all the ways of acquiring books, writing them oneself is regarded as the most praiseworthy method....Writers are really people who write books not because they are poor, but
because they are dissatis…ed with the books which they could buy but do not like.
Walter Benjamin, Unpacking My Library.
The only teaching worthy of the name is scholarly, not personal; analogies between
teaching and various aspects of show business or guidance counseling are more often
than not excuses for abdicating the task. Scholarship has, in principle, to be eminently
teachable.
Paul de Man, The Resistance to Theory.
xiii
xiv
LIST OF TABLES
Chapter 1
Introduction
1.1
The Questions
1. Growth.
2. Business Cycles.
3. Economic Policy: E¤ects and Optimal Design
1.2
The Answers
Although it is impossible to summarize the analysis in the next chapters in a few paragraphs we can In order to do so it is instructive, even at this early stage, to introduce
one of the main ingredients of the models we will present in this book, the neoclassical
production function. This function relates the total production Yt of goods and services
in an economy (and thus the total income generated from that production) at a given
point in time t to the use of factor input capital Kt and labor Lt ; as well as the prevailing
level of technology Bt :
Yt = Bt Kt L1t
(1.1)
Here the number 2 (0; 1) is a …xed production parameter. We will discuss this production function in much greater detail in chapter (4):
Under the assumption that the relation between output and factor inputs in an
economy can indeed be described accurately by the function in (1:1); we can then ask
what share of output (income) growth in the long run is due to growth in the amount
of capital and labor used, and what share is due to improvements in technology (which
we will term technological progress). Similarly we can ask what share of business cycle
‡uctuations is due to variation in capital and labor inputs over the business cycle, and
what share is due to short-run ‡uctuations in technology. Of course the answers will not
tell us why capital and technology grow in the long run and why labor inputs ‡uctuate
over the business cycle, but it will give us a clear indication on which of the three factors
growth theory and business cycle theory should place its modeling emphasis.
1
2
CHAPTER 1. INTRODUCTION
1.2.1
Long Run Growth
Population growth makes output grow, but now per capita output (income)
Per capita income growth due to growth of capital per capita and technological
progress. Thus study of growth will focus on explaining growth of capital (chapter
XXX) and technological progress
1.2.2
Business Cycle Fluctuations
Not much from capital (its a stock and thus can’t move too much over short horizon)
Smaller part from movements in technology
Most part comes from ‡uctuations in labor input (so in principle we should study
the labor market in this part).
1.3
The Approach and the Structure of the Book
1. Dynamic equilibrium macro. Uni…ed view
2. Growing consensus.
3. Math.
Part I
Measurement and Stylized Facts
3
Chapter 2
How We Measure Facts
Macroeconomics is built around the interface of theory and data. Therefore, a basic
knowledge of existing data is an important pre-requisite for understanding the basic
behavior of the economy and to evaluate the empirical performance of economic models.
Economic data comes in many shapes and forms. For macroeconomics, the most
important data source is national accounting. Thus, we will spend some time learning
about how national accounts are built and what they mean with an special attention
at the case of the United States. Then, we will study how we can compare data across
di¤erent countries. This is important because di¤erences across countries have much to
tell us about theory and about the performance of economic policy. International comparisons are complicated, however, by a number of issues like di¤erences in methodology,
structure of the economy, and availability of consistent data over time. Our next step will
be to introduce some microeconomic data sources that report observations at the level
of individuals, households, and …rms. Over the last years, these micro databases have
become a crucial tool in the evaluation of macroeconomic models. Micro data have their
own set of the advantages and drawbacks. We will close this chapter with a discussion
of the interaction between data and theory. In the same way that is di¢ cult to think
about models without data, we will argue it is challenging to interpret data (or even to
de…ne data) without the help of explicit economic models.
2.1
National Income and Product Accounts
National accounting is the process of measuring the output and income of an economy
(or parts of it like a region or an industry) and its distribution and use among the
agents. Di¤erent countries implement slightly di¤erent national accounts systems. For
example, the states members of the European Union follow the system known as the
European System of Accounts as de…ned in the 1995 update (and often abbreviated as
“1995 ESA”).1 However, most the systems share their fundamental structure of doubleentry books and basic concepts. Since 1953, the United Nations have published an
1
See the link http://circa.europa.eu/irc/dsis/nfaccount/info/data/ESA95/en/titelen.htm.
5
6
CHAPTER 2. HOW WE MEASURE FACTS
international standard system of national accounts called the United Nations System
of National Accounts (“SNA” or “UNSNA”). The most recent version of the SNA
methodology dates from 1993 (with some minor posterior amendments).2 Countries
are suggested to comply with the guidelines of SNA and to deliver information to the
United Nations to facilitate the collection of homogeneous data for their inclusion on the
National Accounts Main Aggregates Database.
In the United States, the national accounts are composed by three major elements:
the National Income and Product Accounts (NIPA) and the Industry Accounts, both
elaborated by the Bureau of Economic Analysis, an agency of the U.S. Department of
Commerce3 and the Flows of Fund Accounts, elaborated by the Federal Reserve Board.4
The National Income and Product Accounts (NIPA) are one of the fundamental
sources of information about the economy. NIPA attempts at answering the questions:
1) What is the size of the U.S. economy and of its di¤erent components?, 2) How is the
output of the economy used for di¤erent purposes? and 3) What is the income of the
U.S. economy and how it is distributed among agents?
2.1.1
Gross Domestic Product and its Components
The building block of NIPA is the concept of Gross Domestic Product (GDP). The GDP
is the market value of …nal goods and services produced by labor and property in the
United States, regardless of nationality. Several points of this de…nition deserve further
discussion.
First, GDP is an aggregate measure of all the …nal goods and services produced by the
economy. Hence, it is the most comprehensive measure of economic activity. However,
we only compute …nal goods and not the intermediate goods used in production. For
instance, to manufacture a car, we require, among other materials, steel. This steel is
not consider a …nal good but an intermediate one.
Second, GDP is based on the idea that the relevant valuation of a good is its market
value and not other criteria as its social worth or its desirability. Consequently, an
expensive cigar contributes more to GDP than a cheap antibiotic pill which may save a
human life. Later, when we discuss the limitations of national accounting concepts, we
will examine this issue in more detail. At this moment, su¢ ce it say that market value
provides us with an operative and relatively noncontroversial valuation criteria while
concepts like social worth or its desirability are open to endless discussion and controversy
(for example, what is the social value of a concert classical music in comparison with a
concert of pop music? Higher? The Same? Lower?).
Third, GDP re‡ects the presumption that the most relevant information is about
the location of the economic activity and not about the nationality status of the persons
involved on it. Hence, a good produced in the U.S. by a German citizen is part of the U.S.
GDP but a good produced by a U.S. citizen in Germany is not. Otherwise, mere legal
2
See the link http://unstats.un.org/unsd/nationalaccount/.
See the link http://www.bea.gov/national/.
4
See the link http://www.federalreserve.gov/releases/Z1/.
3
2.1. NATIONAL INCOME AND PRODUCT ACCOUNTS
7
acts (like the naturalization of an immigrant) would change GDP while nothing tangible
in the economy has changed. Similarly, if we are studying the e¤ects of a change in
income taxes in the U.S., we are probably most interested in the responses of workers in
the U.S. whether or not they are U.S. citizens than in the behavior of an expatriate which
is not a¤ected by the new tax. In the past, much attention was devoted to an alternative
concept, Gross National Product (GNP), that accounts all goods and services produced
by national labor and property. Our argument above suggests that this concept is less
useful for macroeconomics. In practice, for most countries, GNP and GDP are quite
similar. In the U.S., the GNP was 0.7 percent higher in 2007 than GDP, a negligible
di¤erence. Exceptions include countries that own substantial assets in other economies
(like Kuwait, where GNP in 2007 was around 12 percent higher than GNP) or where a
large proportion of assets are owned by foreigners (like Ireland, where GNP in 2007 was
around 15 percent lower than GDP).
There are three alternative ways to compute GDP, which all give the same result,
absent measurement mistakes. First, GDP can be computed by adding up the value
added of all sectors of the economy, where value added (VA) is the di¤erence between
the value of …nal sales and the purchase of intermediate inputs. Thus, this approach to
calculating GDP is called the production approach or value added approach. Second, the
goods and services produced in an economy can be used for consumption or investment
purposes, by domestic households, …rms or the government, or shipped to the rest of
the world. Calculating GDP by summing up its uses is called the expenditure approach.
Finally, the production and sales of goods and services generates income for the owners of
production factors (e.g. wages for workers, pro…ts for business owners etc.). Calculating
GDP by adding up the incomes generated in an economy is called the income approach
to GDP.
We now discuss these three approaches in greater detail and present US data on the
components of GDP following these three approaches
Value Added
we have the production (or value added) approach where GDP is computed as the sum of
all gross production (all goods and services produced in the economy) minus intermediate
good consumption. In our previous example, gross production includes the value of the
car and the value of steel in the car while intermediate consumption is the steel used
in the car. Since the di¤erence between the value of the goods and services sold by an
industry and the value of the goods and services used by the industry is known as the
value added by the industry, the sum of all values added across industries equals GDP.
[Add a table with distribution of value added by sector]
[Add a time series with the shares of value added by coarse sectors, to display sectoral
transformation away from agriculture and towards services]
Remark 1 In this remark we will collect basic mathematical facts about growth rates
that we will exploit repatedly later on. We de…ne the growth rate of a variable Y (say
8
CHAPTER 2. HOW WE MEASURE FACTS
real GDP) from period t to t + 1 as
gY;t+1 =
Yt+1 Yt
Yt
(2.1)
Note that this de…nition implies that
(1 + gY;t+1 )Yt = Yt+1 :
Now take natural logs on both sides of this equation to obtain
log(1 + gY;t+1 ) + log(Yt ) = log(Yt+1 )
log(1 + gY;t+1 ) = log(Yt+1 )
log(Yt ):
As long as gY;t+1 is not too far fom zero (recall that growth rates of GDP are typically in
the order of 0%-5% = 0:01-0:05) we can use the approximation5
gY;t+1
log(1 + gY;t+1 ) = log(Yt+1 )
log(Yt )
and thus the growth rate of a variable is approximately equal to its log-di¤erence from one
period to the next. This approximation will be useful in many applications in the chapters
to come. Particularly interesting is the case where a variable grows at a constant rate,
say g, over time. Suppose at period 0 GDP equals some number Y0 and GDP grows at a
constant rate of g% a year. Then in period t GDP equals
Yt = (1 + g)t Y0
(2.2)
Sometime it is interesting to do the reverse calculation. Suppose you know GDP at time
0 and at time t and want to know at what constant rate GDP must have grown to reach
Yt ; starting from Y0 in t years. We can use the formula (2:2) to solve for g
Yt = (1 + g)t Y0
Yt
(1 + g)t =
Y0
(1 + g) =
Yt
Y0
g =
Yt
Y0
1
t
1
t
1
5
You can either trust us on this, or recognize that a …rst order Taylor series approximation of
log(1 + gY;t+1 ) about gY;t+1 = 0 yields
log(1 + gY;t+1 )
log(1 + 0) + (gY;t+1
= gY;t+1 :
0)
1
1+0
2.1. NATIONAL INCOME AND PRODUCT ACCOUNTS
9
Finally, we might be interested in the following question: Suppose we know the GDP of
a country in period 0 and its growth rate g and we want to know how many time periods
it takes for GDP in this country to double (to triple and so forth). Again we can use the
formula, but this time we solve for t :
Yt = (1 + g)t Y0
Yt
(1 + g)t =
Y0
(2.3)
Now we need a little mathematical fact about logarithms: if a and b are arbitrary positive
numbers, then
log ab = b log(a)
Using this fact and taking (natural) logarithms on both sides of equation (??) yields
log (1 + g)t
= log
t log(1 + g) = log
log
t =
Yt
Y0
Yt
Y0
Yt
Y0
log(1 + g)
Now suppose we want to …nd the number of years it takes for GDP to double, i.e. the t
such that Yt = 2 Y0 or YY0t = 2: We get
t=
log(2)
log(1 + g)
0:7
70
=
g
g in %
So once we know the growth rate of our country, we can answer our question. For
example with a growth rate of g = 1% it takes about 70 years, with a growth rate of
g = 2% it takes about 35 years, with a growth rate of g = 5% it takes about 14 years for
a country to double its GDP.
Expenditure
The second approach is the expenditure approach, we can sum the goods and services
sold to persons, businesses, the government, and the rest of the world. The components
of the expenditures are:
Personal consumption expenditures, which is the total amount of goods bought by
persons. Examples of items in this category include foods and haircuts.
Gross private …xed investment, which is the total additions and replacements to
the stock of …xed private assets. Gross private …xed investment is subdivided in:
10
CHAPTER 2. HOW WE MEASURE FACTS
1. Nonresidential …xed investment, which includes investment in nonresidential
structures, equipment, and software.
2. Residential …xed investment, which includes investment in single-family and
multifamily homes.
Change in private inventories which measures the change in the physical volume
of inventories owned by private businesses.
Government consumption expenditures and gross investment. Among government
consumption we …nd items like police o¢ cers or teachers wages while the construction of a new public school or university is gross investment. Government includes
all levels of the polity (federal, state, and local) as well as government enterprises
(for example, a government owned local utility).
Net exports of goods and services de…ned as exports minus imports. Exports
appear with a positive number since they are part of the uses of GDP while imports
appear with a minus because they are already included in other expenditures of
the economy.
The previous discussion is often represented as the accounting identity:
Y
C + I + IC + G + (X
M)
where Y is GDP, C is personal consumption, I is gross private …xed investment, IC is
inventories changes, G is government consumption expenditures and gross investment,
and X M is net exports (exports X minus imports I). Note that we use the symbol “ ”
to emphasize that we are dealing with an accounting identity and not with an equation
or functional form.
[Add a table with expenditure decomposition]
[Add a …gure with evolution over time, perhaps a separate plot that shows time trend
of (X + M )=Y to show increase in trade, also time trend in trade balance and perhaps
in the current account, to talk about global imbalances -could also mention Gourinchas
and Rey stu¤]
Public Expenditures vs. Public Consumption With only a small simpli…cation,
we can divide the total government expenditure of modern states in three cathegories:
1. Public consumption.
2. Public investment.
3. Transfers.
2.2. INTERNATIONAL DATA SOURCES
11
Public Consumption Public consumption are all the goods and services consumed by the government in the period. The consumption of the goods and services
can be done directly by the government or delivered by the government to the public
without a fee (or a nominal one). These goods can be produced by the government itself
or purchased in the market.
An example of the …rst case public education (with some nuances that we will mention
momentarily). In a public high school, the government (in the use the local authority),
hires teachers and administrative sta¤ that deliver a good that is consumed by the students, who do not pay for the education (or perhaps only a nominal fee). Other examples
of public consumption produced directly by the government is police, …re…ghters, justice system, national defense, etc. In general, these activities are recorded by NIPA by
the cost of producing them excluding the investment in capital goods. For instance, in
public education, we record as public consumption directly produced by the government
the cost is the wages of teachers and other sta¤. However, the cost of a new public
high school building will show up, as we will describe in the next subsection, as public
investment.
An example of public consumption purchased in the market is the electricity that
the high school consumes or the bluebooks used for exams. Other examples of public
consumption purchased in the market include the gas used by the police cars or the sodas
o¤ered by the Courts to those citizens awaiting for jury duty.
Public consumption is di¤erent from the goods and services produced by the public
sector that are sold in the market. For example, in many countries, the railroads are operated by a government corporations, like Amtrak in the U.S.. However, these companies
sell the train tickets in the market for a price (even if this price is di¤erent from what a
privately owned company would charge). Consequently, Amtrak tickets are recorded as
private consumption, not public consumption.
In practice, however, there are cases where the line between public consumption and
goods and services produced by the public sector sold in the market is di¢ cult to draw.
For example, in state universities there is a part of the education service produced for
the market, which corresponds to the tuition paid by students, and a part subsidized by
the state.
Income
The third approach to compute GDP is to
[Add a table with US data on income distribution]
[Add a plot with time trends, perhaps discussing declining labor share, and brie‡y
digressing towards income inequality trends
2.2
International Data Sources
Summers-Heston and other.
12
CHAPTER 2. HOW WE MEASURE FACTS
2.3
Micro Data
PSID, CEX, SCF, ....
2.4
Interaction between Theory and Data.
Data is itself in‡uenced by theory
2.5
Dichotomy Growth and Cycles: A Good Idea?
2.6
Appendix: A Primer on Filtering
2.7
Exercises
2.7.1
Decomposing the National Defense Budget
In the section xxx, we decomposed the budget of a public high school in three components:
1. The wages of teachers and the sta¤ that is counted the cost of a good produced
directly for the government.
2. The cost of electricity, computers, bluebooks, and all other goods and services
purchased by the government from the market.
3. The public investment in capital goods, like buildings, computers, the school bus,
etc.
Component 1. and 2. add to public consumption while component 3. was public
investment.
Can you decompose the budget for national defense (money employed in Armed
Forces, intelligence, etc.) in three similar components?
Chapter 3
Basic Growth and Business Cycle
Facts
I would suggest to scrap this chapter and leave the more elaborate and treated facts to
the corresponding introduction section in the growth, cycles part (DK)
[Not clear we should separate the two. Having one chapter just with conceptual stu¤
is too dry to swallow. It might also be good to have as one part basic facts and basic
model (to parallel the interplay between data and model in the growth and the business
cycle part.
13
14
CHAPTER 3. BASIC GROWTH AND BUSINESS CYCLE FACTS
Part II
The Basic Model
15
Chapter 4
Set-Up of the Basic Model
4.1
Introduction
In this chapter we will develop the basic dynamic macroeconomic model that, with
very simple extensions, we will use to explain both long run economic growth as well
as business cycle ‡uctuations.1 In the model only two types of actors populate the
economy, households and …rms. Households represent all the families in the economy.
They decide how much to consume, how much to save, and how much to work. Firms
represent the production side of the economy. They hire labor and rent capital from
households and use these inputs to manufacture goods. Finally, they sell these goods
back to the households. For the moment, and to keep the analysis simple, we will ignore
the government sector and the relations of the economy with the rest of the world. The
absence of a foreign sector makes our economy what is called is a closed economy. Later
in the book, we will generalize our model by introducing a government and a foreign
sector.
We are interested in studying the evolution of the economy over time. Thus, we need
to specify how time evolves. We will assume that time proceeds in discrete periods.2
Depending on the application of our model, we can think of a period as a month, a
quarter or a year, although other units would be admissible, too. The concrete choice
of a period length depends on the particular question that we are interested in. The
economy lasts for T periods. Note that we allow the case T = 1, in which case the
economy lasts forever. Perhaps counterintuitively, many models are easier to handle
when T = 1 than when T is …nite. We will point out in the text how to take care of
T = 1 whenever a special treatment of this case is needed. A typical time period is
1
Many economists refer to this model as the neoclassical growth model, although the simplest version
of the model we discuss in this part will not permit long-run economic growth. As the next part will
unveal, it is straightforward to incorporate long-run growth into this model.
2
Instead, we could work with continuous time, as a signi…cant share of modern macroeconomics does.
However, the use of continuous time time requires more involved and less familiar mathematical tools
than the ones required to deal with discrete time. In addition, real world data are typically measured
in discrete intervals, so the mapping between theory and data is easier when employing models set in
discrete time.
17
18
CHAPTER 4. SET-UP OF THE BASIC MODEL
denoted by t:
In this chapter, we study only deterministic models, that is, models where there is no
source of uncertainty and where households and …rms know the current state of the world
and can perfectly forecast future values of the quantities and prices in the economy. In
chapter 13, we will relax this assumption by introducing uncertainty and discussing how
it a¤ects our analysis.
The structure of this chapter is a follows. First, we describe in detail the structure and behavior of private households and …rms in our simple economy. Second, we
discuss the aggregate resource constraint of the economy. Third, we specify how …rms
and households interact in competitive markets. We close the chapter by de…ning and
characterizing the outcome of this interaction as a competitive equilibrium.
4.2
4.2.1
Households
Basic Assumptions
Households live for T periods (that is, as long as the economy lasts). If T = 1 households
live forever. This assumption is often convenient and, as shown by Robert Barro in
1974, is justi…ed by thinking about the household as a dynasty of generations linked by
altruistically motivated bequests instead of an single individual.
In this chapter, we assume that all households are completely identical. This assumption violates the most basic observations of households in the data. In the real world,
households di¤er in size (some include several members, like parents and children, some
include only one person living alone), education levels, innate abilities, wealth, preferences, and so on. However, assuming homogeneous households drastically simpli…es the
analysis and provides insightful answers to many of the central questions of interest in
macroeconomics. Furthermore, it is a necessary …rst step to understand how to work
with models with heterogeneous households.3 Finally, one can show, under admittedly
strong assumptions, that even in an economy with heterogenous households, macroeconomic aggregates will evolve as if the economy is populated by a single household that
behaves competitively (that is, the household takes market prices as given and beyond
her control).4 Since all households in our economy are identical we will often use the
term the representative household, as the one household that stands in for (“represents”)
all other households in the economy.
We also assume that there is large number of households in the model, so that each
household perceive herself as small relative to the rest of the economy. This assumption is
3
One of the most active areas of research in macroeconomics is the study of models with heterogeneous
households. Much of this work is computational in nature, and thus not suitable for an undergraduate
text in macroeconomics. The interested reader can …nd an introduction to this topic in Ríos-Rull (1995)
and a progress report in Heathcote et al. (2008).
4
The crucial assumptions required for this result are that …nancial markets allow households to fully
insure against household-speci…c risk and that all households’period utility functions belong to either
the class of Constant Relative Risk Aversion (CRRA), which we will discuss later in this chapter, or the
Constant Absolute Risk Aversion (CARA) class.
4.2. HOUSEHOLDS
19
crucial because it allows us to credibly treat households as behaving competitively. With
competitive behavior, all households in the economy believe that their own actions do not
a¤ect market prices. They rationally think that there are so many other households in the
economy that their individual weight in the population is negligibly small. If households
did not behave competitively, we would need to employ the tools of Game Theory and
model the strategic interactions of households. In other areas of economics, such as
Industrial Organization, modelling this strategic behavior is crucial to interpret market
outcomes. For example, Game Theory helps us to understand industries populated
by a small number of …rms. In comparison, macroeconomists typically believe that
competitive behavior is a plausible assumption for single households in the economy.
After all, there are over 112 million households in the United States, and even the
wealthiest of them, such as the Gates or the Bu¤ett family, command only a small share
of overall income, consumption and wealth in the aggregate economy (Bill Gates’ net
worth is estimated by the Fortune Magazine to be around $59 billions in 2008, which is
only around 0.18 percent of the total wealth of households in the U.S. His consumption
is likely an even smaller share of aggregate consumption, although no reliable data exists
to substantiate this conjecture).
For convenience, we normalize the total number of households to 1: This means that
if we want to study a country with 100 households, we think of each household as having
a size of 0.01. This normalization is completely innocuous; it simply avoids to divide all
aggregate variables by the number of households to derive per-capita values (remember
that all households are identical). With our assumption economy-wide and per-capita
variables coincide.
4.2.2
Household Preferences
In this chapter, we present a version of the general model where households in each
period only have one decision to make, namely how much to consume and how much
to save out of their income. That is, for now, we abstract from the decision of the
household what fraction of her time to work. Technically, this is accomplished by having
the household not valuing leisure in the utility function. Finally, we assume that the
total time available to each household in every period is 1 unit of time. Again, this is
an innocuous normalization: think about the 1 unit of time as representing perhaps 40
hours per week, or 2000 hours per year (40 hours per week times 50 work weeks). As
a direct consequence of our assumptions, households …nd it optimal to work the entire
time they have available.
Let ct denote the household’s consumption at time t and let fct gTt=0 = fc0 ; c1 ; :::; cT g
be the path of consumption from time 0 to period T: We assume that the preferences of
households over di¤erent paths of consumption fct gTt=0 can be represented by a utility
20
CHAPTER 4. SET-UP OF THE BASIC MODEL
function of the form:
U (c0 ; c1 ; : : : ; cT ) = u(c0 ) + u(c1 ) +
T
X
t
=
u(ct )
2
u(c2 ) + : : : +
T
u(cT )
(4.1)
t=0
where
2 (0; 1) is the time discount factor that measures how much consumption
is valued in the next period in comparison with consumption in the current period.
The function u is the period utility function and determines how much satisfaction
consumption ct in period t gives; u(ct ). In contrast, U (c0 ; c1 ; : : : ; cT ) = U (fct g) measures
lifetime utility from the entire consumption stream fct g during a households’life.5 In the
case that T = 1, we assume, without further discussion, that the sum in (4.1) converges
to a …nite number. Given that < 1, this will be the case if u(ct ) does not grow to fast.
The upper bound for of 1 is not only convenient to calculate the in…nite sum of
period utilities. More importantly, it indicates that our consumer is impatient: she
derives less utility from the same consumption level if that consumption occurs later
rather than earlier in life. Although one may envision circumstances where is bigger
than one (and macroeconomists do sometimes study models where is bigger than one,
especially when allowing for the possibility that the household dies in a given period
with positive probability), impatience is an intuitive hypothesis.6
Sometimes, we will …nd it convenient to measure the degree of the household’s impatience by the time discount rate : The time discount rate and the time discount factor
are related via the equation:
1
:
=
1+
Later, we will make more detailed assumptions on the period utility function u.
For the moment, it su¢ ces to assume that is at least twice di¤erentiable and strictly
increasing, u0 (c) > 0 for all c: This assumption implies that more consumption is better
than less. Furthermore, we posit that u is strictly concave, u00 (c) < 0 for all c: With this
assumption, the additional utility derived from an extra unit of consumption is lower the
more consumption one already has. Finally, we assume that the utility function satis…es
the so-called Inada conditions
lim u0 (c) = +1
c!0
lim u0 (c) = 0
c!+1
5
Typically, microeconomists model household preferences over consumption bundles (in our case,
streams of consumption over time) by fairly abstract and general preference orderings. For a preference
ordering to have the speci…c utility representation assumed here (where period utilities are discounted
with factors t and summed up to obtain lifetime utility), we require speci…c assumptions on the
preference ordering that, for simplicity, we do not discuss.
6
Moreover, there are reasons to believe that evolution will select individuals with this type of preferences. See Robson and Samuelson (2007).
4.2. HOUSEHOLDS
21
The …rst Inada condition tells us that, as consumption goes to zero, marginal utility
becomes in…nite. Technically, this condition ensures that households never …nd it optimal
to set consumption ct = 0 because a little bit of consumption will generate an in…nite
additional (“marginal”) utility. The second Inada condition implies that, as consumption
grows larger marginal utility goes to zero. This condition is often used to rule out
consumption paths that grow without bounds and make the lifetime utility function
in…nite. These two conditions are named in honor of the Japanese economist Ken-Ichi
Inada who highlighted their importance.
An example of utility function that satis…es our assumptions and that we will often
use in applications is the logarithmic utility function, u(c) = log(c); where log denotes
the natural logarithm for the rest of this book. The log utility function is particularly
convenient partially because it gives us simple solutions to the household maximization
problem, partly because the solution we will obtain has some very plausible properties.
Interestingly, the log utility function belongs to a larger class of utility function: the
Constant Relative Risk Aversion (CRRA) utility functions.
Remark 2 (CRRA Utility Functions) The most commonly used period utility function in macroeconomics is the so-called Constant Relative Risk Aversion (CRRA) utility
function, which takes the form:
c1
1
(4.2)
u(c) =
1
where > 0 is a parameter that has several important interpretations.
First, we note that although for = 1 the utility function (4.2) is not well-de…ned, we
can still take limits of the function as goes to 1: To do so, and since both the numerator
and denominator go to zero as goes to 1; we can use l’Hopital’s rule to …nd that
c1
!1 1
lim
1
= lim
!1
c1
log(c)
= c1
1
1
log(c) = log(c)
Thus, the log utility function is simply a special case of the CRRA utility function with
= 1: Henceforth, whenever we use the utility function (4.2), we therefore take it as
understood that the case = 1 means the log utility function.
The period CRRA utility function has several desirable features that make it attractive in many applications. First, this utility function satisfy our general assumption
about period utility functions: it is continuous, twice di¤erentiable, the …rst derivative, u0 (c) = c , is positive for positive consumption levels, and the second derivative,
1
u00 (c) =
c
is negative for all c: Moreover, the CRRA utility function satis…es the
Inada conditions.
Our presentation of the CRRA utility function has quietly introduced a concept,
relative risk aversion, that we have not yet de…ned. To remedy this gap, we now explain
risk aversion and a concept, the intertemporal elasticity of substitution (IES), that turns
out to be closely related to relative risk aversion, for the case of the CRRA utility
function.
22
CHAPTER 4. SET-UP OF THE BASIC MODEL
Figure 4.1: Strictly Concave Utility Function and Risk Premium
Even if we have not yet talked about risk, you may recall from your microeconomics
course (or you may know from introspection) that economists typically assume that
households do not like risk. Economists call this phenomenon risk aversion.
Consider a single period. A household is said to be strictly risk averse if she prefers
safe consumption c in that period to a consumption lottery that gives an expected value
of c; but is risky. For example, when presented with a gamble that entails a 50 percent
chance of losing a fraction 1/3 of her current consumption c and a 50 percent chance of
winning the same fraction 1/3, a strictly risk-averse household household strictly prefers
safe consumption c over this gamble.
Given a period utility function u, there is a simple way to check whether the household
is risk averse. We state without proof the fact that a household is risk averse if and only
if she has a strictly concave utility function, that is, if and only if u00 (c) < 0 for all
consumption levels. The de…nition of concavity of u implies that, elaborating on our
previous example, that
0:5 u (4=3 c) + 0:5 u (2=3 c) < u (0:5 4=3 c + 0:5 2=3 c) = u (c)
In words, the expected utility of consumption under the 50/50 lottery described above,
0:5 u (4=3 c) + 0:5 u (2=3 c), is lower than the utility from expected consumption,
0:5 4=3 c + 0:5 2=3 c = c.
Thus checking whether the utility function is strictly concave is su¢ cient to evaluate
whether the household is risk averse. It does not tell us, however, how strongly (in a
quantitative sense) the household dislikes risk The strength of risk aversion is typically
measured by the households’willingness to pay in order to avoid risk. Formally, the risk
4.2. HOUSEHOLDS
23
premium is the amount of consumption the household is willing to give up and still be
indi¤erent between the risky lottery and the safe consumption c
: For the example,
is de…ned by
0:5 u (4=3 c) + 0:5 u (2=3 c) = u (c
)
Figure 4.1 shows the risk premium graphically and suggests that as long as the utility
function is strictly concave, the risk premium is greater than zero.
In celebrated work, Kenneth Arrow (1965) and John Pratt (1965) showed that the
risk premium is, to a …rst approximation, equal to the amount of risk, as measured
in the lottery above by the fraction of consumption lost or gained (1/3 in our example),
times the coe¢ cient of relative risk aversion, de…ned as
RRA(c) =
u00 (c)
c:
u0 (c)
Therefore since the risk premium is a measure of the extent of risk aversion and since it
is proportional to the coe¢ cient of relative risk aversion, RRA(c) itself is an appropriate
measure how risk averse a household is.
Remark 3 (An Alternative Measure of Risk: Absolute Risk Aversion) There is
an alternative, yet related, measure of risk aversion, called the coe¢ cient of absolute risk
aversion. This is given by
u00 (c)
:
ARA (c) =
u0 (c)
If instead of losing or gaining a fraction of current consumption the lottery entails losing
or gaining an absolute amount, the risk premium associated with this lottery is proportional to ARA(c) rather than RRA(c):
For the case of the ARA(c) we can show the Arrow-Pratt result directly. Suppose the
household starts at c and has a 50% chance of losing or winning an amount h: As before
de…ne the risk premium as
u(c
) = 0:5u(c
h) + 0:5u(c + h):
Using a …rst order Taylor series approximation of the left hand side (since c
risk-free, a …rst order approximation is good enough) about = 0 gives
u(c
)
u(c)
is
u0 (c)
whereas a second order Taylor series approximation of the right hand side (since c
is risky, a second order approximation is needed) around h = 0 gives
0:5 [u(c h) + u(c + h)]
0:5 [u(c) + u(h)] + 0:5h(u0 (c))
0:5
+ h2 (u00 (c)) + u00 (c))
2
1
= u(c) + h2 u00 (c)
2
u0 (c))
h
24
CHAPTER 4. SET-UP OF THE BASIC MODEL
Equating the approximation for the left hand side and the right hand side gives
u(c)
and thus
=
u0 (c)
1
u(c) + h2 u00 (c)
2
1
1 2 u00 (c)
h 0
= h2 ARA (c)
2 u (c)
2
where h2 measures the variance (and thus the risk) of the gamble. Thus we have precisely
related the size of the risk premium to the coe¢ cient of absolute risk aversion.
A few remarks about RRA(c) are in order. Since the utility function is strictly
increasing, u0 (c) > 0 and thus the coe¢ cient of relative risk aversion is positive if and
only if u00 (c) < 0; that is, if the utility function is strictly concave, and thus the household
is risk averse.7 Second, RRA(c) typically depends on the level of consumption. How
much we are willing to pay to avoid a risky consumption lottery may plausibly depend
on the level of consumption.
Remark 4 (Why CRRA Utility Functions?) The CRRA utility function derives
its name from the fact that households with this utility function have constant relative
risk aversion. A simple calculation shows that for CRRA utility functions
RRA(c) =
u00 (c)
c=
u0 (c)
(
c
c
1
)
c= ;
that is, for the CRRA utility function, relative risk aversion is constant and independent
of the level of c; and equal to the parameter : Therefore,
serves as a measure of
how risk averse a household is, with higher values of indicating larger risk aversion.
Starting from part IV, we will work with models in which households face risk and where
the interpretation of as risk aversion will be central.
Constant relative risk aversion is an appealing property because, over the last two
centuries, the levels risk premia associated to …nancial assets traded in the U.S. and
7
So why do we divide u00 (c) by u0 (c) if the sign of u00 (c) is enough to determine whether the household
is risk averse? You may remember from microeconomics that any linear transformation
u
e(c) = a + bu(c)
of the utility function u represents the same preferences over consumption lotteries. Here a and b > 0
are arbitrary numbers. Since u and u
~ represent the same preferences, it is desirable that they measure
the extent of risk aversion in the same way. Simple calculations reveal that
u
e00 (c) = bu00 (c)
u
e00 (c)
bu00 (c)
u00 (c)
c =
c= 0 c
0
0
u
e (c)
bu (c)
u (c)
and thus the RRA of both utility functions is the same, due to the fact that in the de…nition we divide
u00 (c) by u0 (c); that is, because we normalize the curvature of u by its slope.
4.2. HOUSEHOLDS
25
the United Kingdom (the two countries with the deepest and most liquid markets) have
‡uctuated around a (roughly) constant mean, despite the roughly constant growth of per
capita income and consumption in both economies. We will see in chapter 20 how asset
prices and their risk premia are related to the utility function of the households. Su¢ ce
it to say at the moment, that only a CRRA utility function can deliver stationary risk
premia when consumption grows steadily.
CRRA utility also has the implication (with appropriate additional assumptions) that
a households’ portfolio choice (what share of her wealth to invest in risky stocks versus
risk free bonds) does not change with the wealth position of the household.
Our previous discussion of risk aversion only used the period utility function u; but
not at all the fact that u is employed in our model to de…ne the lifetime utility function
U (c0 ; c1 ; : : : ; cT ) =
T
X
t
u(ct ):
(4.3)
t=0
Similarly, the de…nitions of concavity and the Inada conditions are properties of the
period utility function exclusively and do not make any reference to time. Now, we
present two properties of the lifetime utility function that are centered around the idea
of time. First, we de…ne the marginal rate of substitution between consumption at any
two arbitrary dates t and t + s as the ratio of the marginal utility of the lifetime utility
function U with respect to consumption ct and consumption ct+s :
M RS(ct+s ; ct ) =
@U (c)
@ct+s
@U (c)
@ct
The function U is said to be homothetic if multiplying both consumption levels by a
…xed number > 0 does not change the marginal rate of substitution. Mathematically:
M RS(ct+s ; ct ) = M RS( ct+s ; ct )
for all > 0 and all c (formally, the function M RS is homogeneous of degree zero).8
Why is this property useful? When we study long-run growth in part III of this book,
8
A function f is said to be homogeneous of degree d if
f ( x) =
for all
which
d
f (x)
> 0: The most important special cases of this de…nition are homogeneity of degree d = 1; for
f ( x) = f (x)
and homogeneity of degree d = 0; for which
f ( x) = f (x):
26
CHAPTER 4. SET-UP OF THE BASIC MODEL
we often want to look at situations in which interest rates (and relative prices in general)
are constant over time while per capita consumption (and other quantities in the model)
grow at positive rates. This can only happen jointly if the lifetime utility function U is
homothetic. Second and related, suppose that the income of a household in all periods of
her life double (say, because we measure income in $ rather than British $), it would be
desirable if consumption in all periods of the households’life doubles, too. Homotheticity
of U guarantees this independence of units of measurement.
Remark 5 (The MRS of CRRA Utility Functions) It is easy to verify that, for
the CRRA utility function u de…ned in (4:2) in conjunction with the lifetime utility
function (4:3); we have
t+s
M RS( ct+s ; ct ) =
t
s
=
ct+s
( ct+s )
= s
ct
( ct )
ct+s
= M RS(ct+s ; ct )
ct
and, hence, it follows directly from the de…nition that lifetime U is homothetic if period
utility u is of the CRRA form.
Homotheticity requires that, if the ratio of consumption at two dates remains the
same, the MRS remains the same, too. It does not explicitly deal with the question by
how much the MRS changes if the consumption ratio between two dates changes. This
entity, named the intertemporal elasticity of substitution (IES), measures how households
feel about consumption changes over time.
Formally, de…ne the IES as the inverse of the percentage change in the MRS, relative
to the percentage change in the ratio of consumption:
ies(ct+1 ; ct ) =
2
6d
6
4
@U (c)
@ct+1
@U (c)
@ct
@U (c)
@ct+1
@U (c)
@ct
,
d
ct+1
ct
ct+1
ct
3
7
7
5
1
=
2
6d
6
4
d
@U (c)
@ct+1
@U (c)
@ct
ct+1
ct
@U (c)
, @ct+1
@U (c)
@ct
ct+1
ct
3
1
7
7
5
Typically the IES, a normal elasticity, is de…ned as the percentage change of the consumption ratio, relative to the percentage change of the MRS. We …nd it easier to think
of its inverse. Since elasticities are often negative, they are typically multiplied by a
minus sign to turn them positive.
Remark 6 (CRRA Utility Functions and the IES) For the CRRA utility function
note that
@U (c)
ct+1
@ct+1
= M RS(ct+1 ; ct ) =
@U (c)
ct
@ct
4.2. HOUSEHOLDS
27
and taking derivatives with respect to
@U (c)
@ct+1
@U (c)
@ct
d
d
ct+1
ct
ct+1
ct
=
yields
1
ct+1
ct
:
Therefore, we can calculate the intertemporal elasticity of substitution as
2
3 1
ies(ct+1 ; ct ) =
6
6
6
4
ct+1
ct
ct+1
ct
ct+1
ct
1
7
7
7
5
=
1
:
Thus, for the CRRA period utility, the intertemporal elasticity of substitution is constant
and equal to 1= . The parameter measures not only attitudes of the household towards
risk, but also its attitudes towards consumption changes over time. The higher is ;
that is, the lower is 1= ; the less willing are households to tolerate consumption changes
over time. Graphically speaking, drawing an indi¤erence curve between ct and ct+1 ; the
smaller is the IES (the larger is ), the more curved the indi¤erence curve becomes.
Macroeconomists are divided about whether the fact that controls both risk aversion
and the IES is a positive or a negative feature of the CRRA utility function. Some
researchers like the parsimony of just one parameter controlling two somewhat related
phenomena, risk aversion and intertemporal substitution. Other macroeconomists, in
comparison, feel that risk aversion and intertemporal substitution are clearly distinctive
concepts; with one governing a household’s attitudes towards consumption changes over
time (IES) and the other governing its attitudes towards consumption changes associated
with di¤erent realizations of risk in a given period. In part IV, we will introduce a class
of utility functions where the tight relation between risk aversion and IES is broken.
For the CRRA period utility function there are three important special cases that we
want to mention
1. (a) If = 0; then the IES is in…nite, consumption in two adjacent periods are
perfect substitutes and households are perfectly willing to shift consumption
over time. The indi¤erence curves are straight lines. A = 0 also constitutes the case where households are risk-neutral: they do not care about risk
and simply maximize the present discounted value of their consumption (with
discount factor t ). The utility function has the linear form:
U (c0 ; c1 ; : : : ; cT ) = c0 + c1 + : : : +
T
cT :
(b) If ! 1; then the IES is zero, households are completely unwilling to substitute consumption over time. The indi¤erence curves become L-shaped with
a kink at the 450 line and the lifetime utility function is of Leontie¤ form:
U (c0 ; c1 ; : : : ; cT ) = minfc0 ; c1 ; : : : ; cT g:
(c) Finally, for
= 1; the log-case discussed above, households have an IES of 1.
28
4.2.3
CHAPTER 4. SET-UP OF THE BASIC MODEL
The Budget Constraint
Now that we know what people like, namely consumption in all periods of their life, we
have to discuss what people can a¤ord to buy. Besides working one unit of time in every
period and earning a wage wt for supplying this labor, households are born with initial
…nancial assets (the initial balance of their bank account, say) a0 : These assets serve
as an initial condition: they are predetermined and not chosen by the household, but
rather taken as given.
Households in every period choose consumption ct and the amount of assets carried
over into the next period, at+1 : The level of assets can be positive, at+1 > 0 (the household
has savings), or negative, at+1 < 0 (the household has debts). The budget constraint in
period t reads as:
ct + at+1 = wt + (1 + rt )at
(4.4)
This equation tells us several things. First, we are taking the consumption good as the
numeraire good (the good in terms of whose price the prices of all remaining goods are
quoted), and thus, we normalize its price to 1: Consequently, assets are real assets, that
is, they pay out in terms of the consumption good, rather than in terms of money (the
term money will be absent in this chapter since we can present the entire model in real
terms). Similarly, wt is the real wage and rt is the real net interest rate. Since, as
explained before, the household behaves competitively, she takes the time path of wages
and interest rates fwt ; rt gTt=0 as given and beyond her control.
Equation (4.4) says that expenditures for consumption plus expenditures for purchases of assets that pay out in period t + 1; at+1 ; have to equal labor income wt plus
the principal plus interest of assets purchased yesterday and coming due today, (1 + rt )at
(capital income that can be positive or negative depending on whether at is positive or
negative). Another way of writing equation (4.4) is:
ct + at+1
at = wt + rt at
(4.5)
with the interpretation that total income, comprised of labor income wt and capital
income rt at ; is spent on consumption ct and saving at+1 at . Saving in turn is nothing
else but the change in the asset position of the household between today and tomorrow.
We allow the household to purchase only one asset, a real risk-free bond with maturity
of one period. Such as asset is also often referred to as a risk-free one period bond. In
our simple model discussed in this chapter, the introduction of other, more complicated
assets would not change consumption and saving choices of households, nor equilibrium
wages and interest rates. For example, even if households are not allowed to trade a bond
with maturity of two periods, they can synthesize that bond by buying a one period bond
and rolling it over to the second period (by buying another one period bond). In chapter
20, we will introduce a larger set of assets into the model and discuss how to derive
the market prices for these assets. In that chapter, we will carefully prove our previous
statement concerning the redundancy of additional assets beyond the one period bond
in our basic model. Also, we will introduce an alternative trading arrangement for goods
and assets.
4.2. HOUSEHOLDS
29
Finally, even if we allow households to borrow, we impose the terminal condition that
they cannot die in debt, or formally, we require aT +1
0 if T < 1. If we allow the
household to die in debt, she would certainly decide to do so. Without this restriction,
the household would run up debt without bounds, increase her consumption while alive
to in…nite amounts, and the household optimization problem would have no solution.
Furthermore, since the household is completely sel…sh and does not care about potential
descendants, knows exactly when she is dying, and likes more consumption better than
less consumption, there is no point in leaving any assets unspent at the end of life.
Therefore, we immediately have that the optimal choice of the household is to set aT +1 =
0.
If the household lives forever, that is, T = 1; the terminal condition for assets is
slightly more complicated. A condition that imposes no borrowing after some …nite time
T (requiring at+1 0 for all periods t T ) would be too restrictive. On the other hand
one has to rule out that a household can run so-called Ponzi schemes.9 A household could
borrow $1; pay back the debt tomorrow by simply borrowing (1 + r1 ) $1; pay back that
debt in period 2 by borrowing (1 + r1 )(1 + r2 ) $1 and so forth. By continuing to roll
over the debt, the household can consume more today without any a reduction of future
consumption and the the household maximization problem would have no solution. In
order to rule out such schemes, it is su¢ cient to assure that household debt does not grow
at a faster rate than the interest rate. Note that for a Ponzi scheme to work it is crucial
that the household lives forever. For …nite life T the terminal condition aT +1 = 0 rules
out such strategies. More precisely, the so-called no Ponzi condition, in our particular
model, takes the form:10
at+1
0
(4.6)
lim t
t!1
i=1 (1 + ri )
that says that, in the limit, the present discounted value of debt must be non-negative.11
By a similar argument as before the household will not …nd it optimal to have assets
with positive present discounted value in the limit and hence any optimal solution to the
household problem will satisfy
at+1
0:
(4.7)
lim t
t!1
i=1 (1 + ri )
This condition is an optimality condition and often termed the transversality condition.
Combining (4:6) and (4:7) result in the appropriate terminal condition for a household
9
The no-Ponzi condition is called that way in honor of Boston business man and criminal Charles
(Carlo) Ponzi who e¤ectively tried to borrow without bounds. His Ponzi scheme eventually exploded,
landing him in jail and the people that entrusted their money with him losing most of their investments.
For further details, see http://en.wikipedia.org/wiki/Charles_Ponzi.
10
The symbol ti=1 (1 + ri ) is a shorthad notation for the product
(1 + r1 ) (1 + r2 ) : : : (1 + rt ):
11
It should be clear that this condition does not require that debt is zero after some …nite period t:
For example, as long as all interest rates are strictly positive and bounded away from zero, ri r > 0;
then a constant debt at+1 = a < 0 for all t; satis…es this condition.
30
CHAPTER 4. SET-UP OF THE BASIC MODEL
that lives forever:
lim
t!1
at+1
t
i=1 (1 + ri )
(4.8)
= 0:
Note that the terminal condition is, strictly speaking, already a condition an optimal
solution has to satisfy, rather than a condition that de…nes the household maximization
problem.12
4.2.4
The Household Optimization Problem and the Euler Equation
Our previous discussion leaves us with the following household maximization problem:
given a time path of wages and interest rates fwt ; rt gTt=0 and given initial assets a0 ; the
household solves:
max
fct ;at+1 gT
t=0
ct + at+1
T
X
t
u(ct )
t=0
subject to
= wt + (1 + rt )at for all t
ct 0 for all t
terminal condition =
aT +1 = 0 if T < 1
= 0 if T = 1
limt!1 t at+1
(1+ri )
i=1
initial condition: a0 given
(4.9)
This is a maximization problem subject to equality and non-negativity constraints. In our
subsequent analysis, we will ignore the non-negativity constraints on consumption, since
for all utility functions u we will use, the Inada conditions ensure that these constraints
are never binding. For the moment, we also ignore the initial and terminal condition for
assets, although we will use them later on.
To derive the necessary (and as long as the household is …nitely lived, T < 1; these
are also the su¢ cient13 ) conditions for an optimal consumption choice, we set up the
Lagrangian function
L=
12
T
X
t=0
t
u(ct ) +
T
X
t
(wt + (1 + rt )at
ct
at+1 )
(4.10)
t=0
The foundations and mathematics of both the no Ponzi condition and the transversality condition
are beyond the scope of this text and we skip the details here. Below, however, we will always assure
take that the solution to the household problem we identify satis…es (4:8). The interested reader is
referred to Ljungqvist and Sargent (2004), chapter 8, for a thorough treatment of this topic.
13
In fact, when T = 1 the crucial role of condition (4:8) is to select between two solutions of the
necessary conditions the true solution of the household maximization problem. Again, interested reader
is referred to graduate textbooks in macroeconomics for the technical details, e.g. Stokey and Lucas
(1989), chapter 4.
4.2. HOUSEHOLDS
31
where t denotes the Lagrange multiplier on the period t budget constraint. Note that
there are as many Lagrange multipliers t as there are budget constraints, namely T + 1:
Furthermore, these Lagrange multipliers have an economic interpretation: t measures
the increase in lifetime utility that a household can realize if the budget constraint is
relaxed by one unit, that is, if the household has one extra unit of wealth (either in the
form of a higher wage or higher assets) to spend at time t: Thus, we will also refer to
t as the shadow value of wealth at time t: Equivalently, t measures the marginal cost
of consumption because an increase of consumption ct by one unit tightens the budget
constraint by one unit, the cost of which is measured by t : We will see below that the
’s are intimately related to market prices that arise in a competitive equilibrium.
Taking …rst order conditions of the Lagrangian (4:10) with respect to ct ; with respect
to ct+1 ; and with respect to at+1 yields
t 0
u (ct )
t
=0
t+1 0
u (ct+1 )
t+1 = 0
t + t+1 (1 + rt+1 ) = 0
Rearranging terms, we get:
t 0
u (ct ) =
t
t+1 0
t
u (ct+1 ) = t+1
= t+1 (1 + rt+1 )
(4.11)
(4.12)
(4.13)
The …rst two equations tell us that the marginal utility of consumption at period t,
u0 (ct ); discounted into present value by t , must be equal to the shadow price of wealth
at time t, which is exactly the Lagrangian multiplier t : In other words, we should
consume in period t until the marginal utility from consumption equals the marginal
cost of consumption, given by t : The second equation has the same interpretation, but
applied to period t + 1:
The third equation describes how the shadow value of wealth, or the marginal cost
of consumption, evolves over time. As long as the real interest rate rt+1 is positive, the
shadow value of wealth decreases between periods t and t + 1: Intuitively, one unit of
wealth today has a bigger value than one unit of wealth tomorrow because the household obtains an interest payment out of saving one unit of wealth between today and
tomorrow. Alternatively, this equation can be written as:
t
= 1 + rt+1
(4.14)
t+1
and states that the relative shadow cost of consumption in period t; relative to that
in period t + 1 is equal to the relative market price of consumption today, relative to
tomorrow, given by the gross real interest rate 1 + rt+1 :
By dividing (4:11) by (4:12); using (4:14) to eliminate t = t+1 and simplifying we
obtain:
u0 (ct ) = (1 + rt+1 )u0 (ct+1 )
(4.15)
32
CHAPTER 4. SET-UP OF THE BASIC MODEL
This equation, known as the intertemporal consumption Euler equation or sometimes
more simply as the Euler equation, is one of the cornerstones of macroeconomics. It
has the interpretation that if the household chooses consumption optimally, she exactly
equates the cost from consuming one small unit less today and saving one more unit,
the loss of t u0 (ct ) utils, to its bene…t. Saving one more unit of consumption today gives
1 + rt+1 more units of consumption tomorrow, and thus (1 + rt+1 ) t+1 u0 (ct+1 ) more utils.
This argument is true when we envision the change in consumption and the corresponding
change in saving to be marginal, that is, so small that its e¤ects on lifetime utility can
be obtained by using derivatives.
The Euler equation will show up in this form, or in slightly di¤erent variants, throughout the entire book, since it is the key implication of the main choice problem of the
household, namely how to allocate consumption optimally over time. Hence, it will be
at the center of our models of long run growth, the business cycle, and asset pricing.
The Euler equation tells us that the main economic forces determining the time path
of consumption are the individual time discount factor that measures how impatient
the household is, and the market return rt+1 a household can obtain from postponing
consumption and saving instead.
Remark 7 (More on the IES) It is interesting to note that:
@U (c)
=
@ct
t 0
u (ct )
that is, the derivative of the life time utility to consumption in period t is just the discounted marginal utility of the period utility at time t: Thus, an alternative way to write
the Euler equation
@U (c)
@ct+1
@U (c)
@ct
M RS(ct+1 ; ct ) =
1
;
1 + rt+1
This expression suggests an alternative interpretation of the Euler equation: the household equates the marginal rate of substitution between consumption tomorrow and today
to its relative price, the inverse of the gross real interest rate 1 + rt+1 : This interpretation is useful because it relates the intertemporal choice problem in this chapter with
the standard static choice problem between two goods in microeconomics, whose solution
is precisely to equate marginal rates of substitution to relative prices. The intimate link
between micro- and macroeconomics is a recurrent theme throughout this book.
With this alternative interpretation of the Euler equation, the intertemporal elasticity
of substitution IES can alternatively be written as (in fact, some authors de…ne the IES
4.2. HOUSEHOLDS
33
that way)
iest (ct+1 ; ct ) =
=
2
d
2
d
4
4
ct+1
ct
ct+1
ct
ct+1
ct
ct+1
ct
2
3,
6d
5 6
4
3 ,2
5
4
d
@U (c)
@ct+1
@U (c)
@ct
@U (c)
@ct+1
@U (c)
@ct
1
1+rt+1
1
1+rt+1
3
7
7
5
3
5:
In this form, we can see how the IES measures the percentage change in the consumption
growth rate in response to a percentage change in the gross real interest rate.
Remark 8 (The Euler Equation with a CRRA Utility Function) For the CRRA
utility function, the Euler equation reads as
ct+1
ct
(1 + rt+1 )
(4.16)
= 1:
Taking logs on both sides and rearranging one obtains
ln(1 + rt+1 ) + log( ) =
or
ln(ct+1 )
ln(ct ) =
1
[ln(ct+1 )
ln( ) +
1
ln(ct )]
ln(1 + rt+1 ):
With data on consumption and the real interest rate one can thus run a simple linear
regression
ln(ct+1 ) ln(ct ) = 0 + 1 ln(1 + rt+1 )
and use the estimate 1 as an estimate for the IES, 1 : Many authors, starting from
Hall (1988), have indeed run this regression to obtain estimates of 1 : Estimates based on
aggregate consumption data tend to …nd low values of 1
0:1; whereas estimates based
on using household level consumption data tend to …nd higher values 1 2 [0:3; 0:8]; and
even higher values for selected subgroups of the population. See Attanasio (1999) for a
survey.
4.2.5
Consumption Levels and the Permanent Income Hypothesis
The Euler equation (4.15) relates consumption ct today to consumption ct+1 tomorrow
and thus determines the time path of consumption, but it does not yet pin down the
level of consumption. To …nd an explicit solution for consumption, we need further work.
34
CHAPTER 4. SET-UP OF THE BASIC MODEL
First, we will combine the period budget constraints into a single intertemporal budget
constraint. Take the budget constraints for the …rst two periods:
c0 + a1 = w0 + (1 + r0 )a0
c1 + a2 = w1 + (1 + r1 )a1
Note that we can write the second budget constraint as:
c1
(1 + r1 )
w1
a2
+
= a1
(1 + r1 ) (1 + r1 )
and plug it into the …rst one:
c0 +
c1
a2
w1
+
= w0 +
+ (1 + r0 )a0
(1 + r1 ) (1 + r1 )
(1 + r1 )
(4.17)
In this equation, we have eliminated the gross receipts (or payments if the household in
debt) from asset holdings a1 ; due to the fact we are expressing all the variables in present
discounted value.
We can repeat our procedure and use the period 3 budget constraint
c2 + a3 = w2 + (1 + r2 )a2
to substitute out a2 in equation (4:17) to obtain
c1
c2
a3
+
+
=
(1 + r1 ) (1 + r1 )(1 + r2 ) (1 + r1 )(1 + r2 )
w2
w1
+
+ (1 + r0 )a0
w0 +
(1 + r1 ) (1 + r1 )(1 + r2 )
c0 +
Iterating, on this procedure, and using the terminal condition
aT +1 = 0 or lim
t!1
at + 1
t
i=1 (1 + ri )
= 0;
we arrive at:
c0 +
|
T
X
t=1
X
ct
wt
=
w
+
+ (1 + r0 )a0
0
t
t
(1
+
r
)
i
i=1 (1 + ri )
i=1
t=1
{z
} |
{z
}
T
P re se nt D isc o u nte d Va lu e
P re se nt D isc o u nte d Va lu e
o f L ife tim e C o n su m p tio n
o f L ife tim e L a b o r In c o m e
Y0 :
|{z}
(4.18)
L ife tim e
In c o m e
This expression tells us that, the present discounted value of lifetime consumption must
be equal to the present discounted value of lifetime labor income (also known as the
households’human capital) plus the gross income from the initial amount of assets a0 .
Note that the value of Y0 does not depend on any choices the household makes, a fact
4.2. HOUSEHOLDS
35
that is due to our current assumption that the household does not value leisure and thus
supplies all her available time to market work.14
In general, the Euler equations, together with the intertemporal budget constraint
can be used to determine the levels of consumption in all periods. However, often this can
only be done with the help of a computer. But if the period utility function is of CRRA
form an analytical solution is available. We present it here for the log utility function
( = 1), leaving the algebraically more burdensome (but conceptually straightforward)
general case as an exercise.
Example 9 If u(c) = log(c) the Euler equation (4:16) becomes
1
ct+1
ct
(1 + rt+1 )
=1
and thus
ct+1 = (1 + rt+1 ) ct :
Moving this equation one period back (by replacing t + 1 by t and t by t
ct = ((1 + rt ) ) ct
1) we obtain:
1
By repeated substitution
ct = ((1 + rt ) ) ct 1
= ((1 + rt ) ) ((1 + rt 1 ) ) ct
= : : : = ti=1 (1 + rt ) t c0
2
Plugging this expression into the intertemporal budget constraint (4:18) yields:
c0 +
T
X
(
t=1
t
t
c0
i=1 (1 + rt ))
t
i=1 (1 + ri )
or
1+
T
X
t=1
t
!
= Y0
c 0 = Y0
14
In the case where T = 1 we need to assume that the sums on the left hand side and the right hand
side of the intertemporal budget constraint are well de…ned. This will be the case if
lim
t!1
ct
t (1 + r )
i
i=1
= lim
t!1
wt
t (1 + r )
i
i=1
=0
This condition is satis…ed if consumption and wages do not grow too fast relative to the interest rate.
Fortunately this will be the case in the competitive equilibrium that we study below.
36
CHAPTER 4. SET-UP OF THE BASIC MODEL
Solving this equation for c0 gives15
1
c0 =
ct =
1
T +1
(1
t
i=1 (1
Y0 if T < 1
) Y0 if T < 1
+ rt )
t
(4.19)
c0 for t > 0
(4.20)
This is, in a nutshell, Milton Friedman’s (1957) celebrated permanent income model of
consumption, often also called the permanent income hypothesis (PIH). When choosing
consumption the household does not only look at current income, as postulated by Keynes
(1936) in his General Theory, but at the whole present discounted value of lifetime income Y0 : Franco Modigliani and Albert Ando (1957) obtained the same result in a setup
that explicitly modeled how income changes over the life cycle, and called it the life cycle
model of consumption. As Friedman they emphasized that consumption decisions are
made taking into account the whole path of income of the household over the life cycle.
The permanent income hypothesis is the starting point of a large empirical and theoretical literature on individual household consumption choices, and their implications for
aggregate consumption and the macro economy as a whole. We will return to this topic
in greater detail in chapter 17.
Remark 10 Equation (4:19) displays another important property, besides the fact that
consumption in all periods depends on lifetime income Y0 ; but not on current income wt
over and above the in‡uence wt has on Y0 : We see that c0 does not depend on the real
interest rate r1 , apart from the in‡uence it has on the present discounted value calculation
of Y0 : This is surprising, given that a change in 1 + r1 is nothing else but a change in the
real relative price of consumption at periods t = 0 and t = 1: From basic microeconomic
theory, we would expect that a change in the relative price 1 + r1 triggers an income and
a substitution e¤ect and thus changes c0 : It is easiest to discuss these e¤ects in a model
with only two periods (and the discussion perfectly carries over to the general model).
For T = 2 and CRRA utility, the two equations characterizing consumption are the Euler
equation (4:16) and the intertemporal budget constraint (4:18), which read as
(1 + r1 )
c0 +
c1
c0
= 1
c1
w1
= w0 +
1 + r1
1 + r1
Y0
Note again that the relative price of consumption in period 1, relative to consumption in
1
period 0, is given by p = 1+r
: An increase in the interest rate r1 has three e¤ects on
1
consumption allocations.
15
Recall the mathematical fact that for any < 1 we have
! (
T +1
T
1
X
if T < 1
t
1
1+
=
:
1
if T = 1
1
t=1
4.2. HOUSEHOLDS
37
1. An increase in r1 (and thus reduction in p) makes period 1 consumption relatively
less expensive compared to period 0 consumption and hence reduces current consumption and increases future consumption. This is the substitution e¤ect.
2. An increase in r1 reduces the price of second period consumption in absolute terms,
thus e¤ectively increasing the real value of income. This is the income e¤ect.
3. Provided that the household has income in the second period16 , an increase in r1
reduces the present value of lifetime income Y0 and hence reduces current consumption. This is the wealth e¤ect or human capital e¤ect.
Solving the two equations for c0 and c1 we obtain (after some tedious algebra):17
c0 =
Y0
1+
1
(1 + r1 )
1
(4.21)
1
Therefore the e¤ect of a change in the interest rate 1 + r1 on current consumption is
given by
@c0
=
@(1 + r1 )
1+
=
1
(1 + r1 )
1
1
1
(1 + r1 )
1+
+
@Y0
@(1+r1 )
1+
1
@
@Y0
@(1+r1 )
(1+r1 )
1
1
Y0
@(1 + r1 )
1
1
1
1
1
(1 + r1 )
1
2
1
2
Y0
(4.22)
1
1+
1
(1+r1 )
1
1
The …rst term is the wealth e¤ect, which is negative (since an increase in r1 reduces Y0 ):
The second term is the combination between income and substitution e¤ect. Whether
the net e¤ect from the combined income and substitution e¤ect is positive exclusively
depends on the sign of 1 1 ; since all other terms in the expression are positive. We
immediately have the following results which again highlight the importance of the IES
for consumption choice (and the special properties of log-utility):
1. If = 1 (the log-case), income and substitution e¤ect exactly cancel out. Absent
wealth e¤ect, the interest rate has no impact on current consumption.
2. If 1 > 1 (high intertemporal elasticity of substitution), then the substitution e¤ect
dominates the income e¤ect and hence current consumption declines as reaction to
an increase in the interest rate (a reduction of price of second period consumption).
16
17
We could imagine the household retiring, in which case Y0 = w0 :
The reader may notice that for = 1 we have
c0 =
as already derived in (4:19):
Y0
1+
=
1
1
2
Y0
38
CHAPTER 4. SET-UP OF THE BASIC MODEL
3. If 1 < 1 (low intertemporal elasticity of substitution), then the positive income
e¤ect dominates the negative substitution e¤ect and absent the wealth e¤ect current
consumption increases.
4.2.6
The General Case
The optimal solution to the household decision problem can be derived analytically
in closed form only under special assumptions (such as CARA or CRRA utility). For a
general utility function, a given sequence of market prices frt ; wt gTt=0 , and an initial value
of assets a0 either mathematical tricks or a computer are required to characterize the
entire dynamics of consumption and asset holdings. Here we discuss the basic structure
of the problem.
Solving the budget constraint for consumption in period t and t + 1 yields:
ct = wt + (1 + rt )at at+1
ct+1 = wt+1 + (1 + rt+1 )at+1
at+2
If we insert these equations for ct and ct+1 into the Euler equation (4:15), we obtain
u0 (wt + (1 + rt )at
at+1 ) = (1 + rt+1 )u0 (wt+1 + (1 + rt+1 )at+1
at+2 )
(4.23)
Remember that the household takes wages and interest rates fwt ; rt gTt=0 as given numbers.
Thus, the only choice variables in this equation are at ; at+1 ; at+2 : Mathematically, this
is a second order di¤erence equation (unfortunately a nonlinear one in general). We
have an initial condition (since the number a0 is exogenously given) and a terminal
t
= 0); so in principle we could explicitly solve
condition (aT +1 = 0 or limt!1 T a(1+r
i)
i=1
this second order di¤erence equation. However, the nonlinear structure of the problem
makes …nding a solution by hand infeasible in most instances. Macroeconomists rely
on two basic approaches to make progress. First, many researchers analyze a linear
approximation of equation (4:23): Linear di¤erence equations are much easier to handle
than nonlinear ones. Alternatively, one can switch on a computer and program an
algorithm (or, more simply, use already existing software) that solves this di¤erence
equation boundary problem. We will come back to both these approaches in part IV of
the book.
4.2.7
The Steady State
Solving for the consumption path is considerably easier if we restrict attention to a
situation where all economic variables of interest are constant over time. We now de…ne
such a situation formally:
De…nition 11 A Steady State is a time path for all economic variables (here consumption, interest rates and wages fct ; at+1 ; rt ; wt gTt=0 ) that is constant over time.
4.3. FIRMS
39
Focusing on steady states is justi…ed if we have reasons to believe that the economy,
in the long run, will reach a steady state and if we are mainly concerned with the long-run
behavior of the economy.
In the current context, a steady state requires that ct = ct+1 = c and rt+1 = r: From
the previous equation (4:15); we obtain that in the steady state:
u0 (c) = (1 + r)u0 (c)
and thus the steady state interest rate satis…es
(4.24)
1 = (1 + r):
Now remember that we de…ned the time discount rate by = 1+1 : We immediately
deduce from (4:24) that in a steady state the subjective time discount rate necessarily
has to equal the market interest rate, = r; because only if this condition is satis…ed
will households …nd it optimal to choose a consumption path that is constant over time.
When = r the households’ impatience, as measured by ; is exactly o¤set by the
return r the household gets for saving.18 The constant level of consumption is again
given by (4:19). Note the di¤erence to the analysis in section 4.2.5. There we assumed
log-utility to derive an explicit solution for arbitrary interest rates. In this section we
restricted attention to steady states, derived that rt = for all period and deduced
the constant optimal consumption level, without making speci…c assumptions about the
utility function.
We will encounter a steady state and its generalization, a balanced growth path in
which all economic variables grow at constant rates (rather than being constant), in part
III of the book, when we discuss economic growth.
4.3
4.3.1
Firms
Basic Assumptions
We now turn our attention to the study our second type of agents: …rms. As with
households, we assume that all …rms have access to the same production technology and
normalize the number of …rms to 1: Therefore we envision a representative …rm that
stands in for the entire production sector in the economy.
We continue to assume that …rms believe to be so small that their hiring decisions
do not a¤ect the wages they have to pay their workers and the rental price they have
to pay for their machines. The representative …rm in each period produces the single
…nal good. This good can be used by households both for current consumption and
for investment into the stock of capital (assets) they own. Think of the …nal good as
potatoes or grain: you can eat it or put into the ground so that it yields production
tomorrow. Let yt denote the output of the …rms’…nal good. In period t; the …rm hires
18
The reader should work out how the time path of consumption looks like if
> r or if
< r:
40
CHAPTER 4. SET-UP OF THE BASIC MODEL
a number of workers lt and an amount of physical capital (machines, buildings) kt to
produce this …nal good. The production technology that transforms the inputs lt and kt
into the …nal good is described by a neoclassical Cobb-Douglas production function f :
yt = f (kt ; lt ) = kt (At lt )1
where 2 (0; 1) is a parameter and At is the current level of technology.19 We comment
on both entities further below.
The Cobb-Douglas production function has several important and useful properties.
First, it displays constant returns to scale . That is, if we multiply each input by a
constant > 0, total output also increases by a factor of (in the terminology introduced
in section 2, the production function is homogeneous of degree 1). To see this, note that:
( kt ) (At lt )1
= kt (At lt )1
= yt :
Constant returns to scale formalize the intuitive idea that if one has a blueprint
technology for production (a factory say), one can simply build exactly the same factory
in a di¤erent location, use the same machines and same number of workers and double
output.20 In comparison, decreasing returns to scale (production multiplies by less than
when we multiply all inputs by ) rule out exactly the exact replication of an existing
production blueprint.21 Lastly, increasing returns to scale (output more than multiplies
by when inputs increase by a factor of ) enjoy some empirical support on the aggregate
level, but pose some theoretical di¢ culties that we would like to avoid at this point of
the book (notably, a competitive equilibrium may fail to exist). We will return to them
when we talk about economic growth.
Second, both inputs are essential in production. If either the capital input kt or the
labor input lt input is zero, output is also equal to zero.
To explain the next property of the Cobb-Douglas production function, we need
to introduce the concept of marginal products. The marginal product of capital is the
additional production the …rm gets out of using an extra unit of capital, holding the labor
input constant. Mathematically, the marginal product of capital is given by the partial
derivative of the production function with respect to capital. The marginal product of
labor is de…ned analogously. For our production function:
fk = kt
fl = (1
19
1
(At lt )1 > 0
)At kt (At lt ) > 0
The Cobb-Douglas production function takes its name from Charles Cobb, a mathematician and
Paul Douglas, an economist, both at the University of Chicago, who …rst proposed it in 1928. Later in
life, after joining the U.S. Marine Corps at age 50 and …ghting in the Paci…c War with high distinction,
Paul Douglas served as a Democratic U.S. Senator from Illinois from 1949 to 1967. In the Senate,
Douglas became a champion of Civil Rights for all Americans. His autobiography, In the Fullness of
Time: The Memoirs of Paul H. Douglas (1972) is an excellent read.
20
This example implicitly assumes that the …rm has free access to other inputs not discussed here,
like land or managerial skills.
21
Decreasing returns to scale may be plausible if a scarce factor of production (land, managerial
talent) is present. Industrial organization which studies market structure and production in particular
industries, often studies …rms with decreasing returns to scale.
4.3. FIRMS
41
but decreasing (additional inputs produce additional output but at a falling rate). Fourth,
the marginal products diverge to in…nity as the corresponding production factors go to
zero
lim fk = lim fl = 1
kt !0
lt !0
while they go to zero as inputs grow to in…nity:
lim fk = lim fl = 0:
kt !1
lt !1
This last condition (sometimes together with the essentiality of inputs) is known as the
Inada condition for production functions, and is the natural counterpart to the Inada
condition on the utility function we introduced earlier.
In the production function, At is a technology parameter that determines, for given
inputs, how much output is being produced. Macroeconomists think about At as embodying not only the pure scienti…c and technological knowledge (for instance, how to
build a car or how to make a hard drive to store data) but also all the institutions and
arrangements that increase our ability to produce goods and services. For example, we
can think about the development of double-book accounting, the appearance of the limited liability corporation, or the just-in-time inventory control techniques as increments
in At :22
Note also that in yt = kt (At lt )1 ; At pre-multiplies lt ; and then whole product is
raised to the power 1
. This form of technology is called labor-augmenting, because a
higher At is equivalent to using higher labor input. Later in the book, when we talk about
economic growth, we will explain why we specify technology with this labor-augmenting
form.23
In di¤erent parts of the book we will make di¤erent assumptions about At : In this part
we assume that At = 1, because it simpli…es the initial analysis of the model. When we
turn to the analysis of long run economic growth, we will often assume that At = (1+g)t ;
which means that production technologies becomes more productive over time and that
the growth rate of technological progress is constant and given by g: Finally, when we
study business cycles, we will introduce random ‡uctuations in the level of technology
At which will then become a (in fact crucial) source of ‡uctuations in aggregate output
(GDP) in the model.
The parameter measures the importance of the capital input in production. Formally, is the elasticity of output with respect to the capital input and thus measures
the percentage increase in yt in response to an increase in kt by a one percent. Take logs
of the production function to obtain:
log yt = log At +
22
log kt + (1
) log lt
Alfred Chandler’s classical work, The Visible Hand (1977), describes how many of the modern
managerial techniques that are now commonly used are relatively new innovations from the second half
of the 19th century.
23
In fact, for the Cobb Douglas production function which production factor is premultiplied by At is
inconsequential. We can always rewrite the production function to make technology labor-augmenting.
42
CHAPTER 4. SET-UP OF THE BASIC MODEL
and thus24
d log(yt )
= :
d log(kt )
Similarly, the elasticity of output with respect to labor is 1
d log(yt )
=1
d log(lt )
because:
:
Momentarily, we will link the parameter to the capital share of income in the economy.
Finally, we assume that a constant share of the capital stock kt used in production
wears out, a process called depreciation.
Remark 12 (CES Production Functions) The Cobb-Douglas production function is
a special case of a more general case of production functions called Constant Elasticity
of Substitution (CES) production functions. The elasticity of substitution between the
two production factors kt and lt is de…ned as the percentage change in the ratio of kt =lt
relative to the percentage change in the marginal rate of substitution fk (kt ; lt )=fl (kt ; lt );
or25
(kt ; lt ) =
d(kt =lt )
kt =lt
d[fk (kt ;lt )=fl (kt ;lt )]
fk (kt ;lt )=fl (kt ;lt )
=
fk (kt ;lt )=fl (kt ;lt )
kt =lt
d[fk (kt ;lt )=fl (kt ;lt )]
d(kt =lt )
:
We will see in the next subsection that it is optimal for the …rm to hire capital and labor
up to the point where their marginal products are equated to their corresponding factor
prices. Therefore, we can replace fk by the rental rate of capital t and fl by the wage
rate wt in the expression above. Viewed this way, the elasticity of substitution is simply
given by
d(kt =lt )
kt =lt
d[ t =wt ]
rt =wt
(kt ; lt ) =
and measures by how many percents the …rm changes its relative inputs kt =lt as the
relative price for its inputs t =wt changes by one percent. Since an increase in t =wt is
24
In general, the elasticity of output yt with respect to the capital input kt (like any other elasticity)
is given by
dy
dy=y
"y;k =
= dk
:
k
dk=k
y
But note that
d log y
=
d log k
d log y
dy
d log k
dk
dy
dk
=
1
y
1
k
dy
dk
=
dy
dk
k
y
= "y;k
and thus the derivative of the log of the varible y with respect to the log of the variable k gives the
elasticity. Note that this is a very general trick to compute the elasticity.
25
Recall the concept of the intertemporal elasticity of substitution (IES) de…ned in the previous
subsection. Instead of measuring the willingness of households to substitute between consumption at
two points of time, the elasticity de…ned here measures the technological ability of the …rm to substitute
the two factors of production at a given point of time.
4.3. FIRMS
43
typically associated with a fall in kt =lt economists multiply the elasticity by 1 to express
it as a positive number. Now consider the production function of the form
h
i 1
1
1
f (kt ; lt ) =
(kt )
+ (1
) (At lt )
where
0 is an additional parameter. We call this production function a CES production function for reasons that will become clear immediately. Tedious but straightforward
calculations show that the marginal products are given by:
h
i 1 1
1
1
1
fk (kt ; lt ) =
(kt )
+ (1
) (At lt )
(kt )
h
i 1 1
1
1
1
1
fl (kt ; lt ) =
(kt )
+ (1
) (At lt )
(1
) (At )
(lt )
and thus
fk (kt ; lt )=fl (kt ; lt ) =
(1
) (At )
1
kt
lt
1
(1
1
) (At )
1
kt
lt
1
d [fk (kt ; lt )=fl (kt ; lt )]
=
d (kt =lt )
1
:
Putting all these results together, we …nd that for the CES production function the elasticity of substitution is given by
(kt ; lt ) =
fk (kt ;lt )=fl (kt ;lt )
kt =lt
d[fk (kt ;lt )=fl (kt ;lt )]
d(kt =lt )
(1
)(At )
1
kt
lt
1
1
kt =lt
=
1
1
(1
)(At )
1
kt
lt
1
= :
That is, for the CES production function the elasticity of substitution between the two
production factors (kt ; lt ) is constant and equal to the parameter ; independent of the
amount of inputs (kt ; lt ) being used, and therefore also independent of the scale of operation of the …rm. There are three important special cases of CES production functions:26
1. If
= 0; then the production function takes the Leontie¤ form
f (kt ; lt ) = minfkt ; At lt g
Even large changes in the relative price of machines versus labor do not result
in changes in the proportion of labor and machines being used in this case. An
example is the technology to drive a bus: we need a bus driver for each bus, and no
matter what the relative price for buses and drivers is, the bus company will need
(exactly) one driver and one bus to provide bus rides.
26
The derivation of the fact that these are indeed special cases is not trivial mathematically, and we
proceed without proving it. The interested reader may consult the discussion in Barro and Sala-i-Martin
(2003).
44
CHAPTER 4. SET-UP OF THE BASIC MODEL
2. If
= 1; then the production function takes the form
f (kt ; lt ) = kt + (1
)At lt ;
and capital and labor are perfect substitutes. Even small changes in relative input
prices lead to dramatic adjustments of the input mix used by the …rm.
3. Finally, if
= 1; the production function becomes Cobb-Douglas
f (kt ; lt ) = kt (At lt )1
In other words, the Cobb-Douglas production function is a special case of a CES
production function with elasticity of substitution of 1. Therefore the Cobb-Douglas
function implies that …rms adjust relative inputs by one percent whenever relative
input prices change by one percent.
4.3.2
The Firm Problem
The …rm hires workers at a wage wt per unit of time. For simplicity, we also assume
that the …rm rents the capital it uses in the production process from the households,
rather than owning the capital stock itself. This turns out to be an inconsequential
assumption (all the quantities and prices in an equilibrium of the economy would be the
same if …rms owned capital themselves) and yet it makes our life easier when stating the
…rm’s problem. Thus, from now on, we identify the asset the household saves in with
the physical capital stock of the economy. Hence, the return to the asset the households
own, rt ; and to physical capital must be the same.
We denoted the rental price per unit of capital by t above. Note that because of
depreciation, whenever a household rents one machine to the …rm, she receives t
as
e¤ective rental payment (since a fraction of the machine disappears in the production
process and thus is not returned back to the household). The rental rate of capital and
the real interest rate therefore satisfy the relation:
rt =
t
:
From the previous two paragraphs, and in comparison with the household problem,
we observe that the …rm problem is static. In every period the …rm rents inputs (kt ; lt )
and produces output yt that is immediately sold on the market. The …rm takes wages
and rental rates of capital as given and maximizes period by period pro…ts:
max fyt
lt ;kt
wt lt
subject to
yt = kt (At lt )1
kt ; lt 0
t kt g
4.3. FIRMS
45
We explained before that, given the form of the production function, both inputs
were essential. Therefore, we can safely ignore the nonnegativity constraints on inputs
in any solution of the …rms’problem that implies positive production. The maximization
problem therefore simpli…es to an unconstrained problem:
max kt (At lt )1
wt lt
lt ;kt
t kt
with associated …rst order conditions:
wt = (1
kt
At lt
=
t
)At
kt
At lt
(4.25)
1
(4.26)
:
These two equations state that it is optimal for the …rm to set its inputs such that it
equates the wage rate (which it takes as exogenously given) to the marginal product of
labor. Likewise, the rental rate of capital is equated to the marginal product of capital.
Dividing both …rst order conditions by each other we obtain:
wt
=
1
At
t
kt
At lt
or
kt
= At
lt
1
1 wt
At t
1
:
Consequently the capital/labor ratio is a function of the ratio of input prices. When wages
are relatively high in comparison with the rental price of capital, the …rm will produce
more capital intensive and it will produce labor intensive when wages are relatively low.
Furthermore, the ratio of used capital to labor kt =lt is the same for all …rms in the
economy since wages wt and rental rates t are the same for all …rms (they are market
prices that all …rms take as given) and the level of technology At is assumed to be the
same for all …rms.
Remark 13 (Indeterminacy of the Number of Firms) The fact that all …rms in
equilibrium have to have the same kt =lt also implies that it is completely inconsequential
whether we assume a single representative …rm or a large number of small …rms, as long
as all …rms behave competitively.
Suppose the economy consists of …rms with only one worker lt = 1 each. Since each
…rm faces the same wages and rental rates, they each have the same kt =lt : The total
number of such small …rms is given by lt ; the number of workers. Total output of these
…rms together equals
yt = lt
kt
lt
= kt (At lt )1
(At 1)1
46
CHAPTER 4. SET-UP OF THE BASIC MODEL
identical to that of the representative …rm that uses all lt workers in the economy and
the entire capital stock kt : The same argument applies if there are many …rms of unequal
size.
In this sense the only entity that the model pins down is the capital-labor ratio of
…rms. How many …rms there are and how much output they produce, given equilibrium
prices, is completely indeterminate, and without restrictions we can focus on the situation
of a single, representative …rm behaving competitively.27
Competitive behavior and a production function with constant returns to scale has
strong implications for pro…ts of …rms. These are given by:
t
= kt (At lt )1
1
= kt (At lt )
wt lt
t kt
(1
)At
kt
At lt
lt
kt
At lt
1
kt
= 0:
and thus equal zero. In fact, knowing this result beforehand we never included pro…ts
of …rms in the household budget constraint above. Likewise, we never even discussed
who owns the …rms in the economy. Given that their pro…ts happen to equal zero,
this is irrelevant. Note that the zero pro…t result does not hinge on the Cobb-Douglas
production function: any constant returns to scale production function, together with
competitive, price taking behavior by …rms will deliver this result.28
When interpreting this result, we need to remember, however, that economists de…ne
pro…ts after payments to capital. Using standard accounting practices, in the business
world the pro…t of a …rm is de…ned before payments to equity (the capital owned by the
…rm). Since we are assuming that the equity of the …rm is zero (all capital is borrowed),
both concepts of pro…ts are consistent if we are careful with their interpretation. We
only need to think about the accounting pro…ts of the business world as payments to the
capital “borrowed”by the …rm from shareholders.
For the Cobb-Douglas production function, it is also straightforward to compute the
labor share and the capital share. De…ne the labor share as that fraction of output (equal
to income in the economy) that is paid as labor income, that is, the ratio of total labor
income wt lt (the product of the wage and the amount of labor used in production) to
output yt :
wt lt
labor share =
yt
27
Because of the indeterminacy of the level of inputs being hired and output being produced by a
given …rm, given prices wt ; t ; some authors prefer to state the problem of a …rm as minimizing cost
wt lt + t kt ; subject to being able to produce at least a given level of output yt : The optimality conditions
arising from this cost minimization problem are identical to the ones stated in (12:4) and (12:5).
28
This result is consequence of the Euler’s Theorem that states that for any function f (kt ; lt ) = yt
that is homogeneous of degree one we have:
yt = fk kt + fl lt
Since price taking behavior implies that fk =
t
and fl = wt , the result follows.
4.4. AGGREGATE RESOURCE CONSTRAINT
47
A simple calculation shows that:
(1
)At Akttlt
wt lt
=
yt
kt (At lt )1
lt
=1
:
Similarly, capital income is given by the product of the rental rate of capital and the
total amount of capital used in production, and thus the capital share equals yt kt t = :
Thus, if the production technology is given by a Cobb-Douglas production function, the
labor share and capital share are constant over time and pinned down by the parameter
: Since we can measure capital and labor shares in the data, this relationship will be
helpful in choosing a concrete number for the parameter when applying the model to
explain the real world.
4.4
Aggregate Resource Constraint
With our (as argued above, completely innocuous) assumption of a single representative
…rm we can think of total output being produced as given by yt = kt (At lt )1 ; the
output of the representative …rm. The entities kt and lt stand now for the total amount
of capital and labor employed in the economy.
Output can only be used for two purposes: for consumption and for private investment. Remember that, for the moment, we are ignoring the government and foreign
sector of the economy. Above, we denoted private consumption by ct (remember, there is
a representative household in this economy and thus aggregate and individual consumption coincide). Let private investment be denoted by it : Then, the resource constraint in
this economy becomes:
c t + it = y t
We want to investigate investment a bit further. In the data, investment takes two
forms: a) the replacement of depreciated capital, called replacement investment or depreciation, and b) net investment, that is, the net increase of the existing capital stock.
In our model, depreciation is given by kt ; since by assumption a fraction of the capital
stock breaks in production or becomes economically obsolete. The net increase in the
capital stock between today and tomorrow, on the other hand, is given by kt+1 kt ; so
that total investment equals:
it = kt + kt+1
= kt+1 (1
kt
)kt
Substituting out it ; the aggregate resource constraint reads as:
ct + kt+1 = kt (At lt )1
+ (1
)kt
(4.27)
48
4.5
CHAPTER 4. SET-UP OF THE BASIC MODEL
Competitive Equilibrium
Our ultimate goal is to study how allocations (consumption, investment, output, labor,
etc.) and prices in the model compare to the data. But, as we have seen, allocations are
chosen by households and …rms, taking prices as given. Therefore, we have to …gure out
how prices (that is, wages and interest rates) are determined. We have assumed above
that all agents in the economy behave competitively, that is, take prices as given. So,
it is natural to have prices be determined in what is called a competitive equilibrium.
In a competitive equilibrium, households and …rms maximize their objective functions,
subject to their constraints, and prices are such that markets clear. The markets in this
economy consist of a market for labor, a rental market for capital, and a market for
goods. In a competitive equilibrium, all these markets have to clear in all periods.
Let us start with the goods market. We ignore the possibility of inventories for the
time being.29 The supply of goods by …rms then equals its output yt . Demand in the
goods market is given by consumption demand and investment demand of households,
ct + it : Thus, the market clearing in the goods markets boils down exactly to equation
(4:27): The labor market clearing condition simply states that the demand for labor by
our representative …rm, lt ; equals the supply of labor by our representative household.
Since we have assumed above that the household can and does supply one unit of labor
in each period, the labor market clearing condition reads as lt = 1. Finally, we have to
state the equilibrium condition for the capital rental market. Firms’demand for capital
rentals is given by kt : The household’s asset holdings at the beginning of period t are
denoted by at : But since physical capital is the only asset in this economy, the assets
held by households have to equal the capital that the …rm desires to rent, or at = kt .
All these equilibrium conditions, of course, have to hold for all periods t = 0; : : : ; T:
But how does an equilibrium in all these markets come about? That is the role of prices
(wages and interest rates): they adjust such that markets clear. We now can de…ne a
competitive equilibrium, one of the key concepts in this book, as follows:
De…nition 14 Given initial assets a0 ; a competitive equilibrium consists of allocations
for the representative household, fct ; at+1 gTt=0 ; allocations for the representative …rm,
fkt ; lt gTt=0 and prices frt ; t ; wt gTt=0 such that:
29
Holding inventories is suboptimal for the …rm as long as the real interest rate is always strictly
positive, which it always will be in this part of the book. Alternatively one could assume that output,
in contrast to capital, is not storable, or that there are su¢ ciently high costs for output storage.
4.5. COMPETITIVE EQUILIBRIUM
49
1. Given frt ; wt gTt=0 ; the household allocation solves the household problem
max
fct ;at+1 gT
t=0
T
X
t
u (ct )
t=0
subject to
ct + at+1 = wt + (1 + rt )at
ct 0
aT +1 = 0 if T < 1
t
= 0 if T = 1
limt!1 T a(1+r
i)
terminal condition =
i=1
given initial condition a0
2. Given f t ; wt gTt=0 ; with
the …rm problem
t
for all t = 0; : : : ; T; the …rm allocation solves
= rt
max (yt
nt ;kt
wt lt
t kt )
subject to
yt = kt (At lt )1
kt ; lt 0
3. Markets clear: for all t = 0; : : : ; T
ct + kt+1
(1
)kt = kt (At lt )1
lt = 1
at = kt
We want to emphasize that we, as most economists, use the name equilibrium to
denote a situation where a) the agents in the model follow a particular behavioral assumption, in this case maximization of either their utility or of pro…ts (in other models,
the behavioral assumption may well be very di¤erent) and b) their decisions are consistent with each other, in the sense of market clearing. In our previous de…nition, points
1. and 2. correspond to the …rst requirement, behavior, and point 3. satisfy the second
requirement, consistency. Nothing in our de…nition of equilibrium implies that prices or
allocations are constant over time (they may be increasing, decreasing, or ‡uctuating
in rather arbitrary ways), or that the economy is in some sort of a rest point, which is
the sense in which equilibrium is often used in other …elds, particularly in the natural
sciences.30 Many confusions about the implications and restrictions that the concept of
equilibrium in economics imposes arise because of a failure to appreciate this subtle difference in the use of language across …elds, and sometimes within the …eld of economics
itself.
30
In fact, we have termed such a special situation steady state above, see de…nition 11.
50
CHAPTER 4. SET-UP OF THE BASIC MODEL
Our de…nition of equilibrium is completely silent about the actors or mechanisms that
bring about equilibrium prices. The de…nition simply states that, at the given equilibrium
prices, markets clear. Also the de…nition implies a surprising property: there are 3(T +1)
market clearing conditions (there are T + 1 time periods and 3 markets open per period),
but we have only 2(T + 1) prices that can be used to bring about market clearing (wages
wt and interest rates rt for each period t = 0; : : : ; T ; note that the t do not count, since
rt = t
always holds).
The resolution of this apparent paradox is, however, rather straightforward. It turns
out that, whenever, in a given period, two markets clear, then the third market clears
automatically. In fact, this is an important general result in general equilibrium theory,
as the research …eld that deals with competitive equilibria and their properties is called.
The result is commonly referred to as Walras’ law, in honor of León Walras, a french
economist from the late 19th century who pioneered the study of general equilibrium in
his book Elements of Pure Economics, …rst published in 1874.
Theorem 15 (Walras’Law) Suppose that at prices frt ; t ; wt gTt=0 ; allocations fct ; at+1 gTt=0
and fkt ; lt gTt=0 solve the household problem and the …rm problem and suppose that in all
periods t the labor and the asset markets clear, lt = 1 and at = kt for all t: Then, the
goods market clears as well, for all t:
Proof. Since the allocation solves the household problem, it has to satisfy the household
budget constraint
ct + at+1 = wt + (1 + rt )at
By market clearing in the asset market, at = kt and at+1 = kt+1 ; and thus
ct + kt+1 = wt + (1 + rt )kt
or
ct + kt+1
But since rt =
t
kt = wt + rt kt
; we have
ct + kt+1
kt = wt + (
t
)kt
or
ct + kt+1
(1
)kt = wt + t kt
= 1 wt + t kt
= lt wt + t kt
(4.28)
where the last equality uses the labor market clearing condition lt = 1: But from the
…rst order conditions of the …rms’problem (remember we assumed that the allocation
solves the …rms problem)
wt = (1
t
=
)At
kt
At lt
kt
At lt
1
4.6. CHARACTERIZATION OF EQUILIBRIUM
51
and thus
wt lt +
t kt
= (1
kt
At lt
)At
lt +
= (1
)kt (At lt )1
= kt (At lt )1
1
kt
At lt
kt
+ kt (At lt )1
(4.29)
Combining (4:28) and (4:29) yields
ct + kt+1
(1
)kt = kt (At lt )1
that is, the goods market clearing condition is satis…ed in period t.
4.6
Characterization of Equilibrium
The de…nition of equilibrium does not tell us much about how this equilibrium looks like
(in fact, it does not even tell us that the equilibrium exists and that there is only one
such equilibrium!). The task of understanding the features of an equilibrium is know as
characterization of the equilibrium. Once we have characterized the equilibrium, we can
assess how well the model describes reality. For the rest of this part we now assume, to
ease the exposition, that At = 1: As we mentioned before, we will bring back growth and
technology shocks in parts III and IV of the book, respectively.
By inserting the period budget constraints into the Euler equation we already found
the optimality condition of the household as:
u0 (wt + (1 + rt )at
at+1 ) = (1 + rt+1 )u0 (wt+1 + (1 + rt+1 )at+1
at+2 )
Now, we can make use of further optimality conditions to simplify matters further. First,
we realize that from the asset market clearing condition, we have kt = at ; kt+1 = at+1
and kt+2 = at+2 : Furthermore we know that lt = lt+1 = 1 and thus equilibrium wages
and interest rates are given by
wt = (1
rt = t
)kt
= kt
(4.30)
(4.31)
1
and similar results hold for wt+1 and rt+1 : Inserting all this in the Euler equation of the
household yields
u0 (1
=
1 + kt+11
)kt + (1 + kt
u0 ((1
1
)kt+1 + (1 +
)kt
kt+1
(kt+1 )
1
)kt+1
kt+2 )
or
u0 (kt + (1
)kt
kt+1 ) = (1 + kt+11
)u0 (kt+1 + (1
)kt+1
kt+2 )
(4.32)
52
CHAPTER 4. SET-UP OF THE BASIC MODEL
This expression is arguably messy, but this equation has as arguments only the capital stocks (kt ; kt+1 ; kt+2 ); since we have substituted out all prices. All the remaining
elements in this equation are the parameters ( ; ; ) and of course the derivative of
the utility function that needs to be speci…ed (and by doing so we may need additional
parameters). We already saw in section 4.2.3, equation (4:23), that an equation like
(4.32) is, mathematically speaking, simply a second order di¤erence equation.31 What
is more, we have an initial condition k0 = a0 ; equal to some pre-speci…ed number, and
our usual terminal condition. In chapter 7, we will study techniques (mostly numerical
in nature) to solve an equation like this. The solution will be the equilibrium path for
capital fkt gTt=0 ; from which we can …nd consumption:
ct = kt (At lt )1
+ (1
)kt
kt+1
and derive equilibrium wages and interest rates from equations (4.30) and (4.31). It
is important to remember at this point that we have neither shown that a solution to
these equations exists nor, that if exists, it is unique. In the next chapter, we will show
existence and uniqueness of such a solution using an indirect but powerful argument.
However, even if we do not solve for the path of capital explicitly, and thus for the
values of all other variables, expressions such as (4:23) or (4:32) imply predictions of
the economic theory that can be tested in the data. For example, if we specify some
utility form u ( ) and we have observations on capital for the U.S. economy, we can ask
several questions. First, do exist parameter values ( ; ; ) that make equation (4.32)
hold in the data? If not, how big is the di¤erence between the left hand side of the
equation and the right hand side? If the di¤erence is big, can we investigate why is the
theory failing to account for the data? Is the theory ‡awed? Or is the data at fault
(perhaps because of measurement errors, or because the variables are not measured in
the way the theory tells us they should be measured)? Another set of questions deal
with observations regarding asset prices. If we have data on consumption over time, we
know that an equilibrium implies
u0 (ct ) = (1 + rt+1 )u0 (ct+1 )
or
1 + rt+1 =
u0 (ct+1 )
u0 (ct )
1
(4.33)
that is, there is tight relation between returns to assets and marginal utilities (in fact
0
)
the term uu(c0 (ct+1
is known as the asset pricing kernel because it is at the heart of asset
t)
pricing theory). Do bond and stock prices that we observe in …nancial markets conform
to the predictions of the theory implicit in equation (4:33)? If not, why?
31
Note, however, despite their similarities, that there is a subtle yet important di¤erence between
equation (4.23) and the equation (4.32). While the former is just an optimality condition for the
household that applies whether prices are equilibrium prices or not, the latter incorporates all equilibrium
conditions as well as equilibrium prices.
4.6. CHARACTERIZATION OF EQUILIBRIUM
53
Equilibrium models have a rich set of implications for objects that we can measure
in the real world and thus open the door for a fruitful dialogue between theory and data.
This interplay between theory and data is the guiding principle of most of good economic
research and certainly the guiding principle of the remainder of this book.
Before using the theory developed so far to speak to the data, in the next chapter
we establish the connection between competitive equilibrium and socially optimal allocations, that is, choices of consumption and capital that a benevolent social planner
would make. This will allow us to provide both a normative justi…cation for studying
competitive equilibria as well as a widely applicable technique to actually solve for them
in practice.
54
CHAPTER 4. SET-UP OF THE BASIC MODEL
Chapter 5
Social Planner and Competitive
Equilibrium
In this chapter, we will show something that, at …rst sight, should be fairly surprising.
Namely, that we can solve for the allocation of a competitive equilibrium by solving
the maximization problem of a benevolent social planner. In fact, this is a very general
principle that often applies (and as such is called a theorem, in fact, two theorems,
the …rst and second fundamental welfare theorems of general equilibrium). Envision a
social planner that can tell agents in the economy (households, …rms) what to do: how
much to consume, how much to work, how much to produce etc. The social planner is
benevolent, that is, she likes the households in the economy and thus maximizes their
lifetime utility function. The only constraints the social planner faces are the physical
resource constraints of the economy (even the social planner cannot make consumption
out of nothing).1
1
Often economists use the notion of Pareto e¢ ciency to de…ne what a socially optimal allocation
is. An allocation is Pareto e¢ cient if it is feasible (it satis…es the resource feasibility constraints and
all nonnegativity constraints on consumption and capital) and there does not exist another feasible
allocation that is weakly preferred by all households and strictly preferred by at least one household.
In our model with a representative household it is straightforward to show that an allocation is Pareto
e¢ cient if and only if it solves the social planner problem. Therefore we focus on the social planner
problem directly, without separately discussing Pareto e¢ cient allocations.
55
56
CHAPTER 5. SOCIAL PLANNER AND COMPETITIVE EQUILIBRIUM
5.1
The Social Planner Problem
As before, we continue assume that At = 1: All arguments straightforwardly generalize
to the case where At = A 6= 1: Then the problem of the social planner is given by
max
fct ;kt+1 ;nt gT
t=0
T
X
t
u(ct )
t=0
subject to
ct + kt+1 (1
)kt = kt lt1
ct 0; 0 lt 1
k0 > 0 given
We make two simple observations before characterizing the optimal solution to the social
planner problem. First, households only value consumption in their utility function,
and do not dislike working. Since more work means higher output, and thus higher
consumption today, or via higher investment tomorrow, it is always optimal to set lt = 1:
Second, we have not imposed any constraint on kt+1 ; but evidently kt+1 < 0 can never
happen since then production is not well-de…ned. In addition, even kt+1 = 0 can be
ruled out if the utility function satis…es the Inada conditions since kt+1 = 0 implies that
output in period t + 1 is zero, thus consumption in that period is zero and kt+2 = 0
and so forth. Thus it is never optimal to set kt+1 = 0 for any time period, unless the
household does not mind too much consuming 0 (in all our applications this will never
happen). Exploiting these facts the social planners problem becomes
max
fct ;kt+1 gT
t=0
T
X
t
u(ct )
t=0
subject to
ct + kt+1 (1
)kt = kt
ct 0 and k0 > 0 given
(5.1)
Note that this maximization problem is an order of magnitude less complex than solving for a competitive equilibrium, because in the later we have to solve maximization
problems of households and …rms and …nd equilibrium prices that lead to market clearing. The social planner problem in contrast is a simple maximization problem.2 Thus
it would be really useful to know that by solving the social planner problem we have in
fact also found the competitive equilibrium.
Remark 16 Before proceeding to characterize the solution of the social planner problem,
we want to discuss what we know a priori about the capital stock chosen in the last period.
If T is …nite, it is obvious that the planner should choose kT +1 = 0; since any capital
stock brought into period T + 1 (the household is dead by then) could instead be consumed
in period T and deliver additional utility.
2
The problem still has many choice variables, namely 2(T + 1); and in…nitely many if T = 1.
5.2. CHARACTERIZATION OF SOLUTION
57
Remark 17 For T = 1 the appropriate terminal condition is less transparent. Since
the capital stock is required to be nonnegative, Ponzi schemes are never possible for the
social planner. On the other hand, there is a transversality condition similar to the one
the we imposed on the household. For the household, it stated that the present discounted
value of assets goes to zero as time goes to in…nity, see equation (4:7): Similarly, for the
social planner the transversality condition is given by
lim
t kt+1
t!1
(5.2)
=0
where the factor t is a discount factor that discounts capital saved in period t back
to to period 0: We characterize t further below. As in the household problem, the
transversality condition (5:2) is a condition that a solution to the social planner problem,
for the case T = 1 has to satisfy. See the appendix of this chapter for the exact details.
5.2
Characterization of Solution
The Lagrangian, ignoring the non-negativity constraints for consumption (which will not
be binding if the period utility function u satis…es the Inada conditions), is given by
L=
T
X
t
u(ct ) +
T
X
t
[kt
ct
kt+1 + (1
)kt ]
t=0
t=0
where t is the Lagrange multiplier on the resource constraint3 in period t: The …rst
order conditions, set to 0; are
@L
=
@ct
@L
=
@ct+1
@L
=
@kt+1
t 0
u (ct )
t
=0
t+1 0
u (ct+1 )
t
+
t+1
=0
kt+11 + (1
t+1
) =0
Rewriting these conditions yields4
t 0
u (ct ) =
u (ct+1 ) =
1
kt+1 + (1
) =
t+1 0
t+1
3
t
(5.3)
t+1
t
It is of course no accident that we denote the Lagrange multiplier by the same symbol as the
discount factor in the transversality condition (5:2):
4
Equation (5:3) makes clear that t measures how much, in terms of discounted additional utility,
one unit of capital saved in period t is worth. The equation allows us to rewrite the transversality
condition of the social planner as
lim t u0 (ct )kt+1 = 0:
t!1
58
CHAPTER 5. SOCIAL PLANNER AND COMPETITIVE EQUILIBRIUM
and thus
t 0
u (ct ) =
u0 (ct ) =
= t+1 kt+11 + (1
u0 (ct+1 ) Akt+11 + (1
t
) =
)]
t+1 0
u (ct+1 )
kt+11 + (1
)
(5.4)
which is, of course, the intertemporal Euler equation, but now for the social planner. The
social planner equates the marginal rate of substitution
of the representative household
u0 (ct )
between consumption today and tomorrow, u0 (ct+1 ) to the the marginal rate of transformation between consumption today and tomorrow. Letting the household consume one
unit of consumption less today allows for one more unit investment and thus one more
unit of capital tomorrow. But this additional unit of capital yields additional production
equal to the marginal product of capital, kt+11 ; and after production 1
units of the
capital are still left over. Thus the marginal rate of transformation between consumption
today and tomorrow equals kt+1 + (1
):
Making use of the resource constraints
ct = kt kt+1 + (1
ct+1 = kt+1 kt+2 + (1
)kt
)kt+1
the Euler equation becomes
u0 (kt
kt+1 + (1
)kt ) = u0 (kt+1
kt+2 + (1
)kt+1 )
kt+11 + (1
)
(5.5)
Comparing equation (5:5) to the Euler equation (4:32) obtained in the household problem
we see that they are exactly the same. This observation will be the basis of our proof
of the two welfare theorems. Note again that this is a second order di¤erence equation,
and we have an initial condition, because k0 is given to us. As long as T < 1 we also
have the terminal condition kT +1 = 0; which is replaced by the transversality condition
(5:2) if T = 1:
5.3
The Welfare Theorems
Now we can state the two fundamental theorems of welfare economics.5
Theorem 18 (First Welfare Theorem) Suppose we have a competitive equilibrium with
allocation fct ; kt+1 gTt=0 : Then the allocation is socially optimal, in the sense that it solves
the social planner problem.
Proof. Since fct ; kt+1 gTt=0 is part of a competitive equilibrium, it has to satisfy the
necessary conditions for household and …rm optimality. We showed that these implies
that the allocation solves the Euler equation (4:32): But then the allocation satis…es the
Euler equations (5:5); which for T < 1 are the necessary and su¢ cient conditions for
an optimal solution of the social planner problem.
5
We will restrict the proof to the case that T < 1 since in this case the Euler equations are not
only necessary, but also su¢ cient conditions for an optimal solution. We comment further on the case
T = 1 in the appendix.
5.4. STEADY STATE ANALYSIS
59
Theorem 19 (Second Welfare Theorem) Suppose an allocation fct ; kt+1 gTt=0 solves the
social planner problem and hence is socially optimal. Then there exist prices frt ; t ; wt gTt=0
that, together with the allocation fct ; kt+1 gTt=0 and flt ; at+1 gTt=0 ; where lt = 1 and at+1 =
kt+1 for all t; forms a competitive equilibrium.
Proof. The proof is by construction. First we note that all market clearing conditions
for a competitive equilibrium are satis…ed by the solution to the social planner problem:
the labor market and asset market equilibrium conditions are satis…ed by construction.
The goods market clearing conditions are satis…ed since the allocation, by assumption,
solves the social planner problem and thus satis…es the resource feasibility conditions
(5:1), for all t: Now construct prices as functions of the allocation, as follows
wt = (1
) (kt )
rt =
(kt ) 1
t = rt +
(5.6)
(5.7)
(5.8)
It remains to be shown that at these prices the allocation solves the …rms’and households’
maximization problem. Looking at the …rms …rst order conditions, they are obviously
satis…ed. And the necessary and su¢ cient conditions for the households’maximization
problem were shown to boil down to (4:32); which the allocation satis…es, since it solves
the social planner problem and hence the conditions (5:5):
These two results come in handy, because they allow us to solve the much simpler
social planner problem and be sure to have automatically solved for the competitive
equilibrium allocations (and prices, using (5:6)-(5:8)), the ultimate object of interest,
too.
The welfare theorems can be read as an endorsement of a market economy: the
competitive equilibrium achieves the same allocation than the social planner. Yet, at
the same time and as we will see in many di¤erent parts of the book, the welfare theorems
also highlight how fragile are the conditions that allow a market economy to achieve the
“right”allocation. Small changes in the assumptions of the model will lead to situations
where the competitive and social planner allocations di¤er.
5.4
Steady State Analysis
In the context of this part of the book a steady state is a competitive equilibrium or a
solution to the social planners problem where all variables are constant over time (see
de…nition 11). Let (c ; k ) denote the steady state consumption level and capital stock.
Then, if the economy starts with the steady state capital stock k0 = k , it never leaves
that steady state. And even if it starts at some k0 6= k ; it may over time approach the
steady state (and once it hits it, of course it never leaves again).6
6
We will see below that the steady state may also an important starting point of the dynamic analysis
of the model; if there is hope that the solution of the model never gets too far away from the steady state,
one can linearize the dynamic system around the steady state and hope to obtain a good approximation
to the optimal decisions outside, but close to the steady state, too.
60
5.4.1
CHAPTER 5. SOCIAL PLANNER AND COMPETITIVE EQUILIBRIUM
Characterization of the Steady State
Since the previous section showed that we can interchangeably analyze socially optimal
and equilibrium allocations, let us focus on equation (5:5); or equivalently, equation (5:4)
u0 (ct ) = u0 (ct+1 )
kt+11 + (1
)
(5.9)
In the steady state we require ct = ct+1 = c and kt+1 = k : But then u0 (ct ) = u0 (ct+1 )
and thus
1 = a (k ) 1 + (1
)
or, remembering
+
1
1+
;
= a (k )
= a (k )
+
1
1
(5.10)
This rule for choosing the steady state capital stock is sometimes called the modi…ed
golden rule: the optimal steady state capital stock is such that the associated marginal
product of capital net of the depreciation rate equals the time discount rate. We will
see very soon why this is called the modi…ed golden rule. Obviously we can solve for the
steady state capital stock explicitly as
k =
+
1
1
1
1
=
+
That is, the optimal steady state capital stock is the higher the more important capital is
relative to labor in the production process (the higher is ); the lower is the depreciation
rate of capital (the lower ) and the lower the impatience of individuals (the lower ).
Also note that the steady state capital stock is completely independent of the utility
function of the household and its parameters characterizing it (e.g. the IES 1= in the
CRRA utility function), as long as u is strictly concave).
The steady state consumption level can now be determined from the resource constraint (4:27)
ct + kt+1 (1
)kt = kt
In steady state this becomes
c + k = (k )
c = (k )
k
that is, steady state consumption equals steady state output (k ) minus depreciation,
since in the steady state capital is constant and thus net investment kt+1 kt is equal to
0:
5.4. STEADY STATE ANALYSIS
5.4.2
61
Golden Rule and Modi…ed Golden Rule
Now we can also see why the modi…ed golden rule is called that way. In the steady state,
consumption equals
c=k
k:
Let k g denote the so-called traditional golden rule capital stock that maximizes steady
state consumption7 , that is, the capital stock that solves
max k
k:
k
Taking …rst order conditions and setting to 0 yields
(k g )
1
or
=
(5.11)
1
kg =
1
(5.12)
Let cg denote the (traditional) golden rule consumption level
cg = (k g )
kg :
Furthermore, in the steady state investment (which equals saving) simply replaces the
depreciated capital stock, i = k: Thus the implied investment (savings) rate associated
with the traditional golden rule capital stock is the fraction of output k not consumed,
or
k
i
=
= k1 :
s=
k
k
Using (5:12) we …nd that the golden rule savings rate is given by s = : This result, derived by Ed Phelps in 1961 (at Columbia University and Nobel prize winner in economics
in 2006), means that in order to maximize per capita consumption in the long run the
economy should adopt a saving rate s equal to the elasticity of output with respect to
its capital input .
Comparing the modi…ed and the traditional golden rule capital stock and consumption we have the following
Proposition 20 The modi…ed golden rule capital stock and consumption level are strictly
lower than the traditional golden rule capital stock and consumption level.
Proof. Obviously
1
1
g
k =
1
1
>k =
+
since > 0 (i.e. < 1). But k g is the unique capital stock that maximizes steady state
consumption, whereas k does not maximize steady state consumption. Consequently,
cg > c .
7
This objective is grounded in the principle, present in many ethical and religious traditions, that
one should treat others (in these case future households) in the same way as oneself.
62
CHAPTER 5. SOCIAL PLANNER AND COMPETITIVE EQUILIBRIUM
What is the intuition for the result? Why is the perfectly benevolent planner choosing
a steady state capital and consumption level lower than the one that would maximize
steady state consumption. Note that the planners objective is to maximize lifetime
utility, not steady state consumption. And since the household is impatient (that is,
> 0) the planner should take this into account by letting the household consume more
today, as the expense of a lower steady state capital stock and resulting consumption in
the future.
The analysis of the steady state of the model determines the long run implications
of the theory. What happens in the short run, and whether or not the capital stock and
consumption actually approach their steady state variables requires the analysis of the
entire dynamic path of the economy, starting from the initial capital stock k0 : We carry
out this analysis in the next section. As we will see, the main conclusions from this
analysis lend strong support to the importance of the steady state. We will show that
from any initial capital stock k0 the economy’s capital stock monotonically converges
over time to its steady state value.
5.5
Dynamic Analysis
Now we want to analyze how the economy, from an arbitrary starting condition k0 ;
evolves over time. Again we exploit the welfare theorems and characterize the solution
of the social planner problem directly. It is somewhat easier to do this analysis for the
case T = 1; that is, for the case in which the economy runs forever. The reason is
simple: if the economy ends at a …nite T; we know that kT +1 = 0: On the other hand
it is never optimal to have kt = 0 for any time t
T; because otherwise consumption
from that point on would have to equal 0; which is never optimal if u satis…es the Inada
conditions. Therefore it is unlikely that in the …nite time case the economy will settle
down to a steady state with constant capital and consumption. In the case T = 1 this
will happen, as we will see below.
We already know that if by coincidence we have k0 = k ; then the capital stock would
remain constant over time in the T = 1 case, so would be the interest rate, consumption
and all other variables of interest. This can be seen from the Euler equation of the social
planners problem, equation (5:5):
u0 (kt
kt+1 + (1
)kt ) =
=
u0 (kt+1
0
u (kt+1
kt+2 + (1
kt+2 + (1
)kt+1 )
)kt+1 )
1+
kt+11 + (1
kt+11
+ (1
)
)
(5.13)
Plugging in kt = kt+1 = kt+2 = k we see that this equation holds.8 Thus kt = k
for all periods is a solution to the social planners problem; in fact the only solution for
T < 1 since the optimal solution to the social planner problem is unique, because it is
1
Simply realize that (k ) 1+ +(1 ) = 1 from equation (5:10); and that the left had side for kt =
kt+1 = k equals the right hand side for kt+1 = kt+2 = k ).
8
5.5. DYNAMIC ANALYSIS
63
a …nite-dimensional maximization problem with strictly concave objective function and
a convex constraint set.
But what happens if we start with k0 6= k ? We assume that k0 > 0; since otherwise
there is no production, no consumption and no investment in period 0 or in any period
from that point on. What we are looking for is a sequence of numbers fkt g1
t=1 that solves
equation (5:13): This is in general hard to do, so let’s try to simplify. The economy starts
with initial capital k0 and the social planner has to decide how much of this to let the
agent consume, c0 and how much to accumulate for tomorrow, k1 : But in period 1 the
planner has exactly the same problem: given the current capital stock k1 ; how much to
let the agent consume, c1 and how much to accumulate for tomorrow, k2 :
This is in fact a general principle: we can try to …nd our solution fkt g1
t=1 by …nding the
unknown function g giving capital tomorrow as a function of capital today, kt+1 = g(kt ):
If we would know this function, then we can determine how the capital stock evolves
over time. Simply start with the given k0 ; and then proceed by repeatedly applying the
function g(:):
k1 = g(k0 )
k2 = g(k1 )
k3 = g(k2 )
and so forth. Of course the challenge is to …nd this function g:
There are three approaches to do this. First, for a very limited set of examples, we
can guess a particular form of g and then verify that our guess was correct by substituting
the guessed functions in the equilibrium conditions of the model and checking that those
hold indeed. Thus, this approach to solve problems is often called guess and verify.9 .
Since it allows us to use “paper and pencil”, it is particularly well suited to learn the
main economic insights of the models. However, as we mentioned, it only applies to a
few cases and it limits the quantitative analysis that we can undertake. Second, and
this is a very general approach that (almost) always works and for which the use of
a computer comes in handy, one can try to approximate equation (5:13) by a linear
equation, which one then can always solve. Third, one can tackle the original nonlinear
problem embedded in equation (5:13) by more sophisticated numerical techniques (and
a reasonably powerful computer). We will demonstrate the …rst approach explicitly in
the next subsection, and then give a brief preview of the second and third approaches.
We will return to these issues on several occassions later in the book.
5.5.1
An Example with Analytical Solution
Suppose that the utility function is logarithmic, u(c) = log(c) (and thus u0 (c) = 1c ) and
also assume that capital completely depreciates after one period, that is = 1 (this may,
with good reason, sound unrealistic, but remember that we can always think about one
9
Guess and verify leaves open, however, the question of how to come up with a good guess! Beyond
experience and some good luck, there is no much that we can add about how to get this guess.
64
CHAPTER 5. SOCIAL PLANNER AND COMPETITIVE EQUILIBRIUM
period as 10 or 20 years, where full depreciation is a much more sensible assumption).
Then, equation (5:13) becomes
1
kt
kt+1
=
kt+11
kt+1 kt+2
(5.14)
Remember that we look for a function g telling us how big kt+1 is, given today’s capital
stock kt : Now lets make an arbitrary guess by conjecturing that the social planner …nds
it optimal to take current output kt and split it between consumption ct and capital
tomorrow, kt+1 in …xed proportions, independent of the level of the current capital stock
and thus output of the economy. That is, we guess that
kt+1 = g(kt ) = skt
where s > 0 is a …xed, but unknown number. Thus, applying the same guess for period
t + 1 yields
kt+2 = g(kt+1 ) = skt+1 :
Given these guesses we have
kt
kt+1
kt+1 = (1
kt+2 = (1
s)kt
s)kt+1
Using these two equations in (7:36) yields
1
(1 s)kt
=
=
(1
kt+11
s)kt+1
(1
s)kt+1
and thus
kt+1 =
kt =
yt
But we had guessed
kt+1 = skt
and thus with s =
and therefore kt+1 =
kt : That is, no matter how big kt is, if we set
kt+1 =
kt+2 =
kt equation (7:36) is satis…ed for every
kt
kt+1
the Euler equation is satis…ed.
Consequently, we found exactly what we were looking for, namely a function that tells
us that if the planner gets into the period with capital kt ; how much of the implied output
she should let the households consume, and what fraction to save for tomorrow. Since
we know the initial capital stock k0 ; we can compute the entire sequence of capital stocks
5.5. DYNAMIC ANALYSIS
65
from period 0 on (of course always conditional on having speci…ed concrete parameter
values ; ). Note that a computer is very good carrying out many of such a calculations
in very short time.
Finally, it is obvious that from the resource constraint
c t = kt
kt+1 + (1
)kt
we can compute consumption over time, once we know how capital evolves over time.
We now want to brie‡y analyze the dynamics of the capital stock, starting from k0
and being described by the so-called policy function
kt+1 =
kt
There are two steady states of this policy function in which kt+1 = kt = k: The …rst
is, trivially, k = 0: The second is our steady state from equation (5.10) above, k = k .
Remembering that we have assumed = 1 above, we can solve
k
=
(k ) or
1
k
= (
)1
1
1
=
+1
In order to describe the dynamics of the capital stock it is best to plot the policy
function kt+1 =
kt . Figure 5.1 does exactly that. In addition it contains the line
kt+1 = kt : We now use this …gure to deduce the time path of capital in the economy,
starting from k0 :
Given an initial capital stock k0 on the x-axis, the graph gives k1 =
k0 on the
y-axis. Going back from the y-axis to the kt+1 = kt 450 line gives k1 on the x-axis. Then
the graph gives k2 =
k1 on the y-axis. Going again to the kt+1 = kt line gives k2 on
the x-axis. Now continue this procedure ad in…nitum.
The important lesson feature from this …gure is that if the initial capital stock is below
its steady state value, k0 < k ; over time the capital stock kt increases and approaches
the steady state k : This is true for any k0 > 0 with k0 < k : In contrast, if the initial
capital stock lies above the steady state, k0 > k ; then the capital stock approaches the
steady state from above. In either case there is monotonic convergence to the steady
state: the capital stock is either monotonically increasing or decreasing and over time
approaches the steady state, which justi…es our focus on steady states in the previous
section. Technically speaking, the unique positive steady state is globally asymptotically
stable.
These last results are very general and do not depend on full depreciation or logutility. As long as u is strictly concave, from any initial capital stock k0 > 0 the path
of capital converges monotonically to the unique positive steady state capital stock, and
consumption does the same.
66
CHAPTER 5. SOCIAL PLANNER AND COMPETITIVE EQUILIBRIUM
Figure 5.1: Dynamics of the Capital Stock
5.5.2
How to Analyze the General Model
Unfortunatelu, guessing and then verifying a solution does not work in general. With
the exception of very few cases no closed form solution to equation (5:13) exists. For the
general case, researchers typically follow one of two approaches.
First, through a suitable approximation about the steady state (which we carry out
in detail in part III of the book) one can turn the nonlinear di¤erence equation into a
linear di¤erence equation which one then either can solve by hand, or can easily solve
on a computer. Second, one can attempt to …nd the unknown function g directly from
the nonlinear equation, by solving for g in the equation (5:13) by numerical brute force.
Plugging in g(kt ) for kt+1 and g(g(kt )) for kt+2 in (5:13) we obtain
u0 (kt g(kt ) + (1
)kt )
0
= u (g(kt )
g(g(kt )) + (1
)g(kt ))
g(kt )
1
+ (1
) :
The computational problem is to …nd the unknown function g(:) that solves this equation
for every possible kt
0: Mathematically, an equation whose solution is an unknown
function is called a functional equation. Functional equations and methods to solve
them are an active area of study in mathematics and economics. The most frequent
functional equations in economics arise in the context of dynamic programming, a very
5.6. APPENDIX A: MORE RIGOROUS MATH
67
general technique to solve dynamic optimization problems of the sort described in this
chapter. A treatment of dynamic programming or functional equations in general is
beyond the scope of this book; the interest economics reader is referred to the treatment
by Stokey and Lucas (1989), Ljungqvist and Sargent (2004), and the mathematical and
computational references therein.
5.6
Appendix A: More Rigorous Math
In this appendix we comment more rigorously on the connection between a solution of
the social planner problem and a competitive equilibrium, …rst for T < 1:
1. The social planner problem has a unique solution since it is a …nite-dimensional
maximization with a strictly concave objective function and a convex constraint
set. The Euler equations are necessary and su¢ cient conditions that the solution
has to satisfy.
2. The second welfare theorem tells us that we can make this solution into a competitive equilibrium, and the proof of the second theorem even tells us how to construct
the prices.
3. Can there be another competitive equilibrium allocation? No, since the …rst welfare
theorem tells us that any other equilibrium allocation would also be a solution to
the social planner problem, contradicting the fact that the social planner problem
has a unique solution.
What are the di¢ culties in applying steps 1-3 if T = 1: First, it is somewhat
harder to argue that an in…nite-dimensional maximization problem has a unique solution.
Second, the Euler equations for T = 1 are only necessary, but not su¢ cient. The Euler
equations, together with the transversality condition, are jointly su¢ cient for an optimal
solution.10 Thus one potentially has to use the transversality condition to select among
the many solutions to the Euler equations the true optimal solution to the social planner
problem. Finally, for steps 2. and 3. we have to argue that the transversality condition of
the household in the competitive equilibrium (after interest rates have been substituted
out) boils down to the transversality condition of the social planner problem (see the
proofs of the welfare theorems for T < 1 in the main text). This equivalence is fairly
straightforward to establish.
Thus while the mathematics becomes more involved in the case T = 1; under appropriate assumptions on u the relationship between the social planner problem and
competitive equilibria is the same as in the case T < 1:
10
Under some conditions, the transversality condition is also necesary for an optimum.
68
CHAPTER 5. SOCIAL PLANNER AND COMPETITIVE EQUILIBRIUM
Part III
Economic Growth
69
Chapter 6
Motivation and Facts about
Economic Growth
6.1
The World Income Distribution in 2010
[Cross-sectional facts from newest Summers-Heston data set]
A quick look at international data reveals that the average American’s income is
around 5 bigger than the average Mexican’s, around 14 times bigger than the average
Indian’s, and 35 times bigger than the average African’s, even when we have corrected all
the …gures to accommodate di¤erences in price levels across countries. This di¤erences
in income levels do not a¤ect only how many consumption goods each person has access
to, what is re‡ected in other, perhaps more worrisome, statistics. For example, life
expectancy in rich countries is 77 years, 67 years in middle income countries, and 53
years in poor countries. Similarly, out of 6.4 billion people, 0.8 do not have access to
enough food, 1 to safe drinking water, and 2.4 to sanitation.
A striking fact is that some of these di¤erences in income are quite recent. North
Korea versus South Korea.
Di¤erences Across Time
1. The average modern American is around 20 times richer than the average colonial
American.
2. An American worked 61 hours per week in 1870, today 34.
3. Japanese boy born in 1880 had a life expectancy of 35 years, today 81 years.
Robert Lucas, in what has become one of the most quoted sentences in economics,
put together all these concerns in 1985 much better than we could aspire to:
I do not see how one can look at …gures like these without seeing them as representing
possibilities. Is there some action a government could take that would lead the Indian
economy to grow like Indonesia’s or Egypt’s? If so, what exactly? If not, what is it about
the “nature of India” that makes it so? The consequences for human welfare involved
71
72
CHAPTER 6. MOTIVATION AND FACTS ABOUT ECONOMIC GROWTH
Figure 6.1:
in questions like these are simply staggering: Once one starts to think about them, it is
hard to think about anything else.
Five Parts of the Question
1. Why are there so large di¤erences in income per capita and worker productivity in
the world?
2. Why do some countries grow quickly while other stagnate?
3. What explains growth over long periods of time?
4. Why did sustained economic growth start around 200 years ago?
5. Why did it start in England?
Di¤erent Hypothesis
1. Luck (a.k.a. multiple equilibria).
2. Economic policy: taxes, education, health,...
3. Institutions: property rights, market versus planned economies,...
4. International Trade (or absence of!).
6.2. CHANGES IN INCOME OVER TIME
73
5. Geography.
6. Culture.
7. Exploitation of some countries by others.
6.2
6.2.1
Changes in Income over Time
The Very Long Run
Industrial revolution as crucial event in human history.
Why England/U.S. and not China in 13th century
Very small income di¤erences at the end of the 18th century. Since then take-o¤ in
some countries, much later (or not at all) in other countries, leading to substantial di¤erences in growth rates across countries in the 19th and …rst half of the 20th century. Thus
large di¤erences across countries by 1960 that have persisted since, although di¤erent
countries have changed their position in the world income distribution.
First talk about the growth experience of those countries that did edxperience the
industrial revolution …rst (balanced growth, Kaldor’s growth facts). Then talk about
international di¤erences in income per capita and growth rate di¤erences, then about
absolute and conditional convergence (or lack thereof)
Diamond: Proximate vs. Ultimate Causes
6.2.2
Kaldor’s Growth Facts for Industrialized Countries (Time
Series for a Given Country)
Classical growth facts motivating neoclassical growth model
6.2.3
6.3
Convergence
Accounting for Income Growth: Standard Growth
Accounting
Production function
Derive growth rate formulation
Putting analysis to work, two examples a) productivity slowdown in the U.S. (and
around the world) starting in the 1970’s b) growth miracles in Asia.
[I think this should come after the theory, since it does involve the production function
with growth, which we have not introduced at that point.]
74
CHAPTER 6. MOTIVATION AND FACTS ABOUT ECONOMIC GROWTH
Chapter 7
The Neoclassical Growth Model
In the next pages, we will use the basic model that we presented in part II of the book
as the theoretical framework to understand the facts about economic growth that we
documented in the previous chapter. Our choice of the basic model for this task is not
a coincidence. The central theme of this book, discussed on various occasions already, is
that (variations of) the same basic model can be used to provide a uni…ed understanding
of economic growth, business cycles, and the impact that economic policy has on these.
Because at the core of the production side of the model stands the neoclassical production
function introduced in equation () of chapter 1 which relates total output produced in
the economy to the use of factor inputs, the entire model constructed in part II (and used
for the study of economic growth momentarily) is often called the neoclassical growth
model.
For the time being we will work with the version of the model in which households
and the economy live forever, that is we assume that T = 1: Since we are interested in
the long run performance of economies a model with a terminal date appears unnatural.
More importantly, a model with a …nite time horizon misses important features of the
data since household resort to rather extreme behavior as the end of their life is near.
Besides an in…nite horizon, we will make two crucial assumptions. First, we postulate
that the population of the economy grows at a exogenous and constant rate n: Then,
given some initial number of households, N0 ; (each of them endowed with one unit of
time per period), the total number of households evolves over time as:
Nt = (1 + n)t N0
Without a¤ecting our results at all we normalize the initial level of population, N0 , to 1
and thus we can write:
Nt = (1 + n)t
which will result in less messy algebraic expressions below.1 We introduce population
growth into the model because it is an important determinant of total output and income
1
The choice of N0 can be thought of as choosing the units in which population size is measured
(thousands, millions, billions).
75
76
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
in actual economies, and because we want to study how di¤erences in population growth
rates across countries or across time in the same country dynamics a¤ect economic growth
of per capita income in the short and in the long run. The relationship between economic
growth and population growth is a classical question in macroeconomics that, as we saw
in the previous chapter, dates back at least to Thomas Malthus (1766-1834). The two
crucial assumptions about population growth we make is that n is exogenous and that
it is constant.2 These assumptions simplify the analysis considerably and allow us to
obtain a closed form solution of the model that is easily interpretable and thus serves as
a natural starting point of our analysis.
Second, we assume that the level of technology At grows at a constant rate g: This
constant improvement in our ability to produce goods with a given amount of capital and
labor input is also taken as given and occurs without any e¤ort or intervention by the
agents in the model. This is why, often, economists call it manna from heaven, after the
Biblical history about how the Israelites obtained food from heaven during their travels
in the Sinai desert.3 Thus, we have:
At = (1 + g)t A0
Since we can pick the measurement units for the …nal good in the economy as we please
(for example, we can measure the good in kilograms or in tons without any real change
in the economics of the problem), we select the measurement units such that A0 = 1 and
At = (1 + g)t
As with our normalization of N0 , this choice simpli…es the algebra below. We introduce
technological progress because it helps us to show that, within the context of our basic
model, the only source of long run economic growth will be technological progress measured by g. The justi…cation of why we take g to be constant and exogenous is very
2
This modelling choice likely violates two observations from the real world. The asusmption that
the population growth rate is exogenous means that it is not a¤ected by any variable whose value
is determined by the model. It thus amounts to assuming that n is not a¤ected by income levels of
households. In reality population growth is determined by the di¤erence between fertility and mortality
rates, both of which are strongly a¤ected by household choices that may depend on their income and
consumption levels. Households may choose to have, or at least try to have, another child if household
income increases. Mortality rates are likely a¤ected by actions such as following a healthy diet, exercising, and more generally, by the amount of resources spent on health care and other consumption goods,
all of which in turn depend on household income. Economists have explored many of these feedback
mechanisms between income levels and fertility and mortality rates in detail and a large literature has
aimed at endogeneizing fertility and mortality, and thus the population growth rate. We will return to
some of these study later in the book.
Second, we assume that the population growth rate is constant over time. This assumption can be
relaxed for much of the analysis, at the expense of a substantial increase in the clutter of the algebraic
expression. It is required, however, for the convergence of the model to a rest point, discussed in the
next section.
3
Book of Exodus 16:14-15 (King James Version): “And when the dew that lay was gone up, behold,
upon the face of the wilderness there lay a small round thing, as small as the hoar frost on the ground.
And when the children of Israel saw it, they said one to another, It is manna: for they wist not what it
was. And Moses said unto them, This is the bread which the LORD hath given you to eat.”
7.1. AGGREGATE AND PER CAPITA VARIABLES
77
similar to the reasons we o¤ered for considering the population growth rate, n; as given:
it is the simplest possible case. Chapter 9 will deal with models where technological
progress is an endogenous phenomenon induced by the conscious investment decisions of
economic agents.
7.1
Aggregate and Per Capita Variables
In chapters 4 to 5, we avoided the di¤erence between aggregate and per capita variables
by normalizing the population size (assumed to be constant in these chapters) to 1. However, in the presence of population growth we can only normalize the initial population
level N0 since the population is not constant. Therefore now we need to distinguish
between aggregate and per capita (or, more precisely, per household values). To do so,
we will use capital letters for the former and lower case letters (as we have implicitly
done so far in the book) for the latter. For example the aggregate production function
is now written as:
Yt = Kt (At Lt )1
where Yt is aggregate, economy-wide output, Kt is aggregate capital, and Lt are aggregate
hours worked in the economy.
Dividing the previous expression by Nt , the number of households in the economy,
we obtain the per capita production function:
K (At Lt )1
Yt
= t
Nt
Nt Nt1
=
Kt
Nt
At
Lt
Nt
1
and thus
yt = kt (At lt )1
where yt = Yt =Nt is output per capita, kt = Kt =Nt is capital per capita and lt = Lt =Nt
is hours worked per capita.
Similarly, we can transform the aggregate resource constraint:
Ct + Kt+1 = Yt + (1
) Kt
into a per capita one by dividing by Nt :
Yt
Kt
Ct Kt+1
+
=
+ (1
)
Nt
Nt
Nt
Nt
Ct Nt+1 Kt+1
Yt
Kt
+
=
+ (1
)
Nt
Nt Nt+1
Nt
Nt
or
ct + (1 + n) kt+1 = yt + (1
where we have used the fact that
Nt+1
= 1 + n:
Nt
) kt
78
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
The term (1 + n) in front of capital per capita tomorrow appears because part of our
current investment compensates for population growth. In order to keep capital per
capita kt+1 at the same level as kt ; since there are n more people in period t + 1 than in
period t; society has to spend an extra nkt+1 units of resources.
7.2
The Social Planner’s Problem
We are now ready to study how our basic model, augmented for population dynamics
and technological progress accounts for economic growth observed in the real world. To
most easily do so, we appeal to the welfare theorems that we laid down in chapter 5.
These allow us to use the social planner’s problem to determine how output, capital, and
hours worked evolve over time. After …nding the optimal allocation of these quantities,
we will show how to …nd the prices for inputs and outputs such that these prices, together
with the allocations from the social planner problem, form a competitive equilibrium of
a market economy.
7.2.1
Setup of the Model
Since the aggregate production function and the resource constraint are already in percapita terms, we present the social planner problem in per capita terms. The social
planner aims at picking a consumption, capital and labor path fct ; kt+1 ; lt g1
t=0 to maximize the lifetime utility of the representative household:
1
X
t
u (ct )
t=0
subject to the per capita resource constraint that we derived above (and substituting
the production function in for yt ):
ct + (1 + n) kt+1 = kt At1
The initial level of capital k0 =
social planner).4
4
K0
N0
+ (1
) kt
= K0 > 0 is given (that is, beyond the control of the
As before in chapters 4 to 5 a transversality condition of the form
lim
t!1
t 0
u (ct ) kt+1 = 0
(7.1)
provides a terminal condition for the capital stock that an optimal solution to the social planner has
to satisfy. Together with the necessary …rst order conditions derived below the transversality condition
form a set of su¢ cient conditions for an optimal solution to the planner problem.
The transversality condition simply states that the value of capital in the very long run (as t ! 1)
must go to zero, where the value is measured by the marginal utility of consumption t u0 (ct ) that an
extra unit of resouces would give. If this condition were not satis…ed, the social planner would make the
economy save too much capital in the long run. Note that both marginal utility and kt+1 are positive,
and thus t u0 (ct ) kt+1
0 for all t: Therefore the condition really only imposes that kt+1 is growing
slower than t u0 (ct ) is falling, but it certainly does not requires that kt+1 itself goes to zero in the long
run.
7.2. THE SOCIAL PLANNER’S PROBLEM
79
Several points deserve elaboration. First, the social planner maximizes per capita lifetime utility. Alternatively, some economists postulate that the social planner maximizes
the total lifetime utility of the entire population:
1
X
t
Nt u (ct ) =
t=0
1
X
[(1 + n) ]t u (ct )
t=0
In this alternative the time discount factor is simply multiplied by (1 + n) to account for
population growth. Thus our subsequent analysis goes through completely unchanged
if instead of
we use a modi…ed time discount factor e = (1 + n) : Second, since
households do not value leisure in the utility function it is optimal for the social planner
to let households work full-time. Therefore we have substituted lt = 1 directly in the
production function. Furthermore, since more consumption is better for households
(recall that u() is strictly increasing) no resources will ever be wasted and thus the
resource constraint always holds with equality. since the household has positive marginal
utility, the resource constraint holds with equality.
7.2.2
Analysis of the Model: The Lagrangian and its First Order Conditions
Since the social planner problem is a maximization problem subject to constraints (for
each time period t the resource constraint has to hold), we now formulate the Lagrangian,
with t denoting the Lagrange multiplier associated with the resource constraint in period
t:
1
1
X
X
t
1
u (ct ) +
+ (1
) kt ct (1 + n) kt+1
L=
t k t At
t=0
t=0
with k0 > 0 given. The necessary conditions for an optimal solution, for ct ; ct+1 ; kt+1 and
t ; respectively, are:
ct : t u0 (ct )
t = 0
t+1 0
ct+1 :
u (ct+1 )
t+1 = 0
1
kt+1 :
kt+11 At+1
+1
=0
t (1 + n) + t+1
1
+ (1
) kt ct (1 + n) kt+1 = 0:
t : k t At
Replacing
t
and
t+1
(7.2)
(7.3)
(7.4)
(7.5)
in equation (7:4), using (7:2) and (7:3) yields
(1 + n) u0 (ct ) = u0 (ct+1 ) kt+11 A1t+1 + 1
ct + (1 + n) kt+1 = kt At1 + (1
) kt
(7.6)
(7.7)
We can think about the …rst equation as the Euler equation of the social planner that
equates marginal utility today with marginal utility tomorrow, multiplied by the gross
marginal product of capital. The second equation is just the resource constraint. If we
80
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
solve the second equation for ct and plug it into the …rst (and do the same for ct+1 ), we
obtain:
(1 + n) u0 kt A1t
u
0
kt+1 A1t+1
+ (1
+ (1
) kt+1
) kt
(1 + n) kt+1 =
(1 + n) kt+2
kt+11 A1t+1 + 1
(7.8)
We saw in chapter 4 that this equation is a second order di¤erence equation in kt . Thus,
to solve it, we need two additional conditions. One is given by k0 , the initial and given
level of capital, the second one is a terminal condition (in the form of a transversality
condition) that ensures that the social planner does not make the economy accumulate
too much capital in the long run.
We now proceed to determine the predictions of the model for economic growth in
the real world on two levels of generality (and di¢ culty). First we analyze a situation
where all variables in the economy grow at a constant rate over time, a situation we call
a balanced growth path. This case is very relevant since, as we will show momentarily, in
the long run the economy will approach this situation. We then proceed to the general
(and more technically demanding) analysis of the dynamics of the economy outside the
balanced growth path.
7.3
Balanced Growth Path Analysis
A balanced growth path (BGP) is the natural generalization of a steady state that we
de…ned in chapter 4.
De…nition 21 A Balanced Growth Path is a solution to the social planner problem (or
a competitive equilibrium) in which all the variables in the model grow at constant rates
(where the growth rates may di¤er across variables). That is, an arbitrary variable xt
evolves in the BGP according to
xt = (1 +
where
x
t
x)
x
is the grow rate of xt and x = x0 is the level of the variable at time 0:
Note that a steady state de…ned in chapter 4 is a special case of a balanced growth
path where the growth rate of all variables is x = 0: We now demonstrate that the
neoclassical growth model has a BGP. Later, we will show that the economy will indeed
approach this BGP over time regardless of where it starts from, that is independent of
the initial capital stock k0 .
7.3.1
Determination of Growth Rates in the BGP
First we determine the growth rates that consumption and capital have to have along a
BGP. The resource constraint reads as
ct + (1 + n) kt+1 = kt (1 + g)t
1
+ (1
) kt
7.3. BALANCED GROWTH PATH ANALYSIS
81
where we used our assumption that the level of technology grows at constant rate g; that
is, At = (1 + g)t : Along a BGP consumption ct = (1 + c )t c grows at a constant rate c
and capital kt = (1 + k )t k at a constant rate.
(1 +
t
c)
c + (1 + n) (1 +
t+1
k)
k =
(1 +
= (1 +
t
k) k
t
k)
(1
)t
(1 + g)
where the second line used the mathematical fact that (1 +
ing both sides by (1 + k )t we obtain
1+
1+
t
c
c + (1 + n) (1 +
k) k =
k
(1 +
t
k)
(1 +
1
(1 + g)t
k)
t
) (1 +
k + (1
t
k)
(1 + g)(1
(1 +
+ (1
(1
k)
) (1 +
= (1 +
k)
t
k)
t
k)
t
k
k
: Divid-
)t
)t
k + (1
)k
and thus
1+
1+
t
c
c + [(1 + n) (1 +
k)
(1
)] k =
k
"
1+g
1+ k
1
#t
k
(7.9)
This since is an equation that has to hold for all time periods t and the only terms that
depend on t are
1+
1+
t
c
k
and
1+g
1+ k
1+
1+
c
1
t
we need to have that
=
k
1+g
1+ k
1
= 1:
Otherwise the left and side and the right hand side of (7:9) would grow at di¤erent rates
(note that the second term on the left hand side is a constant) and thus cannot be equal
to each other in all times t:
(1
)t
1+g
(7.10)
1+ k
But equation (7:10) immediately implies that in a BGP it needs to be the case that
g=
k
=
c
That is, in the BGP the growth rate of capital and consumption per capita is the same
and equal to the growth rate of technology. Moreover, output per capita
yt = kt (1 + g)(1
)t
= (1 + g) t k (1 + g)(1
)t
= (1 + g)t k
also grows at rate g: This result is remarkable: regardless of the levels of c and k (and
hence, of the level of savings), or for that matter, regardless of any other parameter in
the model than g, the growth rate of output, consumption and investment in the BGP
is determined exclusively by the rate of technological progress g.
Therefore along a BGP g = k = c and equation (7:9) reads as
c + [(1 + n) (1 + g)
(1
)] k = k :
(7.11)
82
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
Output per capita k is split between consumption per capita c and e¤ective investment
i = [(1 + n) (1 + g) (1
)] k: Note that as long as the growth rate of the population
and of technology are small (say, in the order of 1-2%, then g n = 0:02 0:02 0 and
thus
i = [1 + n + g + ng
(n + g + )k:
(1
)] k
Thus investment in the steady state replaces physical depreciation k; and augments the
capital stock by (n + g)k so that the capital stock per person and level of technology,
k = NKt At t remains constant over time.
Remark 22 (The Limits to the Accumulation of Capital ) The previous discussion also indicates that the neoclassical growth model puts strong limits to the extent to
which capital accumulation can generate long run growth. The most capital the economy
can accumulate is by setting consumption to zero in the long run, c = 0, and use all
output for investment.5 Then equation (7:9) becomes:
#t
"
1
1+g
k
[(1 + n) (1 + k ) (1
)] k =
1+ k
which still implies that
g=
k
=
y;
that is, in the long run capital per capita and out put still do not grow faster than at the
rate of productivity growth g: The level of the capital stock will be higher when setting
c = 0; but not its growth rate. Since the Cobb-Douglas production function implies a
marginal product of capital that decreases with the level of capital, no matter how much
the economy saves, eventually the return of an extra unit of capital falls to the point
where it is equal to the amount required to compensate for population growth, productivity
growth, and depreciation.
The neoclassical growth model therefore imposes severe constraints on how much
growth in per capita output and consumption capital accumulation can generate: nothing
in the long run. The only source of long run growth of output, income and consumption per capita is productivity growth. Furthermore long run growth is exogenous in the
neoclassical growth model, in the sense that the growth rate of technology g is taken as
exogenously given, rather than its origin being modeled. Furthermore, according to this
model, any policy geared towards increasing per capita growth needs to target productivity
growth as it is the sole determinant of the long run growth rate in the economy. This
conclusion contrasts, for example, with the emphasis placed by many international development agencies in the 1950s and 1960s on the rapid accumulation of capital as a path
5
Of couse such a choice would never be optimal since the planner can increase lifetime utility by
letting households consume a little more (due to the Inada condition that implies that the extra utility
from consuming a bit more than zero is very large). But setting c = 0 generates maximum capital
accumulation, so our results are even stronger if we assume c > 0:
7.3. BALANCED GROWTH PATH ANALYSIS
83
to growth, or on the strategy of countries like the Soviet Union of accelerated industrialization. After a period of fast capital accumulation in the short run decreasing returns
to capital set in and growth comes to a halt unless the economy enjoys technological
progress. This prediction of the model is not a bad description, to a …rst approximation, of the experience of the Soviet Union of rapid growth from the 1930s to the 1960s
and stagnation afterwards (see the exposition by Robert Allen in his 2003 book Farm to
Factory, A Reinterpretation of the Soviet Industrial Revolution).
The conclusion (and the implied policy conclusions) that long-run per capita income
growth is driven exclusively by productivity growth is shared by any model with a neoclassical production section. In chapter 9 we will demonstrate that phenomena such as
externalities or increasing returns to scale in production may invalidate this conclusion.
These phenomena are central to Endogenous Growth Theory, as the class of models discussed in that chapter has become to be known.
7.3.2
Wages and Interest Rates in the BGP
Above we showed that the BGP implied by the solution of the social planner’s problem
implies that output, consumption, and capital per capita grow at the common growth
rate g. The welfare theorems tell us that these quantities are also part of a competitive
equilibrium. But what about the input prices of the competitive equilibrium associated
with this solution for consumption, capital and output?
An identical argument as in chapter 5.3 shows that wages and interest rates in the
competitive equilibrium must equal to the marginal products implied by the allocations
(see equations (5:6) and (5:7); but add At to the production function:
rt = k t
wt = (1
1
At1
) kt A1t
= k 1
= (1 + g)t (1
)k
where we made use of the fact that At = (1 + g)t and kt = (1 + g)t k along the BGP. Thus
we see that the real wage wt grows at the same rate g as all the other variables in the
economy, but the interest rate is constant along a BGP. The increases in the marginal
product of capital induced by technological progress At = (1 + g)t are exactly o¤set by
the fall in the marginal product that higher levels of capital kt = (1 + g)t k imply. Note
that a variable that is constant grows at a constant rate of zero, and thus the fact that
interest rates are constant does not contradict that the economy is in a BGP.
7.3.3
Existence of Balanced Growth Path
The previous section showed that if the economy has a BGP, all variables grow at the
rate g of technological progress and the real interest rate is constant. However, we have
not demonstrated that indeed a BGP exists for the neoclassical model, and it turns out
that further assumptions are needed to assure this. Apart from the resource constraint,
the only equation we exploited in the previous section, the necessary (and with the
84
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
transversality condition, su¢ cient) conditions for an optimal solution of the planner
problem include the Euler equation (7:6); which we reproduce here
(1 + n) u0 (ct ) = u0 (ct+1 )
kt+11 A1t+1 + 1
:
1
In a balanced growth path ct = (1 + g)t c and kt+1 = (1 + g)t+1 k and thus kt+11 At+1
=
At+1
kt+1
1
=
1 1
k
=k
1
: Plugging both into the Euler equation yields
(1 + n) u0 (1 + g)t c = u0 (1 + g)t+1 c
or
u0 (1 + g)t+1 c
u0
t
(1 + g) c
=
1
k
(1 + n)
( k 1+1
+1
)
(7.12)
Since the right hand side of this equation is constant, a balanced growth path exists only
if the marginal rate of substitution between consumption at two adjacent dates t and
t+1
u0 (1 + g)t+1 c
(7.13)
u0 (1 + g)t c
is also constant. Thus for a BGP to exist the period utility function u must have the
property that the ratio of marginal utilities is constant along the BGP. Thus a necessary
condition for the model to have a BGP (conditional on all the other assumptions we have
already made) is that (7:13) is constant despite the fact that the expression contains a
t: Reversely, if u satis…es this condition, then a balanced growth path exists.6
Now remember from section 4.2.2 that a lifetime utility function was de…ned to be
homothetic if multiplying consumption levels in two periods, ct and ct+1 by a …xed
number > 0 does not change the marginal rate of substitution. But if the lifetime
utility function is homothetic, then
u0 (1 + g)t+1 c
u0 (1 + g)t c
=
u0 ((1 + g) c)
u0 (c)
1
where we used = (1+g)
t : Note that the expression on the right hand side is constant
and independent of t: Thus homotheticity of the lifetime utility function is the missing
necessary and su¢ cient assumption on the utility function that insures the existence of
a balanced growth path.
Remark 23 As we saw in section 4.2.2, if the period utility function u is of CRRA
1
form, u(c) = c1 ; then the lifetime utility function U is homothetic. In this case
u0 (1 + g)t+1 c
u0 (1 + g)t c
6
=
(1 + g)t+1 c
(1 + g)t c
=
(1 + g)
There are two additional technical conditions that have to be satis…ed, insuring that the balanced
path allocation satis…es the transversality condition (7:1) and that the objective function
P1 growth
t
u(c
)
is
a …nite number when ct = (1 + g)t c: We return to this issue below.
t
t=0
7.3. BALANCED GROWTH PATH ANALYSIS
85
1
Without proof we state that the CRRA utility function u(c) = c1 is essentially the only
period utility function that makes the lifetime utility function U homothetic. Thus any
model that is meant to have a balanced growth path needs to adopt the CRRA utility
function, which partially explains its widespread use in modern macroeconomics.7
7.3.4
Full Characterization of the Balanced Growth Path
Now we can give a full solution of the balanced growth path. Since a CRRA utility
function is required for the BGP to exist, we restrict attention to this utility function.
With this utility function we can rewrite the Euler equation (7:12) as
(1 + g)
=
(1 + n)
( k 1+1
(7.15)
)
From this expression we can, with some algebra, determine the initial level of capital k
in the BGP as
1
1
(7.16)
(1 + n) (1 + g)
(1
)
Once we have determined k, from the resource constraint (7:11) along a BGP, we determine the initial levels of consumption and investment (saving) as
k=
c = k
((1 + g) (1 + n)
i = ((1 + g) (1 + n) (1
(1
)) k
)) k
(7.17)
(7.18)
and output from the production function
1
y=k =
(1 + n) (1 + g)
(1
)
(7.19)
7
We now return to the issue of making sure that lifetime utility is …nite if consumption follows a balanced growth path and that the balanced growth path consumption allocation satis…es the transversality
condition. Lifetime utility is given by
1
X
t (c(1
t=0
=
1
X
c1
1
1
+ g)t )
1
t
(1 + g)1
t=0
1
and thus is …nite as long as (1 + g)
< 1: Incidentally the same condition guarantees that the
transversality condition (7:1) is satis…ed since
t 0
lim
t!1
lim
t!1
t
c(1 + g)t
ck(1 + g) lim
u (ct )kt+1
=
0
k(1 + g)t+1
=
0
=
0
(1 + g)1
t
which is true as long as
(1 + g)1
< 1:
(7.14)
86
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
as well as the input prices that decentralize the equilibrium:
r = k
w = (1
1
)k
Thus we have determined the level of all variables (k; c; y; r; w). Since per capita variables
in the BGP grow at the rate (1 + g) and aggregate variables grow at rate (1 + g)(1 + n)
we have
kt
ct
yt
rt
wt
=
=
=
=
=
(1 + g)t k and Kt = (1 + n)t (1 + g)t k
(1 + g)t c and Ct = (1 + n)t (1 + g)t c
(1 + g)t y and Yt = (1 + n)t (1 + g)t y
k 1
(1 + g)t (1
)k
(7.20)
(7.21)
(7.22)
(7.23)
(7.24)
The net growth rate of the aggregates variables is therefore given by
(1 + g) (1 + n)
1
g+n
and thus by the sum of growth rate of technology g and the growth rate of population
n, plus the cross-product term g n which is very close to zero as long as n and g are
not too large. Just compare n + g and (1 + g) (1 + n) 1 if, say, n = 0:01 = 1% and
g = 0:02 = 2%:
From the Euler equation (7:15); using (7:23) we can derive a simple and easy to
interpret expression for the interest rate along a BGP:
(1 + g)
and thus
1+r =
1
=
(1 + n)
(1 + r)
(1 + n) (1 + g)
(7.25)
From this expression, we can see that the interest rate depends positively on the
growth rate g of the economy. As g increases, income yt of the representative household
grows faster, the household is richer and wants to consume more today. However, since
higher g only means higher output in the future, the household has to be persuaded not
to consume more despite feeling richer. This can be done by making consumption today
more expensive relative to future consumption, that is, by raising the interest rate. Note
that the bigger is , the lower is the Intertemporal Elasticity of Substitution (IES), the
more strongly the household wants to smooth consumption and consume more today.
Thus a stronger increase in the interest rate r for a given increase in g is required to
persuade the household not to consume more today. Thus, as the formula shows, a given
increase in g raises r the more, the higher is (that is, the lower is the IES).
7.3. BALANCED GROWTH PATH ANALYSIS
87
Remark 24 (The Solow Model) In the neoclassical growth model presented so far
households (or the social planner for them) decide optimally how much to save. In
the Solow model, named after economist Robert Solow from MIT the saving rate s; the
fraction of income (output) that is being saved is a …xed number that is taken as given.8
It is now straightforward to relate our model with optimal saving to the Solow model.
Recall the resource constraint along a BGP, equation (7:11)
c=k
((1 + g) (1 + n)
(1
)) k:
In the Solow model a fraction s of output k is saved by assumption, and thus a fraction
1 s is consumed. Thus c = (1 s)k : Plugging this in yields
(1
s)k = k
((1 + g) (1 + n) (1
((1 + g) (1 + n) (1
)) k = sk
)) k
and thus the BGP initial capital stock and output in the Solow model is given by
k
Solow
y
Solow
=
sSolow
(1 + g) (1 + n)
=
sSolow
(1 + g) (1 + n)
1
1
(1
(7.26)
)
1
(1
(7.27)
)
Comparing (7:26) with (7:16) we see that they are exactly the same as long as the assumed
savings rate sSolow in the Solow model is equal to the optimally chosen savings rate by
the households (the planner) in our model
s=
(1 + g) (1 + n)
(1 + n) (1 + g)
(1
(1
)
)
:
Thus the only di¤erence in the two models, along a BGP, is that our neoclassical growth
model the fraction of income being saved is not simply assumed, but derived from the
underlying technology and preference parameters ( ; ; ; g; n; ) characterizing the economy. The simplest expression is derived when capital depreciates fully within one period,
and thus = 1: In that case
s=
(1 + g)1
and the savings rate is the higher the more patient households are (the higher ) and the
more important is capital in the production function (the higher is ). The impact of the
growth rate g on the savings rate depends on the size of and thus the size of the IES.
For = 1 (log-utility) the growth rate g has no impact on the savings rate. If the IES is
8
Historically, the Solow model comes …rst (in a famous paper by Robert Solow in 1957). Solow
emphasis was on the study of the role of capital accumulation in economic growth and he considered
that, for that purpose, one could take the savings rate as given. Only later in the 1960s economists such
as David Cass or Tjalling Koopmans extendeded the Solow model to have intertemporal optimization
and create, in that way, the neoclassical growth model.
88
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
smaller than 1 (i.e. > 1) then and increase in the growth rate reduces the savings rate
and if the IES is larger than 1 an increase in the growth rate increases the savings rate.
Intuitively, for small IES households want to have very smooth consumption. An increase in the growth rate of income, making households richer, thus triggers a big desired
increase in current consumption, reducing the savings rate. Reversely, if households are
happy to tolerate strongly growing consumption over time (high IES, < 1) then current
consumption increases less strongly than current income, and the savings rate s increases
as g increases.
Remark 25 (The Social Planner and the Golden Rule - Again) The social planner picks consumption and investment levels to maximize the utility of the representative
household along the whole path of the economy. As already shown in section 5.4.2, the
choices made by the social planner do not maximize the level of consumption of the
household in the long run. We now revisit this issue in our model with long-run growth.
Recall that from the resource constraint in the BGP, equation (7:11), BGP consumption is given by
c=k
((1 + g) (1 + n) (1
)) k:
In order to determine the traditional golden rule capital stock we take …rst order conditions with respect to k on the right hand side and set it to zero, which yields:
1
k
((1 + g) (1 + n)
(1
)) = 0
and thus
1
1
kGR =
(1 + g) (1 + n)
(1
)
:
Note that without population growth and technological progress, n = g = 0 we obtain our
1
1
: As before we can derive the saving
previous result from section 5.4.2, kGR =
rate associated with the golden rule capital stock as follows. Investment and thus saving
in the economy is given by (see equation (7:18)):
i = ((1 + g) (1 + n)
(1
)) k
and thus the saving rate associated with the golden rule capital stock is given by
((1 + g) (1 + n) (1
yGR
kGR
kGR
1
= ((1 + g) (1 + n) (1
)) kGR
=
sGR =
i
=
i
=
)) kGR
where the last equality follows from plugging in the value of kGR : As before in the model
without growth the savings rate that maximizes consumption in the long run (the BGP)
is equal to the elasticity of output with respect to capital, : Comparing the golden rule
7.3. BALANCED GROWTH PATH ANALYSIS
89
capital stock kGR to the optimal capital stock k in the BGP we …nd that
1
1
kGR =
k =
"
(1 + g) (1 + n)
(1+g)
(1 + n)
(1
(1
)
)
#1 1
and thus kGR > k as long as
(1 + g)
> (1 + g)
1 >
(1 + g)1
:
But this is exactly the condition (see equation (7:14)) that we need to insure that the
social planner problem has a solution in the …rst place. Thus we conclude that, as in
the model without growth, the social planner …nds it optimal, in the long run, to select
a capital stock that is lower than the one that maximizes per capita consumption in the
long run. Note that the golden rule and the optimal saving rate satisfy
sGR = ((1 + g) (1 + n)
s = ((1 + g) (1 + n)
(1
(1
1
)) kGR
)) k 1
and since kGR > k it follows that sGR > s; that is, the social planner saves a smaller
share of output in the BGP than the share that maximizes output (and consumption) per
capita.
But why is the social planner doing this. Remember this social planner is benevolent
(and very good at solving maximization problems). But by selecting a lower value of k
than the golden rule kGR in the long run means that starting from a level of capital k0 < k
less capital will need to be accumulated in the short run than if a capital stock kGR is
built up. But less capital accumulation means less investment and more consumption in
the short run, which is bene…cial for households, especially if they are impatient. Thus
the consumption and utility losses in the BGP from having less consumption than the
golden rule is (more than) compensated by the higher utility along the path towards the
BGP (the so-called transition path).
Remark 26 (Growth Accounting and the BGP of the Neoclassical Growth Model)
Using the Cobb-Douglas production function we can perform a simple decomposition of
income growth into its underlying sources. Such an exercise is called growth accounting.
Output and income in the economy is given by the Cobb-Douglas production function:
Yt = Kt (At Lt )1
:
Taking logs on both sides yields
log(Yt ) =
log(Kt ) + (1
) log(Lt ) + (1
) log(At ):
90
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
Similarly for period t
1
log(Yt 1 ) =
log(Kt 1 ) + (1
) log(Lt 1 ) + (1
) log(At 1 ):
Subtracting the second equation from the …rst yields
log(Yt )
log(Yt 1 ) =
[log(Kt ) log(Kt 1 )]
+(1
) [log(Lt ) log(Lt 1 )]
+(1
) [log(At ) log(At 1 )] :
(7.28)
By our assumptions
log(Lt )
log(At )
log(Lt 1 ) = log (1 + n)t
log(At 1 ) = log (1 + g)t
log (1 + n)t
log (1 + g)t
1
1
= log(1 + n)
= log(1 + g)
Now we make use of the mathematical facts that the growth rate of an arbitrary variable
xt is given by
xt xt 1
gx =
log(xt ) log(xt+1 )
xt 1
and
log(1 + gx ) gx
as long as gx is not too large.9 Then we can rewrite (7:28) as:
gY = gK + (1
) n + (1
)g
A naïve interpretation of growth accounting will interpret this result as indicating that
technological progress, g, only contributes (1
) g towards economic growth and not g,
as implied by the neoclassical growth model.
How can this be? The simple resolution of this puzzle is that along a BGP the growth
rate of capital is given by gK = g + n and thus
gY
gY
= (g + n) + (1
= g + n:
) n + (1
)g
9
As long as gx is not too di¤erent from zero we can precisely approximate log(1 + gx ) by a …rst order
Taylor series expansion about gx = 0:
log(1 + gx )
log(1 + 0) +
1
(gx
1+0
0) = gx :
Furthermore
log(xt )
log(xt+1 )
=
=
xt
= log
xt 1
log(gx + 1) gx :
log
xt
xt
xt
1
+1
1
If the math does not convince you, take a pocket calculator and di¤erent values for gx ; say gx =
0:001 = 0:1%; gx = 0:01 = 1% and gx = 0:1 = 10% and compare gx to log(xt ) log(xt+1 ) and
log(1 + gx ):
7.4. CALIBRATION
91
But why is Kt growing? Because of technological progress (g) and population growth (n).
Thus, technological progress contributes (1
) g to growth directly through making labor
more productive, and g indirectly through inducing capital accumulation. This example
shows how the results of even such a relatively atheoretical exercise as growth accounting
requires the input of economic theory, in order to be interpreted adequately. In other
words, it is hard to …nd cases in economics where the “data speaks for itself” without the
need for help from economic theory.
7.4
Calibration
So far we have derived very precise predictions from our model about how consumption,
capital and output (per capita) as well as real wages and real interest rates should grow
in the long run in the economy. However, all these predictions are phrased in terms of
the six parameters describing the economy, two describing preferences ( and ), three
describing technology (g; ; and ), and one specifying the growth rate of population,
n. So far we did not specify concrete values for these parameters. Hence, it is di¢ cult
to interpret the exact predictions of the model and compare them to the real world.
Therefore, a natural next step is to select particular parameter values and exploit the
consequences of these choices. Such exploration is sometimes called quantitative theory
since it aims at evaluating the quantitative predictions of the theoretical model relative
to the real-world data.
There is a large literature in macroeconomics on how to select parameter values.
Much of this literature that we cannot review within the scope of this book employes
sophisticated econometric techniques in order to select parameter values so that the
associated competitive equilibrium (or solution to the social planner problem) …ts the
data best.10
Instead we introduce a simple yet powerful, and equally frequently used technique
to select parameter values based on observed data, known as calibration, that does not
rely on elaborated statistical tools. The basic idea of calibration is to choose parameters
such that the quantitative predictions of the model match the data along a selected set
of dimensions. Of course these dimensions can then not be used to evaluate the success
of the model to explain the data, since the model does so by construction, that is, by
choice of the parameter values.
More concretely, calibration selects parameter values based on three di¤erent, but
related criteria. First, certain parameter values in the model have counterparts in the
data. that we can directly measure. For example, we can consult U.S. population data
and measure directly population growth rates. Similarly we can consult the U.S. NIPA
to measure growth rates of per capita output. The measured observations from the
data correspond directly to n and g in our model. In order to make this procedure
operational we …rst have to take a stand how long one time period is in the model, since
10
The notion of …t and best of course have to be made operational, which is often done by maximizing
the likelihood function of the model or by miniming the weigted sum of deviations of model-implied
moments from their empirical counterparts.
92
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
obviously population and output grow by more from one period to the next the longer is
the time period. For growth models, a natural unit of time is one year, since quarterly
observations (the highest frequency that we have of most economic series) are relatively
rare outside the U.S. and even for the U.S. don’t go back too far in history. In contrast,
reasonably good annual data are available for many countries, and for the U.S. for much
longer time periods.
Once we agree that a period in the model (and data) should last one year, we have
to decide how to deal with the fact that in the data population and per capita income
growth rates vary signi…cantly from one year to the next, while they are assumed to be
constant in the model (at least along the BGP). Since the neoclassical growth model is
meant to explain long-run growth rather than short-run ‡uctuations it is natural to use
as empirical targets for n and g the long-run average population and per-capita output
growth rates in the data.
The average population growth rate in the U.S. for the last century has been roughly
1% per year, and thus we choose n = 0:01 for our model. Similarly, the per capita
growth rate in the period 1865-2007 has been around 1:9% on average, also per year.
Therefore we set g = 0:019: Note that this choice for our model implicitly assumes that
the U.S. economy has been traveling through its BGP for the last century and a half,
since in our model the per capita growth rate of output equals to g only in the BGP.
We argued in chapter 6, though, that the U.S. per capita income has been growing quite
steadily around a 1:9% trend per year over that period. Consequently, thinking about
the U.S. economy as being in its BGP is a good …rst approximation that we impose for
our calibration. Summarizing, we choose the values of the two growth rates n and g in
the model equal to their direct counterparts in the data.
A second criterion to calibrate a subset of the model parameters is to exploit (less
direct) predictions implied by the model for variables we can also measure in the data.
For example, one direct result from the model, already discussed in chapter 4, that is
implied by the Cobb-Douglas production function and competitive behavior of …rms is
that the labor share of income,
wL
=1
Y
is constant over time and equals 1
in the model. The labor share of income can
be readily measured for the actual U.S. economy. Doing so we observe that in the data
labor share has been roughly constant around 0.7, and thus we choose = 0:3 for the
model.
Following the same principle to exploit model predictions to choose model parameters,
we observe from the resource constraint that investment i plus consumption c equals
output k : Thus
c+i=k
i=k
c
and substituting for c from (7:17); we obtain investment as
i = ((1 + g) (1 + n)
(1
)) k:
7.4. CALIBRATION
93
Therefore
i
= (1 + n) (1 + g) 1 + :
k
This expression relates the investment to capital ratio to the parameters n; g; . Dividing
both i and k by output y we obtain
i=y
= (1 + n) (1 + g)
k=y
or:
=
i=y
k=y
1+
(1 + n) (1 + g) + 1:
This expression is useful because it relates the yet unchosen parameter to the share of
output that is being invested, i=y, to the capital-output ratio k=y and to the (already
calibrated) parameters n; g: The U.S. data show that roughly 20% of GDP is made up
by investment (including residential …xed investment), i=y ' 0:2 and that the value of
the U.S. capital stock (including residential homes) is about 2.5 times annual GDP, i.e.
k=y ' 2:5: Note that we omit the discussion of important details about how exactly i and
k are measured; the interested reader should review chapter I for further details. Using
these observations from the data and the calibrated values n = 0:01 and g = 0:019, we
obtain for the depreciation rate
=
0:2
2:5
(1:01) (1:019) + 1 = 0:05
a value of 5% per year.
Having chosen values for the technology parameters ( ; ; g) and the population
growth rate n leaves us with the preference parameters ( ; ) to be determined. The crucial implication from the household maximization problem is the Euler equation, which
in the BGP reads as (see (7:25)):
r=
1
(1 + n) (1 + g)
1
(7.29)
The number r is the annual real interest rate that households face in the economy. The
model does not include risk, and thus r is a risk-free interest rate. At the same time,
r is also the net (of depreciation) return on capital in the model. Picking an empirical
counterpart for r and thus a target value to be matched is not straightforward. On one
hand T-Bills issued by the U.S. Treasury carry an average annual real return of roughly
1%, but even this return is not entirely risk-free (since it carries a trivial default risk
and a more considerable in‡ation risk11 ). On the other hand, interpreting the value of
the capital stock as the value of all …rms owning the capital stock, and thus as the stock
market value of …rms, the annual average real return on capital, as measured by the
return on the stock market, is about 7%; but clearly this return is not risk-free. Thus
11
We will come back to this issue in much more detail when we deal with asset prices in chapter 20.
94
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
even ignoring the issue of capital income taxes, the range of possible choices for a target
value for r includes at least the interval [0:01; 0:07]:
For concreteness and as a compromise let us choose a target of r = 4%: Then equation
(7:29) becomes12
1
1
0:04 = (1:01) (1:019)
which generates a relation between
and
of the form:
(1:019)
=
1:04=1:01
(7.30)
Unfortunately, this expression does not provide us with a value for each parameter
separately. In these situations macroeconomists often use a third source of information to calibrate parameters: independent microeconometric evidence. Over the years,
economists have used microeconomic household-level consumption and savings data to
estimate the values of di¤erent parameters, in particular the intertemporal elasticity of
substitution 1 . Based on these estimates many researchers have concluded that the plausible range of values for ranges from roughly 1 to 3 (that is, the IES ranges from 1=3 to
1): Once one chooses a value for from this range, one can then exploit the relationship
(1:019)
= 1:03
to determine the associated value for : Equivalently, one may use (less frequent) microeconometric evidence on and use the relationship above to pin down :
Remark 27 (Microeconometric Evidence for Parameters in Macro Models) Using
the third source of information for the calibration of our macroeconomic model, borrowing
from microeconometric evidence, is the most controversial among economists. A group of
researchers, like Victor Ríos-Rull, at the University of Minnesota, holds the strong view
that parameters are only de…ned within the context of a particular economic model. They
argue that just because we call parameters in two di¤erent models with the same greek
letter, this does not imply that the parameter should take the same value. Parameters are
not model-free. This argument becomes clear by looking again at equation (7:29): Suppose
we consider two models, the …rst one being the one we have considered so far, the second
12
When choosing parameters through calibration it also matters how we choose the objective function
of the planner. If the planner maximizes total lifetime welfare
X
t
(1 + n)t u(ct )
t
instead of per capita welfare, equation (7:29) becomes
r=
1
(1 + g)
1
and the parameter values discussed below have to be adjusted appropirately.
7.4. CALIBRATION
95
one being identical, but abstracting from population growth (perhaps because the model
builder thinks it is irrelevant for the question at hand). Letting = 1 and aiming for the
same target of r = 4% in the model without population growth a = 0:98 is required.
Thus it may make little sense to use an estimate of derived from a individual data and
microeconomic model that incorporates population growth for a macro model that does
not. In general, this group of economists tends to rely, for calibration, exclusively on the
…rst source (directly observed parameters) and the second source (restrictions implied by
the theory on observables) of information for calibration.
A second group of critics on use of microeconometric parameter estimates for macro
models, such as James Heckman, at the University of Chicago and Nobel prize winner in
2000, base their rejection of the idea on the observation of substantial heterogeneity of behavior at individual household level which is not explained by standard models and perhaps
driven by heterogeneity of preference parameters between households. When households
have very di¤erent preference parameters, it is unclear which households should serve
to determine parameter values of a macroeconomic model. Simply averaging the values
may be a bad idea. For example, if our current model would be populated by households
that di¤er only along their patience (one group very patient, with a time discount factor
high , one with low time discount factor
low ), one can show that the aggregate economy (at least in the long run) will behave nearly the same as an economy in which all
households have high time discount factor high . Hence knowing the value low or even
the mean of low and high is inconsequential for a macroeconomist, and certainly should
not be used in the analysis. The practice of using microeconometric evidence to calibrate
macro models is typically defended by pointing out that often microeconomic evidence is
the only source of information readily available for particular parameters, and that the
use of limited or potentially biased information is better than simply choosing arbitrarily.
Moreover, micro evidence often be used as a plausibility check of choices made using information sources 1 and 2. At a more formal econometric level, defenders of the practice
also argue that microeconometric evidence can be used to form priors used for estimating
our macroeconomic models using econometric methods. The interested reader is referred
to Sungbae An and Frank Schorfheide (2007) for further details.
Remark 28 (IES and International Interest Rate Di¤erences) Robert Lucas (1986),
at the University of Chicago and Nobel prize winner in 1996, presents an additional powerful argument in favor of low values of (high IES) based on macroeconomic data. He
notes that there are large di¤erences in growth rates across di¤erent countries. According
to the BGP of the model (see equation 7.25):
1+r =
1
(1 + n) (1 + g) :
Taking logs on both sides yields
log(1 + r) = log(1 + n) + log(1 + g)
log( )
and using the approximation that for a variable x close to zero, log(1 + x)
r =n+ g
log( ):
x we have
96
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
Now take two countries i and j: Assuming that the preference parameters ;
across countries we have that
ri
rj = n i
nj + (gi
don’t vary
gj ):
Since per capita growth rates di¤er substantially across countries, yet di¤erences in population growth rates is fairly small, under the assumption that all countries under investigation are in the BGP (and thus their per capita income growth rates equal gi and
gj ; respectively) large di¤erences gi gj are associated with large di¤erences in interest
rates across countries if is large. In the data, di¤erences in interest rates (returns
on capital) across countries are modest. But these modest di¤erences across countries
in interest rates are only consistent with large di¤erences in per capita income growth
rates, according to the model, if is not too large. Based on the empirical evidence
for cross-country growth rate and interest rate di¤erentials Lucas argues for a value of
= 1:
If we postulate that countries that grow fast may not be in their BGP but still approaching it from below, the argument is even stronger. As we will show in the next
section, a country with income per capita below its BGP not only has a higher growth
rate than in the BGP, but also a higher interest rate r. Thus the currently observed
interest rate di¤erentials are than an upper bound for the eventual (once every country
is in the BGP) di¤erences in r.
This argument so far is based on the assumption that every country is a closed economy. If capital can ‡ow between countries (that is, if we give up the assumption of closed
economies) then these international movements of capital may equate interest rates r
across countries, regardless of the value takes. The amount of capital that would need
to ‡ow across countries to equate interest rates is much larger than the ‡ows we actually
observe in the data, according to Lucas. The actual ‡ows are not nearly enough to wash
out the large di¤erences in r that the model would predict if households had a high IES.
Now suppose we take a stand on and choose the familiar case = 1, that is,. a
logarithmic period utility function. We then obtain
0:99, which nicely …ts intuition
and evidence regarding . If we pick larger values of the implied values for also
rise, as evident from (7:30): The intuition for this result was explained in the previous
subsection. A higher means lower a IES, and thus less willingness to tolerate consumption growth. Thus either the interest rate has to rise to persuade the household to forgo
higher consumption today, or, if we want to target a given interest rate (say r = 4%) in
the calibration, the household has to be made more patient, that is has to rise. But
note that as we increase , the associated discount factor approaches 1. Note that
even if we are willing to push all the way to 1, the upper bound that we set when
we presented the basic model, the neoclassical growth model together with the desired
target for the interest rate imply an upper bound for : To see this, plug in = 1 into
equation (7.30), which gives, after taking logs:
=
log (1:03)
= 1:57
log (1:019)
7.4. CALIBRATION
Thus, any value of
than 4 percent.
97
above 1.57 will generate a BGP interest rate in the model higher
Remark 29 (More on the Discount Factor and IES) Alternatively, we can allow
> 1; which will not cause further theoretical problems as long as lifetime utility
P1
t
u (ct ) is a …nite number. We showed above that the condition (see equation (7:14))
t=0
(1 + g)1
<1
is necessary and su¢ cient for this. Equivalently, we need:
< (1 + g)
1
(7.31)
Consequently, we can make almost as big as (1 + g) 1 and still satisfy condition (7:31).
To see the range of target interest rates we can match now, recall that the interest rate
is given from (7:29) by
1
1:
r = (1 + n) (1 + g)
Now impose that < (1 + g) 1 ; which gives a restriction on the target interest rate as
(since r depends negatively on )
(1 + n) (1 + g)
(1 + g) 1
= (1 + n) (1 + g)
r >
1
1
or
1 + r > (1 + g)(1 + n)
Note that this expression does not depend on ; but purely on the readily observable real
interest rate one wants to match in the data, and the output per capita growth rate g and
the population growth rate. With the observed g = 1:9% and n = 1%; the equation says
that the lowest BGP real interest rate that can be attained in our model with population
growth and technological progress is
r = (1 + g)(1 + n)
1
3%
Thus it is impossible to generate real interest rates as low as 1%, the average real interest
rate on government treasury bills (so call T-bills) in the data.13 In addition, if we want
13
Note that letting the planner maximize total instead of per capita lifetime utility does not relax this
constraint. On one hand the BGP interest rate in this version of the model is given by
1+r =
1
(1 + g)
but on the other hand the requirement that the objective function remains …nite requires
(1 + n)(1 + g)1
< 1:
98
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
Calibrated Parameter Values
g
n
0:97 1 0:019 0:3 0:05 0:01
Table 7.1: Parameter Values
to generate interest rates as low as r = 3% with ’s signi…cantly bigger than one (say,
with = 5) one needs implausibly large ’s (with = 5 we need = 1:077). There
is nothing wrong mathematically choosing such a high ; but it means that households
are happier consuming the same amount of consumption tomorrow than today. For most
economists this de…es basic plausibility and introspection, but again, from a formal model
point of view there is nothing wrong with that choice.
To summarize our most preferred calibration of the neoclassical growth model, used
to deduce the quantitative implications of the model Table 7.1 below lists our chosen
parameter values.
7.5
Comparative Statics of the BGP
Now that we have an explicit solution for the BGP of the model, and a sense of reasonable
choices of the parameters of the model we can investigate how the BGP of the model
changes as the model parameters change. This is particularly instructive if we want to
use the model to explain di¤erences in income per capita levels and growth rates across
countries in the data with the model.
Recall from section 7.3, equations (7:19) and (7:22); that per-capita and total output
of a country, according to the model, is given by
yt = (1 + g)t y = (1 + g)t
Yt = (1 + g)t (1 + n)t
1
(1 + n) (1 + g)
(1
(7.32)
)
1
(1 + n) (1 + g)
(1
)
:
(7.33)
The …rst prediction of the model is that once a country has reached its balanced growth
path, the grow rate of per capita income yt is exclusively determined by the growth
rate of technological progress g: Furthermore, as long as all countries have access to the
Imposing this requirement leads to
1
(1 + g) > (1 + n)(1 + g)1
1+r
=
1+r
> (1 + n)(1 + g)
(1 + g)
and the lower bound on the interest rate that can be attained via choices of the parameter values remains
the same.
7.5. COMPARATIVE STATICS OF THE BGP
99
same production technologies (and thus face the same g), the model predicts that all
countries’ per capita incomes eventually (once they all have reached the BGP) should
grow at the same rate, the common rate of technological progress g: Put another way,
di¤erences in long run growth rates or per capita income across countries can only be
due to cross-country di¤erences in their growth rates of technology. The grow rates of
total GDP (income) can of course di¤er because of di¤erences in population growth rates
n across countries. The fact that in the model economic growth eventually is completely
determined by a force (technological progress) that itself is assumed exogenously, rather
than explained, within the model makes the neoclassical growth model an exogenous
growth model. In section 9 we present an endogenous growth model in which long run
economic growth is still driven by technological progress, but in which this progress is
the outcome of conscious and costly research and development decisions by …rms.
This result, on …rst sight, is astonishing. It says that how patient households are
(the size of their ), how willing they are to tolerate consumption swings over time (the
size of their ), how important is capital in production and how fast capital depreciates
(measured by and ; respectively), and how fast the population is growing (the size of
n) is irrelevant for long run growth in per capita income.
However, this does not mean that these parameters are irrelevant. As equation (7:32)
shows they crucially determine the level of per capita income and thus economic living
standards in a country. If two countries always grow at the same rate g; but one country
starts with double the initial level than another country, then it will be double as rich
as the other country forever. Thus studying what determine income levels y across
countries is important.
Rewriting
1
y = k =
=
1
(1 + n) (1 + g)
(1 + n) (1 + g) +
1
(1
!1
)
we see that the income level of a country changes positively with the capital intensity
of its production and the degree of patience of its households (note that a higher
reduces the denominator and thus increases y). It falls with the depreciation rate ;
the population growth rate n and declines with (that is, it rises as the intertemporal
elasticity of substitution 1 increases).14
Since the level of per capita output is determined the by the level of per capita
capital, whatever is good for capital accumulation increases the income level in the BGP.
If households are more patient, they are more willing to forgo consumption today for
higher saving and investment, and thus a higher long run capital stock and consumption
14
We don’t stress the importance of g for the income level since g also a¤ects the growth rate of
output per capita. Once can show that increasing g increases yt ; because the e¤ect on (1 + g)t more
than outweighs the e¤ect on y:
100
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
in the (long run) future. A lower (a higher IES) increases the willingness to tolerate a
consumption pro…le that is uneven and increases over time, towards the BGP.
On the production technology side, the slower capital depreciates, the better it is
to accumulate more of it. Thus a lower increases the steady state capital stock and
output. The same is true the more important is capital in production (that is, the higher
is ). Finally, a higher population growth rate n requires more capital accumulation just
to keep per capita capital and output the same. A higher n thus e¤ectively acts as a
higher depreciation rate and reduces per capita capital and income levels.
In order to assess whether the neoclassical growth model provides a good description
of reality we can check whether real world data about per capita income levels and
growth rates across countries bear out these predictions. We return to this in chapter
8. Before doing so we now verify that our focus on the BGP in most of this chapter is
warranted, by showing that the model economy in the long indeed approaches this BGP,
independent from what initial capital stock it starts.
7.6
Transitional Dynamics
In this section we expand on our discussion from chapter 5.5 and discuss the dynamic
transition path that the economy will take from an arbitrary initial capital stock k0
towards the BGP discussed above. The two crucial equations describing the dynamics
of the economy are the Euler equation and the resource constraint (7:6) and (7:7) which
we reproduce here. Since we have shown above that the existence of a BGP requires a
1
utility function of the constant relative risk aversion form u(c) = c1 : We assume this
utility function from now on. Then the Euler equation and resource constraint read as
(1 + n) (ct ) = (ct+1 )
kt+11 A1t+1 + 1
ct + (1 + n) kt+1 = kt At1 + (1
) kt
(7.34)
(7.35)
Together with the initial condition for the capital stock k0 and a terminal condition (the
mysterious transversality condition) these equations determine the paths of per capita
consumption and capital fct ; kt+1 g:
As we discussed in chapter 5.5, there are three ways of trying to obtain a solution to
these equations:
1. For speci…c examples we can guess that the solution takes a simple form (e.g.
that households consume and save a constant fraction of output (income) in every
period) and then verify that this solution indeed satis…es equations (7:34)-(7:35):
The next section presents an extension of the example from chapter 5.5 where this
can be done.
2. The key mathematical di¢ culty is that the equations (7:34)-(7:35) are not linear in
the variables of interest (ct ; ct+1 ; kt ); that is, these variables appear in the equations
raised to powers di¤erent from 1. The second approach is to replace these nonlinear
7.6. TRANSITIONAL DYNAMICS
101
equations with linear approximations, that is, make them linear. Of course, these
linear equations and their solutions do not exactly solve the true equations (7:34)(7:35) and thus are not the exact solutions to the social planner problem, but the
hope is that this approximation is “not too bad”. We will discuss the general idea
and the results from this approach in the subsection 7.6.2 below qand relegate the
somewhat tedious (but conceptually straightforward) details of the procedure to
the appendix of this chapter. Note that this approach is very frequently used in
applied research these days, and typically implemented on a computer (although,
as we demonstrate, it often can be done by pencil and paper and a su¢ ciently
patient researcher).
3. For relatively simple models such as the one in this chapter, one can use numerical approximation techniques and a computer to solve for the exact solution to
the original nonlinear equations (7:34)-(7:35): A discussion of these computational
techniques is beyond the scope of this book. The interested reader is refered to the
advanced textbooks by Judd () or Heer and Maussner (). Note that although these
techniques …nd solutions to the correct equations, these solutions are numerical and
hence they do not yield an analytic expression.
7.6.1
An Example with Analytical Solution
In this subsection we continue our example from section 5.5 in our current context,
but we extend it with population growth and technical progress to talk about economic
growth.
Remember that we have:
(1 + n) (ct ) = (ct+1 )
kt+11 A1t+1 + 1
ct + (1 + n) kt+1 = kt At1 + (1
) kt
t
At = (1 + g)
Then, we make two assumion, one on preferences,
nology, = 1. Therefore, we get:
= 1 (log utility) and one on tech-
1
1+n
=
k 1 (1 + g)t+1
ct
ct+1 t+1
ct + (1 + n) kt+1 = kt (1 + g)t
1
1
or
1+n
=
ct
1
ct+1
kt+1
(1 + g)t+1
ct + (1 + n) kt+1 = (1 + g)t
1
kt
(1 + g)t
102
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
Now, we de…ne, for an arbitrary variable xt
x
et =
xt
(1 + g)t
which is the variable xt divided by the trend of the economy along the BGP at the time.
Basically with these step, we transform the problem from one where variables such as xt
grow over time into one where variables such as x
et do not (more precisely, we have made
the problem stationary), a step that enormously simplify the analysis. We will refer to
x
e as the re-scaled value of xt .
Then:
e
kt+11
(1 + g)t+1e
ct+1
t
t+1
ct + (1 + g) (1 + n) e
kt+1 = (1 + g)t e
kt
(1 + g) e
1+n
=
(1 + g)te
ct
Thus, simplifying the previous expressions
e
kt+11
1
=
e
ct
(1 + n) (1 + g) e
ct+1
e
ct + (1 + n) (1 + g)e
kt+1 = e
k
t
or
1
e
kt
(1 + n) (1 + g)e
kt+1
=
(1 + n) (1 + g) e
kt+1
which closely resambles the result in section 5.5:
1
kt
kt+1
=
e
kt+11
(1 + n) (1 + g)e
kt+2
kt+11
kt+1 kt+2
(7.36)
except for the terms taking care of technology and population growth and that we are
dealing with a problem in re-scaled variables. In that section 5.5, we guessed and veri…ed
that the solution was
kt+1 =
kt :
Thus, as a natural extension, we now guess:
e
kt+1 =
(1 + n) (1 + g)
and, by the resource constraint of the economy,
e
kt
e
ct + (1 + n) (1 + g)e
kt+1 = e
kt )
e
ct = e
k
((1 + n) (1 + g)e
kt+1
t
e
ct = (1
)e
kt
(7.37)
7.6. TRANSITIONAL DYNAMICS
103
Our guess is motivated by the idea that the social planner is still going to save a fraction
kt+1 ,
of output e
kt (in the re-scaled variable), but that to accumulate future capital e
it needs to compensate for the increase in population (term 1 + n) and technological
improvement (term 1 + g) that appears in the de…nition of the transformed capital.
To verify that this indeed a solution, we plug it in
e
kt+11
1
=
)
e
ct
(1 + n) (1 + g) e
ct+1
e
kt+11
1
=
)
(1 + n) (1 + g) (1
)e
kt
)e
kt+1
(1
e
kt+1 =
(1 + n) (1 + g)
and our guess is veri…ed.
Note that our solution yields a steady state:
e
kt
1
e
k=
(1 + n) (1 + g)
e
k =
1
(1 + n) (1 + g)
in the re-scaled variable e
k. Compare, for instance, with equation (7.16) that gave us the
BGP initial level of capital k
1
1
k=
(1 + n) (1 + g)
(1
)
:
Obviusly, in the case when = = 1 that we are considering, both expressions are the
same and therefore e
k = k. The result is not surprising: the BGP initial level of capital k
is the level that gets multiplied by (1 + g)t to deliver kt along the BGP. But e
kt is nothing
t
e
more that kt divided by (1 + g) . Thus, we must have that k = k. A similar reasoning
holds for all the other variables in the model.
The great advantage of the expression
e
kt+1 =
(1 + n) (1 + g)
e
kt
is that we can study situations where we are away from the BGP. We will now show the
properties that we can derive from this solution.
Monotone Convergence
The …rst property that it is important to understand is that equation (7.37) ensures
that, when the economy is not at the BGP, it converges monotonically to it. To see this,
divide both sides of the equation by k
!
!
e
e
e
kt+1
k
k
t
t
=
k 1
=
k
(1 + n) (1 + g)
k
k
104
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
that is, the ratio of (re-scaled) capital over initial capital at the BGP in period t + 1 is
a function exclusively of the same ratio in period t. Since < 1, we have that:
1. when
e
kt
k
>
2. when
e
kt
k
<
3. when
e
kt
k
> 1, that is when (re-scaled) capital is above its steady state value,
e
kt+1
.
k
e
kt
k
> 1, that is when (re-scaled) capital is below its steady state value,
e
kt+1
.
k
e
kt
k
= 1, that is when (re-scaled) capital is at its steady state value,
e
kt
k
=
e
kt+1
.
k
e
In other words, when kkt is di¤erent from 1, we get closer to 1 in the next period
in a monotone fashion given by . Economists often expressed this same idea in an
alternative way. Note that the log of the ratio
e
kt
b
kt
kt = log = log e
k
log k
is approximately the percentage deviation of kt with respect to its value along the BGP:
b
kt = log e
kt
log k
kt
= log
log k
(1 + g)t
= log kt log(1 + g)t k
= log kt log ktBGP
kt ktBGP
ktBGP
where the last line uses the fact that log (1 + x)
x for small values of x. That is, if
b
kt = 0:05; then kt is 5% above its BGP value, and if b
kt = 0:03; then kt is 3% below its
BGP value. Therefore:
!
e
e
kt+1
kt
)
=
k
k
log
e
e
kt+1
kt
= log )
k
k
b
b
kt+1 = kt
that tells us that the percentage deviation of capital with respect its BGP value is smaller
(in absolute terms) over time. Also, we can see how the speed of convergence is given by
1
. For instance, if b
kt is 10%, b
kt+1 will be b
kt and in our period the gap between kt+1
t
and (1 + g) k would have been close by a percentage 1
. In section 7.4, we argued
that = 0:3 and thus 70% of the gap between kt+1 and (1 + g)t dissappears each period
7.6. TRANSITIONAL DYNAMICS
105
(we need to remember, though, that since we are assuming = 1, it is probably best to
think about one period as 20 or so years, more than a quarter or a year).
The convergence is, however, only asymptotic, as the percentage of the reduction in
b
kt is constant. For an initial b
kt 6= 0, we have
and by repited substitution:
that only implies
b
kt+2 = b
kt+1 =
b
kt+j =
jb
kt
2b
kt
kt+j = 0
lim b
j!1
but
b
kt+j 6= 0
for any …nite j.
The economics of this result is straightforward. When there is too much capital in
the economy with respect to its BGP (b
kt > 0), its rate of return is too low and the social
planners decides to save less and consume more. This pushes the economy back into its
BGP. The rate of return is low because, when capital is high, its marginal productivity is
low. In the opposite direction, when there is too litle capital (b
kt < 0), the social planners
wants to accumulate capital fast and get the economy closer to the BGP because its
marginal productivity and its return is high.
Growth Rates
Taking advantage of our previous derivations, output per capita is given by
yt = kt A1t
or, in logs,
= (1 + g)t k e
log yt = t log(1 + g) + b
kt +
b
kt
log k
Then, its growth rate is just:
log yt+1
log yt = log(1 + g) + b
kt+1
= log(1 + g)
g
(1
(1
)b
kt
b
kt
)b
kt
(7.38)
which is composed by two terms, the BGP growth rate g and a negative term on b
kt
b
with coe¢ cient (1
). When kt < 0, capital is below its BGP level, the growth rate
is higher than in the BGP because the social planner is accumulating capital fast and,
conversely, when b
kt > 0, capital is above its BGP level, the growth rate is lower than in
the BGP because the social capital is reducing the level of (re-scaled) capital. Since we
showed before that b
kt converges asymptotically to 0, the growth rate also converges to
g asymptotically.
106
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
Remark 30 (Transitional dynamics in action I) Our previous results allow us to
apply the neoclassical growth model to the study of di¤erent historical events. Imagine,
for instance, that due to a war or some natural disaster, a country loses 20% of its
capital. In other words, all of a sudden, b
kt = 0:2. Equation (7.38) tells us that, after
the initial decline caused by the loss of capital, the growth rate of the country will go up.
For a few periods, the growth rate will stay high but it will converge back to the BGP rate.
Indeed, the historical experience of countries such as Germany or Japan after world war
two, when much of their capital was destroyed, was one of unusually high growth rates
that eventually came back to normal after a few decades. Far from being a miracle, the
neoclassical growth model suggests that fast growth after a large loss of capital should be
the expected outcome and that the real puzzle is to understand why some countries do not
recover fast from wars or natural disasters.
[TO BE DONE: a …gure here?]
It is important to highlight, though, that this period of fast growth rate is not a sign
that welfare is high. First, the economy has lost a 20% of its capital and, thus, it can
produce substantially less output for many periods. Losses of capital are, …rst and foremost, bad events even if the mechanics of the neoclassical growth model deliver a recovery
from them.
Remark 31 (Transitional dynamics in action II) We can reverse our previous example. Imagine, for instance, that country A defeats country B in a war, takes over its
capital, and it ships it back to country A. This can be done directly, as the Soviet Union
did after World War Two, when it shipped many factories from the former East Germany
back to its territory as war reparations, or somewhat more subtly, by levying on country
B an indemnity that must be paid in some generally accepted mean of payment (such as
gold) that country A can use to buy capital in the market. This is a better description of
the plundering that the Spanish undertook in the Americas after the conquest of the Aztec
and Incan empires. Either way, capital in country A jumps over its BGP level, b
ktA > 0
and country A will have a period of unusually low growth (interestingly enough, Spain
su¤ered remarkably dissapointing levels of the growth in the century after the conquest
of its american empire). Again, welfare analysis is somewhat more subtle: country A is
growing less than on its BGP, but it is enjoying several periods of high output and high
consumption as it converges back to its BGP.
[TO BE DONE: cases where we change one parameter? Should we have them as an
exercise?]
7.6.2
Linearizing the Problem
General Idea of Linearization
As already noted in chapter 5.5, the system of …rst-order di¤erence equations that caractherize the solution of the neoclassical growth model is di¢ cult to solve analytically,
beyond special cases as the one we just saw. The technical problem in dealing with the
system of di¤erence equations is that it is nonlinear in the variables we are interested
7.6. TRANSITIONAL DYNAMICS
107
in, kt ; kt+1 ; ct ; ct+1 : That is, these terms enter not only by themselves, but raised to some
power, as they do in kt ; kt+11 ; or ct+1 : And nonlinear di¤erence equations are hard to
deal with.15 There are essentially two approaches to deal with the problem. The …rst is
to use advanced computational methods to solve the nonlinear system directly. We will
not pursue this approach in this book, as the techniques are beyond its scope and the
second approach is currently more widely used in practice anyway.
This second approach consists in replacing the system (7:34)-(7:35) with a linear
approximation that is easier to analyze and that provides an accurate description of the
behavior of the original system, at least for values that consumption and capital typically
take.16
You might remember from your calculus class that a linearization of a function (or
a system of functions) provides an accurate approximation of the function if one stays
within a neighborhood of the point around which the function is approximated. Thus the
choice of the point around which the approximation is carried out is crucial. Basically,
we want to choose a point at the center of the “action”. Since eventually consumption
and capital will reach their BGP values, we will choose as our point of approximation
the BGP of the model.
Preliminary Steps
Recall from our previous analysis that along a BGP all per capita variables grow at the
rate of technological progress, that is, for an arbitrary variable xBGP
= (1 + g)t x; where
t
xt may stand for consumption ct ; output yt ; etc. Recall also that a variable x without
time index x stands for the initial level of the variable at time zero (and corresponds to
the steady state in earlier chapters). Now, and as we did in subsection 7.6.1, we de…ne a
new variable x
bt as the percentage deviation of the variable xt from its BGP value xBGP
:
t
That is, if x
bt = 0:05; then xt is 5% above its BGP value, and if x
bt = 0:03; then xt is
3% below its BGP value.
Formally we de…ne
log(xBGP
)
t
log((1 + g)t x)
xt
= log
(1 + g)t x
xt xBGP
t
xt
x
bt = log(xt )
= log(xt )
(7.39)
15
The trick in the analytic example in chapter 5.5 used a transformation of variables to make the
system linear in the saving rate:
kt+1
st = 1
:
At kt
16
More generally, the approach of replacing the original nonlinear system with an “easier to handle”
system that provides an accurate approximation is called perturbation theory, and is widely employed
in applied mathematics to solve a large class of problems involving non-linearity and dynamics. Perturbation has been used in economics for several decades with much sucess.
108
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
where we again made use of the fact that as long as xt is not too far away from its
x xBGP
BGP value xBGP
= (1 + g)t x; then its percentage deviation from the BGP, t xtt ; is
t
very well approximated by its log-di¤erence log(xt ) log(xBGP
): See remark 26 for the
t
details. Since we will do the approximation in terms of log-deviations (because of its
nice interpretation as percentage deviations from the BGP) our approach is often also
called log-linearization. An additional advantage in working with variables in percentage
deviations from their BGP values (in addition to the easy interpretation) is that it makes
a variable that is growing, xt ; stationary, by dividing by a variable, xBGP
; that is growing
t
as well. We will take as point around which we do the linear approximation x
bt = 0; that
is, the situation in which the variable of interest is exactly equal to its BGP value. Thus
our approximation will be accurate as long as xt is not too far away from its BGP value
:
xBGP
t
Carrying Out the Linear Approximation
We will proceed in two steps. First, we will simply rewrite the system (7:34)-(7:35) in
terms of the ^-variables. Second, we will linearize the system around c^t = k^t = c^t+1 =
k^t+1 = 0: In the appendix we show that these two steps result in the linear system
1
(1 + g)
k 1(
1)b
kt+1
1+n
cb
ct + (1 + n)(1 + g)kb
kt+1 = k b
kt + (1
)kb
kt :
b
ct+1
b
ct =
(7.40)
(7.41)
The …rst equation is the linearized Euler equation which has the following useful interpretation. The left had side gives the change in consumption between two periods,
expressed in percentage deviations from the BGP. The equation says that this change
in consumption equals the intertemporal elasticity of substitution 1 times the change
in the interest rate (net marginal product of capital) in response to a small percentage
change in the capital stock:
@( k
1
+1
@k
)
b
kt+1 = (
1) k
1
b
kt+1 ;
adjusted by the time discount factor and the e¤ect of population and technological
: For any given change in the interest rate, the larger is the IES 1= ;
growth, (1+g)
1+n
the stronger does consumption change. This was precisely our interpretation of the IES
in chapter 4: it is a measure of how sensitive consumption is to changes in the real
interest rate.17 The second equation is simply the linearized version of the economy wide
resource constraint.
17
See Campbell (1994) for an example of how to obtain many economic insights from the neoclassical
growth model through the application of linearization (in his paper augmented with technology shocks,
exactly the model we discuss in the business cycle part of the book).
7.6. TRANSITIONAL DYNAMICS
109
Analysis of the Linear System
We now analyze the system of equations derived through linearization above, On …rst
sight, the system (12:25)-(12:26) is as complicated as the original system of equations
(7:34)-(7:35): This is, however, very misleading, since the system derived through linearization is linear in the variables b
ct ; b
ct+1 ; b
kt ; b
kt+1 ; whereas the original system was nonlinear in the original variables. But linear systems of equations are both much easier to
solve computationally than nonlinear system, and in contrast to the latter solutions to
the former can be characterized analytically.
In order to pursue this we now rewrite (12:25) and (12:26) more compactly as
b
ct = b
ct+1 + Ab
kt+1
b
ct + B b
kt+1 = C b
kt
(7.42)
where we have de…ned the coe¢ cients A; B; C
(1 + g)
1+n
k
B = (1 + g) (1 + n)
c
k + (1
)k
C =
c
A =
1
k
1
(7.43)
to conserve notation. These constants are composed of parameters of the model ( ; ; ; ; g; n)
and the BGP values for consumption and capital (c; k); which are themselves functions
of parameters (see equations (7:16) and (7:17)). Also, given that all parameters are
positive and
1; the coe¢ cients A; B; C are all strictly positive as well.
The system of equations (7:42) therefore is a system of two linear di¤erence equations
with constant (positive) coe¢ cients 1; A; B; C in the two variables consumption b
ct and
capital b
kt . However, the characteristic of the two variables is very di¤erent. While
consumption b
ct can be controlled (chosen) by the social planner at time t; capital b
kt
is predetermined from choices in the previous period, and cannot be chosen at time t
anymore. Since b
kt determines what consumption today, capital for tomorrow and thus
future consumption is feasible it fully determines the current state of the economy. Thus
b
kt is called a state variable, and in the simple neoclassical growth model it is the only
state variable. More generally, the state variables of a dynamic system are those variables
of the system whose values fully determine the behavior of the system from the current
period on.
The other variable in the system, consumption, is a control or jump variable since it
can be set, in the current period, to any desired value that is feasible at the start of the
period. More generally, the control variables of a dynamic system are those variables
that be chosen in the current period. Note that capital is also a control variable, since
we can pick b
kt+1 . We will, however, with a slight abuse of language, reserve the word
control to those jump variables that are not states themselves.
110
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
Determining the Policy Functions
We now use, without proof, a basic result from mathematics that states that a system
of di¤erence equations such as the one de…ned in (7:42) has a solution of the form:
b
kt+1 = P b
kt
b
b
ct = R kt
(7.44)
where P and R are unknown constant coe¢ cients that we yet need to determine, or in
the language of economics, undetermined coe¢ cients.
Remark 32 (Coe¢ cients versus Parameters) We have called P and R (and A; B; C)
coe¢ cients and not parameters. We reserve the word parameters for constants that
describe the utility function (preferences) of households and production technologies of
…rms. The word coe¢ cients, in contrast, is used for those constants that are functions
of the underlying fundamental parameters of the model, in our case ( ; ; ; ; g; n).
The linear functions in (7:44) are called policy functions. They relate the current
state of the economy, b
kt ; to the future state b
kt+1 , and the control b
ct . The name policy
functions comes the fact that these function describe how the social planner optimally
chooses consumption today and capital for tomorrow (the social planners’policy).
Even though we will not prove that the policy functions must take the form in (7:44);
the intuition for their form is easy to understand. First, since by de…nition that state
variable(s) of the model encodes all relevant information for the behavior of the economy,
the policy functions can be functions only of b
kt : Secondly since the system we are trying
to solve is linear it is perhaps not surprising that the solution itself (the policy functions)
is linear, too (note that while this second argument is intuitively true it does require some
work to prove it).
Our …nal task is to …nd the values of P and R in terms of the coe¢ cients A; B; C,
which themselves are functions of the underlying parameters of the model, see equations
(7:43): A simple method, sometimes called the method of undetermined coe¢ cients, it to
simply plug in the guessed policy functions (7:44) with their yet undetermined coe¢ cients
P and R into the system (7:42) and see whether by doing so we can somehow determine
the values of P and R: We carry out this method in the appendix where we show that
the coe¢ cients (P; R) of the decision rules we seek are given by
A+C+B
P =
2B
R = C BP
r
(A + B + C)2
4B 2
4BC
where we recall from equations (7:43) that the coe¢ cients A; B; C are themselves determined by the underlying parameters of the model ( ; ; ; ; g; n) and the BGP values
for consumption and capital (c; k):
7.6. TRANSITIONAL DYNAMICS
111
The optimal policy functions for capital and consumption is then given by
b
kt+1 = P b
kt
b
ct = (C BP ) b
kt
Given the constants A; B; C and thus the coe¢ cient P just depend on the underlying
parameters of the model, ( ; ; ; ; g; n); we have now fully solved the (linearized) neoclassical growth model for all admissible parameter values. Alternatively, we can use the
speci…c parameter values chosen (i.e. calibrated) in section 7.4 and compute the exact
numerical values for A; B; C, thus the exact value for P and R = C BP and therefore
deduce the exact full quantitative solution of the model, from an arbitrary initial capital
stock k0 to its BGP in the long run.
Interpretation and Implications of the Policy Function
When we use the parameter values calibrated in section 7.4 we calculate P = 0:85 and
R = 0:52 and thus
b
kt+1 = 0:85b
kt
b
b
ct = 0:52kt
Together with the initial condition b
k0 = log(k0 ) log(k); where k0 is the arbitrary initial
capital stock and k is the BGP capital stock, these equations fully describe the dynamic
behavior of the capital stock and consumption (in percent deviation from their BGP)
over time.
Figure 7.1 plots the policy function b
kt+1 = 0:85b
kt with b
kt on the x-axis, taking values
between -10% and +10% of its BGP value. It also shows how, starting from a b
k0 = 0:08
(the per capita capital stock being below its BGP by 8%), the capital stock over time
approaches its BGP (represented by the b
k = 0 line). The …gure shows that eventually
the capital stock approaches its BGP level, with the speed dictated by the slope of the
policy function, that is, by the value of P1 : Note that, mathematically speaking, the
economy never gets to the BGP in …nite time, although it gets very close to the BGP
quite quickly.
Note that once we know how capital, in percent deviation from the BGP, evolves,
we can reverse the transformation and easily obtain the time path of capital per capita
itself. The same is true for all other variables. Recall that the hatted variables were
de…ned as
x
bt = log(xt ) log(xBGP
):
t
Solving this for xt and using the fact that xBGP
= (1 + g)t x we …nd that
t
log(xt ) = x
bt + log(xBGP
)
t
t
x
bt
xt = (1 + g) xe :
(7.45)
Thus once we have computed the BGP level of capital k and the path for b
kt in sections
7.3 and ?? we can put them together and derive the entire time series for capital per
112
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
Figure 7.1: Dynamics of k^t in the Neoclassical Growth Model
capita kt implied by the model. The same of course applies to consumption per capita
and all other variables: once we have their BGP levels x and the path for x
bt ; we simply
use (7:45) to …nd the path for xt :
Once we have derived the dynamics of capital and consumption we can easily derive
how all other variables in the economy behave over time. We can either do this directly,
since once kt and ct are known all other variables are given by
yt
it
wt
rt
= kt A1t
= yt ct
= (1
) kt At1
= kt 1 A1t :
Alternatively, we can solve for the other variables in b-form and then use (7:45) to
obtain their paths. For example, output per capita was given by
yt = kt A1t
= kt (1 + g)t(1
)
= (1 + g)t
The BGP level of output was given by, see (7:22)
ytBGP = (1 + g)t k :
kt
(1 + g)t
7.7. APPENDIX A: DETAILS OF THE LINEARIZATION PROCEDURE
113
Thus
ybt = log(yt )
log(ytBGP )
= log (1 + g)t +
=
log
=
log
=
b
kt
log
kt
(1 + g)t
kt
k (1 + g)t
kt
(1 + g)t
log (1 + g)t
log(k)
log(k)
=
log(kt )
log(ktBGP )
Similarly (after some algebra)
bit = ybt b
ct
b
w
bt = kt
rbt = (
1)b
kt
and from these equations and (7:45) we can readily compute the time path of GDP,
investment, wages and interest rates once we have computed the time paths of capital
and consumption, as discussed above.
7.7
Appendix A: Details of the Linearization Procedure
7.7.1
Preliminary Steps
Before we carry out the linearization of the system (7:34)-(7:35) in the main text it is
useful to note that from equation (7:39):
Thus
log(xt ) = x
bt + log(xBGP
):
t
BGP
BGP )
elog(xt ) = exbt +log(xt ) = exbt elog(xt
xt = xBGP
exbt
t
xt = (1 + g)t xexbt
(7.46)
And since we will have to deal with our variables being raised to some power, we note
that for an arbitrary power p 0 we have, raising both sides of (7:46) to the power p;
(xt )p = (1 + g)t
or simply
p
xp exbt
xpt = (1 + g)pt xp epbxt :
p
(7.47)
114
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
7.7.2
Carrying Out the Linear Approximation
We will proceed in two steps. First, we will simply rewrite the system (7:34)-(7:35) in
terms of the ^-variables. Second, we will linearize the system around c^t = k^t = c^t+1 =
k^t+1 = 0:
Step 1: Rewriting the System
Let’s start with the simpler equation, equation (7:35):
ct + (1 + n) kt+1 = kt At1
+ (1
) kt :
Now recall that At = (1 + g)t : Thus At1 = (1 + g)t (1 + g)
(7:47) we can rewrite the previous equation as
b
(1+g)t cebct +(1+n)(1+g)t+1 kekt+1 = (1+g) t k e
b
kt
t
: Using this and (7:46) and
(1+g)t (1+g)
t
+(1
b
) (1+g)t kekt
Dividing both sides by (1 + g)t and simplifying yields
b
cebct + (1 + n)(1 + g)kekt+1 = k e
b
kt
b
) kekt :
+ (1
(7.48)
Unfortunately equation (7:35)
(1 + n) (ct )
1
kt+11 At+1
+1
= (ct+1 )
(7.49)
is slightly more messy. We note that
kt+11 A1t+1
(ct )
(ct+1 )
=
= (1 + g)
= (1 + g)
b
1
kt+1
At+1
(1 + g)t+1 kekt+1
(1 + g)t+1
=
t
b
ct
c
e
(t+1)
c
b
ct+1
e
!
1
=k
1 (
e
1)b
kt+1
:
Plugging these expressions into (7:49) yields
(1 + n) (1 + g)
t
c
e
b
ct
t
Dividing both sides by (1 + n) (1 + g)
e
b
ct
=
(t+1)
= (1 + g)
(1 + g)
1+n
e
c
c
e
b
ct+1
k
1 (
e
1)b
kt+1
+1
:
to simplify yields
b
ct+1
k
1 (
e
1)b
kt+1
+1
(7.50)
Thus we have expressed both dynamic equations in terms of the BGP levels of consumption and capital, c and k; as well as the percentage deviations b
ct ; b
ct+1 ; b
kt ; b
kt+1 from the
b
BGP. Now we are ready to linearize both equations around b
ct = b
ct+1 = kt = b
kt+1 = 0:
7.7. APPENDIX A: DETAILS OF THE LINEARIZATION PROCEDURE
115
Step 2: Linearizing the System
Again we start with the simpler equation, equation (7:48): To get practice in linearization,
we go term by term. So let’s …rst linearize the term cebct around b
ct = 0: In general, the
linear approximation of a function f (xt ) around a particular point x0 is given by
f (x0 ) + f 0 (x0 ) (x
f (x)
x0 ):
ct = 0: Noting
Here the function is f (b
ct ) = cebct and the point of approximation is x0 = b
b
ct
that that c is simply a constant and that the derivative of e equals again ebct ; we …nd
f (b
ct ) = cebct
ce0 + ce0 (b
ct
Thus the linear approximation of cebct is
cebct
0) = c + cb
ct
(7.51)
c + cb
ct
c+cb
ct : Similarly, the linear approximations of the other terms are given by (using exactly
the same reasoning):
b
(1 + n)(1 + g)kekt+1
(1
b
kt
) ke
b
(1 + n)(1 + g)k + (1 + n)(1 + g)kb
kt+1
(1
) k + (1
)kb
kt
(7.52)
(7.53)
The linear approximation of k e kt is slightly more di¢ cult. First remember that the
b
b
derivative of e kt is equal to e kt : Then
k e
b
kt
k e0 + k e
= k + k b
kt :
0
(b
kt
0)
(7.54)
Thus we have linearized all terms of equation (7:48): Plugging the result in yields
c + cb
ct + (1 + n)(1 + g)k + (1 + n)(1 + g)kb
kt+1 = k + k b
kt + (1
) k + (1
)kb
kt :
) k] + cb
ct + (1 + n)(1 + g)kb
kt+1 = k b
kt + (1
)kb
kt :
Rearranging yields
[c + (1 + n)(1 + g)k
k
(1
But c; k are the BGP levels of consumption and capital, and the term in [] brackets is
nothing else but the resource constraint in the BGP, see equation (7:11): Thus the entire
[] term equals to zero, and the remaining equation reads as
cb
ct + (1 + n)(1 + g)kb
kt+1 = k b
kt + (1
)kb
kt
(7.55)
The equation is simply the aggregate resource constraint, now expressed in terms of
variables measured as percentage deviation from their long-run BGP values.
116
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
In fact, whenever we linearize around the BGP and use as variables percentage deviations from the steady state (the x
b-variables), the constants in the linearization always
drop out.18 We will also make use of this fact when linearizing the second equation,
equation (7:50):
e
b
ct
e
b
ct
(1 + g)
1+n
(1 + g)
1+n
=
=
b
ct+1
e
h
k
b
ct+1
e
k
1)b
kt+1
1 (
e
1)b
kt+1
1 (
e
+1
b
ct+1
+e
(1
i
)
(7.56)
For this equation we then make the use of the following “replacements”, using the =)
symbol instead of the
symbol because of the omission of the constant terms in the
…rst order approximation19
b
ct
e
e
e
b
ct+1
k
1 (
e
=)
=)
b
ct+1
1)b
kt+1
=)
b
ct
b
ct+1
b
ct+1 k
1
+ k
1
1)b
kt+1
(
Substituting into (7:56) and simplifying yields
b
ct =
b
ct =
(1 + g)
1+n
(1 + g)
1+n
h
h
k
b
ct+1 k
1
+1
1
+ k
)(
1
(
1)b
kt+1
b
ct+1 ) + k
1
(
b
ct+1 (1
1)b
kt+1
i
i
)
The reason is simple. At b
ct = b
ct+1 = b
kt = b
kt+1 = 0 all variables are equal to their BGP values. But
in the BGP both the resource constraint and the Euler equation holds. Thus in a full linearization of
either (7:48) or (7:50) which we did piece by piece, the constant term of the approximation equals
18
f (x0 ) = f (b
ct = b
ct+1 = b
kt = b
kt+1 = 0) = 0:
In fact, this is another reason why we chose theb-variables and as point of linearization b
ct = b
ct+1 = b
kt =
b
kt+1 = 0.
19
For the last of the three terms we need a …rst order approximation of a term that involves two
variables. In general, the linear approximation of a function of two variables f (x; y) around the point
(x0 ; y0 ) = (0; 0) is given by
f (x; y)
f (0; 0) +
@f (0; 0)
@f (0; 0)
x+
y
@x
@y
or, leaving out the constant term
f (x; y) )
where the term
point (0; 0):
@f (0;0)
@x
@f (0; 0)
@f (0; 0)
x+
y;
@x
@y
means the partial derivative of the function f with respect to x; evaluated at the
7.7. APPENDIX A: DETAILS OF THE LINEARIZATION PROCEDURE
117
But now we note that that in the BGP the Euler equation reads as, see equation (7:15);
(1 + g)
1+n
1=
( k
1
+1
):
Using this in the above equation yields
and thus …nally
b
ct =
b
ct+1
(1 + g)
1+n
b
ct+1 +
b
ct =
1
(1 + g)
1+n
1
k
k
1
(
(
1)b
kt+1
1)b
kt+1
(7.57)
Equations (7:55) and (7:57) constitute the system (12:25)-(12:26) of equations analyzed in the main text
7.7.3
Determining the Policy Functions
In this section we describe how to …nd the unknown coe¢ cients P and R of the policy
functions b
kt+1 = P b
kt and b
ct = R b
kt . We want to …nd the values of P and R in terms
of the coe¢ cients A; B; C, which themselves are functions of the underlying parameters
of the model, see equations (7:43): The method of undetermined coe¢ cients used here
works as follows. First, plug in the guessed policy functions of the form speci…ed in
(7:44) with their yet undetermined coe¢ cients P and R into the system (7:42):Then see
whether from the resulting equations we can determine the values of P and R:
We can readily plug in (7:44) into (7:42) for b
ct and b
kt+1 ; but what about b
ct+1 ? Here
b
we use the fact that if b
ct = Rkt then in period t + 1 the same policy function implies
b
ct+1 = Rb
kt+1
and now substituting out b
kt+1 from (7:44) yields
b
ct+1 = RP b
kt :
Now, using (7:44) and (12:32) in (7:42) yields:
Rb
kt = RP b
kt + AP b
kt
Rb
kt + BP b
kt = C b
kt :
Rearranging and factoring out b
kt yields
1) + AP ) b
kt = 0
(R + BP C) b
kt = 0:
(R (P
(7.58)
118
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
But these equations have to hold no matter which value the current state b
kt takes (remember, the economy starts at some given arbitrary capital k0 and implied b
k0 and we
want to characterize the dynamic behavior of consumption and capital for all possible
initial conditions, and thus for all possible values of b
kt : But the above two equations can
b
only hold for all possible values of kt if the unknown coe¢ cients R; P satisfy the two
equations:
R (P 1) + AP = 0
R + BP C = 0
(7.59)
(7.60)
Thus the remaining step in solving for the optimal policy functions that fully characterize
the solution of the (linearized) model it to solve these two equations for P; R:
Solving this system of equations is relatively easy, but more importantly, it will be
very informative about the economic forces at work in the neoclassical growth model.
The second equation (which came from the economy-wide resource constraint (??)),
directly implies that the coe¢ cient R (which tells what fraction of the capital stock b
kt
the planner should optimally let households consume), is given by:
R=C
BP
(7.61)
Remark 33 (The Resource Constraint and Equation (7.61)) This restriction has
an intuitive interpretation. Let us go back to the resource constraint, the second equation
of system (7:42):
b
ct + B b
kt+1 = C b
kt or
b
b
c t = C kt B b
kt+1
(7.62)
Total output available for consumption and capital accumulation is given by C times the
capital stock b
kt : According to the policy function capital tomorrow b
kt+1 equals P times
capital today, b
kt : But each unit of capital tomorrow costs B units of resources, so that of
the total of C units of output a total of BP units go to capital tomorrow. Equation (7:61)
then simply says that the remaining R units go to consumption, and (7:61) is nothing
else but a restatement of the resource constraint.
Plugging back for R from equation (7:61) into (7:59) yields
(C
BP ) (P
1) + AP = 0
which now simply one (quadratic) equation in the one unknown coe¢ cient P: Multiplying
out the terms and simplifying yields
CP C BP 2 + BP + AP = 0
BP 2 + (A + C + B) P C = 0
C
A+B+C
P+ =0
P2
B
B
7.7. APPENDIX A: DETAILS OF THE LINEARIZATION PROCEDURE
119
and thus the typical form of a quadratic equation in the unknown coe¢ cient P .
As any quadratic equation, this one has two solutions (often called roots) de…ned by:
P1
P2
r
(A + B + C)2
A+C+B
=
2B
4B 2
r
(A + B + C)2
A+C+B
+
=
2B
4B 2
4BC
4BC
(7.63)
So in principle we have two options of how to choose P: To summarize the somewhat
technical discussion to follow, we will show that both P1 and P2 are positive real numbers,
but that P1 < 1 while P2 > 1: We will then argue that if we were to choose P1 as the
solution, the capital stock would deviate further and further from its BGP path, which
would violate the terminal condition (the transversality condition). Thus P2 cannot be
the right solution. As a consequence we will choose as our solution P = P1 in the main
text. Since P1 < 1; we will show below that over time the capital stock gets closer and
closer to its steady state value. Since the dynamics of the capital stock when using P1 is
stable (it settles down to the BGP over time), the number P1 is often called the stable
root of the system.
The optimal policy function for capital is then given by
b
kt+1 = P1 b
kt
where is the stable root determined above and the optimal policy function for consumption is given by
b
ct = (C BP1 ) b
kt :
Given that P1 and the constants A; B; C it depends on are all just expressions that
depend on the underlying parameters of the model, ( ; ; ; ; g; n); we have now fully
solved the (linearized) neoclassical growth model for all admissible parameter values.
Alternatively, we can use the speci…c parameter values chosen (i.e. calibrated) in section
7.4 and compute the exact numerical values for A; B; C, thus the exact value for P1 and
R = C BP2 and therefore deduce the exact full quantitative solution of the model,
from an arbitrary initial capital stock k0 to its BGP in the long run. We will do so after
having discussed the mathematics behind selecting the stable root of the system. Those
not interested in the details can without loss of continuity proceed to section 7.6.2.
7.7.4
Selecting and Characterizing the Stable Root
We now return to a formal discussion of the two roots in (7:63): First we need to make
2 4BC
sure that the term under the square root (sometimes called discriminant), (A+B+C)
4B2
is positive, because if not, then taking the square root of a negative number does not
120
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
give a real number.20 So we want to check under what conditions
(A + C + B)2 4BC
(A + C + B)2
0 or
4BC:
since the denominator 4B 2 is always positive. First remember that all A; B; C are strictly
positive numbers. Then, by multiplying out the term on the left hand side, we …nd
(A + C + B)2
= A2 + C 2 +B 2 + 2AC + 2AB + 2BC
> B 2 + C 2 + 2BC
2BC + 2BC
= 4BC.
In this sequence the …rst equality simply comes from multiplying out the square and
the second inequality comes from the fact that we simply dropped the positive number
A2 + 2AC + 2AB. Finally, the fact that B 2 + C 2
2BC (which was used when going
from the third to the forth line) can be shown as follows:
B
2
(B C)2
2BC + C 2
B2 + C 2
0 and thus
0
2BC.
Thus we conclude that (A + C + B)2 > 4BC and both roots P1 and P2 are real numbers.
Second, it is obvious that P1 < P2 (since for P1 we are subtracting the square root
term, while for P2 we are adding it). Third, we note that P1 > 0 since
r
A+B+C
(A + B + C)2 4BC
P1 =
2B
4B 2
r
A+B+C
(A + B + C)2
<
2B
4B 2
A+C+B A+B+C
=
2B
2B
= 0
where the inequality between the …rst and the set equation comes from the fact that by
leaving out the term 4BC under the square root we make the square root bigger, and
since the square root enters with a sign, it makes the entire expression in the second
line larger than the one in the …rst line. Thus we conclude that 0 < P1 < P2 :
20
If the term under the square root is negative, the solutions P1 and P2 are complex numbers. While
we will not pursue this possibility further (mainly because we can show that the term under the squre
root is positive), even complex roots can be handled mathematically and potentially give rise to cycles
in the economic variables of interest that could be interpreted as business cycles.
7.7. APPENDIX A: DETAILS OF THE LINEARIZATION PROCEDURE
121
Now we want to show that P2 > 1 (which then rules out P2 as a viable candidate, as
we will show below). For this we …rst note from the BGP budget constraint (??) that
B = (1 + g) (1 + n)
k
c
k + (1
c
)k
=C
since the resources spent on capital accumulation, (1 + g) (1 + n) k cannot be larger than
all resources available for capital accumulation and consumption, k + (1
) k. But
since C B we have that
P 1 + P2
A+C+B
C+B
=
>
2
2B
2B
B+B
= 1;
2B
that is, the average of the two roots is bigger than one. Since both roots P1 ; P2 are
positive and their average is larger than one, it must be that the larger root, P2 > 1:
Thus the only viable candidate for the coe¢ cient P in the policy function is P1 : We
…nally want to show that P1 < 1: To see this note that, using messy but straightforward
algebra
p1 =
,
,
,
,
,
,
,
r
A+B+C
(A + B + C)2 4BC
<1
2B
4B 2
r
(A + B + C)2 4BC
A+B+C
1<
2B
4B 2
r
A B+C
(A + B + C)2 4BC
<
2B
4B 2
2
(A + B + C)2 4BC
(A B + C)
<
4B 2
4B 2
4B 2
2
2
(A B + C) < (A + B + C)
4BC
2
2
2
A + B + C + 2AC 2AB 2BC < A2 + B 2 + C 2 + 2AC + 2AB + 2BC
2AB < 2AB
0 < 4AB
4BC
which is true since A and B are positive. Thus P1 < 1: In summary, we have shown that
the two roots P1 ; P2 satisfy
0 < P1 < 1 < P2 :
But why can’t we choose the larger root P2 > 1 (sometimes called the unstable root)?
Suppose we did, and thus have the policy function for capital
b
kt+1 = P2 b
kt :
Let us start from an arbitrary capital stock k0 > 0: The BGP capital stock at time zero
is k0BGP = (1 + g)0 k = k > 0: There are two possible situations.
122
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
Case 34 If k0 > k the economy is initially above its BGP and b
k0 > 0: But then what
happens over time? Using the policy function
b
k 1 = P2 b
k0
b
b
k0
k2 = P2 k1 = (P2 )2 b
and in general
b
kt = (P2 )t b
k0
(7.64)
As time t increases, since P2 > 1; the term (P2 )t explodes (mathematically, limt!1 (P2 )t =
1), and thus b
kt explodes as well, limt!1 b
kt = 1. Over time, the capital stock kt exceeds
its BGP path more and more.
Case 35 If k0 < k the economy is initially below its BGP and b
k0 < 0: Equation (7:64)
t
b
kt collapses
still applies and (P2 ) still explodes. But now since k0 < 0 the capital stock b
b
without bounds, limt!1 kt = 1:
An imploding or exploding capital stock seems unlikely to be optimal, but note that
P2 is a valid solution and together with the implied R satis…es both equations (7:59) and
(7:60), and thus the implied policy functions satisfy the linearized resource constraint
and linearized Euler equation. But it turns out that the policy associated with P2 does
not satisfy is the transversality condition (7:1) and thus can indeed be discarded.21
21
The transversality condition reads as, for the CRRA utility case
t
lim
t!1
ct kt+1 = 0:
Replacing ct and kt+1 by their hatted counterparts yields
t
lim
t!1
c
t
(1 + g)
c
(1 + g)k lim
e
(1 + g)t+1 kekt+1
1
b
ct + b
kt+1
t
b
ct b
kt+1
e
and thus the transversality condition becomes
c
(1 + g)k lim
t!1
(1 + g)
Ignoring constants irrelevant for the limit, the term
t+1 b
and replacing b
kt+1 = (P2 )
k0 yields
lim
t!1
b
b
ct
(1 + g)
t!1
Linearizing around b
ct = b
kt+1 = 0 yields
e
(1 + g)
t
b
ct b
kt+1
e
0
=
0:
b
ct + b
kt+1 = 0:
1
b
ct (which may only help to violate the condition)
P2
But this condition is violated if
(1 + g)
e
=
P2
t
P2 b
k0 = 0:
1
which can be shown to be the case [really? or can we demonstrate violation of the TVC in a simpler
way TBC].
7.7. APPENDIX A: DETAILS OF THE LINEARIZATION PROCEDURE
123
In contrast, using the smaller root P1 and the same reasoning to derive (7:64) we …nd
that
b
kt = (P1 )t b
k0 :
The big di¤erence now is that since P1 < 1 the term (P1 )t over time shrinks to zero, and
so does b
kt = (P1 )t b
k0 ; no matter with what k0 and thus b
k0 the economy starts. Formally
b
limt!1 kt = 0: But remember that the variables withbwere percentage deviations from
their BGP values. So if over time b
kt goes to zero, this means that the capital stock kt
itself over time approaches its BGP path. Thus in the long run the economy is well
described by the BGP, and in the short run, according to the (linearized) model, it
approaches the BGP no matter where it starts from. We say that there is convergence in
the model to the BGP. The speed at which kt approaches the BGP ktBGP is exclusively
determined by P1 ; with smaller values implying faster convergence (the term (P1 )t falls
to zero faster).22
Remark 36 (Computation in Economics) Our previous steps, although conceptually straightforward, were cumbersome algebraically. For models with a larger number
of state and/or control variables (say, with shocks leading to business cycles, or with
multiple sectors), the algebraic derivations just become too tedious and complex too long
be practical. In current research, it is not uncommon to have models with 50 or even
more states and 30 or 40 controls. Just imagine having to take a linear approximation
involving 30 or more variables. Fortunately, all the previous steps we took manually can
be automated with computers. We just have to program them to undertake the linear
approximation, plug in the guessed policy functions, and solve for the undetermined coe¢ cients. A good computer does this without mistakes in a matter of seconds for models
with dozens of state and control variables. This one of the key reasons why computational techniques are now so commonly used in macroeconomics, with the linearization
technique we have discussed here just being one of them. In fact, even though our linear approximation is very accurate for the neoclassical growth model we could have used
higher order (quadratic, cubic etc.) approximations around the BGP or even approximations that are very precise not only close to the BGP. But linearization is by far the
22
We can be even more precise. How long does it take for the economy to cut its distance to the
steady state in half? If the distance is 10% at time zero, how long does it take to get to a distance of
5% from the BGP. This time interval solves
T
0:05 = (P1 ) 0:1
or
T
1 = 2 (P1 )
log(1) = log(2) + T log(P1 )
log(2)
T =
log(P1 )
where we remember that since P1 < 1; log(P1 ) < 0: And the smaller is P1 ; the more negative is log(P1 )
and thus the smaller is T: Therefore a smaller P1 makes the economy go faster to its BGP.
124
CHAPTER 7. THE NEOCLASSICAL GROWTH MODEL
fastest method, by far the most commonly used and the only one we can describe without
introducing mathematical concepts beyond the scope of this book.
Chapter 8
Neoclassical Growth Model:
Confronting the Facts
8.1
Using the Neoclassical Production Function: Growth
Accounting
In chapter 1 and 4 we introduced the aggregate production function
Yt = Bt Kt L1t
(8.1)
relating total output (income) in a country to the country-wide labor and capital input
Lt and Kt : The factor Bt captures the level of technology and equals the technology term
A1t
from chapter 4. Bt is called total factor productivity, and a production function
in which technological progress enters the way as shown is said to have Hicks-neutral
technological progress.1
Now we take logs of both sides of (8:1) to obtain
log(Yt ) = log(Bt ) +
log(Kt ) + (1
(8.2)
) log(Lt ):
Similarly, for period t + 1 we obtain
log(Yt+1 ) = log(Bt+1 ) +
log(Kt+1 ) + (1
) log(Lt+1 ):
(8.3)
Subtracting (8:2) from (8:3) yields
log(Yt+1 ) log(Yt ) = log(Bt+1 ) log(Bt )+ [log(Kt+1 )
log(Kt )]+(1
) [log(Lt+1 )
log(Lt )] :
Now we recall from XXX that for any variable
gY;t+1 =
Yt+1 Yt
Yt
log(Yt+1 )
1
log(Yt )
There are several reasons of why we make the change from At ; multiplying labor, to Bt ; multiplying
K(t) L(t)1 : First, the growth rate of B is a widely used productivity measure by economists. Second,
in the Cobb-Douglas case both ways are equivalent, but for more general production functions this is
not true anymore.
125
126CHAPTER 8. NEOCLASSICAL GROWTH MODEL: CONFRONTING THE FACTS
and thus the growth rate of output between period t and t + 1; gY;t+1 ; equals
gY;t+1 = gB;t+1 + gK;t+1 + (1
)gL;t+1 :
Thus we can decompose the growth rate of output into the sum of the growth rate of
technology, the growth rate of the capital stock and the growth rate of labor input.
The growth rate of Bt ; gB;t+1 is called total factor productivity (TFP) growth or multifactor productivity growth. We can use these formulas to perform our basic accounting
exercise for a particular country: …rst we have to take a stand on what is. Since it
turns out to be the capital share, an = 31 is quite popular among economists. Next we
measure the growth rate of real GDP, gY the growth rate of the aggregate capital stock
gK and the growth rate of labor input gL from the data.2 We then use the formula above
to compute gB as the residual
gB;t+1 = gY;t+1
gK;t+1
(1
)gL;t+1
Computed this way, gB is also called the Solow residual, it is that part of output growth
that cannot be explained by the growth in inputs capital and labor.3
Before actually carrying out the accounting exercise one word of caution is in order.
We will only measure TFP growth correctly if we measure the growth in output and in
labor and capital inputs correctly. Measuring gY and gL is relatively straightforward, but
measuring the growth rate of the capital stock may be tricky. An example: suppose the
capital stock of an economy consists only of Pentium 3 computers and now the economy
invests in a new Pentium 4, which is double as fast as the Pentium 3’s. Did the capital
stock go up by 10% (as the number of computers went up by 10%) or did it go up
by 20% (as the computing power went up by 20%)? In practice a lot of assumptions
and simpli…cations are needed when measuring the growth rate of the capital stock and
this variable is probably one of the most poorly measured economic variables. The
consequences of this problem for measuring TFP growth are enormous: suppose we
measure gK as 3% but it was in fact 6%: Then we attribute
(6% 3%) = 1%
of output growth to productivity growth when it was in fact due to growth in capital
input. Computing productivity as a residual leads to mismeasurement of productivity
whenever inputs or output are not measured correctly.
[This will have to be redone with new data]
But now let’s go ahead and perform the accounting exercise for US data from 1960
to 1990. In Table 7 we report averages of growth rates for output and factor inputs for
several subperiods. We assume that = 31 : In parenthesis is the percent that capital,
labor and TFP growth contribute to GDP growth
Table 7
2
Labor input is usually measured by the total number of manhours worked in the economy in a given
period. This is a more precise measure of labor input than the number of workers as the number of
hours a worker works per year may change over time.
3
In some sense it measures our ignorance in explaining growth. In the light of our previous discussion,
we hope that the Solow residual does indeed measure technological progress.
8.1. USING THE NEOCLASSICAL PRODUCTION FUNCTION: GROWTH ACCOUNTING127
Period
1960
1960
1970
1980
90
70
80
90
GDP gY Capital gK Labor (1
)gL TFP (gB ) GDP p. worker gy
3:1
0:9 (28%)
1:2 (38%) 1:1 (34%)
1:4
4:0
0:8 (20%)
1:2 (30%) 1:9 (50%)
2:2
2:7
0:9 (35%)
1:5 (56%) 0:2 (8%)
0:4
2:6
0:9 (34%)
0:7 (26%) 1:1 (41%)
1:5
We see that real GDP grew strongest in the 60’s, at 4% a year, and at about 2 12 %
since then. Approximately 1 percentage point of this growth is due to accumulation of
physical capital. Between 0:7 and 1:5 percentage points is due to growth of labor input.
We see the dramatic decline of total factor productivity in the 70’s: from 1:9% in the
60’s to just about 0: This productivity slowdown is one of the most studied and least
understood phenomena of recent economic history; it is an international phenomenon in
that a lot of countries experienced a productivity slowdown at approximately the same
time. The 80’s showed somewhat of a recovery of TFP growth to 1:1%; and the latest
numbers indicate that for the last four years TFP growth was again at the speed of the
60’s.
Remember that GDP per worker is de…ned as y = YL : Sometimes this variable is also
referred to as labor productivity, as the ratio of output to labor input. We immediately
have that gy = gY gL ; hence from the formulas above
gy = gB + gk
gy = (1
)gA + gk
and we see the direct impact of TFP growth on per worker income growth (or labor
productivity). As predicted by this equation, the productivity slowdown of the 70’s led
to a sharp decline of income per worker in that period, with the growth rate of per worker
income recovering in the 80’s (and even more so in the late 90’s).
What are possible reasons for the productivity slowdown? As mentioned it is still
somewhat of a puzzle. Here are some explanations
1. Sharp increases in the price of oil which made companies use inferior technology
that didn’t require oil. Problem: oil prices (adjusted for in‡ation) are lower in the
late 80’s than in the 60’s.
2. Structural changes: as the economy produced more and more services and less and
less manufacturing goods the high productivity sectors (manufacturing) become
less important than the low productivity sectors (services).
3. Slowdown in resources spent on R&D in the late 60’s.
4. TFP was abnormally high in the 50’s and 60’s since all the new technologies developed for the war became available for private business sector use. So the 70’s
and 80’s are the “normal”situation.
128CHAPTER 8. NEOCLASSICAL GROWTH MODEL: CONFRONTING THE FACTS
5. Information technology (IT) revolution in the 70’s. Computers swept into business
o¢ ces and for the last 10 to 15 years a lot of time was spent learning how to use
them (instead of producing output). Hence the productivity slowdown. Once the
new technology is …gured out, TFP should boom again.
Probably the truth is that all these factors contributed to the slowdown, although
I personally …nd the last explanation very intriguing, in particular given that TFP has
been extraordinarily high in the last …ve years, possibly showing the e¤ects of investment
in IT started in the 70’s and 80’s.
We can use the same framework for the analysis of the growth process in other
countries. In particular, what determinants are responsible for the growth miracles in
East Asia, the Singapores, Japans, Koreas, Hong Kongs and Taiwans? There exists a
somewhat heated discussion about this issue, with one group of economists attributing
most of the fantastically high growth rates from the 60’s to the mid 90’s to TFP growth,
whereas others attribute most of it to the fast accumulation of physical (and human)
capital. In Table 8 we show results from growth accounting for the Asian growth miracles,
and, as a comparison, data for some industrialized and some Latin American countries.
The calculations are done with country-speci…c ’s, where the for a particular country
is matched to that country’s average capital share during the relevant time period.
8.2. BALANCED GROWTH PATH PREDICTIONS
129
Table 8
Country
Germany
Italy
UK
Argentina
Brazil
Chile
Mexico
Japan
Hong Kong
Singapore
South Korea
Taiwan
Time Per. GDP gY Cap. Sh.
Cap. gK
1960 - 90
3:2
0:4 1:9(59%)
1960 - 90
4:1
0:38 2:0(49%)
1960 - 90
2:5
0:39 1:3(52%)
1940 - 80
3:6
0:54 1:6(43%)
1940 - 80
6:4
0:45 3:3(51%)
1940 - 80
3:8
0:52 1:3(34%)
1940 - 80
6:3
0:63 2:6(41%)
1960 - 90
6:8
0:42 3:9(57%)
1966 - 90
7:3
0:37 3:1(42%)
1966 - 90
8:5
0:53 6:2(73%)
1966 - 90
10:3
0:32 4:8(46%)
1966 - 90
9:1
0:29 3:7(40%)
Labor (1
)gL TFP (gB )
0:3( 8%)
1:6(49%)
0:1(3%)
2:0(48%)
0:1( 4%)
1:3(52%)
1:0(26%)
1:1(31%)
1:3(20%)
1:9(29%)
1:0(26%)
1:5(40%)
1:5(23%)
2:3(36%)
1:0(14%)
0:2(29%)
2:0(28%)
2:2(30%)
2:7(31%)
0:4( 4%)
4:4(42%)
1:2(12%)
3:6(40%)
1:8(20%)
Although there is always the issue of mismeasurement (which is very important in
these exercises) it does not appear to be the case that the bulk of East Asia’s growth
miracle is due to particularly strong TFP growth. Fast capital accumulation (a high
growth rate of the capital stock) seems to be at least as important.
8.2
Balanced Growth Path Predictions
Evaluation of the Model: Kaldor’s Facts
1. Output and capital per worker grow at the same constant, positive rate in BGP of
model.
2. Capital-output ratio
kt
yt
constant along BGP.
3. Interest rate constant in balanced growth path.
4. Capital share equals
along BGP).
; labor share equals 1
in the model (always, not only
Success of the model along these dimensions, but source of growth, technological
progress, is left unexplained.
International income di¤erences?
Evaluation of the Model: Development Facts
1. Di¤erences in income levels across countries explained in the model by di¤erences
in ; n; g; ; and .
2. Variation in growth rates: in the model permanent di¤erences can only be due to
di¤erences in rate of technological progress g: Temporary di¤erences can only be
explained by transitional dynamics.
130CHAPTER 8. NEOCLASSICAL GROWTH MODEL: CONFRONTING THE FACTS
3. That growth rates are not constant over time for a given country can be explained
by transition dynamics and/or shocks to ; n; g; ; and .
4. Changes in relative position need to be explained with changes in ; n; g; ; and :
Most economist take ; g; and
should they be di¤erent?)
as being (roughly) the same across countries (why
Compare U.S. and Chad, where
yt;us
yt;chad
25
Then
yt;us = (1 + g)t
1
1+g 1
1 + nus us
yt;chad = (1 + g)t
1
1+g
1 + nchad
1
1+
1
chad
or:
1+g 1
1+nus us
1+g
1
1+nchad chad
yt;us
=
yt;chad
nus
0:01 and nchad
yt;us
=
yt;chad
Assume
output?
us
0:02. Also,
1+
1+
!
1
1=3, g = 0:02; and
1:02 1
1:01 us
1:02 1
1:02 chad
0:9
0:9
= 0:96. How big must
!
chad
1
2
0:9
0:152
1
0:9
0:9
0:1
! 12
be to explain the observed di¤erences in
chad
1
chad
1:01
us
1:01
0:96
chad
1
=
1
yt;us
=
yt;chad
International Comparisons III
1
1+
0:9
0:9
! 12
= 25 )
= 625 )
chad
= 95:9 )
chad
= 0:01!!!!
8.3. CONVERGENCE
131
An alternative Aus 6= Achad .
Then, assuming all the parameters are the same:
yt;us
=
yt;chad
Aus
Achad
2=3
25 )
Aus
= 253=2
Achad
What where do di¤erences in technology levels come from?
Arguments of Ed with respect to international di¤erences of TFP
A Summary
We can take the absence of growth beyond g as a positive lesson.
Illuminates why capital accumulation has an inherit limitation as a source of economic growth:
1. Soviet Union.
2. Development theory of the 50’s and 60’s.
3. East Asian countries today?
Tells us we need to look some place else: technology.
8.3
Convergence
Is there convergence? Absolute/conditional convergence.
The Convergence Hypothesis
Fact: Enormous variation in incomes per worker across countries.
Question: Do poor countries eventually catch up?
Convergence hypothesis: They may do, but in the right sense!
8.4
Absolute Convergence
For those countries with the same s; n; ; ; and g:
1. Eventually same growth rate of output per worker and same level of output
per worker.
132CHAPTER 8. NEOCLASSICAL GROWTH MODEL: CONFRONTING THE FACTS
2. Countries starting further below the balanced growth path (poorer countries)
should grow faster than countires closer to balanced growth path.
Let us understand why.
Same Growth Rate and Same Level of Output
The reason for the …rst part of the hypothesis is pretty obvious.
In the long run, ouput per worker is given by:
y(t) = A(t)
s
n+g+
1
and the growth rate by n:
If two countries have the same s; n; ; ; and g, the will converge to the same level
of output per worker and the same growth rate.
Now, remember that we found ybt = b
kt
Chapter 9
Endogenous Growth Theory
9.1
Introduction
Our study of the neoclassical growth model highlighted the role of technological progress
as the source of economic growth in the long run. However, we did not postulate a
theory of technological progress. We simply assumed the existence of a continuous ‡ux
of innovations that was exogenous to the model. This was unsatisfactory because it
did not allow us to understand what drives technological change and what governments
can do to increase it. One of the big contribution of growth theory starting with Paul
Romer’s (1990) path breaking paper has been the development of operative models where
technological change is endogenous to the economy. This literature is, consequently,
known as endogenous growth theory.
There are many ways to model technological progress, but a particularly fruitful way
to analyze it is to think about technological advance as the appearance of new ideas.
Economists call ideas to all abstract concepts that increase our productivity given the
same level of inputs like capital and labor. As we emphasized in chapter 4, this de…nition
implies that we want to understand ideas in an extremely expansive way. Ideas are a
new product, such as the steam engine or the personal computer, but ideas are also
scienti…c discoveries such as calculus or quantum mechanics, managerial techniques, such
as cost accounting or human resources management, and business models such the credit
card or internet commerce. The common factor is that all these concepts make capital
and workers more productive. The issue at hand is not, therefore, to catalogue ideas
what to understand why and under what circumstances, are resources are spent on the
development of new ideas.
However, …rst we will present a simple model where growth does not come because
of technological change but because of the absence of decreasing returns to scale. This
model will highlight several results which will be useful later one.
133
134
9.2
CHAPTER 9. ENDOGENOUS GROWTH THEORY
The Simplest Endogenous Growth Model: The
AK Model
So far, we use a production function of the form:
Yt = Kt (At Lt )1
Since we wanted to overcome the e¤ects of decreasing returns to scale in capital and
generate sustained growth, we assumed that the level of technology grew at a constant
positive rate g; so that At = (1 + g)t : In some sense, we assumed growth rather than
explained it.
Thus, a …rst natural step is to eliminate these decreasing returns to scale. The
simplest endogenous growth model to do this, the so-called AK-model, departs from
the neoclassical growth model in only a single, small, but hugely important detail: it
assumes that = 1; that is, there are no decreasing returns to scale to capital in the
aggregate production function.
The production function, with = 1; is given by
Y t = Kt
Now there is no diminshing return to capital. Often researchers slightly generalize this
production function by writing it as Yt = AKt ; where now A gives the constant level of
technology. This formulation justi…es the name “AK-model”. We proceed by assuming
A = 1; although the analysis below straightforwardly generalizes to any arbitrary A > 0:
Clearly the production function can be written in per capita terms as
y t = kt :
9.2.1
Social Planner Problem
We proceed to analyze the social planner problem. A competitive equilibrium is de…ned
as in section 4.5, and we can establish the equivalence between equilibrium consumption
and capital allocations and those chosen by the social planner exactly as in section 5.3.
The social planner maximizes
1
X
t
u (ct )
t=0
ct + (1 + n) kt+1
subject to
= kt + (1
)kt
k0 given
(9.1)
A balanced growth path is de…ned as before as a consumption and capital allocation
such that ct = (1 + c )t c and kt = (1 + k )t k:
9.2. THE SIMPLEST ENDOGENOUS GROWTH MODEL: THE AK MODEL
135
The Euler equation now reads as
(1 + n)u0 (ct ) = u0 (ct+1 ) [1 + 1
]
The existence of a BGP again requires a CRRA utility function. Under this assumption
the Euler equation becomes
(1 + n)ct
ct+1
ct
= ct+1 [A + 1
[A + 1
]
=
1+n
]
(9.2)
Note that this equation implies that in the AK model the growth rate of consumption
ct+1
is always constant in the optimal solution of the social planner problem (since the
ct
right hand side of the equation is a constant). This constant growth rate is given by
1+
c
=
1
[2
]
1+n
(9.3)
:
or taking logs on both sides
log(1 +
c)
=
=
1
2
log
1
(1 + )(1 + n)
[log(2
)
log(1 + )
log(1 + n)]
where we recall that the time discount rate was de…ned by = 1+1 : First we observe
that as long as 1
> + n; then the growth rate of consumption in the the economy
satis…es c > 0; that is, there is positive growth even though the level of technology
A = 1 is constant over time. No exogenous technological progress is needed to generate
sustained growth in the AK model. Furthermore, as long as 1
; ; n and thus are
small (note that these are percent numbers, so we expect them to rarely exceed 10% or
0:1; we can replace log(1+ )
; log(1+1 ) 1 , log(1+ )
and log(1+n) n:
1
Then the above equation becomes
c
=
1
[1
n]:
We have seen from equation (9:2) that the growth rate of consumption is always
constant in the AK model. This suggests that in this model there is no transition period
towards the BGP; we conjecture that for any given initial capital stock k0 = K0 the
optimal solution to the social planner problem is to go to the BGP with consumption
growth rate c given above immediately in period 0: Thus we guess that
ct = (1 +
kt = (1 +
1
t
c ) c0
t
k ) k0
(9.4)
Had we worked with continuous time this equation would be exact, rather than just an approximation.
136
CHAPTER 9. ENDOGENOUS GROWTH THEORY
where the growth rate of capital per capita k and the initial consumption level c0 are
yet to be determined. To do so we now exploit the other necessary condition for an
optimal solution (the Euler equation was one), the resource constraint (9:1): Replacing
ct ; kt and kt+1 by their corresponding guesses from (9:4); we obtain
(1 +
t
c ) c0
+ (1 + n) (1 +
k )(1
+
(1 +
t
k ) k0
t
c ) c0
= [2
= [2
](1 + k )t k0
(1 + n)(1 +
k )](1
+
t
k ) k0
First, this equation can only hold for all periods t if the left and the right hand side grow
at the same rate, which requires k = c = ; as in the standard neoclassical growth
model. Second, the consumption level in period 0 is given by
c0 = [2
where the BGP growth rate
given.
9.2.2
(1 + n)(1 + )]k0 :
was determined in (9:3); and the initial capital stock is
Empirical Predictions of the Model
In contrast to the neoclassical growth model in which the long run growth rate of per
capita income, consumption and capital was exclusively determined by the growth rate
g of technology, which itself was exogenous and not explained. In the AK-model the
long-run growth rate
1
[2
]
1+ =
1+n
depends on preference parameters ; 1 ; the technology parameter and population
growth n: The driver of long-run growth in this model is the accumulation of physical capital, since, in contrast to the neoclassical growth model, the marginal product of
capital does not decline as capital increases. Long-run growth increases as households
become more patient ( increases) and more willing to tolerate nonsmooth consumption
pro…les (the intertemporal elasticity 1 increases). A higher physical depreciation rate
and a higher population growth rate n decrease the long-run growth rate.
Another important di¤erence in predictions, relative to the neoclassical growth model,
is that in the AK-model the ecomomy immediately enters the BGP and there is no
transition period. Thus according to this model changes in the grwoth rate of per capita
income should only occur if ; ; n or change. Thus while the most interesting and
testable implications of the neoclassical growth model pertained to levels of income per
capita, those of endogenous growth models in general and the AK-model spe…cally,
pertain to growth rates of income per capita.
One undesirable feature of the the AK-model is its prediction with respect to wages.
In the competitive equilibrium of the model the representative …rm, as in section 4.5
maximizes pro…ts, given by
max Kt wt Lt
t Kt
Kt ;Lt
9.3. AN ENDOGENOUS GROWTH MODEL WITH HUMAN CAPITAL
137
and the equilibrium wages wt and rental rates t are given by the corresponding marginal
products of labor and capital, see equations (4:30) and (4:31): But given the production
function Yt = Kt ; this implies
t
wt
= 1
= 0:
Since labor is not essential in the production of goods, its price (the wage) equals zero.
All revenues from production are paid out to the owners of the essential production
factor, capital. This in turn implies that the capital share of income implied by this
model is 1, and the labor share is zero. Taken at face value, this prediction of the model
is grossly at odds with the data which show a labor share of income between 60%-80%
for most countries.
One might argue that the capital stock Kt ought to be interpreted broadly to include
human capital, so that total capital income t Kt includes labor income. Still, the income
that raw labor generates in this model is zero. In addition human capital (the brains
of engineers) and the machines they construct are prefect substitutes in this model,
lumped together into the overall capital stock Kt : The next section shows that it is
fairly straightforward to construct a model with a plausible labor share and imperfectly
substitutability between human and physical capital that inherits all predictions of the
AK-model with respect to economic growth.
9.3
An Endogenous Growth Model with Human Capital
The previous model had the undesirable feature that all income accrued to capital income. We now re-introduce labor as an essential production factor. But instead of
envisioning labor as the number of physical bodies Lt ; we now think of it as the human
capital Ht used in production, and replace Lt by Ht in the production function. The
aggregate production function now becomes
Yt = Kt (AHt )1
There is still no exogenous progress (since A is constant). The key substantive di¤erence to standard neoclassical growth model is that while raw labor simply grows at the
exogneously given population growth rate n; we assume that households can build up
human capital through investing in education, much in the same way they could build
up their physical capital stock by investing in it.
In particular, we assume that human capital follows the law of motion
Ht+1 = (1
h )Ht
+ Ith
(9.5)
where h is the depreciation rate on human capital (the rate at which old knowledge is
forgotten or becomes obsolete) and Ith is the amount of the …nal output good used for
138
CHAPTER 9. ENDOGENOUS GROWTH THEORY
investment in human capital. As before, physical capital accumulates as
Kt+1 = (1
(9.6)
)Kt + It
and the aggregate resource constraint reads as
Ct + Kt+1 + Ht+1 = Kt (AHt )1
+ (1
)Kt + (1
(9.7)
h )Ht
As in the previous chapters the welfare theorems apply and we can solve for competitive
equilibrium allocations by solving the social planner problem. After inserting equations
into and dividing by the number of households Nt = (1 + n)t we obtain the resource
constraint in per capita terms
ct + (1 + n)kt+1 + (1 + n)ht+1 = kt (Aht )1
+ (1
)kt + (1
h )ht
)kt + (1
h )ht
Social planner problem
max
fct ;kt+1 ;ht+1 g1
t=0
1
X
t
u (ct ) s.t.
t=0
ct + (1 + n)kt+1 + (1 + n)ht+1 = kt (Aht )1
ct ; kt+1 ; ht+1 0
k0 ; h0 > 0 given
+ (1
We can ignore all three nonnegativity constraints since zero consumption is never optimal
because of the Inada conditions on the utility function. If one of the production factors
kt+1 ; ht+1 is set to zero, production is zero, too.
First order conditions
t
t+1
(1 + n)
(1 + n)
t
t
t 0
=
=
u (ct )
t+1 0
u (ct+1 )
=
A1
t+1
=
t+1
ht
kt
)A1
(1
1
+1
ht
kt
!
+1
h
!
Combining the …rst two equations gives the standard Euler equation
u0 (ct ) =
t
u0 (ct+1 )
t+1
and combing the second two equations yields
t
t+1
A1
=
ht+1
kt+1
1
1+n
+1
(1
=
)A1
ht+1
kt+1
1+n
+1
h
9.3. AN ENDOGENOUS GROWTH MODEL WITH HUMAN CAPITAL
139
The second equation in turn delivers that at any period t > 0 the optimally chosen ratio
between human and physical capital, t+1 = kt+1 =ht+1 is determined by the equation
A1
(
1
t+1 )
= (1
Now for simplicity let us assume that
given by2
=
h;
)A1
(
t+1 )
h
(9.8)
in which case the optimal capital ratio is
1
1
=
:
1
We observe that the higher the capital share ; the larger is the optimal ratio of physical
to human capital.
Note that we did not assume that the optimal capital-human capital ratio is constant,
nor did we impose BGP restrictions. It is optimal for the planner to keep the kt+1 =ht+1
ratio constant at all times. This also immediately implies that kt and ht must grow at
the same rate at all times, at least from period 1 on.3
Now we can determine
t
=
1
A1
1
+1
1+n
t+1
(1
=
)A1 + 1
1+n
(9.9)
In order to insure the existence of a balanced growth path we, as always, need to assume
a CRRA utility function. Doing so yields the Euler equation
ct
ct+1
ct
=
t
t+1
=
ct+1
t
t+1
and using the above result
ct+1
ct
=
(1
)A1 + 1
1+n
Thus the growth rate of consumption along the BGP is given by4
1+
2
c
=
[(1
)A1 + 1
1+n
]
1
:
(9.10)
Even if 6= h equation (9:8) has a unique positive solution ; albeit one that we cannot determine
in closed form, but rather have to determine with a computer. The rest of the discussion to follow goes
through completely unchanged, though.
3
Since k0 and h0 are arbitrary initial conditions, it will in general not be the case that k0 =h0 = :
In this case the growth rates from period t = 0 to t = 1 of k and h will di¤er. The planner will choose
(per capita) investment i and ih such that k1 =h1 = : Note that this may require negative investment
in either physical or human capital, which we have permitted. If one imposes nonnegativity constraints
on i; ih ; then it may take longer than one period before the planner equates k=h to :
4
Note that if = 1; we are back in the simple AK-model. In this case it is optimal to set the stock
140
CHAPTER 9. ENDOGENOUS GROWTH THEORY
From resource constraint we can show, using an argument identical to that for the simple
AK model, that the growth rate of consumption c equals the growth rate of physical
and human capital, c = k = h : The economy hits the BGP immediately (again there
is no transition period), and the growth rate is given by (9:10):
Essentially same qualitative …ndings. It is crucial to have constant returns to scale in
factors that can be accumulated. Here labor, in the form of human capital, is noting else
but physical capital. Growth does not have to be assumed, it happens endogenously.
This model successfully resolves the problem of the AK-model to deliver a plausible
capital share. In the current model, as always, wages (returns to human capital) and
returns to physical capital are given by their corresponding marginal products
wt = (1
Kt
t =
)Kt Ht (A)1
1
(AHt )1 :
Thus the labor share in total income is given by
)Kt Ht (A)1
Kt Ht1 (A)1
(1
wt Ht
=
Yt
Ht
=1
;
as before in the standard neoclassical growth model. Furthermore, while the wage per
unit of human capital
wt = (1
)Kt Ht
= (1
)
kt
ht
Kt
Ht
(A)1
= (1
)
(A)1
(A)1
= (1
) ( ) (A)1
is constant over time, since human capital per capita is growing at the constant rate
; so are wages per household. Households get smarter at a constant rate, and this is
re‡ected in rising wages.
9.4
Models with Externalities
So far we could solve for the competitive equilibrium in our model by solving the associated problem of a social planner that maximizes lifetime utility of the representative
household. We now consider a class of endogenous growth models where this equivalence
of human capital to zero, and thus
= 1: Then from (9:9)
t
=
1
A1
+1
1+n
t+1
0
=
1
A
1 0
1
+1
1+n
=
2
1+n
and we recover exactly the same BGP growth rate as in the previous section.
9.4. MODELS WITH EXTERNALITIES
141
breaks down, because there are externalities in production. The model we discuss is due
to Romer (1986). The model di¤ers from the previous models in its speci…cation of
the aggregate production function. We again start with a production function for the
representative …rm given by
Yt = Kt (At Lt )1
where Kt is the total amount of capital used by the …rm. But now we suppose that
the level of technology is determined by the average amount of capital …rms use in
production, that is, At = Kt =Lt ; where Kt is the average capital used across all …rms.
Thus the level of technology equals the per capita capital stock in the economy, and the
production function reads as
Y t = Kt
Kt
Lt
1
Lt
= Kt Kt1
(9.11)
Note that we, always, assume that the total number of …rms is equal to 1, and that
all …rms are identical. Thus in equilibrium the average capital Kt in the economy equals
the capital stock Kt that individual …rms optimally choose, Kt = Kt . But crucially, the
individual …rm does not take into acount that when it uses more capital Kt ; it not only
gets more productive itself, but makes all other …rms more productive by raising Kt :
This positive side e¤ect is unintended by …rms and is called an externality: the choice
of one …rm changes the production technology of other …rms. Since a …rm does not get
compensated for this positive externality in a competitve equilibrium, it does not take
it into account when choosing the individually optimal capital stock for production. A
social planner, on the other hand, that can dictate consumption and production decisions
in the economy understands and internalizes the e¤ect a larger capital stock in one …rm
has on the technology of other …rms. It is this di¤erence that leads to a divergence
between socially optimal and competitive equilibrium allocations: the welfare theorems
do not apply in this model.
In the next section we analyze the solution to the social planner problem that determines, as a benchmark against which the competitive equilibrium is compared, the
socially optimal allocation of consumption and capital. We then turn to a characterization of the competitive equilibrium and compare it to the social optimum. We …nally
discuss what policies a government could adopt so that the competitive euqilibrium under
these policies is socially optimal.
9.4.1
Social Planner Problem
The social planner chooses consumption and capital, internalizing that capital used by
each production unit Kt equals the average capital stock in the economy, Kt : Using the
fact that Kt = Kt in (9:11) we obtain the aggregate production function faced by the
social planner problem as
Yt = Kt ;
which is exactly the same as the production function in the basic AK-model. Given
that all other elements of the model remain unchanged, we can readily use all results
142
CHAPTER 9. ENDOGENOUS GROWTH THEORY
from section 9.2. In particular the BGP growth rate that the social planner chooses
for per-capita consumption and capital is given by (the superscript SP stands for social
planner):
1
[2
]
SP
1+
=
:
(9.12)
1+n
As in all endogenous growth models so, there is no transition period and the economy
immediately reaches the BGP. The resource constraint at period t = 0 determines initial
consumption (exactly as in the simple AK-model)
cCE
= [2
0
9.4.2
(1 + n)(1 +
SP
)]k0 :
(9.13)
Competitive Equilibrium
As in section 4.5 the representative household maximizes lifetime utility,
T
X
t
(9.14)
u (ct )
t=0
subject to a sequence of budget constraints. With population growth we have to be
slightly careful. A household maximizes per capita lifetime utility, but the budget constraint for the household has to be stated in terms of overall household consumption and
assets:
Ct + At+1 = wt Lt + (1 + rt )At
where Ct is total consumption, At are total asset holdings and Lt = (1 + n)t is total
number of members in the household. Dividing the budget constraint by Lt and de…ning
t
as per capita consumption and assets, we obtain
ct = CLtt and at = A
Lt
Ct At+1
At
+
= wt + (1 + rt ) :
Lt
Lt
Lt
t+1
But by realizing that ALt+1
= A
Lt+1
t
constraint in per capita terms as
Lt+1
Lt
= at+1 (1 + n) we obtain the household budget
ct + (1 + n)at+1 = wt + (1 + rt )at
(9.15)
Households maximize (9:14) subject to (9:15) and subject to the nonnegativity constraints on consumption and the No-Ponzi conditions on assets. Attaching Lagrange
multiplier t to the budget constraint (9:15); forming the Lagrangian (exactly as in
section 4.5) and taking …rst order conditions with respect to ct ; ct+1 and at+1 yields
t 0
u (ct ) = t
u (ct+1 ) = t+1
(1 + n) t = (1 + rt+1 )
t+1
t+1 :
9.4. MODELS WITH EXTERNALITIES
143
Divididing the second by the …rst equation and using the third equation to eliminate
t+1 = t yields the standard Euler equation (see equation (4:15)), but now with population
growth:
(1 + n)u0 (ct ) = (1 + rt+1 )u0 (ct+1 ):
In order to obtain a balanced growth path we again assume CRRA utility, and then the
Euler equation becomes
(1 + n)ct
ct+1
ct
=
(1 + rt+1 )ct+1
(1 + rt+1 )
1+n
=
and thus consumption growth is given by
ct+1
=
ct
1
(1 + rt+1 )
1+n
(9.16)
:
In order determine the competitive equilibrium growth rate, we now have to determine
the real interest rate in the economy. To do so, we have to study the …rm’s maximization
problem of the …rm, and the market clearing condition for assets.
As in section 4.5 the asset households save in, at is the phyiscal capital stock in the
economy, that is at = kt : The real interest rate rt then equals the rental rate t …rms pay
for renting out one unit of capital, minus the depreciation rate ; so that rt = t
: The
representative …rm maximizes pro…ts by choosing own labor Lt and capital Kt , taking
as given wages wt , rental rates t and the productivity At in the economy
max Kt (At Lt )1
wt Lt
Lt ;Kt
t Kt
:
The …rst order condition with respect to capital reads as
At L t
Kt
1
=
(9.17)
t:
But now we recall that productivity At = Kt =Lt equals the average (across …rms) capital
per worker, and that, since all …rms are identical, Kt = Kt : Thus At = Kt =Lt : Using this
result in (9:17) we …nd that
t
=
(Kt =Lt )Lt
Kt
1
=
which implies a real interest rate of rt =
: Using this result in the Euler equation
gives the growth rate in the competitive equilibrium (the superscript CE stands for
competitive equilibrium):
1+
CE
ct+1
=
=
ct
(1 +
1+n
)
1
(9.18)
144
CHAPTER 9. ENDOGENOUS GROWTH THEORY
Note that because the real interest rate is constant over time, the growth rate of consumption per capita is constant over time. Thus the economy, as in the social planner
problem, reaches a balanced growth path immediately. An argument identical to the one
presented in section 9.2 shows that capital and output per capita also grow at the same
rate CE : The initial level of consumption is given by
cCE
= [2
0
9.4.3
(1 + n)(1 +
CE
)]k0
(9.19)
Comparison and Policy Implications
In all models studied so far the competitive equilibrium is Pareto e¢ cient, and thus
coincides with the solution to the social planner problem. This is not the case in the
Romer (1986) model presented in this section. However, the socially optimal and the
competitive equilibrium consumption, capital and output allocations share a number of
crucial features. In both cases the BGP is reached immediately, and the growth rate is
constant and depends on all preference parameters ; ; the technology parameters ;
and the population growth rate n:
But this is where the similarities stop. Comparing equations (9:11) and (9:18) we
see that since < 1; economic growth is slower in the competitive equilibrium than is
socially optimal (as in the social planner problem). Since faster growth in the model
can only be accomplished by saving more and consuming less in the short run, it is not
surprising that consumption in period 0 is smaller in the solution to the social planner
problem than in the competitive equilibrium (compare equations (??) and (9:19); and
realize that SP > CE ). Clearly (since the social planner maximizes lifetime welfare
of households) the solution to the social planner problem yields higher welfare. Stated
di¤erently, Adam Smith’s invisible hand in the competitive equilibrium does not produce
a welfare maximizing consumption and capital allocation.5
9.5
Endogenizing Technological Progress: The Romer
Model
9.6
Ideas
Our previous assertion deserves further explanation. Economists think about the appearance of ideas from a very di¤erent perspective than many historians of science.
Those historians focus on the autonomous role of science (possibly mediated by social
institutions) in developing inventions and progress. This framework is sometimes called
the “Newton paradigm”, for the most likely apocryphal history of how Isaac Newton
discovered gravitation inspired by the fall of an apple of his head.6 Without denying the
possibility of random ideas, economists emphasize the role of pro…t as a main driving
5
6
While this result is obvious in principle, it is not ...TBA
Or, in the more poetic words of Don Juan (1821), Canto 10, Verse I, by Lord Byron:
9.6. IDEAS
145
force behind the development of new concepts. After all, even if Newton could have been
inspired by the fall of an apple, he had to assess the situation, derive a mathematical
representation of gravitation, write up his results, and publish them, all of which are
costly steps that require purposeful action on his part.
Classical study of Schmokler (Invention and Economic Growth, 1963): innovation
is determined by the size of the market.
Examples:
1. Horseshoe, many innovations in the late XIXth century and early XXth century, stop afterwards.
2. Air conditioners sold at Sears, between 1960 and 1980 and between 1980 and
1990.
3. Drugs for Malaria versus drugs for male impotence.
Ideas
What is an idea?
What are the basic characteristics of an idea?
1. Ideas are nonrivalrous goods.
2. Ideas are, at least partially, excludable.
Di¤erent Types of Goods
1. Rivalrous goods that are excludable: almost all private consumption goods, such
as food, apparel, consumer durables fall into this group.
2. Rivalrous goods that have a low degree of excludability: tragedy of the commons.
3. Nonrivalrous goods that are excludable: most of what we call ideas falls under this
point.
4. Nonrivalrous and nonexcludable goods: these goods are often called public goods.
When Newton saw an apple fall, he found
In that slight startle from his contemplation –
’Tis said (for I’ll not answer above ground
For any sage’s creed or calculation) –
A mode of proving that the earth turn’d round
In a most natural whirl, called "gravitation;"
And this is the sole mortal who could grapple,
Since Adam, with a fall or with an apple.
146
CHAPTER 9. ENDOGENOUS GROWTH THEORY
Nonrivalrousness and Excludability of Ideas
Nonrivalrousness: implies that cost of providing the good to one more consumer,
the marginal cost of this good, is constant at zero. Production process for ideas
is usually characterized by substantial …xed costs and low marginal costs. Think
about software.
Excludability: required so that …rm can recover …xed costs of development. Existence of intellectual property rights like patent or copyright laws are crucial for
the private development of new ideas.
Intellectual Property Rights and the Industrial Revolution
Ideas engine of growth.
Intellectual property rights needed for development of ideas.
Sustained growth recent phenomenon.
Coincides with establishment of intellectual property rights.
Data on Ideas
Measure technological progress directly through ideas.
Measure ideas via measuring patents.
Measure ideas indirectly by measuring resources devoted to development of ideas.
Important Facts from Data
Number of patents issued has increased: in 1880 roughly 13,000 patents issued in
the US, in 1999 150,000.
More and more patents issued in the US are issued to foreigners. The number of
patents issued to US …rms or individuals constant at 40,000 per year between 1915
and 1991.
Number of researchers engaged in research and development (R&D) in the US
increased from 200,000 in 1950 to 1,000,000 in 1990.
Fraction of the labor force in R&D increased from 0.25% in 1950 to 0.75% in 1990.
9.7. THE MODEL
9.7
147
The Model
We present a basic model of endogenous technological change. In comparison with our
development of the neoclassical growth model, the literature has not coalesced in a
canonical model of this class. Our choice of a concrete formulation should be consider
more as an example of the results that can be obtained in this line of research more than
a exhaustive explanation of the theory.
Our particular choice of a model of research and growth is based on Paul Romer’s
seminal paper (1990).7 The basic structure of the model is relatively straightforward.
We have three sectors, a …nal goods sector, an intermediate goods sector, and a research
sector. The …nal good sector produces the good that the households consume and invest
to accumulate capital. The intermediate good sector produces intermediate goods that
are employed in the production of the …nal good sector. The research sector develops
news ideas or blueprints for new intermediate good sectors. The intermediate good
sector will have monopoly power and generate pro…ts in equilibrium while the …nal good
sector and the research sector are perfect competitors and their pro…ts are zero. The
intermediate good sector pro…ts is distributed among the shareholders, in this case the
households of the economy.
Remark 37 (Inventors versus Managers) The model distinguishes between a sector
that invents new ideas and a sector that produces them. We make this distinction mainly
for convenience, since it is easier to understand each margin better in isolation. We could
redo the model with an integrated sector that invents new products and manufactures
them at the cost of more complicated expressions. Moreover, there is empirical evidence
gathered by Tom Holmes and Jim Schmitz (1990) that those that invent a new product and
those that sell it are quite often very di¤erent agents. A simple comparative advantage
argument justi…es this behavior: some people are extremely innovative and some other
are (comparatively) good managers. Then, the market will reward those that focus on
innovation and sell their ideas shortly after being developed to managers that implement
them at large scale. Think, for example, in the case of successful rock musicians. While
they are quite creative, over the years we have seen that a large number of them are not
particularly good managers of their products. That is why the most careful of them hire
a competent and reputable company to handle their copyrights and royalties.
Our model emphasizes technological progress through the introduction of new intermediate goods. This is a relatively straightforward channel for purposeful innovation
activity that is simple to model. However, we could think about many other forms of
technological innovation, some of which are easier to incorporate in the model, some of
which are more complicated.
Remark 38 (Other Technological Innovations) In the data, there is a di¤erence
between the invention of a new product (for instance, when the DVD was introduced) from
a new process (for example, a better inventory management practice by an electronics
7
A nearly identical formulation was proposed by Grossman and Helpman (1991a,b).
148
CHAPTER 9. ENDOGENOUS GROWTH THEORY
store). We will equate new products and new processes as new ideas that increase our
ability to produce the …nal good that the household enjoys. If we think about the …nal
good as watching movies at home, the household requires both the DVD player and the
availability of the DVD player in the electronics store when she goes to purchase it. Our
model does not capture, however, general purposes technologies, i.e., those technologies
that a¤ect a large class of activities. Examples of these technologies include electricity
or the internet. See Helpman (1998) for details on how to model these general purposes
technologies.
A crucial di¤erence between our model of research and growth and the neoclassical
growth model is that we cannot solve for the social planner’s problem and map the
results into the competitive equilibrium. The producers in the intermediate good sector
will have market power and charge a mark-up over marginal cost that will break the
…rst welfare theorem. Hence, we need to work further by searching directly for the
competitive equilibrium. We will explore later how the competitive equilibrium and the
social planner’s problem di¤er.
9.7.1
The Households
We will assume a constant population:
Nt = N0 = 1
Contrary to previous models in this book, normalizations are not innocuous. We will
discuss below that this model displays a scale e¤ect. However, it will be more convenient
algebraically specify that this scale e¤ect comes from the total numbers of hours, lt ,
which could be di¤erent from 1 and changing over time. The simple trick of keeping
population constant but changing available hours spares us the cumbersome distinction
between aggregate and per capita variables while keeping all the interesting economics
in. We still assume, though, that the household does not dislike to work and, hence,
that it will work its full amount of hours lt .
The representative household has preferences that are representable with the usual
CRRA utility function:
1
1
X
t ct
1
t=0
and a budget constraint:
ct + at+1 = wt lt + (1 + rt )at +
Z
At
t
(i) di
0
where all the notation should be known and with ct ; at
term:
Z At
t (i) di
0
0. Note that we have an extra
9.7. THE MODEL
149
that corresponds to sum of the pro…ts t (i) of the intermediate …rms that are distributed
back to the households since they are the owners of the …rms. The integration limits are
between 0; the very …rst idea, and At , the last idea existing in the economy at period t.
We have a continuum of intermediate goods …rms for all this range of ideas and we have
to integrate over all their pro…ts. The terminal condition that precludes Ponzi schemes
is given by:
at
=0
lim T
t!1
i=1 (1 + ri )
that is equivalent to the terminal condition in the neoclassical growth model.
The optimality condition for the household is given by an otherwise standard Euler
equation
ct = ct+1 (1 + rt )
9.7.2
Final Good Sector
We start by dealing with the …nal good sector. In that sector, there is a representative
…rm that stands for a large number of competitive producers that take the price of the
…nal good, the price of all the intermediate goods pt (i), and wages wt as given. We will
select the …nal good as the numeraire in the economy and normalize its price to 1.
The representative …nal good …rm has access to a production function of the form:
yt =
1
Z
At
xt (i) di lt1
0
where 0 < < 1. This production function takes a continuum of intermediate goods,
xt (i) and combines them with labor lt in a Cobb-Douglas function with elasticity coef…cient .
We highlight two points regarding this production function. First, we assume the
existence of a continuum of intermediate goods instead of a …nite number of them. This
is done basically for technical reasons: with a continuum of goods we have an integral,
which is easier to work with than the sum
!
At
1 X
yt =
xt (i) di lt1
i=0
that we would have in the case of a …nite number At of intermediate goods. Second, the
elasticity of substitution among di¤erent intermediate goods is just:
=
1
)
=
1
1
[proposed exercise: show that this is the elasticity]
150
CHAPTER 9. ENDOGENOUS GROWTH THEORY
The optimization problem of the …nal good …rm is:
Z At
pt (i) xt (i) di
max yt wt lt
lt ;xt (i)
0
yt =
1
subject to
Z At
xt (i) di lt1
0
xt (i) ; lt
0
The …rst order conditions for this problem are given:
Z At
1
xt (i) di lt = (1
wt = (1
)
0
pt (i) = xt (i)
1 1
lt
)
yt
lt
for 8 i 2 [0; A]
where we emphasize that the second necessary condition is taken for every intermediate
good i.
We can rewrite the second equation in terms of xt (i)
xt (i) = pt (i)
1
lt
1
This expression gives the demand of xt (i) as a function of labor lt and the price pt (i) :
Note that the price elasticity of xt (i)
d log xt (i)
=
d log pt (i)
1
1
is exactly the negative of the elasticity of substitution. When the intermediate good
price goes up by 1 percent, the quantity demanded goes down by 1 1 percent. This
result is intuitive: goods that are very good substitutes of each other will su¤er huge
variations in demand when one of the prices changes in relation to the other prices.
9.7.3
Intermediate Goods Sector
The intermediate goods sector is populated by a continuum of monopolists. Each intermediate good …rm i produces good i and nobody else does. You can think, for example,
that the intermediate good …rm holds a patent for the blueprints of the good or it is privy
to a trade secret. These intermediate good producers can set up the price they desire
for the good (or more precisely, the quantity sold), but they take lt and the demand
function that we derived in the previous subsection as given.
Intermediate goods …rms only use capital for production with a extremely simple
production function: a linear function that requires units of the …nal good to produce
one unit of intermediate good. We do not include labor in their production function to
ease the algebraic derivations. Also, note that the cost of production, , is the same
9.7. THE MODEL
151
that the elasticity in the production function. This simple normalization can be achieved
with the right choice of units in which we measure intermediate goods, while notably
simplifying the derivations below.
Then, the optimization problem of the intermediate good …rm is to set quantities to
maximize pro…ts:
t
(i) = max pt (i) xt (i)
xt (i)
xt (i)
subject to
1 1
lt
pt (i) = xt (i)
By substituting the constraint inside the objective function, we have:
t
(i) = max xt (i) lt1
xt (i)
xt (i)
that has a …rst order condition:
xt (i)
1 1
lt
= 0 ) pt (i) = 1
Using again the de…nition of pt (i)
pt (i)
=0
or:
pt (i) = 1
The interpretation of the price chosen by the monopolist is equal to a (gross) markup, 1= ; over marginal cost . The mark-up depends on the elasticity of substitution.
As the good becomes easier to substitute, the mark-up falls to 1. This is a standard
result in Industrial Organization.
Remark 39 (Modern Growth Theory and Industrial Organization) Macroeconomics
and Industrial Organization often deal with similar problems. As soon as we depart from
perfect competition in the …rms side, macroeconomists need to rely on the models developed by researchers in Industrial Organization to understand how …rms will behave. In
our model, we have built a production function for the …nal good producer that follows a
Dixit-Stiglitz structure, one of the workhorses of the analysis of non-competitive markets.
Thus, it is not a surprise that, in the pricing decision of the intermediate good producer,
we …nd the classical result of constant mark-up over marginal cost that depends on the
elasticity of substitution.
The particular formula is that the price of a good i is equal to
mc
1
is the elasticity of substitucion and mc is the marginal cost. Since
p (i) =
where
1
1
1
1
=
1
1
=
1
and mc = , we get the result in the main text pt (i) = 1:
152
CHAPTER 9. ENDOGENOUS GROWTH THEORY
The markup over marginal cost is at the center of the breakdown of the …rst welfare
theorem in this economy. Because of it, rates of transformation are not equal to relative
prices and the market allocation is not e¢ cient.
The total demand for the intermediate good is:
1 1
lt
pt (i) = 1 = xt (i)
) xt (i) = lt
The pro…t of the monopolist is then:
t
9.7.4
(i) = (pt (i)
) xt (i) = (1
) lt
Aggregation
An interesting property of the solution of monopolist problem is that it is independent
of i:
xt (i) = xt = lt and t (i) = t for 8 i 2 [0; A]
that is, all the intermediate good producers will behave exactly in the same way. Consequently, we can write:
yt =
Z
1
At
xt (i) di lt1
0
=
Z
1
At
xt di lt1
0
=
=
1
1
At xt lt1
At lt
This result is most remarkable: after aggregating, we have a production function that is
linear in labor. Since we can think about this production function as a Cobb-Douglas
where capital does not play a role, this will allow us to defeat the curse of the decreasing
marginal productivity of acculable factors.
The total consumption of intermediate goods is given by
xt =
Z
At
xt (i) di
0
=
At lt
It will also be useful to de…ne output net of the consumption of intermediate goods
(which is closer to the measure of GDP in NIPA) as
ybt =
1
At lt = ct + zt
9.7. THE MODEL
9.7.5
153
Production Function for Ideas
There is a production function that takes …nal good zt to generate new ideas with the
form:
At+1 = At + zt
or, in growth rates:
A
=
At+1
At
1=
zt
At
for A0 = 1, which is an innocuous normalization. This production function is linear in
the resources used to produce new ideas and it re‡ects the view that old ideas are not
forgotten. A simple generalization to allow for this forgetting would be
At+1 = (1
A ) At
+ zt
where A is the depreciation rate of old ideas. It is easier, though, to handle our benchmark case A = 0.
9.7.6
Research Sector
There is a large number of researchers, each of which takes At are given. By applying the
…nal good into the research sector production sector, scientists can produce new ideas,
which have a patented blueprint (or a know-how that is di¢ cult to back-engineered) that
they can sold to new intermediate good producers at price PA;t . Researchers also take
PA;t ; the price of a new idea or design as given.
Remark 40 (Patents versus Trade Secrets) Patents or copyrights are not the only
way to protect intellectual property. We can design products that are di¢ cult to reproduce.
A classic example is the formula for Coca Cola, which was never patented to avoid being
disclosed and fall into the public domain after the patent expires. Petra Moser (2005)
…nds strong evidence that patent laws in‡uence the distribution of innovation across
industries. In her sample, countries without e¤ective patent laws experienced a lot of
innovations in those industries where trade secrets were a stronger barrier to imitation
like food processing and scienti…c instruments. This is the result we would expect in a
world where innovation is a purposefully activity of rational inventors that respond to
incentives.
If there is free entry into the research sector, we have an equivalent formulation of
the sector as a perfectly competitive environment where pro…ts must be zero:
PA;t = 1
or:
PA;t =
1
154
CHAPTER 9. ENDOGENOUS GROWTH THEORY
The interpretation of this result is simple: if we put 1 unit of the …nal good into the
research sector, we will get new ideas that we can sell for a revenue PA;t . Then,
PA;t
1. Otherwise, a su¢ cient number of resources will ‡ow into the research sector
and lower the price of new ideas, PA;t , until the previous inequality holds. The strict
inequality, PA;t < 1, could happen in equilibria where there is no investment in new
ideas. Since for the relevant parameters this will not be the case, we simplify and just
write PA;t = 1.
By non-arbitrage, an investor must not obtain any pro…t from buying an idea today
at price PA;t , get a pro…t t+1 from operating that idea for one period, and selling
it tomorrow at price PA;t+1 (with the payments discounted at the interest rate rt+1 ).
Otherwise, if the pro…t is positive, investors will bid for the idea until the price goes up
enough, or conversely, if the pro…t is too low, they will short the patent of the idea until
the price falls enough. This is just a particular example of the price of an asset (in this
case the patent) that we will study in more detail in chapter xxx. Formally:
PA;t +
+ PA;t+1
=0
1 + rt+1
t+1
Then:
(1 + rt+1 ) PA;t =
t+1
+ PA;t+1
Since we know that PA;t = 1, we have
rt+1 =
t+1
or (with a lag)
rt =
t
= (1
) lt
This expression is interesting because it pins down the interest rate in the economy. By
lack of arbitrage in the patents, the return on patents and with it the interest rate must
be such that new scientists are indi¤erent about entering or not the research sector.
9.7.7
Characterizing the Equilibrium
We will look a symmetric market equilibria where all the intermediate good …rms produce
the same amount of the intermediate good xt , charge the same price pt , and have the
same pro…t t . We showed before that, indeed, this will be the case.
De…nition 41 Given initial ideas A0 = 1; a market equilibrium for our research and
development models consists of allocations for the representative household, fct ; lt g1
t=0 ;
1
allocations for the …nal good producer, fYt ; lt ; xt g1
,
a
sequence
of
ideas
fA
g
;
and
t t=0
t=0
prices frt ; wt ; pt ; PA;t g1
such
that:
t=0
9.7. THE MODEL
155
1. Given frt ; wt g1
t=0 ; the household allocation solves the household problem:
max
fct ;kt+1 gT
t=0
1
X
t
t=0
c1t
1
subject to
ct + at+1 = wt lt + (1 + rt )at +
Z
At
t
(i) di
0
ct
0
at
lim T
=0
t!1
i=1 (1 + ri )
given initial condition A0
2. Given fwt ; pt g1
t=0 ; the …nal good producer solves the problem:
max
lt ;xt (i)
yt
wt lt
Z
At
pt (i) xt (i) di
0
yt =
1
subject to
Z At
xt (i) di lt1
0
xt (i) ; lt
0
3. Given demand function of …nal good producer, each intermediate good …rm i sets
quantities xt (i) to maximize pro…ts:
t
(i) = max pt (i) xt (i)
xt (i)
t xt
(i)
subject to
pt (i) = xt (i)
1 1
lt
4. Given prices fwt ; PA;t g1
t=0 , the research sector has zero pro…ts:
PA;t = 1
5. Ideas evolve following the law of motion:
At+1 = At + zt
where A0 = 1:
156
CHAPTER 9. ENDOGENOUS GROWTH THEORY
6. Markets clear:
ct + xt + zt = yt
Z At
xt (i) di
xt =
0
Z At
1
xt (i) di lt1
yt =
0
(1 + rt+1 ) PA;t =
t+1
+ PA;t+1
We can also write the equations that characterize this equilibrium. First, we have
the Euler condition for the household derived in the same way than in the neoclassical
growth model:
ct = ct+1 (1 + rt )
Second, we can write the interest rate
rt = (1
) lt
Third, we have net output from the supply
ybt =
and demand side
1
At lt
ybt = ct + zt
and the law of motion of ideas
At+1 = At + zt
9.7.8
Balanced Growth Path Analysis
We will divide the analysis of the BGP of the model in two parts. First, we will construct
a BGP and …nd the rate of growth of the economy. Second, we will determine the level
of the di¤erent variables in the BGP. Through most of the following paragraphs we will
concentrate on the case in the case where lt = l, a constant not necesarily equal to 1.
Building a BGP
We build a BGP as follows. We start by noticing that the interest rate is always constant:
rt = r = (1
)l
Then, from the Euler equation of the representative household, along the BGP:
(1 +
c)
t
ct
= ct+1 (1 + r) )
c
=
(1 +
c)
(t+1)
c
(1 + r)
9.7. THE MODEL
157
we can …nd
r=
(1 +
c)
1 = (1
Then:
c
From ybt =
1
= ( (1 + (1
)))
)l
1
1
At l, ybt and At must grow at the same rate. Since At+1 = At + zt , zt
and At must also grow at the same rate. Finally, from ct + zt = ybt , we …nd that
g=
c
=
k
=
=
z
y
= ( (1 + (1
)))
1
1
which the growth rate of the BGP that justify all our previous equations.
Finally, from the innovation production function:
g=
zt
g
) zt = At
At
and
ct = ybt
Transitional Dynamics
zt
Now, from our previous derivations, when A0 = 1,
r = (1
z0 =
y0 =
yb0 =
)l
g
1
l
1
l
c0 = yb0 z0
x0 = l
and all variables will grow at rate
g = ( (1 + (1
For instance
yt = (1 + g)t
)))
1
1
1
l
The most interesting property of the model is that we do not have any transitional
dynamics. Given some l and some A0 , all the variables except the interest rate grow at
the same rate g in all periods.
158
CHAPTER 9. ENDOGENOUS GROWTH THEORY
9.7.9
Market Equilibrium versus Social Planner
We can set up the problem of the social planner that maximizes the utility of the representative household:
1
1
X
t ct
max
1
t=0
subject to
1
ct + xt + zt =
Z
At
xt (i) di
0
At+1 = At + zt
given initial condition A0 = 1.
Note: for the social planner’s problem we can already impose that lt = 1.
We can start with the problem of maximizing net output
Z At
Z
1 At
SP
xt (i) di
ybt = ct + zt =
xt (i) di
0
0
At+1 = At + zt
FOCs:
1
xt (i)
and then:
ytSP
=
1
1
= 0 ) xt (i) =
Z
At
1
1
xt (i) di =
1
At
0
and
ybtSP =
1
1
1
At
1
At = (1
)
1
1
At
Compare with equilibrium output:
ytSP =
1
1
At > y t =
1
At
Why is bigger?
A New Formulation
With our previous results, the problem of the social planner is just:
max
1
X
t
t=0
s.t. At+1 = At +
(1
ct1
1
)
1
1
At
ct
9.7. THE MODEL
159
FOC:
= ct+1 1 + (1
ct
1
)
1
Then (already imposing a BGP)
1+
SP
t
c
=
SP
1+
=
(t+1)
SP
1 + (1
c
1 + (1
1
1
)
)
1
1
)
1
1
Compare with equilibrium:
SP
>
= ( (1 + (1
)))
1
1
Too too little research.
given initial condition K0 and A0 = 1. We have already written all the terms
in aggregate values for ease of notation. Note that this social planner still has
three technologies (one for the …nal good, one for the intermediate goods, and
one for research). However, it does not have to follow prices among them, just
the resource constraints. Finally, the terminal condition is written in terms of the
marginal utility of per capita consumption times capital. As before, we impose
that the value of capital as measured by the marginal utility of consumption at an
arbitrarily distant future must be zero.
The …rst thing to note is that, given LY;t , the social planner will distribute capital
among the di¤erent intermediate goods as to solve:
Z At
1
max
xt (i) di LY;t
xt (i)
0
Z At
xt (i) di = Kt
s.t.
0
The …rst order condition of this problem is:
xt (i)
where
t
1
1
LY;t
=
t
is the multiplier of the constraint. Then:
xt (i) =
1
1
1
t
LY;t
which is independent of i. Therefore,
xt = xt (i) =
Kt
At
160
CHAPTER 9. ENDOGENOUS GROWTH THEORY
This part of the solution is equivalent to the market allocation: the capital is distributed
evenly among di¤erent intermediate good producers. However, the level of total capital
and the level of technology may be di¤erent. To see that, we plug it in our …rst order
condition in the production function to get back a value for the Lagrangian:
Z
At
1
1
t
1
LY;t diLY;t
=
0
1
1
At LY;t = Kt (At LY;t )1
t
t
= Kt
1
=
(At LY;t )1
and we get:
Yt
Kt
xt =
1
1
LY;t
With our previous results, the problem of the social planner is just:
max
fCt ;Kt+1 ;LY;t ;LA;t gT
t=0
1
X
Ct
Nt
t
1
1
t=0
subject to
Ct + Kt+1 = Kt (At LY;t )1
+ (1
) Kt
At+1 = At + BAt LA;t
LY;t + LA;t = Nt
Nt+1
=1+n
Nt
Ct 0
lim
t!1
t
Ct
Nt
Kt = 0
The Euler condition for this problem is given by:
CSP;t =
(1 + n)1
YSP;t+1
KSP;t+1
CSP;t+1 1 +
where we have added the subindex SP to emphasize that the variables are evaluated
at the social planner’s allocation. We can compare this Euler equation with the Euler
equation of the market equilibrium:
CE;t =
(1 + n)1
CE;t+1 1 +
2
YE;t+1
KE;t+1
9.7. THE MODEL
161
We can see that there is a di¤erence in the term generated by the wedge created by
the market power of intermediate good producers. But there is a second di¤erence in
the output-capital ratio. The social planner is deciding how much e¤ort to concentrate
on research, LA;t . At a …rst pass, the social planner internalizes the reduction in future
productivity of researchers implied by a change in At that the market equilibrium does
not. This will tend to reduce the e¤ort in research.
However, the social planner will have a di¤erent intensity of capital per idea, xt :
Remember that in the market equilibrium, the total demand for the intermediate good
was:
1
YE;t 1
LE;Y;t
xE;t =
KE;t
while in the social planner problem,
xSP;t =
YSP;t
KSP;t
1
1
LSP;Y;t
which may be quite di¤erent. This may increase or reduce the amount of labor devoted
to research and make the Euler equations di¤er.
Unfortunately, we cannot sign the di¤erence between the market and the social planner allocation. There will be parameter values for which the social planner will invest
more on research than the market and parameter values for which it will invest less. The
important conclusion is that, due to market power, there is no presumption that the
market will generate the “right”level of investment in new ideas.
Remark 42 (Intellectual Property Rights and Innovation) Our previous argument
illuminates a paradoxical property of market power granted by intellectual property rights.
On one hand, market power creates distortions and reduces welfare (either by investing
too little or too much on research). On the other hand, absent market power, the pro…ts
from a new idea will be zero and no investor will spend any resources in producing a new
idea, which will stop innovation. The way in which this result is traditionally read is that,
as policy makers, we need to provide legal protection to intellectual property right but not
more than necessary to induce the socially desirable level of research. But it is important
to realize that this protection must be granted ex ante: once the innovation exists, there
is no good reason to protect it any longer (except for a concern for reputation that we
discuss in the next remark), and certainly there is no reason to increase the protection
ex post.
This insights casts a particularly bad light on the Copyright Term Extension Act
of 1998, also know as the Sonny Bono Act (for his main proponent), or the Mickey
Mouse Protection Act. This act extended the copyright terms in the United States by
20 years, including both existing and new works (and hence protecting Disney’s Mickey
Mouse until 2023 as it was going to enter in the public domain in 1998). If there might
been an economic case for extending copyright terms for new works, there is close to
no argument in favor of extending existing terms beyond, of course, improving Disney’s
pro…ts or Sonny Bonno’s heirs royalties.
162
CHAPTER 9. ENDOGENOUS GROWTH THEORY
Remark 43 (Intellectual Property Rights and Time Inconsistency) The presence
of intellectual property rights creates one example of a pervasive problem in economic
policy: time inconsistency. The government has an incentive to announce that it will
protect intellectual property rights in the future to induce innovation today. However,
once the innovation has been created, there is no reason for the government to stick with
its promise: by eliminating the property rights over the idea, the market power of the
intermediate good producer disappears and the social welfare increases (except, of course,
for the investor who holds the patent). But the agents will foresee this problem and, in
the absence of a commitment device that will bind the government, they will not believe
the announcement of the government and they will not invest in innovation. Therefore,
time inconsistency generates a particularly suboptimal allocation. In real life, time inconsistency can be tempered by a concern on part of the government for their reputation.
We will have more to say about how to model reputation and how it interacts with time
inconsistency in chapter xxx.
Remark 44 (Competitive Innovation) The view that market power is required for
innovation has been recently challenged by Boldrin and Levine (2008) who argue that the
pro…ts made by an innovator during the time the competitors try to replicate the good and
push the price to its marginal cost may be enough to induce innovation in equilibrium.
They document many examples of competitive innovation and defend that, in particular
in industries like movies and music, the current length of copyrights is excessive.
Ct + Kt+1 = Yt + (1
If we are moving along a BGP with growth rates
(1 +
Dividing by (1 +
t
C)
C + (1 +
t
Y)
t+1
K)
K = (1 +
) Kt
C,
t
Y)
K,
and
Y + (1
Y,
we must have that
t
K)
) (1 +
K
; we get:
1+
1+
t
C
C + (1 +
K) K
1+
1+
=
K
t
Y
Y + (1
)K
K
This equation can only hold if consumption, capital, and output all grow at the same
rate:
C = K = Y
or in per capita terms:
g=
c
=
k
=
y
Similarly, from the Euler equation of the representative household:
(1 + g)
t
ct
=
(1 + n) ct+1 (1 + rt+1 ) )
c
=
(1 + n) (1 + g)
(t+1)
c
(1 + rt+1 )
9.7. THE MODEL
163
which will hold if rt+1 is constant and equal to:
r=
(1 + g)
1+n
1
1
Hence, if we can …nd a g such that consumption, capital, and output grow at that rate
and we can …nd a constant r, we would have shown the existence of a BGP.
We start our construction of the BGP of the model by imposing that LY;t =Lt is
constant. This must be the case or, otherwise, we cannot have a BGP as the number of
workers in at least one sector will go asymptotically to zero. Hence, both LY;t and LA;t
must grow at the rate of population growth:
LY;t = (1 + n)t LY
LA;t = (1 + n)t LA
where
L Y + LA = 1
The second element in our construction of a BGP comes from the growth rate of
ideas:
A
=
Then, substituting also At = (1 +
A
At+1
At
t
A)
= B (1 +
1 = BAt
1
LA;t
A and LA;t = (1 + n)t LA , we get:
t(
A)
1)
A (1 + n)t LA
The right hand side of the equation is a constant. Hence, all the terms that depend
on t on the left hand side must cancel each other for that part of the equation to be a
constant as well:
(1 + A )1 = (1 + n)
or:
1+
A
= (1 + n) 1
which gives us the remarkable result that the growth rate of ideas depends on the growth
rate of population.
We argued above that the interest rate must be constant in the BGP. Then,
rt =
t
= xt
1
L1Y;t
= (1 +
t+1
x)
x
1
(1 + n)t+1 LY;0
1
Again, the left hand side is constant, which requires that the terms on t cancel each
other on the right hand side
(1 +
1
x)
= (1 + n)1
or:
1+
x
=1+n
164
CHAPTER 9. ENDOGENOUS GROWTH THEORY
the amount of intermediate good grows at the rate of population growth. This gives us
that since:
Kt
xt =
At
and:
(1 + K )t K
(1 + x )t x = (1 + n)t x =
(1 + n) 1 t A
which implies
1 + K = (1 + n)1+ 1
or, in per capita terms,
1+
k
= (1 + n) 1
i.e., capital per capita grows at the same rate than ideas. But we argued before that
g=
c
=
=
k
y
and hence, the growth rate of the BGP is
1 + g = (1 + n)1+ 1
It is easy to check that, in fact, this rate holds for the growth of production of the
…nal good grows:
(1 +
Since
x
t
Y)
= n and 1 +
A
1
Yt = At xt LY;t
)
Y = (1 +
t
A)
t
x)
A (1 +
xt
(1 + n)t LY
1
= (1 + n) 1 :
(1 +
t
Y)
Y = (1 +
t
A)
(1 + n)t Ax LY1
which means that
1+
Y
= (1 +
A ) (1
+ n) = (1 + n)1+ 1
or in per capita terms
1+
y
= (1 + n) 1
=1+g
as we wanted to show (the result for consumption comes directly now from the resource
constraint of the economy).
Remark 45 (Growth Rate of xt and Kt ) The gross growth rate of xt is 1 + n while
capital grows at (1 + n)1+ 1 . The result can be read as follows. In every period, the
…nal good producer uses the same level of each intermediate good per household but the
number of households is growing at gross rate n (which requires also a growth in the
amount of capital). All the growth in per capita consumption appears because we have
(1 + n) 1 new varieties of intermediate goods.
9.7. THE MODEL
165
The last variable of interest, wages,
wt = (1
Since Yt grows at rate (1 + n) 1
(1 + n) 1 .
+1
)
Yt
LY;t
and LY;t at rate (1 + n), wt will also grow at rate
Remark 46 (BGP under the Social Planner) We can redo most of our previous
argument for the social planner. From the resource constraint of the economy, we will
also get:
g= c= k= y
(we did not use any element in that derivation except accounting identities). Then, the
social planner will obtain, from its Euler equation:
CSP;t =
(1 + n)1
cSP;t (1) =
(1 + g)
t
YSP;t+1
KSP;t+1
YSP;t+1
1+
KSP;t+1
)
CSP;t+1 1 +
(1 + n) cSP;t+1
cSP =
(1 + n) (1 + g)
1=
(1 + n) (1 + g)
(t+1)
cSP
1+
)
YSP
KSP
)
ySP
kSP
1+
which implies that a BGP will hold if kySP
is constant. But since, this follows directly
SP
from the fact that k = y :
>From the law of motion for ideas, we must have that also for the social planner
1+
A
= (1 + n) 1
and from the production function of the …nal good grows
g=
A
Therefore the social planner’s problem also has a BGP with the same rate of growth than
the market equilibrium. The di¤erences between the market allocation and the social
planner allocation will all concentrate in the level of the variables.
Solving for the BGP
Now we want to solve for the level of the variables in the BGP. From the Euler equation
of the household:
(1 + g) =
(1 + n) 1 + x
x
K
=
=
LY
ALY
1
(1 + g) 1
1+n
1
L1Y
)
1
1
1+
166
CHAPTER 9. ENDOGENOUS GROWTH THEORY
Remark 47 (Relation with Neoclassical Growth Model) The previous expression
for capital closely resembles the level of capital in the BGP of the neoclassical growth
model kN EO :
kN EO =
1
1
(1 + g) 1
1+n
1
1+
except that we divide by A times the number of workers in the …nal good sector, ALY ,
instead of the total number of workers.
Production is simply:
Y = K (ALY )1
= K L1Y
Now, we can go back to the resource constraint:
C + (1 + K ) K = Y + (1
)K )
1
C = K LY
( K + )K
>From the research sector …rst order condition:
PA;t BAt LA;t1 = (1
Yt
)
LY;t
t
) LYY;t
(1
PA;t =
)
BAt LA;t1
As we saw before, the numerator grows at rate (1 + n) 1 , while the denominator grows
at gross rate:
+
(1 + n) 1
1
(the gross rate of At times the gross rate of LA;t1 ). Therefore the ratio grows at rate
(1 + n)( 1
1
+1
) = (1 + n)
which tell us that PA;t grows at rate n, the same rate than x (this is, of course, not
a coincidence, since the price of the idea re‡ects the intensity at which the idea is
employed).
Now pro…ts:
t
= (1
)
Yt
= (1
At
)
(1 + n)(1+ 1 )t Y
= (1
(1 + n)( 1 )t
) (1 + n)t Y
also grow at rate n (we are using more of each intermediate good x).
9.7. THE MODEL
167
>From the BGP, we can …nd the value of LA that generates the right growth rate of
ideas :
1+
= (1 + n) 1
A
(1 + n) 1
1
(1 + n) 1
BA
1
which implies:
(1 + n) 1
BA
LY = 1
LA )
LA )
!1
A
= A + BA
LA =
1
= A + BA
A
1
!1
Finally, from the non-arbitrage condition for the price of the patent:
(1 + r) PA = (1 + n) ( + PA ) )
1+n
1+n
PA =
=
(1
) Y
r n
r n
Then, using the de…nition of pro…ts:
Y
1+n
(1
) Y BLA 1 = (1
)
)
r n
LY
1+n
1
BLA 1 =
)
r n
LY
r n
BLA 1 (1 LA ) =
)
1+n
2 1
1)
B LA 1 LA = (1 + n) 1
0
! 1
!1
1
1
(1 + n)
A
A A
(1 + n)
B@
= (1 + n)
1
1
BA
BA
At this moment we remember that:
(1 + g) 1
r=
1+n
to continue writing:
1
BLA (1
LA ) =
(1 + n)
B@
(1 + n)
BA
A
1
1
1 1
(1 + n)
1+n
B LA
0
1
1 = (1 + n)
!
1
LA = (1 + n)
1
1
(1 + n)
BA
1
1
1
1
1
1
!1
1
1
1
= (1 + n)
2
2
1
1
2
1
1)
1)
A A
= (1 + n)
1
2
1
1
which gives us the last equation on A (which unfortunately we cannot solve analytically).
168
CHAPTER 9. ENDOGENOUS GROWTH THEORY
Comparative Statics
To be completed.
9.8
A More General Model
There is a production function for new ideas of the form:
At+1 = At + BAt LA;t
or, in growth rates:
A
=
At+1
At
1 = BAt
1
LA;t
for A0 = 1, which is an innocuous normalization. New ideas depend on how much time
we spend thinking (LA;t ), our current stock of ideas, At , and how good society is in
producing new ideas, regardless of the level of inputs employed, as controlled by the
parameter B > 0.
The parameter 0 < < 1 tells us that there are decreasing returns to more time
spent on research: The parameter nets out “standing on shoulders” e¤ect and the
“…shing out” e¤ect. The “standing on shoulders” e¤ect denotes the advantages that
we obtain from previous ideas. For example, we could not write good economic models
without calculus. Hence, the presence of calculus enhances the production of new ideas
in economics. The “…shing out”e¤ect captures the depletion of possible ideas caused by
previous researchers. Coming back to our example in economics, after the neoclassical
growth model and the endogenous growth model have been developed, it is much more
di¢ cult for a young researcher to create original models that o¤er additional insights.
If > 1, the “standing on shoulders”e¤ect dominates the “…shing out”e¤ect. This
will imply that we will get better all the time at producing new ideas. While there is an
argument to be made that progress in science and technology has been accelerating over
time, > 1 will eventually take us to a situation where we create an in…nite number
of new ideas every period, a situation which is di¢ cult to see as empirically plausible.
In the case = 1, the “standing on shoulders” e¤ect exactly cancels out the “…shing
out” e¤ect. Even if possible, this hedge case feels more a divine coincidence than a
environment we need to spend much time thinking about. Most economists, hence, …nd
that the case < 1; where the “…shing out” e¤ect dominates, the most relevant. We
will then take this case as our parameterization of interest. Interestingly enough, even
this case will deliver growth in the long run. The case for endogenous growth for
1
are just stronger (technically, it also makes sense to focus on the case < 1 since it is
the only situation where we will have a BGP).
[To be completed].
9.9. PRODUCTIVE PUBLIC INFRASTRUCTURE IN ENDOGENOUS GROWTH MODELS (BARRO
9.9
Productive Public Infrastructure in Endogenous
Growth Models (Barro’s 1990 model)
170
CHAPTER 9. ENDOGENOUS GROWTH THEORY
Chapter 10
Endogenous Growth Models:
Confronting the Facts
10.1
Balanced Growth Path Predictions
171
172CHAPTER 10. ENDOGENOUS GROWTH MODELS: CONFRONTING THE FACTS
Part IV
Business Cycle Analysis
173
Chapter 11
Business Cycle Facts
[to be completed]
175
176
CHAPTER 11. BUSINESS CYCLE FACTS
Chapter 12
The Real Business Cycle Model
In this chapter, we build a prototype model to study business cycle ‡uctuations. In
previous chapters, we saw how the neoclassical growth model and some extensions to
it could account for much of the time series and cross sectional evidence of long-run
economic growth. Now, we examine whether the same neoclassical growth model can
also be used to analyze the short-run movements of the economy. To do so, we introduce
two key additions to the model.
First, we make labor supply endogenous. So far, we have mainly dealt with models
where the representative household supplies all its available time to the market. However,
in real life, households can choose between working longer or shorter hours. For example,
many households are composed by professionals or independent contractors, who enjoy
at least some ‡exibility at determining how many hours they work per week. Other
households may decide to work more or less overtime, take or leave a part-time position,
drop a job to go to school, take early retirement, or perhaps just moonlighting (or, if you
are an economics professor) do some consulting for a few hours. Even if many households have little ‡exibility in their labor supply, our representative household framework
means that all we need is that some households in the economy make decisions at the
margin regarding their labor supply. By allowing an endogenous labor supply, our goal
is to let the model capture the ‡uctuations on employment that we documented in the
previous chapter and to study how households react to di¤erent shocks to the economic
environment.
These shocks are precisely the second extension of our model. The very …rst type of
shocks that we study -and where this chapter concentrates on- are random technology
(also called productivity) shocks. Instead of having a determinist process for technology,
as we assumed when we studied long-run growth, we will postulate that technology
evolves over time in a random way. Shocks will make technology grow faster than
average in some periods and less than average (or even fall) in others. We will discuss
later how we can identify these productivity shocks in the data, which are their possible
causes, and how plausible they are as a modelling device. Nevertheless, before getting
into those issues, we can already adventure that focusing on productivity shocks makes
our life as builders of a business cycle model particularly simple. Therefore, tracing the
177
178
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
e¤ects of productivity shocks is an outstanding warm-up exercise before incorporating
other shocks to the economy, such as shocks to the preferences of the agents, shocks to
…scal policy, or shocks to monetary policy.
Because the only shock present in the model is a shock to technology, which many
observers interpret as a “real” factor, the stochastic version of the neoclassical growth
model that we present in this chapter is also known as the real business cycle model. The
surprising result is that, when we impose that the technology evolves over time in ways
similar to how the measured technology evolves in the U.S., the real business cycle model
accounts for a large share of aggregate ‡uctuations and co-movements across variables.
Despite this partial success, the …t to the data will not be perfect and we will have
much room for improvement. We will answer to these empirical challenges by extending
the model along several important dimensions, such as including more shocks and, in
chapter xxx, nominal rigidities that limit how quickly households and …rms can respond
to the shocks. But since these much richer models still have at their core the stochastic
neoclassical growth model, we …rst need to acquire a thorough understanding of the basic
real business cycle model.
12.1
The Environment
12.1.1
Households Preferences
In this chapter, we assume that the representative household has preferences over random
sequences of consumption and work, fct ; lt g1
t=0 ; of the form:
E0
1
X
t
u (ct ; lt )
(12.1)
t=0
where u (ct ; lt ) is the period utility function. As we did in the part of the book on
economic growth, we focus on cases where the time horizon is in…nite.
The most important new element in equation (12.1) is the presence of the conditional
expectation operator E0 ( ) in front of the sum of discounted period utilities. The expectation appears because future technology is random from the perspective of period
0, and thus, the household is uncertain about the amount of consumption and work
that she will enjoy in the future. Under standard assumptions described in any good
microeconomic textbook, agents rank those uncertain outcomes through the expectation
of the utility function. The conditional expectation operator E0 ( ) in equation (12.1) is
taken with respect to the beliefs that the household has regarding the probabilities of
each of the di¤erent events that can happen.
Remark 48 (Rational Expectations) We assume that, in this model, agents have
rational expectations: the probabilities that the agents use to evaluate the expectation in
(12.1) and the probabilities implied by the equilibrium of the model are identical. This
feature is also known as having model-consistent expectations because the beliefs of the
12.1. THE ENVIRONMENT
179
agents are con…rmed by the outcomes of the model. Rational expectations were …rst proposed by John F. Muth (1960 and 1961) and popularized in the early 1970s by Robert
Lucas, who was awarded the Nobel Prize in 1995 for this work. The introduction of rational expectations in macroeconomics during the 1970s and 1980s represented a gigantic
leap in the understanding of economic dynamics. Consequently, many economists call
this time the rational expectations revolution.
The rational expectations hypothesis has one great advantage and one serious drawback. The advantage is that it eliminates arbitrariness in the selection of expectations:
the probabilities of events happening and the beliefs of the agents about the probabilities
of events happening is one and the same and the researcher does not need to worry about
free parameters (Thomas Sargent, an economist at NYU, a Prize Nobel winner in 2011,
and one of the early leaders in the development of the idea of rational expectations, likes
to call this situation a “communism” of expectations). The disadvantage is that rational
expectations preclude us from talking about situations where agents do not have those
beliefs (perhaps because they are still learning about the probabilities or perhaps because
they have cognitive constraints on their ability to process information) or where di¤erent agents in the economy have di¤erent beliefs. All these cases are both plausible and
important to understand.
There is a large literature examining the assumption of rational expectations and evaluating where we can and we cannot apply it con…dently (Sargent, 2008). For example,
Evans and Honkapohja (2001) summarize conditions under which agents learning the
probabilities in the model will converge to rational expectations. These conditions show
that, in some contexts, it is natural to use rational expectations because they are the
limit of a process where agents learn about the objective probabilities of events. However, there are also cases where learning may not converge to rational expectations and
even situations where heterogeneous beliefs (that is, situations where di¤erent agents in
the model have di¤erent beliefs) will survive in equilibrium. These situations and their
consequences are, however, beyond the scope of the book.
Another promising line of research is the study of environments where the agents are
not fully sure about the world in which they live (that is, they cannot pin down exactly the
probabilities in the conditional expectation in equation 12.1) and want to make decisions
that take explicit account of the ambiguity in the world (Epstein and Schneider, 2010) or
that are “robust” to mistakes in their perception of the situation (Hansen and Sargent,
2007). Again, a more detailed study of these models would divert us too far away from
the main message of this chapter.
The second new element in equation (12.1) is the presence of both consumption and
work as arguments of the period utility function. We assume that households do not
like to work, that is, ul ( ; ) < 0: The reason households do actually work is to generate
income to …nance consumption. Given the wage and interest rate in the economy, the
household will decide how much to work out of its available time (that we still normalize
to 1) and how much to consume in each period in order to maximize its expected utility.
The household choice generates an endogenous labor supply, where the word “endogenous” emphasizes that the labor supply curve is determined by the agents within the
180
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
model. As we mentioned in the introduction to this chapter, all that we need for this
mechanism to matter in the real world is that a percentage of households in the economy,
those close to the margin, change their labor supply decisions when wages or interest
rates change, even if the large majority of household do not.
Remark 49 (Intensive versus Extensive Labor Supply) Labor economists often distinguish between an intensive and an extensive margin of the labor supply decision. The
intensive labor supply decision is about how many hours an agent works, conditional on
her working; it refers to how “intensively” the agent works in a job. For example, an
intensive labor supply decision is the choice of a lawyer of how many cases to take in
her practice or of a medical doctor of how many patients to meet. Overtime or changing
vacation time are other examples of choices at the intensive margin.
The extensive labor supply decision corresponds to the choice of whether to work. For
instance, a young agent may choose between staying one more year at college or getting
a full time job; an older agent may decide between retiring this year or waiting one more
year; an agent may opt for going back to work after a few years of focus on household
work.
Economists like to distinguish between these two margins because they involve somewhat di¤erent issues. A common argument by many researchers, and one that resonates
with most of us, is that working implies a substantial …xed cost, such as the time involved
in commuting every day. In comparison, staying for an extra hour in the o¢ ce may
not involve any additional …xed cost. Therefore, the intensive and extensive margin can
respond in a very di¤erent way to the same changes in wages. For instance, a small
increase in her wage per hour may induce a practicing medical doctor to see one more
patient a day (intensive margin), but it may not induce a non-practicing medical doctor
to come back from retirement (extensive margin). On the other hand, having access to
company-subsidized health insurance is a reason why many people decide to get a job
(extensive margin), but it may have very little impact on whether, conditional on having
a job, to take or not a few more days of vacation a year (intensive margin).
Our discussion in this chapter will omit the distinction between the intensive and extensive labor supply decision because, in the context of a representative household model,
this distinction is di¢ cult (although not impossible) to make. Since the representative
household is a stand-in for everyone in the economy, her labor supply decisions will summarize both margins. Thus, a reduction of, for instance, a 10 percent in hours worked by
the representative household may correspond in the data with a reduction of 10 percent of
the number of employees (while the remaining employees still work the same number of
hours as before), a reduction of 5 percent of the number of employees and 5 percent of the
hours worked by those still working, or a reduction of 10 percent in hours worked by all
the employees without any change in their numbers. We will consider that all those cases
are equivalent, a convenient feature for analysis, although one that can easily be removed
in richer but more advanced models such as those often used to analyze unemployment
over the business cycle.
There are many possible parametric forms for the utility function with consumption
12.1. THE ENVIRONMENT
181
and leisure. Among the most popular, we have, the CRRA-Cobb Douglas:
u (ct ; lt ) =
ct (1
1
lt )1
1
1
that embodies a Cobb-Douglas aggregator of consumption and leisure (1 minus labor),
ct (1 lt )1 ; inside a standard CRRA utility function with EIS 1= ; the log-log:
u (ct ; lt ) = log ct + (1
) log (1
lt )
that is nothing more than the limit of the …rst utility function as
CRRA:
lt1+
u (ct ; lt ) = log ct
1+
! 1; and the log-
The last two parametric forms are separable in consumption and labor (that is, consumption and labor enter in di¤erent terms in the utility function), but not the …rst. Separability is often useful to derive analytic insights (it make partial derivatives simpler!),
although it is not clear whether is a good description of the preferences of households in
the data.
Remark 50 (Are Leisure and Consumption Substitutes or Complements?) The
empirical literature has not reached a conclusion regarding whether leisure and consumption are substitutes, complements, or separable. Casual observation suggests that we see
situations where leisure and consumption are substitutes (we go to a nice restaurant after
a long day at the o¢ ce instead of spending the whole day at home resting), complements
(you need a lot of leisure to enjoy the consumption of a sea cruise), or separable (wearing
an expensive watch). Which is of these three possibilities prevails across the whole basket
of consumption goods is, thus, an open question.
In this part of the book, we will extensively use one class of preferences proposed by
Greenwood, Hercowitz and Hu¤man (1988) and, consequently usually called GHH:
lt1+
1+
u (ct ; lt ) = log ct
GHH preferences are most convenient because, as we will see below, they lead to extremely simple equilibrium conditions that we can easily manipulate to get explicit solutions.1
1
We have already imposed a log function outside the aggregator ct
have a CRRA structure of the form:
ct
u (ct ; lt ) =
lt1+
1+
1
1
1
:
lt1+
1+
. More generally, we could
182
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
To ease on notation, we will assume that population growth, n, is zero and that total
population is normalized to one. This allows us to substitute aggregate variables for per
capita variables. Nothing of substance is lost by this assumption and we could rewrite
the model with population growth at the cost of messier algebra.
12.1.2
Budget Constraint and First Order Conditions of the
Household
The budget constraint of the household requires some additional care because of the
presence of random shocks in the economy. Instead of a single asset, a one-period bond
as we had in chapter 4, the household will be able to trade a much larger class of
…nancial assets that will deliver payo¤s in particular events (these assets are called Arrow
securities, after Kenneth Arrow, who de…ned them precisely for the …rst time in 1953).
Imagine, as an illustration, the case when productivity can be either high or low.
Thus, there is potential asset to be traded that will pay 1 unit of the …nal good if
productivity is low and 0 otherwise, but there is also a second asset that will pay 1 unit
of the …nal good if productivity is high and 0 otherwise.
Contingent assets of this shape are common in daily life. For example, most households in the U.S. buy many types of insurance, which is nothing else than …nancial assets
that only pay if a concrete event ocurrs (for instance, we need to go to the hospital or
our car is stolen). At the same time, an asset that always pays 1 unit of the …nal good
regardless of the level of productivity is nothing more than a bundle of the asset that
pays if productivity is low and the asset that pays if productivity is high. De…ning all
these di¤erent …nancial assets, keeping track of them, and pricing them will be the task
of chapter xxx. Presenting that material here will distract us from the main thrust of
the building the real business cycle model.
Instead, here we will implement a simple trick that will make our task much easier.
Since there is only one representative household in the economy, the net value of all the
…nancial assets must be equal to the only real asset in the economy, physical capital,
kt . Think about it in this way. In the real world, you may have a debt, let’s say a
mortgage, but on the opposite side, there is another household that owns that mortgage
or the shares of the …nancial company that owns the mortgage. In other words, for each
debtor, there is a creditor. When we net out these two positions, as we do by assuming
that we have a representative household (and no foreign sector), the debt and the asset
cancel each other and we only need to worry about the amount of physical capital in
the economy, in this case, the house that backs the mortgage. In chapter xxx, we will
see how the equilibrium prices of …nancial assets will be such that the equality of all the
…nancial assets portfolios and physical capital holds exactly event by event.
Thanks to our simpli…cation, the household receives as income wage payments equal
to the wage per unit of time, wt , times the amount of labor supplied to the market, lt ,
plus the rental rate of physical capital, t , times the amount of physical capital, kt , it
holds. The household uses this income for consumption, ct , or to invest on more physical
12.1. THE ENVIRONMENT
183
capital, it .2 Therefore, the budget constraint of the household is given by:
ct + it = wt lt +
t kt
and its holdings of capital evolve as:
kt+1 = (1
) k t + it :
Putting these two equations together:
ct + kt+1 = wt lt +
t kt
+ (1
) kt :
Therefore, we can summarize the problem of the household as:
max E0
fct ;lt g1
t=0
s.t. ct + kt+1 = wt lt +
1
X
t
u (ct ; lt )
t=0
t kt
) kt for all t
+ (1
for some given k0 .
The Lagrangian of this problem is given by:
(1
1
X
X
t
u (ct ; lt ) +
L = E0
t (wt lt +
t kt
+ (1
) kt
0
ct
t=0
t=0
)
kt+1 )
with …rst order conditions:
ct :
ct+1 :
lt :
kt :
u1 (ct ; lt ) = t
Et u1 (ct+1 ; lt+1 ) = Et t+1
u2 (ct ; lt ) = t wt
t + Et
t+1
t+1 + 1
=0
If we combine the …rst, second, and fourth equation, we get:
u1 (ct ; lt ) = Et u1 (ct+1 ; lt+1 )
t+1
+1
that is nothing more than our usual Euler equation, except that now we have an expectation operator and the marginal utility of consumption with respect to consumption
depends on labor (if the utility function is separable in consumption and labor, this
dependence on labor disappears).3 As always, the Euler equation tells us that the household equates the marginal utility of consumption today with the discounted marginal
2
We do not impose that it is positive, that is, we can have negative investment. In this case, the
household transforms back physical capital into the …nal good costlessly. Even if not a very realistic
assumption, the possibility of negative investment allows us to avoid worring about inequality constrainst
of the form it 0 that will complicate our task at this introductory level of exposition.
3
The combination of the second and fourth equation is a bit subtle, because because in general
Et u1 (ct+1 ; lt+1 )
t+1
+1
6= Et u1 (ct+1 ; lt+1 ) Et
In exercise xx, we will ask you to work out the details explicitly.
t+1
+1
184
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
utility of consumption tomorrow times the rate of return of an investment in additional
physical capital, except that now all the right hand side is in expected terms because of
the presence of risk regarding the possible state of the world at time t + 1.
Another way to write the Euler equation is as:
1 = Et
u1 (ct+1 ; lt+1 )
u1 (ct ; lt )
t+1
+1
This expression tells us that, the discounted return of the investment in one extra marginal unit of physical capital,
must be equal to its opportunity cost, 1,
t+1 + 1
when we weight it with the ratio of marginal utilities
u1 (ct+1 ; lt+1 )
u1 (ct ; lt )
and take expectations with respect to every possible state of the world in period t + 1.
The ratio of marginal utilities values the return ‡ow of investment for each possible
situation of consumption and labor supply. For example, when ct+1 is low, and therefore
u1 (ct+1 ; lt+1 ) is high, we value the return from investment more. This is optimal because
an additional ‡ow of resources when we are consuming little and marginal utility is high
is particularly welcomed. In comparison, when ct+1 is high, and therefore u1 (ct+1 ; lt+1 ) is
low, we value the return from investment less because it adds little to our marginal utility.
In chapter xxx, when we talk about asset pricing, we will elaborate in this interpretation
of the Euler equation in much more detail and relate it with issues like risk aversion or
risk premia.
If we combine the …rst and second equation, we get:
u2 (ct ; lt )
= wt
u1 (ct ; lt )
that tells us that the (negative of) the marginal rate of substitution between labor
and consumption must be equal to the ratio of the price of labor (the wage wt ) over
consumption. Since the price of consumption was normalized to one because it is our
numeraire, we get that this ratio is just equal to wt .
Therefore, the maximizing behavior of the household can be neatly summarized by
a dynamic optimality condition:
u1 (ct ; lt ) = Et u1 (ct+1 ; lt+1 )
t+1
+1
a static one,
u2 (ct ; lt )
= wt
u1 (ct ; lt )
and the budget constraint:
ct + kt+1 = wt lt +
t kt
+ (1
) kt :
12.1. THE ENVIRONMENT
12.1.3
185
Technology
Technology is given by a familiar Cobb-Douglas production function:
yt = kt (At lt )1
that depends on capital, kt , labor, lt , and technology, At . Recall that, when we studied
long-run growth, we proposed that the level of technology At grows at a constant rate g:
At = (1 + g)t A0 :
Now, we will assume instead that technology is random over time:
At = (1 + g)t ezt A0 ;
(12.2)
where zt follows an autoregressive process of order 1 (abbreviated AR(1)) of the form:
zt = zt
1
(12.3)
+ "t
where 0 < < 1, "t is a productivity shock that comes from a standard normal distribution N (0; 1), and (for simplicity) we assume initial conditions A0 = 1 and z 1 = 0.
Equation (12.3) is called an autoregressive process because zt depends on its own values
from the past. In particular, in an order 1, it only depends on zt 1 .
The speci…cation for equations (12.2) and (12.3) captures well much of the properties
of At in the U.S. data. Equation (12.2) combines a deterministic trend (1 + g)t that captures the long-run evolution of At with a transitory component ezt .4 The interpretation
of this component is that technology sometimes improves faster than average (ezt > 1,
which happens when zt > 0) and sometimes it improves less than average (ezt < 1,
which happens when zt < 0). When ezt > 1, At is above its long-run trend and when
ezt < 1, At is below this long-run trend. If ezt is su¢ ciently small, for example because
the economy su¤ered a particularly large negative productivity shock, we will have that
At < At 1 , that is, technology worsens in that period. We will revisit in a few pages the
interpretation of zt and of the productivity shock "t and we will discuss the strengths
4
To account for the data, it is important to combine an exponential part, ezt , with a linear equation:
zt = z t
1
+ "t
Some authors prefer to write the process for technology as:
At = (1 + g)t t A0
where
log
t
= log
t 1
+ "t :
By de…ning
ezt =
t
) zt = log
t
we can see the alternative formulation of the evolution of technology and the one in the text are
equivalent.
186
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
and shortcoming of this approach to think about the evolution of technology. But before
doing so, and in order to understand the behavior of the model, we need to lay down
some of the technical properties of equation (12.3).
The AR(1) process (12.3) has an unconditional expectation of zero (remember that
z 1 = 0 and all the shocks have zero mean, thus at time 0, all expected future values of
zt are trivially equal to zero). However, the conditional expectation at time t 1 will be:
Et 1 zt = zt
1
that is, in general, di¤erent from zero. A way to distinguish between unconditional and
conditional expectations is that, while over long periods of time, we should see that the
mean of zt is zero (its unconditional expectation), in the short run we will deviate from
zero and that, since such deviation will persist for some time, zt is partially predictable
(its conditional expectation).
Remark 51 (Conditional Expectations) To understand better our previous statement, note that the expectation is a linear operation:
Et 1 zt = Et 1 ( zt 1 + "t )
= Et 1 zt 1 + Et 1 "t
= Et 1 zt 1 + Et 1 "t :
By the rules of conditional expectations:
Et 1 zt
1
= zt
1
(the conditional expectation of a predetermined variable is the value of the variable itself)
and
Et 1 "t = 0
because "t is a normal random variable with zero mean. Therefore:
Et 1 zt = zt 1 :
We can iterate on this procedure. Note that:
zt+1 =
=
zt + "t+1
2
zt 1 + "t + "t+1 :
Thus:
Et 1 zt+1 = Et 1 2 zt 1 + "t + "t+1
= 2 Et 1 zt 1 + Et 1 "t + Et 1 "t+1
= 2 zt 1 :
In general:
Et 1 zt+j =
j+1
zt
1
12.1. THE ENVIRONMENT
for an integer j > 0: Since
187
< 1 and zt
1
Et 1 zt+j =
is given, we have
j+1
zt
1
!0
as j ! 1:
How much zt deviates from zero depends on , a parameter that controls the volatility
of the shock (a large means high volatility and a small a low volatility), and , a
parameter that controls the persistence of those deviations (a closer to 1 means high
persistence, a closer to 0 implies small persistence).
Figure 1: Simulations from 4 AR(1) Processes
We illustrate the e¤ects of these two parameters in …gure 1, where we have plotted
four sample paths of the process for technology zt for 200 periods. In all four panels, we
keep the same sample path of productivity shocks "t .5 However, we change and in
5
A sample path means a particular realization of productivity shocks for 200 periods. If we simulated
the process a second time, we would have a di¤erent graph but the qualitative patterns would be the
same.
188
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
each panel. In the top left panel, low persistence and low volatility, we set = 0:05 and
= 0:001. In the top right panel, low persistence and high volatility, we set = 0:05
and = 0:007. Similarly, the bottom panels correspond to the cases high persistence,
low volatility ( = 0:95 and = 0:001) and high persistence, high volatility ( = 0:95
and = 0:007).
Low persistence appears in these panels as the two jagged graphs of the top row of
the …gure: the deviations of zt are very transitory and zt crosses zero many times. High
persistence generates the two smoother plots of the bottom row: the deviations of zt last
for a long time and zt crosses zero only a few times. This is the case because we can
write the process for technology as:
zt = zt 1 + "t
= 2 zt 2 + "t 1 + "t
= :::
= t "0 + ::: + "t 1 + "t
that is, as a weighted sum of the productivity shocks f"0 ; :::; "t g, where the weight is
smaller for shocks far in the past than for shocks nearer to current time. When is close
to zero, nearly all the weight is in the shock "t . Since "t changes sign very often (that is,
it crosses zero), zt also changes signs very often and we have a jagged graph. Instead,
when is close to 1, the new shock "t only has a small e¤ect in comparison with the sum
t
"0 + ::: + "t 1 : Thus, zt moves less widly and it does not cross zero that often. The
volatility a¤ects the the size of the ‡uctuations, the smaller , the smaller the range
of variation of zt as we can see if we compare the …rst column of panels with the second
column (note the di¤erent scales).
At risk of getting ahead of ourselves: we did not pick = 0:95 and = 0:007 by
chance. Those two parameter values are the ones that roughly result from estimating
the process for zt using U.S. data from 1948 until today. We will revisit this issue later,
but at this moment su¢ ce it to say that the panel at the bottom right resembles the
process for zt in the U.S.
Remark 52 (Unit Roots) We assumed in the main text that
something interesting happens. The process for zt becomes:
zt = zt
1
< 1. When
= 1,
+ "t
which is an example of a process known as a random walk. Random walks have many
intriguing properties. For example, if we substitute recursively, as we did above:
zt = zt 1 + "t
= zt 2 + "t 1 + "t
= :::
= "0 + ::: + "t 1 + "t
12.1. THE ENVIRONMENT
189
an equation that shows that shocks do not die over time. In our language before, we
have total persistence (sometimes called in…nite memory). Charles Nelson and Charles
Plosser argued in an in‡uential article in 1982 that many aggregate series in the U.S. can
be well described by random walks. A large literature followed Nelson and Plosser’s insight
and, quite often, models in macro are written assuming that zt follows a random walk.
However, random walks generate some non-trivial technical problems (in the jargon of
probability theory, they are non-stationary processes). For simplicity, we prefer to avoid
these problems in this textbook and we stick with out assumption that < 1.
Remark 53 (Going Beyond AR(1) Processes) There is nothing special about modelling the evolution of zt as an AR(1) process beyond its simplicity. We can easily extend
our real business cycle model to more general processes that belong to the Autoregressive
Moving Average (ARMA) class (of which AR(1) is an element). A basic example would
be to assume that zt follows an AR(2) process:
zt = zt
1
+ zt
2
+ "t
where both zt 1 and zt 2 matter for today’s zt : A slightly more complicated case is when
zt follows an ARMA(1,1) process:
zt = zt
1
+ "t +
"t
1
where the presence of the term "t 1 means that the stochastic component is a moving
average between the shock today, "t , and the shock yesterday, "t 1 with relative weight
given by the parameter :
Recently, macroeconomists have paid attention to cases where news about productivity
changes arrive before productivity actually varies (Jaimovich and Rebelo, 2009). Think,
for example, about reading the news that, in two years from today, we will have a more
productive computer. Households face an interesting dilemma: news have already arrived (and hence, they change the conditional expectation of utility in the future), but the
currently existing technology has not improved yet. A possible response is to start working and investing harder to have more capital when the new technology comes on-line.
This will generate a boom even before the arrival of the productivity advances. Stephanie
Schmitt-Grohé and Martín Uribe, two economists at Columbia University, have argued
that anticipated shocks account for more than two thirds of predicted aggregate ‡uctuations (Schmitt-Grohé and Uribe, 2008).
Finally, another interesting variation is to assume that the parameter that controls
volatility, , instead of being a constant, is a random variable itself. Therefore, we can
write:
zt =
log t =
zt 1 +
log t
t "t
1
+
t
where t is a volatility shock that comes from a standard normal distribution N (0; 1).6
This process is known as stochastic volatility. It allows researchers to represent the idea
6
Note that we write the process for
positive.
t
in logs to ensure that
t,
a standard deviation, is always
190
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
that sometimes we live in times of high risk, subject to large shocks ( t is high), such as
the 1970s, when the economy su¤ered large oil price shocks, and sometimes we live in
times of low risk, subject to small shocks such as during the 1990s.
All of the previous examples are more general representations that allow us to capture
richer dynamics of the data. However, an AR(1) is always a good “default” choice: it
works remarkably well and it is extremely simple to work with.
12.1.4
The Problem of the Firm
Everything we learned about the …rm in chapter 4 is unchanged in the real business cycle
model. The problem of the representative …rm is still de…ned by:
max fyt
lt ;kt
t kt g
wt lt
subject to
yt = kt (At lt )1
kt ; lt 0
with associated …rst order conditions:
wt = (1
)At
=
kt
At lt
t
12.2
kt
At lt
(12.4)
1
(12.5)
:
Equilibrium
The de…nition of competitive equilibrium in the real business cycle is a straightforward
extension of previous de…nitions of equilibrium that we have seen before.
De…nition 54 Given initial capital k0 ; a competitive equilibrium consists of allocations
for the representative household, fct ; lt ; kt+1 g1
t=0 ; allocations for the representative …rm,
1
fkt ; lt g1
and
prices
fr
;
;
w
g
such
that:
t
t t=0
t
t=0
1. Given frt ; wt g1
t=0 ; the household allocation solves the household problem
max1 E0
fct ;lt gt=0
1
X
t
u (ct ; lt )
t=0
s.t. ct + kt+1 = wt lt + t kt + (1
) kt for all t
given initial condition k0
0
12.2. EQUILIBRIUM
191
2. Given f t ; wt g1
t=0 ; the …rm allocation solves the …rm problem
max (yt
wt lt
nt ;kt
t kt )
subject to
yt = kt (At lt )1
At = (1 + g)t ezt A0
zt = zt 1 + "t , "t N (0; 1)
kt ; lt 0
3. Markets clear: for all t = 0; : : : ; 1
ct + kt+1
)kt = kt (At lt )1
(1
:
We proceed now to characterize this competitive equilibrium. From the problem of
the household, we found the Euler equation
u1 (ct ; lt ) = Et u1 (ct+1 ; lt+1 )
and
t+1
(12.6)
+1
u2 (ct ; lt )
= wt :
u1 (ct ; lt )
(12.7)
From the problem of the …rm, we have
wt = (1
t
=
)At
kt
At lt
kt
At lt
1
=
= (1
)
yt
:
kt
yt
lt
(12.8)
(12.9)
Thus, from (12.6) and (12.9), we get:
u1 (ct ; lt ) = Et u1 (ct+1 ; lt+1 )
yt+1
+1
kt+1
(12.10)
and from (12.7) and (12.8), we derive:
u2 (ct ; lt )
= (1
u1 (ct ; lt )
yt
) :
lt
(12.11)
In exercise xxx at the end of the chapter, we ask the reader to show that these two
last equations are also the optimality conditions of the Social Planner. This should not
a surprise since the …rst welfare theorem still holds in this economy, regardless of the
presence of random technological shocks. Thus, we can jump between the competitive
equilibrium and the social planners’s problem depending on which formulation is more
convenient in each moment.
192
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
Remark 55 (Utility function and existence of a BGP) The …rst three utility functions that we presented in section 12.1.1 share an important property. The marginal rate
of substitution between labor and consumption is linear in consumption. We write this
linearity as:
u2 (ct ; lt )
= f (lt ) ct
u1 (ct ; lt )
where f is a function that depends on u and the optimality condition that equates the
marginal rate of substitution to wage as
f (lt ) ct = wt :
If both consumption and the wage grow over time at a rate g, as they do along the BGP,
we will have:
f (lt ) ct = f (lt ) c (1 + g)t = wt = w (1 + g)t
where c and w are the values of consumption and wages at the start of the BGP. Cancelling terms:
f (lt ) c = w
which implies that lt is constant along a BGP, as it should be the case (otherwise we
do not have a BGP: since labor is bounded between 0 and 1, it cannot grow or fall at a
constant rate without getting out of these bounds).
We can see this linearity on ct for the CRRA-Cobb Douglas and the log-log cases:
ul
1
=
uc
and for the log-CRRA:
ct
1
lt
ul
= lt c t
uc
Then, we have the BGP conditions:
1
c
1
=w
l
and:
l c = w:
Unfortunately, the linearity condition is not satis…ed for GHH preferences:
lt
ul
=
uc
ct
1+
lt
1+
= lt
1
ct
1+
lt
1+
since consumption disappears from the marginal rate of substitution (more technically,
GHH preferences do not have a wealth e¤ect). A simple trick to …x this problem is to
assume a generalized form of GHH preferences:
log ct
t
(1 + g)
lt1+
1+
12.3. SOLVING FOR THE EQUILIBRIUM
193
where the disutility of labor depends on the long-run growth of technology (for example,
because technology also improves how much we enjoy leisure: think about how much better
is to watch a football game on a brand new HDTV). Then, we have
ul
= (1 + g)t lt
uc
and labor is constant over the BGP.
12.3
Solving for the Equilibrium
Whe we solved the neoclassical growth model, in chapter 7, we argued that do that we
needed, in general, we required some short of approximation since we did not have a
close-form solution to the set of equilibrium conditions of the model. Given that the
real business cycle is a richer version of the neoclassical growth model (with endogenous
labor supply and productivity shocks), it is not a surprise that we cannot solve for its
equilibrium in a close-form either.
We discussed also in chapter 7, the existence of three approaches for the task of
solving the model: …nding some particular case that accepts a solution with “paper and
pencil,” log-linearizing the model, and using more powerful yet involved computational
non-linear techniques. And, as we did in that chapter, now we will present one case
where, by imposing certain parameter values, we can …nd a particularly simple analytic
expression for the solution with “paper and pencil.”This solution will embody nearly all
the insights from the model and it will only fall short in quantitative terms. We leave
the log-linearization to an appendix and the non-linear algorithms for more advanced
textbooks.
12.3.1
An Expression for Equilibrium
As in chapters 5 and 7, the key to our analytic expression is to make two assumption, one
on preferences -that they are of the GHH form introduced in section 2 of this chapterand one on technology -that we have full depreciation = 1. While the …rst assumption
is empirically plausible (several empirical estimates suggests that the wealth e¤ect on
labor supply is small7 ), the second one is rather unreallistic when dealing with a business
cycle mode. We were not bothered in excess by = 1 when we studied long-run growth
because we could always pick as a period length one or two decades (for which full
depreciation is sensible) and still say something useful about the economic implications
of the model. However, when dealing with business cycles, we need to pick as a period
length a quarter (or, if quarterly national accounts are not available, a year). Otherwise,
the model would be quite useless to discuss high frecuency phenomena as the business
cycle. But our discussion of calibration in chapter 7 suggested that plausible values
are around 0.05 at an annual level (0.0125 at a quarterly level), radically far away from
7
xxxx document that....
194
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
= 1. This lack of empirical realism is the price we should gladly pay to learn the
intuition of the model while avoiding going to the computer.
With this caveat in mind and our two assumptions, we have that equations (12.6)
and (12.9) can be written as:
1
lt1+
1+
ct
= Et
1
yt+1
kt+1
1+
lt+1
1+
ct+1
lt = (1
)
yt
lt
(12.12)
(12.13)
Also note that,since = 1, we have kt+1 = it , that is, capital tomorrow is equal to
investment today. Then, (12.12) becomes:
1
lt1+
ct
= Et
1+
1
yt+1
kt+1
1+
lt+1
ct+1
1+
and the resource constraint of the economy is:
ct = kt (At lt )1
kt+1 = yt
kt+1 :
At this moment we make a third assumption, not strictly necessary, but that simpli…es
algebra, g = 0, the long-run growth of technology is zero8 and thus
At = ezt :
We start by working with equation (12.13):
lt = (1
)
yt
)
lt
lt1+
1+
=
1
yt
1+
and we can substitute in the …rst condition (12.12):
1
ct
1
1+
yt
= Et
1
ct+1
1
1+
yt+1
yt+1
:
kt+1
In chapter 5, we guessed and veri…ed that the solution of the neoclassical growth
model with log utility and full depreciation was
kt+1 =
yt :
We follow this lead, and postulate that the the optimal decision of capital accumulation
is also given by:
kt+1 =
yt
8
You will be ask as an exercise to …nd a solution when g > 0. It only involves de…ned, as in chapter
7, re-scaled variables to deal with their growth.
12.3. SOLVING FOR THE EQUILIBRIUM
195
(the di¤erence being that now, in the production function, labor can change from period
to period). With our guess and the resource constraint of the economy, we …nd that
consumption must be equal to:
ct = yt
kt+1 = yt
yt = (1
) yt :
We verify our guess. In the Euler equation:
1
(1
1
ct
1
) yt
1
1
1+
1
1+
1
1+
yt
yt
= Et
= Et
1
ct+1
(1
= Et
yt
1
1+
yt+1
)
kt+1
yt+1
1
1
1+
yt+1
yt+1
kt+1 )
yt+1 kt+1
) yt+1
1
1+
1
yt+1
)
kt+1
1
1
= Et
)
yt
kt+1
kt+1 =
yt :
where in the last kine we use the fact that, even if kt+1 is dated by period t + 1, it is
determined today and, consequently:
1
Et
1
=
kt+1
kt+1
:
Therefore, we got back our guess and the solution is veri…ed.
To …nd labor, we come back to the static condition:
lt1+
lt =
lt1+
1+
1
=
1
=
1
yt )
1+
kt (ezt lt )1
(1
kt e
)zt
)
1
+
:
and now we have the complete solution for the equilibrium of the model:
lt =
1
(1
kt e
)zt
yt = kt (ezt lt )1
kt+1 =
yt
ct = (1
) yt :
1
+
(12.14)
(12.15)
(12.16)
(12.17)
196
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
To get an alternative yet equivalent expression for output, we can substitute the
expression for lt (12.14) inside (12.15):
ezt
y t = kt
=
1
kt e(1
1
1
+1
+
+
kt
e(1
1
+
)zt
!1
+1
zt
+
)
(12.18)
that depends only on capital, kt , and productivity, zt .9
12.3.2
Interpreting the Dynamics of the Model
The previous set of equations tells us a powerful history about how business cycles appear
and propagate in this economy. Imagine that at time t, we have a productivity shock
"t > 0. This means that zt is higher that it would have been in the absence of this shock.
If we go to equation (12.14)
lt =
1
(1
kt e
)zt
1
+
;
we see that a higher zt means a higher lt (the other variable, kt , is …xed at time t). The
intuition is that marginal labor productivity is higher because "t > 0. Consequently
wages are higher, and the household wants to work longer hours. This is an example of
the substitution e¤ect that you learned about in microeconomics, when the price of a
good goes up (in this case the price of leisure, that is, the wage forgone), the demand
for this good (leisure) goes down (we work more).10
This mechanism may seem farfetched, but it actually holds in many unsuspected
situations. Think about how you decide when to study. If you are morning person,
you learn a lot early in the day and you probably try to …nd time in the morning
to study. In comparison, if you are more of an evening person, you will search for
opportunities to study later in the day. Regardless of your type, your behavior is just
responding to a higher productivity: you work comparatively more when you are more
productive. In the business world, high productivity may translate in that the …rm for
whom you work wants you to work overtime, or perhaps skip a few days of vacations.
The way your employer will induce you to work longer hours is to o¤er you a good
9
The derivation of an expression for labor and output that only depend on capital and productivity
is more general than the form of the policy functions for capital and consumption. Since we only used
equations (12.14) inside (12.15), these expressions for labor and output hold for any < 1 as long as
we keep GHH preferences.
10
You may also remember from microeconomics that, in addition to the substitution e¤ect, price
changes induced an income or wealth e¤ect of opposite sign (the higher the wage, the richer we are,
the more leisure we want to consume). But the GHH preferences that we use have a zero wealth e¤ect.
This feature simpli…es the analysis notably. For more general utility functions, the substitution e¤ect
will still dominate the wealth e¤ect, but the analytics become cumbersome.
12.3. SOLVING FOR THE EQUILIBRIUM
197
overtime supplement, a generous bonus to forgo your vacation, or the promise of a faster
promotion, all of which are equivalent to a higher wage. Finally, the lure of a higher
wage may convince a worker to postpone retirement for a few months or to skip attending
professional school (applications to Law or Business schools are markedly down when
the economy is booming and surge when the economy is in a recession).
Once we have determined that when "t > 0, lt is higher than it would be otherwise,
we can go to (12.15):
yt = kt (ezt lt )1 :
In this expression, there are two forces increasing yt : a higher zt and a higher lt caused
by a higher zt itself. This indirect e¤ect of a higher yt through higher lt is known as an
ampli…cation mechanism: it transforms a original impulse, the shock "t , into a bigger
e¤ect through the endogenous decisions of the agents, in this case, their choice of labor
supply. A general objective in business cycle theory is to search for strong ampli…cation
mechanisms that account for changes in aggregate variables caused by the relatively
small changes in the economic environment that we see from one quarter to the next.
Coming back to our previous example, if you study more in the morning because you
are more productive, your output (how much you learned) will be higher both because
you work more and because you were more productive.
Another way to see the same point is to look at our alternative expression for output
in equation (12.18):
yt =
1
1
+1
+
+
kt
e(1
)
+1
zt
+
:
Note that in this alternative expression, the coe¢ cient in front of zt in equation (12.15):
(1
)
are multiplied by the term:
+1
+
because we have already substituted labor supply out. Since 1 >
+1
> 1:
+
This shows how changes in zt become ampli…ed by the changes in labor supply they
induce (something similar happens with the e¤ects of capital). The ampli…cation is
+1
given precisely by the term +
.
The next step is to look at investment and consumption decisions given by (12.16)
and (12.17):
kt+1 =
ct = (1
yt
) yt :
198
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
Since output is higher, both investment and consumption are higher as well. Consumption is higher because the household is richer. However, part of the extra output is saved
as capital for tomorrow to smooth the marginal utility of consumption over time. This
shows how the real business cycle has, as a built-in, the idea of consumption smoothing
that we developed in the context of the household choice problem of chapter 4.
Investment is higher because of two reasons. First, consumption smoothing implies
that saving must go up and the only channel to save in this economy is through higher
investment in physical capital. Second, higher productivity today implies a higher productivity tomorrow: a shock "t > 0 means, as discussed in subsection 1.3 of this chapter,
that expected productivity tomorrow will be "t > 0 higher. Hence, it is a good time
to invest; the return of such investment will be high on expectation. If we summarize:
after a positive productivity shock, the household works more and output, consumption,
and investment all increase at time t.
What happens at time t + 1? First, we still have a higher expected productivity
tomorrow, "t . Second, the fact that we invested more at time t has the consequence
that we have more capital at time t + 1. Hence, the persistance of the productivity shock
and investment transmit the initial shock over time. This dynamic e¤ect is known as a
propagation mechanism. Both elements, higher expected productivity at time t + 1 and
higher capital, increase the marginal productivity of labor and with it wages, and, by
the same reasoning that we outlined before, the household will also work more at time
t + 1 than it would have otherwise done. Thus, output, consumption, and investment
will be higher than in the absence of the shock at time t: We can continue the study of
the dynamics over time following our previous steps, but after a few periods, it becomes
too boring to do it by hand. Fortunately, it is very easy to trace the dynamics with the
help of a computer.
To do so, macroeconomists use two tools. One is to simulate the model, compute
moments of the simulated data, and compare them with data from the real economy.
The second tool is called an impulse-response function, or IRF for short. The IRF
trace the response of the economy after one particular shock. Loosely speaking, while
the simulation computes the unconditional moments of the model, IRFs look at the
conditional moments (this point will become clearer momentarily). However, in both
cases, the answers that we get are inherently quantitative and for doing so, we require
as a previous step to determine the parameters of the model. Hence, in the next section,
we discuss how we can take our RBC model to the data.
12.4
Bringing the Model to the Data
This section examines how we can take the real business cycle model to the data. First,
we will explain how we can measure productivity shocks from observations of output,
capital, and hours worked. Second, we will discuss how we calibrate the remaining
parameters of the model to match a set of key features of the data.
12.4. BRINGING THE MODEL TO THE DATA
12.4.1
199
Measuring the Productivity Shock
How do we measure productivity shocks in the data? The most direct route is to use
the restrictions that our model imposes on the data to back up a estimated time series
of the shock. In particular, we exploit the following three properties of the technology
of the real business cycle model:
1. Technology is given by a familiar Cobb-Douglas production function:
yt = kt (At lt )1
2. The parameter
:
is equal to capital income share on total income.
3. Technology follows the processes
At = (1 + g)t ezt A0
(12.19)
and
zt = zt
1
+ "t :
(12.20)
Therefore, we can go to the statistical agency that elaborates NIPA for each country
and obtain a time series for output, yt , capital kt , hours worked lt , and labor income. In
the U.S., the Bureau of Economic Analysis and the Bureau of Labor Statistics provide
users this information freely on the web.
First, with the information on labor income share, we determine (point 2) above). In
chapter 7, we saw that the labor share of income can be readily measured for the actual
U.S. economy and that, with our assumptions, it is equal to 1
. Doing so, we also
pointed that in the data labor share has been roughly constant around 0.7, and thus we
choose = 0:3 for the model.
Second, we use point 1) to write:
At =
yt
kt
1
1
1
:
lt
(12.21)
We plug in equation (12.21) the observed yt , kt , and lt for a particular period and the
we just found. This gives us a the value of At . Repeating this exercise for every period
t = 1; :::; T that we are interested in, we get a time series fA1 ; :::; AT g of technology
(note that yt , kt , and lt will be di¤erent in each period but stays the same).
Third, we use fA1 ; :::; AT g to compute the trend g and the path of zt : A simple way
to do so is to take logs of (12.19):
log At = log A0 + [log(1 + g)] t + zt
This expression is a simple linear regression, where log At is the dependent variable,
log A0 is the constant, t is the independent variable, [log(1 + g)] is the coe¢ cient on the
200
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
independent variable (the time trend), and zt are the errors in the linear regression. We
can estimate:
\
\
log At = log
A0 + log(1
+ g)t + zbt
using any standard statistical software (or even a spreadsheet) and the series fA1 ; :::; AT g,
either with ordinary least squares (OLS) or maximum likelihood (ML).11 Part of the
output of the regression is the time series of estimated productivity levels zbt , fb
z1 ; :::; zbT g.
Finally, we come to equation (12.20), we plug it in the zbt from the previous step,
zbt = zbt
1
+ "t
and estimate and also using any standard statistical software and either OLS or
ML. Part of the output of the autoregression is the time series of estimated productivity
\
shocks b
"t , fb
"1 ; :::; b
"T g. The productivity shocks and the estimated log(1
+ g); b; and b
are the inputs that we need to simulate the model. Using quarterly data from the U.S.,
\
many researchers obtain numbers that ‡uctuate around log(1
+ g) = 0:005, b = 0:95,
12
b = 0:007:
The beauty of the process we just described is that there is a fundamental agreement
between the productivity shocks de…ned in the model and the way we back up the
information from the data. We use economic theory to guide how we interpret our
observations. Also, note that even if we used data to back up productivity levels, the
model still needs to be able to account for the behavior of yt , kt , and lt : It is entirely
possible, at least a priori, that the model, even when fed with the productivity shocks
implied by the data, delivers predictions about yt , kt , and lt that are totally at odds with
the observed variables. The empirical procedure just outlined only uses the restrictions on
technology implied by the model, but none of the behavioral implications of equilibrium.
12.4.2
Calibration
The real business cycle model that we have presented is indexed by several parameters:
the discount factor, , the share of capital on production, , depreciation, , the persistence and volatility of the productivity process, and , and the parameters of the
speci…c utility function that we select. We saw in the previous subsection on measuring
the productivity shocks how we set to match the mean of labor income share and how
we …nd and from the backed-up series estimated productivity shocks zbt : Also, and
to …nd an analytic solution, we assumed that = 1. Now we address how to calibrate
the utility function we pick and the discount factor. Our steps will be quite similar to
the calibration exercise that we undertook in chapter 5.
11
Any good textbook on econometrics will teach you more details about how to do this.
If we limit our sample to the years after 1984, the estimated standard deviation is only around half
of that number. This is a manifestation of a phenomenon known as the “great moderation.” Observed
business cycle ‡uctuations in the U.S. were much smaller between 1984 and 2007. In the context of
the simple framework used in this chapter, this shows up as technology shocks with a smaller standard
deviation.
12
12.4. BRINGING THE MODEL TO THE DATA
201
Calibrating the Utility Function
Since we can have many utility functions of interest, it is not feasible to catalogue all the
possible parameters that we may need to calibrate. However, all utility functions will
imply some restrictions on the behavior of labor supply. By selecting some observations
on labor supply and asking how we calibrate our model to match them in the set of the
most commonly used utility functions, we will …nd some general principles applicable to
other utility functions not covered here.
The two basic observations that we will care about is the average percentage of hours
worked by households and how these hours change when wages change. The key tool in
this endeavor is the …rst-order condition relating labor supply and wages:
u2 (ct ; lt )
= wt :
u1 (ct ; lt )
Remark 56 (First-Order Conditions for Di¤erent Utility Functions) For the CRRACobb Douglas:
u (ct ; lt ) =
ct (1
1
lt )1
1
;
1
the …rst order condition becomes:
ct
1
1
lt
= wt :
For a log-log utility
u (ct ; lt ) = log ct + (1
) log (1
lt )
we also have the …rst order condition:
ct
1
1
lt
= wt :
For the log-CRRA:
u (ct ; lt ) = log ct
lt1+
1+
we have
lt ct = wt :
And, …nally, for GHH preferences:
u (ct ; lt ) = log ct
lt1+
1+
we get:
lt = wt :
We now analyze how the …rst-order condition can be put to good use.
202
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
Labor Supply Elasticity We start by determing how the utility function determines
the responses of labor supply to changes in relative prices. A device to think about how
labor supply changes when wages vary is to introduce the concept of Frisch elasticty.13
We de…ne Frisch Elasticity as the percentage change in labor supply when wages change
while we keep consumption level constant. The reason we want to …x consumption is
that, after a change in wages, labor supply will change and with it labor income, and
potentially consumption. Thus, the Frisch Elasticity measures the substitution e¤ect of a
change in wage rate on labor supply without the income e¤ect induced on consumption.
Formally:
d log lt
dlt wt
=
F risch =
dwt lt ct constant
d log wt ct constant
To illustrate the concept of Frisch elasticity, we can compute it for the di¤erent
examples of utility functions that we presented before. For the CRRA-Cobb Douglas:
u (ct ; lt ) =
we have
d log lt
d log wt
1
lt )1
ct (1
1
1
=
1
lt
lt
c constant
:
To see this note that, from the …rst order condition
ct
1
1
lt
can be written as:
lt = 1
1
= wt
ct
wt
Then:
dlt
dwt
d log lt
d log wt
ct
1 lt
=
)
2
1
wt
wt
ct constant
dlt wt
1 lt
=
=
dwt lt ct constant
lt
ct constant
=
As we will discuss below, time-use surveys tells us that lt
1=3. Thus, the Frisch
Elasticity for a CRRA-Cobb Douglas utility function calibrated to match the average
amount of hours worked is 2.
13
Frisch elasticity is named in honor of Ragnar Frisch, a Nobel-Prize winner Norwegian economist.
Frisch was one of the creators of modern economics. He invented several new words, including “econometrics” and “macroeconomics” and he was one of the founders of the Econometric Society, one of
the key international associations of research economists. During the German ocuppation of Norway
in World War II, Frisch forcefully opposed the encroachments of the new authorities in the governance
of Oslo University. In part as a consequence of these activities, Frisch was arrested and sent to a
concentration camp for nearly a year.
12.4. BRINGING THE MODEL TO THE DATA
203
For a log-log utility
u (ct ; lt ) = log ct + (1
we have:
d log lt
d log wt
) log (1
=
ct constant
1
lt
lt
lt )
:
For the log-CRRA:
u (ct ; lt ) = log ct
we have
d log lt
d log wt
lt1+
1+
1
= :
ct constant
For a GHH utility Frisch elasticity is also given by
d log lt
d log wt
1
= :
ct constant
Indeed, for the log-CRRA and GHH utility functions, since consumption does not appear
in the …rst-order condition, leaving consumption constant is irrelevant (the compensated
and uncompensated elasticity are the same).
The calibration of the Frisch elasticity has traditionally been one of the main bones
of contention for researchers that distrusted real business cycle models. Many labor
economists thought that this class of models relied, to match the data successfully, on
what they saw as an unreasonably high labor supply elasticity (Alesina, Glaeser, and
Sacerdote, 2006, is a recent instance of such criticism). For example, MaCurdy (1981),
Browning, Deaton and Irish (1985), or Altonji (1986) documented Frisch elasticities
between 0 and 0.5.
However, the evidence that fed their attacks was gathered mainly for prime age white
males in the U.S. (or a similarly restrictive group). But representative agent models are
not about prime age white males: the representative agent is instead a stand-in for
everyone in the economy. It has a bit of a prime age male and a bit of old woman, a
bit of a minority young and a bit of a part-timer. If much of the response of labor to
changes in wages is done through the labor supply of women and young workers, it is
perfectly possible to have a high aggregate elasticity of labor supply and a low labor
supply elasticity of prime age males. To illustrate this point, Rogerson and Wallenius
(2007) construct an economy where agents go through a life cycle of youth and maturity
and where the micro and macro elasticities are virtually unrelated. But we should not
push the previous example to an exaggerated degree: it is a word of caution, not a
licence to concoct wild values for the aggregate Frisch elasticity. If the researcher wants
to depart in her prior from the micro estimates, she must have at least some plausible
explanation of why she is doing so (see Browning, Hansen, and Heckman (1999) for a
thorough discussion of the mapping between micro and macro estimates).
204
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
Based on these type of arguments (cite Keane), many macroeconomists think that
Frisch elasticites for a representative household model of around 2 are reasonable. We will
take this position while acknowledging the high level of uncertainty about this parameter.
Hence, for a log-CRRA or a GHH utility function, we will get:
= 0:5:
Average Hours We move now to analyze how the utility function determines the
average percentage of the available time of the household worked in the market. We have
observations from a number of time uses surveys that indicate that the average adult
in the U.S. economy works in the market around 1/3 of his available time (that is, 24
hours less time for sleeping and personal care). This observation was …rst documented
by Ghez and Becker (1975). Then, we can search for parameters in the utility function
such that the …rst order condition evaluated in the steady state:
u2 (c; l)
=w
u1 (c; l)
will imply that the household works around 1/3 of time.
This strategy is particularly transparent in the case of GHH preferences. In this
example, we can search for parameter values such that, given:
l =w
we have that l = 1=3: In appendix 1, we show that, in the steady state:
l= (
Then:
=(
)1
)1
1
l
1
1
=(
)1
1
1
3
Since we know from capital income share, from matching the interest rate, and we
computed before, we only need to …nd such that the previous equation holds. As an
example, for the values of = 0:3, = 0:995, and = 0:5 that we suggested before:
= 0:722
Calibrating the Discount Factor
Calibrating the Discount Factor
We start by calibrating the discount factor of the economy. Note, …rst, that, in equilibrium we have
u1 (ct ; lt ) = Et u1 (ct+1 ; lt+1 ) t+1
12.4. BRINGING THE MODEL TO THE DATA
where we have already use the assumption that
205
= 1. Then, in steady state:
u1 (c; l) = u1 (c; l)
or:
r=
=
1
where r is the real interest rate on an uncontingent bond (which, by absence of arbitrage,
must be equal to the rental rate of capital minus ). Now, if we observe the average yield
of a bond in the economy, we can get
=r
1
In the U.S. economy (Campbell, 2003) the average real yield of a short run Treasury
bond (the closest thing we have in the data to a default-free asset) has been, over the
last decades, around 1-2% on an annual basis and 0.25-0.5% on a quarterly basis. Hence,
a plausible calibrated value of will ‡uctuate around 0:99:5 0:9975 if we use a quarter
as the frequency of our model. Often economists call the observation from the data (in
this case, the average R), the calibration target.
Our last calibration parameter is the depreciation . In the case of the model that we
solved analytically, we assumed that = 1, but we already argued that we did this only
to obtain results without the need of a computer. In more realistic models, we would
need to pick a value for . A simple procedure to do this is as follows. Note that, by
market clearing
ct + kt+1 (1
)kt = yt
Then, in steady state:
c+ k =y
and then
=
y
c
=
k
is just the ratio of investment over capital stock.
i
k
Wrapping up
We can collect all the numbers for the calibration of the basic version of the RBC model
with GHH preferences and full depreciation:
Table 1: Calibration of RBC Model
0:3 0:995 0:5 0:722 0:95 0:007
With these numbers, the steady state of the model is
l=
1
3
206
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
(this was a calibration target) and:
k =
(
)1
+
1
1
= 0:059
y = k l1 = 0:199
c = 0:139
Now we are ready to start looking at moments of the model.
12.5
Unconditional Second Moments
Our …rst exercise is to compute the unconditional moments of the model. This can be
easily done as follows. First, we use a random number generator to draw a path of
productivity shocks, f"t gN
t=1 ; from a normal distribution N (0; 1) ; where N is a large
number (for instance, 10,000). Most spreadsheet programs or mathematical software
can easily draw these random numbers. Then, we can compute a simulated path for
fzt gN
t=1 using the formula:
zt = zt 1 + "t
starting with z0 = 0 (its steady state value).
Now, we set k1 = k (that is, capital is at its steady state value) and we …nd:
l1 =
1
(1
k1 e
)z1
1
+
y1 = k1 (ez1 l1 )1
k2 =
y1
c1 = (1
) y1
Then, with the k2 we just found and the z2 from the simulated path, we …nd:
l2 =
1
(1
k2 e
)z2
1
+
y2 = k2 (ez2 l1 )1
k3 =
y2
c2 = (1
) y2
with gives us a new k3 . We can iterate on this procedure until t = N , and obtain, at the
end of it, a sequence fct ; lt ; yt ; kt+1 gN
t=1 .
The next step is to drop some of the initial components of the sequence, for instance
the …rst 1,000 if we are generating a 10,000 observation path. The idea is that these 1,000
…rst components of the sequence are a “burn-in,” that is, a period that depends (even
if very weakly) from the initial conditions (z0 = 0 and k1 = k). Hence, our simulation
12.5. UNCONDITIONAL SECOND MOMENTS
207
depends less from those initial conditions and the concrete value, within a reasonable
range, does not a¤ect our results. Let us denote by J the …rst period after the burn-in,
in our example, J = 1; 001: At this moment, we have left a sequence fct ; lt ; yt ; kt+1 gN
t=J
that is draw from the dynamics of the model.
If N is large enough, a law of large numbers will ensure us that the properties of
1 14
fct ; lt ; yt ; kt+1 gN
Therefore, we
t=J are very close to the properties of fct ; lt ; yt ; kt+1 gt=1 .
can compute simple statistics, such as means, variances, and covariances, and we can
compare how these statistics compare with the statistics of the data.
More technically, the distribution of variables implied by fct ; lt ; yt ; kt+1 g1
t=1 is known
as the ergodic distribution of the model (where “ergodic”means the long-run properties
of dynamic systems that settle down in stationary behavior) and fct ; lt ; yt ; kt+1 gN
t=J is a
draw from that ergodic distribution. If N is su¢ ciently large, the draw from the ergodic
distribution will converge to the exact ergodic distribution itself.
Figure 2: Simulated Path of the Equilibrium of the RBC Model
14
In our economy, we can proof that a law of large numbers apply in the simulation. For more general
economies, such proof may be challenging.
208
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
We show now the results from a simulation as the one just described. First, we plot
in …gure 2 the last 500 observations from the simulation (from 9,501 to 10,000) for labor,
output, investment, and consumption. This type of plots is useful because it provides us
with a “reality check”: do the simulation from the model look like something that could
have come up from the data? If the answer is a resounding no, the model will not be
very useful. Also, this exercise is often a good way to build intution about the behavior
of the model. In this case, …gure 2 teaches us two important lessons:
1. Our simulated economy displays ‡uctuations that resemble, at least at …rst inspection, the ‡uctuations in the data: period of high output are follow by periods of
low output, and high (low) output comes at the same time than high (low) consumption, hours worked, and investment. While this is not a formal statistical test
of the model, it is at least some preliminary evidence that the model is interesting
enough to be analyzed further.
2. The di¤erent variables ‡uctuate around values that are close to the steady state
values (we will expand on this observation in a few paragraphs), but the deviations
with respect to those values can often be large. For example, l9501 = 0:369 (the
…rst observation in the top panel on the left), is nearly 11 percent above the steady
state value of 0.333.
In table 2, we report some selected moment from the simulated data.
Table 2: Moments
Variable Steady State
lt
0:333
yt
0:199
it
0:059
ct
0:139
of the Simulated Data
Mean Standard Deviation
0:334 0:014
0:200 0:013
0:060 0:004
0:140 0:009
You can appretiate a small, positive di¤erence between the steady state values and
the mean of the simulation. This is due to the fact that while zt is symmetric around 0,
its steady state value, exp(zt ) is not centered around 1, its steady state value (exponent
is a non-linear function). In particular, we will have a slight skewness toward the right
and the mean of exp(zt ) will be slightly above 1. That pushes the means of the other
variables also slightly above the steady state means. Even if this e¤ect is very small, it
illustrates that the average properties of stochastic economies may be di¤erent from the
steady state behavior of deterministic economies.
Often Covariances of the Simulated Data. Here pretty boring 1.
12.6
Impulse Response Functions
The idea is to compute and to plot the response of one or several variables in the economy over time after an impulse, in this case the productivity shock, assuming that the
12.6. IMPULSE RESPONSE FUNCTIONS
209
realizations of all the shocks in future periods are exactly equal to zero. Loosely speaking, an IRF is like a partial derivative: it studies how the economy responds to a shock
without worrying about any further shock in the future will a¤ect the dynamics.
To illustrate the shape of these IRFs, we proceed as follows. We start the model at
time t = 1 with the values of capital and productivity at the steady state (which we
derive in appendix 1 of this chapter). This also implies that k2 = k. Then, we assume
that, at time t = 2, the economy is hit by a shock "2 of 1 standard deviation (that is
"2 = 1 and z2 = 0:007"2 = 0:007). At this moment, we …nd c2 , l2 , y2 , and k3 using
equations (12.14) to (12.17) and make
z3 =
"2 = 0:95 0:007
This last steps embodies the idea that we are keeking all future shocks equal to zero and
trace the e¤ects over time of the shock at time t = 2. At this point, we just iterate again
and again using equations (12.14) to (12.17) and the law of motion for productivity.
Figure 2: IRF from RBC model
We plot the result of this exercise in …gure 2. In the top panel we can see the evolution
210
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
of hours worked over time. As we will explain below, we calibrated our model to get
that l = 1=3, the representative household works one third of the available time.
12.7
On the Interpretation of Productivity Shocks
At the core of the real business cycle model we have the productivity that drive the
dynamics of the model. The concept and measurement of productivity shocks raised
several questions.
First, what does it mean to have a productivity shock? More concretely....
12.8
Appendix 1: Steady State of the Model
In this appendix, we derive the steady state of the RBC model presented in the main
body of the chapter. Remeber that we derived the equilibrium conditions
1
lt =
kt e(1
)zt
1
+
yt = kt (ezt lt )1
kt+1 =
yt
ct = (1
) yt
In steady state, it must be the case that:
l=
1
1
+
k
y = k l1
k=
y
c = (1
)y
with z = 0. Then:
y=k l
1
=k
1
1
+
k
=
1
1
+
k
+1
+
12.9. APPENDIX 2: LINEARIZATION
211
Using now the third equation:
k=
y=
k
k=
1
"
+
+
#
+
)
+
1
)
1
1
1
k= (
+1
+
k
1
=
+
1
1
)
+
(1
)
)
1
1
Also,
l =
=
1
(
(
)
)1
1
1
!
1
+
1
+
1
and
y = k l1
=
=
12.9
(
(
)1
+
)1
1
(
( +1)
1
)1
1
1
1
Appendix 2: Linearization
Preliminary Steps
In chapter 5, we linearized the model in terms of variables:
x
bt = log(xt )
log(xBGP
)
t
xBGP
t
xt
xt
In this appendix, and to simplify the algebra, we will assume that g = 0 and, therefore,
we can write more easily
x
bt = log(xt )
log(x)
xt
x
xt
in terms of the steady-state value of the variable (remember that for a variable xt we use
x to denote its steady-state value).
212
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
Also, we need to assume some form of the utility function. While in the main text,
we used a GHH function, we opt now for a log-CRRA
lt1+
:
1+
u (ct ; lt ) = log ct
While GHH utility functions are particularly convinient when = 1 to derived analytic
solutions, they are not easy to linearize because its marginal utilities are somewhat
involved expressions. Also, much of the quantitative work in business cycles is done with
log-CRRA and, hence, it is worthwhile to gain some familiarity with their manipulation.
Then, equations (12:10) and (12:11) become:
1
1
= Et
ct
ct+1
yt+1
+1
kt+1
and:
lt1+ ct = (1
= kt (ezt lt )1
or, being explicit about yt = kt (At lt )1
1
1
= Et
ct
ct+1
)yt :
:
kt+11 (ezt+1 lt+1 )1
+1
(12.22)
:
(12.23)
and:
)kt (ezt lt )1
lt1+ ct = (1
In addition, we have the market clearing condition:
ct + kt+1
(1
)kt = kt (At lt )1
(12.24)
:
and the law of motion for productivity:
zt = zt
1
+ " t , "t
N (0; 1) ;
although this expression is already linear. That means that we will not have to transform
the variable zt below into zbt .
Carrying Out the Linear Approximation
We will proceed in two steps. First, we will simply rewrite the system (12:22)-(12:24) in
terms of the ^-variables:
1
1
= Et
k 1 l1
b
c
b
c
t
t+1
ce
ce
b
1+
l ce(1+ )lt +bct = (1
and:
b
cebct + kekt+1
(1
e(1
)(zt+1 +b
lt+1 +b
kt+1 )
)k l1
b
)kekt = k l1
e
b
kt +(1
e
b
kt +(1
+1
)(zt +b
lt )
)(zt +b
lt )
:
12.9. APPENDIX 2: LINEARIZATION
213
Simplifying, we get
1
1
b
b
= Et bct+1
k 1 l1 e(1 )(zt+1 +lt+1 +kt+1 ) + 1
b
c
t
e
e
b
b
b
e(1+ )lt +bct = e kt +(1 )(zt +lt )
and:
b
cebct + kekt+1
b
)kekt = k l1
(1
e
b
kt +(1
)(zt +b
lt )
where we have used the fact that, in steady state, l1+ c = (1
)k l1 :
Second, we will linearize the system around c^t = k^t = b
lt = c^t+1 = k^t+1 = b
lt+1 = 0: In
the appendix we show that these two steps result in the linear system
b
ct = Et
b
ct+1 + (1
and:
cb
ct + k b
kt+1
)
l
b
kt
( + )b
lt + b
ct =
(y + (1
where we have used, in the …rst equation:
l
+1
(12.25)
(12.26)
zt :
)k) b
kt = (1
1 1
k
zt+1 + b
lt+1 + b
kt+1
1 1
k
) y zt + b
lt
(12.27)
= 1;
and in the third equation:
y = k l1
:
Note that we have a system of three stochastic di¤erence equations on c^t , k^t , b
lt
(plus the law of motion of productivity). Although handling such system is rather
straightforward, it would require to use some results in matrix algebra that we prefer to
avoid in a textbook. The interested reader can check Uhlig....Instead, we use equation
(12:26) to write
1
1
b
b
lt =
kt +
zt
b
ct
+
+
+
and
Et zt+1 = zt :
Then we get a new version of equation (12:25):
b
ct +Et
b
ct+1 + (1
)
k
1 1
l
zt+1 +
and of (12:27)
cb
ct + k b
kt+1
(y + (1
)k) b
kt
(1
+
1
b
kt+1 +
) y zt +
+
+
1
b
ct+1 + b
kt+1
+
zt+1
1
b
kt +
+
zt
1
b
ct
+
=0
=0
214
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
We de…ne some coe¢ cients to conserve notation:
A = (1
B =
)
(1
C =
)
D = 1
E = k
l
+
1 1
k
(1
1
1 1
k
l
)
k
1+
+1
1
+
1 1
l
+
(1
)y
+
F =
(y + (1
)k)
1
G = (1
)y 1
+
y
H = c + (1
)
+
and then we get (using the linearity of the conditional expectation operator and the
fact that Et b
kt+1 = b
kt+1 because b
kt+1 is determined in period t) the much more compact
system:
Ab
kt+1 + Bzt + CEtb
ct+1 + Db
ct = 0
(12.28)
and
plus the auxiliary condition:
Eb
kt+1 + F b
kt + Gzt + Hb
ct = 0
b
lt =
+
1
b
kt +
+
(12.29)
1
b
ct :
+
zt
Determining the Policy Functions
The policy functions that determine the optimal behavior of the agents in the linearized
model have the linear form:15
b
kt+1 = P b
kt + Qzt
(12.30)
and
b
ct = R b
kt + Szt
(12.31)
Our task is to …nd the unknown coe¢ cients P; Q; R; and S: From those, we can get:
b
lt =
=
15
+
+
1
b
kt +
1
b
kt +
+
+
zt
zt
1
b
ct
+
1
Rb
kt + Szt
+
This is a result in optimal control theory that we can take as given in this book. Basically, the
policy functions are linear functions of the states of the model, that is, those variables that determine
the dynamic evolution of the model. In our economy, the state variables are b
kt and zt :
12.10. EXERCISES
215
and any other variable of interest.
The method of undetermined coe¢ cients used here is exactly the same than the one
in chapter 5. First, plug in the guessed policy functions of the form speci…ed in (12:30)
and (12:31) with their yet undetermined coe¢ cients P; Q; R; and S into the system
(12:28) and (12:29):Then see whether from the resulting equations we can determine the
values of P; Q; R; and S:
Since we also need to deal with b
ct+1 (and exactly as we did in chapter 5), we use the
fact that:
b
ct+1 = Rb
kt+1 + Szt+1
and now substituting out b
kt+1 from (12:30) yields
b
ct+1 = RP b
kt + RQzt + Szt+1
or
(12.32)
Etb
ct+1 = RP b
kt + (RQ + S) zt
Now, using (12:30), (12:31); and (12:32) in (12:28) and (12:29) yields:
and
A Pb
kt + Qzt + Bzt + C RP b
kt + (RQ + S) zt + D Rb
kt + Szt = 0
Rearranging:
and
kt + Gzt + H Rb
kt + Szt = 0
E Pb
kt + Qzt + F b
(12.33)
(12.34)
(AP + CRP + DR) b
kt + (Q + B + C (RQ + S) + DS) zt = 0
(EP + F + HR) b
kt + (EQ + G + HS) zt = 0:
But these equations have to hold no matter which value the current value of the state
b
kt and zt . But the above two equations can only hold for all possible values of b
kt and zt
if the unknown coe¢ cients R; P satisfy the two equations:
R (P 1) + AP = 0
R + BP C = 0
(12.35)
(12.36)
donde estan los cuadrados?
12.10
Exercises
12.10.1
Social Planner and the Real Business Cycle
Suppose that we have an economy with a representative household with preferences:
E0
1
X
t=0
t
u (ct ; lt )
(12.37)
216
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
and a Cobb-Douglas technology:
yt = kt (At lt )1
where:
At = (1 + g)t ezt A0
(12.38)
and
zt = zt
1
(12.39)
+ "t
where 0 < < 1, "t is a productivity shock that comes from a standard normal distribution N (0; 1), and A0 = 1 and z 1 = 0.
1. Write down the social planners problem of this economy.
2. Derive the optimality conditions.
3. Show that these optimality conditions are equivalent to the equilibrium conditions
that we derived in section xxx.
4. Discuss how the …rst and second welfare theorem explain your answer to the previous question.
5. The social planner in this economy generates aggregate ‡uctuations. Why? Does
it violate your intuition about what the optimal policy of a government should be?
12.10.2
A Real Business Cycle Model with Stochastic Volatility
(Based on Fernández-Villaverde and Rubio-Ramírez, 2012). Assume now that technology
evolves as
zt =
log t =
zt 1 +
log t
t "t
1
+
t
where t is a volatility shock that comes from a standard normal distribution N (0; 1).
Solve, using “guess-and-verify”, for the equilibrium of the real business cycle model
in section 2 of this chapter assuming GHH preferences and = 1.
12.10.3
Changing Capacity Utilization
We assumed in the main body of the text that capital is fully utilized in every period.
However, in real life, most capital is not utilized 24 hours, 7 days a week without interruption. Even forgetting about maintenance time, few factories operate a continuous
three-shift schedule or trucks are not driven non-stop by a team of drivers. This means
that there is a margin of utilization of capital. Macroeconomists have postulated that a
higher rate of utilization of capital means faster depreciation....
1. De…ne a competitive equilibrium for this economy.
12.10. EXERCISES
12.10.4
217
Investment-Speci…c Technological Change
The recent work of Greenwood, Hercowitz, and Krusell (1997 and 2000) has focused the
attention of economists on the role of investment-speci…c technological change as a main
driving force behind economic growth and business cycle ‡uctuations. Fisher (1999)
documents two key empirical observations that support these conclusions. First, the
relative price of business equipment in terms of consumption goods has fallen in nearly
every year since the 1950s. Second, the fall in the relative price of capital is faster during
expansions than during recessions. Models of investment-speci…c technological change
have also being successfully used to account for the evolution of the skill premium in the
U.S. since the Second World War (Krusell et al., 2000) or the cyclical behavior of hours
and productivity (Fisher, 2003), among several other applications.
We can model investment-speci…c technological change as an economy that produces
a …nal good with our standard Cobb-Douglas production function:
yt = kt (At lt )1
The …nal good can be used either for consumption, ct , or investment, xt . Investment is
transformed into new capital with the law of motion:
kt+1 = (1
) kt + v t x t
where vt is an index of productivity of the investment. We can think of vt as investmentspeci…c technological change. The rest of the model is the same than in our basic real
business cycle model.
1. Show that the relative price of capital is 1=vt : How can we use this fact to account
for the drop in the relative price of capital? Conversely, what does the fall in the
relative price of capital tell us about vt ?
2. Assume that
vt = v t
1
+
t;
t
N (0; 1)
where < 1 and t are investment-speci…c technological shocks. De…ne a competitive equilibrium for this economy.
3. Assuming GHH preferences and full depreciation, …nd the equilibrium dynamics of
the model.
4. Compare the answer you got from 3. with the equilibrium dynamics of the basic
real business cycle model.
[Answer to 3.:
lt =
1
(1
kt e
)zt
yt = kt (ezt lt )1
kt+1 =
v t yt
ct = (1
) yt
1
+
218
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
An investment-speci…c technological shock does not a¤ect output at impact but in the
next period]
12.10.5
Rule-of-Thumb Households
We assumed in the main body of the text that all households optimize intertemporally
and accumulate capital. However, it may be the case that only a fraction of them do
so, while the rest, 1
, just consume whatever they earn in this period. We will call
these consumers rule-of-thumb consumers. Their preferences are still given by:
E0
1
X
t
(ltr )1+
1+
log crt
t=0
!
but now their budget constraint of the household is:
crt = wt ltr
1. Write down the problem of the optimizing households and the problem of the
rule-of-thumb households.
2. Find the equilibrium conditions for this economy.
3. Solve for the equilibrium dynamics assuming that
= 1:
[Answer: Problem of the optimizing households is
max E0
1
X
t=0
s.t. cot
kt+1
t
log cot
(lto )1+
1+
!
+ it = wt lto + rt kt
= (1
) k t + it
The problem of the rule-of-thumb households is:
max E0
1
X
t
log
crt
t=0
s.t. crt = wt ltr
(ltr )1+
1+
!
12.10. EXERCISES
219
The equilibrium conditions are given by:
1
cot
1
1+ (1 + rt+1
o
(lt+1
)
= Et
(lto )1+
cot+1
1+
lt =
)
1+
o
(lt ) = wt
crt = wt ltr
(ltr ) = wt
lto + (1
) ltr
yt
kt
rt =
yt
lt
kt+1 = (1
)kt + it
o
ct = ct + (1
) crt
c t + it = y t
wt = (1
)
yt = ( kt ) (ezt lt )1
zt = zt 1 + z "z;t
To solve for the equilibrium dynamics when
1
(lto )1+
1+
cot
= 1, we start from the Euler equation:
1
1+ rt+1
o
(lt+1
)
= Et
cot+1
1+
Note that:
lto = ltr = lt
(lto ) = wt = (1
and then:
1+
(lto )1+ =
)
yt
lt
1
yt
1+
Also, we guess that:
cot = yt
where
is a parameter to be determined. Then:
1
1
1
1+
yt
= Et
1
=
yt
1
1
1+
1
1
kt+1
yt+1
yt+1
)
kt+1
220
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
or
yt
kt+1 =
Now, income of each optimizing household is:
0
1
1
+ (1
)y A =
y
wt lto + rt kt = @
|{z}t
| {z }t
capital income
+ (1
) yt
labor income
and hence their consumption is:
cot =
+ (1
) yt
=
+ (1
)
=
yt
yt
yt
The interesting …nding is that the dynamics of capital, consumption, and labor are
identical to those of the case where = 1 except for a constant.
12.10.6
Capitalists and Workers
This problem is an extension of the previous one. Now, the optimizing households do
not work, they just accumulate capital. Therefore, we will call them capitalist and the
rule-of-thumb consumers, which only work, workers.
1. Write down the problem of the capitalists and of the workers.
2. Find the equilibrium conditions for this economy.
3. Solve for the equilibrium dynamics assuming that
= 1:
[Answer: Problem of the capitalist is
max E0
1
X
t
log cct
t=0
cct
s.t. + it = rt kt
kt+1 = (1
) k t + it
while the problem of the workers is:
max E0
1
X
t
log
cw
t
t=0
w
s.t. cw
t = wt lt
(ltw )1+
1+
!
12.10. EXERCISES
221
The equilibrium conditions are:
1
1
= Et c (1 + rt+1
c
ct
ct+1
c
c t = r t k t it
cw
t = wt lt
lt = wt
yt
rt =
kt
wt = (1
)
)
yt
(1
) lt
kt+1 = (1
)kt + it
c
ct = ct + (1
) cw
t
) lt )1
yt = ( kt ) (ezt (1
zt = zt 1 +
z "z;t
To solve for dynamics, we look at the Euler equation when
1
1
= Et c
c
ct
ct+1
=1
yt+1
kt+1
and we guess that
cct = yt
where
is a parameter to be determined. Then:
1
(1
) yt
= Et
(1
1
=
yt
1
) yt+1
1
kt+1
yt+1
)
kt+1
or
kt+1 =
Now:
cct = rt kt
kt+1 =
yt
yt
yt = (1
) yt
and again we get that the dynamics of the economy are the same than the one in the
main text except for a constant.
12.10.7
Home Production
Randy....
12.10.8
Lotteries
Show log-crra and lotteries are the same....
222
CHAPTER 12. THE REAL BUSINESS CYCLE MODEL
Chapter 13
Extending the Model I: Fiscal Policy
We saw in the previous chapter that the basic real business cycle model accounts for
a surprisingly large share of the ‡uctuations of the U.S. economy. However, the model
failed to explain the data along many dimensions, in particular, in the labor market, and
relied on productivity shocks that many economists …nd di¢ cult to accept. Moreover,
the basic model was too simple and it could not be used for policy analysis. The basic
real business cycle model does not have a government or anything that resembles …scal
or monetary policy and, consequently, is silent about all those issues.
In this chapter and the next, we will attempt at partially …xing these two concerns.
We will extend the model to introduce new features that will improve the …t of the model
to the data and will extend the topics the model can deal with. We will focus on two
basic extensions. In this chapter, we will introduce a government that taxes and spends.
In the next chapter, we will introduce money. This will allow us to laid the basis of a
formal analysis of …scal and monetary policy.
With the help of the model, we will revisit some of the most classical questions of
macroeconomics: what are the e¤ects of an increase in public consumption? What are
the consequences of a reduction in taxes? What are the e¤ects of government de…cits?
Does it matter how we …nance changes in public consumption?
13.1
Introducing a Government
We can think about a government as a third agent in the economy, besides the households
and the …rms. The government taxes the households and the …rms and uses the resources
obtained in this way to …nance public expenditure.
This elementary framework can become as complicated as we want, since we can
have many di¤erent of taxes and many di¤erent types of public expenditures (remember
our discussion chapter 2 about the di¤erences between public consumption, public investment, and transfers). Moreover, there are many theoretical possibilities about how
to model the evolution of taxes and public expenditures over time. But before we can
undertake a thorough analysis of the government, we need to understand the mechanics
of an environment with a very simple …scal structure.
223
224
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
Remark 57 (Government in Macroeconomics) Economists are interested in the
government (or in more general terms, the public sector of an economy) from at least
two perspectives.
First, we can think about the government as an entity without independent agency,
that is, as an agent that just follows some rules of behavior speci…ed by the researcher from
outside the model. In this chapter, we will concentrate on this view of the government.
We will specify some stochastic rules about how the government taxes and spends and we
will trace the consequences of the shocks that appear in those rules. This approach is the
simplest possible modeling of a government and a natural starting point of the analysis.
Unfortunately, this view of the government leaves many questions unanswered. For
instance, it does not explain why the government follows those particular rules and it
does not have any theory to explain why taxes (or other policies) are di¤erent across
countries. For example, taxes in the United States as a percentage of output are lower
than in Sweden. How can we understand those di¤erences? Are they the product of
historical accidents, of ideology, or of the distribution of political power among di¤erent
political groups? Moreover, some governments seem to follow economic policies that we
argued in the part of the book where we talked about economic growth are likely to boost
income per capita. For example, they respect property right, enforce contracts, encourage
education and research, or protect investments among many others. Other governments,
in comparison, follow disastrous policies that most economists agree increase the poverty
and misery of their citizens. These governments expropriate private property without
due process, con…scate income in arbitrary ways, or exhort unjusti…ed violence on their
citizens. Why are the policies of the government of Switzerland so di¤erent from the
policies of the government of Zimbabwe? Are they a random event, a twist of historical
fate, or the product of a set of circumstances that we can analyze as we study other
problems in economics?
The study of how government make decisions (including the interaction between politicians and bureaucrats) and how the di¤erent social groups in a country in‡uence these
decisions is the …eld of political economy.1 The intersection between political economy
1
You sould beware of other uses of the expression “Political Economy”:
1. Historically, it was the original name of economics (the …rst economists were particularly interested in the economy of nations or “polities”.) That why still today one of the top journals of
the profession is called the Journal of Political Economy.
2. Some economists outside the mainstream (both on the left and at the right!), prefer to call
themselves Political Economists to emphasize their di¤erences with respect to what they see as
a misleading predominant work in economics.
3. Many researchers in Political Science call political economy to the study of the interrelations
between the economy, the institutions, the political environment, ideologies, and law without
necessarily using the standard tools of economics (mathematical models, maximizing behavior,
and equilibrium concepts).
That is why some economists have proposed to call the …eld Political Economics, but the name has
not became widely accepted.
13.1. INTRODUCING A GOVERNMENT
225
and macroeconomics is one of the most exciting areas of current research. Later in the
book, we will come back to some of the issues of how to understand why governments
behave in the way they do.
13.1.1
Taxes and Government Budget
We will assume that the government taxes labor income, wt lt ; at a rate l;t and capital
income, rt kt , at a rate k;t . As it is the case in most countries, capital income is taxed
after an allowance for depreciation. Therefore, the tax is expressed in terms of the
interest rate rt , and not in terms of the rental rate of capital, t (where, as in previous
chapters, absence of arbitrage means that t = rt
). For both rates, we will limit
ourselves to the case:
f l;t ; k;t g 2 [0; 1]2
that is the tax rates positive and bounded by 1. Even if we could think of tax rates outside
these bounds (for example a subsidy to work can be modelled as l;t < 0), allowing for
these cases would force us to worry about issues like whether the government can …nance
the negative tax rate or whether the household has enough income to pay a tax rate
bigger than 1. It is better to avoid these complications at this moment.
The two tax rates are linear, that is, they do not depend on the level of income
of the household. In the real world, taxes on income are progressive: higher income
households pay a higher fraction of their income on taxes than lower income households.
For example, in 2005 and according to the Congress Budget O¢ ce, the total e¤ective
tax rate of the bottom 20 percent of households in the U.S. in terms of income was 4.3
percent while the total e¤ective tax rate of those households in the 99 to 99.5 percentile
of income was 29.7 percent, around six times more in percentage of their income (the
average federal tax rate was 20.5).2 In our model, it is di¢ cult to have progressive
taxation because we only have one representative household.
In addition, the government sends resources to the household in terms of a lump
sum transfer T rt . This transfer can be positive (the government sends resources to
the household, think for instance of a social security or a welfare payment) or negative
(the government gets resources from the household, for example with a poll tax). The
main di¤erence between transfers and taxes is that transfers are not conditional on any
behavior of the household, who needs to pay them no matter what, while taxes are
only accrued when the household decides to work or to accumulate capital. In this
di¤erence, we see the origin of a theme that will appear constantly in this chapter: taxes
are distortionary because they change the incentives of agents toward labor supply or
capital accumulation at the margin while transfers are not. The distortions created by
taxes will appear in the form of wedges in the equilibrium conditions of the market
allocation that will force a deviation with respect to the optimal allocation of the social
planner.
2
These estimates exclude local and state taxes because they are extremely di¢ cult to measure of in
a country such as the U.S. with many variations in local taxes.
226
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
Given our previous description, the tax revenue net of transfers of the government in
this economy, T axt , is equal to:
T axt =
l;t wt lt
+
k;t rt kt
T rt
where the …rst term on the right, l;t wt lt , is the total labor tax (rate, l;t , times labor
income, wt lt ), the second term, k;t rt kt , is the total capital income (rate, k;t , times
capital income after depreciation, rt kt ), and the last term is transfers.
Net taxes T axt are used to …nance public consumption cpt :
cpt = T axt
Note that the government balances its budget in each period, that is, there is no public
debt that carries de…cit or surpluses from one period to the next. We impose a balanced
budget as a simpli…cation in our …rst take on the model.
At this moment we will not say much about how the government determines the tax
rates or the level of public consumption. We will just limit the sequence of …scal policy
3
fcpt ; l;t ; k;t g1
t=0 ; possibly random, to be such that the transfers:
T rt = cpt
l;t wt lt
k;t rt kt
balance the budget and the tax rates are between 0 and 1. Much more important is
that the household will take the sequence of …scal policy fcpt ; l;t ; k;t g1
t=0 as given and
that their actions will not change the behavior of the government. This assumption is
similar, for instance, to the assumption that households are price takers in the markets
for goods and inputs.
Remark 58 (Game Theory and Fiscal Policy) There is a subtle point regarding our
previous assumption that households take the actions regarding taxes and public consumption of the government as given. We allow households to respond to the tax rates. We will
derive in the next subsection the necessary conditions of the household and we will see in
them that the household changes its behavior as a consequence of the presence of taxes.
What we will not allow is the household to think about the consequences of her decisions
regarding, for example, labor supply, will have on total tax raised by the government and
how the government will respond to those. The justi…cation for this assumption is the
same one than in competitive behavior. The representative household stands in for a
large numbers of households. Each of them is too small to think that its behavior will
change any aggregate or have an impact on …scal policy.
Our assumption stops us from studying situations where there is a non-trivial interaction between the actions of the government and the household. This interaction is a
strategic problem that requires the tools of game theory to be analyzed formally. This will
take us further than the scope of this book. Su¢ ce it to say here that …scal policy is one
of the areas of macroeconomics where game theory has had a bigger impact.
3
By random, we mean a sequence that may depend on stochastic shocks. For example, the government
may follow a …scal policy sequence where the tax on labor is at the rate 0.4 if a fair coin yields tails
and 0.3 otherwise. More realistically, the government may decide to raise taxes when the productivity
shock is positive and lower them when the productivity shock is negative.
13.1. INTRODUCING A GOVERNMENT
227
Remark 59 (Commitment and Time Inconsistency) We have assumed that the
government selects a sequence of …scal policy fcpt ; l;t ; k;t g1
t=0 , possibly random, and
it follows it over time. This implies a notable power of commitment on the part of the
government. At each moment in time, the government will implement the …scal policy
that was already known at period 0 (or, in the case that the policy depends on a random
shock, the policy implied by that shock where the mapping between shocks and policy was
known at time 0).
However, in real life, governments change their policies all the time and announcements of policy are constantly reverted at implementation time. Sometimes, this is just
because the president or the prime minister of a country changes. Sometimes it is because the policies being announced at time 0; even if optimal at that time, are not the
best possible polices at time t: This phenomenon is known as time inconsistency.
An example clari…es it. Imagine that a government announces that it will impose the
death penalty to all households that cheat on their taxes, even for one cent. Moreover, the
government also implements a perfect auditing system that can detect even the smallest
irregularity in tax compliance. If households believe the government will carry its threat,
none will dare to evade their tax obligation and the government will be able to raise
revenue in a more equitable and e¢ cient way and avoid to actually have to execute
any household. Therefore, tough enforcement of the death penalty announced at time 0
may be optimal: we get perfect compliance and no executions. Let us now imagine that
the government …nds at time t that one household has not paid, let’s say, $1 owed in
taxes. What should the government do? If it executes the household, it is clearly cruel
and unusual punishment for a extremely small violation of the law (after all, we are
talking about $1). But if the government does not execute the household, it breaks the
commitment to the enforcement policy! Most people, if faced with these choice as Treasury
Secretary, will chose NOT to execute the cheater, therefore showing that the policy that
was optimal when announced at time 0, is not optimal at time t: The punishment of
death penalty for small tax cheating is, in the language of game theory, a non-credible
threat. We can take the example one step further: households will understand that the
government will not execute them at time t if they just cheat for a small quantity, not
matter what the announced policy is, which means that they will have an incentive to
cheat and some will do so, creating the temptation for time inconsistency.4
A detailed analysis of time inconsistency requires a considerable amount of work
that we will delay until chapter xxx. Nevertheless, before engaging in that study, it is
important to understand the e¤ects of …scal policy when governments do have access to
a commitment device.
4
There is yet one more step in the argument since the government may decide to carry a threat like
death penalty for a minor crime to build a reputation as a “tough” enforcer and avoid having to deal
with these issues in the future. How to model reputation and how reputation may be a substitute for
commitment has been one of the central topics in macroeconomics over the last decades.
228
13.1.2
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
Household Preferences
The representative household has preferences that are representable by a expected utility
function:
1
X
t
E0
u (ct ; lt ; cpt )
(13.1)
t=0
that has as arguments private consumption, ct , and labor, lt , as in the basic real business
cycle model of previous chapter, and public consumption, cpt . At this moment we are
not restricting the form in which public consumption enters into the utility function.
The household may like public consumption immensely, dislike it, or just be indi¤erent
towards it.
An empirically plausible utility function is of the log-CRRA form:
lt1+
+ log cpt
1+
where we have added a third term, log cpt , equal to the log of public consumption times
a parameter, , that controls how much the household likes public consumption.
In previous chapter, we showed that GHH preferences are particularly easy to handle
analytically. Consequently, we extend them to the case with public consumption:
u (ct ; lt ; cpt ) = log ct
u (ct ; lt ; cpt ) = log ct
lt1+
1+
+
(13.2)
log cpt
We will use utility (13.2) extensively below.
Remark 60 (Separability of Private and Public Consumption) In our previous
examples, we have assumed that private and public consumption enter in a separable way
in the utility function. Separability simpli…es algebra for our exposition but there is not
theoretical restriction against more general forms. For instance, we could have a utility
function of the form:
u (ct ; lt ; cpt ) =
(&ct + (1
&) cpt ) (1
lt )1
1
1
1
where inside the CRRA structure, we have a Cobb-Douglas form between leisure, 1
and a CES aggregate:
1
(&ct + (1 &) cpt )
lt ,
of private and public consumption (where 1=(1
) is the elasticity of substitution and
& is a share parameter).
Larry Christiano and Martin Eichenbaum (1992) argued, in an in‡uential paper, that
a possible choice for the utility function that does not display separability, but it is still
rather simple, is:
u (ct ; lt ; cpt ) = log (ct + cpt )
log(1 lt )
This speci…cation has the advantage that private and public consumption enter a sum in
the …rst term, making 1 unit of private consumption equivalent, in utility terms, to 1=
units of public consumption.
13.1. INTRODUCING A GOVERNMENT
13.1.3
229
Budget Constraint and First Order Conditions of the
Household
The budget constraint for the household is given by:
ct + kt+1 = (1
l;t ) wt lt
+ (1
k;t ) rt kt
+ kt + T rt
This budget constraint tells us that households can consume, ct , or accumulate capital,
kt+1 in a quantity equal to the sum of:
1. The after tax labor income, (1
l;t ) wt lt .
2. The after tax capital income, (1
3. The undepreciated capital, (1
k;t ) rt kt
+ kt .
) kt .
4. Transfers from the government, T rt (that can be positive or negative).
This budget constraint summarizes how the presence of a government modi…es the
possible choices made by the household. Therefore, we can summarize the problem of
the household as:
max E0
fct ;lt g1
t=0
s.t. ct + kt+1 = (1
l;t ) wt lt
1
X
t
u (ct ; lt ; cpt )
t=0
+ (1
k;t ) rt kt
+ kt + T rt for all t
for some given k0 and a given sequence of f l;t ; k;t ; T rt g1
t=0 :
The Lagrangian of the household’s problem is given by:
(1
1
X
X
t
u (ct ; lt ; cpt ) +
L = E0
t ((1
l;t ) wt lt + (1
k;t ) rt kt + kt + T rt
0
ct
kt+1 )
t=0
t=0
with …rst order conditions:
ct :
ct+1 :
lt :
kt :
u1 (ct ; lt ; cpt ) = t
Et u1 (ct+1 ; lt+1 ; cpt+1 ) = Et t+1
u2 (ct ; lt ; cpt ) = t (1
l;t ) wt
t + Et
t+1 (1 + (1
k;t ) rt ) = 0
If we combine the …rst, second, and fourth equation, we get:
u1 (ct ; lt ; cpt ) = Et u1 (ct+1 ; lt+1 ; cpt+1 ) (1 + (1
k;t+1 ) rt+1 )
This equation is one more version of the Euler equation. The variation is that the return
of investment is corrected by taxes:
1 + (1
k;t+1 ) rt+1
)
230
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
and that marginal utilities are evaluated also taken account of public consumption (although if the utility function is separable in public consumption, this e¤ect disappears).
If we divide the third equation by the …rst one, we get:
u2 (ct ; lt ; cpt )
= (1
u1 (ct ; lt ; cpt )
l;t ) wt
a static condition that tells us that the (negative of the) ratio of marginal utilities of
labor and consumption must be equal to the take home wage. The di¤erence with respect
to the equivalent expression in the basic real business cycle model is the presence of the
term on the tax on labor income (1
l;t ), and the dependence of marginal utilities on
public consumption. This term (1
)
l;t is often known as a wedge since it introduces a
di¤erence between the marginal rate of substitution
u2 (ct ; lt ; cpt )
u1 (ct ; lt ; cpt )
and the wage (that, as we will see momentarily, is the marginal productivity of labor).
13.1.4
The Problem of the Firm
Taxes are paid by the households on their labor and capital income. Therefore, the
problem of the representative …rm is the same than in the previous chapter:
max fyt
lt ;kt
t kt g
wt lt
subject to
yt = kt (At lt )1
kt ; lt 0
with associated …rst order conditions:
wt = (1
)At
=
kt
At lt
t
13.1.5
kt
At lt
(13.3)
1
(13.4)
:
Aggregate Resource Constraint
Now that we have a third agent in the economy, the government, we need to ensure that
the right aggregate resource constraint holds in the economy and that we do not use more
resources than the ones the economy has access to. To …nd this resource constraint, we
add the budget constraint of the household:
ct + kt+1 = (1
l;t ) wt lt
+ (1
k;t ) rt kt
+ kt + T rt
13.2. EQUILIBRIUM
231
to the budget constraint of the government
cpt = T axt =
l;t wt lt
+
k;t rt kt
T rt
and we get:
ct + kt+1 + cpt = wt lt +
t kt
+ (1
(13.5)
) kt
that says that income plus undepreciated capital must …nance private and public consumption and the new capital.
We can push (13.5) one more step, and by noting that given our assumption about
competitive markets for inputs and our Cobb-Douglas production function, we have:
wt lt +
t kt
= kt (At lt )1
which gives us the aggregate resource constraint:
ct + kt+1 + cpt = kt (At lt )1
+ (1
) kt
or, using the law of motion for capital:
kt+1 = (1
) k t + it
in terms of investment it , we get:
ct + it + cpt = kt (At lt )1
This aggregate resource constraint tells us that the uses of the …nal good in the economy,
private and public consumption plus investment, must be equal to the production of this
…nal good.
13.2
Equilibrium
We modify the de…nition of competitive equilibrium in our real business cycle to incorporate the presence of government.
De…nition 61 Given initial capital k0 and the sequence of …scal policy fcpt ; l;t ; k;t ; T rt g1
t=0 ,
a competitive equilibrium consists of allocations for the representative household, fct ; lt ; kt+1 g1
t=0 ;
1
1
allocations for the representative …rm, fkt ; lt gt=0 and prices frt ; t ; wt gt=0 such that:
1. Given frt ; wt g1
t=0 and fcpt ;
household problem:
l;t ;
1
k;t ; T rt gt=0
max1 E0
fct ;lt ;kt gt=0
s.t. ct + kt+1 = (1
1
X
t
; the household allocation solves the
u (ct ; lt ; cpt )
t=0
+ (1
k;t ) rt kt + kt + T rt for all t
given initial condition k0
l;t ) wt lt
0
232
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
2. Given f t ; wt g1
t=0 ; the …rm allocation solves the …rm problem:
max (yt
nt ;kt
wt lt
t kt )
subject to
yt = kt (At lt )1
At = (1 + g)t ezt A0
zt = zt 1 + "t , "t N (0; 1)
kt ; lt 0
3. Fiscal policy …scal policy fcpt ; l;t ; k;t ; T rt g1
t=0 is such that the government satis…es
its budget constraint: for all t = 0; : : : ; 1
cpt =
l;t wt lt
+
k;t rt kt
T rt
4. Market clears and the aggregate resource constraint holds: for all t = 0; : : : ; 1
ct + kt+1 + cpt = kt (At lt )1
+ (1
) kt
One important feature of this competitive equilibrium is that the presence of taxes
will stop the welfare theorems to hold. For example, putting together the static …rst
order condition of the household with the observation that the wage is equal to the
marginal productivity of labor, we get:
u2 (ct ; lt ; cpt )
= (1
u1 (ct ; lt ; cpt )
l;t ) wt = (1
l;t ) (1
13.3
Solving for the Equilibrium
13.3.1
An Expression for Equilibrium
)
kt (At lt )1
lt
The presence of a government makes even more di¢ cult to solve for the equilibrium of
our economy. We will resort then to a similar set of assumptions than in the previous
chapter. First, we will assume that the utility function of the household is given by
equation (13.2):
lt1+
u (ct ; lt ; cpt ) = log ct
+ log cpt
(13.6)
1+
Second, we will assume that we have full depreciation, that is, = 1: Third, we set
transfers equal to zero, that is, the government spends all the resources raised by taxation
on public consumption (or equivalently, it sets the tax rates to …nance the level of public
consumption it desires). Finally, we will assume that the tax rates of labor and capital
are equal:
t = l;t = k;t 2 [0; 1]
13.3. SOLVING FOR THE EQUILIBRIUM
233
Moreover, the government does not provide an allowance for depreciation. This is counterfactual (and contrary to our assumption before!) but it simpli…es the algebra dramatically (on the other hand, we do not need to impose any structure in the way taxes
evolve over time).
The reason for the last assumptions is simple. When we have equal taxation to
labor and (gross) capital income at rate t , we would be able to substitute our original
economy by a similar, …ctitious economy that is much easier to handle but that has the
same implications for allocations. This …ctitious economy has a modi…ed production
function:
1
yet = (1
t ) kt (At lt )
where productivity is lower by a term (1
e represents that we are
t ) and where the x
working with the …ctitious economy equivalent of variable x.
In this …ctitious economy, the input prices are given by:
kt (At lt )1
= (1
(13.7)
t ) wt
lt
kt (At lt )1
et = (1
= (1
(13.8)
t)
t) t
kt
that is, they are equal to the after-tax input prices of the original economy.
This basically means that if we look at the …rst order conditions of the household in
the original economy, the Euler equation:
1
1
= Et
(1
t+1 ) t+1
1+
1+
lt
lt+1
ct
ct+1
1+
1+
w
et = (1
t ) (1
)
lt = (1
t ) wt
-where the term (1
t+1 ) t+1 is the return on capital after depreciation and taxes- and
the …rst order conditions of the …ctitious economy:
1
1
= Et
1+
1+ e t
lt
lt+1
ct
ct+1
1+
1+
lt = w
et
we can immediately see that they are equivalent.
The great advantage of working with our …ctitious economy is that we know how to
…nd the policy functions, since it is absolutely identical to our basic real business cycle
model except that now we have an extra term (1
t ) reducing productivity. Therefore,
the optimal decision of capital accumulation is given by:
kt+1 =
(1
t ) yt
and, by the resource constraint of the economy:
ct = yt
= yt
= (1
kt+1 gt
(1
t ) yt
) (1
t ) yt
t yt
234
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
where the second line uses the fact that the government balances its budget with zero
transfers:
cpt =
t yt
=
t
(wt lt +
t kt )
To check this result, you can substitute in the Euler equation:
1
ct
1
= Et
lt1+
1+
(1
1+
lt+1
1+
ct+1
yt+1
kt+1
t+1 )
and corroborate that, in fact, the left and the right hand side are equal.
To …nd labor, we come back to the static condition:
lt = (1
1
lt1+ =
(1
1
lt =
(1
t ) wt
)
t ) kt
(ezt lt )1
t ) kt
(1
)
1
+
)zt
e
and therefore we have the complete solution for the equilibrium of the model:
1
lt =
(1
t ) kt
(1
1
+
)zt
e
yt = kt (ezt lt )1
kt+1 =
(1
t ) yt
ct = (1
) (1
t ) yt
cpt = t yt
A couple of expressions that will be useful below are:
yt = kt (ezt lt )1
= kt
1
ezt
= (1
t)
1
+
(1
kt
+ 1+
t ) kt
e
1+
+
e(1
zt
)zt
!1
1
+
1
+
1
!1
and:
cpt = T axt =
t yt
=
t
(1
t)
1
+
kt
+ 1+
e
1+
+
zt
1
1
+
!1
13.3. SOLVING FOR THE EQUILIBRIUM
13.3.2
235
Interpreting the Dynamics of the Model
Transitional Dynamics
First, note that in a steady state, the optimality conditions of the household at the
steady state are:
1
1
(1
) k 1 l1
=
l1+
l1+
c
c
1+
1+
and:
l = (1
) (1
)k l
where variables without time subindex denote steady state values. Simplifying:
l
=
k
(1
1
)
and
l=
(1
1
+
1
)
k
Then:
(1
1
+
1
)
k = ((1
k
)
= (1
)
+
1+ +
)
=
+
(1
(
)
1
(1
1
1
)
1
1
+
)
)
and we get that
y = k l1
k
=
l
= ((1
= (1
1
k
)
)
)
+ +1
1
((1
(
)
)
1+
)
+
+
(1
)
1
1
Also,
l=
(1
)
1
1
+
k
1
1
236
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
and
c =
= (1
(1
)
)
+ +1
+ +1
(
(
)
1+
)
+
+
1
1
+
1
1
From these expressions, we can see how changes in the steady state level of taxes
change output. In particular, higher taxes mean lower output, labor supply, and consumption.
Note that to compute the transition, we can still use the equilibrium dynamics derived
in the main text with constant productivity and constant taxes
lt =
1
1
+
(1
) kt
yt = kt lt1
kt+1 =
(1
) yt
ct = (1
) (1
) yt
cpt = yt
Then, for any k0 , we can …nd
l0 =
1
1
+
(1
) k0
y0 = k0 l01
k1 =
(1
) y0
c0 = (1
) (1
) y0
cp0 = y0
and iterate until convergence.
13.3.3
A Permanent Increase in the Tax Rate
Las mismas funciones de politica.....
13.3.4
A Temporay Increase in the Tax Rate
The interpretation of the dynamics of the model with our assumptions is surprisingly
simple: a change in the tax rate t plays exactly the same role as a productivity shock.
When taxes are high, that is, t is high, labor supply is lower because it reduces the
after-tax wage. The household works less hours in equilibrium. This lowers output, yt :
Private consumption and investment are reduced even further, not only because yt is
13.3. SOLVING FOR THE EQUILIBRIUM
237
lower, but because the term (1
t ) is smaller. The net e¤ect on tax revenue and public
consumption is ambiguous. On one hand, a higher rate raises more revenue per unit of
output. However, a higher tax also lowers output because it decreases labor supply and
capital accumulation. The interaction between these two e¤ects is summarized in the
La¤er curve that we discuss in the next remark.
Remark 62 (La¤er Curve) We derive before an expression for the total amount of
tax revenue:
!1
1
+
1+
1
+ 1+
1
+ k
e + zt
T axt = t (1
t)
t
to maximize tax revenue T axt with respect to t given some level of capital kt and productivity zt , we can take logs of the previous expression:
!1
1
+
1+
+ 1+
1
1
log (1
e + zt
log T axt = log t +
t ) + log kt
+
and …nd the …rst order condition:
1
=
t
where
t
1
1
+
1
t
is the tax rate that maximizes tax revenue. Then:
t
=
+
+1
For t < t , tax revenues rise as we increase t . The interesting point is that, for
value of t > t , tax revenues fall as we increase the tax rate even further because labor
supply falls so much that it overcomes the rise in the tax rate. Note that in our particular
parametric case, the tax rate t is independent of the level of capital or of the productivity
shock, but this might not be the case in more general cases, for example, if < 1.
We illustrate our discussion by plotting in …gure xxx the total tax revenue for values
of t ranging from 0 to 1 in the case where = 0:5, = 1=3 and
!1
1
+
1+
+ 1+
1
kt
e + zt
=1
(this last assumption is just a normalization since it is a multiplicative term that does
not depend on ).
This graph is known as the La¤er curve, in honor of Arthur La¤er who popularized
the concept in the late 1970s and early 1980s.5 The maximum of the …gure is at t =
5
Although the basic insight goes back in time to the past. With his usual insight, Keynes wrote
in his 1933 pamphlet The Means to Prosperity: “Nor should the argument seem strange that taxation
may be so high as to defeat its object, and that, given su¢ cient time to gather the fruits, a reduction
of taxation will run a better chance, than an increase, of balancing the Budget.”
238
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
Figure 13.1:
0:553: This suggests that most western economies, where average tax rates on income are
below 55% are probably at the left hand side of the La¤er curve and that, consequently,
reductions in tax rates will lead to reductions in tax revenue. Mathias Trabandt and
Harald Uhlig (2006) examine the shape of the La¤er curve in the U.S. and the European
Union in a richer, calibrated model. They conclude that the US and the EU-15 are located
on the left side of their labor and capital tax La¤er curves, but that the EU-15 economy
is actually quite close to the peak of the curve.
A rise in the tax rate is propagated over time through lower capital as dictated by
dynamics of investment:
kt+1 =
(1
t ) yt
and the associated drop in labor supply (because with lower capital, the marginal productivity of labor is also lower). Think about the case where the tax rate goes up at
time t from its mean rate to + "; where " > 0, and falls back to its original position
in all subsequent periods (we call this a one period deviation). Then, in period t, output
is lower because labor supply falls and investment is lower both because t is higher and
because yt is lower. In period t + 1, even if the tax rate has come back to its original
value, a lower capital means lower labor supply and lower output. This e¤ect disappears
only asymptotically and, more relevant for the discussion, is quantitatively signi…cant
for many periods.
13.3. SOLVING FOR THE EQUILIBRIUM
239
Figure 13.2:
To gain further insights, we plot the IRFs of a one period increase in the tax rate from
0.2 to 0.25, where the parameter values of the economy are the same than in the model
of chapter xxx. Capital share, = 0:33, the inverse of the Frisch elasticity, = 0:5; and
the parameter of labor in the utility function = 0:67 (for this exercise we do not need
to specify parameter values for the process for technology). From the graph, we can
see how at impact of the tax increase, labor supply goes from 0:171 to 0:158 and stays
below 99% the original level for three additional quarters. Output falls from 0:088 to a
minimum of 0:082 in two quarters and requires another 4 additional quarters to return
to 99% of its original value. Similarly, investment and consumption drop at impact and
recover in a few quarters. Public consumption rises at impact, but it falls later, as we
revert to the original tax rate but with a lower output base to tax.
Remark 63 (Static versus Dynamic Scoring) Since a change in tax rates implies
responses of households in terms of labor supply and capital accumulation, we can evaluate the e¤ects of the modi…cation of policy from two perspectives. First, we can quantify
the e¤ects assuming that the behavior stays constant. This is known as static scoring.
In our previous example, raising the tax rate from 0.2 to 0.25 in one period will imply,
with static scoring, an increase in revenue of 25% (0.25 is 1.25 of 0.2). Static scoring
provides a useful benchmark because it states the costs of a reform from a pure accounting
position and hence, it is a natural point of departure for the discussion. However, we
240
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
have argued that the behavioral responses of agents are qualitatively and quantitatively
important. Dynamic scoring captures those e¤ects and, hence, it is a better assessment
of the e¤ects of a tax change (and hence, the adjective “dynamic”). However, dynamic
scoring requires the use of an economic model, and like all models, the answers will be
sensitive to details like the concrete speci…cation of the model or the parameter values that
we select. For example, the responses of labor will be pin down by the Frisch elasticity
and we argued before that there is little consensus about the best value for this elasticity.
In our example, a 25% raise in the tax rate only generates a 18.75% increase in revenue
in period t and it causes a fall in revenue in the next period of nearly 7% with respect to
the original situation of a 0.2 tax rate. The fall in revenue asymptotically disappears but
it is still noticeable for several period. If we discount the total ‡ow of revenues generated
by a 25% raise in the tax rate for 1 period using the discount factor of the households,
we get that the raise in present value of tax revenue is only 2.18%.6
An important point to remember is that, even if output and consumption fall after
an increase in tax rates, the household welfare may increase. First, because part of the
reduction in consumption is compensated by more leisure (although not all of it, after
all with taxes we distort the decisions of households and distortions tend to increase at
an approximate quadratic rate with respect to the tax level). Second, because public
consumption appears in the utility function of the household. If the level of public
consumption is too low, its marginal utility is very high and shifting resources from
private consumption to public consumption increases welfare (of course, the problem
being that each individual household for itself does not have an incentive to …nance
public consumption and therefore we need a government to solve this collective action
problem). Whether the increase in public consumption will help welfare or not will
depend on the parameter values of the term in the utility function involving public
consumption and the levels of private and public consumption at the moment of the
change in the tax rate.
An Increase in Public Consumption
Before, we analyzed the dynamics of the model in terms of a change in the tax rate t that
is spent on public consumption cpt . Instead, we could have discussed the model in terms
of a change of public consumption cpt that causes a change in the tax rate t . All the
analysis would have reach the same conclusions except for a subtle interpretation issue
that we discuss in the next remark. Our model then predicts that temporary increments
in public consumption are contractionary: they need to be …nanced by distortionary
taxes and a raise in taxes decreases the incentives to work and to save, and therefore
output.
How general is the …nding that temporary increases in public consumption lower
output? Not much. There are at least three possible channels through which a temporary
6
We actualize future revenue by the discount factor to avoid the problem that the market interest
rate is a¤ected itself by the change in taxes. However, given the size of the policy change that we
evaluate, using the market interest rate will get us nearly the same number.
13.3. SOLVING FOR THE EQUILIBRIUM
241
raise in government spending may increase output:
1. Wealth e¤ects. If we had a utility function with wealth e¤ects, like a log-log utility,
a temporary increase in taxes generates two e¤ect: a substitution e¤ect (work
is now less attractive, hence we work less) and a wealth e¤ect (households are
poorer because they need to pay taxes, hence they work more). If the wealth e¤ect
dominates, output expands as a response to an increase in public consumption.
This correlation between increases in public consumption and output have been
documented by many empirical studies (see, for instance, Mountford and Uhlig,
2008). Interestingly, even if output rises, the welfare of the household may decrease
(we are producing more output but part of it goes to public consumption that may
appear with a small parameter in the utility function and we are working longer
hours). The classic analysis of these richer situations are in Rao Aiyagari, Larry
Christiano, and Martin Eichenbaum (1992) and in Marianne Baxter and Robert
King (1993).
2. Productive Public Consumption. Some forms of public consumption may increase
the productivity of the economy. For example, imagine that we increase the amount
of resources available to courts to enforce contracts (more policeman, more administrative support, and so on). If better contract enforcement allows households to
engage in more productive activities (they are less afraid of the other party defaulting on them), total output may increase. In a more general model, we can
also think about public investment as accumulating public capital that increases
the productivity of the economy (for example, more public highways make transportation of goods easier and cheaper).
3. Frictions. If we have some type of frictions like nominal rigidities, an increase in
government consumption may increase output by raising the demand for goods and
services in the economy and employing at a higher rate the resources available. We
will come back to this point when we introduce nominal rigidities in chapter xxx.
The previous arguments show a discomforting reality of …scal policy: instead of clear
theoretical predictions, dynamic equilibrium models may imply very di¤erent consequences of …scal policy depending on the particular assumptions that we make. The
important goal is, therefore, to understand the di¤erent mechanisms at play and to evaluate, given the data, which of them are more relevant in any given moment of time to
forecast the e¤ects of …scal policy in the real world.
Remark 64 (Tax changes versus Public Consumption changes) In our explanation in the main text, the government raises some taxes and uses for public consumption.
Therefore, regardless of what the household do, the government always satis…es its budget constraint. However, in the second case, when the government …xes cpt and raises
taxes to …nance it, there is the possibility that households will do something di¤erent
than the government has foreseen and therefore, public consumption cannot be …nanced.
For example, the government had foreseen that, in equilibrium, households were going to
242
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
supply lte units of labor but it turns out that they only supply lto < lte , which means that
tax revenue is also lower. In the case that the government consumes after the revenue
has been raised, this is not an issue: the government will just lower cpt . In the case
that the government has somehow committed to some level of cpt , we face the question
of what would happen in this situation. Would the government default on its previous
commitments on cpt ? Would it change the tax level ex-post? The di¢ culty in the study of
the situation lies on the fact that it involves out-of-equilibrium behavior: the equilibrium
labor supply is lte and not lto and therefore there is no reason to expect lto . One possibility
is to ignore the possibility of this out-of-equilibrium behavior. But this position ignores
that it is easy to build situations where di¤erent assumptions regarding out-of-equilibrium
behavior have consequences on equilibrium dynamics (for instance, Bassetto, 2005, builds
examples of how di¤erent speci…cations of what the government does outside equilibrium
change the equilibrium allocations). The second possibility is to be explicit about how
the government would act in each possible situation, regardless of this situation is ever
reached. This is the position we take in the main text, since the government just sets
cpt = t yt for any level of yt .
13.4
Government De…cits
Our analysis so far of …scal policy has assumed that the government balances its budget
every period. We observe, however, that this rarely happens in the real world. Modern
states run large de…cits and, sometimes, like the U.S. in the late 1990s surpluses. These
de…cits are …nance either by the printing of money (an topic that we will revisit in the
next chapter) or by borrowing from the households in the economy.7 Consequently, we
need to extend our analysis to allow for de…cits and surpluses. This will help us answer
questions like: What are the e¤ects of government de…cits? Are government bonds net
wealth?
Our …rst answer will be surprising: government de…cits do not matter. This result,
known as Ricardian equivalence, holds only under an extremely restrictive set of assumptions. However, by studying when the Ricardian equivalence holds, we will isolate several
mechanisms upon which we can build a more thorough study of government de…cits. But
…rst, we introduce government bonds (which we will also call public debt) in our model.
13.4.1
Government Debt
We start by assuming that we have a government that has to …nance a given exogenous
stream of government expenditures (in real terms) denoted by fcpt g1
t=1 through distorsionary taxes and lump-sum taxes (negative transfers). Note that in this part of the
analysis we are …xing the sequence fcpt g1
t=1 that is not any longer a choice of the government. Think for example, of the need to …nance a war or the rescue operation after
7
In the real world, there is a third possibility: to borrow from abroad. Chapter xxx on open
macroeconomics will deal with the presence of a foreign sector.
13.4. GOVERNMENT DEFICITS
243
a natural disaster. Also, the government has initial outstanding real debt (that is, the
government owes real consumption goods to its citizens) of b0 that is held by the households and can issue future new debt bt . The possibility of issuing debt opens the door
to intertemporal trade by the government: do we want to …nance public consumption
today with taxes today or with taxes tomorrow?
Remark 65 (Sovereign Wealth Funds) Although we will center most of our explanations around the case where debt is positive, bt > 0, like in the U.S. and most other
OECD countries, we allow for the case that bt < 0, that is, debt is negative and the government is owed money by the households. Some governments, however, have positive
asset positions. Usually, these governments have experienced a large windfall in terms
of oil royalties (like several Arab countries or Norway) or have a strong strategy interest
in keeping large reserves (like Singapore). Those assets are often organized in Sovereign
Wealth Funds, which are nothing more than a government-owned investment funds. The
Abu Dhabi Investment Authority, the Government Pension Fund of Norway, or the Government of Singapore Investment Corporation are some of the biggest of those funds. In
the U.S., the state of Alaska created the Alaska Permanent Fund in 1976 to manage the
large royalty income from new oil …elds. Some of these funds are large. For example, the
Abu Dhabi Investment Authority holds assets for a value of around one million dollars
for each citizens of its country (this is an educated guess by experts since the fund has
never been open about its balance sheet).
At any period t, the government budget constraint reads as:
bt+1
= cpt
Rt+1
T rt
l;t wt lt
k;t rt kt
+ bt
that says that the new debt issued by the government, bt+1 , multiplied by its price,
1=Rt+1 , is equal to the di¤erence between expenses and revenues in the period, cpt
T rt
l;t wt lt
k;t rt kt , plus the debt carried from the previous period, bt : We divide the
total issue of debt bt+1 by the gross risk-free interest rate, Rt+1 , because the government
sells debt at a discount. For a bond that will pay one unit of good at time t + 1,8 the
household pays 1=Rt+1 , which nearly always will satisfy:
1
<1
Rt+1
(and hence the name “selling at a discount”). The di¤erence between 1=Rt+1 and 1 is
the yield that the household gets from holding the debt (to save on notation, we assume
that the interest generated by the public debt are tax-free).
8
This debt is known as uncontingent debt because the government pays one unit of the good in all
possible events at time t + 1. Later, we will introduce contingent debt, where the government only pays
in certain events of the world (for example, if productivity is high). Most debt observed in the real
world is, indeed, uncontingent debt.
244
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
We can determine this yield by looking at the optimality conditions of the household.
Formally, the household’s problem becomes:
max
fct ;lt ;kt ;bt g1
t=0
E0
1
X
t
u (ct ; lt ; cpt )
t=0
bt+1
s.t. ct + kt+1 +
= (1
l;t ) wt lt + (1
k;t ) rt kt + kt + T rt for all t
Rt+1
given initial conditions k0 ; b0
0
If we take …rst order conditions with respect to the choices of the household, ct ; lt ; kt ; bt ,
we …nd:
t
ct :
lt :
kt+1 :
t
u1 (ct ; lt ; cpt ) = t
u2 (ct ; lt ; cpt ) = t (1
t + Et t+1 (1 + (1
t
bt+1 :
Rt+1
+ Et
t+1
(13.9)
(13.10)
(13.11)
l;t ) wt
k;t ) rt+1 )
=0
(13.12)
=0
The three …rst order conditions are standard and we do not elaborate on them. The last
condition tells us that the household should be indi¤erent at the margin between holding
one unit of debt where the marginal valuations are given by the lagrangian multiplier
t . Then, the yield of the public debt must be:
Et
t+1
Rt+1 = 1
t
or, since Rt+1 is known at time t
1
Rt+1 =
Et
t+1
t
=
Et
u1 (ct+1 ; lt+1 ; cpt+1 )
u1 (ct ; lt ; cpt )
1
which tells us that the gross return on the public debt must be equal to the inverse of
the expected discounted ratio of marginal utilities.
Also, we can use equations (13.11) and (13.12) to build a simple non-arbitrage argument that tells us that the household should be indi¤erent between buying 1=Rt+1 bond
from the government and investing one extra unit of capital. Formally:
Et
Et
t+1
t
Rt+1 = 1 = Et
t+1
(1 + (1
k;t ) rt+1 )
t
u1 (ct+1 ; lt+1 ; cpt+1 )
u1 (ct+1 ; lt+1 ; cpt+1 )
Rt+1 = Et
(1 + (1
u1 (ct ; lt ; cpt )
u1 (ct ; lt ; cpt )
)
k;t ) rt+1 )
In this expression, both returns, Rt+1 and 1 + (1
k;t ) rt+1 , are weighted by the discounted ratio of marginal utilities that equates the valuation of the return in each possible
state of the world at period t + 1.
13.4. GOVERNMENT DEFICITS
245
Another way to think about the previous result is that the government cannot force
households to buy its debt (otherwise, we will not call it debt but some type of tax).
Therefore, it has to o¤er a yield that is competitive in comparison with the alternatives
uses of resources by the household (in this case, investment in physical capital). At the
same time, the government does not want to waste resources and it will not o¤er a return
that is higher than the return of physical capital.
13.4.2
Intertemporal Budget Constraint of the Government
In chapter xxx, we gained a lot of insight about the household problem by consolidating
the di¤erent period budget constraints into a single intertemporal budget constraint. We
will do the same with the government period budget constraint and also learn a lot about
the limits to …scal policy that budget constraints impose.
Our point of depart is the government budget constraint in period 0:
b1
= cp0
R1
T r0
l;0 w0 l0
k;0 r0 k0
(13.13)
+ b0
and in period 1:
b2
= cp1 T r1
l;1 w1 l1
k;1 r1 k1 + b1
R2
Note that we can write the second budget constraint as:
b2
1
=
(cp1
R2 R1
R1
T r1
l;1 w1 l1
k;1 r1 k1 )
(13.14)
+
b1
R1
and we can subsitute the last term bt+1 =Rt+1 by (13.13):
b2
1
=
(cp1
R2 R1
R1
T r1
l;1 w1 l1
k;1 r1 k1 )
+ cp1
T r1
l;1 w1 l1
k;1 r1 k1
+ b1
Rearranging terms:
b2
cp1
= cp1 +
R2 R1
R1
l;1 w1 l1
l;1 w1 l1
R1
k;1 r1 k1
k;1 r1 k1
R1
T r1
T r1
+ b1 (13.15)
R1
This equation says that the debt at the end of period 2; b2 , multiplied by its price:
1
R2 R1
is equal to the initial debt b0 plus the di¤erence between the discounted expenditure
cp0 +
cp1
R1
and the discounted revenues:
l;0 w0 l0
+
l;1 w1 l1
R1
+
k;0 r0 k0
+
k;1 r1 k1
R1
+ T r0 +
T r1
R1
246
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
where the discounting is done with the relevant price for the government, the return on
the public debt R1 .
We can repeat our procedure for all periods. For example, we take the period 2
budget constraint
b3
= cp2 T r2
l;2 w2 l2
k;2 r2 k2 + b2
R3
and use it to eliminate the term bt+2 =Rt+2 Rt+1 from equation (13.15). This will give us
an expression that depends on b3 , which we substitute with period 3 budget constraint
and so on. After repeated substitutions we get the expression:
1
X
cpt
+ b0 = l;0 w0 l0 +
cp0 +
t
|{z}
i=1 Ri
t=1
t=1
In
itia
l D ebt
{z
}
{z
|
|
1
X
l;t wt lt
+ r;0 r0 k0
t
i=1 Ri
P re se nt D isc o u nte d Va lu e
P re se nt D isc o u nte d Va lu e
o f G ove rn m e nt C o n su m p tio n
o f L a b o r In c o m e Ta x e s
} |
+
1
X
t=1
{z
k;t rt kt
+T r0
t
i=1 Ri
P re se nt D isc o u nte d Va lu e
o f C a p ita l In c o m e Ta x e s
} |
+
1
X
t=1
{z
t!1
bt
t
i=1 Ri
o f Tra n sfe rs
=0
that ensures that the presented discounted value of debt at in…nity is zero. The interpretation of this transversality condition is very similar to the interpretation of the
transversality condition of the household. First, it prevents governments from running
Ponzi schemes. Second, it forces the government to spend all the tax revenue it rises.
Remark 66 (Default) Our previous derivation of the intertemporal budget constraint
of the government implicitely assumes that the government will always honor its debt
and pay it back. However, default by governments is a common phenomenon, contracts
non-enforceable....
Finally, we repeat our procedure of chapter xxx to …nd the intertemporal budget
constraint of the household:
1
1
X
X
ct
T rt
c0 +
+ T r0 +
t
t
i=1 Ri
i=1 Ri
t=1
t=1
= (1
l;0 ) w0 l0 +
1
X
(1
t=1
l;t ) wt lt
t
i=1 Ri
+ (1
r;0 ) r0 k0 +
1
X
(1
t=1
k;t ) rt kt
t
i=1 Ri
}
P re se nt D isc o u nte d Va lu e
that says that the present discounted value of government consumption plus the initial
debt must be equal to the presented discounted value of tax revenues.
Note that there is a term missing in the previous equation, the one involving debt at
in…nity. This is because we impose the transversality condition:
lim
T rt
t
i=1 Ri
+ b0 + k0
that tells us that the present discounted value of consumption and transfers must be
equal to the present discounted value of after tax labor and capital income and the
initial endowment of public debt and capital. The derivation is left as an exercise at the
end of the chapter.
13.5. A RICARDIAN WORLD
13.5
247
A Ricardian World
The idea of Ricardian equivalence is that, given a …xed sequence of public consumption
fcpt g1
t=0 , if this public consumption is …nanced through non-distorsionary taxes (a negative transfer, that is a transfer from households to the government), then the timing of
taxes is irrelevant. In other words, when taxes are non-distorsionary, it does not matter
if we tax today or tomorrow: the allocations and prices of the competitive equilibrium
are una¤ected.
Note that Ricardian equivalence does NOT say that public consumption is irrelevant
or that changes to the sequence of fcpt g1
t=0 are irrelevant. What it says is that when
we …nance that sequence is irrelevant (as long as we respect the intertemporal budget
constraint and the taxes are non-distorsionary).
Remark 67 (Ricardian Equivalence and War Financing) The idea of Ricardian
equivalence takes its name from the classical economist David Ricardo (1772-1823). Ricardo lived in the middle of the long sequence of wars between the United Kingdom and
France that lasted, with only brief interruptions, between 1792 and 1815. Thus, Ricardo
was keenly interested in how the government should …nance the war (or, for the matter,
any other government expenditure).9
Ricardo understood that there are two principal ways to levy revenues for the war,
namely to tax current generations or to issue government debt in the form of government
bonds the interest and principal of which has to be paid later.10 The question then is what
are the macroeconomic consequences of using these di¤erent instruments, and which
instrument is to be preferred from a normative point of view. The Ricardian equivalence
claims that it makes no di¤erence, that a switch from one instrument to the other does
not change real allocations and prices in the economy. Therefore this concept is also
called Modigliani-Miller theorem of public …nance.11
Ricardo proposed the following example. The government has to …nance a war with
annual expenditures of $20 millions. Does it makes a di¤erence to …nance the $20
millions via current taxes or to issue government bonds with in…nite maturity (so-called
consols) and …nance the annual interest payments of $1 million in all future years by
9
Thinking about public debt was closer to Ricardo’s mind that many other issues in economics. At
21, and after breaking with his family because of his marriage, he set up a trading company that dealt
with government bonds. He was so immensly sucessful in this business that in 1814, at the age of 42,
he thought he was “su¢ ciently rich to satisfy all my desires and the reasonable desires of all those
about me” (Letter to James Mill, 1815, the father of other important classical economist, John Stuart
Mill), and retired. Ricardo bought Gatcomb Park, an estate in Gloucestershire, became a member of
the country gentry, and got himself elected to the United Kingdom Parliament in 1819 where he served
until his untimely death in 1823. Of course, as an MP he focuseed on economic policy. A remarkable fact
about Ricardo is that he only started thinking about economics late in life (1799) when, on a vacation,
he read Adam Smith’s The Wealth of Nations.
10
We have not incorporated …at money in the model (we will do that in the next chapter). Hence the
government cannot levy revenue via seignorage.
11
We will derive in chapter xxx the Modigliani-Miller theorem in its original formulation regarding
the value of a …rm.
248
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
future taxes (at an assumed interest rate of 5%)? His conclusion was (in “Funding
System”) that:
....in the point of the economy, there is no real di¤erence in either of the modes; for
twenty millions in one payment [or] one million per annum for ever ... are precisely of
the same value.
Here Ricardo formulates and explains the equivalence hypothesis, but immediately
makes clear that he is sceptical about its empirical validity
...but the people who pay the taxes never so estimate them, and therefore do not
manage their a¤airs accordingly. We are too apt to think, that the war is burdensome
only in proportion to what we are at the moment called to pay for it in taxes, without
re‡ecting on the probable duration of such taxes. It would be di¢ cult to convince a man
possessed of $20; 000, or any other sum, that a perpetual payment of $50 per annum
was equally burdensome with a single tax of $1; 000:
Ricardo doubts that agents are as rational as they should, according to “in the point
of the economy”, or that they rationally believe not to live forever and hence do not have
to bear part of the burden of the debt. Since Ricardo did not believe in the empirical
validity of the theorem, he has a strong opinion about which …nancing instrument ought
to be used to …nance the war
...war-taxes, then, are more economical; for when they are paid, an e¤ort is made to
save to the amount of the whole expenditure of the war; in the other case, an e¤ort is
only made to save to the amount of the interest of such expenditure.
Ricardo thought of government debt as one of the prime tortures of mankind. Not
surprisingly he strongly advocates the use of current taxes. We will, after having discussed
the Ricardian equivalence hypothesis, brie‡y look at the long-run e¤ects of government
debt on economic growth, in order to evaluate whether the phobia of Ricardo (and almost
all other classical economists) about government debt is in fact justi…ed from a theoretical
point of view. Now we turn to a model-based discussion of Ricardian equivalence.
Before further analysis, note that the absence of distorsionary taxes implies that the
period government budget constraint is:
bt+1
= cpt
Rt+1
T rt + b t
and its intertemporal budget constraint is:
cp0 +
1
X
t=1
X
cpt
+
b
=
T
r
+
0
0
t
i=1 Ri
t=1
1
T rt
t
i=1 Ri
For the household, the intertemporal budget constraint is:
c0 +
1
X
t=1
w0 l0 +
ct
t
i=1 Ri
+ T r0 +
1
X
t=1
T rt
t
i=1 Ri
=
1
1
X
X
wt lt
rt k t
+
r
k
+
+ b0 + k0
0 0
t
t
R
R
i
i
i=1
i=1
t=1
t=1
13.5. A RICARDIAN WORLD
249
Our strategy to show how the Ricardian equivalence holds is to start by de…ning a
competitive equilibrium for a given sequence of transfers fT rt g1
t=0 :
De…nition 68 Given initial capital k0 and the sequence of …scal policy fcpt ; T rt g1
t=0 , a
competitive equilibrium consists of allocations for the representative household, fct ; lt ; kt+1 g1
t=0 ;
1
1
allocations for the representative …rm, fkt ; lt gt=0 and prices frt ; t ; wt ; Rt gt=0 such that:
1
1. Given frt ; wt g1
t=0 and fcpt ; T rt gt=0 ; the household allocation solves the household
problem:
max E0
fct ;lt g1
t=0
1
X
t
u (ct ; lt ; cpt )
t=0
bt+1
= wt lt + rt kt + kt + T rt for all t
Rt+1
given initial condition k0 ; b0
s.t. ct + kt+1 +
0
2. Given f t ; wt g1
t=0 ; the …rm allocation solves the …rm problem:
max (yt
nt ;kt
wt lt
t kt )
subject to
yt = kt (At lt )1
At = (1 + g)t ezt A0
zt = zt 1 + "t , "t N (0; 1)
kt ; lt 0
3. Fiscal policy …scal policy fcpt ; l;t ; k;t ; T rt g1
t=0 is such that the government satis…es
its budget constraint: for all t = 0; : : : ; 1
bt+1
= cpt
Rt+1
T rt + b t
4. Market clears and the aggregate resource constraint holds: for all t = 0; : : : ; 1
ct + kt+1 + cpt = kt (At lt )1
+ (1
) kt
We state now a formal de…nition of Ricardian equivalence:
Theorem 69 Take as given a sequence of government spending fcpt g1
t=1 and initial government debt b0 . Suppose that the allocations for the representative household, fct ; lt ; kt+1 g1
t=0 ;
1
1
allocations for the representative …rm, fkt ; lt gt=0 and prices frt ; t ; wt ; Rt gt=0 are such
250
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
c 1
that they form a competititive equilibrium given a sequence of taxes fT rt g1
t=1 . Let fT r t gt=1
be an arbitrary alternative tax system satisfying:
T r0 +
1
X
t=1
fct ; lt ; kt+1 g1
t=0 ,
fkt ; lt g1
t=0 ,
X
T rt
cr0 +
=
T
t
i=1 Ri
t=1
1
1
t ; wt ; Rt gt=0
Tcrt
t
i=1 Ri
formn a competititive
equilibrium
o1
given the initial capital k0 and the sequence of …scal policy cpt ; Tcrt
.
Then
and frt ;
t=0
Proof. First, note that, with the new tax sequence,
is still feasible since all components of:
ct + kt+1 + cpt = kt (At lt )1
fT rt g1
t=1 ,
+ (1
the allocation fct ; lt ; kt+1 g1
t=0
) kt
are unchanged.
Second, by construction, the government still satis…es its intertemporal budget constraint, since
1
1
X
X
T rt
Tcrt
c
T r0 +
=
T
r
+
0
t
t
i=1 Ri
i=1 Ri
t=1
t=1
Third, given the price system frt ; t ; wt ; Rt g1
t=0 , the …rm still maximizes its pro…ts
1
by hiring fkt ; lt gt=0 (its …rst order conditions are una¤ected).
Fourth, given the price system, the household can still a¤ord its all allocation fct ; lt ; kt+1 g1
t=0 :
To see this, note that the intertemporal budget constraint of the household under the
original tax sequence is
c0 +
1
X
t=1
ct
t
i=1 Ri
+ T r0 +
1
X
X wt lt
X r t kt
T rt
=
w
l
+
+
r
k
+
+ b0 + k0
0 0
0 0
t
t
t
R
R
i
i
i=1 Ri
i=1
i=1
t=1
t=1
1
X
1
1
X
X
Tcrt
wt lt
rt kt
= w0 l0 +
+ r0 k0 +
+ b0 + k0
t
t
t
i=1 Ri
i=1 Ri
i=1 Ri
t=1
t=1
t=1
1
1
and for the new tax sequence is:
c0 +
1
X
t=1
ct
t
i=1 Ri
+ Tcr0 +
t=1
Since the present value of transfers is unchanged, the sequence fct ; lt ; kt+1 g1
t=0 still satis…es the constraint. Given that fct ; lt ; kt+1 g1
satis…ed
the
old
…rst
order
conditions
and
t=0
1
those are una¤ected by the change in taxes, fct ; lt ; kt+1 gt=0 still solves the problem of
the household.
The previous theorem tells us that, when we change the transfers over time in such a
way that their present discounted value is still the same (using the original equilibrium
public debt returns), we still have the same equilibrium allocation and prices. The key
of the proof is the observation that the intertemporal budget constraint of the household
is una¤ected: lower taxes today mean higher taxes tomorrow and both e¤ects cancel
exactly. This is why Barro, an economist who revived the idea of Ricardian equivalence
called his 1974 article: “Are Government Bonds Net Wealth?” His answer was that,
under the conditions we speci…ed before, government bonds are not net wealth: they are
just receipts for future taxes.
13.6. EXERCISES
251
13.6
Exercises
13.6.1
Long Run E¤ects of Fiscal Policy
The chapter has focused on the e¤ects of temporary changes in …scal policy. What often
the discussion is about the e¤ects of permanent changes in …scal policy.
1. Use the simple model with GHH preferences and full depreciation to study how
permanent changes in tax rates a¤ect the steady state value of the economy.
2. Describe the transition from any initial level of capital to the steady state if we
keep over time (you can also assume that productivity is constant over time and
ez = 1).
Answer
13.6.2
A Dynamic La¤er Curve
In the main text we saw that, for our model with GHH preferences and full depreciation,
the maximum of tax revenue is achieved when
+
t =
+1
However, our derivation assumed that kt was given. However, kt will depend on taxes
itself. This generates a dynamic La¤er curve: an increase in taxes today may lower
the presented discounted value of future tax revenue even if it raises tax revenue in the
current period.
While a full analysis of a dynamic La¤er curve is complicated, a simple characterization of the La¤er curve in the steady state is not.
1. Find the tax rate
that maximizes tax revenue in the steady state of our model.
2. Compare the tax rate
with the tax rate that maximizes tax revenue within the
period given some capital and technology. Provide a quantitative assement and
intuition.
Answer: We derive in exercise xxx that:
y = (1
)
+ +1
(
)
1+
1
1
+
Therefore, tax revenue is:
T ax =
=
y
(1
)
+ +1
(
)
1+
+
1
1
252
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
To maximize tax revenue T ax with respect to , we can take logs of the previous
expression:
log T ax = log +
+
+1
log (1
) + log (
)
1+
1
1
+
and …nd the …rst order condition:
1
where
+
=
1
+1
1
is the tax rate that maximizes steady state tax revenue. Then:
=
+
+ +1
The tax rate that maximizes revenue in the current period is
t
=
+
+1
We can see how
< t . For the calibration in the main text, = 0:5 and = 0:33, we
have that t = 0:553 while
= 0:25, a much lower number. This number suggests that
many advanced economies may not be maximizing the steady state level of tax revenue
(since taxes range between 0.25 to 0.50 of output for most OECD countries).
The intuition is that, when we allow capital to respond to taxes, we need to account
for the fact that investment is reduced when we increase the tax rate.
13.6.3
Investment-Speci…c Technological Change
We can revisit investment-speci…c technological change when we have …scal policy.....
13.6.4
Intertemporal Budget Constraint of Household
Using the same steps than in chapter xxx, derive the intertemporal budget constraint of
the household:
c0 +
1
X
t=1
= (1
l;0 ) w0 l0
+
1
X
(1
t=1
ct
t
i=1 Ri
l;t ) wt lt
t
i=1 Ri
+ T r0 +
+ (1
1
X
t=1
r;0 ) r0 k0
T rt
t
i=1 Ri
+
1
X
(1
t=1
k;t ) rt kt
t
i=1 Ri
+ b0 + k0
Extending the Model II: Money
Fiat money, the small pieces of otherwise useless green paper called dollars (or with some
other colors and names if you live outside the U.S.) that all of us exchange daily, is one of
most common and intriguing of economic institutions. We learn to deal with …at money
as children and we rarely stop to think about how remarkable is the social arrangement
by which we trade pieces of intrinsically worthless paper for goods and services that we
desire.
More remarkable even is the idea that, by printing more or less of these small pieces
of paper, a monetary authority like a Central Bank may have an impact on real variables
like output, consumption, or investment (in fact, in modern economies, we can even skip
the printing part and just debit an electronic account). How can it be that, by having
more pieces of paper or a di¤erent number in an electronic account, a market economy
will produce more real output? But this exactly how many macroeconomists interpret the
empirical evidence. A long tradition of thought that goes back at least to David Hume,
an 18th Scottish philosopher and economist who …rst outlined this link in a clear and
forceful way, a large group of economists have defended the existence of a transmission
mechanism from increases in …at money supply to business cycle ‡uctuations.12
In this chapter, we will attempt at making sense of the previous two paragraphs. First,
we will document some empirical facts regarding money and the economy in the long
and short run and we will explain which type of observations have led many economist
to emphasize a monetary explanation to the business cycle.
Second, we will think about why …at money exist and the role it plays in modern
economies. We will emphasize that …at money may allow the economy achieve allocations
that would otherwise impossible to implement. To do so, we will step back for a few
pages from our basic model and present a extremely simple search model of money that
accounts for the presence of money. This section will also explain why some of the
quickest answers for the existence of money are not so convincing upon closer inspection.
Third, we will introduce money in our basic dynamic equilibrium model. So far in
the book, …at money has not played any role in our explanations. This was because
the basic equilibrium model that we have developed in previous chapters did not need
money: agents could undertake all their transactions in a centralized market where their
12
Some economists have gone even further and defended the existence of a link between long-run
growth and …at money. However, this position has been much more minoritary and we will not discuss
it further. Also, other thinkers before Hume wrote about the links between money and economic
‡uctuations, but none as clearly and precisely.
253
254
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
net positions at end of the trade were liquidated in terms of goods or …nancial assets. But,
while being able to dispense with money was a convenient simpli…cation that allowed
us to focus on the real aspects of the environment and the equilibrium, it generates a
fundamental problem. Namely, the model cannot explain why we observe …at money in
the world as a mean of exchange and why it is not displaced by other assets like treasury
bills that provide nearly as much liquidity while yielding a positive interest rate.
The presence of money leads itself naturally to examine the role of the authority that
prints this money and to ask some basic questions about monetary policy: Why do we
need a central bank? What are the e¤ects of monetary policy? What is the optimal rule
to follow? What is the best way to implement monetary policy? We will address some
of those basic topics of monetary economics in section 4 of the chapter.
We will conclude the chapter by examining how the presence of …at money by itself is
unlikely to produce much of a relation between money and the business cycles. This was
to be expected since …at money does not increase the technological level of the economy
or the amount of physical capital available for production. There are two possible answers
to that …nding. One is to revisit the empirical evidence and question if the link between
money and output exist after all or at least, whether we got the causality direction right
(is perhaps the link from increases in output to increases in …at money?). The second
answer is to add further elements in the model, nominal rigidities, that prevent agents
from adjusting prices when money supply changes. We will incorporate nominal rigidities
in the analysis in chapter xxx.
An additional advantage from incorporating money in our model is that we will be
able to talk about a nominal price level and develop a theory that accounts for the
observed changes in the price level of the economy. Up to this moment, we only talked
about relative prices, expressed in terms of one good, usually the consumption good,
that we took as our numeraire. Once we have money, the price level will be equal to
the quantity of cash that we need to exchange for one unit of our numeraire good. Since
this amount can increase over time, the phenomenon that we call in‡ation (or de‡ation
in the more rare cases when it falls), we will index the price level, pt , by the period t.
13.7
Empirical Evidence
In the introduction, we talked about …at money without explaining what we add the
adjective “…at”. The word “…at”comes from latin and means “let it be done.”13 It refers
to those examples of money that are embodied into some worthless or nearly worthless
object, like paper, plastic chips, or an electronic debit. Fiat money is, then, created by
some decision of an authority of by the common use of society. Fiat money stands in
opposition with commodity money, like gold or silver, where the object used as money is
valuable in itself.
13
The origin of the expression comes from Genesis 1,3, the …rst book of the Bible, in the old latin
Vulgate translation: “Dixitque Deus: Fiat lux. Et facta est lux", or “And God said, “Let there be
light.”And light became.”The analogy is that, by some creation act, …at money appears out of nothing
(or just a piece of paper).
13.7. EMPIRICAL EVIDENCE
255
For most of human history, the predominant form of money was commodity money.
Commodities that could easily be transported and stored and that were valued by a
large part of the population became generally accepted media of exchange. Gold and
silver were perfect candidates for this role. Gold, for instance, is the most malleable and
ductile pure metal, which made it ideal for uses in jewelry and sculpture. Moreover, gold
is long-lasting and easy to maintain. But many other objects can become commodity
money if the circumstance arises. In prisoners-of-war camps during the Second World
War, cigarettes quickly became the commodity money (Radford, 1946). Even those
prisoners who did not smoke accepted them as a mean of payment since cigarettes allowed
them to buy goods in the future from other prisoners. More recently, after U.S. prisons
banned smoking in 2004, mackerel …llets, highly appreciated by those prisoners like
weight lifters who want an extra supply of protein, have become the standard currency in
many correctional facilities (see Scheck, 2008, who reports many other of the fascinating
details of economic life within prisons).
From the late 19th century to 1971, the world quickly evolved from a monetary system
based on commodity money, basically the Gold standard (with some smaller uses of silver
and copper) to a system nearly exclusively based on …at money. The change is breathtaking in its speed and thoroughness. Before the late 19th century, we observed only a
few isolated examples of …at money.14 Among those, one of the most cited is the bills
of credit issued by the Continental Congress, therefore called “Continentals”, during the
Revolutionary War of the American Colonies against the British Empire. Continentals
were not explicitly backed by any commodity and circulated widely for many years even
if they su¤ered a high rate of in‡ation and depreciation with respect to other currencies.
Even as late as 1914, the eve of the First World War, an attachment to the Gold standard
was considered the basic proof of soundness of economic policy. John Maynard Keynes,
one of the most brilliant economists of all times, summarized better than anyone else this
feeling (free trade and the Gold standard)....The economic turmoil caused by the two
world wars and by the Great Depression nearly completely eliminated the Gold standard.
After the Second World War, only the U.S. government kept the convertibility of U.S.
dollars into gold at a …xed rate but only to other governments. This system survived,
with many di¢ culties, until 1971. In that year, the U.S. ended the convertibility of
dollars into gold and all the residuals of commodity money disappeared.
Consequently, this chapter will focus on the study of …at money, with only some
passing references to commodity money. Now, the issue at hand is to establish the
empirical relation between money and other variables, both in the long run and in the
short run, to have a framework of reference to evaluate our models of money.
13.7.1
Facts about Money in the Long Run
hello
14
During the Song dinasty in the 11th century, paper money circulated in China, but this paper money
were convertible into some amount of gold, silver, or silk. Even if convertibility was often impaired, the
Song paper money was just a shortcut to reduce the transportation costs of commodity money.
256
13.7.2
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
Facts about Money in the Short Run
Economists have searched for evidence on the relation between money and other economic
variables in the short run using many di¤erent tools. We will review three of the most
popular ones: the narrative approach, Vector Autoregresions, and the Phillips curve.
The Narrative Approach
hola....
Vector Autoregresions
In chapter xxx, we postulated that the process for productivity followed an autoregressive
process of order 1 (AR(1)) of the form:
zt = zt
1
(13.16)
+ "t
This equation was the particular case of the more abstract process:
xt =
x xt 1
+
(13.17)
" "t
where xt is an arbitrary variable.
A natural generalization of the previous process is to assume that, instead of regressing a variable in itself, we are regressing a vector of variables Xt in itself:
Xt = AXt
1
(13.18)
+ "t
Given a vector Xt with n variables, A in equation (13.18) is a n
vector of n shocks distributed as a multivariate normal:
"t
n matrix and "t a
N (0; )
where 0 is a vector of n zeros and is a n n matrix. Equation (13.18) is a Vector
Autoregresion of order 1 (VAR(1), often the order is dropped when no confusion arises
or when we talk about Vector Autoregresions in general).
An example will clarify the notation. Imagine that we have three variables, money
supply, mt , real output, yt , and the price level, pt . We can then postulate a VAR(1) with
these three variables:
1 0
1
0
10
1 0
a11 a12 a13
mt 1
"mt
mt
@ yt A = @ a21 a22 a23 A@ yt 1 A + @ "yt A
a31 a32 a33
pt 1
"pt
pt
{z
}|
{z
} | {z }
| {z } |
Xt
A
Xt
"t
1
We can look at the …rst row of the system. It reads:
mt = a11 mt
1
+ a12 yt
1
+ a13 pt
1
+ "mt
13.8. DEEP MODELS OF MONEY
257
and it tells us that money supply this period, mt , is a function of money supply last
period, mt 1 , output yesterday last period, yt 1 , and the price level last period, and of
a shock "mt : We can interpret the other two rows in similar ways.
This example illustrates how a VAR is a ‡exible procedure to summarize the dynamic
of a vector of variables and the interactions among them. Better still, we can estimate
the parameters in A and using Ordinary Least Squares and any standard statistical
software package. Flexibility in capturing dynamic behavior and simplicity in estimation
explain why VARs are one of the most popular empirical techniques in macroeconomics.
13.8
Deep Models of Money
The existence of commodity money is easy to justify. In a complex world, doublecoincidence of wants is rare: agent A produces a shirt and wants to buy a lecture on
macroeconomics and agent B produces lectures on macroeconomics and wants to buy
a shirt. Instead, commodity money allows agent A to sell a shirt to agent B iwho can
sell lectures on macroeconomics to agent C who sells a car to agent A, all transaction
realized with commodity money. Since commodity money is valuable in itself, each agent
is not particularly worried about getting stuck with the commodity. In the worst case
scenario, the agent can consume the good itself.15
In comparison, the existence of …at money is more problematic. Why do we accept
these pieces of paper in exchange for goods and services? Some of the arguments that
…rst come to mind are ‡awed when we examine them at a closer range. For example,
many would argue that …at money exists because is created by governments that enforce
its acceptance. Legally, we say that money is legal tender. However, this explanations
misses the fact that we observe many situations where …at money exists without being
legal tender or even with the opposition of the local authorities. U.S. dollars circulate in
daily transactions in many poor countries and the dollar is not a legal tender in any of
these countries. In fact, many governments persecute the users of dollars and impose …nes
or send to prison those found employing them. In comparison, local currencies that are
legal tender are not accepted by the local citizens except for the smallest transactions.
This example is not de…nitive because, even if not legal tender in the poor country,
American dollars are legal tender in the U.S. and one could argue that the only thing
that matters is that the currency is legal tender somewhere.
That is why the example of the Iraqi Swiss dinar is so compelling. After the Gulf
War of 1991, Saddam Hussein’s regime printed new dinar notes in substitution of the old
dinar notes that, because they had been printed with plates made in Switzerland, were
know as Swiss dinars.16 However, the substitution of Swiss dinars did not happen in the
15
When the agent does not like the good (for example, a non-smoking prisoner-of-war and his cigarettes), the reasoning becomes more complicated because the agent needs to worry about the possibility
of getting trapped in a large commodity money position that he cannot unload. How we can get around
this issue will become clear when we talk about …at money since, in some sense, for the agent that does
not like the good, the commodity money is similar to …at money.
16
Although others argue that the name comes from the fact that the pre-Gulf war Iraq had a tradition
258
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
region at the north of Iraq controlled by the Kurds who were opposed to the regime,
and consequently a percentage of the old notes continued in circulation. Over the next
decade and a half, the Swiss dinars, which were not longer legal tender, continued to
be used by many Iraqis, until the point that after 13 years of circulation, many notes
were so spent that they had to be kept together with staples and tape (Foote et al.,
2004). Moreover, as Hussein’s regime printed more and more notes to …nance its huge
government de…cits, the Swiss dinar started to appreciate. By around 1998, the Swiss
dinar was exchanging for around 100 of the new dinars.
Others argue that …at money circulates because it is backed by the full credit of the
government that issues it. But this reasoning runs into the problem that, being backed
by the full credit of the government is a rather empty statement: the government is not
commited to reedem the note for any real good (and governments did during the Gold
standard, where the Bank of England was ready to exchange Sterling Pounds for a …xed
quantity of gold at any time.17 )
13.9
Money in the Utility Function
Money in the utility function (MIU, or money-in-utility) is one of the simplest tricks to
incorporate money in the model. We will assume that the real cash-balances mt 1 =pt
enter in the utility function as an additional argument:
E0
1
X
t=0
t
u ct ; lt ;
mt 1
pt
(13.19)
The real cash-balances are the ratio of the nominal cash-balance, mt 1 , the amount of
currency that the household carry into the period, divided by pt , the price level.
The justi…cation for having real balances appear in the utility function is that cash
facilitates transaction and makes life easier. For example, if you are going to pay in the
co¤ee shop for your morning tea with a credit card, it takes a few extra second. Or
when you are tipping a hotel attendant or a theater usher, you have to give her cash. In
most vending machines or parking meters, you cannot pay for your soda with a credit
card. The examples of situations in life where carrying cash facilitates transactions are
plentiful.
In a more general interpretation, cash not only includes the notes in your wallet, but
also the balance in checking accounts and other highly liquid can be interpreted as cash,
since they do not yield any interest (or perhaps a trivial amount). Checking accounts
are a form of using your cash where the …nancial institution arranges the payment for
you in exchange for a low . Similarly, you can think about prepaid or charge cards that
you may use in your university or even debit cards as cash since they are just a more
of such a low in‡ation that the dinar was said to be “as solid as the Swiss franc.”
17
A position that the British honored for nearly two centuries and only most reluctantly gave up
during the Napoleonic Wars, the First World War (1914), and …nally, during the Great Depression
(check time and references).
13.9. MONEY IN THE UTILITY FUNCTION
259
convenient form of carrying your cash where a …nancial institution handles the payment
for you, the important aspect being that the money in prepaid card is not yielding any
interest)
Now, we modify the utility function to include money:
E0
1
X
t
lt1+
1+
log ct
t=0
+
mt
pt
1
1
and budget constraint:
pt ct + pt xt + mt +
1
bt+1 = pt wt lt + pt rt kt + mt
Rt+1
1
+ bt + pt Tt + pt
t
or dividing by pt :
ct + xt +
mt
1 bt+1
m t 1 bt
+
= wt lt + rt kt +
+ + Tt +
pt
Rt+1 pt
pt
pt
t
The nominal interest rate:
Rt+1
=
R
where
zero:
t
=
pt
pt 1
Rt
R
R
yt
y
t
1
y
R
e't
…nanced with lump-sum transfers Tt such that the de…cit are equal to
1 bt+1
bt
mt mt 1
+
pt
pt
Rt+1 pt
pt
The steady state R is not a target because this is G.E. model. The central bank either
targets R or but it cannot target both. Instead:
Tt =
R=
F.O.C. of household
1
lt1+
1+
ct
1
1+
lt+1
1+
ct+1
=
t
=
t+1
lt = wt
mt
pt
t
1
+ Et
pt
1
t+1
pt+1
1
1
= Rt+1 Et t+1
pt
pt+1
= Et t+1 (rt+1 + 1
)
t
t
=
260
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
and …rm:
yt = kt (ezt lt )1
yt
rt =
kt
yt
wt = (1
)
lt
Then:
1
= Et
lt1+
ct
1+
1
lt1+
ct
1
ct+1
1+
= Et
1+
mt
pt
1
pt
pt+1
1+
1+
lt1+ = (1
Make
Rt+1
1
1
+ Et
pt
ct+1
lt1+
ct
1+
lt+1
ct+1
1
=
yt+1
kt+1
1+
1+
lt+1
1
1+
lt+1
pt+1
1+
) yt
= 1 and guess that:
kt+1 =
ct = (1
yt
) yt
In the FOCs:
1
1
1+
1
yt
1
1
1+
1
mt
pt
yt
yt+1
kt+1
yt+1
1
1
1
1+
1
1
+ Et
yt pt
1
1
1+
1
1
1+
1
= Rt+1 Et
1
=
1
= Et
yt+1
t+1
1
1
1
1+
yt+1 pt+1
Simplifying:
1
=
yt
1
kt+1
1
yt
Et
t+1 yt+1
Rt+1 =
0
mt = @
1
1
1
1+
yt
pt
) kt+1 =
1
+ Et
yt
1
1
1
1
1+
yt+1
1
pt A
pt+1
1
13.9. MONEY IN THE UTILITY FUNCTION
261
Emphasize classical dichotomy and Fisher equation. From the Euler equations:
Et
yt+1
kt+1
yt
yt+1
yt Rt+1
yt+1 t+1
= Et
which tells you that expected return on capital is nominal rate less expected in‡ation
yt
:
with the valuation kernel yt+1
Now, for in‡ation and nominal interest rate:
Rt+1
=
R
Rt
R
R
Rt+1 =
R
R
yt
y
t
e
R
1
R
't
1
e't
yt
t+1 yt+1
1
y
y
1
Et
or:
Rt
R
yt
y
t
1
yt
Et
t+1 yt+1
=
1
which gives us a path for in‡ation (if
> 1 is unique and determined). Note that that
the R.H.S. is a function of states today (except for Rt+1 showing up in the expectation
of t+1 ), and hence it pins down Rt+1 : Then, in Taylor rule, t must be such that the
equation holds. It is simpler to see in the case R = y = 0:
1
t
=
1
e't
1
Et
1
t+1
yt
yt+1
!!
1
Shock 't appears in denominator: positive shock to nominal interests is negative shock
to money, in‡ation falls.
Taylor principle:
> 1 with = 1
t
=
=
't
e
yt
Et
Et
yt+1
e't
yt
Et
Et
yt+1
1
1
t+1
yt+1
Et
Et
yt+2
e't+1
which converges only if
>1
Note that this shows why we cannot put Et
R
Rt
R
Et
R
which does not pin down
R
Rt
R
R
t.
t+1
yt
y
t+1
1
y
t 1
yt
y
't
e
y
1
1
t+2
!!!
1
in the Taylor rule. We would have:
R
We cannot put down
1
=
1
1
yt
Et
t+1 yt+1
t 1:
R
e't =
because there is no reason why this equality must hold.
Et
1
yt
t+1 yt+1
1
262
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
New Keynesian Model
Outlook
Monopolistic Competition (Blanchard and Kiyotaki, 1987, and Horstein, 1993).
Final good producer. Competitive behavior.
Continuum of intermediate good producers with market power.
Alternative formulations: continuum of goods in the utility function.
The Final Good Producer
Production function:
Z
yt =
1
"
" 1
" 1
"
yit di
0
where " controls the elasticity of substitution.
Final good producer is perfectly competitive and maximize pro…ts, taking as given
all intermediate goods prices pti and the …nal good price pt .
Maximization Problem
Thus, its maximization problem is:
Z
max pt yt
yit
1
pit yit di
0
First order conditions are:
pt
"
"
1
Z
1
" 1
"
"
" 1
1
yit di
"
1
"
0
" 1
yit "
Working with the First Order Conditions I
263
1
pit = 0
8i
264
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
Dividing the …rst order conditions for two intermediate goods i and j, we get:
pit
=
pjt
1
"
yit
yjt
or:
1
"
yit
yjt
pjt =
pit
Interpretation.
Hence:
1
" 1
pjt yjt = pit yit" yjt"
Integrating out:
Z
1
1
"
pjt yjt dj = pit yit
Z
1
" 1
1
" 1
yjt" dj = pit yit" yt "
0
0
Input Demand Function
R1
By zero pro…ts (pt yt = 0 pjt yjt dj), we get:
" 1
1
1
1
"
pt yt = pit yit" yt " ) pt = pit yit" yt
Consequently, the input demand functions associated with this problem are:
yit =
"
pit
pt
yt
8i
Interpretation.
Price Level
By the zero pro…t condition pt yt =
tions:
Z 1
pit
p t yt =
pit
pt
0
Thus:
pt =
R1
pit yit di and plug-in the input demand func-
0
"
yt di )
Z
p1t "
=
Z
1
0
1
1 "
1
p1it " di
0
13.10
Money in Equilibrium Theory
Money in the utility function, Cash-in-advance
Remark 70 (Search Models of Money) Randy and friends
pit1 " di
13.11. PRICE AND WAGES RIGIDITIES
13.11
Price and Wages Rigidities
13.12
A Benchmark New Keynesian Model
13.13
Coordination Failures
265
266
CHAPTER 13. EXTENDING THE MODEL I: FISCAL POLICY
Part V
Economic Policy
267
Chapter 14
Stabilization Policy
14.1
Fiscal Policy
14.2
Monetary Policy
269
270
CHAPTER 14. STABILIZATION POLICY
Chapter 15
Long Run E¤ects of Policy
15.1
Monetary Policy
Derive long-run neutrality
15.2
Fiscal Policy
15.2.1
Taxation
Labor Income Taxation and Labor Supply
Capital Income Taxation and Aggregate Saving
15.2.2
Social Security
15.2.3
Social Insurance
271
272
CHAPTER 15. LONG RUN EFFECTS OF POLICY
Chapter 16
An Introduction to Optimal
Monetary and Fiscal Policy
1. Ramsey for dummies.
2. Basic problem of time inconsistency
3. Dynamic Games and Markov Equilibria
273
274CHAPTER 16. AN INTRODUCTION TO OPTIMAL MONETARY AND FISCAL POLICY
Part VI
Miscellaneous Topics
275
Chapter 17
Consumption
[A summary of partial equilibrium consumption theory, essentially a nontechnical version
of the consumption savings manuscript]
277
278
CHAPTER 17. CONSUMPTION
Chapter 18
Investment
[Q Theory]
279
280
CHAPTER 18. INVESTMENT
Chapter 19
The Labor Market
[Baby Search and Mortensen-Pissarides].
281
282
CHAPTER 19. THE LABOR MARKET
Chapter 20
Asset Pricing
[Very basic facts and pricing simply using the stochastic Euler equation].
283
284
CHAPTER 20. ASSET PRICING
Chapter 21
Open Economy Macroeconomics
[Baby open macro models].
285
286
CHAPTER 21. OPEN ECONOMY MACROECONOMICS
Part VII
Epilogue
287
Chapter 22
The History and Future of
Macroeconomics
Now that we approach the end of our book is a good time to recap the history of
macroeconomics and to outline some areas of intensive research.
22.1
History of growth theory
Adam Smith, David Ricardo, Robert Malthus, and John Stuart Mill had little hope for
sustained growth. Forgotten for a while. Ill attempted in UK (Roy Harrod and Evsey
Domar). Robert Solow (MIT, Nobel 1987): two main papers, 1956 and 1957. Completed
by David Cass (Penn) and Tjalling Koopmans (Nobel 1971). 1980’s and 1990’s: Paul
Romer (Stanford, Nobel 20??), Robert Lucas (Chicago, Nobel 1995), ...Recent: Daron
Acemoglu (MIT, Clark Medal 2005): political economy, institutions,...
Thomas Malthus (1766–1834) was an English economist that undertook the …rst
modern systematic study of the relation between economic growth and population dynamics in his famous An Essay on the Principle of Population, which he re…ned in six
editions (the last one published in 1826). Malthus’view, even if more subtle and nuance
that the simpli…ed accounted common in many textbooks, was relatively pessimistic as
he thought that population dynamics imposed serious constraints in per capita income
growth in the long run.
289
290
CHAPTER 22. THE HISTORY AND FUTURE OF MACROECONOMICS
Bibliography
[1] Arrow (1965)
[2] Barro (1974) Are Government Bonds Net Wealth?
[3] Heathcote, J., K. Storesletten and G. Violante (2008)
[4] Ljungqvist, L. and T. Sargent (2004), Recursive Macroeconomic Theory, 2nd ed.,
MIT Press, Cambridge, MA
[5] Pratt (1965)
[6] Rios-Rull, V. (1995) in Marimon Scott
[7] Stokey and Lucas (1989)
[8] Trabandt, Mathias and Harald Uhlig (2006). “How Far Are We From the Slippery
Slope? The La¤er Curve Revisited.”CEPR Discussion Papers 5657.
291