Undergraduate study in Economics,
Management, Finance and the Social Sciences
Macroeconomics
K. Sheedy
EC2065
2022
Macroeconomics
K. Sheedy
EC2065
2022
Undergraduate study in
Economics, Management,
Finance and the Social Sciences
This subject guide is for a 200 course offered as part of the University of London
undergraduate study in Economics, Management, Finance and the Social Sciences.
This is equivalent to Level 5 within the Framework for Higher Education Qualifications
in England, Wales and Northern Ireland (FHEQ).
For more information, see: london.ac.uk
This guide was prepared for the University of London by:
Dr Kevin Sheedy, Assistant Professor of Economics, Department of Economics, The
London School of Economics and Political Science.
This is one of a series of subject guides published by the University. We regret that
due to pressure of work the authors are unable to enter into any correspondence
relating to, or arising from, the guide. If you have any comments on this subject
guide, favourable or unfavourable, please use the form at the back of this guide.
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EC2065 Macroeconomics
Contents
Chapter 1: The supply side of the economy ........................................................................... 8
1.1 Production functions and factors of production........................................................... 8
1.1.1 Factors of production............................................................................................ 8
1.1.2 The production function ....................................................................................... 9
1.1.3 Neoclassical production functions....................................................................... 10
1.1.4 The Cobb-Douglas production function ............................................................... 12
1.1.5 The per worker production function ................................................................... 12
1.2 Factor markets and the distribution of income .......................................................... 13
1.2.1 The Cobb-Douglas production function ............................................................... 15
Box 1.1: Understanding inequality in wages ................................................................ 15
1.3 Population growth according to Malthus ................................................................... 17
1.3.1 Demographics ..................................................................................................... 17
1.3.2 An agricultural economy ..................................................................................... 18
1.3.4 Dynamics of a Malthusian economy.................................................................... 19
1.3.5 What does (or does not) help? ............................................................................ 20
Box 1.2: A 14th-century pandemic ............................................................................... 21
1.4 Hours of work and the supply of labour ..................................................................... 22
1.5 The effects of wages on labour supply ....................................................................... 26
1.5.1 Effects on those already participating in the labour market ................................ 26
1.5.2 Effect on the labour-market participation decision ............................................. 27
1.5.3 The labour supply curve ...................................................................................... 28
1.5.4 Do higher tax rates raise more revenue? ............................................................ 30
1.6 Equilibrium and efficiency ......................................................................................... 31
1.6.1 A static macroeconomic model ........................................................................... 31
1.6.2 Equilibrium in labour and goods markets ............................................................ 31
1.6.3 Economic efficiency ............................................................................................ 32
Box 1.4: Should wages or rents be taxed to pay for public expenditure? ..................... 34
1.6.4 Taxing wages but not rents ................................................................................. 35
1.6.5 Taxing rents but not wages ................................................................................. 35
1.7 Capital accumulation ................................................................................................. 36
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EC2065 Macroeconomics
1.8 The Solow model ....................................................................................................... 38
1.8.1 The Solow diagram ............................................................................................. 39
Box 1.5: The ‘Asian tiger’ economies ........................................................................... 41
Box 1.6: Interest rates in the long run .......................................................................... 44
Chapter 2: Economic growth ............................................................................................... 46
2.1 Evidence on economic growth ................................................................................... 46
2.1.1 Measuring economic growth .............................................................................. 47
2.1.2 Why growth matters ........................................................................................... 47
2.1.3 Economic growth in historical perspective .......................................................... 48
2.1.4 The distribution of income across countries........................................................ 50
2.1.5 Convergence ....................................................................................................... 50
2.2 Income and growth rates across countries ................................................................ 52
Box 2.1: Can the Solow model explain large income differences across countries? ...... 54
Box 2.2: How long does convergence to the steady state take in the Solow model? .... 56
2.3 Technological progress .............................................................................................. 57
2.3.1 A constant growth rate of technology ................................................................. 58
2.4 International flows of investment .............................................................................. 60
Box 2.3: Institutions and income differences across countries ..................................... 61
2.5 The golden rule ......................................................................................................... 62
2.5.1 Finding the golden rule ....................................................................................... 63
2.5.2 Transitional paths to the golden rule .................................................................. 65
2.5.3 Testing for dynamic inefficiency.......................................................................... 66
Box 2.4: Climate change and the economy .................................................................. 67
2.6 The AK model ............................................................................................................ 68
2.6.1 The AK production function ................................................................................ 68
2.6.2 Endogenous growth ............................................................................................ 70
Box 2.5: Endogenous growth and divergence between countries ................................ 70
2.7 Learning by doing ...................................................................................................... 72
2.8 Human capital ........................................................................................................... 73
2.9 Research and development ....................................................................................... 75
2.9.1 Non-rivalrous but excludable technologies ......................................................... 76
2.9.2 A production function for ideas .......................................................................... 77
2.9.3 Endogenous growth ............................................................................................ 77
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EC2065 Macroeconomics
2.9.4 Constant returns or diminishing returns to R&D? ............................................... 78
2.10 International technology transfer ............................................................................ 78
Box 2.6: How strong should intellectual property rights be? ........................................ 80
Chapter 3: Aggregate demand............................................................................................. 82
3.1 Detrending macroeconomic data .............................................................................. 82
3.1.1 Business cycles.................................................................................................... 82
3.1.2 Detrending.......................................................................................................... 83
3.1.3 Business-cycle stylised facts ................................................................................ 84
3.2 Consumption ............................................................................................................. 86
3.3 A two-period consumption model ............................................................................. 88
3.3.1 Preferences ........................................................................................................ 88
3.3.2 Budget constraint ............................................................................................... 89
3.3.3 Choice of an optimal consumption plan .............................................................. 92
3.4 Bonds, yields, and interest rates ................................................................................ 96
3.5 Interest rates and consumption ................................................................................. 96
Box 3.2: Durables, non-durables, and services ........................................................... 100
3.6 Consumption smoothing in the aggregate ............................................................... 101
Box 3.3: Supply disruptions and real interest rates .................................................... 104
3.8 A two-period model of investment .......................................................................... 106
3.8.1 The production function ................................................................................... 106
3.8.2 Capital accumulation ........................................................................................ 106
3.8.3 Firms’ profits .................................................................................................... 106
3.8.4 Options for financing investment ...................................................................... 107
3.8.5 The optimal investment decision ...................................................................... 108
3.8.6 Does the source of financing matter? ............................................................... 109
3.8.7 The investment demand curve.......................................................................... 109
3.9 The stock market ..................................................................................................... 110
Box 3.4: Stock prices and firms’ investment decisions ............................................... 111
Box 3.5: Should capital be taxed? .............................................................................. 112
3.10 Labour supply over time ........................................................................................ 114
3.11 A dynamic macroeconomic model ......................................................................... 116
3.11.1 A representative household ............................................................................ 116
3.11.2 Firms............................................................................................................... 117
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EC2065 Macroeconomics
3.11.3 Government ................................................................................................... 118
3.12 General equilibrium ............................................................................................... 118
Box 3.6: Growth slowdowns and real interest rates ................................................... 121
Chapter 4: Fiscal policy and credit-market imperfections .................................................. 123
4.1 Taxes and the government’s budget constraint ....................................................... 123
4.2 Ricardian equivalence .............................................................................................. 126
Box 4.2: The effects of a fiscal stimulus ..................................................................... 131
4.3 Credit-market imperfections ................................................................................... 134
Box 4.3: Bequests and intergenerational redistribution ............................................. 136
4.4 Interest-rate spreads ............................................................................................... 139
4.5 Asymmetric information .......................................................................................... 142
Box 4.4: Does the current profitability of firms matter for investment?..................... 143
Box 4.5: Financial crises ............................................................................................. 144
4.6 Limited commitment ............................................................................................... 146
Box 4.6: Interest rates and the value of housing collateral......................................... 149
4.7 Overlapping generations ......................................................................................... 151
Box 4.7: Pay-as-you-go pension systems.................................................................... 153
Box 4.8: Should pensions be fully funded? ................................................................. 155
Box 4.9: Declining population growth rates and pay-as-you-go pensions ................... 157
Box 4.10: Bubbles in financial markets....................................................................... 159
Box 4.11: Does the government have a budget constraint when interest rates are low?
.................................................................................................................................. 161
Chapter 5: Unemployment ................................................................................................ 163
5.1 Introduction to unemployment ............................................................................... 163
5.2 Efficiency wages ...................................................................................................... 166
Box 5.1: Changes in firms’ ability to monitor workers ................................................ 170
5.3 Search and matching in the labour market .............................................................. 171
5.4 A model of job search .............................................................................................. 172
5.5 Stocks and flows in the labour market ..................................................................... 175
5.51 Stock-flow accounting........................................................................................ 175
5.52 The equilibrium unemployment rate ................................................................. 176
5.53 Unemployment in the search-and-matching model ........................................... 177
Box 5.2: The generosity of the welfare state.............................................................. 178
Box 5.3: Wage dispersion and incentives to search for jobs ....................................... 179
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EC2065 Macroeconomics
5.6 Vacancies and unemployment ................................................................................. 180
5.6.1 An equilibrium search model of unemployment ............................................... 181
5.6.2 The matching function ...................................................................................... 182
5.6.3 Explaining the Beveridge curve ......................................................................... 183
5.6.4 Market tightness............................................................................................... 184
5.7 Wage bargaining...................................................................................................... 184
5.8 Job creation ............................................................................................................. 186
Box 5.4: Mismatch ..................................................................................................... 188
Box 5.6: ‘Furlough’ policies in the COVID pandemic ................................................... 189
Box 5.5: The bargaining power of workers ................................................................. 189
Chapter 6: Money ............................................................................................................. 191
6.1 Why does money matter? ....................................................................................... 191
6.1.1 Medium of exchange ........................................................................................ 191
6.1.2 Unit of account ................................................................................................. 192
6.1.3 Different objects that serve as money .............................................................. 192
6.2 A search-theory perspective on money ................................................................... 192
6.2.1 A simple search model of money ...................................................................... 193
6.2.1 Commodity money ........................................................................................... 193
6.2.2 Credit money .................................................................................................... 194
6.2.3 Money and credit ............................................................................................. 195
6.2.4 Fiat money ........................................................................................................ 196
Box 6.1: Cryptocurrencies .......................................................................................... 198
6.3 Money and assets as stores of value ........................................................................ 199
6.3.1 Inflation ............................................................................................................ 199
6.3.2 The Fisher equation .......................................................................................... 200
6.3.3 Ex-ante and ex-post interest rates .................................................................... 201
6.3.4 The opportunity cost of holding money ............................................................ 201
6.3.5 Real and nominal interest rates ........................................................................ 202
6.4 The demand for money ........................................................................................... 203
6.4.1 Economising on holding money ........................................................................ 203
6.4.2 Alternatives to money ...................................................................................... 205
6.4.3 The money demand function ............................................................................ 205
6.5 Money and economic activity .................................................................................. 207
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EC2065 Macroeconomics
6.6 The supply of money ............................................................................................... 208
6.7 Money and prices .................................................................................................... 210
Box 6.2: The instability of money demand ................................................................. 211
6.8 Money and public finance ....................................................................................... 213
6.9 Does monetary policy matter?................................................................................. 216
Box 6.3: Money supply increases that the central bank announces are temporary .... 218
6.10 Optimal monetary policy and the costs of inflation................................................ 220
Box 6.4: Hyperinflations ............................................................................................ 222
Box 6.5: Cash and tax evasion .................................................................................... 223
6.11 Conducting monetary policy by setting interest rates ............................................ 224
6.12 Taylor rules and the Taylor principle ...................................................................... 226
6.13 The liquidity trap and the zero lower bound .......................................................... 227
Box 6.6: A deflation trap ............................................................................................ 228
6.14 Negative nominal interest rates............................................................................. 229
Chapter 7: Banking and finance ......................................................................................... 232
7.1 Fractional reserve banking....................................................................................... 232
7.2 The tools of monetary policy ................................................................................... 234
7.2.1 Open-market operations .................................................................................. 234
7.2.2 Standing facilities.............................................................................................. 235
7.2.3 Reserve requirements....................................................................................... 235
7.3 The interbank market .............................................................................................. 236
7.3.1 The demand for reserves .................................................................................. 236
7.3.2 Equilibrium in the interbank market ................................................................. 237
Box 7.1: The ‘channel’ system of monetary policy ..................................................... 239
Box 7.2: The ‘floor’ system of monetary policy .......................................................... 240
7.4 The supply of bank deposits .................................................................................... 241
7.4.1 Costs of maintaining adequate reserves ........................................................... 242
7.4.2 Bank capital requirements ................................................................................ 243
7.5 Equilibrium in the banking market ........................................................................... 243
Box 7.3: Should central banks pay interest on reserves?............................................ 244
Seigniorage revenue and the profitability of the central bank ................................... 246
7.6 Bond maturity and the yield curve .......................................................................... 246
7.7 The expectations theory of long-term interest rates ................................................ 248
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EC2065 Macroeconomics
Box 7.4: Forecasting from the shape of the yield curve.............................................. 249
7.8 Risk and portfolio choice ......................................................................................... 250
Box 7.5: The typical shape of the yield curve ............................................................. 253
7.9 The functions of banks............................................................................................. 254
7.9.1 The Diamond-Dybvig model .............................................................................. 255
7.9.2 An economy with no financial intermediaries ................................................... 256
7.10 Banking as maturity transformation ...................................................................... 256
7.10.1 Bank deposits ................................................................................................. 256
7.10.2 Competition between banks ........................................................................... 257
7.10.3 The equilibrium deposit contract offered by banks ......................................... 258
7.11 Bank runs .............................................................................................................. 260
7.11.1 Strategic demands for withdrawals and bank failures ..................................... 260
7.11.2 Multiple equilibria and the possibility of bank runs ......................................... 261
7.11.3 The 2007 Northern Rock bank run .................................................................. 261
7.11.4 The ‘shadow’ banking system and the 2008 financial crisis ............................. 261
7.12 Deposit insurance and bank regulation .................................................................. 261
7.12.1 Deposit insurance ........................................................................................... 262
7.12.2 The central bank as ‘lender of last resort’ ....................................................... 262
7.12.3 Bank capital requirements .............................................................................. 262
7.12.4 Reserve requirements ..................................................................................... 263
Box 7.6: The 100 per cent reserve requirements ....................................................... 263
Box 7.7: Central-bank digital currency ....................................................................... 265
Chapter 8: Business cycles................................................................................................. 266
8.1 Nominal rigidity ....................................................................................................... 266
8.2 The new Keynesian model ....................................................................................... 268
8.3 The real effects of monetary policy.......................................................................... 271
Box 8.1: The Volcker disinflation................................................................................ 272
8.4 Business cycles due to demand shocks .................................................................... 273
Box 8.2: Can the new Keynesian model match the business-cycle stylised facts?....... 274
Box 8.3: Labour hoarding ........................................................................................... 280
8.5 The natural rate of interest...................................................................................... 281
8.5.1 Imperfect competition and the output supply curve ......................................... 281
8.5.2 Market clearing in the absence of nominal rigidities ......................................... 282
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EC2065 Macroeconomics
8.5.3 The long run and the short run ......................................................................... 283
8.6 Optimal stabilisation policy...................................................................................... 283
Box 8.4: Modelling monetary policy using Taylor rules and LM curves ....................... 285
8.6.1 Money supply targets, the LM curve and the IS-LM model................................ 286
8.6.2 The Taylor rule .................................................................................................. 287
8.7 Real business cycle theory ....................................................................................... 288
8.8 Business cycles due to supply shocks ....................................................................... 289
8.8.1 Supply shocks ................................................................................................... 289
8.8.2 The predictions of the RBC model ..................................................................... 290
8.8.3 Stabilisation policy? .......................................................................................... 291
Box 8.5: Sources of supply shocks in the RBC model .................................................. 291
8.9 Coordination failure model ...................................................................................... 293
8.9.1 Labour productivity spillover across firms ......................................................... 293
8.9.2 Implications for the output supply curve .......................................................... 295
8.10 Multiple equilibria and business cycles .................................................................. 296
Box 8.6: The strength of strategic complementarities................................................ 299
Chapter 9: Inflation, expectations and macroeconomic policy........................................... 300
9.1 Inflation and the Phillips curve ................................................................................ 300
9.1.1 Firms’ incentives to adjust prices ...................................................................... 300
9.1.2 Price changes and economic activity ................................................................. 301
9.1.3 Expectations ..................................................................................................... 302
9.1.4 The Phillips curve .............................................................................................. 302
9.1.5 Inflation and unemployment ............................................................................ 303
9.2 Expectations and aggregate demand ....................................................................... 303
9.2.1 Inflation expectations and real interest rates .................................................... 303
9.2.2 Expectations of the economy’s future GDP ....................................................... 304
9.3 Aggregate demand with market imperfections ........................................................ 304
9.3.1 Consumption and aggregate demand ............................................................... 305
9.3.2 Consumption and aggregate demand with credit-market imperfections........... 306
Box 9.1: Multiplier and crowding-out effects of fiscal policy ...................................... 307
Box 9.2: Asset prices and the financial accelerator .................................................... 308
Box 9.3: The 2008 financial crisis ............................................................................... 309
9.4 Inflation, aggregate demand and monetary policy................................................... 310
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EC2065 Macroeconomics
9.5 The costs of inflation ............................................................................................... 311
9.5.1 Money being a poor store of value ................................................................... 311
9.5.2 Menu costs and relative-price distortions ......................................................... 311
9.5.3 Inflation and redistribution ............................................................................... 311
Box 9.4: Inflation targeting ........................................................................................ 312
9.6 Time inconsistency .................................................................................................. 314
Box 9.5: The inflation bias problem ........................................................................... 314
9.7 Unconventional monetary policy at the interest-rate lower bound ......................... 316
9.7.1 Quantitative easing........................................................................................... 316
9.7.2 Forward guidance ............................................................................................. 318
Box 9.6: Inflation targeting and the interest-rate lower bound problem .................... 319
Box 9.7: Forward guidance and confidence ............................................................... 320
Box 9.8: Negative interest rate policies...................................................................... 321
Chapter 10: Open-economy macroeconomics ................................................................... 322
10.1 International trade in goods and assets ................................................................. 322
10.2 Gains from trade in assets ..................................................................................... 324
Box 10.1: International risk sharing ........................................................................... 328
Box 10.2: The ‘twin deficits’....................................................................................... 330
10.3 Sovereign default................................................................................................... 331
10.4 Open-economy real dynamic model ...................................................................... 334
10.4.1 Balance-of-payments equilibrium, capital flows and net exports .................... 334
10.4.2 Examples ........................................................................................................ 336
Box 10.3: Global imbalances ...................................................................................... 337
Box 10.4: Capital controls .......................................................................................... 339
10.5 The terms of trade ................................................................................................. 340
10.6 Exchange rates ...................................................................................................... 341
10.7 Exchange-rate regimes .......................................................................................... 343
10.7.1 Intervention in the foreign-exchange market .................................................. 343
10.7.2 A shock to foreign prices ................................................................................. 344
10.7.3 Monetary policy autonomy with a flexible exchange rate ............................... 345
10.8 Open-economy sticky-price model ........................................................................ 346
10.8.1 Competitiveness and output demand ............................................................. 346
10.8.2 Balance-of-payments equilibrium and uncovered interest parity .................... 347
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10.8.3 Monetary policy and the exchange-rate regime .............................................. 348
Box 10.5: The trilemma ............................................................................................. 350
Box 10.6: Currency crises ........................................................................................... 352
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EC2065 Macroeconomics | General information
General information
Module level: 5
Credit: 30
Notional study time: 300 hours
Summary
This course will cover the fundamental principles of macroeconomics at an
intermediate level. Topics include the supply side of the economy and economic
growth, the demand side of the economy, consumption, investment, fiscal policy, the
labour market and unemployment, financial markets, banking, money and monetary
policy, business cycles, inflation, and international macroeconomics.
Conditions
Prerequisite: If taken as part of a BSc degree, the following course(s) must be
passed before this course may be attempted:
EC1002 Introduction to economics and
Either MT105A Mathematics 1 (half course) or MT1174 Calculus or MT1186
Mathematical methods).
Aims and objectives
This course aims to bring you up to date with modern developments in
macroeconomics and to help you analyse the macroeconomic issues of the day.
Learning outcomes
At the end of the course and having completed the essential reading and activities,
you should be able to think about and give answers to key macroeconomic
questions, for example:
•
What are the forces that drive long-term prosperity?
•
Is a growth slowdown in emerging economies inevitable?
•
Why are real interest rates so low?
•
What causes bubbles in financial markets?
•
Does the government have a budget constraint?
•
How does the labour market respond to structural change and shifting
employment patterns?
•
What is the role of banks and why are they inherently fragile?
•
Is it a good idea for central banks to set up new digital currencies?
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EC2065 Macroeconomics | General information
•
Why does economic activity fluctuate?
•
Can and should policymakers seek to ameliorate business cycles?
•
What options do central banks have when nominal interest rates fall to zero?
•
What are the causes of global imbalances?
The approach of the course is to discuss the salient features of the data and then go
on to present macroeconomic models to study these questions.
Essential reading
The textbook for the course is:
Williamson, S.D. Macroeconomics. (London: Pearson, 2018) 6th edition
[ISBN 9780134472119].
Detailed reading references in this subject guide refer to the editions of the set
textbooks listed above. New editions of one or more of these textbooks may have
been published by the time you study this course. You can use a more recent edition
of any of the books; use the detailed chapter and section headings and the index to
identify relevant readings. Also check the virtual learning environment (VLE)
regularly for updated guidance on readings.
Online study resources
In addition to the subject guide and the Essential reading, it is crucial that you take
advantage of the study resources that are available online for this course, including
the VLE and the Online Library.
You can access the VLE, the Online Library and your University of London email
account via the Student Portal.
You should have received your login details for the Student Portal with your official
offer, which was emailed to the address that you gave on your application form. You
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If you have forgotten these login details, please click on the ‘Forgotten your
password’ link on the login page.
The VLE
The VLE, which complements this subject guide, has been designed to enhance
your learning experience, providing additional support and a sense of community. It
forms an important part of your study experience with the University of London and
you should access it regularly.
The VLE provides a range of resources for EMFSS courses:
•
Course materials: Subject guides and other course materials available for
download. In some courses, the content of the subject guide is transferred
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EC2065 Macroeconomics | General information
•
•
•
•
•
•
•
•
•
into the VLE and additional resources and activities are integrated with the
text.
Readings: Direct links, wherever possible, to essential readings in the Online
Library, including journal articles and ebooks.
Video content: Including introductions to courses and topics within courses,
interviews, lessons and debates.
Screencasts: Videos of PowerPoint presentations, animated podcasts and
on-screen worked examples.
External material: Links out to carefully selected third-party resources.
Self-test activities: Multiple-choice, numerical and algebraic quizzes to
check your understanding.
Collaborative activities: Work with fellow students to build a body of
knowledge.
Discussion forums: A space where you can share your thoughts and
questions with fellow students. Many forums will be supported by a ‘course
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clarify difficult topics.
Past examination papers: We provide up to three years of past
examinations alongside Examiners’ commentaries that provide guidance on
how to approach the questions.
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Note: Students registered for Laws courses also receive access to the dedicated
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Some of these resources are available for certain courses only, but we are
expanding our provision all the time and you should check the VLE regularly for
updates.
Making use of the Online Library
The Online Library contains a huge array of journal articles and other resources to
help you read widely and extensively.
To access the majority of resources via the Online Library you will either need to use
your University of London Student Portal login details, or you will be required to
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The easiest way to locate relevant content and journal articles in the Online Library is
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If you are having trouble finding an article listed in a reading list, try removing any
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For further advice, please use the online help pages or contact the Online Library
team.
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EC2065 Macroeconomics | General information
Syllabus
Topic 1: The supply side of the economy
Contents: Factors of production (labour, land, and capital), the production function
and the supply side of the economy, the distribution of income between factors
(wages, rents, and interest), population growth, labour-market participation and
labour supply, taxation, competitive equilibrium and efficiency, the Solow model.
Questions to address
•
Why was economic growth so different before and after the Industrial
Revolution?
•
Why have hours worked declined in advanced economies even though wages
are so much higher than in the past?
•
Why has wage inequality increased in recent decades?
•
Should land-value taxes be used?
•
Is a growth slowdown in emerging economies inevitable?
Topic 2: Economic growth in the long run
Contents: Evidence on economic growth and the income distribution across
countries, convergence, saving rates and the Golden rule, technological progress,
international investment flows, institutions and misallocation, endogenous growth
theory, learning by doing, human capital, research and development, diffusion of
knowledge between countries.
Questions to address
•
What are the forces that drive long-run prosperity?
•
Are we saving enough for the future?
•
Why is the gap between rich and poor countries so large?
•
How strong should intellectual property rights be?
•
What are the implications of climate change for the economy?
Topic 3: The demand side of the economy
Contents: Evidence on macroeconomic fluctuations using detrended data,
consumption, the relationship between consumption and income, durable and nondurable goods and services, interest rates and saving, bond yields, determinants of
real interest rates, investment, the stock market, a dynamic macroeconomic model.
Questions to address
•
Why are purchases of capital goods by firms very volatile while purchases of
services by households are more stable?
•
Why are real interest rates so low?
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EC2065 Macroeconomics | General information
•
Do low interest rates discourage saving and stimulate consumption?
•
Should capital and wealth be taxed to fund public expenditure instead of
wages?
•
Is there a link between the stock market and the amounts invested by firms?
Topic 4: Credit-market imperfections and fiscal policy
Contents: The government budget constraint, Ricardian equivalence, taxes on
consumption, borrowing constraints, interest-rate spreads, asymmetric information
between borrowers and lenders, limited commitment and collateral, overlapping
generations, pension systems.
Questions to address
•
Can tax cuts stimulate demand?
•
Why do house prices affect the economy?
•
How does a financial crisis affect households’ and firms’ spending decisions?
•
Should the government be involved in providing pensions or leave it to the
market?
•
How do demographics affect the pension system?
•
What causes bubbles in financial markets?
•
Does the government have a budget constraint?
Topic 5: Unemployment, vacancies, and wages
Contents: Unemployment and wage rigidity, efficiency wages, the process of search
and matching, wage dispersion and the reservation wage, labour-market flows,
vacancies and the Beveridge curve, wage bargaining, job creation, and labourmarket tightness.
Questions to address
•
Why does unemployment occur, even when job vacancies are unfilled?
•
How does the labour market respond to structural change and shifting
employment patterns?
•
What are the effects of labour-market institutions such as unemployment
insurance?
•
What was the role of job-support schemes during the Covid pandemic?
Topic 6: Money and monetary policy
Contents: The nature and functions of money, money’s role as a medium of
exchange with search and matching, inflation and interest rates, the demand for
money and credit as a substitute for money, the fiat monetary system, the effects of
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EC2065 Macroeconomics | General information
monetary policy on inflation and economic activity, controlling inflation using Taylor
rules, the lower bound on nominal interest rates.
Questions to address
•
Why has the link between money supply growth and inflation been unstable?
•
What advantages do governments derive from being able to create money?
•
What are hyperinflations and why are they so damaging?
•
How should central banks conduct monetary policy?
•
Is it better to have a monetary system without physical cash?
•
Can and should nominal interest rates ever be negative?
•
How do cryptocurrencies differ from existing forms of money?
Topic 7: Banking, finance, and the money markets
Contents: Fractional reserve banking, the tools of monetary policy and how policy
decisions are implemented, the interbank market, reserves and deposit creation by
commercial banks, bonds and the yield curve, the expectations theory of long-term
interest rates, risk and portfolio choice, banking and maturity transformation, bank
runs.
Questions to address
•
How do central banks control interest rates?
•
What information can we learn from the shape of the yield curve?
•
What is the role of banks and why are they inherently fragile?
•
How should the banking system be regulated?
•
Would the monetary system work better if commercial banks were prevented
from creating money by imposing 100 per cent reserve requirements?
•
Is it a good idea for central banks to set up new digital currencies?
Topic 8: Nominal rigidities and business cycles
Contents: Sticky prices, the New Keynesian model, the real effects of monetary
policy, business cycles and the role of stabilization policy, demand and supply
shocks, real business cycle theory, evidence on business-cycle fluctuations,
coordination failure and multiple equilibria.
Questions to address
•
Why does economic activity fluctuate?
•
What are the shocks that cause booms and recessions?
•
Can and should policymakers seek to ameliorate business cycles?
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EC2065 Macroeconomics | General information
•
Can changes in confidence be the driving force of business cycles?
Topic 9: Inflation, expectations, and macroeconomic policy
Contents: Inflation and the Phillips curve, the role of expectations, costs of inflation,
time inconsistency and the inflation bias, inflation targeting, aggregate demand
multipliers and the effectiveness of fiscal policy, unconventional monetary policy at
the interest-rate lower bound, quantitative easing and forward guidance.
Questions to address
•
Can a fiscal stimulus raise GDP by more than the extra government
spending?
•
Should central banks prioritise controlling inflation or focus on trying to
stabilise fluctuations in real GDP?
•
What options do central banks have when nominal interest rates fall to zero?
•
What is forward guidance, and how effective is it as a monetary policy tool?
•
Should inflation targeting be reformed, or abandoned and replaced by targets
for the price level or nominal GDP?
Topic 10: International trade in goods and assets
Contents: The balance of payments, gains from trade in assets, sovereign default
and limits on international lending, determinants of the current account, exchange
rates and exchange-rate regimes, purchasing power parity, the terms of trade,
uncovered interest parity, capital mobility and capital controls, the trilemma.
Questions to address
•
What are the causes of ‘global imbalances’?
•
Do government budget deficits cause current-account deficits?
•
Are monetary and fiscal policy still effective in an open economy where capital
can flow freely?
•
Should exchange rates be fixed or left to float?
•
What are the causes of currency crises and the collapse of fixed exchangerate systems?
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EC2065 Macroeconomics | Chapter 1: The supply side of the economy
Chapter 1: The supply side of the economy
We start by studying the supply side of the economy. This means the economy’s capacity to
produce goods and services. We will defer consideration of demand-side issues until
Chapter 3. In this chapter, we will introduce a basic supply-side theory of gross domestic
product (GDP). GDP is a measure both of production and of income, so our model is a
starting point in understanding how much an economy is able to produce and how much
people within the economy are able to earn.
Using the supply-side model, we will look at what explains the level of GDP and how the
economy’s total income is distributed. This model serves as the foundation for our study of
economic growth in Chapter 2, where we will explore why there is growth in GDP per
person over time and why the level of GDP per person differs so much across countries.
Essential reading
•
Williamson, Chapters 4, 5 and 7.
1.1 Production functions and factors of production
The supply-side model of an economy’s GDP has two ingredients: first, the quantities of
factors of production available and second, the production function.
1.1.1 Factors of production
Factors of production are basic inputs into the production process for goods and services.
Here, we consider three factors:
1. Labour
2. Land
3. Capital.
Capital refers to goods used to produce other goods in the future, for example, machinery,
tools, buildings, computers, vehicles. In our model, suppose that capital goods are
homogeneous and the capital stock, denoted 𝐾𝐾, is the quantity of units of capital available
to use for production. The quantity of capital can be increased by producing new capital
goods and this process of capital accumulation is described in Section 1.7. For now, the basic
supply-side model takes as given the available supplies of each of the factors of production,
including capital.
Land refers to physical space needed to produce goods and services, for example, farmland,
or a city-centre site for a shopping mall. An essential characteristic of land is that it is in fixed
supply. Even when we allow for the supplies of labour and capital to change over time in
later models, the quantity of land remains fixed. Treating units of land as homogeneous for
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EC2065 Macroeconomics | Chapter 1: The supply side of the economy
simplicity, we denote the fixed quantity by 𝐿𝐿. Natural resources can also be included as part
of land, broadly interpreted.
Labour as a factor of production refers to work done by people to produce goods and
services. The supply of labour to the economy depends on how many workers there are and
the number of hours worked by each person. To begin with, we will treat labour as
homogeneous and denote the quantity by 𝑁𝑁 that, depending on the context, could be
measured as a number of people or a number of hours worked. More broadly, we could also
consider the skills, training and experience of the workforce as part of what determines the
supply of labour, although these are often counted as a separate factor of production
known as ‘human capital’, considered in Chapter 2.
1.1.2 The production function
The production function describes how the factors of production are combined to produce
the output of final goods and services. The production function is a high-level summary of
the production process, which may involve many stages and many intermediate inputs that
are not considered explicitly. The production function is the link between the basic inputs of
the factors of production and the output of final goods and services.
The economy’s real GDP 𝑌𝑌 measures output of final goods and services. In practice, this
comprises a large number of different products that are aggregated using their relative
prices. However, for much of the time we will simplify matters by assuming there is just a
single homogeneous good, or a stable basket of goods, produced in the economy.
The production function describes how output 𝑌𝑌 is produced:
𝑌𝑌 = 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝐿𝐿, 𝑁𝑁)
Assuming factors are fully employed, factor supplies 𝐾𝐾, 𝐿𝐿, and 𝑁𝑁 are inputs to production
function 𝐹𝐹 (𝐾𝐾, 𝐿𝐿, 𝑁𝑁). In addition to the factor supplies, output also depends on the overall
productivity of the factors. This is the notion of total factor productivity (TFP), which is
represented in the production function by the coefficient 𝑧𝑧. The value of 𝑧𝑧 can represent the
level of technology, or how efficiently factors of production are allocated to their best uses.
The production function, together with the factor supplies and TFP, determine real GDP 𝑌𝑌.
We typically make some conventional assumptions about the function 𝐹𝐹(𝐾𝐾, 𝐿𝐿, 𝑁𝑁).
Constant returns to scale
The first assumption is that the production function has constant returns to scale. If the
economy were able to double its supplies of all factors of production then it should be able
to double its output. More generally, scaling all inputs of factors of production scales output
in same way. The usual justification for assuming constant returns to scale is the ‘replication
principle’. By using the same production techniques and technologies with the additional
factors organised in the same way as the original factors, it should be possible to make
double the original amount of output if all factors have been accounted for.
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EC2065 Macroeconomics | Chapter 1: The supply side of the economy
Positive, but diminishing, marginal products
The second assumption is that increasing the supply of one factor of production without
changing the amounts of the other factors available raises output but less than
proportionately. The extra output produced by an extra unit of one factor is the marginal
product of that factor. Mathematically, a factor’s marginal product is the partial derivative
of the production function with respect to that factor. For example, the marginal product of
capital, denoted by 𝑀𝑀𝑃𝑃𝐾𝐾 , is:
𝑀𝑀𝑃𝑃𝐾𝐾 =
𝜕𝜕𝜕𝜕
= 𝑧𝑧𝐹𝐹𝐾𝐾 (𝐾𝐾, 𝐿𝐿, 𝑁𝑁)
𝜕𝜕𝜕𝜕
Stated in terms of marginal products, the second assumption is that each factor’s marginal
product is positive but diminishes as the supply of that factor increases. Mathematically,
𝑀𝑀𝑃𝑃𝐾𝐾 is positive but declines as 𝐾𝐾 increases. The same assumptions are made for the
marginal products of land 𝑀𝑀𝑃𝑃𝐿𝐿 = 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 and labour 𝑀𝑀𝑃𝑃𝑁𝑁 = 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕.
Why are factors’ marginal products decreasing? Consider, for example, the use of capital in
an office job. Without a computer, it is very difficult to perform many tasks. Giving an office
worker a computer has a large effect on that worker’s output compared to no computer, so
marginal product of capital initially high. And while a more powerful computer may allow
the worker to produce more, given the nature of the task performed by the worker, extra
computing power is unlikely to raise the worker’s output proportionately. This means that
the marginal product of capital declines.
It is important to note that this argument holds fixed the number of workers, their skills and
state of technology. If the extra computing power were allocated to an additional worker, or
used by workers with higher skills to perform more advanced tasks then there is no
presumption that output will not rise proportionately.
Inada conditions
The third assumption on the production function is known as the ‘Inada conditions’. These
are essentially a stronger version of the assumption of diminishing marginal products.
Rather than just requiring the marginal product of a factor declines as the use of that factor
increases, the Inada conditions require that the marginal product declines all the way to
zero. Similarly, the Inada conditions require that the marginal product of a factor is initial
very high (mathematically, it is said to approach infinity) when the usage of the factor is
close to zero. One consequence of the Inada conditions is that some positive amount of
each factor of production is essential to produce any output.
1.1.3 Neoclassical production functions
In summary, the usual assumptions we make about production functions are:
•
Constant returns to scale:
𝐹𝐹 (𝑠𝑠𝑠𝑠, 𝑠𝑠𝑠𝑠, 𝑠𝑠𝑠𝑠) = 𝑠𝑠𝑠𝑠(𝐾𝐾, 𝐿𝐿, 𝑁𝑁)
A scaling of inputs of all factors of production by 𝑠𝑠 (for example, 𝑠𝑠 = 2 is doubling
inputs) scales output by 𝑠𝑠.
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EC2065 Macroeconomics | Chapter 1: The supply side of the economy
•
Positive but diminishing marginal products of factors:
𝜕𝜕𝜕𝜕
𝐹𝐹𝐾𝐾 (𝐾𝐾, 𝐿𝐿, 𝑁𝑁) =
>0
𝜕𝜕𝜕𝜕
𝜕𝜕 2 𝑌𝑌
𝐹𝐹𝐾𝐾𝐾𝐾 (𝐾𝐾, 𝐿𝐿, 𝑁𝑁) =
<0
𝜕𝜕𝐾𝐾 2
,
These are for capital. The same assumptions hold for other factors. For land, 𝐹𝐹𝐿𝐿 > 0 and
𝐹𝐹𝐿𝐿𝐿𝐿 < 0, and for labour, 𝐹𝐹𝑁𝑁 > 0 and 𝐹𝐹𝑁𝑁𝑁𝑁 < 0.
•
Inada conditions:
lim 𝐹𝐹𝐾𝐾 (𝐾𝐾, 𝐿𝐿, 𝑁𝑁) = ∞
,
𝐾𝐾→0
lim 𝐹𝐹𝐾𝐾 (𝐾𝐾, 𝐿𝐿, 𝑁𝑁) = 0
𝐾𝐾→∞
These are for capital. The same assumptions hold in terms of 𝐹𝐹𝐿𝐿 and 𝐹𝐹𝑁𝑁 for land and
labour.
A production function that satisfies all three assumptions is called a ‘neoclassical production
function’. Figure 1.1 plots the relationship between output 𝑌𝑌 and capital 𝐾𝐾 for a neoclassical
production function 𝑌𝑌 = 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝐿𝐿, 𝑁𝑁). This is not a plot of the entire production function
because land and labour are held constant at 𝐿𝐿0 and 𝑁𝑁0 but similar diagrams can be used to
show the relationship between 𝑌𝑌 and 𝐿𝐿, and 𝑌𝑌 and 𝑁𝑁, holding the other two factors fixed.
However, the constant returns to scale assumption cannot be illustrated in the diagram
because that would require changing all the factor inputs at the same time.
Figure 1.1: A neoclassical production function
The production function is upward sloping because its gradient is the marginal product of
capital, which is positive. The gradient declines as 𝐾𝐾 increases because the marginal product
of capital is diminishing. The Inada conditions imply the production function is extremely
steep for 𝐾𝐾 close to zero and flattens out as 𝐾𝐾 becomes very large. The production function
must also pass through the origin because some capital is essential for production given the
neoclassical assumptions.
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EC2065 Macroeconomics | Chapter 1: The supply side of the economy
1.1.4 The Cobb-Douglas production function
The most commonly used example of a neoclassical production function is the CobbDouglas functional form:
𝑌𝑌 = 𝑧𝑧𝐾𝐾 𝛼𝛼 𝐿𝐿𝛽𝛽 𝑁𝑁 1−𝛼𝛼−𝛽𝛽
The parameters 𝛼𝛼 and 𝛽𝛽 each lie between 0 and 1, and the sum 𝛼𝛼 + 𝛽𝛽 is less than 1. The
Cobb-Douglas production function satisfies the three neoclassical assumptions. First, it has
constant returns to scale because:
𝑧𝑧(𝑠𝑠𝑠𝑠 )𝛼𝛼 (𝑠𝑠𝐿𝐿)𝛽𝛽 (𝑠𝑠𝑁𝑁)1−𝛼𝛼−𝛽𝛽 = 𝑠𝑠𝑠𝑠𝐾𝐾 𝛼𝛼 𝐿𝐿𝛽𝛽 𝑁𝑁 1−𝛼𝛼−𝛽𝛽
The marginal product of capital is:
𝑀𝑀𝑃𝑃𝐾𝐾 = 𝑧𝑧𝑧𝑧𝐾𝐾 𝛼𝛼−1 𝐿𝐿𝛽𝛽 𝑁𝑁 1−𝛼𝛼−𝛽𝛽
This is positive because 𝛼𝛼 > 0. It declines as 𝐾𝐾 increases because 𝛼𝛼 < 1, so the exponent of
capital in the expression for 𝑀𝑀𝑃𝑃𝐾𝐾 is negative. It follows that the marginal product of capital
is positive but diminishing. Using the partial derivatives with respect to land and labour, the
same is true of the marginal products of land and labour.
The expression for the marginal product of capital approaches infinity as 𝐾𝐾 → 0, and
approaches zero as 𝐾𝐾 → ∞. The same can be shown for the other factors. This confirms the
Inada conditions hold for the Cobb-Douglas production function.
1.1.5 The per worker production function
In many contexts, we are interested in how much output is produced per worker, rather
than total production. This is relevant if we want to calculate living standards in an
economy, which are connected to how much is produced per person.
Here, we focus on just two factors of production, capital 𝐾𝐾 and labour 𝑁𝑁. We assume a
neoclassical production function 𝑌𝑌 = 𝑧𝑧𝑧𝑧 (𝐾𝐾, 𝑁𝑁). Output per worker, denoted by 𝑦𝑦 = 𝑌𝑌/𝑁𝑁,
can be explained in terms of capital per worker 𝑘𝑘 = 𝐾𝐾/𝑁𝑁 and TFP 𝑧𝑧.
𝑦𝑦 =
𝑌𝑌 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑁𝑁)
𝐾𝐾 𝑁𝑁
=
= 𝑧𝑧𝑧𝑧 � , � = 𝑧𝑧𝑧𝑧(𝑘𝑘, 1) = 𝑧𝑧𝑧𝑧(𝑘𝑘)
𝑁𝑁
𝑁𝑁
𝑁𝑁 𝑁𝑁
This equation is derived using the constant returns to scale property of a neoclassical
production function which implies a scaling of all factors of production (here by 1/𝑁𝑁) is
equivalent to scaling output in the same proportion. In the above, the function 𝑓𝑓(𝑘𝑘) is
simply used as a shorthand for 𝐹𝐹(𝑘𝑘, 1).
Taking the Cobb-Douglas production function 𝑌𝑌 = 𝑧𝑧𝐾𝐾 𝛼𝛼 𝑁𝑁 1−𝛼𝛼 for example:
𝑌𝑌 𝑧𝑧𝐾𝐾 𝛼𝛼 𝑁𝑁 1−𝛼𝛼
𝐾𝐾 𝛼𝛼
𝛼𝛼 −𝛼𝛼
𝑦𝑦 = =
= 𝑧𝑧𝐾𝐾 𝑁𝑁 = 𝑧𝑧 � � = 𝑧𝑧𝑘𝑘 𝛼𝛼
𝑁𝑁
𝑁𝑁
𝑁𝑁
This confirms that 𝑦𝑦 = 𝑧𝑧𝑧𝑧(𝑘𝑘) with 𝑓𝑓(𝑘𝑘) = 𝑘𝑘 𝛼𝛼 in this case.
If the production function 𝑌𝑌 = 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑁𝑁) is neoclassical, the per worker production function
has an increasing and concave shape. Observe that aggregate output is 𝑌𝑌 = 𝑧𝑧𝑧𝑧𝑧𝑧(𝐾𝐾⁄𝑁𝑁), so
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EC2065 Macroeconomics | Chapter 1: The supply side of the economy
the marginal product of capital is 𝑀𝑀𝑃𝑃𝐾𝐾 = 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 = 𝑧𝑧𝑓𝑓 ′ (𝐾𝐾⁄𝑁𝑁) = 𝑧𝑧𝑧𝑧′(𝑘𝑘). The neoclassical
assumptions state that 𝑀𝑀𝑃𝑃𝐾𝐾 is positive but diminishing in capital 𝐾𝐾, which must also hold for
capital per worker 𝑘𝑘. This implies 𝑓𝑓′(𝑘𝑘) is an increasing and concave function of 𝑘𝑘.
1.2 Factor markets and the distribution of income
The previous section shows a simple model of how the real GDP of an economy is
determined by its production technologies and its supplies of factors of production. As well
as being a measure of production, GDP is a measure of total income in an economy, so the
model also explains aggregate income. The next step is to ask how that aggregate income is
distributed among the factors of production.
Up to this point, we have not considered markets in our model of the economy, although
markets were implicit in how the factors of production were organised and allocated among
different uses. Here, markets are introduced to explain the distribution of income. We use
an analysis with firms in perfectly competitive markets for factors of production. An
example of a factor market is a market for labour, a market where firms can hire the
services of workers for a time. There are also markets for renting land and capital.
For each of the factor markets, there is a factor price. In the labour market, this is a wage 𝑤𝑤
per hour of labour, or per person working for a given amount of time. For land, there is a
rent 𝑥𝑥 and for capital there is a rental price 𝑅𝑅. We do not consider markets for outright
purchases or sales of factors of production at this stage. In the factor markets, households
supply factors of production that they own, which are hired or rented by firms. Households
supply their own labour, together with land and capital goods. In this analysis, firms do not
own factors of production themselves. As in the earlier model of production, all supplies of
factors of production are taken as given, so the supply curves are price inelastic.
On the demand side, firms hire factors of production 𝐾𝐾, 𝐿𝐿 and 𝑁𝑁 to produce output 𝑌𝑌,
taking prices and factor prices as given. The production function is 𝑌𝑌 = 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝐿𝐿, 𝑁𝑁), which
has the same properties assumed earlier. Firms aim to maximise profits 𝜋𝜋:
𝜋𝜋 = 𝑌𝑌 − 𝑅𝑅𝑅𝑅 − 𝑥𝑥𝑥𝑥 − 𝑤𝑤𝑤𝑤
These profits are measured in real terms, making the price of a unit of output equal to 1.
The revenue of the firm is simply the quantity 𝑌𝑌 of output it produces and sells. Its costs are
its spending on hiring factors of production, which are given by the factor prices multiplied
by the quantities hired of each factor.
The first-order conditions for profit maximisation are 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 = 0, 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 = 0 and
𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 = 0, which are equivalent to the following:
𝑧𝑧𝐹𝐹𝐾𝐾 (𝐾𝐾, 𝐿𝐿, 𝑁𝑁) = 𝑅𝑅 , 𝑧𝑧𝐹𝐹𝐿𝐿 (𝐾𝐾, 𝐿𝐿, 𝑁𝑁) = 𝑥𝑥 , 𝑧𝑧𝐹𝐹𝑁𝑁 (𝐾𝐾, 𝐿𝐿, 𝑁𝑁) = 𝑤𝑤
The terms 𝑧𝑧𝑧𝑧𝐾𝐾 , 𝑧𝑧𝑧𝑧𝐿𝐿 and 𝑧𝑧𝑧𝑧𝑁𝑁 denote the marginal products of capital, land and labour
respectively, where 𝐹𝐹𝐾𝐾 , 𝐹𝐹𝐿𝐿 , and 𝐹𝐹𝑁𝑁 are the partial derivatives of the function 𝐹𝐹(𝐾𝐾, 𝐿𝐿, 𝑁𝑁) with
respect to 𝐾𝐾, 𝐿𝐿, and 𝑁𝑁. These marginal products are diminishing in the quantity hired of
each factor, holding the quantities of the other factors constant. Hence, in each factor
market diagram with the factor price on the vertical axis and the quantity hired on the
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EC2065 Macroeconomics | Chapter 1: The supply side of the economy
horizontal axis, the demand curves of firms are the downward-sloping marginal product
curves. The supply curves are vertical, representing given price-inelastic supplies of each
factor.
In competitive markets, the factor prices are determined by market clearing. The
equilibrium rental prices of capital and land 𝑅𝑅∗ and 𝑥𝑥 ∗ and the equilibrium wage 𝑤𝑤 ∗ are
found where the factor demand curves intersect the supply curves. This is illustrated in the
rental market for capital in Figure 1.2. The diagram shows how the amount of real capital
income 𝑅𝑅 ∗ per unit of capital owned is determined. The total amount of capital income is
then 𝑅𝑅 ∗ 𝐾𝐾, where 𝐾𝐾 is the supply of capital. Note that this is gross capital income – there is
no allowance made here for depreciation of capital, consistent with how GDP is a measure
of gross income. Similar diagrams can be used to see how rents of land and wages are
determined.
Figure 1.2: Factor market equilibrium
While firms are maximising profits when they choose their factor demands, perfect
competition and a constant-returns-to-scale production function imply that profits 𝜋𝜋 are
zero in equilibrium. It is a general mathematical property that if 𝑧𝑧𝑧𝑧 (𝐾𝐾, 𝐿𝐿, 𝑁𝑁) has constant
returns to scale then:
𝑧𝑧𝑧𝑧 (𝐾𝐾, 𝐿𝐿, 𝑁𝑁) = 𝑧𝑧𝑧𝑧𝐹𝐹𝐾𝐾 (𝐾𝐾, 𝐿𝐿, 𝑁𝑁) + 𝑧𝑧𝑧𝑧𝐹𝐹𝐿𝐿 (𝐾𝐾, 𝐿𝐿, 𝑁𝑁) + 𝑧𝑧𝑧𝑧𝐹𝐹𝑁𝑁 (𝐾𝐾, 𝐿𝐿, 𝑁𝑁)
Intuitively, constant returns to scale implies that a 1% increase in each of the factors of
production 𝐾𝐾, 𝐿𝐿, and 𝑁𝑁 adds 1 per cent to existing output 𝑌𝑌, that is, this raises output by
0.01 × 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝐿𝐿, 𝑁𝑁). Adding 1 per cent to capital 𝐾𝐾 raises output by 0.01𝐾𝐾 × 𝑧𝑧𝑧𝑧𝐾𝐾 , where 𝑧𝑧𝑧𝑧𝐾𝐾
is the marginal product of capital. Adding 1% to labour 𝐿𝐿 would raise output by
0.01𝑁𝑁 × 𝑧𝑧𝑧𝑧𝑁𝑁 , and so on for all factors. Summing over all factors then confirms the equation
above.
Since profit-maximisation by firms in perfectly competitive markets equalises marginal
products and factor prices for each factor, it follows that
𝑌𝑌 = 𝑧𝑧𝑧𝑧 (𝐾𝐾, 𝐿𝐿, 𝑁𝑁) = 𝑅𝑅𝑅𝑅 + 𝑥𝑥𝑥𝑥 + 𝑤𝑤𝑤𝑤
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This means that payments to factors of production will use up all a firm’s revenues, so
profits 𝜋𝜋 are zero. Strictly speaking, this shows that economic profits are zero. Where firms
own factors of production themselves rather than renting them, which is often the case with
land and capital, some of the payments to factors described above effectively go the owner
of the firm. In that case, accounting profits would not be zero. However, these would not be
true economic profits but would instead represent the implicit rental of the factors of
production owned by the firm.
1.2.1 The Cobb-Douglas production function
As an example, consider the distribution of income when the production function has the
Cobb-Douglas functional form 𝑌𝑌 = 𝑧𝑧𝐾𝐾 𝛼𝛼 𝐿𝐿𝛽𝛽 𝑁𝑁 1−𝛼𝛼−𝛽𝛽 . As seen earlier, the marginal products of
capital, land and labour are 𝑀𝑀𝑃𝑃𝐾𝐾 = 𝛼𝛼𝛼𝛼⁄𝐾𝐾, 𝑀𝑀𝑃𝑃𝐿𝐿 = 𝛽𝛽𝛽𝛽 ⁄𝐿𝐿, and 𝑀𝑀𝑃𝑃𝑁𝑁 = (1 − 𝛼𝛼 − 𝛽𝛽)𝑌𝑌 ⁄𝑁𝑁.
With these being equated to the factor prices 𝑅𝑅, 𝑥𝑥, and 𝑤𝑤, each factor of production
receives a constant share of GDP as income:
𝑅𝑅𝑅𝑅
= 𝛼𝛼
𝑌𝑌
,
𝑥𝑥𝑥𝑥
= 𝛽𝛽
𝑌𝑌
,
𝑤𝑤𝑤𝑤
= 1 − 𝛼𝛼 − 𝛽𝛽
𝑌𝑌
The income shares are given by the exponents of each factor of production in the CobbDouglas formula. For example, the exponent of capital 𝐾𝐾 is 𝛼𝛼, a parameter between 0 and 1.
Total capital income 𝑅𝑅𝑅𝑅 as a fraction of GDP 𝑌𝑌 is equal to the parameter 𝛼𝛼.
Box 1.1: Understanding inequality in wages
The last few decades have seen rising income inequality within many countries. What
might explain why this has occurred? Here, we focus on inequality in wages rather than
income inequality more broadly (which would also consider capital income), or on
wealth inequality.
One important dimension of the rise in wage inequality is the increase in the relative
wages of highly skilled workers compared to those with more basic skills. While a
university or college education is not the only measure of having skills, the ‘collegewage premium’ in the USA and elsewhere has received much attention. The size of the
wage premium from attending university is crucial to the debate on the returns to
higher education. A large premium means the returns might be high even if the cost of
education has risen.
In the USA, prior to 1980, an average college-educated worker earned less than 60 per
cent extra compared to an average worker without a college education. By the 2010s,
this college-wage premium had risen to close to 100 per cent. At first glance, this is
puzzling because there has also been a substantial increase in the fraction of collegeeducated workers during that period, which rose from 20 per cent to 50 per cent. With
diminishing returns to individual factors of production, an increase in supply should
push down the factor payment, all else being equal. These observations suggest
something else must have changed after 1980.
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One hypothesis we will explore is a shift in the relative demand for workers with
different levels of skill. We separate the supply of labour into highly skilled workers 𝐻𝐻
and unskilled workers 𝑁𝑁. The supply of 𝐻𝐻 is related to the concept of ‘human capital’
studied in Section 2.8. We consider the following example of a production function 𝑌𝑌 =
𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑁𝑁, 𝐻𝐻):
𝑌𝑌 = 𝑧𝑧(𝐾𝐾 𝛼𝛼 𝑁𝑁 1−𝛼𝛼 + 𝐵𝐵𝐾𝐾 𝛼𝛼 𝐻𝐻1−𝛼𝛼 )
As with a Cobb-Douglas production function, the parameter 𝛼𝛼, a number between 0 and
1, indicates the importance of physical capital in producing goods and services. The
variable 𝑧𝑧 is total factor productivity and a change in 𝑧𝑧 affects the marginal products of
all factors of production. What is new in the production function above is 𝐵𝐵, an
exogenous variable that represents what is known as ‘skill-biased technology’. A change
in 𝐵𝐵 affects the marginal product of skilled labour 𝐻𝐻 but not the marginal product of
unskilled labour 𝑁𝑁.
The production function above resembles two Cobb-Douglas production functions that
are added together. However, it is not possible to use a standard Cobb-Douglas
production function such as 𝑌𝑌 = 𝑧𝑧𝐾𝐾 𝛼𝛼 𝐻𝐻𝛽𝛽 𝑁𝑁 1−𝛼𝛼−𝛽𝛽 in this exercise. If 𝐻𝐻 were multiplied
by a coefficient 𝐵𝐵 then this would be algebraically equivalent to a change in total factor
productivity 𝑧𝑧. It is not possible to build in skill-biased technological change with a basic
Cobb-Douglas production function.
We now apply our analysis of the distribution of income. Competitive markets imply
wages 𝑤𝑤𝐻𝐻 and 𝑤𝑤𝑁𝑁 for skilled and unskilled labour that are equal to their marginal
products:
𝑤𝑤𝑁𝑁 = 𝑀𝑀𝑀𝑀𝑁𝑁 =
𝜕𝜕𝜕𝜕
= (1 − 𝛼𝛼 )𝑧𝑧𝐾𝐾 𝛼𝛼 𝑁𝑁 −𝛼𝛼
𝜕𝜕𝜕𝜕
,
𝑤𝑤𝐻𝐻 = 𝑀𝑀𝑃𝑃𝐻𝐻 =
𝜕𝜕𝜕𝜕
= (1 − 𝛼𝛼 )𝑧𝑧𝑧𝑧𝐾𝐾 𝛼𝛼 𝐻𝐻 −𝛼𝛼
𝜕𝜕𝜕𝜕
The implications for the relative wage 𝑤𝑤𝐻𝐻 /𝑤𝑤𝑁𝑁 can be deduced from these equations:
𝑤𝑤𝐻𝐻 (1 − 𝛼𝛼 )𝑧𝑧𝑧𝑧𝐾𝐾 𝛼𝛼 𝐻𝐻 −𝛼𝛼
𝐻𝐻 −𝛼𝛼
=
=
𝐵𝐵
�
�
(1 − 𝛼𝛼 )𝑧𝑧𝐾𝐾 𝛼𝛼 𝑁𝑁 −𝛼𝛼
𝑤𝑤𝑁𝑁
𝑁𝑁
The relative wage declines with the relative supply of high-skilled labour 𝐻𝐻/𝑁𝑁 but it
increases with skill-biased technological change, that is, higher 𝐵𝐵.
This logic suggests one explanation for the rising skill premium (the relative wage
𝑤𝑤𝐻𝐻 /𝑤𝑤𝑁𝑁 increasing) alongside an increase in the relative supply of skilled labour 𝐻𝐻/𝑁𝑁 is
skill-biased technological change. Skill-biased technological change is improvements in
technology that disproportionately boost the productivity of skilled workers compared
to unskilled workers. For example, advances in computing, telecommunications, data
science and e-commerce may increase demand for highly skilled workers but not
unskilled workers. These changes can be represented by an increase in 𝐵𝐵 rather than an
increase in total factor productivity 𝑧𝑧. Earlier technological progress that may have been
more uniform in its effects is represented by higher TFP 𝑧𝑧. As we have seen, higher 𝐵𝐵
can raise 𝑤𝑤𝐻𝐻 /𝑤𝑤𝑁𝑁 even though there is an increase in the relative supply 𝐻𝐻/𝑁𝑁 of skilled
workers. Changes in TFP do not affect relative wages 𝑤𝑤𝐻𝐻 /𝑤𝑤𝑁𝑁 .
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Skill-biased technological progress is not the only explanation for the rising skill
premium. Globalisation owing to lower barriers to international trade is another
explanation. Even if labour is not mobile internationally, free trade in goods tends to
equalise the skill premium across countries through equalising the relative prices of
goods that are produced more or less intensively using skilled labour. The skill premium
is then determined by the relative supply of skilled labour at world level, where there
are relatively fewer skilled workers than within advanced economies.
1.3 Population growth according to Malthus
We now extend the basic model of production and distribution so that the supplies of
factors of production can change over time. This allows us to consider the dynamics of
aggregate GDP and individual incomes. We begin by considering the supply of labour,
interpreted as the number of workers. One reason the supply of labour adjusts over time is
because the population rises or falls. Here, we study the theory of population growth put
forward by Malthus in the 18th century and its implications for economic growth.
A key prediction of the theory is that population growth holds down living standards when
production of goods depends on land that is in fixed supply. We will see that a Malthusian
model of the economy can explain the stagnation in per-capita incomes seen in most of the
world prior to the 19th century. Technological advances lead to population growth, but not
rising living standards.
1.3.1 Demographics
Malthus in his Essay on the principle of population argued that per capita income and
consumption affect population growth. Lower consumption per person leads to worse
nutrition and health, hence higher death rates and infant mortality, and lower or negative
population growth rate. Furthermore, lower income per person induces families to have
fewer children they would struggle to support, hence lower birth rates and a lower
population growth rate. Higher income and consumption have the opposite effects and
raise the population growth rate. These effects are larger when people are close to
subsistence.
In the model, assume that all individuals are workers. The current population and number of
workers is denoted by 𝑁𝑁. The future population is denoted 𝑁𝑁′, where the notation ′ refers
to a value of a variable in the next time period (the future). The population growth rate
between the current and future time periods is (𝑁𝑁 ′ − 𝑁𝑁)/𝑁𝑁. If 𝐶𝐶 is aggregate consumption,
𝑐𝑐 = 𝐶𝐶/𝑁𝑁 measures average consumption per person, which is a measure of average living
standards. A mathematical representation of the demographics assumed by Malthus is:
𝑁𝑁 ′
= 𝑔𝑔(𝑐𝑐 )
𝑁𝑁
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The demographic function 𝑔𝑔(𝑐𝑐) is increasing in 𝑐𝑐, an example of which is depicted in Figure
1.3. The population growth rate is 𝑔𝑔(𝑐𝑐 ) − 1, so for levels of 𝑐𝑐 where 𝑔𝑔(𝑐𝑐 ) > 1 the
population is rising , and for levels of 𝑐𝑐 where 𝑔𝑔(𝑐𝑐 ) < 1 the population is falling.
Figure 1.3: Demographics and living standards
1.3.2 An agricultural economy
Malthus’s demographic assumption is particularly relevant for the predominantly
agricultural economies of the past when land 𝐿𝐿 and labour 𝑁𝑁 were the key factors of
production. We assume a neoclassical production function 𝑌𝑌 = 𝑧𝑧𝑧𝑧(𝐿𝐿, 𝑁𝑁). The crucial
feature of land is that the supply 𝐿𝐿 remains constant over time. There is no capital, so no
investment and we ignore government and international trade in the model, hence average
consumption per person 𝑐𝑐 is equal to income per worker 𝑦𝑦 = 𝑌𝑌/𝑁𝑁.
For the same amount of land available, more workers can produce more output but there
are diminishing returns to labour. Think of additional workers needing to use lower quality
land relative to that already being farmed, or instead work more intensively on land already
in use. These do not increase output of crops as much as if additional workers had access to
unlimited land of the same quality as that used by existing workers.
The per-worker production function in the Malthusian model is:
𝑦𝑦 =
𝑌𝑌 𝑧𝑧𝑧𝑧(𝐿𝐿, 𝑁𝑁)
𝐿𝐿 𝑁𝑁
=
= 𝑧𝑧𝑧𝑧 � , � = 𝑧𝑧𝑧𝑧 (𝑙𝑙, 1) = 𝑧𝑧𝑧𝑧 (𝑙𝑙 )
𝑁𝑁
𝑁𝑁
𝑁𝑁 𝑁𝑁
Here, 𝑙𝑙 = 𝐿𝐿/𝑁𝑁 denotes the amount of land available per worker and 𝑓𝑓(𝑙𝑙 ) is simply a
shorthand for 𝐹𝐹(𝑙𝑙, 1). This per-worker production function is illustrated in Figure 1.4, where
output per worker is increasing in land per worker 𝑙𝑙. Hence, as the population 𝑁𝑁 rises,
available land per worker 𝑙𝑙 declines, which reduces average output produced per worker.
This is a reflection of diminishing returns to labour when land is in fixed supply.
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Figure 1.4: Per-worker production function
1.3.4 Dynamics of a Malthusian economy
How do the population and living standards change over time in a Malthusian economy?
Using 𝑁𝑁′⁄𝑁𝑁 = 𝑔𝑔(𝑐𝑐) and 𝑐𝑐 = 𝑦𝑦 = 𝑧𝑧𝑧𝑧 (𝑙𝑙 ) = 𝑧𝑧𝑧𝑧(𝐿𝐿⁄𝑁𝑁), the change in the population over time
is determined by the equation:
𝐿𝐿
𝑁𝑁 ′
= 𝑔𝑔 �𝑧𝑧𝑧𝑧 � ��
𝑁𝑁
𝑁𝑁
For a low current population 𝑁𝑁, 𝑓𝑓(𝐿𝐿⁄𝑁𝑁) is high and g(z𝑓𝑓(𝐿𝐿⁄𝑁𝑁)) is greater than 1, so 𝑁𝑁 ′ >
𝑁𝑁, which means the population is increasing over time. For a high current population 𝑁𝑁,
𝑓𝑓(𝐿𝐿⁄𝑁𝑁) is low and g(z𝑓𝑓(𝐿𝐿⁄𝑁𝑁)) is less than 1, so 𝑁𝑁 ′ < 𝑁𝑁, which means the population is
falling over time. This tells us that for given parameters of this model, such as the level of
technology 𝑧𝑧, the population converges to a steady state 𝑁𝑁 ∗ .
A steady state is a value of a variable such that once the economy reaches that level of the
variable, there is no further change in that variable over time. The steady state of the
Malthusian model is depicted in Figure 1.5. Since the population converges to a steady
state, per capita income and consumption also reach a steady state 𝑐𝑐 ∗ . For this to result in
zero population growth, it must be the solution of equation 𝑔𝑔(𝑐𝑐 ∗ ) = 1. This solution is
shown using the demographic function in the left panel of the figure. Intuitively, if 𝑐𝑐 > 𝑐𝑐 ∗
then the population rises, pushing down 𝑙𝑙 and 𝑦𝑦 = 𝑐𝑐.
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Figure 1.5: Steady state of the Malthusian model
Given the steady state 𝑦𝑦 ∗ = 𝑐𝑐 ∗ for output and consumption per worker, the steady state for
land per worker 𝑙𝑙 ∗ is the solution of 𝑦𝑦 ∗ = 𝑧𝑧𝑧𝑧(𝑙𝑙 ∗ ). This is found using the per worker
production function in the right panel of the figure. Finally, given 𝑙𝑙 ∗ , the population in
steady state is simply 𝑁𝑁 ∗ = 𝐿𝐿⁄𝑙𝑙 ∗ .
As we will see in Section 2.1, the Malthusian model’s prediction of stagnation in living
standards 𝑐𝑐 ∗ is consistent with historical evidence prior to the 19th century.
1.3.5 What does (or does not) help?
The conclusion that living standards 𝑐𝑐 stagnate in the long run continues to hold even if
technology 𝑧𝑧 improves. This can be seen from Figure 1.5 observing that steady-state 𝑐𝑐 ∗ is
independent of 𝑧𝑧. Better technology 𝑧𝑧 ultimately leads only to a larger population 𝑁𝑁 ∗
because an upward shift of the per worker production function with the same 𝑐𝑐 ∗ = 𝑦𝑦 ∗
results in lower 𝑙𝑙 ∗ . There would be higher living standards during the period until population
converges to its new higher steady state but not in the long run.
The discovery of new land 𝐿𝐿 does not help either in the long run. This would simply cause
the population to rise (higher 𝑁𝑁 ∗ ) with no change in 𝑙𝑙 ∗ or 𝑐𝑐 ∗ in the long run, although there
would be temporarily higher living standards before the population reaches its new steady
state.
Weakening the link between the birth rate and living standards would help. This
demographic transition shifts down the 𝑔𝑔(𝑐𝑐 ) line and leads to a higher steady-state for
living standards 𝑐𝑐 ∗ .
Structural transformation of economy of the economy is another way the Malthusian trap
can be escaped. Industrialisation of the economy reduces the dependence of production on
land in fixed supply. Crucially, since capital can be accumulated, this helps to avoid the
problem of diminishing returns to labour.
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Box 1.2: A 14th-century pandemic
Around 1350, Europe, North Africa and Western Asia were struck by a bubonic plague
pandemic (known as the ‘Black Death’). This pandemic is believed to have killed more than
a third of the population of the affected areas. The main economic effect of the pandemic
came from the shortages of labour it created.
In the Malthusian model, a pandemic causes a temporary downward shift of the
demographic function 𝑔𝑔(𝑐𝑐). The population growth rate is 𝑔𝑔(𝑐𝑐 ) − 1, so more deaths can be
represented by a lower 𝑔𝑔(𝑐𝑐) for each value of 𝑐𝑐. Starting from a steady state, lower 𝑔𝑔(𝑐𝑐)
means a falling population. With a fixed supply of land 𝐿𝐿, land per worker 𝑙𝑙 = 𝐿𝐿/𝑁𝑁 rises, so
output and consumption per worker 𝑐𝑐 = 𝑦𝑦 = 𝑧𝑧𝑧𝑧(𝑙𝑙) are higher. This is illustrated in Figure
1.6.
For the survivors, the pandemic leads to higher output per worker because land was
previously more scarce. This is true even though total GDP 𝑌𝑌 declines when the population
falls. Once pandemic is over, 𝑔𝑔(𝑐𝑐) returns to normal and population and living standards
ultimately go back to their former steady state unless something else changes.
Figure 1.6: Population growth and output per worker in a pandemic
The pandemic also has significant distributional effects. Let us consider how total output 𝑌𝑌
is distributed among workers and owners of land in the Malthusian model. Workers do not
receive all of 𝑦𝑦 as income unless they own the land they use to produce. If competition
determines factor payments, wages 𝑤𝑤 and rents 𝑥𝑥 are equal to the marginal products of
labour 𝑀𝑀𝑃𝑃𝑁𝑁 and land 𝑀𝑀𝑃𝑃𝐿𝐿 .
With a neoclassical production function, the marginal product of labour 𝑀𝑀𝑃𝑃𝑁𝑁 is diminishing
in the population 𝑁𝑁. Given a fixed supply of land 𝐿𝐿, a lower population 𝑁𝑁 means the
marginal product of labour is higher, so wages 𝑤𝑤 = 𝑀𝑀𝑃𝑃𝑁𝑁 rise. What about rents 𝑥𝑥 = 𝑀𝑀𝑃𝑃𝐿𝐿 ?
The per worker production function implies 𝑌𝑌 = 𝑧𝑧𝑁𝑁𝑓𝑓 (𝐿𝐿⁄𝑁𝑁), which can be used to obtain
expressions for the marginal products of land and labour using the chain rule.
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𝑀𝑀𝑃𝑃𝑁𝑁 =
𝑀𝑀𝑃𝑃𝐿𝐿 =
𝜕𝜕𝜕𝜕
𝑁𝑁
𝐿𝐿
= 𝑧𝑧 𝑓𝑓 ′ � � = 𝑧𝑧𝑧𝑧′(𝑙𝑙)
𝜕𝜕𝜕𝜕
𝑁𝑁
𝑁𝑁
𝜕𝜕𝜕𝜕
𝐿𝐿
𝐿𝐿
𝐿𝐿
= 𝑧𝑧𝑧𝑧 � � − 𝑧𝑧𝑧𝑧 2 𝑓𝑓 ′ � � = 𝑧𝑧(𝑓𝑓(𝑙𝑙 ) − 𝑙𝑙𝑙𝑙′(𝑙𝑙))
𝜕𝜕𝜕𝜕
𝑁𝑁
𝑁𝑁
𝑁𝑁
These equations imply 𝑤𝑤 = 𝑦𝑦 − 𝑥𝑥𝑥𝑥, which says that wages are equal to output per worker
minus rent times land used per worker.
Both marginal products and hence wages and rents depend on the relative supply of land to
labour as measured by land-per-worker 𝑙𝑙 = 𝐿𝐿/𝑁𝑁. It can be seen that 𝑀𝑀𝑃𝑃𝐿𝐿 is diminishing in 𝑙𝑙
because 𝑓𝑓 (𝑙𝑙 ) = 𝐹𝐹(𝑙𝑙, 1), so 𝑓𝑓 ′′ (𝑙𝑙 ) < 0, whereas 𝑀𝑀𝑃𝑃𝑁𝑁 is increasing in 𝑙𝑙 because derivative of
𝑓𝑓(𝑙𝑙 ) − 𝑙𝑙𝑙𝑙′(𝑙𝑙) with respect to 𝑙𝑙 is 𝑓𝑓 ′ (𝑙𝑙 ) − 𝑓𝑓 ′ (𝑙𝑙 ) − 𝑙𝑙𝑓𝑓 ′′ (𝑙𝑙 ) = −𝑙𝑙𝑓𝑓 ′′ (𝑙𝑙 ) > 0. These relationships
are depicted in Figure 1.7. As the pandemic increases land per worker 𝑙𝑙, it causes a rise in
𝑀𝑀𝑃𝑃𝑁𝑁 and hence higher wages, but a fall in 𝑀𝑀𝑃𝑃𝐿𝐿 and hence lower rents.
Figure 1.7: Wages and rents with a lower population
1.4 Hours of work and the supply of labour
As well as changes in the population, the supply of labour also depends on how many hours
people work, which is related to decisions such as:
•
•
•
Full-time versus part-time work?
Participate or not in the labour market?
Early retirement or continue working?
Here, we take as given the number of workers, also the skills, education and training of
workers, returning later to the issue of human-capital accumulation.
In our analysis of labour supply, the key trade-off is that more hours of work lead to more
income and hence a greater ability to purchase goods and services but also less time for
other things such as leisure.
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We use a simple static model (meaning there is no saving or borrowing) to study the choice
of labour supply by households. The supply of labour by a household (in units of time, e.g.
hours) is denoted by 𝑁𝑁. Time the household enjoys as leisure is denoted by 𝑙𝑙. ‘Leisure’ is a
catch-all term for anything other than time spent earning wages, so as well as leisure in the
usual sense, it also includes cooking, cleaning and childcare, activities known as ‘home
production’. While home production includes a component of work, the household benefits
from it in the sense that otherwise the household would have to pay for equivalent services
in the market, for example, eating out in restaurants, hiring a cleaner or a childminder.
A crucial constraint is that the household has a fixed amount of time ℎ available in a given
day, week, or year. Time used for work cannot also be used for leisure:
𝑁𝑁 + 𝑙𝑙 = ℎ
By working (not counting home production), the household is paid a wage 𝑤𝑤 per unit of
time, e.g. an hourly wage. This wage is specified in real terms, as with all other variables in
this chapter. If a household works for 𝑁𝑁 hours then total wage income is 𝑤𝑤𝑤𝑤. The ultimate
purpose of work is to use the income to buy goods and services. Consumption of goods and
services (not counting home production) is denoted by 𝐶𝐶.
Some households may also be able to use non-wage income such as dividend income from
owning shares to buy goods and services. The amount of non-wage income is denoted by 𝜋𝜋.
More broadly, considering the household as a family, 𝜋𝜋 could also be interpreted as the
income of an individual’s partner separate from the amount the individual earns direct.
Our analysis must also consider taxes, which influence how much households are able to
spend of their pre-tax incomes. For now, taxes are assumed to take a ‘lump sum’ form: an
amount of tax 𝑇𝑇 that must be paid irrespective of how much a household earns or
consumes, for example, a poll tax. While this simplifies matters, most taxes are not like this
and we will see what difference it makes by considering income taxes and consumption
taxes later. Note that we allow 𝑇𝑇 to be negative, indicating the household receives a
transfer payment from the government rather than being a taxpayer.
Given wage income 𝑤𝑤𝑤𝑤, non-wage income 𝜋𝜋, and taxes 𝑇𝑇, the maximum amount of
consumption affordable to a household is:
𝐶𝐶 = 𝑤𝑤𝑤𝑤 + 𝜋𝜋 − 𝑇𝑇
As this is a static model, there is no role for saving for the future in this budget constraint, so
households will consume their income. Instead, the purpose of the model is to analyse how
much labour households will supply, which affects their income and, hence, their
consumption.
Our analysis proceeds by combining the two constraints on time and spending power. Since
𝑁𝑁 = ℎ − 𝑙𝑙, we can write a combined constraint in terms of the consumption 𝐶𝐶 and leisure 𝑙𝑙
that the household ultimately values:
𝐶𝐶 + 𝑤𝑤𝑤𝑤 = 𝑤𝑤ℎ + 𝜋𝜋 − 𝑇𝑇
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This budget constraint is plotted in Figure 1.8 with leisure 𝑙𝑙 on the horizontal and
consumption 𝐶𝐶 on the vertical axis. Given the time physically available, it is not possible to
have more than ℎ hours of leisure. If a household chose the maximum amount of leisure 𝑙𝑙 =
ℎ this would mean supplying no labour and thus the ability to consume goods would depend
solely on non-wage income after tax 𝜋𝜋 − 𝑇𝑇. As leisure falls below ℎ, hours of labour supplied
increase and each extra hour of labour adds 𝑤𝑤 to wage income, increasing the ability to
consume by 𝑤𝑤. Therefore, the budget constraint is a downward-sloping straight line with
gradient −𝑤𝑤 that passes through the point (ℎ, 𝜋𝜋 − 𝑇𝑇).
Figure 1.8: Constraint on consumption and leisure
Households like both more consumption 𝐶𝐶 and more leisure 𝑙𝑙 but the constraints imply
there is a trade-off between them. To study the optimal choice of leisure and, hence, the
supply of labour, we need to say more about preferences.
We describe the household’s preferences over 𝐶𝐶 and 𝑙𝑙 using indifference curves added to
the diagram with the budget constraint. Indifference curves are downward sloping because
less of one thing the household likes requires more of the other to compensate. Indifference
curves are also assumed to be convex to the origin, as depicted in Figure 1.9. This shape
reflects a dislike of extremes where the household has very little consumption or leisure but
much more of the other.
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Figure 1.9: Indifference curves over consumption and leisure
The shape of the indifference curves can be described in terms of a diminishing marginal
rate of substitution 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 between consumption 𝐶𝐶 and leisure 𝑙𝑙. The marginal rate of
substitution is how much extra of one good a household needs to be given to compensate
for the loss of one unit of another. The gradient of an indifference curve is −𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 , with
𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 representing how much extra consumption a household needs to receive to be no
worse off after giving up a unit of leisure by supplying more labour.
Consumption and leisure are also assumed to be normal goods. This means that when
households are better off and able to reach a higher indifference curve, they choose to have
more consumption and more leisure, holding constant the hourly wage 𝑤𝑤. In the diagram,
this means the line joining points on different indifference curves where the tangent lines
have the same gradient is upward sloping.
A household wants to reach the highest indifference curve subject to the constraints. As
shown in Figure 1.10, there are two general cases to consider. First, where it is optimal to
participate in the labour market (𝑙𝑙 < ℎ), in which case the optimal consumption-leisure
choice is where an indifference curve is tangent to the constraint, mathematically, 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 =
𝑤𝑤. The second case is where the household does not find it optimal to participate in the
labour market (𝑙𝑙 = ℎ) and the optimal choice of 𝐶𝐶 and 𝑙𝑙 is at the corner of constraint.
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EC2065 Macroeconomics | Chapter 1: The supply side of the economy
Figure 1.10: Labour-market participation decision
Participation in the labour market is optimal if the marginal rate of substitution 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 at
zero labour supply (𝑙𝑙 = ℎ) is less than the hourly real wage 𝑤𝑤, i.e. the indifference curve
passing through the corner of the constraint is less steep than the budget constraint to the
left of this point.
This logic indicates participation in the labour market is more likely when wages 𝑤𝑤 are high,
which makes the budget constraint steeper. High taxes 𝑇𝑇 or low transfer payments (the
negative of 𝑇𝑇) increase the likelihood of participation, moving the corner (ℎ, 𝜋𝜋 − 𝑇𝑇)
downwards. Similarly, a low level of other income 𝜋𝜋 increases participation, with 𝜋𝜋 being
low because the household has little wealth or an individual’s partner does not have a high
income. Finally, preferences can also matter, with a strong preference for consumption over
leisure (a low marginal rate of substitution 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 ) making participation more likely.
For those choosing to participate, we can also analyse how the number of hours worked
depends on these considerations.
1.5 The effects of wages on labour supply
This section studies how wages affect the supply of labour. We will use our analysis here to
derive a supply curve for labour. There are two aspects of the labour supply response to
wages. First, how do hours worked change for those participating in the labour market?
Second, how do wages affect the decision to participate or not in the labour market?
1.5.1 Effects on those already participating in the labour market
Let us first consider participants in the labour market. An increase in the real wage 𝑤𝑤 pivots
the budget constraint upwards, making a household better off all else being equal. A careful
study of how a household reacts to the wage change requires breaking down the response
into income and substitution effects.
Intuitively, the substitution effect captures the effect of wages on incentives. A higher wage
increases the price of leisure (more consumption is forgone by taking leisure), so a
household substitutes away from leisure towards consumption, which means choosing to
work more. The income effect captures the impact of wages on how well off households
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are. A higher wage makes a household better off and, since consumption and leisure are
both normal goods, there is desire to enjoy more leisure by choosing to work less.
Income and substitution effects are analysed in Figure 1.11. A higher wage makes the
budget constraint steeper, pivoting it around the corner. Formally, the substitution effect
(SE) is found by considering the effects of the steeper budget constraint gradient,
controlling for whether the household is made better off or worse off. Since the higher
wage makes the household better off, the substitution effect can be isolated by also making
a parallel shift downwards of the budget constraint so that it is tangent to the original
indifference curve. This results in a new tangency point north-west of the original tangency
because this is where the indifference curve is steeper. Leisure falls (labour supply
increases) and consumption rises.
Figure 1.11: Income and substitution effects on labour supply
The income effect (IE) is isolated by removing the hypothetical parallel shift of the budget
constraint used to derive the substitution effect. Hence, the income effect results from a
parallel upward shift of the budget constraint in this case, causing a movement on to a
higher indifference curve in a north-east direction. Leisure and consumption both rise, so
labour supply falls.
Overall, combining the substitution effect and income effect to obtain the combined effect,
consumption must rise but leisure may rise or fall. Thus, the effect of wages on labour
supply is ambiguous. In the diagram, income and substitution effects exactly cancel out for
leisure and labour supply but this is a special case. In general, either the substitution effect
or the income effect could dominate. If the substitution effect dominates, leisure falls and
labour supply rises, while if the income effect dominates, leisure rises and labour supply
falls.
1.5.2 Effect on the labour-market participation decision
What about those not already participating in the labour market? A higher wage 𝑤𝑤 makes
the budget constraint steeper, pivoting it around the point of non-participation. Since the
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indifference curve passing through that point needs to be steeper than the budget
constraint for non-participation to be optimal, a sufficiently high wage would cause an
individual to supply some labour. This is shown in Figure 1.12.
Figure 1.12: The effects of wages on labour-market participation
Taking 𝜋𝜋 as given, a higher wage 𝑤𝑤 has no offsetting income effect on the participation
decision because a higher wage does not make non-participants better off if they are not
earning any labour income. However, if 𝜋𝜋 is interpreted more broadly as including family
income from a partner who works, in this case higher wages have an income effect on the
household’s labour supply.
1.5.3 The labour supply curve
The labour supply curve shows the optimal choice of 𝑁𝑁 𝑠𝑠 = ℎ − 𝑙𝑙 for each level of real wages
𝑤𝑤. An example is shown in Figure 1.13 with the real wage 𝑤𝑤 on the vertical axis and the
quantity of labour on the horizontal axis.
Figure 1.13: The labour supply curve
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EC2065 Macroeconomics | Chapter 1: The supply side of the economy
A labour supply curve can be drawn for an individual or for all households together. For
those participating in the labour market, hours worked increase with higher wages if the
substitution effect is larger than the income effect. For such households, the 𝑁𝑁 𝑠𝑠 curve is
upward sloping if the substitution effect dominates the income effect. For those not
participating initially, high wages make participation more likely and there is no offsetting
income effect until some labour is supplied by a household. An upward-sloping labour
supply curve drawn for all households can also represent more participation at higher wages
as well as those already working choosing to supply more hours.
If all households are participating in the labour market, we can think of the labour supply
curve as representing the optimality condition 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 = 𝑤𝑤 with the marginal rate of
substitution between leisure and consumption rising as more labour is supplied because
leisure falls.
The labour supply curve shifts if there is a change in any variable that matters for optimal
labour supply other than the real wage. Lower non-wage income 𝜋𝜋, or higher taxes 𝑇𝑇, would
make a household worse off, reducing demand for leisure as a normal good and increasing
the supply of labour. This would cause 𝑁𝑁 𝑠𝑠 to shift to the right.
A very long-run perspective on labour supply is provided by the 160 years of data for the UK
shown in Figure 1.14. This graph shows time series of real wages, average hours worked per
week for those who have jobs and the fraction of the whole population who have jobs. Over
the 160 years, UK real wages rise by a factor of 20. Hours per worker, though, fall by around
half over this period. A broad measure of labour-market participation, the ratio of workers
to the total population (not adjusting for those of ‘working age’) does not display any clear
trend. One interpretation of this evidence points to the importance of income effects as UK
workers became significantly better off over this period and chose to work fewer hours.
Figure 1.14: UK wages and labour supply in the long run
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1.5.4 Do higher tax rates raise more revenue?
In the household budget constraint, taxes 𝑇𝑇 were assumed to have a ‘lump sum’ form
because the amount 𝑇𝑇 paid to the government does not depend on the household’s
choices. In practice, the amount of tax paid depends on individual behaviour. For example, a
proportional labour income tax has households pay a fixed percentage of labour income as
tax. If the household chooses to work more, more tax will be paid. Different from earlier,
this means that taxes also have effects on incentives.
Assume there is a proportional labour income tax rate of 𝜏𝜏, for example, 𝜏𝜏 = 0.2 if wages
are taxed at a 20 per cent rate. The pre-tax wage is denoted by 𝑤𝑤, and labour supply by 𝑁𝑁 𝑠𝑠 .
The amount of tax revenue collected by the government is 𝑇𝑇 = 𝜏𝜏𝜏𝜏𝑁𝑁 𝑠𝑠 in this case. The aftertax real wage is (1 − 𝜏𝜏)𝑤𝑤, so the households now supply labour up to the point where:
𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 = (1 − 𝜏𝜏)𝑤𝑤
A higher tax rate 𝜏𝜏 reduces the after-tax wage (1 − 𝜏𝜏)𝑤𝑤, so the effects of tax are similar to
those of lower wages.
Since the tax rate 𝜏𝜏 affects tax revenue 𝑇𝑇 = 𝜏𝜏𝜏𝜏𝑁𝑁 𝑠𝑠 indirectly through its impact on
behaviour 𝑁𝑁 𝑠𝑠 as well as directly, the relationship between 𝜏𝜏 and 𝑇𝑇 is not always positive.
This leads to the ‘Laffer curve’ relationship between tax rates and revenue shown in Figure
1.15. To understand the Laffer curve, note that a 0 per cent tax rate obviously generates no
revenue. On the other hand, a 100 per cent tax rate implies no incentive to supply labour
because the after-tax wage is zero, so no tax revenue would be obtained in this case as
𝑁𝑁 𝑠𝑠 = 0. This basic logic indicates there is a tax rate somewhere between 0 per cent and 100
per cent where the Laffer curve peaks and tax revenue is maximised. After this point, higher
tax rates would reduce revenue.
Figure 1.15: A Laffer curve
Although the Laffer curve implies that ever higher tax rates eventually result in lower
revenue, it does not give specific guidance at which tax rate 𝜏𝜏 revenue will start to fall as 𝜏𝜏
rises. This is an empirical question.
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1.6 Equilibrium and efficiency
Having studied firms’ demand for labour in the context of the distribution of income in
Section 1.2 and now households’ supply of labour in Section 1.4, we can put the two
together to set up our first simple macroeconomic model. At this stage, the model is static
with only a single period, which misses out many important macroeconomic issues we will
address later.
1.6.1 A static macroeconomic model
This first macroeconomic model looks at the implications of optimising behaviour of
households and firms in the markets for labour and goods, where those markets will ‘clear’,
i.e. reach an equilibrium between demand and supply. The model will also have a
government that chooses a fiscal policy setting tax and public expenditure.
In the model, assume that all households share the same preferences (same indifference
curves over consumption and leisure) and all have equal claims on non-wage income (which
arises from ownership of capital or land). In this case, there is said to be a ‘representative
household’: the economy is comprised of many households, each of which is small relative
to the size of the economy but all will optimally choose to behave the same way because
they have the same preferences and face the same constraints.
As we have seen, optimisation by households implies a labour supply curve 𝑁𝑁 𝑠𝑠 . Since all
households are the same and will have to participate in the labour market in equilibrium, we
can represent the labour supply curve by the optimality condition 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 = 𝑤𝑤, where
𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 is the marginal rate of substitution between leisure 𝑙𝑙 = ℎ − 𝑁𝑁 𝑠𝑠 and consumption 𝐶𝐶,
and 𝑤𝑤 is the real wage.
We have also seen that profit maximisation by firms implies a labour demand curve. Firms
hire labour up to the point where 𝑀𝑀𝑃𝑃𝑁𝑁 = 𝑤𝑤, where 𝑀𝑀𝑃𝑃𝑁𝑁 is the marginal product of labour
and 𝑁𝑁 is employment. Firms face a neoclassical production function where labour and
capital, and/or land are used to produce goods.
Other factors of production apart from labour, such as land or capital, are assumed to be in
fixed supply. This means we do not consider changes in the capital stock through investment
and there is no depreciation of existing capital. Factors of production are equally owned by
all households.
The government’s fiscal policy sets the level of public expenditure 𝐺𝐺, interpreted as
government purchases of privately produced goods and services. This expenditure is
financed by a lump-sum tax 𝑇𝑇 and every household faces the same tax 𝑇𝑇, so there is no
redistribution. Since the model is static, there is no scope here for government budget
deficits and debt, so the government’s budget constraint is 𝑇𝑇 = 𝐺𝐺.
1.6.2 Equilibrium in labour and goods markets
The labour market of the model is shown in Figure 1.16. Firms’ demand for labour 𝑁𝑁 𝑑𝑑 is
determined by 𝑀𝑀𝑃𝑃𝑁𝑁 = 𝑤𝑤, and households’ supply of labour is determined by 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 = 𝑤𝑤.
The model is based on the real wage 𝑤𝑤 adjusting to clear the labour market, i.e. achieve
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EC2065 Macroeconomics | Chapter 1: The supply side of the economy
𝑁𝑁 𝑑𝑑 = 𝑁𝑁 𝑠𝑠 , so the desired supply of labour is equal to the amount firms want to demand. This
occurs if the real wage 𝑤𝑤 adjusts to 𝑤𝑤 ∗ where the demand and supply curves cross.
Figure 1.16: Labour market equilibrium
Once equilibrium employment 𝑁𝑁 ∗ is known, the amount of goods and services 𝑌𝑌 produced
by firms is determined because all other factors (land, capital) are in fixed supply. This is the
amount firms supply to the goods market. On the demand side, households buy goods for
consumption 𝐶𝐶 and the government to provide public services 𝐺𝐺, so aggregate demand is
𝐶𝐶 + 𝐺𝐺. The government’s fiscal policy sets 𝐺𝐺 and households choose 𝐶𝐶 subject to their
budget constraint when making the consumption-leisure trade-off that underlies labour
supply. Equilibrium of the goods market requires 𝐶𝐶 + 𝐺𝐺 = 𝑌𝑌.
It turns out that another diagram for the goods market is not necessary given that labourmarket equilibrium has already been found. The amount of non-wage income received by
households is 𝜋𝜋 = 𝑌𝑌 − 𝑤𝑤𝑁𝑁 𝑑𝑑 . This is true irrespective of whether households or firms own
factors of production such as land and capital as households will ultimately receive these
factor payments as the owners of firms.
The household budget constraint is 𝐶𝐶 = 𝑤𝑤𝑁𝑁 𝑠𝑠 + 𝜋𝜋 − 𝑇𝑇, and substituting the government
budget constraint 𝐺𝐺 = 𝑇𝑇 and the equation for 𝜋𝜋, it follows that 𝐶𝐶 = 𝑌𝑌 + 𝑤𝑤(𝑁𝑁 𝑠𝑠 − 𝑁𝑁 𝑑𝑑 ) − 𝐺𝐺.
Writing this as 𝑌𝑌 − (𝐶𝐶 + 𝐺𝐺 ) = 𝑤𝑤(𝑁𝑁 𝑑𝑑 − 𝑁𝑁 𝑠𝑠 ), labour-market equilibrium 𝑁𝑁 𝑑𝑑 = 𝑁𝑁 𝑠𝑠 implies
𝐶𝐶 + 𝐺𝐺 = 𝑌𝑌, so the goods market must also be in equilibrium.
1.6.3 Economic efficiency
One important implication of equilibrium, at least for the simple model studied here, is that
the outcomes for employment, output and consumption are economically efficient,
conditional on the government’s choice of public expenditure 𝐺𝐺. This is because 𝑁𝑁 𝑑𝑑 = 𝑁𝑁 𝑠𝑠
means that 𝑀𝑀𝑃𝑃𝑁𝑁 = 𝑤𝑤 ∗ = 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 , so the market-clearing real wage is equal to the marginal
product of labour and the marginal rate of substitution between leisure and consumption.
Intuitively, once the economy reaches equilibrium, the marginal value (measured in goods)
that households put on a unit of their time is equal to what amount of goods firms can
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produce with that unit of time. A basic function of an economy is to allow households to
turn their time as work into income that can be spent on goods and services. Here, the
market economy performs this task as well as possible and we say that the market
equilibrium is economically efficient, or Pareto efficient.
Note that we say nothing here about whether the government’s choice of public
expenditure 𝐺𝐺 is optimal and, hence, whether the allocation of resources between private
spending 𝐶𝐶 and public spending 𝐺𝐺 is optimal. The desirability of a particular amount of
public expenditure is taken as given here.
A more careful way to reach the efficiency result is to imagine a hypothetical world where a
government can control all economic decisions (consumption, employment, etc.) without
the need for markets. The government is said to be the ‘social planner’ in this case. Assume
the government acts benevolently with the aim of making the representative household as
well off as possible.
The economy’s ability to produce goods and services is still limited by the production
function 𝑌𝑌 = 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑁𝑁), where 𝐾𝐾 is denotes a factor of production other than labour, for
example, capital. The supply of this factor is fixed here. Assume that a particular level of
public expenditure 𝐺𝐺 is desirable. The government faces a resource constraint 𝐶𝐶 + 𝐺𝐺 = 𝑌𝑌
and a constraint 𝑙𝑙 + 𝑁𝑁 = ℎ on households’ time. Combining these and the production
function leads to a single constraint 𝐶𝐶 = 𝑧𝑧𝑧𝑧 (𝐾𝐾, ℎ − 𝑙𝑙 ) − 𝐺𝐺 linking consumption 𝐶𝐶 and
leisure 𝑙𝑙, the two things households ultimately care about. The constraint and the
representative household’s indifference curves are illustrated in Figure 1.17.
The ability to raise consumption by reducing leisure and setting households to work longer
is found by differentiating the constraint with respect to leisure 𝑙𝑙:
𝜕𝜕𝜕𝜕
= −𝑧𝑧𝐹𝐹𝑁𝑁 (𝐾𝐾, ℎ − 𝑙𝑙 ) = −𝑀𝑀𝑃𝑃𝑁𝑁
𝜕𝜕𝜕𝜕
The gradient is the negative of the marginal product of labour, so the constraint is
downward sloping and becomes steeper as 𝑙𝑙 rises (because 𝑁𝑁 falls, increasing 𝑀𝑀𝑃𝑃𝑁𝑁 ). To
make representative-household utility as high as possible subject to the constraint, the
social planner would choose a combination (𝑙𝑙, 𝐶𝐶) such that 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 = 𝑀𝑀𝑃𝑃𝑁𝑁 , where an
indifference curve is tangent to the constraint.
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Figure 1.17: The social planner allocation
Since the market equilibrium features 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 = 𝑀𝑀𝑃𝑃𝑁𝑁 and satisfies the same constraints
faced by the government, the market equilibrium and the benevolent planner’s choice
coincide. It is not possible to make the representative household better off than in the
market equilibrium, so the equilibrium is efficient.
More generally, without a representative household, the market equilibrium is said to be
Pareto-efficient if no one can be made better off without making someone else worse off. It
is a general result (the first welfare theorem) that an equilibrium is Pareto efficient if:
•
•
•
Markets are perfectly competitive
There are no externalities or tax distortions
There are no missing markets or restrictions on trade
We will see many examples later where an equilibrium is not efficient. In these cases, the
economy is failing in its basic function to allow households to convert their time into work
and enjoy the fruits of their labours.
Box 1.4: Should wages or rents be taxed to pay for public expenditure?
This application addresses the question of how a government should best pay for a given
amount of public services 𝐺𝐺 it needs to provide. Here we assume that lump-sum taxes are
not available. Instead, the government is restricted to taxing different types of income.
Assume firms produce output of goods and services using land 𝐿𝐿 and labour 𝑁𝑁. The
production function is 𝑌𝑌 = 𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏, which is linear in both 𝐿𝐿 and 𝑁𝑁. This is not a
neoclassical production function but it is useful for illustration and the arguments
developed here apply more generally. The marginal products of labour and land are
𝑀𝑀𝑃𝑃𝑁𝑁 = 𝑎𝑎 and 𝑀𝑀𝑃𝑃𝐿𝐿 = 𝑏𝑏, where 𝑎𝑎 and 𝑏𝑏 are positive constants. With firms hiring labour
and renting land in competitive factor markets, the pre-tax wage 𝑤𝑤 and rent 𝑥𝑥 must be
equal to these constant marginal products:
𝑤𝑤 = 𝑀𝑀𝑃𝑃𝑁𝑁 = 𝑎𝑎
,
34
𝑥𝑥 = 𝑀𝑀𝑃𝑃𝐿𝐿 = 𝑏𝑏
EC2065 Macroeconomics | Chapter 1: The supply side of the economy
The economy has a fixed supply of land 𝐿𝐿. The supply of labour 𝑁𝑁 is chosen by a
representative household that owns an equal share of the economy’s land. The
government can only levy proportional taxes on incomes, setting a tax rate 𝜏𝜏𝑤𝑤 on wage
income and a tax rate 𝜏𝜏𝑥𝑥 on rental income. The total amount of tax revenue raised is
𝜏𝜏𝑤𝑤 𝑤𝑤𝑤𝑤 + 𝜏𝜏𝑥𝑥 𝑥𝑥𝑥𝑥. With 𝑤𝑤 and 𝑥𝑥 equal to the constants 𝑎𝑎 and 𝑏𝑏, the budget constraint of
the government is:
𝑎𝑎𝜏𝜏𝑤𝑤 𝑁𝑁 + 𝑏𝑏𝜏𝜏𝑥𝑥 𝐿𝐿 = 𝐺𝐺
The total amount of income (wages and rents) the representative household receives
after tax is (1 − 𝜏𝜏𝑤𝑤 )𝑤𝑤𝑤𝑤 + (1 − 𝜏𝜏𝑥𝑥 )𝑥𝑥𝑥𝑥 and the budget constraint is 𝐶𝐶 = 𝑎𝑎(1 − 𝜏𝜏𝑤𝑤 )𝑁𝑁 +
𝑏𝑏(1 − 𝜏𝜏𝑥𝑥 )𝐿𝐿. With a given amount of time ℎ available, labour supply is equal to 𝑁𝑁 = ℎ −
𝑙𝑙, where 𝑙𝑙 is the choice of leisure. The combined constraint faced by the household is:
𝐶𝐶 + 𝑎𝑎(1 − 𝜏𝜏𝑤𝑤 )𝑙𝑙 = 𝑎𝑎(1 − 𝜏𝜏𝑤𝑤 )ℎ + 𝑏𝑏(1 − 𝜏𝜏𝑥𝑥 )𝐿𝐿
1.6.4 Taxing wages but not rents
Suppose initially that only wage income is taxed, so 𝜏𝜏𝑥𝑥 = 0. The government’s budget
constraint simplifies to 𝑎𝑎𝑎𝑎𝑤𝑤 𝑁𝑁 = 𝐺𝐺 and the household’s budget constraint is 𝐶𝐶 +
𝑎𝑎(1 − 𝜏𝜏𝑤𝑤 )𝑙𝑙 = 𝑎𝑎(1 − 𝜏𝜏𝑤𝑤 )ℎ + 𝑏𝑏𝑏𝑏. The household chooses 𝐶𝐶 and 𝑙𝑙 at the tangency of the
budget constraint and an indifference curve, i.e. where 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 = 𝑎𝑎 (1 − 𝜏𝜏𝑤𝑤 ). The
gradient of the budget constraint is the after-tax wage 𝑎𝑎(1 − 𝜏𝜏𝑤𝑤 ). Figure 1.18 depicts
this tangency point (𝑙𝑙1∗ , 𝐶𝐶1∗ ) as the economy’s initial equilibrium.
Figure 1.18: Taxes on rents instead of taxes on wages
1.6.5 Taxing rents but not wages
Now suppose the government switches completely to taxing rents instead of wages, so
𝜏𝜏𝑤𝑤 = 0. The government’s budget constraint is now 𝑏𝑏𝜏𝜏𝑥𝑥 𝐿𝐿 = 𝐺𝐺. For this alternative tax
system to be feasible it is necessary that 𝑏𝑏𝑏𝑏 > 𝐺𝐺, which says that there is enough rental
income to tax given the need for public expenditure (the government budget constraint
can hold for a tax rate less than 100 per cent). This is an important limitation on this
analysis that should be borne in mind.
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The household budget constraint in this case is 𝐶𝐶 + 𝑎𝑎𝑎𝑎 = 𝑎𝑎ℎ + 𝑏𝑏(1 − 𝜏𝜏𝑥𝑥 )𝐿𝐿. This has
gradient −𝑎𝑎 instead of −𝑎𝑎(1 − 𝜏𝜏𝑤𝑤 ), so the budget constraint becomes steeper as the aftertax wage rises. The budget constraint also shifts down at 𝑙𝑙 = ℎ, reflecting the reduction in
non-wage income after tax.
A crucial observation is that the choice of 𝐶𝐶1∗ and 𝑙𝑙1∗ under the previous tax system remains
affordable under the new one. This is because 𝐶𝐶1∗ and 𝑙𝑙1∗ satisfy 𝑎𝑎𝜏𝜏𝑤𝑤 (ℎ − 𝑙𝑙1∗ ) = 𝑎𝑎𝜏𝜏𝑤𝑤 𝑁𝑁1∗ =
𝐺𝐺 = 𝑏𝑏𝜏𝜏𝑥𝑥 𝐿𝐿 and 𝐶𝐶1∗ + 𝑎𝑎(1 − 𝜏𝜏𝑤𝑤 )𝑙𝑙1∗ = 𝑎𝑎(1 − 𝜏𝜏𝑤𝑤 )ℎ + 𝑏𝑏𝑏𝑏, so they also consistent with the new
budget constraint 𝐶𝐶1∗ + 𝑎𝑎𝑙𝑙1∗ = 𝑎𝑎ℎ + 𝑏𝑏(1 − 𝜏𝜏𝑥𝑥 )𝐿𝐿. Therefore, the original choice of leisure
and consumption remains on the new budget constraint. Since the budget constraint is now
steeper, it must cut the indifference curve at this point because there is a tangency with a
less steep budget line there.
It is possible to reach a higher indifference curve under the new tax system by choosing less
leisure, a higher labour supply, and more consumption. The analysis indicates a switch from
taxing wages to taxing rents makes the representative household better off. This is because
households’ labour supply choice is distorted by a proportional income tax that
disincentivises work. On the other hand, the supply of land is inelastic and does not respond
to tax. The removal of this distortion to labour supply and production allows the
representative household to reach a higher indifference curve.
As noted, in practice, rental income may not be high enough to shift tax burden completely
away from wages (which would require total pre-tax rents exceed total public expenditure,
𝑏𝑏𝑏𝑏 > 𝐺𝐺). By having a representative household, the analysis also ignores the distributional
consequences of such shifts in the tax system.
1.7 Capital accumulation
Modern industrial or service-based economies produce output mainly using capital and
labour rather than land. Capital is defined as goods used for the production of other goods
and services in the future (in other words, capital is not an intermediate input that is
immediately used up in current production). Capital includes such things as machinery,
buildings, computers, and aeroplanes. An important question we will address is whether
economic growth can be explained through a process of accumulating capital. In Section
2.30, we will also look at whether different levels of capital accumulation across countries
can explain differences in countries’ income levels.
Capital 𝐾𝐾 used for production is a stock variable, not a flow. Adding new capital to the
capital stock is a flow variable known as investment 𝐼𝐼. While capital is not immediately used
up in producing other goods, it does not last forever, in other words, there is depreciation.
Depreciation is the loss of capital from wear and tear or obsolescence, or maintenance costs
incurred to avoid this loss. We assume depreciation of capital takes place at a constant rate
𝑑𝑑 over time. The following equation describes the process of capital accumulation:
𝐾𝐾 ′ = (1 − 𝑑𝑑 )𝐾𝐾 + 𝐼𝐼
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EC2065 Macroeconomics | Chapter 1: The supply side of the economy
Next year’s capital stock 𝐾𝐾′ is equal to capital left over (1 − 𝑑𝑑 )𝐾𝐾 after depreciation from the
current time plus investment 𝐼𝐼.
Focusing on capital and labour as the relevant factors of production and ignoring land, the
production function is:
𝑌𝑌 = 𝑧𝑧𝑧𝑧 (𝐾𝐾, 𝑁𝑁)
GDP is 𝑌𝑌, the labour force is 𝑁𝑁 (the number of workers), the capital stock is 𝐾𝐾, and total
factor productivity (TFP) is 𝑧𝑧. The production function 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑁𝑁) is assumed to be
neoclassical, and the most important of the neoclassical assumptions here is the diminishing
marginal product of capital. The Cobb-Douglas production function 𝑌𝑌 = 𝑧𝑧𝐾𝐾 𝛼𝛼 𝑁𝑁 1−𝛼𝛼 is one
such example of a neoclassical production function. The value of the parameter 𝛼𝛼 could be
estimated using the capital share of income.
How is the capital stock 𝐾𝐾 measured? There are two approaches. First, the perpetual
inventory method. The change in the capital stock from one year to the next can be
calculated using 𝐾𝐾 ′ − 𝐾𝐾 = 𝐼𝐼 − 𝑑𝑑𝑑𝑑. Hence, given an estimate of 𝐾𝐾, the capital stock 𝐾𝐾′ can
be estimated by adding investment 𝐼𝐼 from the national accounts and subtracting an
estimate of the depreciation rate 𝑑𝑑 multiplied by 𝐾𝐾. Starting from some conjectured initial
value, this method can be applied iteratively to construct a time series of capital-stock
estimates.
A second method is based on imputation from capital income. Suppose we have an estimate
of the gross percentage return on capital 𝑅𝑅. Then given a measure of GDP 𝑌𝑌 and the capital
share of GDP 𝛼𝛼, the implied capital stock is 𝐾𝐾 = 𝛼𝛼𝛼𝛼⁄𝑅𝑅 .
In studying capital accumulation, we will mainly be concerned with the output per worker
𝑦𝑦 = 𝑌𝑌/𝑁𝑁, not total output 𝑌𝑌. Since the production function 𝑌𝑌 = 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑁𝑁) has constant
returns to scale, output per worker is given by:
𝑦𝑦 =
𝐾𝐾 𝑁𝑁
𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑁𝑁)
= 𝑧𝑧𝑧𝑧 � , � = 𝑧𝑧𝑧𝑧 (𝑘𝑘, 1) = 𝑧𝑧𝑧𝑧(𝑘𝑘)
𝑁𝑁
𝑁𝑁 𝑁𝑁
This shows that 𝑦𝑦 depends on capital per worker 𝑘𝑘 = 𝐾𝐾/𝑁𝑁 and TFP 𝑧𝑧. The equation 𝑦𝑦 =
𝑧𝑧𝑧𝑧(𝑘𝑘) is called the ‘per worker production function’, where the function 𝑓𝑓 (𝑘𝑘) is defined by
𝑓𝑓 (𝑘𝑘) = 𝐹𝐹(𝑘𝑘, 1). It is an increasing and concave function as shown in Figure 1.19 because
𝑧𝑧𝑓𝑓′(𝑘𝑘) is the marginal product of capital.
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EC2065 Macroeconomics | Chapter 1: The supply side of the economy
Figure 1.19: Output per worker and capital per worker
1.8 The Solow model
The Solow model explains the level of capital accumulation in an economy with a
neoclassical production function 𝑌𝑌 = 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑁𝑁) and its implications for the level and growth
rate of real GDP. To begin with, we will assume that TFP 𝑧𝑧 is constant over time. Later in 0,
we consider a version of the Solow model where 𝑧𝑧 is increasing over time owing to
technological progress.
The Solow model focuses on capital accumulation, so the supply of labour 𝑁𝑁, which is both
the labour force and the population, is taken to be exogenous. Assume that 𝑁𝑁 grows at rate
𝑛𝑛 over time. The Solow model assumes a closed economy with no government sector, which
implies investment 𝐼𝐼 is equal to household saving 𝑆𝑆 in equilibrium. This follows from the
definition 𝑆𝑆 = 𝑌𝑌 − 𝐶𝐶 with no taxes, and the goods-market equilibrium condition 𝑌𝑌 = 𝐶𝐶 + 𝐼𝐼
with no public expenditure or international trade. Households’ saving behaviour is
exogenous – specifically, households save a given fraction 𝑠𝑠 of income 𝑌𝑌.
Mathematically, the assumptions of the Solow model are the following equations:
•
•
•
•
•
The neoclassical production function: 𝑌𝑌 = 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑁𝑁)
The capital accumulation equation: 𝐾𝐾 ′ = (1 − 𝑑𝑑 )𝐾𝐾 + 𝐼𝐼
Investment equals saving condition: 𝐼𝐼 = 𝑆𝑆
Households’ saving behaviour: 𝑆𝑆 = 𝑠𝑠𝑠𝑠
The labour force grows at a constant rate: 𝑁𝑁 ′ = (1 + 𝑛𝑛)𝑁𝑁
Putting together the production function, the requirement that investment equals saving
and the fixed saving rate, the implications for next year’s capital stock 𝐾𝐾′ are:
𝐾𝐾 ′ = (1 − 𝑑𝑑 )𝐾𝐾 + 𝑠𝑠𝑠𝑠𝑠𝑠(𝐾𝐾, 𝑁𝑁)
Together with the demographic assumption of constant population growth, we have
equations for 𝐾𝐾′ and 𝑁𝑁′ in terms of 𝐾𝐾 and 𝑁𝑁, so we can calculate how the supplies of factors
of production evolve over time in the economy.
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The per worker production function 𝑦𝑦 = 𝑧𝑧𝑧𝑧(𝑘𝑘) implies output per worker depends on the
ratio of 𝐾𝐾 to 𝑁𝑁. As 𝑦𝑦 is the ultimate variable of interest, we only need to keep track of the
ratio of capital per worker 𝑘𝑘 = 𝐾𝐾/𝑁𝑁 over time. By combining the equations for 𝐾𝐾 ′ and 𝑁𝑁 ′ :
𝑘𝑘 ′ =
𝐾𝐾 ′ (1 − 𝑑𝑑 )𝐾𝐾 + 𝑠𝑠𝑠𝑠 (1 − 𝑑𝑑 )𝑘𝑘 + 𝑠𝑠𝑠𝑠
=
=
𝑁𝑁′
(1 + 𝑛𝑛)𝑁𝑁
1 + 𝑛𝑛
Using 𝑦𝑦 = 𝑧𝑧𝑧𝑧(𝑘𝑘), the right-hand side of the equation depends on capital per worker 𝑘𝑘 only:
𝑘𝑘 ′ =
(1 − 𝑑𝑑 )𝑘𝑘 + 𝑠𝑠𝑠𝑠𝑠𝑠(𝑘𝑘)
1 + 𝑛𝑛
Subtracting 𝑘𝑘 from both sides leads to an equation for the change in 𝑘𝑘 over time:
𝑘𝑘 ′ − 𝑘𝑘 =
𝑠𝑠𝑠𝑠𝑠𝑠 (𝑘𝑘) − (𝑑𝑑 + 𝑛𝑛)𝑘𝑘
1 + 𝑛𝑛
We conclude from this equation that changes in the amount of capital accumulated per
worker are explained by the difference between two terms. First, 𝑠𝑠𝑠𝑠𝑠𝑠 (𝑘𝑘), the amount of
saving and hence of investment per worker, which is the saving rate 𝑠𝑠 multiplied by the perworker production function 𝑦𝑦 = 𝑧𝑧𝑧𝑧(𝑘𝑘). Second, (𝑑𝑑 + 𝑛𝑛)𝑘𝑘, the amount of investment per
worker needed to sustain the same level of capital per worker next year. This interpretation
comes from a fraction 𝑑𝑑 of all capital depreciating, so an amount of capital per worker 𝑑𝑑𝑑𝑑
must be replaced to keep capital per worker unchanged. Furthermore, the number of
workers increases by a percentage 𝑛𝑛 each year, so if existing workers use capital 𝑘𝑘 each,
there needs to be investment 𝑛𝑛𝑛𝑛 per existing worker to give future workers the same capital
𝑘𝑘 each as current workers.
1.8.1 The Solow diagram
We can use a diagram to study the evolution over time of capital per worker 𝑘𝑘 and its
implications for output per worker 𝑦𝑦. Figure 1.20 is this key diagram of the Solow model
that plots:
•
•
•
The per worker production function 𝑦𝑦 = 𝑧𝑧𝑧𝑧(𝑘𝑘), which is an increasing and concave
function of 𝑘𝑘.
The ‘saving line’ 𝑠𝑠𝑠𝑠𝑠𝑠(𝑘𝑘), a scaled-down version of 𝑧𝑧𝑧𝑧(𝑘𝑘) because of 0 < 𝑠𝑠 < 1 and,
hence, an increasing and concave function of 𝑘𝑘.
The ‘effective depreciation line’ (𝑑𝑑 + 𝑛𝑛)𝑘𝑘, an upward-sloping straight line with
gradient given by the effective depreciation rate of capital per worker: the sum of
the depreciation rate plus the growth rate of the labour force.
Starting from a low level of capital per worker, the saving line is above the effective
depreciation line because the production function and saving line are initially very steep,
reflecting a high marginal product of capital. This means 𝑘𝑘 ′ − 𝑘𝑘 > 0, so capital per worker 𝑘𝑘
is increasing over time. This leads to higher output per worker 𝑦𝑦 since 𝑦𝑦 = 𝑧𝑧𝑧𝑧(𝑘𝑘), with the
economy moving along the per worker production function over time. Starting from this
position, by adding more capital per worker, workers’ productivity can be increased and the
economy experiences growth in output per worker coming from capital accumulation.
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EC2065 Macroeconomics | Chapter 1: The supply side of the economy
Figure 1.20: The Solow model diagram
The steady state of the Solow model
It is an important implication of the Solow model that such growth in output per worker
cannot continue indefinitely. We can see this from the Solow model diagram because there
is a steady state where the saving line crosses the effective depreciation line. If the economy
reaches this level of capital per worker 𝑘𝑘, it will remain at that level of 𝑘𝑘 unless something
changes. Mathematically, if 𝑠𝑠𝑠𝑠𝑠𝑠 (𝑘𝑘) = (𝑑𝑑 + 𝑛𝑛)𝑘𝑘 then 𝑘𝑘 ′ = 𝑘𝑘.
With no further change in 𝑘𝑘, there is no further growth in output per worker 𝑦𝑦 unless
something else changes because 𝑦𝑦 = 𝑧𝑧𝑧𝑧(𝑘𝑘). Notice that both GDP 𝑌𝑌 and the total capital
stock 𝐾𝐾 are still growing in line with the labour force 𝑁𝑁 at rate 𝑛𝑛 but the more important
variable is how much is produced and earned per worker.
Does the basic Solow model always have such a steady state where growth in per worker
incomes grinds to a halt? If the production function is neoclassical, the gradient of 𝑓𝑓(𝑘𝑘) is
extremely large for 𝑘𝑘 close to zero but declines as 𝑘𝑘 increases and approaches zero as 𝑘𝑘
becomes large. The saving line 𝑠𝑠𝑠𝑠𝑠𝑠(𝑘𝑘), as a multiple of 𝑓𝑓(𝑘𝑘), shares the same properties.
On the other hand, the gradient of the effective depreciation line is constant. Consequently,
there exists a positive steady state 𝑘𝑘 ∗ (and only one) where the saving and effective
depreciation lines cross.
Moreover, the saving line is above the effective depreciation line for 𝑘𝑘 below 𝑘𝑘 ∗ , which
means that 𝑘𝑘 rises over time when it is below 𝑘𝑘 ∗ (and would fall over time if above 𝑘𝑘 ∗ ). The
economy thus converges to 𝑘𝑘 ∗ in the long run, so this point of stagnation is eventually
reached.
The Solow model also always has a steady state with zero capital per worker because some
capital is essential for production with a neoclassical production function. However, the
economy would diverge from this steady state no matter how close it gets, so this does not
need to be taken seriously and is ignored in what follows.
In summary, while the Solow model can explain how a country can become richer starting
from a low level of capital accumulation, it cannot explain long-run growth. Intuitively,
returns to capital are high when capital is initially scarce, so investment leads to large
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EC2065 Macroeconomics | Chapter 1: The supply side of the economy
increases in income. After that, diminishing returns to capital means that further investment
has lower returns and, with less extra output generated per unit of capital while the
maintenance cost of capital increases proportionately, a point is reached where the capital
stock cannot rise any further.
Box 1.5: The ‘Asian tiger’ economies
The so-called ‘Asian tiger’ economies (Singapore, Taiwan, Hong Kong, South Korea)
experienced very fast economic growth in the period from 1960 to 1990 but their growth
rates subsequently declined. The process of development in these economies provides a
good example of the mechanisms at work in the Solow model.
Let us consider Singapore for illustration. Figure 1.21 shows that Singapore has had a very
high national saving rate, which was above 40 per cent since the 1980s and sometimes
even above 50 per cent. While not all of this saving was channelled into domestic
investment, the share of investment in GDP was also very high in Singapore. Consequently,
Singapore’s economy experienced rapid capital accumulation, although it started from a
low base in the 1960s. The time series of the average amount of capital per worker in
Singapore is plotted in Figure 1.22. The amount of capital per worker increases by
approximately 400 per cent (in real terms) over the 40-year period after 1970.
Figure 1.21: Singapore national saving rate
This process of rapid capital accumulation is what the Solow model predicts for an
economy with a high saving rate and a low initial level of capital per worker. The model
predicts this would lead to a significant increase in income per worker and income per
person as the economy moves along the per worker production function. Data on real
income per person are shown in Figure 1.23 and we see that there is a dramatic
improvement in prosperity.
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EC2065 Macroeconomics | Chapter 1: The supply side of the economy
Figure 1.22: Singapore capital per worker
Another prediction of the Solow model is that the rate of progress, specifically, the growth
rate of income per person would gradually slow down over time, even if the economy
maintained very high rates of saving. It is not easy to read growth rates from a graph of
income plotted on an ordinary scale against time. By taking logarithms of the data, or
using a logarithmic scale, the gradient of the time series is informative about the growth
rate.
Figure 1.23: Singapore real GDP per person
Figure 1.24 plots the natural logarithm of real income per person against time. Here we
see that the gradient of the graph tends to decrease over time, indicating that economic
growth is slowing down in Singapore. This happens even though the saving rate does not
fall over this period but actually rises. Such a growth slowdown as capital per worker rises
is in line with the Solow model’s prediction, which is a consequence of diminishing returns
to capital.
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EC2065 Macroeconomics | Chapter 1: The supply side of the economy
Figure 1.24: Singapore log real GDP per person
The growth slowdown can be seen more directly in Figure 1.25, which plots growth rates of
real income per person averaged over each decade.
Figure 1.25: Singapore average growth rates by decade
While these predictions are consistent with the basic Solow model, we do not necessarily see
evidence that growth rates of income per person are falling all the way to zero. Even in
developed economies that have been experiencing growth for more than a century, we still
typically see growth being positive. In contrast, the basic Solow model predicts that the longrun growth rate of income per person is zero. The inability to explain long-run growth is one
of the major weaknesses of the Solow model and we will return to this issue in Chapter 2.
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Box 1.6: Interest rates in the long run
Capital accumulation in the Solow model is financed by households’ saving and,
moreover, in a closed economy with no government debt that investment is the only
outlet for those savings. What then are the Solow model’s implications for the return
received by savers?
The real interest rate 𝑟𝑟 is real return earned from owning capital. In an economy with
competitive markets, owners of capital are able to rent it to firms at price 𝑅𝑅 = 𝑀𝑀𝑃𝑃𝐾𝐾 , the
marginal product of capital. The real return 𝑟𝑟 on capital is equal to the rental price 𝑅𝑅
minus the cost of replacing capital lost through depreciation, which is 𝑑𝑑 per unit of
capital.
𝑟𝑟 = 𝑅𝑅 − 𝑑𝑑
Firms are producing output according to the production function 𝑌𝑌 = 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑁𝑁). In per
worker terms, this production function is 𝑦𝑦 = 𝑧𝑧𝑧𝑧(𝑘𝑘), where 𝑓𝑓 (𝑘𝑘) = 𝐹𝐹(𝑘𝑘, 1). Since
aggregate output can be written as 𝑌𝑌 = 𝑧𝑧𝑧𝑧𝑧𝑧(𝐾𝐾⁄𝑁𝑁), the marginal product of capital is
𝑀𝑀𝑃𝑃𝐾𝐾 = 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 = 𝑧𝑧𝑧𝑧′(𝐾𝐾⁄𝑁𝑁) = 𝑧𝑧𝑧𝑧′(𝑘𝑘). Therefore, the real return on capital is
𝑟𝑟 = 𝑧𝑧𝑓𝑓 ′ (𝑘𝑘) − 𝑑𝑑
The Solow model predicts that capital per worker 𝑘𝑘 converges to a steady state in the
long run. Starting from a low initial level, the stock of capital rises over time relative to
the number of workers as the economy converges to its steady state. With a diminishing
marginal product of capital, 𝑓𝑓′(𝑘𝑘) is decreasing in 𝑘𝑘, so this means 𝑟𝑟 declines as capital
per worker increases. The Solow model predicts the real return 𝑟𝑟 received by savers falls
over time, ultimately converging to a steady state. This argument is illustrated
graphically in Figure 1.26. As we will see, this steady state for 𝑟𝑟 could be positive or
negative.
Figure 1.26: Real interest rates over time in the Solow model
What can we say about real returns empirically over time? It is not easy to measure 𝑟𝑟 in
a consistent way over very long periods. Figure 1.27 plots a time series of real interest
rates on UK government bonds over a 300-year period. This real bond yield is taken as a
proxy for the real return earned by savers (even though government bonds are
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EC2065 Macroeconomics | Chapter 1: The supply side of the economy
not included in the Solow model, if they were, savers would have a choice of holding
bonds or capital, so the returns on the two assets would be linked).
Real interest rates in the UK have fluctuated over a wide range during those three
centuries, sometimes being as high as 8 per cent and sometimes turning negative.
Looking at the picture as a whole, it appears there is a moderate downward trend in
real interest rates over time.
Figure 1.27: UK real interest rates in the long run
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EC2065 Macroeconomics | Chapter 2: Economic growth
Chapter 2: Economic growth
This chapter delves deeper into the study of economic growth that was introduced in
Chapter 1. Economic growth refers to increases over time in the level of real GDP per
person, which are a crucial component of living standards in an economy. In Chapter 1, we
have seen several models that fail to explain sustained economic growth. The Malthusian
model predicts living standards can stagnate at extremely low levels of income close to
subsistence. The Solow model, while it can explain economic growth through capital
accumulation, for example, when an economy industrialises, it fails to explain why economic
growth will continue in the long run.
This chapter explores alternative models that help to understand long-run growth. These
models focus on human capital and technological progress rather than physical capital
accumulation. We will also look at the implications of growth models for levels of income
per person across countries.
Essential reading
•
Williamson, Chapters 7 and 8.
2.1 Evidence on economic growth
We will use real GDP per person, or sometimes per worker, as a measure of living standards.
This variable is defined by 𝑦𝑦𝑡𝑡 = GDP / Population and the growth rate of GDP per person is
𝑔𝑔𝑡𝑡 = (𝑦𝑦𝑡𝑡 − 𝑦𝑦𝑡𝑡−1 )/𝑦𝑦𝑡𝑡−1 . Figure 2.1 plots a time series of real GDP per person in the USA
between 1870 and 2018, which is measured in units of 2011 dollars per person. We see that
per person real income has risen by more than 10 times over this period of almost 150
years.
Figure 2.1: Economic growth in the USA
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EC2065 Macroeconomics | Chapter 2: Economic growth
2.1.1 Measuring economic growth
While we can see from the graph that incomes have increased over time in the USA, the
growth rate of real income per person is not simply the gradient of this plot of 𝑦𝑦𝑡𝑡 against
time 𝑡𝑡. Instead, the gradient of a plot of the natural logarithm of real GDP per person is
approximately equal to 𝑔𝑔𝑡𝑡 . The gradient of ln 𝑦𝑦𝑡𝑡 plotted against time 𝑡𝑡 is
(ln 𝑦𝑦𝑡𝑡 − ln 𝑦𝑦𝑡𝑡−1 )⁄(𝑡𝑡 − (𝑡𝑡 − 1)) = ln 𝑦𝑦𝑡𝑡 − ln 𝑦𝑦𝑡𝑡−1 . By using ln(1 + 𝑔𝑔) ≈ 𝑔𝑔 for small 𝑔𝑔, this
gradient is ln 𝑦𝑦𝑡𝑡 − ln 𝑦𝑦𝑡𝑡−1 = ln 𝑦𝑦𝑡𝑡 ⁄𝑦𝑦𝑡𝑡−1 = ln(1 + 𝑔𝑔𝑡𝑡 ) ≈ 𝑔𝑔𝑡𝑡 . Hence, when studying long-run
economic growth, it makes sense to plot time series data in logarithms. This is done for US
real GDP per person in Figure 2.2.
Figure 2.2: US real GDP per person in logarithms
2.1.2 Why growth matters
Taking the period from 1870 to 2000 in the USA, real GDP per person reached 𝑦𝑦2000 =
$36,000 in the year 2000 (measured in year-2000 dollars). The growth rate over this 130year period was 𝑔𝑔 =1.75% per year on average. Small differences in 𝑔𝑔, if they had been
sustained over this long period of time 1870-2000, would have made a huge difference to
the resulting level of 𝑦𝑦. If US growth rates had been a percentage point lower on average at
𝑔𝑔 = 0.75% then we would have had 𝑦𝑦2000 = $10,000, leaving it as merely a middle-income
country. On the other hand, if 𝑔𝑔 = 2.75% instead then we would have had 𝑦𝑦2000 =
$120,000, a level of income per person not yet attained in any country.
These numerical examples demonstrate why long-run economic growth is so important. A
small difference in growth rates compounded over a long period of time leads to
dramatically different economic outcomes.
One way to appreciate the magnitude of the difference between growth rates is known as
the ‘rule of 70’. It states that if economic growth occurs at 𝑔𝑔 per cent then it takes
approximately 70/𝑔𝑔 years for income to double. For example, the 2 per cent growth seen as
normal in a developed country implies a doubling of income in 35 years, with each
generation enjoying living standards at twice the level of their parents’ generation.
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EC2065 Macroeconomics | Chapter 2: Economic growth
Some countries have experienced much faster growth. In the ‘Asian tiger’ economies,
growth rates were often above 5 per cent in the 1960–90 period. This implies income
doubles in less than 15 years. Other countries have experienced even faster growth rates.
China grows at a rate of 6.3 per cent on average between 1980 and 2000, and even more
rapidly in the following decade. Figure 2.3 shows this led China’s income per person, which
in 1980 was only 5 per cent of the level in the USA, to rise to more than 20 per cent of the
US level by 2010. Income per person in India was also close to 5 per cent of the US level in
1980 but, owing to a lower growth rate than China’s, it rises only to 10 per cent of the USA
by 2016.
Figure 2.3: Why growth matters
2.1.3 Economic growth in historical perspective
Although sustained economic growth has come to be taken for granted in advanced
economies, it is a relatively recent phenomenon from a historical perspective. For much of
history, most countries experienced stagnation in income per person. This stagnation is not
because there was no growth in total GDP but because increases in population offset any
rise in GDP. However, starting in the 18th and 19th centuries, some countries began to
experience ongoing growth in income per person.
Economic historians have estimated levels of real GDP per person over the last 2000 years
using historical sources. Figure 2.4 shows these estimates for a number of countries (strictly
speaking, the geographical areas spanned by the modern countries).
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Figure 2.4: Stagnation for much of history
For much of the last 2000 years, the graphs of real GDP per person over time are close to
flat lines as economies stagnated. Only in the last 300 years do we see sustained economic
growth in some countries – modern economic growth – with others only beginning to see
significant growth in the last 40–50 years.
Although GDP per person stagnates until relatively recently, there is sustained population
growth in some parts of the world over a much longer period of time. Figure 2.5 shows plots
of estimated populations over time for a number of countries. This evidence is in line with a
version of the Malthusian model from Section 1.3 where there are improvements in
technology. This technological progress does not lead to sustained rises in living standards
but does increase the population.
Figure 2.5: Rising world population
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2.1.4 The distribution of income across countries
Prior to the 19th century, differences in income across the countries of the world were
much smaller than today. Research suggests the more successful countries had incomes that
were only around twice those of the less successful countries. As countries began to
experience modern economic growth at different times, they diverged in the 19th century.
Continuing through the 20th century, huge differences in income per person have opened
up across countries. It is not unusual to have difference of 10–20 times between average
incomes per person in rich and poor countries.
Figure 2.6 shows the world income distribution across countries in 2017. Average incomes
per person are expressed in comparable units across countries by adjusting for purchasing
power parity and converting into a US dollar equivalent. Ordering countries by their income
levels, real GDP per person is plotted against the percentiles of the world population
(ignoring any income differences within countries). We see that much of the world
population lives in countries that are far below the income levels achieved in advanced
economies.
Figure 2.6: World income distribution across countries
2.1.5 Convergence
Given the large income differences that have arisen between countries, an important
question is whether poor countries will remain much poorer than rich countries in the
future, or whether they will catch up. For poorer countries to close the gap with richer
countries, the poor must grow faster than the rich. If this happens, we say that those
countries are converging.
We distinguish between two notions of convergence. First, absolute convergence, which
means simply that poor countries grow faster than the rich. Second, conditional
convergence, which means that among a group of countries with similar fundamentals,
poorer ones grow faster than richer ones. Data from the post-Second-World-War period
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suggest convergence among some groups of countries but certainly not worldwide. This
means there is evidence of conditional convergence, but not absolute convergence.
Figure 2.7 is a scatterplot of the annual average growth rates of real GDP per person
achieved by a large number of countries over the period 1960–2017 against their initial
levels of real GDP per person in 1960 (measured in comparable 2017 US dollars). If there
were absolute convergence, we would expect to see a clear negative relationship between
initial income levels and subsequent growth rates as poorer countries grow faster than
richer countries. However, this is not the case. While some poorer countries grow faster,
there are many examples of poorer countries that grow at slower rates than richer
countries. Overall, the correlation between growth rates and initial incomes is close to zero.
Focusing on a narrower set of countries, Figure 2.8 restricts the sample to OECD (mainly
Europe, North America and Australasia) and East Asian countries. Now, there is a clear
negative relationship between growth rates and initial income levels, providing evidence of
convergence within this group. To the extent that these countries can be seen as having
similar fundamentals, this indicates that conditional convergence is taking place. Observe
that the growth rates in East Asian countries are even higher than the relationship between
growth and initial incomes for the OECD would suggest. Restricting attention to a narrower
group of countries would provide even clearer evidence of conditional convergence within
that group.
Figure 2.7: Limited evidence of convergence worldwide
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EC2065 Macroeconomics | Chapter 2: Economic growth
Figure 2.8: Convergence (OECD and East Asia)
One characteristic of some groups of countries is that they save and invest a greater share
of their GDP than the global average. For example, high levels of saving and investment are
a feature of many East Asian countries. Figure 2.9 is a scatterplot of countries’ investment
shares of GDP and their levels of real GDP per person. While the relationship is far from
perfect, overall, we can say that investment rates and levels of real GDP per person are
positively correlated. Differences in the amount of resources countries devote to capital
accumulation is thus one fundamental reason why they might not converge to the same
level of income per person in the long run.
Figure 2.9: Investment rates and income
2.2 Income and growth rates across countries
In Section 1.8, we saw that the Solow model can explain growth for a period in an economy
that begins with a low level of capital per worker but also that the model fails to explain
growth in the long run. Before turning to alternative models of long-run economic growth,
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EC2065 Macroeconomics | Chapter 2: Economic growth
we will first explore the implications of the Solow model for the cross-section of countries.
In particular, does the Solow model help in understanding the large differences in growth
rates and levels of income per person across countries?
In the cross-section of countries, the Solow model makes two key predictions. First, if
countries share the same fundamentals such as technologies and saving rates, then they
should converge to the same level of output per worker in the long run, even if they start
from different initial conditions. This means that controlling for a group of countries with
the same fundamentals, the initially poorer countries in the group should grow faster than
the initially richer countries. The Solow model therefore predicts conditional convergence.
The second key prediction is that if saving rates differ across countries, then there will be
differences in output per worker even in the long run. This means that the Solow model
does not predict absolute convergence (except in the unrealistic case where every country
in the world shares the same fundamentals).
We now consider the logic for the first prediction. If countries share the same fundamentals,
then they have the same lines in the Solow model diagram introduced in Section 1.8. A
version of this diagram is reproduced in Figure 2.10. There is a per worker production
function 𝑧𝑧𝑧𝑧(𝑘𝑘), a saving line 𝑠𝑠𝑠𝑠𝑠𝑠(𝑘𝑘), and an effective depreciation line (𝑑𝑑 + 𝑛𝑛)𝑘𝑘. The
fundamentals are the saving rate 𝑠𝑠, the population growth rate 𝑛𝑛, the level of total factor
productivity (TFP) 𝑧𝑧 and the per worker production function 𝑓𝑓(𝑘𝑘) itself.
As the fundamentals are the same for the countries we are considering, all have the same
per worker production function, saving line and effective depreciation line. This implies they
share the same steady state for 𝑘𝑘 and 𝑦𝑦, which is found where the saving line intersects the
depreciation line.
Figure 2.10: Conditional convergence
With the fundamentals being the same, the only differences between countries are the
amounts of capital per worker they have accumulated owing to different histories.
Specifically, those countries that have accumulated less capital begin poorer because a
lower level of 𝑘𝑘 implies less 𝑦𝑦 = 𝑧𝑧𝑧𝑧(𝑘𝑘) with 𝑧𝑧 being the same. The more 𝑘𝑘 is below the
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EC2065 Macroeconomics | Chapter 2: Economic growth
steady state 𝑘𝑘 ∗ , the larger the gap between the saving and effective depreciation lines
relative to the level of 𝑘𝑘. Since this gap determines the change in capital per worker over
time and, hence, output per worker, poorer countries grow faster than richer countries. All
end up converging to the same steady-state income level in the long run.
Now suppose countries are identical in all fundamentals except for their saving rates 𝑠𝑠.
Different levels of 𝑠𝑠 imply different saving lines 𝑠𝑠𝑠𝑠𝑠𝑠(𝑘𝑘) and different saving lines result in
different steady-state levels of capital per worker, hence, output per worker. This is
illustrated in Figure 2.11. A higher saving rate allows a higher stock of capital per worker to
be sustained for the same production technology. With more capital, output per worker is
higher, although not necessarily consumption (see the analysis of the golden rule in Section
2.5). Long-run growth is always zero and is not affected by the saving rate.
Figure 2.11: Differences in saving rates
The logic of the second prediction of the Solow model is that if countries are converging to
different steady states, there is no presumption that starting from a lower level of income
per worker leads to faster subsequent growth. A poor country with a low saving rate might
be closer to its steady state than a richer one with a high saving rate, so the poorer country
grows more slowly. The countries might thus move further apart, indicating a failure of
absolute convergence. However, given the difference in saving rates, the ratio of incomes
across the two countries must stop widening further in the long run.
Box 2.1: Can the Solow model explain large income differences
across countries?
In principle, differences in saving rates and the resulting levels of capital
accumulation in the Solow model could explain why some countries are rich and
others are poor. How large are the differences in income levels predicted by the
Solow model for realistic differences in saving rates across countries? We will focus
on the income differences that will prevail in the long run, i.e. where economies
have reached their steady-state levels of capital per worker.
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Performing this exercise quantitatively requires some specific assumptions. We will
work with a Cobb-Douglas production function 𝑌𝑌 = 𝑧𝑧𝐾𝐾 𝛼𝛼 𝑁𝑁 1−𝛼𝛼 , which implies the
per worker production function 𝑦𝑦 = 𝑧𝑧𝑘𝑘 𝛼𝛼 . With competitive markets, the parameter
𝛼𝛼 is equal to the capital share of income, so we set 𝛼𝛼 = 1/3 as a reasonable value.
We will consider differences in saving rates 𝑠𝑠 across countries, but 𝑧𝑧 and 𝛼𝛼, and the
depreciation and population growth rates 𝑑𝑑 and 𝑛𝑛, are assumed to be the same
everywhere.
The Solow model’s long-run prediction for levels of income per person across
countries is found by solving for steady-state capital per worker 𝑘𝑘 ∗ . This is
determined by the equation:
𝑠𝑠𝑠𝑠𝑘𝑘 ∗𝛼𝛼 = (𝑑𝑑 + 𝑛𝑛)𝑘𝑘 ∗
This can be rearranged to obtain 𝑘𝑘 ∗1−𝛼𝛼 = 𝑠𝑠𝑠𝑠/(𝑑𝑑 + 𝑛𝑛) and hence the following
solution:
1
𝑠𝑠𝑠𝑠 1−𝛼𝛼
𝑘𝑘 ∗ = �
�
𝑑𝑑 + 𝑛𝑛
Substituting this into the per-worker production function 𝑦𝑦 ∗ = 𝑧𝑧𝑘𝑘 ∗𝛼𝛼 yields the
solution:
𝑦𝑦
∗
1
= 𝑧𝑧 1−𝛼𝛼 �
𝛼𝛼
𝑠𝑠 1−𝛼𝛼
�
𝑑𝑑 + 𝑛𝑛
If we consider two countries 𝐴𝐴 and 𝐵𝐵 where all parameters above are the same
except for the saving rates 𝑠𝑠𝐴𝐴 and 𝑠𝑠𝐵𝐵 , then dividing 𝑦𝑦𝐴𝐴∗ by 𝑦𝑦𝐵𝐵∗ results in:
𝛼𝛼
𝑦𝑦𝐴𝐴∗
𝑠𝑠𝐴𝐴 1−𝛼𝛼
∗ =� �
𝑦𝑦𝐵𝐵
𝑠𝑠𝐵𝐵
In a closed economy with no government sector, the saving rate 𝑠𝑠 is equal to the
ratio of investment to GDP. Looking at the evidence in Figure 2.9, investment
shares of GDP in countries with very high investment ratios are approximately five
times higher than those with very low investment ratios (for example, 35 per cent
in high-investment countries to 7 per cent in low-investment countries). Hence, we
suppose that country A saves a fraction of income five times higher than in country
B, that is, 𝑠𝑠𝐴𝐴 ⁄𝑠𝑠𝐵𝐵 = 5. Noting 𝛼𝛼 = 1/3 implies 𝛼𝛼⁄(1 − 𝛼𝛼 ) = 1/2, it follows that
𝑦𝑦𝐴𝐴∗ ⁄𝑦𝑦𝐵𝐵∗ = 51/2 ≈ 2.2. A saving rate five times higher thus implies income per person
only slightly more than double. Intuitively, although country 𝐴𝐴 accumulates more
capital than country 𝐵𝐵, diminishing returns to capital imply that the resulting
income difference is much smaller than the difference in saving rates.
With this reasonable parameterisation of the Solow model, very large difference in
rates of saving and investment lead only to modest difference in income levels. This
means the Solow model has a very limited capacity to explain the income
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EC2065 Macroeconomics | Chapter 2: Economic growth
differences of an order of magnitude of 10–20 between rich and poor countries seen in
Figure 2.6. There must be some other cause of these large income differences, perhaps
total factor productivity 𝑧𝑧.
Box 2.2: How long does convergence to the steady state take in the
Solow model?
A key prediction of the basic Solow model is that there is no growth in income per
person in the long run, i.e. when the economy has converged to its steady-state value of
capital per worker. The relevance of this prediction depends on how long it takes to get
close to the steady state – if this were to take hundreds of years then the basic Solow
model would have the ability to explain economic growth over quite long periods of
time. This exercise presents an example to illustrate the likely timeframe for
convergence to the steady state in the Solow model.
We assume the production function is 𝑌𝑌 = 𝑧𝑧√𝐾𝐾 √𝑁𝑁, which is a Cobb-Douglas form with
𝛼𝛼 = 1/2, implying that the capital and labour income shares are 50 per cent each if
markets are competitive. This is larger than the typical value of 𝛼𝛼 = 1/3, and the use of
a larger 𝛼𝛼 turns out to slow down convergence to the steady state. The per-worker
production function in this case is 𝑦𝑦 = �𝑧𝑧√𝐾𝐾 √𝑁𝑁�⁄𝑁𝑁 = 𝑧𝑧�𝐾𝐾⁄𝑁𝑁 = 𝑧𝑧√𝑘𝑘. The marginal
product of capital can be obtained by differentiating 𝑦𝑦 with respect to 𝑘𝑘, thus 𝑀𝑀𝑃𝑃𝐾𝐾 =
𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 = 𝑧𝑧⁄�2√𝑘𝑘� = 𝑦𝑦⁄(2𝑘𝑘). The steady state for capital per worker 𝑘𝑘 ∗ is the solution
of the equation 𝑠𝑠𝑦𝑦 ∗ = (𝑑𝑑 + 𝑛𝑛)𝑘𝑘 ∗ :
𝑠𝑠𝑠𝑠√𝑘𝑘 ∗ = (𝑑𝑑 + 𝑛𝑛)𝑘𝑘∗ ⇒ √𝑘𝑘 ∗ =
𝑠𝑠𝑠𝑠
𝑑𝑑 + 𝑛𝑛
⇒
𝑘𝑘 ∗ = �
𝑠𝑠𝑠𝑠 2
�
𝑑𝑑 + 𝑛𝑛
The steady state for output per worker 𝑦𝑦 ∗ is then found using 𝑦𝑦 ∗ = 𝑧𝑧√𝑘𝑘 ∗ :
𝑦𝑦 ∗ = 𝑧𝑧 �
𝑠𝑠𝑠𝑠
𝑠𝑠𝑧𝑧 2
�=
𝑑𝑑 + 𝑛𝑛
𝑑𝑑 + 𝑛𝑛
Since the marginal product of capital 𝑀𝑀𝑃𝑃𝐾𝐾 is the effect on output per worker of a
marginal increase in capital per worker, the change in output 𝑦𝑦 ′ − 𝑦𝑦 is approximately
equal to 𝑀𝑀𝑃𝑃𝐾𝐾 multiplied by 𝑘𝑘 ′ − 𝑘𝑘. Hence, using 𝑘𝑘 ′ − 𝑘𝑘 = (𝑠𝑠𝑠𝑠 − (𝑑𝑑 + 𝑛𝑛)𝑘𝑘)⁄(1 + 𝑛𝑛):
𝑦𝑦 ′
𝑦𝑦 𝑠𝑠𝑠𝑠 − (𝑑𝑑 + 𝑛𝑛)𝑘𝑘
𝑠𝑠𝑦𝑦 2 ⁄𝑘𝑘 − (𝑑𝑑 + 𝑛𝑛)𝑦𝑦
(𝑘𝑘 − 𝑘𝑘) =
𝑦𝑦 − 𝑦𝑦 ≅
�
�=
1 + 𝑛𝑛
2(1 + 𝑛𝑛)
2𝑘𝑘
2𝑘𝑘
′
The per-worker production function implies 𝑦𝑦 2 ⁄𝑘𝑘 = 𝑧𝑧 2 , so 𝑦𝑦 ′ − 𝑦𝑦 can be expressed as:
𝑦𝑦 ′ − 𝑦𝑦 ≅
(𝑑𝑑 + 𝑛𝑛) 𝑠𝑠𝑧𝑧 2
�
− 𝑦𝑦�
2(1 + 𝑛𝑛) 𝑑𝑑 + 𝑛𝑛
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Recalling that the steady state is 𝑦𝑦 ∗ = 𝑠𝑠𝑧𝑧 2⁄(𝑑𝑑 + 𝑛𝑛), the change in income per
person is 𝑦𝑦 ′ − 𝑦𝑦 ≅ ((𝑑𝑑 + 𝑛𝑛)⁄(2(1 + 𝑛𝑛)))(𝑦𝑦 ∗ − 𝑦𝑦). The gap between 𝑦𝑦 and its
steady-state value 𝑦𝑦 ∗ is 𝑦𝑦 ∗ − 𝑦𝑦, and the change in this gap over time is (𝑦𝑦 ∗ − 𝑦𝑦 ′ ) −
(𝑦𝑦 ∗ − 𝑦𝑦) = 𝑦𝑦 − 𝑦𝑦 ′ , so:
(𝑦𝑦 ∗ − 𝑦𝑦 ′ ) − (𝑦𝑦 ∗ − 𝑦𝑦) ≅ −
(𝑑𝑑 + 𝑛𝑛) ∗
(𝑦𝑦 − 𝑦𝑦)
2(1 + 𝑛𝑛)
Therefore, the dynamics of the gap to steady state 𝑦𝑦 ∗ − 𝑦𝑦 are determined by:
(𝑦𝑦 ∗ − 𝑦𝑦 ′ ) ≅ �1 −
(𝑑𝑑 + 𝑛𝑛)
� (𝑦𝑦 ∗ − 𝑦𝑦)
2(1 + 𝑛𝑛)
Taking 𝑑𝑑 ≈ 9 per cent and 𝑛𝑛 ≈ 1 per cent as reasonable values of the depreciation
rate of capital and the population growth rate results in (𝑦𝑦 ∗ − 𝑦𝑦 ′ ) ≈ 0.95(𝑦𝑦 ∗ − 𝑦𝑦).
This means 5 per cent of the gap between 𝑦𝑦 and its steady state 𝑦𝑦 ∗ is closed each
year. The rule-of-70 then implies approximately half of the gap closed in 70⁄5 = 14
years and three quarters of the gap in approximately 28 years. With only a small gap
remaining after four decades, the zero steady-state growth prediction of the Solow
model is reached after an economy has been accumulating capital for a relatively
short space of time.
2.3 Technological progress
With the basic Solow model being unable to generate long-run growth and the long run of
the model being reached in just a few decades, it is necessary to turn to other explanations
of why advanced economies have continued to experience sustained economic growth for
more than a century. We will now explore the possibility of growth in total factor
productivity (TFP) 𝑧𝑧.
Technological progress is one reason why total factor productivity 𝑧𝑧 can increase over time.
New ideas provide new uses for capital and labour, or lead to efficiency gains in producing
existing goods. In the Solow model diagram, a one-off increase in 𝑧𝑧 shifts up the per worker
production function 𝑧𝑧𝑧𝑧(𝑘𝑘) and saving line 𝑠𝑠𝑠𝑠𝑠𝑠(𝑘𝑘) as shown in Figure 2.12.
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Figure 2.12: Technological progress
The improvement in technology results in higher steady-state values of 𝑘𝑘 and 𝑦𝑦. It is
possible to sustain a higher level of capital and output per worker with the same saving rate
𝑠𝑠 because technological progress raises the marginal product of capital. However, a one-off
improvement in technology ultimately leads only to a one-off improvement in output per
worker. This means explaining long-run growth requires ongoing technological progress.
2.3.1 A constant growth rate of technology
In studying ongoing technological progress, we assume the following production function:
𝑌𝑌 = 𝐹𝐹(𝐾𝐾, 𝐴𝐴𝐴𝐴)
Instead of TFP 𝑧𝑧, a variable 𝐴𝐴 represents labour-augmenting technology, labour-augmenting
in that 𝐴𝐴 scales up labour input 𝑁𝑁. With labour-augmenting technology, effective labour
input into production is not 𝑁𝑁 but 𝐴𝐴𝐴𝐴. The function 𝐹𝐹(𝐾𝐾, 𝑁𝑁) has the usual neoclassical
properties. Technological progress is represented by 𝐴𝐴 growing at a constant rate 𝑔𝑔:
𝐴𝐴′ = (1 + 𝑔𝑔)𝐴𝐴
In analysing this version of the Solow model, it is convenient to define variables 𝑘𝑘𝑒𝑒 and 𝑦𝑦𝑒𝑒
representing capital and output per effective labour input 𝐴𝐴𝐴𝐴:
𝑘𝑘𝑒𝑒 =
𝐾𝐾
𝐴𝐴𝐴𝐴
and
𝑦𝑦𝑒𝑒 =
𝑌𝑌
𝐴𝐴𝐴𝐴
A per-effective-worker production function is derived in the same way as the usual perworker production function, with 𝑓𝑓(𝑘𝑘𝑒𝑒 ) used as a shorthand for 𝐹𝐹(𝑘𝑘𝑒𝑒 , 1):
𝑦𝑦𝑒𝑒 =
𝐹𝐹(𝐾𝐾, 𝐴𝐴𝐴𝐴)
𝐾𝐾 𝐴𝐴𝐴𝐴
= 𝐹𝐹 �
,
� = 𝐹𝐹 (𝑘𝑘𝑒𝑒 , 1) = 𝑓𝑓(𝑘𝑘𝑒𝑒 )
𝐴𝐴𝐴𝐴
𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴
Understanding the dynamics of 𝑘𝑘𝑒𝑒 is central to the analysis. By using the equations for 𝐾𝐾′,
𝑁𝑁′, and 𝐴𝐴′ and 𝑦𝑦𝑒𝑒 = 𝑓𝑓(𝑘𝑘𝑒𝑒 ), the future-period value of 𝑘𝑘𝑒𝑒 is:
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𝑘𝑘𝑒𝑒′ =
(1 − 𝑑𝑑 )𝐾𝐾 + 𝑠𝑠𝑠𝑠
(1 − 𝑑𝑑 )𝑘𝑘𝑒𝑒 + 𝑠𝑠𝑠𝑠(𝑘𝑘𝑒𝑒 )
𝐾𝐾 ′
=
=
𝐴𝐴′ 𝑁𝑁′ (1 + 𝑔𝑔)(1 + 𝑛𝑛)𝐴𝐴𝐴𝐴
(1 + 𝑔𝑔)(1 + 𝑛𝑛)
Subtracting 𝑘𝑘𝑒𝑒 yields an equation for the change in capital per effective worker:
𝑘𝑘𝑒𝑒′ − 𝑘𝑘𝑒𝑒 =
𝑠𝑠𝑠𝑠 (𝑘𝑘𝑒𝑒 ) − (𝑑𝑑 + 𝑛𝑛 + 𝑔𝑔 + 𝑛𝑛𝑛𝑛)𝑘𝑘𝑒𝑒 𝑠𝑠𝑠𝑠(𝑘𝑘𝑒𝑒 ) − (𝑑𝑑 + 𝑛𝑛 + 𝑔𝑔)𝑘𝑘𝑒𝑒
≈
(1 + 𝑔𝑔)(1 + 𝑛𝑛)
(1 + 𝑔𝑔)(1 + 𝑛𝑛)
The approximation in this equation uses that 𝑛𝑛𝑛𝑛 is highly likely to be small relative to 𝑛𝑛 and
𝑔𝑔 themselves. The resulting equation has the same form as that for 𝑘𝑘 ′ − 𝑘𝑘 in the basic
Solow model: the numerator is the difference between a ‘saving line’ 𝑠𝑠𝑠𝑠(𝑘𝑘𝑒𝑒 ) and an
‘effective depreciation line’ (𝑑𝑑 + 𝑛𝑛 + 𝑔𝑔)𝑘𝑘𝑒𝑒 . These lines have the same shapes as in the basic
Solow model because 𝑓𝑓(𝑘𝑘𝑒𝑒 ) is also a concave function here, and (𝑑𝑑 + 𝑛𝑛 + 𝑔𝑔)𝑘𝑘𝑒𝑒 is linear.
Capital per effective worker 𝑘𝑘𝑒𝑒 is rising when the saving line is above the depreciation line,
and is at a steady state where the two lines cross. Once 𝑘𝑘𝑒𝑒 is known, output per effective
worker is found using the per-effective-worker production function 𝑦𝑦𝑒𝑒 = 𝑓𝑓(𝑘𝑘𝑒𝑒 ). The
equivalent of the usual Solow model diagram here is shown in Figure 2.13.
Figure 2.13: Modified Solow diagram
The figure reveals there is a steady state for capital and output per effective worker 𝑘𝑘𝑒𝑒 and
𝑦𝑦𝑒𝑒 , and the economy will converge to this steady state in the long run using the same logic
as in the basic Solow model. But this does not mean that there is no economic growth in the
long run. Output per worker is by definition 𝑦𝑦 = 𝑌𝑌/𝑁𝑁 = 𝐴𝐴𝑦𝑦𝑒𝑒 . Since 𝑦𝑦𝑒𝑒 converges to a steady
state while 𝐴𝐴 grows at a constant rate 𝑔𝑔, the long-run growth rate of 𝑦𝑦 is 𝑔𝑔. Similarly, capital
per worker 𝑘𝑘 has long-run growth rate 𝑔𝑔.
Intuitively, this Solow model with constant growth in labour-augmenting technology is able
to generate long-run economic growth because growth in the capital stock in line with
labour-augmenting technology is able to scale up inputs of both capital and labour per
person in the economy. Hence, by constant returns to scale, output per person grows at the
rate of labour-augmenting technological progress. These improvements in technology are
effectively raising the return to capital and offsetting the diminishing returns to capital that
limit long-run growth in the basic version of the Solow model. The economy has a balanced
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growth path where all per-worker variables grow in line with technological progress. On this
balanced growth path, returns to capital remain constant over time.
This version of the Solow model is consistent with the occurrence of long-run growth.
However, that growth comes only from technological progress, which is exogenous in the
model. The Solow model here is not actually explaining the source of long-run economic
growth – it is simply assuming it. In Section 2.6, we look at theories that make long-run
economic growth endogenous.
2.4 International flows of investment
In the Solow model we have so far assumed a closed economy. The consequence of this
assumption is that investment must be financed by domestic saving (𝐼𝐼 = 𝑆𝑆). But if crosscountry income differences are explained by differences in capital accumulation, then
returns to capital have diminished more in some countries than others. This implies
differences in returns to capital across countries. To the extent that economies are actually
open to trade, there would be an incentive for savings to flow from rich countries with
abundant capital to poor countries with scarce capital where returns are higher. We present
here an exercise to calculate how large are the incentives for capital flows implied by the
Solow model given the observed differences in income levels across countries.
Suppose all countries have a Cobb-Douglas production function 𝑌𝑌 = 𝑧𝑧𝐾𝐾 𝛼𝛼 𝑁𝑁 1−𝛼𝛼 , which in
per-worker terms is 𝑦𝑦 = 𝑧𝑧𝑘𝑘 𝛼𝛼 . Further, assume the same TFP 𝑧𝑧 across countries and the
same parameter 𝛼𝛼. If markets are competitive and absent distortions in the economy, the
gross return 𝑅𝑅 received by owners of capital is equal to the marginal product of capital 𝑀𝑀𝑃𝑃𝐾𝐾 ,
and 𝛼𝛼 is the share of capital income in total income. We will set 𝛼𝛼 = 1/3 to match a
reasonable value of the capital share.
Taking as given an observed value of income per person 𝑦𝑦, the required level of capital per
worker can be found conditional on TFP 𝑧𝑧 by rearranging the per worker production
function 𝑦𝑦 = 𝑧𝑧𝑘𝑘 𝛼𝛼 to give 𝑘𝑘 = (𝑦𝑦⁄𝑧𝑧)1⁄𝛼𝛼 . Differentiating the per worker production, the
marginal product of capital is 𝑀𝑀𝑃𝑃𝐾𝐾 = 𝛼𝛼𝛼𝛼𝑘𝑘 𝛼𝛼−1 and substituting for 𝑘𝑘 in terms of 𝑦𝑦 and 𝑧𝑧, the
implied capital return 𝑅𝑅 as a function of income per person 𝑦𝑦 is:
𝛼𝛼−1
1−𝛼𝛼
1
𝑦𝑦 𝛼𝛼
𝑅𝑅 = 𝛼𝛼𝛼𝛼 � �
= 𝛼𝛼𝑧𝑧 𝛼𝛼 𝑦𝑦 −� 𝛼𝛼 �
𝑧𝑧
We now compare implied returns to capital in two countries 𝐴𝐴 and 𝐵𝐵, supposing 𝐴𝐴 has an
income per person 10 times higher than in 𝐵𝐵, that is, 𝑦𝑦𝐴𝐴 ⁄𝑦𝑦𝐵𝐵 = 10. The relative (gross)
return on capital between two countries with the same 𝑧𝑧 and 𝛼𝛼 is:
1
1−𝛼𝛼
�
−�
1−𝛼𝛼
𝑅𝑅𝐵𝐵 𝛼𝛼𝑧𝑧 𝛼𝛼 𝑦𝑦𝐵𝐵 𝛼𝛼
𝑦𝑦𝐴𝐴 𝛼𝛼
=
=� �
1−𝛼𝛼
1 −�
𝑅𝑅𝐴𝐴
𝑦𝑦𝐵𝐵
�
𝛼𝛼𝑧𝑧 𝛼𝛼 𝑦𝑦𝐴𝐴 𝛼𝛼
Using parameter 𝛼𝛼 = 1/3, note that (1 − 𝛼𝛼)⁄𝛼𝛼 = 2, so the implied capital returns across
two countries with 𝑦𝑦𝐴𝐴 ⁄𝑦𝑦𝐵𝐵 = 10 are 𝑅𝑅𝐵𝐵 ⁄𝑅𝑅𝐴𝐴 = 102 = 100. This numerical exercise suggests
the gross return to capital would be 100 times higher in the poorer country compared to
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the rich country. Returns after depreciation 𝑟𝑟 = 𝑅𝑅 − 𝑑𝑑 would also differ by a large amount,
with 𝑟𝑟𝐵𝐵 = 𝑅𝑅𝐵𝐵 − 𝑑𝑑 = 100𝑅𝑅𝐴𝐴 − 𝑑𝑑 = 100𝑟𝑟𝐴𝐴 + 99𝑑𝑑.
However, if the difference in capital returns suggested by this exercise for realistic income
differences were really true then there would be huge incentives for rich countries to lend
to poor countries for investment. But international capital flows have not occurred to the
extent that would be expected if the calculation were correct. We conclude that some
assumptions made in this exercise must be incorrect. There are a number of possibilities:
•
•
•
Some other factor of production that differs across countries has not been
accounted for, e.g. human capital. If human capital is higher in some countries, then
that raises their marginal product of physical capital 𝑀𝑀𝑃𝑃𝐾𝐾 .
TFP 𝑧𝑧 differs across countries and differences in TFP imply differences in the
marginal product of capital 𝑀𝑀𝑃𝑃𝐾𝐾 . We explore in Box 2.3 what a difference in TFP
across countries might capture.
Property rights are weaker in some countries, implying that investors’ actual return
is less than 𝑀𝑀𝑃𝑃𝐾𝐾 .
Box 2.3: Institutions and income differences across countries
Differences in total factor productivity (TFP) 𝑧𝑧 across countries help to resolve a number of
difficulties in reconciling the Solow model with the observed distribution of income levels
around the world. We have seen in Box 2.1 that differences in saving rates and capital
accumulation can explain only a small proportion of cross-country income differences.
Moreover, there would be very large gaps in the implied rate of return to capital across
countries if income differences arise from different stocks of capital per worker.
If richer countries benefit from higher TFP then this can account for some of the income
differences across countries without having to rely on differences in saving rates.
Moreover, if richer countries have higher TFP than poorer countries, it does not follow
that they necessarily have a much lower return on capital and making it less of a puzzle
that capital does not flow from rich to poor countries.
However, if TFP represents technology, which is what we assumed when explaining longrun growth through technological progress, how can very large differences in TFP exist
when knowledge can be copied? While there may be some costs of imitation, which we
consider later in Section 2.10, it is nonetheless a challenge to explain the size of the TFP
differences across countries needed to account for income levels purely in terms of
differences in technology.
Equating TFP with technology is, though, too narrow an interpretation. In a production
function 𝑌𝑌 = 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑁𝑁), a change in TFP 𝑧𝑧 simply represents any change in 𝑌𝑌 that cannot
be explained by changes in the quantities of capital 𝐾𝐾 or labour 𝑁𝑁. The efficiency with
which an economy’s stock of capital and labour are allocated to the best uses to produce
aggregate output 𝑌𝑌 would also be part of TFP 𝑧𝑧. It is more plausible that this broader
notion of TFP differs across countries, perhaps owing to differences in institutions.
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Institutions
Institutional differences across countries are one hypothesis for why there are large
differences in TFP. The ‘institutions’ of a country mean the set of rules that govern the
process of decision making and resource allocation in a country. These include the country’s
political and legal systems and economic institutions such as its markets, financial system,
and taxes and regulations. Institutions can affect both incentives to accumulate factors of
production and the efficiency with which those factors of production are allocated to
different uses.
A country with weak property rights, severe corruption and a poorly designed tax system
does not provide strong incentives to save, invest, or start a new business. For example, the
risk of expropriation reduces the expected return on any long-term investments, lowering
capital accumulation. This problem would also make it harder to attract foreign direct
investment, helping to explain the puzzle of why capital does not flow from rich to poor
countries.
Misallocation
Institutions can also affect the efficiency with which factors of production are allocated. For
example, in making loans, corruption might lead banks to favour those with political
connections over entrepreneurs with the best business ideas. Consequently, the economy’s
supply of capital 𝐾𝐾 and labour 𝑁𝑁 is not allocated to the businesses that can add the most to
GDP 𝑌𝑌. This shows up as a lower level of TFP 𝑧𝑧 in a country.
More generally, institutions affect incentives to pursue ‘rent-seeking’ rather than productive
activities. Productive activities are those that increase the total size of the ‘pie’, that is,
investments in physical and human capital, new businesses, and new technologies. Rentseeking refers to activities that give someone a larger share of the pie but do not increase its
size. For example, someone might seek a position of political power to benefit from
corruption. Institutions that reward rent-seeking activities thus divert effort from growing
the pie, which leads to lower TFP.
2.5 The golden rule
We have seen in the Solow model that a higher saving rate allows the economy to sustain
more capital per worker in steady state and thus obtain a permanently higher level of
income per worker. Does this mean that more saving is always a good thing? Although
income is higher, since a greater fraction of it is saved, consumption per worker might end
up being lower. Underlying this is the need to maintain in steady state the capital
accumulated through more saving and these resources used for investment cannot also be
used for consumption.
This logic suggests we should look for a saving rate 𝑠𝑠 that maximises consumption per
worker 𝑐𝑐 = 𝐶𝐶/𝑁𝑁. Since 𝐶𝐶 = 𝑌𝑌 − 𝑆𝑆 and 𝑆𝑆 = 𝑠𝑠𝑠𝑠, consumption is 𝐶𝐶 = (1 − 𝑠𝑠)𝑌𝑌 and the
equation for consumption per worker is 𝑐𝑐 = (1 − 𝑠𝑠)𝑦𝑦. Taking 𝑦𝑦 as given in the short run, 𝑐𝑐
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is maximised by saving nothing. However, that level of income 𝑦𝑦 cannot be sustained with
zero saving because the capital stock needed to produce output would eventually be
depleted.
Instead, we ask what is the highest sustainable level of 𝑐𝑐, and which saving rate is needed to
attain it? A sustainable value of 𝑐𝑐 is one that is a steady state for a particular saving rate 𝑠𝑠.
For extreme values of 𝑠𝑠, we can say immediately what the steady state for 𝑐𝑐 results. Having
𝑠𝑠 = 1 results in 𝑐𝑐 = 0, even though steady-state income 𝑦𝑦 might be very high. Having 𝑠𝑠 = 0
results in 𝑐𝑐 = 0 because the steady-state value of 𝑘𝑘 with zero saving is zero and this implies
that 𝑦𝑦 = 0 because some capital is essential for production. With the extreme values of 𝑠𝑠
leading to the worse possible outcomes for 𝑐𝑐 in steady state, it follows that an intermediate
saving rate with 0 < 𝑠𝑠 < 1 is optimal.
2.5.1 Finding the golden rule
To find which saving rate is best, we first ask what saving rate 𝑠𝑠 is necessary to sustain a
particular level of capital per worker 𝑘𝑘 in steady state. The required saving rate must satisfy
𝑠𝑠𝑠𝑠𝑠𝑠(𝑘𝑘) = (𝑑𝑑 + 𝑛𝑛)𝑘𝑘. Given 𝑘𝑘, output per worker is 𝑦𝑦 = 𝑧𝑧𝑧𝑧(𝑘𝑘), which implies consumption
per worker is 𝑐𝑐 = (1 − 𝑠𝑠)𝑦𝑦 = 𝑧𝑧𝑧𝑧(𝑘𝑘) − 𝑠𝑠𝑠𝑠𝑠𝑠(𝑘𝑘). By substituting the equation with the saving
rate 𝑠𝑠, steady-state consumption per worker 𝑐𝑐 is only a function of the steady-state 𝑘𝑘:
𝑐𝑐 = 𝑧𝑧𝑧𝑧(𝑘𝑘) − (𝑑𝑑 + 𝑛𝑛)𝑘𝑘
This equation can be interpreted as saying steady-state consumption can be found
geometrically in the Solow model as the difference in height between the per-worker
production function and the effective depreciation line. The logic is that consumption is the
difference between the production function and the saving line by definition but, in steadystate, the effective depreciation line has the same height as the saving line.
Figure 2.14 illustrates this argument, showing the steady-state levels of capital per worker
and consumption per worker associated with low, medium and high saving rates. Note that
it is not necessary to know the saving rate associated with a steady-state for 𝑘𝑘 to calculate
the implied steady state for 𝑐𝑐.
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Figure 2.14: Steady states for different saving rates
The golden rule level of capital per worker is the steady state where consumption per
worker is maximised. Mathematically, this can be found by differentiating the expression for
steady-state 𝑐𝑐 with respect to steady-state 𝑘𝑘, and the first-order condition is 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 = 0:
𝑧𝑧𝑓𝑓 ′ (𝑘𝑘) = 𝑑𝑑 + 𝑛𝑛
Therefore, the golden rule calls for capital to be accumulated up to the point where the
marginal product of capital 𝑀𝑀𝑃𝑃𝐾𝐾 = 𝑧𝑧𝑓𝑓 ′ (𝑘𝑘) is equal to the effective depreciation rate 𝑑𝑑 + 𝑛𝑛.
This first-order condition is illustrated geometrically in Figure 2.15. The golden rule is the
steady state for 𝑘𝑘 where the tangent to the per worker production function has the same
gradient as the effective depreciation line. This maximises the difference between the
heights of the production function and the effective depreciation line, with this difference
being the steady-state consumption per worker.
Figure 2.15: Golden rule diagram
Intuitively, the golden rule equation reflects a comparison of the marginal benefits and costs
of a higher level of capital per worker in steady state. The marginal benefit of maintaining an
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extra unit of capital per worker is the additional output per worker 𝑀𝑀𝑃𝑃𝐾𝐾 each year. The
marginal cost of maintaining a stock of capital per worker permanently one unit higher is
the depreciation cost 𝑑𝑑 and the new capital 𝑛𝑛 required for new workers each year. The
difference between the benefit and the cost is extra output that can sustainably be
consumed, so the highest steady state for consumption per worker is where the marginal
benefit equals the marginal cost.
Once the golden rule level of 𝑘𝑘� is found, the saving rate 𝑠𝑠̂ that brings the economy to that
steady state is found from the equation 𝑠𝑠𝑠𝑠𝑠𝑠 (𝑘𝑘) = (𝑑𝑑 + 𝑛𝑛)𝑘𝑘:
𝑠𝑠̂ =
(𝑑𝑑 + 𝑛𝑛)𝑘𝑘�
𝑧𝑧𝑧𝑧(𝑘𝑘�)
2.5.2 Transitional paths to the golden rule
While the golden rule maximises steady-state consumption per worker, it does not
necessarily follow that households should choose that saving rate because there may be a
cost in terms of lower consumption during a transitional period to reach the golden rule.
Starting from a steady state with a saving rate below the golden rule, the required increase
in saving initially lowers consumption because the higher long-run capital and output that
results from more saving does not become available instantaneously – there is always a
transitional period in moving from one steady state to another. This case of a short-term
sacrifice of consumption for a long-term gain is illustrated in the left panel of Figure 2.16.
Figure 2.16: Transitional paths to the golden rule
However, starting from a steady state with a saving rate above the golden rule, the required
reduction in saving to reach the golden rule raises consumption during the transitional
period as well as the long run. In the short run, consumption per worker is actually higher
than in the long run because capital and output per worker are falling as the economy
approaches the golden rule steady state. This case is depicted in the right panel of Figure
2.16.
An economy with more capital in steady state than the golden rule level is said to be
dynamically inefficient because extra consumption in the long run can be obtained without
any short-run sacrifice. It is possible to raise consumption per worker in all subsequent years
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by saving less. Where this is not true, so that some sacrifice now is necessary for a gain in
the future, the economy is said to be dynamically efficient.
If all households in the economy were alike (a representative household), then it is irrational
to continue to save more than the golden rule because everyone could be better off by
saving less. However, if saving is done by a young generation to accumulate assets for
retirement in old age, then it is possible to have a dynamically inefficient economy where
savers do not want to save less. We discuss dynamic inefficiency further in the ‘overlapping
generations’ economy introduced in Section 4.7.
2.5.3 Testing for dynamic inefficiency
How is it possible to know whether or not an economy is dynamically inefficient? Here, we
briefly discuss two possible tests. The first test compares the real interest rate to the growth
rate of the economy. The second test compares the capital share of income to the share of
investment expenditure in GDP.
In an economy with competitive markets, the gross return on capital is equal to the
marginal product of capital, that is, 𝑅𝑅 = 𝑀𝑀𝑃𝑃𝐾𝐾 . The real interest rate received by savers
would be equal to the real rate of return on capital after allowing for depreciation costs,
which means that 𝑟𝑟 = 𝑅𝑅 − 𝑑𝑑. In a steady state for capital per worker, 𝑘𝑘 and 𝑦𝑦 are constant
over time, so total GDP 𝑌𝑌 = 𝑁𝑁𝑁𝑁 grows at the same rate 𝑛𝑛 as the population 𝑁𝑁. If the
economy’s real GDP growth rate 𝑛𝑛 exceeds the real interest rate 𝑟𝑟 then:
𝑛𝑛 > 𝑀𝑀𝑃𝑃𝐾𝐾 − 𝑑𝑑
This implies 𝑀𝑀𝑃𝑃𝐾𝐾 < 𝑑𝑑 + 𝑛𝑛, and therefore the economy is dynamically inefficient because it
has more capital per worker in steady state than the golden rule level that satisfies 𝑀𝑀𝑃𝑃𝐾𝐾 =
𝑑𝑑 + 𝑛𝑛 (recalling that 𝑀𝑀𝑃𝑃𝐾𝐾 diminishes with 𝑘𝑘).
The second test for dynamic inefficiency compares the capital share of income to the
investment share of expenditure. In a competitive economy, gross capital income is equal to
𝑅𝑅𝑅𝑅 = 𝑀𝑀𝑃𝑃𝐾𝐾 𝐾𝐾, so the capital share of all gross income is 𝑀𝑀𝑃𝑃𝐾𝐾 𝐾𝐾/𝑌𝑌. By definition of 𝑘𝑘 = 𝐾𝐾/𝑁𝑁
and 𝑦𝑦 = 𝑌𝑌/𝑁𝑁, the capital-output ratio can also be written as 𝐾𝐾⁄𝑌𝑌 = 𝑘𝑘⁄𝑦𝑦. Using 𝑦𝑦 = 𝑧𝑧𝑧𝑧(𝑘𝑘)
and 𝑀𝑀𝑃𝑃𝐾𝐾 = 𝑧𝑧𝑧𝑧′(𝑘𝑘), it follows that the capital share of income is 𝑧𝑧𝑓𝑓 ′ (𝑘𝑘)𝑘𝑘⁄(𝑧𝑧𝑧𝑧(𝑘𝑘)).
The investment share of total expenditure is 𝐼𝐼/𝑌𝑌, which is equal to 𝑆𝑆/𝑌𝑌 and to the saving
rate 𝑠𝑠 in the basic Solow model. The steady-state level of capital per worker 𝑘𝑘 satisfies the
equation 𝑠𝑠𝑠𝑠𝑠𝑠 (𝑘𝑘) = (𝑑𝑑 + 𝑛𝑛)𝑘𝑘, so the investment share of GDP is 𝑠𝑠 = (𝑑𝑑 + 𝑛𝑛)𝑘𝑘/(𝑧𝑧𝑧𝑧 (𝑘𝑘)). If
the investment share is greater than the capital income share:
(𝑑𝑑 + 𝑛𝑛)𝑘𝑘 𝑧𝑧𝑓𝑓 ′ (𝑘𝑘)𝑘𝑘
>
𝑧𝑧𝑧𝑧(𝑘𝑘)
𝑧𝑧𝑧𝑧(𝑘𝑘)
By cancelling terms in 𝑘𝑘 and 𝑧𝑧𝑧𝑧(𝑘𝑘) from both sides, we that this holds if 𝑧𝑧𝑓𝑓 ′ (𝑘𝑘) < 𝑑𝑑 + 𝑛𝑛,
which is the condition for dynamical inefficiency.
In conclusion, dynamically inefficient economies have low real interest rates relative to real
GDP growth, or high levels of investment expenditure relative to capital income.
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Box 2.4: Climate change and the economy
Climate change has become an increasing concern for the welfare of future
generations. How might climate change affect long-run economic outcomes
according to the Solow model?
One way is through disrupting agriculture and generally raising the costs of economic
activity. This means less output is produced from a given amount of factor inputs,
which is equivalent to lower total factor productivity 𝑧𝑧. We can analyse effects of this
kind as the opposite of the better technology studied earlier in Section 2.3.
Another way climate change matters is that more frequent natural disasters such as
flooding raise the costs of maintaining the capital stock, for example, repairing
damage and rebuilding. Furthermore, existing capital becomes obsolete faster when
greener technologies are adopted, most notably creating a need to replace capital
used in the transportation and energy sectors of the economy. Both of these
considerations point to a higher depreciation rate 𝑑𝑑 of capital.
The effects of a higher depreciation rate 𝑑𝑑 on long-run economic outcomes are
studied in Figure 2.17. The effective depreciation line becomes steeper and pivots to
the left. This implies the economy’s new steady state has less capital per worker 𝑘𝑘 ∗ .
Intuitively, if capital is more costly to maintain due to climate change, or needs to be
replaced by greener capital more frequently, a given saving rate sustains less capital
per worker. With less capital per worker, output per worker is reduced. This negative
effect on 𝑦𝑦 is in addition to any direct negative effects through lower 𝑧𝑧.
Figure 2.17: Higher depreciation rate of capital
How might economies mitigate these negative economic effects? In particular, should
economies save more to compensate? One benchmark for an appropriate level of the
saving rate 𝑠𝑠 is the golden rule, which says that the goal of the highest sustainable
level of consumption per person is at steady state for capital per worker with 𝑀𝑀𝑃𝑃𝐾𝐾 =
𝑑𝑑 + 𝑛𝑛. A higher depreciation rate 𝑑𝑑 thus reduces the golden rule capital stock. With
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no change in the saving rate, we have already seen that steady-state capital per
worker will decline with higher 𝑑𝑑, so it is not clear whether the golden rule calls for a
higher or lower saving rate 𝑠𝑠.
A higher depreciation rate 𝑑𝑑 reduces steady-state output per worker 𝑦𝑦 ∗ but how large
is this effect? Consider an example we used earlier in Box 2.1 to study the effect of
differences in saving rates across countries. The production function is the CobbDouglas form 𝑌𝑌 = 𝑧𝑧𝐾𝐾 𝛼𝛼 𝑁𝑁 1−𝛼𝛼 with 𝛼𝛼 = 1/3. With a Cobb-Douglas production function,
steady-state output per worker is:
𝑦𝑦
∗
1
= 𝑧𝑧 1−𝛼𝛼 �
𝛼𝛼
𝑠𝑠 1−𝛼𝛼
�
𝑑𝑑 + 𝑛𝑛
We can find the effect of going from depreciation rate 𝑑𝑑1 to 𝑑𝑑2 by taking the ratio of
the resulting steady-state levels of 𝑦𝑦2∗ and 𝑦𝑦1∗ holding 𝑧𝑧 fixed:
1 2
𝑦𝑦2∗
𝑑𝑑1 + 𝑛𝑛 3�3
𝑑𝑑1 + 𝑛𝑛
=
�
�
=�
∗
𝑦𝑦1
𝑑𝑑2 + 𝑛𝑛
𝑑𝑑2 + 𝑛𝑛
Assuming a population growth rate 𝑛𝑛 = 1%, and a depreciation rate that rises just
one percentage point from 𝑑𝑑1 = 9% to 𝑑𝑑2 = 10%, we obtain:
0.1
𝑦𝑦2∗
�
≈ √0.91 ≈ 0.95
∗ =
𝑦𝑦1
0.11
Therefore, when capital becomes obsolete or is destroyed 11% faster than before
(going from 9% to 10%), output per worker is 5% lower in the long run.
2.6 The AK model
The diminishing marginal product of capital lies behind many of the troubling predictions of
the Solow model that we have seen earlier. First, its failure to generate long-run economic
growth. Second, the modest differences in income levels it predicts even when there are
large differences in saving rates. Third, the extremely large differences in the implied return
to capital that result from the observed distribution of income across countries.
To try to overcome these failures of the Solow model, we now look at an alternative model
with constant returns to capital. As there are now no diminishing returns to capital, the
production function does not satisfy the neoclassical assumptions. All other assumptions of
the Solow model are maintained for comparison.
2.6.1 The AK production function
We assume that 𝐹𝐹 (𝐾𝐾, 𝑁𝑁) = 𝐾𝐾, so the production function is 𝑌𝑌 = 𝑧𝑧𝑧𝑧. This is often written
with 𝐴𝐴 denoting TFP 𝑧𝑧:
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𝑌𝑌 = 𝐴𝐴𝐴𝐴
The resulting model is known as the ‘AK’ model after the equation for its production
function. In this model, TFP 𝐴𝐴 is assumed to be constant over time, so there is no exogenous
source of long-run growth.
Although the AK production function is not neoclassical, it still has constant returns to scale,
so it can be represented by a per-worker production function as in the Solow model. With
𝑓𝑓(𝑘𝑘) = 𝐹𝐹(𝑘𝑘, 1) = 𝑘𝑘, it follows that the per worker production function is 𝑦𝑦 = 𝐴𝐴𝐴𝐴, which can
also be seen directly by dividing 𝑌𝑌 = 𝐴𝐴𝐴𝐴 by 𝑁𝑁.
The marginal product of capital in the AK model is 𝑀𝑀𝑃𝑃𝐾𝐾 = 𝐴𝐴𝑓𝑓 ′ (𝑘𝑘) = 𝐴𝐴, which is a constant.
This does not change with the amount of capital accumulated because there are no
diminishing returns to capital. It is also constant over time because there is no growth in TFP
𝐴𝐴 in this model.
As all of the other assumptions of the Solow model are maintained, the same equation for
the change over time in capital per worker is applicable here, the only difference being that
the per-worker production function 𝑦𝑦 = 𝐴𝐴𝐴𝐴 is used in place of 𝑦𝑦 = 𝑧𝑧𝑧𝑧(𝑘𝑘):
𝑘𝑘 ′ − 𝑘𝑘 =
𝑠𝑠𝑠𝑠𝑠𝑠 − (𝑑𝑑 + 𝑛𝑛)𝑘𝑘
(1 + 𝑛𝑛)
The dynamics of capital per worker and output per worker can thus be analysed using a
diagram similar to the one in the Solow model. Crucially, since the per worker production
function is linear, it and the saving line in the equivalent of the Solow diagram are now
straight lines. The AK model diagram is shown in Figure 2.18.
Figure 2.18: The AK model
The key difference compared to the Solow model diagram is that there is no intersection
between the saving and effective depreciation lines and, hence, no steady state (apart from
the uninteresting steady state at zero). This is because the saving line is a straight line,
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reflecting the absence of diminishing returns to capital, unlike its concave shape in the
Solow model.
If 𝑠𝑠𝑠𝑠 > 𝑑𝑑 + 𝑛𝑛 then the saving line is steeper than the effective depreciation line and
therefore the change in capital per worker 𝑘𝑘 ′ − 𝑘𝑘 is always positive. This implies there is
perpetual growth with no convergence to a steady state. Moreover, the production function
𝑦𝑦 = 𝐴𝐴𝐴𝐴 implies output per worker 𝑦𝑦 grows at same rate as capital per worker 𝑘𝑘:
𝑦𝑦 ′ − 𝑦𝑦 𝑘𝑘 ′ − 𝑘𝑘 𝑠𝑠𝑠𝑠 − (𝑑𝑑 + 𝑛𝑛)
=
=
𝑦𝑦
𝑘𝑘
1 + 𝑛𝑛
2.6.2 Endogenous growth
The AK model is able to generate long-run growth endogenously through capital
accumulation if the saving rate is high enough. Intuitively, more capital allows more output
to be produced, some of which is saved, which is ploughed back into capital accumulation,
leading to more output. Unlike the Solow model, there are no diminishing returns to capital
that weaken this feedback loop.
Moreover, the economy’s rate of growth depends on the saving rate 𝑠𝑠. In contrast to the
Solow model, changing the saving rate has a permanent effect on growth rate, not only on
level of income. An economy can enjoy a faster long-run growth rate by choosing to save a
higher fraction of income.
Although the AK model makes some interesting predictions, the AK production function
itself is problematic. As it completely excludes labour input 𝑁𝑁, the implied capital share of
income is 100 per cent – the AK production function is a special case of a Cobb-Douglas with
𝛼𝛼 = 1. However, including labour input 𝑁𝑁 in the production function while preserving
linearity in capital input 𝐾𝐾 would result in a production function with increasing returns to
scale. Nonetheless, there are alternative models of endogenous growth with similar
predictions to the AK model that make more plausible assumptions about the production
function.
Box 2.5: Endogenous growth and divergence between countries
Theories that generate endogenous growth through physical capital accumulation
such as the AK model imply that the saving rate has a permanent effect on the
economy’s long-run growth rate of output per worker. Higher saving rates lead to
faster long-run growth. This means that a higher saving rate always leads to higher
consumption per worker eventually in the long run. Figure 2.19 shows the trajectories
of (log) income and consumption per worker for an economy where the saving rate
rises. Recall that a constant growth rate corresponds to a straight-line trajectory when
variables are plotted as logarithms.
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Figure 2.19: Effects of a higher saving rate
In contrast, the Solow model predicts that a higher saving rate only raises the long-run
steady-state level of income per worker, without any effect on the long-run growth
rate. This means saving too much has a negative effect on long-run consumption and
there is a golden rule saving rate that maximises long-run consumption. In the AK
model, saving more (as long as the saving rate is less than 100 per cent) always
increases consumption in the long run.
Another important feature of endogenous growth models is that they do not generate
convergence among economies. For countries with the same TFP level, same
population growth and depreciation rates and the same saving rate, the AK model
implies the growth rate of income per worker is the same irrespective of whether a
country starts with more or less capital per worker than others. This is illustrated in
the left panel of Figure 2.20.
Figure 2.20: No convergence, or divergence, across countries
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Moreover, endogenous growth also means that small but persistent differences in
saving rates can explain large gaps in income per worker across countries. This is
because different saving rates imply different growth rates, which when compounded
over time can open up very large gaps in income, as shown in the right panel of Figure
2.20. We have seen that the Solow model could not explain large income differences
even with large differences in saving rates.
Finally, observe that the AK model predicts there is no difference between the return on
capital in poor and rich countries. If countries share the same level of TFP, the marginal
product of capital is the same everywhere. This would resolve the puzzle of why capital
does not flow from rich to poor countries.
2.7 Learning by doing
We now consider some alternative routes to generating endogenous economic growth
beyond the direct assumption of an AK production function. We do this because the AK
production function itself does not seem very plausible. The first of these alternative routes
to endogenous growth is known as ‘learning by doing’.
Learning-by-doing is when skills or knowledge are accumulated as a by-product of the
production process. We will look at a model with the feature that when new capital is
installed, workers using the capital discover through use how to deploy it most effectively.
Learning-by-doing models claim that capital accumulation leads to a positive ‘spillover’ or
externality that increases the economy’s stock of ideas and knowledge. As knowledge and
ideas are non-rival, all firms can benefit from greater productivity in using capital. This
supposes firms cannot have intellectual property rights over any knowledge discovered
through learning by doing. We return to that issue in Section 2.9 when discussing research
and development.
Suppose production is done by perfectly competitive firms using production function 𝑌𝑌𝑖𝑖 =
𝐹𝐹(𝐾𝐾𝑖𝑖 , 𝐴𝐴𝑁𝑁𝑖𝑖 ). Firm 𝑖𝑖 uses capital 𝐾𝐾𝑖𝑖 and hires labour 𝑁𝑁𝑖𝑖 to produce output 𝑌𝑌𝑖𝑖 . There is labouraugmenting technology 𝐴𝐴 that is common to all firms because knowledge is a public good
(non-rival and non-excludable). Unlike the AK model, here the production function
𝐹𝐹(𝐾𝐾𝑖𝑖 , 𝐴𝐴𝑁𝑁𝑖𝑖 ) at the level of individual firms is neoclassical, so there are diminishing returns to
capital.
Each firm 𝑖𝑖 hires labour 𝑁𝑁𝑖𝑖 up to the point where the wage 𝑤𝑤 equals the marginal product of
labour 𝑀𝑀𝑃𝑃𝑁𝑁𝑖𝑖 . As all competitive firms face the same wage, the marginal products of labour
and capital end up being the same across all firms. These marginal products then determine
the distribution of income between labour and capital in the usual way. Given knowledge 𝐴𝐴,
the production function for whole economy 𝑌𝑌 = 𝐹𝐹(𝐾𝐾, 𝐴𝐴𝐴𝐴) is the same as the one for an
individual firm but with 𝑌𝑌 being GDP, and 𝐾𝐾 and 𝑁𝑁 being the total capital stock and labour
force. Assume the aggregate supply of labour 𝑁𝑁 is constant because there is no population
growth (𝑛𝑛 = 0).
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The key assumption of the learning-by-doing model is that economy-wide knowledge 𝐴𝐴 rises
with the economy’s capital stock 𝐾𝐾. Specifically, we assume the two variables are
proportional with 𝜆𝜆 > 0 being a constant:
𝐴𝐴 = 𝜆𝜆𝜆𝜆
By substituting 𝐴𝐴 = 𝜆𝜆𝜆𝜆 into 𝑌𝑌 = 𝐹𝐹(𝐾𝐾, 𝐴𝐴𝐴𝐴), the aggregate production function becomes
𝑌𝑌 = 𝐹𝐹(𝐾𝐾, 𝜆𝜆𝜆𝜆𝜆𝜆). The constant-returns-to-scale property of the production function then
implies 𝑌𝑌 = 𝐹𝐹(1, 𝜆𝜆𝜆𝜆)𝐾𝐾. As 𝐹𝐹(1, 𝜆𝜆𝜆𝜆) is a constant, this has the same form as an ‘AK’
production function 𝑌𝑌 = 𝑧𝑧𝑧𝑧, where the constant 𝑧𝑧 is 𝑧𝑧 = 𝐹𝐹(1, 𝜆𝜆𝜆𝜆).
Intuitively, the spill over effect from the aggregate capital stock to knowledge offsets the
usual diminishing returns to capital. If labour-augmenting technology 𝐴𝐴 is proportional to
capital 𝐾𝐾, the spill over cancels out diminishing returns and implies constant returns to
capital overall. This makes it equivalent to an ‘AK’ production function in the aggregate,
even though individual production functions at the level of firms are neoclassical. Therefore,
it is possible to generate endogenous growth without having directly to assume the unusual
features of the AK production function itself.
However, even if there is a spill over from capital accumulation to knowledge, the model’s
implications can change substantially if knowledge is not proportional to capital. If following
a 1 per cent increase in 𝐾𝐾, knowledge 𝐴𝐴 rises by less than 1 per cent then there are
diminishing returns to aggregate capital and it is not possible to generate endogenous longrun growth. On the other hand, if 𝐴𝐴 rises more than 1 per cent following the 1 per cent rise
in 𝐾𝐾 then there are increasing returns to aggregate capital. This would imply endogenous
growth but, unrealistically, the economy’s growth rate would increase over time.
Unfortunately, the version of the model with stable endogenous growth resembles a special
case for which there is no clear support.
Even in the case where 𝐴𝐴 and 𝐾𝐾 are proportional, the learning-by-doing model implies what
are known as ‘scale effects’. This means that the economy’s growth rate of income per
person depends positively on the size of the population, a prediction that has a lack of
empirical support. This feature of the learning-by-doing model results from a larger
economy with more people producing more non-rival knowledge from which everyone
benefits. Mathematically, it can be seen from the aggregate production function displaying
increasing returns to capital 𝐾𝐾 and labour 𝑁𝑁 together.
2.8 Human capital
In the models we have seen so far, labour input 𝑁𝑁 is simply the number of workers, or
sometimes the number of hours worked, as in the model of the choice of labour supply. But
in practice, the amount of effective labour input that goes into producing goods and
services depends on ‘human capital’. This concept refers to the education, skills and training
of workers that affects their productivity. Crucially, like physical capital, human capital is
something that can be accumulated. By devoting time and resources to improve education
or training, an economy can increase its stock of human capital.
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How might an economy’s stock of human capital be measured? It is possible to look at data
on average years of schooling and the fraction of college graduates in the population.
Combined with research on wages that estimates the effect of, for example, an extra year of
education on workers’ pay, a measure of human capital based on the wage premiums
associated with education/training can be constructed.
Although we have seen models earlier where effective labour input depended on
technology, it is important to note that human capital is different from the notions of ideas
and technology. Human capital is embodied in a worker who has learned a particular skill –
it is not the abstract concept or discovery of the knowledge for the first time.
We now consider a model of human capital accumulation as a potential source of
endogenous long-run growth. The economy’s stock of human capital is denoted by 𝐻𝐻.
Assume the labour force 𝑁𝑁 has a constant size, so a change in 𝐻𝐻 reflects a change in average
human capital per person. Output of final goods and services 𝑌𝑌 is produced with physical
capital 𝐾𝐾 and human capital according to the neoclassical production function:
𝑌𝑌 = 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑢𝑢𝑢𝑢)
In the production function, 𝑢𝑢 denotes fraction of human capital 𝐻𝐻 used to produce goods.
As we will see, some existing human capital is dedicated to producing future human capital,
not goods and services in the present. We assume TFP 𝑧𝑧 is constant over time, and continue
to maintain all the other assumptions about capital and goods from the Solow model.
Human capital itself needs to be produced using existing human capital. Given 𝑢𝑢, the
amount of existing 𝐻𝐻 producing new human capital is (1 − 𝑢𝑢)𝐻𝐻. We assume a production
function 𝐺𝐺(∙) for producing new human capital. The only input is (1 − 𝑢𝑢)𝐻𝐻, although it is
possible to generalise this making production of human capital depend on both physical and
human capital.
The equation for next time period’s human capital 𝐻𝐻′ is:
𝐻𝐻′ = (1 − 𝑑𝑑𝐻𝐻 )𝐻𝐻 + 𝐺𝐺((1 − 𝑢𝑢)𝐻𝐻)
The first term represents undepreciated current human capital that remains usable in the
future, assuming the rate at which human capital depreciates is a constant 𝑑𝑑𝐻𝐻 . The
dynamics of human capital are therefore determined by:
𝐻𝐻′ − 𝐻𝐻 = 𝐺𝐺�(1 − 𝑢𝑢)𝐻𝐻� − 𝑑𝑑𝐻𝐻 𝐻𝐻
The difference between the terms on the right-hand side is the difference between a ‘saving
line’ 𝐺𝐺�(1 − 𝑢𝑢)𝐻𝐻�, the shape of which depends on the function 𝐺𝐺(∙), and a depreciation line
𝑑𝑑𝐻𝐻 𝐻𝐻, which is a straight line. A diagram for human capital accumulation analogous to the
Solow model diagram is displayed in Figure 2.21.
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Figure 2.21: Human capital accumulation
The predictions of the human capital model depend on whether the returns to producing
human capital are diminishing or constant. The case of diminishing returns is shown in the
right panel of the figure. The human capital production function 𝐺𝐺(∙) is concave, which
implies it intersects the depreciation line and there exists a steady state 𝐻𝐻 ∗ for the stock of
human capital. As the 𝐻𝐻 entering the goods production function does not grow in the long
run, diminishing returns to physical capital in producing goods imply that there is no longrun economic growth by the usual Solow model logic.
But what happens if there are constant returns to producing human capital? In this case, the
function 𝐺𝐺 (∙) is linear. There is no intersection between the saving and depreciation lines
for human capital (except at zero), so it is possible to have a constant positive growth rate of
𝐻𝐻 in the long run. The implications for the production of goods and services is then exactly
equivalent to the Solow model with labour-augmenting technological progress studied in
Section 2.3. Mathematically, constant growth in workers’ human capital 𝐻𝐻 plays exactly the
same role as constant growth of labour-augmenting technology 𝐴𝐴.
An increase in the amount of resources devoted to education and training can be
represented by a decline in 𝑢𝑢 and an increase in 1 − 𝑢𝑢. If the human capital model
generates positive long-run growth, then higher 1 − 𝑢𝑢 increases the economy’s long-run
growth rate. Although there is certain to be a benefit in the long run, this is not a free lunch
because there is a cost during a transitional period with lower 𝑢𝑢 implying lower output 𝑌𝑌
starting from the initial level of 𝐻𝐻.
If the model does not generate long-run growth, then reducing 𝑢𝑢 has two conflicting effects
on the long-run level of output. First, it leads to a higher steady-state stock of human
capital, which boosts output. Second, it reduces the share of human capital used to produce
goods, which lowers output of goods and services. There is an optimal choice of 𝑢𝑢 to
maximise long-run output level analogous to the Golden Rule for capital accumulation.
2.9 Research and development
Advances in ideas, technologies, and production techniques are central to explaining longrun economic growth in several of the models we have seen so far. The Solow model with
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exogenous improvements in labour-augmenting technology from Section 2.3 showed how
technological progress can drive long-run growth but the improvements in technology there
were treated as exogenous. The learning-by-doing model from Section 2.7 featured
endogenous discovery of new ideas and techniques as a by-product of installing new capital.
capital accumulation. However, since ideas are non-excludable public goods in that model,
firms did not have any incentive consciously to try to make new discoveries.
But we see that firms and governments devote considerable resources to research and
development (R&D). R&D activity directly aims to discover new ideas and technologies, with
countries at the frontiers of research often spending 2–3 per cent of GDP on R&D. The R&D
model introduced in this section studies growth through the deliberate accumulation of
ideas and technologies rather than of physical or human capital.
2.9.1 Non-rivalrous but excludable technologies
For private firms to undertake R&D activity, there needs to be some protection of
intellectual property rights such as patents. Some excludability of newly discovered ideas
and technologies is necessary to earn profits from R&D otherwise other firms would copy
any innovations and compete away the profits. This is what happens in the learning-bydoing model, where technology only improves because some ideas arise as a by-product of
using new capital.
Physical and human capital, along with most consumption goods, are rivalrous and
excludable private goods. Rivalrous means the same capital or good cannot be enjoyed by
multiple users simultaneously. Excludable means that private ownership of the capital or
good is enforced. Ideas are by their nature are non-rivalrous. One firm or person using a
new idea or technology does not prevent others benefitting from using the same idea.
However, ideas may be excludable to some extent if there are legal restriction on using
others’ discoveries, for example, patent protection.
The stock of ideas and knowledge in the economy is represented by labour-augmenting
technology 𝐴𝐴. The production function for final goods and services is:
𝑌𝑌 = 𝐹𝐹(𝐾𝐾, 𝑢𝑢𝑢𝑢𝑢𝑢)
The fraction of the labour force 𝑁𝑁 who produce goods and services is 𝑢𝑢, so 𝑢𝑢𝑢𝑢 is labour
input in the production function above, augmented by technology 𝐴𝐴. The labour force 𝑁𝑁 is
assumed to have a constant size (no population growth, 𝑛𝑛 = 0).
The production function has the neoclassical properties, including constant returns to scale
with respect to inputs of capital 𝐾𝐾 and effective labour 𝐴𝐴𝐴𝐴. Note that this implies constant
returns to scale with respect to 𝐾𝐾 and 𝑁𝑁 but not with respect to 𝐾𝐾, 𝑁𝑁 and 𝐴𝐴 together. Since
technology 𝐴𝐴 is non-rivalrous, a doubling of the rivalrous inputs of capital 𝐾𝐾 and labour 𝑁𝑁 is
sufficient to double output. Ideas do not need to be discovered again for additional capital
and labour to use them in producing more output.
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2.9.2 A production function for ideas
New ideas are produced in the R&D sector, which adds to the stock of knowledge 𝐴𝐴. We will
write down a production function for ideas, analogous to the production functions we have
used for goods and human capital. For simplicity, only labour input is used in the R&D
sector, though generalising this to have capital used as well to produce ideas would not
fundamentally change this model.
The R&D sector employs (1 − 𝑢𝑢)𝑁𝑁 workers, a fraction 1 − 𝑢𝑢 of the labour force. We assume
the production function for new knowledge 𝐴𝐴′ − 𝐴𝐴 has the following form:
𝐴𝐴′ − 𝐴𝐴 =
(1 − 𝑢𝑢)𝐴𝐴𝐴𝐴
𝜇𝜇
The parameter 𝜇𝜇 measures the cost of innovation, in that labour input (1 − 𝑢𝑢)𝑁𝑁 must be
0.01𝜇𝜇 to get 𝐴𝐴 to grow by 1 per cent. It is inversely related to the productivity of
researchers in growing the stock of knowledge. An important contrast to earlier equations
for the dynamics of physical capital 𝐾𝐾 ′ − 𝐾𝐾 or human capital 𝐻𝐻′ − 𝐻𝐻 is that there is no
depreciation of ideas. Knowledge, once discovered, is never lost.
2.9.3 Endogenous growth
The production function for new ideas implies that future technology 𝐴𝐴′ is:
𝐴𝐴′ = �1 +
(1 − 𝑢𝑢)𝑁𝑁
� 𝐴𝐴
𝜇𝜇
This means the growth rate (𝐴𝐴′ − 𝐴𝐴)/𝐴𝐴 of the stock of ideas 𝐴𝐴 is (1 − 𝑢𝑢)𝑁𝑁⁄𝜇𝜇. The linearity
of the production function for new ideas implies constant returns to R&D activity, i.e. the
same workforce (1 − 𝑢𝑢)𝑁𝑁 in the R&D sector can produce new ideas at a constant rate.
Given 𝑢𝑢, 𝜇𝜇, and 𝑁𝑁, the R&D model predicts a constant positive growth rate of 𝐴𝐴. The
implications for real income per worker can then be analysed using the framework from
Section 2.3 where there was exogenous labour-augmenting technological progress. With a
constant growth rate of 𝐴𝐴, there is positive long-run growth in output 𝑌𝑌 (and, hence, output
per worker here) at the same rate (1 − 𝑢𝑢)𝑁𝑁⁄𝜇𝜇, so the R&D model generates growth
endogenously. Moreover, shifting resources to the R&D sector, i.e. lowering 𝑢𝑢 and raising
1 − 𝑢𝑢, raises the economy’s long-run growth rate. This comes with the short-run cost of
lower output of goods when workers are diverted from the goods-producing sector to the
R&D sector but before they have produced any new knowledge.
We have not analysed the determinants of the fraction 1 − 𝑢𝑢 of labour allocated to R&D
using the model but we can think about this as resulting from a comparison of benefits and
costs to firms engaged in R&D. The cost to a firm of having more R&D workers is the wages
they need to be paid. The benefit of having them depends on the value to the firm itself of
the knowledge the R&D workers discover, for example, the value of new patents registered.
This value is affected by the strength of patent protection. Stronger protection of
intellectual property rights raises the private benefit of R&D and should lead to a lower
value of 𝑢𝑢.
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2.9.4 Constant returns or diminishing returns to R&D?
The conclusion that the R&D model is able to generate endogenous growth rests on the
assumption of constant returns in the R&D sector. However, there are reasons to be
sceptical of this. We see that the fraction of resources allocated to R&D in advanced
economies has risen substantially over the course of the 20th century. The model suggests
this should lead to faster rates of economic growth but that has not been observed.
We can allow for increasing or decreasing returns in R&D by making the cost of discoveries
𝜇𝜇 depend on the stock of existing knowledge 𝐴𝐴. Increasing returns would mean 𝜇𝜇(𝐴𝐴) is a
decreasing function of 𝐴𝐴. One justification for this assumption might be that past discoveries
provide the tools and foundation for future discoveries – ‘standing on the shoulders of
giants’. On the other hand, decreasing returns in R&D can be represented by 𝜇𝜇(𝐴𝐴) being an
increasing function of 𝐴𝐴. This could be because more advanced ideas are inherently harder
to discover – the ‘low-hanging fruit have been picked’.
The standard version of the model with 𝜇𝜇 being a constant can be seen as assuming these
two forces are approximately in balance. However, the case of 𝜇𝜇(𝐴𝐴) being an increasing
function of 𝐴𝐴 is more consistent with the observation that 𝑢𝑢 has declined over time in
advanced countries while growth rates (1 − 𝑢𝑢)𝑁𝑁/𝜇𝜇(𝐴𝐴) have not increased. This case with
decreasing returns in R&D means that if 𝜇𝜇(𝐴𝐴) keeps rising, rates of long-run economic
growth will fall to zero.
2.10 International technology transfer
If endogenous growth is coming from the discovery of new ideas, such as with learning-bydoing or as in the R&D model, can those models be applied to understand differences in
income levels across countries? These models do not imply convergence to a steady state,
which seems to suggest they might be consistent with large income differences around the
world through the logic developed in Box 2.5.
However, a key feature of models that generate endogenous growth through the discovery
of knowledge is that ideas are non-rivalrous and this non-rivalrous nature of ideas should
also apply across countries, not only across firms within a country. Therefore, countries
should be able to copy the ideas and knowledge of those at or closer to the frontier of
knowledge. If ideas can be copied at no cost, this would severely limit the ability of
endogenous growth models based on knowledge to explain large cross-country income
differences.
This section explores whether this logic also applies when there is a cost of imitating ideas
discovered in other countries. Take two countries labelled 1 and 2, and assume both have
the same labour force 𝑁𝑁. Suppose country 1 is at the frontier of knowledge, while country 2
lags behind. Using the notation from the R&D model in Section 2.9, we have 𝐴𝐴1 > 𝐴𝐴2 , which
is due to 𝑢𝑢1 < 𝑢𝑢2 since there are no other differences between them. That is, country 1
allocates a greater fraction of its labour force to R&D than country 2. Country 1 innovates as
described by the R&D model with constant returns. Its growth rate of 𝐴𝐴1 and its GDP is
(1 − 𝑢𝑢1 )𝑁𝑁/𝜇𝜇𝑖𝑖 , where 𝜇𝜇𝑖𝑖 is the cost parameter for innovation in country 1’s R&D sector.
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Country 2 can copy ideas from country 1 at a cost. Some labour input is required to
understand and apply the discoveries of country 1 but less than is needed to ‘reinvent the
wheel’ through independent R&D. We write down a ‘production function’ for ideas newly
imitated in country 2 that is also of the form 𝐴𝐴′2 − 𝐴𝐴2 = (1 − 𝑢𝑢2 )𝑁𝑁/𝜇𝜇𝑐𝑐 , but with 𝜇𝜇𝑐𝑐 being
the cost of copying ideas rather than innovating. This cost 𝜇𝜇𝑐𝑐 is lower than the cost of
innovation 𝜇𝜇𝑖𝑖 .
While 𝜇𝜇𝑖𝑖 is a constant parameter, reflecting the assumption of constant returns in R&D for
the frontier country, it is reasonable that the cost of imitation 𝜇𝜇𝑐𝑐 should depend on the size
of the knowledge gap between countries 1 and 2. Hence, 𝜇𝜇𝑐𝑐 is assumed to be a function
𝜇𝜇𝑐𝑐 = 𝑐𝑐(𝐴𝐴1 ⁄𝐴𝐴2 ) of the knowledge ratio 𝐴𝐴1 /𝐴𝐴2 . The cost function is decreasing in the size of
the knowledge gap 𝐴𝐴1 /𝐴𝐴2 because it is easier to copy more basic ideas discovered by
country 1 further in the past. The cost is close to zero when 𝐴𝐴1 /𝐴𝐴2 is extremely large
because when country 2 starts far behind, picking up some simple knowledge from country
1 should be almost free. Finally, the cost approaches the cost of innovation 𝜇𝜇𝑖𝑖 when the
knowledge gap becomes small (𝐴𝐴1 /𝐴𝐴2 is close to 1). This says that cost of quickly imitating
the most recent discoveries of country 1 is close to the cost of doing innovation itself. A cost
function with these features is depicted in Figure 2.22.
Figure 2.22: Imitation cost function
The growth rate of knowledge 𝐴𝐴1 and GDP in country 1 is (1 − 𝑢𝑢1 )𝑁𝑁/𝜇𝜇𝑖𝑖 , and the growth
rate of knowledge 𝐴𝐴2 and GDP in country 2 is (1 − 𝑢𝑢2 )𝑁𝑁/𝑐𝑐(𝐴𝐴1 ⁄𝐴𝐴2 ). The two growth rates
are plotted in Figure 2.23 as a function of the knowledge ratio 𝐴𝐴1 /𝐴𝐴2 . For country 1, its
growth rate is independent of how far country 2 is behind. For country 2, its growth rate is
faster when it is further behind because it becomes cheaper to imitate, i.e. 𝑐𝑐(𝐴𝐴1 ⁄𝐴𝐴2 ) is
lower. Over time, the ratio 𝐴𝐴1 /𝐴𝐴2 declines to the extent that the growth rate of 𝐴𝐴2 exceeds
the growth rate of 𝐴𝐴1 . We can see from the diagram that there is convergence to a steady
state for the knowledge ratio 𝐴𝐴1 /𝐴𝐴2 where the growth rate lines intersect. Mathematically,
this occurs where:
𝜇𝜇𝑐𝑐 = 𝑐𝑐 �
(1 − 𝑢𝑢2 )
𝐴𝐴1
�=
𝜇𝜇 < 𝜇𝜇𝑖𝑖
𝐴𝐴2
(1 − 𝑢𝑢1 ) 𝑖𝑖
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Since 𝐴𝐴1 /𝐴𝐴2 converges to a steady state in the long run, the follower country’s long-run
growth rate equals the growth rate of the frontier country. This means that even if 𝑢𝑢1 and
𝑢𝑢2 are permanently different, the gap between the countries eventually stabilises. The
international technology transfer model thus suggests it is difficult to explain very large
income differences across countries through knowledge gaps because those countries that
have not yet adopted much knowledge from the rest of the world can improve at a
relatively low cost. Note that this does not mean that knowledge gaps will shrink to zero in
the long run. With fewer resources allocated to obtaining knowledge, country 2 cannot
imitate the newest technologies of country 1.
Figure 2.23: The steady-state knowledge gap across countries
If country 2 were to lower 𝑢𝑢2 , allocating more workers to imitating country 1’s ideas, this
will shrink the knowledge gap between the two in the long run but it would not change
country 2’s long-run growth rate (assuming country 2 remains the follower). This reflects
the diminishing returns to imitation because the stock of existing knowledge to copy from
country 1 is necessarily finite.
Box 2.6: How strong should intellectual property rights be?
The strength of intellectual property rights such a patent protection affects incentives
to undertake research and development. If owners of patents can benefit from their
innovations for longer then there is a greater incentive to carry out R&D. However,
while excludability encourages the discovery of new ideas, ideas are fundamentally
non-rivalrous, so putting obstacles in the way of those who can benefit from using
them has a welfare cost. Here we study the international dimension of this issue using
the model of international technology transfer.
Suppose there is greater protection of intellectual property rights across countries’
borders. In the international technology transfer model, the follower country (2) now
faces higher costs of adopting new technologies developed by the innovating country
(1).
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The imitation cost 𝑐𝑐(𝐴𝐴1 ⁄𝐴𝐴2 ) rises for each level of the knowledge ratio 𝐴𝐴1 /𝐴𝐴2 . This
implies the growth rate (1 − 𝑢𝑢2 )𝑁𝑁/𝑐𝑐(𝐴𝐴1 ⁄𝐴𝐴2 ) of 𝐴𝐴2 is lower for each value of 𝐴𝐴1 ⁄𝐴𝐴2 ,
shifting down the growth rate function for country 2 as shown in the left panel of
Figure 2.24.
Firms in the innovating country (1) now gain more from discovering new technologies
and have a greater incentive to do R&D. This increases the fraction of workers
employed in the R&D sector, i.e. 𝑢𝑢1 is lower. The growth rate (1 − 𝑢𝑢1 )𝑁𝑁⁄𝜇𝜇𝑖𝑖 of 𝐴𝐴1 is
now higher, which shifts up the growth rate line for country 1 in the figure (it is a
horizontal line because the growth rate is independent of the levels of 𝐴𝐴1 and 𝐴𝐴2 ).
Figure 2.24: Effects of stronger intellectual property rights
The right panel of the figure depicts the time paths of knowledge (in logarithms) for the
two countries. The gradient of the path for country 1 becomes steeper as the growth
rate of knowledge 𝐴𝐴1 increases. From the perspective of the follower country 2,
stronger intellectual property rights slow down the adoption of new technologies from
the frontier country, which means 𝐴𝐴2 is below the path it would have followed for
some time after the strengthening of intellectual property rights. This implies its output
per worker grows more slowly for some time.
However, since there is a steady state for relative technology 𝐴𝐴1 /𝐴𝐴2 , albeit at a higher
level than before, eventually 𝐴𝐴2 will rise above the path it would have followed
because of the faster growth in 𝐴𝐴1 now country 1 has a greater incentive to innovate. It
follows that country 2 gains from greater protection of intellectual property rights in
the long run, although loses out during a transitional period (which could last for a
considerable amount of time).
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Chapter 3: Aggregate demand
This chapter focuses on the demand side of the economy. We look at the determinants of
expenditure on goods and services by households and firms. These components of
aggregate expenditure, consumption and investment, are the largest components of GDP
broken down by expenditure. An analysis of the effects of the government’s fiscal policies
on demand is deferred until Chapter 4. The links between international trade and aggregate
expenditure are analysed in Chapter 10 but, for now, we continue to consider a closed
economy.
The focus on the demand side means we will give more emphasis to fluctuations in the
economy compared to the long-run trends that were the subject of the previous two
chapters. We will look at how macroeconomic data can be detrended to zoom in on the
business-cycle, i.e. the fluctuations of macroeconomic variables around their long-run
trends.
After having studied the determinants of consumption and investment, we will see how to
integrate our analysis of the demand side of the economy with the supply side from earlier
chapters. This entails setting up a dynamic macroeconomic model where demand and
supply factors both matter in understanding changes in GDP.
Essential reading
•
Williamson, Chapters 9 and 11.
3.1 Detrending macroeconomic data
Many macroeconomic variables are rising over time for reasons discussed in the analysis of
economic growth. These long-run trends dominate the data but sometimes we would like to
emphasise the fluctuations seen in data. This is done by detrending time series data.
First, a trend line is estimated. These data points are typically plotted as logarithms so that a
constant growth rate would correspond to the straight line plotted against time.
The estimated trend is then removed from the data by subtracting the height of the trend
line from each data point. This leaves a time series of deviations from trend. If the data were
in logarithms, these deviations from trend can be interpreted as percentage deviations. The
deviations from trend are then taken as a measure of the business-cycle component of a
variable. A stylised representation of this way of measuring business cycles is shown in
Figure 3.1.
3.1.1 Business cycles
Applying this methodology to real GDP, we identify business cycles with fluctuations around
the trend in real GDP. Persistent positive deviations from trend are referred to as ‘booms’
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and persistent negative deviations from trend are ‘recessions’. The turning points of the
business cycle are the peaks and troughs in the deviations of real GDP from its trend. Note
that the definition of a recession here differs from the conventional two consecutive
quarters of decline in real GDP and that an economy can still be in recession if it is still
below its trend while growing.
Figure 3.1: Stylised representation of business cycles
Some terms are widely used to describe the patterns seen in diagrams like Figure 3.1. The
amplitude of the fluctuations in a variable refers to how far above or below trend a variable
goes. The persistence of the fluctuations in a variable refers to how long it typically takes for
the variable to return to its trend line. The frequency of fluctuations refers to how often the
variable switches from being above trend to below trend, or below to above.
3.1.2 Detrending
How is the detrending done to construct a version of the figure with real data? It is not
necessary to know the technical details in this course but it essentially involves putting a line
of best fit through the data points plotted as a time series. If the long-run percentage
growth rate is stable, the trend line for data plotted in logarithms can be estimated as the
best-fitting straight line. In practice, long-run growth rates are not entirely stable, so the
best trend line is not a completely straight line. Researchers often use the Hodrick-Prescott
(HP) filter, which is essentially a trend line that can change ‘smoothly’ over time.
Figure 3.2 shows the HP-filter trend line for the logarithm of US real GDP, plotted as a
quarterly time series from 1947 to 2021. The estimated HP-filter trend line is not completely
straight. For instance, the gradient is steeper in the 1960s and 1990s than the 1970s,
reflecting the ‘productivity slowdown’ experienced by the US economy in the 1970s. We see
that the trend dominates the data, so that even events such the recessions after the 2008
financial crisis and during the COVID pandemic appear as relatively small dips compared to
the secular progress in the US economy over many decades.
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Figure 3.3 shows the percentage deviations from US real GDP from its HP-filter trend line,
and it is calculated as the difference between actual and trend log real GDP multiplied by
100. This is our measure of the business cycle in the USA. The business cycle is typically
ranges from +4% to −4% of trend real GDP, although the recession owing to COVID is
much larger. The period from the mid-1980s up to the 2008 financial crisis is known as the
‘great moderation’ owing to the small ±2% range of fluctuations in real GDP.
Figure 3.2: HP filter applied to US real GDP
Figure 3.3: US business cycles as measured by detrended real GDP
3.1.3 Business-cycle stylised facts
After detrending real GDP and other macroeconomic variables, we document some typical
patterns seen in the fluctuations. While each business-cycle episode has unique features,
nonetheless, we observe some relationships across different variables that generally hold
true in different booms and recessions. We refer to these patterns as ‘stylised facts’.
For a particular macroeconomic variable 𝑥𝑥, we document:
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•
•
•
Co-movement: the correlation of 𝑥𝑥 with real GDP
Relative volatility: the size of fluctuations in 𝑥𝑥 relative to fluctuations in real GDP
Leads or lags: the dynamics of 𝑥𝑥 relative to real GDP.
The pattern of co-movement between a variable 𝑥𝑥 and real GDP can be determined either
by plotting time series of percentage deviations from trend of both on the same axis, as is
done in Figure 3.4, or using a scatterplot of the deviations from trend in Figure 3.5. The
variable 𝑥𝑥 is said to be ‘procyclical’ if 𝑥𝑥 and real GDP have peaks and troughs at similar
times, or if the scatterplot reveals a positive relationship. 𝑥𝑥 is said to be ‘countercyclical’ if
peaks of 𝑥𝑥 are associated with troughs of real GDP and vice versa, or if the scatterplot
relationship is negative. A variable is said to be ‘acyclical’ if it displays neither a procyclical
nor countercyclical pattern.
Figure 3.4: Co-movement in data plotted over time
Figure 3.5: Co-movement shown in scatterplots
The relative volatility of a variable 𝑥𝑥 can be judged by plotting a time series of its percentage
deviations from trend against those of real GDP. If the amplitude of the fluctuations in 𝑥𝑥 is
typically larger than those of real GDP then 𝑥𝑥 has a higher relative volatility. If 𝑥𝑥 has smaller85
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amplitude fluctuations than real GDP, as shown for example in Figure 3.6, then 𝑥𝑥 has a
lower relative volatility.
Finally, the dynamic relationship between 𝑥𝑥 and real GDP can be described in terms of leads
or lags. If the fluctuations in 𝑥𝑥 anticipate those of real GDP that follow then 𝑥𝑥 is said to be a
‘leading’ variable. On the other hand, if fluctuations in 𝑥𝑥 follow those of real GDP with some
delay than 𝑥𝑥 is said to be ‘lagging’. These two cases are illustrated in Figure 3.7. If 𝑥𝑥 neither
leads nor lags real GDP then it is said to be ‘coincident’.
Figure 3.6: Relative volatility
Figure 3.7: Leading and lagging variables
3.2 Consumption
Consumption refers to purchases of final goods and services by households. Consumption is
the ultimate purpose of all economic activity and a key determinant of households’ welfare.
We begin by documenting some simple facts about consumption. In advanced economies,
consumption expenditure is typically around two-thirds of GDP. This makes it the largest
component of aggregate expenditure. While it makes up a large fraction of GDP,
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consumption is generally less volatile than GDP. This observation is important in motivating
the theories of consumption we will study.
Figure 3.8 shows detrended real consumption expenditure in the United States alongside
detrended real GDP for the period 1947–2021. The peaks and troughs of consumption
closely align with those of real GDP, so consumption is said to be a procyclical and
coincident variable because no leads or lags can be discerned relative to GDP. The most
notable observation is that the absolute value of the peaks and troughs of consumption are
almost always smaller than those of GDP. Consumption therefore has a lower relative
volatility compared to GDP.
Figure 3.8: Empirical evidence on consumption
We now turn to explaining the determinants of households’ consumption decisions. While it
is obvious that a household’s income puts limits on its consumption, note that households
can use saving or borrowing to transfer purchasing power from one point in time to
another. Hence, it is not necessarily a household’s current income that constrains its current
consumption but a notion of total income over time. Income from the past that has been
saved can be used to finance current consumption, as can borrowing that could be repaid
using future income the household anticipates receiving.
Moreover, a key idea in studying consumption is that households are forward-looking and
care about consumption in the future. The central theory of consumption we will study is
based on the idea that households have a desire for ‘consumption smoothing’, i.e. avoiding
fluctuations in how much goods and services they purchase. To do this, the theory supposes
households make use of saving or borrowing to smooth out income fluctuations.
However, as we will see, it may not always be possible to achieve this consumption
smoothing. In particular, borrowing against the expectation of receiving income in the
future might be very costly or difficult. This is a particular example of what we will refer to
as ‘credit-market imperfections’ and we will see how they place limits on the extent to
which households are able to smooth consumption even if they want to.
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3.3 A two-period consumption model
The basic principles underlying what are known as dynamic theories of consumption can be
illustrated using a model with two periods 1 and 2. Period 1 refers to the present and period
2 to the future. A more realistic version of this framework might have a separate period for
each year but ideas can be illustrated with just two periods.
The key features of the model are that households make plans for the future with a
preference for smoothing out fluctuations in consumption expenditure. In doing so, they
can use saving or borrowing to transfer purchasing power across time subject to an
intertemporal budget constraint.
3.3.1 Preferences
In the model, households choose consumption plans (𝑐𝑐, 𝑐𝑐 ′ ), where 𝑐𝑐 is current consumption
and 𝑐𝑐′ is future consumption. Preferences over consumption plans can be represented in a
diagram with 𝑐𝑐 and 𝑐𝑐′ on the axes using indifference curves as shown in Figure 3.9.
Indifference curves are downward sloping because more consumption is preferred to less at
each date. Importantly, indifference curves are drawn as convex to origin. This indicates a
dislike of extreme consumption plans, for example, the household does not like to have high
𝑐𝑐 but 𝑐𝑐′ being very low and would prefer a greater balance between 𝑐𝑐 and 𝑐𝑐′. Similarly, the
household would prefer 𝑐𝑐 and 𝑐𝑐′ closer together than very low 𝑐𝑐 but high 𝑐𝑐′. This is how we
represent a preference for consumption smoothing.
The absolute value of the gradient of the indifference curve at a point is known as the
marginal rate of substitution between 𝑐𝑐 and 𝑐𝑐′, denoted by 𝑀𝑀𝑀𝑀𝑆𝑆𝑐𝑐,𝑐𝑐 ′ . Since the indifference
curve gradient indicates how much extra future consumption 𝑐𝑐′ the household would
require to be as well off after losing a unit of current consumption 𝑐𝑐, the marginal rate of
substitution indicates the relative value households put on a unit of current consumption
compared to future consumption. The convex shape of the indifference curves is equivalent
to assuming the marginal rate of substitution is diminishing as current consumption 𝑐𝑐 rises.
Another assumption on preferences is that both 𝑐𝑐 and 𝑐𝑐′ are normal goods. A normal good
is a good the household wants more of when better off. Geometrically, this assumption
means that if points on different indifference curves with the same marginal rate of
substitution were joined up then they would trace out an upward sloping line. This
assumption captures the idea that if more consumption is affordable, the household would
like to spread out the extra expenditure over time.
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Figure 3.9: Household indifference curves
3.3.2 Budget constraint
The dynamic consumption model works by putting together the preferences described
above with a budget constraint that spans the two periods. The link between the two
periods comes from saving or borrowing decisions.
Suppose the household receives real income 𝑦𝑦 in the current period and 𝑦𝑦′ in the future
period. These incomes are not explained by the model: consumption is endogenous but
income is exogenous. In Section 3.11, we will see how a larger model can explain
consumption and aggregate income simultaneously. Income here should be interpreted as
non-financial income. Wages are included but not any interest income, which will be
accounted for separately.
The incomes here are before tax. The government collects tax revenue 𝑡𝑡 and 𝑡𝑡′ from the
household in the two periods, so disposable incomes in the two periods are 𝑦𝑦 − 𝑡𝑡 and 𝑦𝑦 ′ −
𝑡𝑡′. Here, these taxes are assumed to have a ‘lump sum’ form: the amount paid to the
government does not depend on any decisions the household makes, for example, a poll
tax. More realistically, the government typically sets tax rates on earned income and
consumption expenditure, and we will see examples of what difference this makes. For now,
maintain the assumption of lump-sum taxes.
Saving 𝑠𝑠 done by the household refers to current disposable income that is not currently
consumed, that is, 𝑠𝑠 = 𝑦𝑦 − 𝑡𝑡 − 𝑐𝑐. Saving can put in a bank account to earn interest or used
to purchase financial assets. For now, suppose savings are held in a bank account or as
bonds that pay a known real rate of interest 𝑟𝑟. This real interest rate adjusts the payment of
interest in terms of money for inflation that is expected to occur between the current and
future periods. The distinction between different types of interest rate is analysed further in
Chapter 4. It is possible for real interest rates to be negative, so it is not necessarily assumed
that 𝑟𝑟 > 0.
Saving, as defined, can be either positive or negative. Negative saving 𝑠𝑠 < 0 represents
borrowing, where consumption expenditure exceeds disposable income in the current
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period. In cases where the household is a borrower, interest has to be paid on debts.
Assume for now that the household can borrow at the same interest rate 𝑟𝑟 that savers
receive. Later financial-market imperfections will explain why borrowers face higher interest
rates than savers in practice.
In the simple two-period model, we suppose that the household begins from a blank slate in
terms of not having any initial assets or debts at the beginning of the current period. Given
saving 𝑠𝑠, the household has financial wealth (1 + 𝑟𝑟)𝑠𝑠 in the future period, which includes
the saving 𝑠𝑠 plus real interest income 𝑟𝑟𝑟𝑟. Note that interest income is counted separately
from future non-financial income 𝑦𝑦′.
In the future period, the household has disposable income 𝑦𝑦 ′ − 𝑡𝑡′ and net financial wealth
(1 + 𝑟𝑟)𝑠𝑠. Net wealth is negative for borrowers. Taking disposable income and financial
wealth together, the household can afford future consumption:
𝑐𝑐 ′ = 𝑦𝑦 ′ − 𝑡𝑡 ′ + (1 + 𝑟𝑟)𝑠𝑠
Given the nature of the two-period model, the future period comprises the whole future the
household is planning for, so there is no reason to save for anything after the future period.
Hence, 𝑐𝑐’ is given by the equation above conditional on the choice of current saving 𝑠𝑠. As
mentioned, saving 𝑠𝑠 can be negative and the only limit imposed on borrowing here is that
debts including interest do not exceed the ability to repay as measured by future disposable
income 𝑦𝑦 ′ − 𝑡𝑡 ′ . It may be reasonable to consider stricter limits on borrowing with financialmarket imperfections and examples of these are considered in Chapter 4.
Dividing both sides of the equation for 𝑐𝑐′ by 1 + 𝑟𝑟 implies
𝑦𝑦 ′ − 𝑡𝑡′
𝑐𝑐 ′
=
+ 𝑠𝑠
1 + 𝑟𝑟
1 + 𝑟𝑟
and substituting the definition of saving 𝑠𝑠 = 𝑦𝑦 − 𝑡𝑡 − 𝑐𝑐 and grouping consumption and
disposable income terms together:
𝑐𝑐 ′
𝑦𝑦 ′ − 𝑡𝑡′
𝑐𝑐 +
= 𝑦𝑦 − 𝑡𝑡 +
1 + 𝑟𝑟
1 + 𝑟𝑟
Consumption plans (𝑐𝑐, 𝑐𝑐 ′ ) must satisfy this equation to be affordable to the household.
Geometrically, the budget constraint is a straight line with gradient −(1 + 𝑟𝑟) passing
through the ‘endowment’ point (𝑦𝑦 − 𝑡𝑡, 𝑦𝑦 ′ − 𝑡𝑡 ′ ) as illustrated in Figure 3.10.
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Figure 3.10: Lifetime budget constraint
The household can always choose neither to be a saver nor a borrower, meaning that 𝑐𝑐 =
𝑦𝑦 − 𝑡𝑡 and 𝑐𝑐 ′ = 𝑦𝑦 ′ − 𝑡𝑡′, so disposable income is simply consumed in both time periods. This
means choosing a consumption plan at the endowment point in the diagram (labelled E).
Choosing a consumption plan to the left of the endowment point means choosing to be a
saver (𝑠𝑠 > 0), while choosing a plan to the right of the endowment point means choosing to
be a borrower (𝑠𝑠 < 0).
For each unit less of income consumed in the current period, an extra unit can be saved,
which means an extra amount 1 + 𝑟𝑟 of financial wealth in the future and thus an extra
amount 1 + 𝑟𝑟 of future consumption is affordable. This is why the budget constraint has
gradient −(1 + 𝑟𝑟). In the absence of financial-market imperfections, the budget line is a
straight line (it has the same gradient in the saving and borrowing regions) and extends all
the way down to the horizontal axis.
The budget constraint is described as a ‘lifetime budget constraint’ because it includes
consumption and income in both periods. However, 𝑐𝑐 and 𝑐𝑐′ and 𝑦𝑦 − 𝑡𝑡 and 𝑦𝑦 ′ − 𝑡𝑡′ cannot
simply be added together because they occur in different periods. Instead, present
discounted values (PDVs) of future consumption 𝑐𝑐 ′ /(1 + 𝑟𝑟) and future income (𝑦𝑦 ′ −
𝑡𝑡 ′ )/(1 + 𝑟𝑟) are added to current 𝑐𝑐 and 𝑦𝑦 − 𝑡𝑡. The present value of a future amount 1 is
equal to 1/(1 + 𝑟𝑟) because if 1/(1 + 𝑟𝑟) were saved in the current period, after including
interest, it would be worth (1 + 𝑟𝑟)/(1 + 𝑟𝑟) = 1 in the future period. The lifetime budget
constraint shows that the present discounted value of all consumption across the two
periods is limited by the present value of all current and future disposable income.
The present discounted value of all income after tax can be interpreted as ‘lifetime’ or
‘human’ wealth, which is denoted by ℎ:
𝑦𝑦 ′ − 𝑡𝑡 ′
ℎ = 𝑦𝑦 − 𝑡𝑡 +
1 + 𝑟𝑟
This is what an asset making payments equal to disposable income in the two periods would
be worth to the household. If the household had any initial financial assets in the current
period, the value of these would be added to ℎ to obtain total wealth in all forms.
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3.3.3 Choice of an optimal consumption plan
Having described households’ preferences and constraints, we now turn to analysing the
optimal choice of consumption plan (𝑐𝑐, 𝑐𝑐 ′ ). Analogous to similar constrained maximisation
problems in microeconomics, the optimal consumption plan is found on the highest
indifference curve that can be reached from the lifetime budget constraint.
The optimal plan is at the tangency point between an indifference curve and the lifetime
budget constraint, i.e. where an indifference curve has the same gradient as that of the
budget constraint, which is −(1 + 𝑟𝑟) all along the budget line. The indifference curve
gradient is −𝑀𝑀𝑀𝑀𝑆𝑆𝑐𝑐,𝑐𝑐′ , where 𝑀𝑀𝑀𝑀𝑆𝑆𝑐𝑐,𝑐𝑐′ is the marginal rate of substitution between 𝑐𝑐 and 𝑐𝑐′.
Hence, the optimal plan is the point on the budget constraint where 𝑀𝑀𝑀𝑀𝑆𝑆𝑐𝑐,𝑐𝑐′ = 1 + 𝑟𝑟 as
shown in Figure 3.11.
Figure 3.11: Optimal consumption plan
Intuitively, the budget constraint gradient 1 + 𝑟𝑟 (in absolute value) represents the market
price of current consumption in terms of future consumption, while the indifference curve
gradient represents how much extra 𝑐𝑐′ is needed to compensate the household for the loss
of one unit of 𝑐𝑐. The tangency point is thus where the subjective value the household puts
on current consumption relative to future consumption is equal to the relative cost of
current consumption given the market interest rate 𝑟𝑟.
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Box 3.1: The consumption puzzle
How does the empirical evidence on consumption and income compare to the predictions
of the model? We can look at two types of data:
•
•
Cross-sectional data: observations of consumption and income for many individual
households in a particular year
Time-series data: observations of aggregate consumption and income for a
number of years in a particular economy.
Interestingly, the relationship between consumption and income seen in these two types
of data is not the same. This apparent contradiction is known as the ‘consumption puzzle’.
Figure 3.12 shows the typical shape of a line of best fit drawn through data on
consumption and income in a cross-section of households. There is evidence of
consumption smoothing in that different levels of income are typically associated with
smaller differences in consumption. In other words, there are high levels of saving for
those households with high incomes, and low or negative saving levels for those with low
incomes.
Figure 3.12: Consumption and income in cross-sectional data
A relationship between consumption and current income consistent with the crosssectional empirical evidence is sometimes directly assumed in macroeconomics. This is
the Keynesian consumption function relating aggregate consumption 𝐶𝐶 to aggregate
disposable income 𝑌𝑌 − 𝑇𝑇, where 𝑇𝑇 denotes taxes:
𝐶𝐶 = 𝑎𝑎 + 𝑏𝑏(𝑌𝑌 − 𝑇𝑇)
The terms a and b are parameters with a>0 and 0<b<1. The marginal propensity to
consume ∂C/∂Y is the constant b. The average propensity to consume C/Y declines as
income rises because a is positive, that is to say, a higher fraction of income is saved when
income is high. Although this equation is consistent with the cross-sectional evidence, it
actually posits that the same relationship holds between aggregate consumption and
income over time. As we will see, this is not the case. Moreover, the equation is not
derived from a model of rational behaviour by households.
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To understand the consumption-income relationship seen in the cross-sectional data using
the dynamic consumption model, let us think first about the sources of the variation seen
in income levels across households. Some, but not all, of this variation is due to:
•
•
Stage of life cycle: the young and old have lower incomes on average compared to
the middle-aged
Temporary shocks: some people experience periods of unemployment, others earn
large bonuses/overtime payments.
First, there are large predictable differences between current income and future income
because households are of different ages. However, the preference for consumption
smoothing means that the optimal consumption plan has 𝑐𝑐 and 𝑐𝑐′ closer together than are
𝑦𝑦 − 𝑡𝑡 and 𝑦𝑦 ′ − 𝑡𝑡′. This is consistent with the cross-sectional empirical evidence.
The left panel of Figure 3.13 illustrates the consumption choice diagram for a young person
with a low current income but expectations of higher income in the future, so the
endowment point 𝐸𝐸 is at a relatively extreme north-west position in the diagram. The right
panel shows a middle-aged person with a high current income but expectations of lower
income in the future owing to retirement, so the endowment point is at a relatively
extreme south-east position. Households with quite different levels of current income
could choose very similar levels of consumption.
Figure 3.13: Life-cycle differences in income
If the life cycle were the only reason for income differences across households then the
predicted consumption-income relationship would be even weaker than found in the data.
However, there are plenty of income differences that do not average out over the life
cycle, for example, people having different careers.
A second source of income variation comes from temporary shocks, such as receiving a
bonus, or a period of unemployment. We can show the predicted effects of such shocks in
the dynamic consumption model by shifting the endowment point 𝐸𝐸 with the change in
income. A temporary shock affects only current income 𝑦𝑦, leaving 𝑦𝑦 ′ unchanged, which
moves the endowment point horizontally. A movement of 𝐸𝐸 causes a parallel shift of the
lifetime budget constraint. Taking the case of a temporary rise in income 𝑦𝑦, Figure 3.14
shows the endowment point move to the right from 𝐸𝐸0 to 𝐸𝐸1 . The budget constraint shifts
to the right, and both 𝑐𝑐 and 𝑐𝑐′ rise in response because they are normal goods. The desire
for consumption smoothing reflected in the rise94of both 𝑐𝑐 and 𝑐𝑐′ means that 𝑐𝑐 rises by less
than 𝑦𝑦. This is also consistent with cross-sectional data on consumption and income.
EC2065 Macroeconomics | Chapter 3: Aggregate demand
Figure 3.14: Temporary and permanent shocks to income
Our analysis is different for income changes expected to be permanent. With a
permanent rise in income, both 𝑦𝑦 and 𝑦𝑦 ′ increase, and the endowment point moves
horizontally and vertically. In the figure, this is shown as 𝐸𝐸0 to 𝐸𝐸2 . Unlike with a
temporary income shock, 𝑐𝑐 and 𝑦𝑦 now adjust by similar amounts, which could be
exactly the same.
Turning now to time series data on aggregate consumption and income, Figure 3.15
shows the typical shape of a line of best fit drawn through the time series data in
comparison to the cross-sectional data relationship already discussed. In contrast to
the cross-sectional data, time-series data suggest that aggregate consumption is
approximately proportional to aggregate income. Over long periods, the fractions of
income consumed and saved are fairly stable.
The relationship between consumption and income in time-series data is quite
different from what is found in cross-sectional data. The apparent inconsistency
between these findings is known as the ‘Kuznets consumption puzzle’. Simply
assuming a Keynesian consumption function that matches the cross-sectional data for
a model of aggregate consumption and income is therefore inconsistent with what we
know from time series data. However, the puzzle can be resolved using the dynamic
model of consumption.
The crucial difference between time series and cross-sectional data is the source of the
variation in income levels. In time series data, most of the variation comes from longrun trends in income that are explained through the analysis of long-run economic
growth in 0. Increases in income over time are expected to be permanent because
long-run growth is not expected to be reversed. In contrast, cross-sectional data
includes many changes in income that are only temporary.
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3.4 Bonds, yields, and interest rates
For households to save, they must hold an asset that transfers purchasing power to the
future. For example, deposits in bank account, bonds, or shares. The real return on an asset
is how much purchasing power over goods grows by holding that asset over time.
The simplest case is where savings are held as bank deposits in an account that pays
interest. The real return on such savings deposits is the interest rate adjusted for inflation.
Households can also save by holding bonds. Bonds are a particular type of financial asset
that promise a sequence of fixed future payments. For now, assume these payments are
fixed in real terms, rather than in units of money. This would correspond to what is called an
inflation-indexed bond.
The yield or interest rate on a bond (these terms are interchangeable) is the discount rate
that makes the present value of all the bond’s future fixed payments equal to the actual
market price it currently trades at. Since the bond payments are fixed, high bond prices are
equivalent to low yields/interest rates and low bond prices are equivalent to high
yields/interest rates.
The simplest form of bond is a discount bond that makes only one payment (its face value)
at the maturity date of the bond. If a bond is held to maturity, the real return received is
equal to the bond’s real yield.
Consider a discount bond maturing in the next period and assume the face value is worth
one unit of goods. If the bond’s price currently is worth 𝑝𝑝 units of goods, then the real yield
or real interest rate 𝑟𝑟 on the bond is the discount rate that makes the present value of 1
equal to the bond price 𝑝𝑝:
𝑝𝑝 =
1
1 + 𝑟𝑟
If the bond has no risk of default, an investment of 𝑝𝑝 gives a payoff of 1 in the future. An
investor therefore makes a profit of 1 − 𝑝𝑝 on an investment of 𝑝𝑝, so the percentage real
return is (1 − 𝑝𝑝)/𝑝𝑝 = 𝑟𝑟.
3.5 Interest rates and consumption
The real interest rate 𝑟𝑟 matters for households’ consumption plans because it determines
the market price of current consumption relative to future consumption. If a household
buys 1 unit more of 𝑐𝑐, saving 𝑠𝑠 is reduced by 1, which means future financial wealth
(1 + 𝑟𝑟)𝑠𝑠 is lower by 1 + 𝑟𝑟, and so future consumption must fall by 1 + 𝑟𝑟, all else equal.
Hence, the price of current consumption in terms of future consumption is 1 + 𝑟𝑟. This
relative price appears in the consumption choice diagram as the gradient of the lifetime
budget constraint in absolute value.
Changes in the real interest rate 𝑟𝑟 affect the gradient of the lifetime budget constraint. For
example, higher 𝑟𝑟 makes the budget constraint steeper as shown in Figure 3.16. When its
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gradient changes, the lifetime budget constraint pivots around the endowment point 𝐸𝐸,
which is always affordable.
To understand how the household’s optimal consumption plan (𝑐𝑐 ∗ , 𝑐𝑐 ′∗ ) adjusts, we break
down the response into income and substitution effects. The substitution effect is found by
looking at the response to the change in the budget constraint gradient with a hypothetical
income transfer that shifts the position of the budget constraint so that the original
indifference curve can be reached. This leaves the household no better or worse off. The
substitution effect obtain in this way captures the pure incentive effect of changes in
interest rates, controlling for whether this makes the household better off or worse off.
The income effect is the response to removing the hypothetical income transfer used to
derive the substitution effect. This holds the gradient of the budget constraint constant, so
it is a response to a parallel shift of the budget constraint. The income effect thus captures
the household’s response to being made better off or worse off by the interest rate change.
Figure 3.15: The lifetime budget constraint with a higher interest rate
Figure 3.17 analyses the effects of higher interest rates on a saver. A saver is a household
that initially chooses a consumption plan to the left of the endowment point 𝐸𝐸. The increase
in 𝑟𝑟 pivots the budget constraint around 𝐸𝐸, making it steeper. The household can now reach
a higher indifference curve, so savers are made better off by higher interest rates because
they earn a higher return on their financial wealth. The substitution effect is found by
making a parallel downward shift of the true budget constraint to where it is tangent to the
original indifference curve. This leaves the household no better off. Even so, the household
still has an incentive to shift towards less consumption in the current period and more in the
future because the relative price of current consumption has risen. This substitution effect is
labelled 𝑆𝑆𝑆𝑆 in the diagram. It is movement north-west to where the indifference curve is
steeper.
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Figure 3.16: Effects of higher interest rates on a saver
The income effect is found by removing the parallel downward shift of the true budget
constraint used to derive the substitution effect. Thus, the income effect is the response to
a parallel upward shift of the budget constraint. Since both current and future consumption
are normal goods, the income effect leads the saver to increase both 𝑐𝑐 and 𝑐𝑐′. Intuitively,
the household wants to spread out over time its gains from higher interest rates. The
income effect is labelled 𝐼𝐼𝐼𝐼 in the diagram and is a movement in a north-east direction.
Figure 3.18 performs the same exercise for a borrower. This is a household that initially
chooses a consumption plan to the right of the endowment point. When 𝑟𝑟 rises, the budget
constraint becomes steeper, pivoting around 𝐸𝐸. The initial consumption plan is no longer
affordable, so the borrower is made worse off when access to credit becomes more
expensive. While the hypothetical income transfer used for the substitution effect (𝑆𝑆𝑆𝑆) now
shifts the true budget constraint upwards, the analysis is qualitatively the same as for the
saver. The 𝑆𝑆𝑆𝑆 is a north-west movement in the diagram with lower 𝑐𝑐 and higher 𝑐𝑐′. On the
other hand, the income effect (𝐼𝐼𝐼𝐼) is the response to a parallel downward shift of the
budget constraint. The 𝐼𝐼𝐼𝐼 is therefore a south-west movement that reduces both 𝑐𝑐 and 𝑐𝑐′.
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Figure 3.17: Effects of higher interest rates on a borrower
For savers, the income and substitution effects of 𝑟𝑟 on current consumption 𝑐𝑐 go in opposite
directions so the overall response is ambiguous. The diagram shows the special case where
𝑆𝑆𝑆𝑆 and 𝐼𝐼𝐼𝐼 exactly cancel out. For future consumption 𝑐𝑐′, 𝑆𝑆𝑆𝑆 and 𝐼𝐼𝐼𝐼 are reinforcing, so 𝑐𝑐′
rises unambiguously. For borrowers, the 𝑆𝑆𝑆𝑆 and 𝐼𝐼𝐼𝐼 are reinforcing for 𝑐𝑐, which
unambiguously declines, while 𝑆𝑆𝑆𝑆 and 𝐼𝐼𝐼𝐼 are conflicting for 𝑐𝑐′ and the overall response is
ambiguous. The diagram shows the special case where 𝑆𝑆𝑆𝑆 and 𝐼𝐼𝐼𝐼 exactly cancel out for 𝑐𝑐′.
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Box 3.2: Durables, non-durables, and services
The business-cycle properties of consumption documented in Section 3.2 are obtained
for overall consumption expenditure. However, consumption can be broken down into
durables, non-durables and services components. Durable goods are products that can
be used for a long period (more than three years) after being purchased, for example,
a car or a refrigerator. Non-durables are physical goods that are used up or worn out
sooner after being purchased and hence, not classified as durable, for example, food
or clothing. Services are intangibles and the benefit of purchasing them is typically
obtained in a short period after a purchase, for example, a haircut.
Figure 3.19 shows the percentage deviations from trend of consumption of durables
alongside those of real GDP. An important aspect of the behaviour of aggregate
consumption is that it is less volatile than aggregate income, which motivated our
emphasis on consumption smoothing in the dynamic consumption choice model.
However, durables consumption is much more volatile than income, so we do not
observe smoothing of households’ expenditure on durable goods. This is not
surprising, though, because durable goods do not need to be purchased all the time to
enjoy the benefit of using them.
Figure 3.18: Consumption of durable goods
In justifying the desire of households to smooth consumption, we have implicitly
assumed that the benefit of current consumption 𝑐𝑐 comes only in the current period,
which explained why households do not want to have too great an imbalance between
𝑐𝑐 and 𝑐𝑐′. The assumptions of the consumption choice model therefore make more
sense for consumption of non-durables and services. Figures 3.20 and 3.21 show the
percentage deviations from trend of non-durables and services alongside those of real
GDP. Here, we see that these categories of consumption are indeed significantly less
volatile than real GDP. Durables consumption is better understood as a form of
investment, which we will analyse in Section 3.8.
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3.6 Consumption smoothing in the aggregate
The desire for consumption smoothing is a key feature of the dynamic model of
consumption in Section 3.3. We have seen that it helps to explain patterns seen in crosssectional data and resolve the ‘consumption puzzle’. But in time series data, consumption
and income are approximately proportional over long periods of time, so it appears at first
glance that consumption smoothing does not have much relevance when analysing
aggregate consumption. However, by removing trends from the data to focus on
fluctuations, we have seen in Figure 3.8 that consumption is less volatile than income, a
pattern that is more pronounced when focusing on non-durables and services.
Figure 3.19: Consumption of non-durables
Figure 3.20: Consumption of services
However, while aggregate consumption is less volatile than income, it is not much less
volatile. Although we cannot investigate this issue quantitatively in the simple two-period
model, for very short-term fluctuations in income relative to the whole future over which
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households are making plans, we would expect households to smooth consumption to a
much greater extent.
Why then is there not more smoothing of aggregate consumption? We will investigate two
explanations. In Chapter 4, we will look at credit-market imperfections such as borrowing
constraints. If income were to fall, some households might want to borrow against future
income to smooth consumption. But if they do not have access to credit, their current
consumption will fall by more than desired. Hence, it may be that households would like to
smooth consumption to a greater extent, but are prevented from doing so by credit-market
imperfections.
Another explanation is that we know changes in interest rates create an incentive to deviate
from a plan where current and future consumption are close together. This is because the
substitution effect of interest rates moves current and future consumption in different
directions. This matters for the relationship between aggregate consumption and income in
time series data as interest rates are also changing over time. However, in cross-sectional
data where all households are observed at the same point in time, interest rates are not
changing and so are of less concern in understanding the relationship between consumption
and income.
Not only can interest rates influence aggregate consumption, there are reasons to believe
that interest rate changes might be systematically linked to changes in aggregate income.
This is because the equilibrium level of income must be consistent with consumption
choices and other demands for goods in aggregate.
To understand this point, consider the following example. The economy is comprised of 𝑁𝑁
representative households each with exogenous income 𝑦𝑦. Since income is treated as
exogenous, this implicitly assumes no adjustment of households’ supply of labour.
Furthermore, we ignore the use of capital goods in production, which also means
investment is zero (𝐼𝐼 = 0). Real GDP in the economy is 𝑌𝑌 = 𝑁𝑁𝑁𝑁, which is exogenous here.
The government’s fiscal policy is an exogenous level of expenditure is 𝐺𝐺.
Households choose a consumption plan (𝑐𝑐, 𝑐𝑐 ′ ) as in the two-period model. Given 𝑐𝑐,
aggregate consumption is 𝐶𝐶 = 𝑁𝑁𝑁𝑁 because all 𝑁𝑁 households make the same choices given
that they have the same preferences and face the same income and interest rates.
Suppose the economy is closed, so market clearing in the goods market requires 𝑌𝑌 = 𝐶𝐶 + 𝐺𝐺.
The aggregate supply of goods, as given by the economy’s real GDP, must equal aggregate
expenditure 𝐶𝐶 + 𝐺𝐺 (there is no investment or net exports). With 𝑌𝑌 and 𝐺𝐺 being exogenous,
equilibrium is reached through adjustment of the real interest rate 𝑟𝑟. The equilibrium real
interest rate 𝑟𝑟 ∗ ensures that the consumption plan chosen by the representative household
is 𝑐𝑐 ∗ = (𝑌𝑌 − 𝐺𝐺)⁄𝑁𝑁 and 𝑐𝑐 ′∗ = (𝑌𝑌 ′ − 𝐺𝐺 ′ )⁄𝑁𝑁, which are equivalent to goods-market clearing
in the current and future time periods.
Figure 3.22 below shows how 𝑟𝑟 ∗ is determined graphically. At an arbitrary real interest rate
𝑟𝑟, the lifetime budget constraint of the representative household has gradient −(1 + 𝑟𝑟) and
passes through the point ((𝑌𝑌 − 𝐺𝐺)⁄𝑁𝑁 , (𝑌𝑌 ′ − 𝐺𝐺 ′ )⁄𝑁𝑁), labelled 𝐸𝐸. A detailed justification of
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the second of these claims is found in Section 4.1 but, for now, just note that the
government could pay for its expenditure using lump-sum taxes 𝑡𝑡 = 𝐺𝐺/𝑁𝑁 and 𝑡𝑡 ′ = 𝐺𝐺 ′ /𝑁𝑁
levied on each household.
The optimal consumption plan (𝑐𝑐 ∗ , 𝑐𝑐 ′∗ ) for a given 𝑟𝑟 will generally not be at
((𝑌𝑌 − 𝐺𝐺 )⁄𝑁𝑁 , (𝑌𝑌 ′ − 𝐺𝐺 ′ )⁄𝑁𝑁), which means that interest rate is not the equilibrium interest
rate. There is a particular interest rate 𝑟𝑟 ∗ at which each household willingly chooses
((𝑌𝑌 − 𝐺𝐺)⁄𝑁𝑁 , (𝑌𝑌 ′ − 𝐺𝐺 ′ )⁄𝑁𝑁). Since the optimal consumption plan is at a tangency point
between an indifference curve and the lifetime budget constraint, the value of 1 + 𝑟𝑟 ∗ is
pinned down by the gradient of the indifference curve passing through the point
((𝑌𝑌 − 𝐺𝐺 )⁄𝑁𝑁 , (𝑌𝑌 ′ − 𝐺𝐺 ′ )⁄𝑁𝑁).
Figure 3.21: Market-clearing real interest rate
It is also possible to understand the determination of 𝑟𝑟 ∗ in terms of balancing saving and
investment. There is no investment in the example and suppose the government runs a
balanced budget, so 𝑡𝑡 = 𝐺𝐺/𝑁𝑁. Aggregate saving by households is 𝑁𝑁(𝑦𝑦 − 𝑡𝑡 − 𝑐𝑐 ) = 𝑌𝑌 − 𝐺𝐺 −
𝑁𝑁𝑁𝑁 and saving equals investment is equivalent to zero saving here, which occurs when 𝑐𝑐 =
(𝑌𝑌 − 𝐺𝐺)/𝑁𝑁.
In this example, if there were a change in income 𝑌𝑌, the equilibrium real interest rate 𝑟𝑟 ∗
would adjust, all else being equal. The example is an extreme one because, in equilibrium,
consumption must adjust one-for-one with any change in income. Although households
would want to smooth consumption through saving or borrowing, this is not possible in
equilibrium. The mechanism by which individual households are dissuaded from trying to
smooth consumption is adjustment of the real interest rate.
In less extreme cases considered later in this chapter we can ask how the economy as a
whole can adjust its level of saving or borrowing. There are two ways this is possible. First,
through adjustment of investment. Second, in an open economy, through adjustment of the
current account, which we will study in Chapter 10.
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Box 3.3: Supply disruptions and real interest rates
To illustrate how equilibrium real interest rates are affected by macroeconomic shocks,
consider an economy hit by a supply shock that disrupts its ability to produce goods.
Assume here that real GDP is exogenously determined by the economy’s capacity to
produce goods and services. We look at a closed economy with a representative
household and zero investment and we suppose that fiscal policy remains unchanged
after the shock.
The supply shock reduces the economy’s current GDP 𝑌𝑌. The shock is assumed to be
temporary, so that expected future GDP 𝑌𝑌′ is unaffected. The direct consequence of the
shock is to reduces all households’ incomes temporarily. Faced with this temporary
decline in income, households would aim to smooth consumption, spreading out
adjustment of consumption between the present and future time periods. But with zero
investment and international trade, it is not possible for all households to save less or
borrow more simultaneously. Therefore, the real interest rate must adjust for the
economy to reach equilibrium.
The representative household’s lifetime budget constraint passes through the point
((𝑌𝑌 − 𝐺𝐺 )⁄𝑁𝑁 , (𝑌𝑌 ′ − 𝐺𝐺 ′ )⁄𝑁𝑁) and has gradient −(1 + 𝑟𝑟). Equilibrium in the goods market
requires 𝐶𝐶 + 𝐺𝐺 = 𝑌𝑌, so the real interest rate 𝑟𝑟 must adjust until the consumption plan
𝑐𝑐 = (𝑌𝑌 − 𝐺𝐺)⁄𝑁𝑁 and 𝑐𝑐 ′ = (𝑌𝑌 ′ − 𝐺𝐺 ′ )⁄𝑁𝑁 is willingly chosen. This is illustrated in Figure 3.23
below. Starting from the initial real interest rate, the decline in 𝑌𝑌 causes a parallel
leftward shift of the lifetime budget constraint. With no change in 𝑟𝑟, households would
choose a consumption plan where both 𝑐𝑐 and 𝑐𝑐′ decline and 𝑐𝑐 falls by less than (𝑌𝑌 −
𝐺𝐺)/𝑁𝑁 does.
Figure 3.22: Temporary reduction in GDP and equilibrium real interest rate
This is not consistent with equilibrium. In order for the chosen consumption plan to move
horizontally to the left as much as the lifetime budget constraint shifts, the real interest
rate must rise so that the budget constraint is tangent to the indifference curve passing
through that point. The indifference curve is steeper here than at the initial consumption
plan because of the diminishing marginal rate of substitution. Therefore, the equilibrium
real interest rate will need to rise to achieve equilibrium in the goods market.
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Intuitively, after the negative supply shock, at an unchanged real interest rate,
households would smooth consumption by borrowing against unchanged future
income. However, all households are hit by the shock in this example, so all want to
increase borrowing and none wants to save more. The interest rate must rise to
dissuade households from borrowing. More generally, as seen later in economies
where investment or net exports can adjust, the shock causes investment or net
exports to fall in addition to, or in place of, the rise in the equilibrium real interest rate.
3.7 Investment
In the national accounts, investment comprises:
•
•
•
Business fixed investment: firms purchasing new capital for use over time in
producing goods and services
Residential investment: purchases of new housing units
Inventory investment: changes in value of stock of finished goods.
In the USA, investment is typically 15–20 per cent of GDP. Of this total amount of
investment, business fixed investment is around two thirds, residential investment is around
one third, and inventory investment is small but volatile.
Figure 3.24 shows a time series of detrended investment in the USA alongside detrended
real GDP. We see that investment is procyclical, approximately coincident with GDP but is
much more volatile. The percentage deviations of investment from its trend are around 3–4
times larger than those of GDP. This is very different from the pattern of consumption
expenditure (except durables, which behaves more like investment) being less volatile than
GDP.
Figure 3.23: US investment over the business cycle
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3.8 A two-period model of investment
We now consider model of investment decisions with two periods, as we did for our
dynamic model of consumption. Although investment expenditure also includes residential
investment and inventories, here, we will focus on purchases of new capital by firms, i.e.
business fixed investment, the largest component of total investment.
3.8.1 The production function
Our two-period model of business fixed investment assumes firms face a neoclassical
production function 𝐹𝐹(𝐾𝐾, 𝑁𝑁). Firms produce output using capital 𝐾𝐾 and labour 𝑁𝑁. The
amounts of output produced in the current and future time periods are:
𝑌𝑌 = 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑁𝑁)
and
𝑌𝑌 ′ = 𝑧𝑧 ′ 𝐹𝐹(𝐾𝐾 ′ , 𝑁𝑁 ′ )
Total factor productivity (TFP) in the current and future periods is 𝑧𝑧 and 𝑧𝑧′, respectively,
where 𝑧𝑧 and 𝑧𝑧′ may differ, for example, technological progress makes 𝑧𝑧′ larger than 𝑧𝑧. The
level of TFP in each time period is exogenous. Labour input is obtained by firms hiring labour
in a competitive market, as was studied in Section 1.2. Unlike the Chapter 1 analysis, firms
are assumed to own the capital 𝐾𝐾 that they use for production. There is no rental market for
capital, though this assumption does not matter greatly here.
3.8.2 Capital accumulation
Firms can change their stock of capital through purchases of new capital. Current
investment 𝐼𝐼 adds to the future stock of capital 𝐾𝐾′ that is used to produce output in the
future. This means it takes time for new capital to be produced and installed and be ready
for use in production (this assumption is known as ‘time-to-build’). The capital accumulation
equation is the following, which is the same one seen in the Solow model from Section 1.8:
𝐾𝐾 ′ = 𝐼𝐼 + (1 − 𝑑𝑑 )𝐾𝐾
The future capital stock is the sum of undepreciated current capital (1 − 𝑑𝑑 )𝐾𝐾 and
investment 𝐼𝐼. Investment is purchases of new capital goods, and we treat capital goods as
equivalent to any other goods for simplicity. This means a new unit of capital costs one unit
of goods (its price is 1), and in principle, undepreciated capital goods can be sold off and
used in the same way as newly produced goods.
As in the Solow model, capital depreciates at a constant rate 𝑑𝑑 over time. Note also that the
current capital stock 𝐾𝐾 is determined by decisions made in the past and is not affected by
current investment. Since the model has only two time periods, it is assumed firms sell off
any undepreciated capital (1 − 𝑑𝑑 )𝐾𝐾 ′ that remains after production in the future time
period.
3.8.3 Firms’ profits
The firm’s profits are the difference between its revenues and its costs of production, which
here are simply payments to factors of production. The firm owns its capital, so factor
payments are just the wage bill. Real wages are 𝑤𝑤 and 𝑤𝑤′ in the current and future periods,
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hence, current and future real costs are 𝑤𝑤𝑤𝑤 and 𝑤𝑤′𝑁𝑁′. Real revenues are the same as
production 𝑌𝑌 and 𝑌𝑌′ because all goods are homogeneous. Real profits are thus given by:
𝜋𝜋 = 𝑌𝑌 − 𝑤𝑤𝑤𝑤
and
𝜋𝜋 ′ = 𝑌𝑌 ′ − 𝑤𝑤 ′ 𝑁𝑁′
Note that these profits implicitly include the return on capital owned by the firm, so the
argument in Section 1.2 that perfect competition and a constant-returns-to-scale
production function push profits to zero does not apply here. Observe that the cost of any
investment 𝐼𝐼 is not deducted from profits because purchases of capital mean acquisition of
an asset that is not immediately used in current production. As profits are gross profits,
there is also no deduction of any cost of depreciation. However, both the cost of purchasing
capital and depreciation will be relevant when analysing the optimal level of investment.
3.8.4 Options for financing investment
Suppose a firm decides to purchase an amount of new capital 𝐼𝐼, which entails real
investment expenditure of the same amount. What are the options for financing this
expenditure?
The first possibility is borrowing. Suppose the firm can take out a loan of size 𝐿𝐿 = 𝐼𝐼 to pay
for the whole cost of purchasing the capital. If this loan has interest rate 𝑟𝑟𝑙𝑙 , the firm must
repay (1 + 𝑟𝑟𝑙𝑙 )𝐿𝐿 in the future period. We allow for 𝑟𝑟𝑙𝑙 to differ from the interest rate 𝑟𝑟 on
government bonds received by savers because of credit-market imperfections discussed
further in Chapter 4. As the loan is sufficient to pay the whole cost, the firm can pay out all
its current profits 𝜋𝜋 as dividends 𝑣𝑣 = 𝜋𝜋. However, in the future, the debt repayment must
be made out of future profits before these can be distributed to shareholders. The future
dividend 𝑣𝑣′ is:
𝑣𝑣 ′ = 𝜋𝜋 ′ + (1 − 𝑑𝑑 )𝐾𝐾′ − (1 + 𝑟𝑟𝑙𝑙 )𝐼𝐼
Note that firms can also distribute the proceeds (1 − 𝑑𝑑 )𝐾𝐾′ from selling off undepreciated
capital as future dividends.
Another possible option is to use retained earnings to pay for the investment. Retained
earnings refers to profits the firm has made but has not distributed to shareholders. In this
case, the firm would pay a current dividend of 𝑣𝑣 = 𝜋𝜋 − 𝐼𝐼, distributing only profits 𝜋𝜋 in
excess of investment expenditure 𝐼𝐼. The future dividend would be 𝑣𝑣 ′ = 𝜋𝜋 ′ + (1 − 𝑑𝑑 )𝐾𝐾′,
which is future profits plus the proceeds of selling off undepreciated capital. There are no
future debts to repay in this case.
The use of retained earnings to pay for investment requires current profits to be sufficiently
large, that is, 𝜋𝜋 ≥ 𝐼𝐼. A third option is the issuance of new equity, which we will treat as
being equivalent to a negative dividend 𝑣𝑣 (the firm receives new funds from shareholders,
rather than a distribution of funds to shareholders). In this case, the current dividend is 𝑣𝑣 =
𝜋𝜋 − 𝐼𝐼 < 0 and the future dividend is 𝑣𝑣 ′ = 𝜋𝜋 ′ + (1 − 𝑑𝑑 )𝐾𝐾′, which is mathematically
equivalent to using retained earnings, the only difference being that 𝑣𝑣 is negative. Although
algebraically the same, financial-market imperfections may limit firms’ ability to issue new
equity when current profitability is low.
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3.8.5 The optimal investment decision
We assume that the managers of the firm making investment decisions act in interests of
the firm’s shareholders. If those shareholders are not credit-constrained, we have seen that
they care only about their lifetime wealth. This means the owners of the firm would want
managers to maximise the present value of dividends discounted using the interest rate 𝑟𝑟:
𝑉𝑉 = 𝑣𝑣 +
𝑣𝑣′
1 + 𝑟𝑟
The managers of the firm choose employment levels 𝑁𝑁 and 𝑁𝑁′ and investment 𝐼𝐼 to
maximise 𝑉𝑉. There is no dynamic dimension of the employment decisions: these are
equivalent to choosing 𝑁𝑁 and 𝑁𝑁′ to maximise 𝜋𝜋 and 𝜋𝜋 ′ respectively. The equations for
labour demand are 𝑀𝑀𝑃𝑃𝑁𝑁 = 𝑤𝑤 and 𝑀𝑀𝑃𝑃𝑁𝑁′ = 𝑤𝑤′ in the two periods. At all times, firms hire
labour up to the point where the marginal product of labour is equal to the real wage. For
the choice of investment 𝐼𝐼, note that the capital accumulation equation implies this is
equivalent to choosing the future capital stock 𝐾𝐾′, with investment then given by 𝐼𝐼 = 𝐾𝐾 ′ −
(1 − 𝑑𝑑 )𝐾𝐾, where 𝐾𝐾 is the initial capital stock.
First, suppose a firm finances investment using borrowing. The present discounted value of
the dividends it distributes to shareholders in this case is
𝑧𝑧 ′ 𝐹𝐹 (𝐾𝐾 ′ , 𝑁𝑁 ′ ) − 𝑤𝑤 ′ 𝑁𝑁 ′ + (1 − 𝑑𝑑 )𝐾𝐾 ′ − (1 + 𝑟𝑟𝑙𝑙 )(𝐾𝐾 ′ − (1 − 𝑑𝑑 )𝐾𝐾)
𝑉𝑉 = 𝑧𝑧𝑧𝑧 (𝐾𝐾, 𝑁𝑁) − 𝑤𝑤𝑤𝑤 +
1 + 𝑟𝑟
The first-order condition to maximise 𝑉𝑉 with respect to 𝐾𝐾 ′ is:
𝑀𝑀𝑃𝑃𝐾𝐾′ + (1 − 𝑑𝑑 ) − (1 + 𝑟𝑟𝑙𝑙 )
=0
1 + 𝑟𝑟
This can be rearranged to obtain the following condition for the optimal level of investment:
𝑀𝑀𝑃𝑃𝐾𝐾′ − 𝑑𝑑 = 𝑟𝑟𝑙𝑙
Alternatively, suppose that retained earnings are available and used to finance investment
instead. The present discounted value of dividends is:
𝑉𝑉 = 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑁𝑁) − 𝑤𝑤𝑤𝑤 − (𝐾𝐾 ′ − (1 − 𝑑𝑑 )𝐾𝐾 ) +
𝑧𝑧 ′ 𝐹𝐹 (𝐾𝐾 ′ , 𝑁𝑁 ′ ) − 𝑤𝑤 ′ 𝑁𝑁 ′ + (1 − 𝑑𝑑 )𝐾𝐾 ′
1 + 𝑟𝑟
The first-order condition to maximise 𝑉𝑉 with respect to 𝐾𝐾′ is:
−1 +
𝑀𝑀𝑃𝑃𝐾𝐾′ + (1 − 𝑑𝑑 )
=0
1 + 𝑟𝑟
This yields an equation for the optimal investment decision:
𝑀𝑀𝑃𝑃𝐾𝐾′ − 𝑑𝑑 = 𝑟𝑟
In all cases, the optimal investment decision equates the marginal benefit and marginal cost
of investment. More investment allows the firm to hold an extra unit of capital. The
marginal benefit of this is the extra production of future output it makes possible, which is
worth 𝑀𝑀𝑃𝑃𝐾𝐾′ , the marginal product of capital in the future.
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The marginal cost of holding an extra unit of capital includes the loss of a fraction 𝑑𝑑 of the
capital through depreciation. There is also the financing cost of holding more capital. If the
funds to purchase it are borrowed, the financing cost is the interest rate 𝑟𝑟𝑙𝑙 on the loan. If
the funds come from retained earnings, the financing cost is an opportunity cost to the
firm’s shareholders. Funds are returned to shareholders in the future period rather than in
the current period and these funds could have earned shareholders a return 𝑟𝑟 if received
and saved.
3.8.6 Does the source of financing matter?
If there are no credit-market imperfections then firms can borrow at interest rate 𝑟𝑟, the
same interest rate as received by savers. With 𝑟𝑟𝑙𝑙 = 𝑟𝑟, the direct cost of borrowing equals
the opportunity cost to shareholders of delaying distribution of profits if retained earnings
are used to pay for investment. This means the condition for optimal investment is 𝑀𝑀𝑃𝑃𝐾𝐾′ −
𝑑𝑑 = 𝑟𝑟 in both cases and hence, the source of financing does not matter for investment. As
we will see in Chapter 4, this may not be true once we allow for credit-market
imperfections.
3.8.7 The investment demand curve
The equation 𝑀𝑀𝑃𝑃𝐾𝐾′ − 𝑑𝑑 = 𝑟𝑟 implies a downward-sloping investment demand curve. A
neoclassical production function has diminishing returns to capital, so 𝑀𝑀𝑃𝑃𝐾𝐾′ decreases with
𝐾𝐾′. Given initial capital 𝐾𝐾, the equation 𝐾𝐾 ′ = 𝐼𝐼 + (1 − 𝑑𝑑 )𝐾𝐾 shows that 𝐾𝐾′ moves one-forone with investment 𝐼𝐼. The left-hand side of the optimal investment equation thus falls with
𝐼𝐼. The right-hand side is the real interest rate 𝑟𝑟. Hence, the investment demand curve is
depicted in Figure 3.25 with 𝑟𝑟 on the vertical axis. The demand curve is given by plotting the
marginal product of capital as a function of 𝐼𝐼 shifted down by the depreciation rate 𝑑𝑑.
Figure 3.24: The investment demand curve
The Inada conditions for a neoclassical production function imply that 𝑀𝑀𝑃𝑃𝐾𝐾′ − 𝑑𝑑 is negative
for very high levels of investment, so 𝐼𝐼 𝑑𝑑 cuts the horizontal axis and falls as low as 𝑟𝑟 = −𝑑𝑑.
Observe also that 𝑀𝑀𝑃𝑃𝐾𝐾′ is finite if investment were zero, even though the Inada conditions
hold. This is because there is existing capital, so 𝐾𝐾′ is not zero even if there were no
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investment currently. In principle, investment 𝐼𝐼 could be negative, which would mean selling
off capital goods for other uses. However, much investment may be irreversible in practice,
hence, this possibility is ignored when drawing the investment demand curve.
3.9 The stock market
The analysis of investment assumes firms are owned by shareholders. Thus, savers are
holding shares in firms in addition to other assets such as bonds. Buying shares in firms is
often described as ‘investment’ but such financial investment is logically distinct from
investments in physical capital made by firms themselves.
Each share in a firm is an equal claim on the profits of the firms after other creditors, for
example, bondholders, have been paid. Shareholders are the residual claimants on the firm.
If shares can be traded in a stock market, what determines the price of those shares? Is
financial investment and the stock-market value of firms related to firms’ decisions to invest
in physical capital?
Assume shareholders expect to be paid dividends 𝑣𝑣 and 𝑣𝑣′ in current and future time
periods but none after that in our two-period model. Suppose these shares trade at a price
𝑝𝑝 at the beginning of the current period, with any purchaser obtaining the right to both
dividends 𝑣𝑣 and 𝑣𝑣′. Taking the dividend 𝑣𝑣 and reinvesting in financial assets results in a
future payoff (1 + 𝑟𝑟)𝑣𝑣. Assume the investor expects to be able to sell shares at price 𝑝𝑝′ in
the future before the dividend 𝑣𝑣′ is paid. The rate of return for an investor who buys shares
at current price 𝑝𝑝 is the sum of the dividend yield and any capital gains or losses as a
percentage of 𝑝𝑝:
(1 + 𝑟𝑟)𝑣𝑣 𝑝𝑝′ − 𝑝𝑝
+
𝑝𝑝
𝑝𝑝
An investor who does not care about risk (said to be ‘risk-neutral’) would choose to hold the
asset with the highest expected return. Bonds have real interest rate 𝑟𝑟, which is their real
return. For both bonds and shares to be willingly held by savers, their expected returns must
be the same:
𝑟𝑟 =
(1 + 𝑟𝑟)𝑣𝑣 𝑝𝑝′ − 𝑝𝑝
+
𝑝𝑝
𝑝𝑝
Solving this equation for 𝑝𝑝 shows that the share price must be 𝑝𝑝 = 𝑣𝑣 + 𝑝𝑝′ ⁄(1 + 𝑟𝑟). Owning
the share in the future simply gives a claim to the final dividend 𝑣𝑣′, hence, the anticipated
future share price must be 𝑝𝑝′ = 𝑣𝑣 ′ . Therefore, the equilibrium share price 𝑝𝑝 and stockmarket value of the firm is equal to the present discounted value of dividends 𝑉𝑉:
𝑝𝑝 = 𝑉𝑉 = 𝑣𝑣 +
𝑣𝑣′
1 + 𝑟𝑟
With equilibrium share price found here, firms choosing investment to maximise the
present value of dividends is the same as aiming to maximise their stock-market value.
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Box 3.4: Stock prices and firms’ investment decisions
We have seen that optimal investment by firms can be understood in terms of maximising
their stock-market value. There is also a more specific prediction of the theory of
investment about the relationship between stock prices and investment in terms of a
variable known as Tobin’s Q.
For a firm that uses both debt and equity financing, Tobin’s Q is defined as follows:
Tobin's Q =
Market value of firm
Replacement cost of capital
The market value of the firm in the numerator of Tobin’s Q is the sum of the value of
bonds issued and the value of its shares, the latter known as the firm’s stock-market
capitalisation. The replacement cost of capital in the denominator is what the physical
capital owned by the firm would cost to buy as new.
Working in the context of the two-period model of investment from Section 3.8, we will
calculate Tobin’s Q in the first period after firms have paid their current dividend 𝑣𝑣. Let 𝑆𝑆
denote a firm’s stock-market value at this point where its shares are ex dividend
(purchasing them gives a claim to future dividends only). With only one period remaining
in the future, the expected future value of the shares is 𝑝𝑝′ = 𝑣𝑣 ′ , where 𝑣𝑣′ is the one
remaining future dividend. If 𝑟𝑟 is the rate of interest and buyers of the shares must
receive the same expected return, then 𝑟𝑟 = (𝑝𝑝′ − 𝑆𝑆)/𝑆𝑆, which implies the current exdividend value of the shares is:
𝑆𝑆 =
𝑝𝑝′
𝑣𝑣′
=
1 + 𝑟𝑟 1 + 𝑟𝑟
If the firm borrows an amount 𝐿𝐿, for example by issuing bonds, with (1 + 𝑟𝑟)𝐿𝐿 repayable
in the future period (assuming it can borrow at rate 𝑟𝑟 absent financial-market
imperfections) then the future dividend it can pay is 𝑣𝑣 ′ = 𝜋𝜋 ′ + (1 − 𝑑𝑑 )𝐾𝐾 ′ − (1 + 𝑟𝑟)𝐿𝐿,
conditional on the capital 𝐾𝐾′ it will have available. The firm’s future profits are 𝜋𝜋 ′ = 𝑌𝑌 ′ −
𝑤𝑤 ′ 𝑁𝑁 ′ = 𝑌𝑌 ′ − 𝑀𝑀𝑃𝑃𝑁𝑁′ 𝑁𝑁 ′ because a profit-maximising choice of future employment implies
𝑀𝑀𝑃𝑃𝑁𝑁′ = 𝑤𝑤′. With constant returns to scale, 𝑌𝑌 ′ = 𝑀𝑀𝑃𝑃𝐾𝐾′ 𝐾𝐾 ′ + 𝑀𝑀𝑃𝑃𝑁𝑁′ 𝑁𝑁 ′ , so future profits
are 𝜋𝜋 ′ = 𝑀𝑀𝑃𝑃𝐾𝐾′ 𝐾𝐾 ′ , from which the stock-market value of the firm can be deduced:
𝑆𝑆 =
𝑀𝑀𝑃𝑃𝐾𝐾′ 𝐾𝐾 ′ + (1 − 𝑑𝑑 )𝐾𝐾′
− 𝐿𝐿
1 + 𝑟𝑟
If a firm issues debt 𝐿𝐿 and has stock-market capitalisation 𝑆𝑆 then its total market value is
𝑆𝑆 + 𝐿𝐿, which is given by:
𝑀𝑀𝑃𝑃𝐾𝐾′ 𝐾𝐾 ′ + (1 − 𝑑𝑑 )𝐾𝐾 ′ (𝑀𝑀𝑃𝑃𝐾𝐾′ + 1 − 𝑑𝑑 )𝐾𝐾 ′
𝑆𝑆 + 𝐿𝐿 =
=
1 + 𝑟𝑟
1 + 𝑟𝑟
If the firm chooses a level of investment that gives it capital stock 𝐾𝐾′ then the
replacement cost of the capital stock 𝐾𝐾 ′ is simply 𝐾𝐾 ′ because each unit of physical capital
costs one unit of goods. Tobin’s Q can therefore be calculated as the ratio of 𝑆𝑆 + 𝐿𝐿 to 𝐾𝐾′:
𝑆𝑆 + 𝐿𝐿 (𝑀𝑀𝑃𝑃𝐾𝐾′ + 1 − 𝑑𝑑 )𝐾𝐾 ′ 𝑀𝑀𝑃𝑃𝐾𝐾′ + 1 − 𝑑𝑑
𝑄𝑄 =
=
=
(1 + 𝑟𝑟)𝐾𝐾 ′
𝐾𝐾 ′
1 + 𝑟𝑟
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Observe that Tobin’s Q is greater than 1 when 𝑀𝑀𝑃𝑃𝐾𝐾′ + 1 − 𝑑𝑑 > 1 + 𝑟𝑟, which is
equivalent to 𝑀𝑀𝑃𝑃𝐾𝐾′ − 𝑑𝑑 > 𝑟𝑟 and would mean investment is too low according to the
theory developed in Section 3.8. If firms were to invest up to the point where 𝑀𝑀𝑃𝑃𝐾𝐾′ −
𝑑𝑑 = 𝑟𝑟, this would mean the same as investing until Tobin’s Q has declined to 1. Thus, the
theory of investment developed earlier is equivalent to the prediction that investment is
higher the further Tobin’s Q is above 1.
In principle, Tobin’s Q is measurable and can be compared to data on investment. In
practice, for many modern firms, intangible capital such as patents and brand reputation
has become more important than physical capital. The replacement cost of intangible
capital is harder to quantify, and makes measurement of the denominator of Tobin’s Q
challenging.
Nonetheless, if we look at just the numerator of Tobin’s Q and focus on stock-market
capitalisation, we see that stock market values are often correlated with investment.
Figure 3.26 plots the detrended S&P500 index and investment in the USA, between which
there is a strong positive correlation.
Figure 3.25: S&P 500 and investment over the business cycle (USA)
Box 3.5: Should capital be taxed?
This section looks at the implications of taxes on capital or capital income for the level of
investment in an economy and investigates how large the distortions from such taxes
might be. To simplify the analysis, we focus on the implications of capital taxes for the
steady-state capital stock 𝐾𝐾 ′ = 𝐾𝐾.
Suppose that net capital income (𝑅𝑅 − 𝑑𝑑 )𝐾𝐾 is taxed at a proportional rate 𝜏𝜏, where 𝑅𝑅 =
𝑀𝑀𝑃𝑃𝐾𝐾 is the marginal product of capital and 𝑑𝑑 is the depreciation rate. For example,
suppose a firm with profits 𝜋𝜋 = 𝑀𝑀𝑃𝑃𝐾𝐾 𝐾𝐾 distributes dividends 𝑣𝑣 = 𝜋𝜋 − 𝑑𝑑𝑑𝑑 after using
retained earnings to finance investment 𝐼𝐼 = 𝑑𝑑𝑑𝑑 to maintain the steady-state capital stock
𝐾𝐾. These dividends might then be subject to an income tax rate 𝜏𝜏.
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Let 𝑟𝑟 denote the equilibrium after-tax return that investors receive, which we will take as
given here. In the absence of tax, investment would be determined by 𝑅𝑅 − 𝑑𝑑 = 𝑟𝑟. When
the tax on capital income is 𝜏𝜏, firms invest in capital up to the point where:
(1 − 𝜏𝜏)(𝑅𝑅 − 𝑑𝑑 ) = 𝑟𝑟
Here, the cost of depreciation reduces the tax liability, although we can also do the
analysis under the assumption that gross capital income is taxed. Given the required
after-tax real rate of return 𝑟𝑟 on capital, investment and the stock of capital are
determined by:
𝑀𝑀𝑃𝑃𝐾𝐾 − 𝑑𝑑 =
𝑟𝑟
1 − 𝜏𝜏
Assume the production function is 𝑌𝑌 = 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑁𝑁), which is 𝑦𝑦 = 𝑧𝑧𝑧𝑧(𝑘𝑘) in per worker
terms. The marginal product of capital is 𝑀𝑀𝑃𝑃𝐾𝐾 = 𝑧𝑧𝑧𝑧′(𝑘𝑘). In a competitive economy, the
real wage received by workers is 𝑤𝑤 = 𝑀𝑀𝑃𝑃𝑁𝑁 . A neoclassical production function has the
feature that 𝑌𝑌 = 𝑀𝑀𝑃𝑃𝐾𝐾 𝐾𝐾 + 𝑀𝑀𝑃𝑃𝑁𝑁 𝑁𝑁, which implies 𝑤𝑤 = 𝑦𝑦 − 𝑀𝑀𝑃𝑃𝐾𝐾 𝑘𝑘 = 𝑧𝑧�𝑓𝑓(𝑘𝑘) − 𝑘𝑘𝑓𝑓 ′ (𝑘𝑘)�.
Taking a given positive value of 𝑟𝑟, a higher tax rate 𝜏𝜏 raises 𝑟𝑟⁄(1 − 𝜏𝜏) and thus 𝑀𝑀𝑃𝑃𝐾𝐾 =
𝑧𝑧𝑧𝑧′(𝑘𝑘) must be higher in equilibrium. Since 𝑓𝑓′(𝑘𝑘) is decreasing in 𝑘𝑘, this means there will
be less investment and lower steady-state 𝑘𝑘. The tax thus reduces output per worker 𝑦𝑦 =
𝑧𝑧𝑧𝑧(𝑘𝑘) and also lowers wages 𝑤𝑤 = 𝑀𝑀𝑃𝑃𝑁𝑁 because 𝑀𝑀𝑃𝑃𝑁𝑁 is lower when there is less capital
per worker 𝑘𝑘. This can be shown by noting 𝑤𝑤 = 𝑧𝑧(𝑓𝑓(𝑘𝑘) − 𝑘𝑘𝑓𝑓 ′ (𝑘𝑘)) is increasing in 𝑘𝑘 since
𝑓𝑓 ′′ (𝑘𝑘) < 0.
Suppose capital income is taxed and used to subsidise the wage income of workers. More
generally, we could ask whether capital income or wages should be taxed to finance
public expenditure. The amount of tax raised per worker is 𝜏𝜏(𝑅𝑅 − 𝑑𝑑 )𝐾𝐾⁄𝑁𝑁 =
𝜏𝜏(𝑧𝑧𝑓𝑓 ′ (𝑘𝑘) − 𝑑𝑑 )𝑘𝑘, so the wage including the subsidy is 𝑤𝑤 + 𝜏𝜏(𝑧𝑧𝑓𝑓 ′ (𝑘𝑘) − 𝑑𝑑 )𝑘𝑘. Using 𝑤𝑤 =
𝑧𝑧(𝑓𝑓 (𝑘𝑘) − 𝑘𝑘𝑓𝑓 ′ (𝑘𝑘)) the wage plus the subsidy is:
𝑧𝑧𝑧𝑧(𝑘𝑘) − 𝑘𝑘𝑘𝑘𝑓𝑓 ′ (𝑘𝑘) + 𝜏𝜏(𝑧𝑧𝑓𝑓 ′ (𝑘𝑘) − 𝑑𝑑 )𝑘𝑘 = 𝑧𝑧𝑧𝑧 (𝑘𝑘) − 𝑘𝑘 (1 − 𝜏𝜏)𝑧𝑧𝑓𝑓 ′ (𝑘𝑘) − 𝜏𝜏𝜏𝜏𝜏𝜏
With 𝑘𝑘 determined by 𝑧𝑧𝑧𝑧′(𝑘𝑘) = 𝑑𝑑 + 𝑟𝑟⁄(1 − 𝜏𝜏), the subsidised wage is given by the
formula 𝑧𝑧𝑧𝑧(𝑘𝑘) − (1 − 𝜏𝜏)𝑑𝑑𝑑𝑑 − 𝑟𝑟𝑟𝑟 − 𝜏𝜏𝜏𝜏𝜏𝜏 = 𝑧𝑧𝑧𝑧(𝑘𝑘) − (𝑟𝑟 + 𝑑𝑑 )𝑘𝑘.
Taking account of the capital tax’s effect on investment, each worker receives 𝑧𝑧𝑧𝑧(𝑘𝑘) −
(𝑟𝑟 + 𝑑𝑑 )𝑘𝑘, where 𝑘𝑘 depends on 𝜏𝜏. This is maximised over 𝑘𝑘 where 𝑀𝑀𝑃𝑃𝐾𝐾 = 𝑧𝑧𝑓𝑓 ′ (𝑘𝑘) = 𝑟𝑟 +
𝑑𝑑. But since 𝑘𝑘 must satisfy 𝑀𝑀𝑃𝑃𝐾𝐾 − 𝑑𝑑 = 𝑟𝑟/(1 − 𝜏𝜏) given 𝜏𝜏, the value of 𝜏𝜏 that is in the
interests of workers is 𝜏𝜏 = 0. Even though the revenue from taxing capital is directly
redistributed to workers, workers prefer no capital tax. This is because the negative effect
of the capital tax on investment and wages is too large.
One important assumption in this analysis is that there is a perfectly elastic supply of
funds for investment at a constant required rate of return 𝑟𝑟, which makes the economy’s
supply of capital perfectly elastic. This makes capital the exact opposite of land, which is
perfectly inelastic in supply.
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3.10 Labour supply over time
In the remainder of the chapter, we will see how to put together the various aspects of
macroeconomics that we have analysed separately: labour demand; labour supply;
consumption; saving and investment. Doing this requires considering one other aspect of
households’ plans, namely, the dynamics of households’ labour supply.
Our analysis of households’ plans has so far considered labour supply at a given time with
reference to the trade-off between leisure and consumption. This was studied in the static
macroeconomic model from Section 1.4. We have also explored the decision to save or
borrow in this chapter through the trade-off between consumption now or future
consumption in the two-period model. This analysis of saving was done taking as given the
income the household expected to earn in the current and future periods. However, income
is affected by the decision to supply labour. This means that saving, defined as disposable
income minus consumption, can be increased by raising income as well as by reducing
consumption. Hence, there is an interaction between the decision to save and the decision
to supply labour that we should take account of.
This section studies the labour supply decision or choice of leisure over time. In a twoperiod model, a household can choose current and future consumption 𝑐𝑐 and 𝑐𝑐′ and current
and future leisure 𝑙𝑙 and 𝑙𝑙′. The household’s preferences are assumed to feature diminishing
marginal rates of substitution between any pair of 𝑐𝑐,𝑐𝑐′,𝑙𝑙,𝑙𝑙′, and all are assumed to be normal
goods.
With ℎ hours available in each period, the choice of 𝑙𝑙 and 𝑙𝑙′ determines the current and
future labour supplies 𝑁𝑁 𝑠𝑠 = ℎ − 𝑙𝑙 and 𝑁𝑁 𝑠𝑠 ′ = ℎ − 𝑙𝑙′. Current and future real wages are 𝑤𝑤
and 𝑤𝑤′, and the household also receives dividends 𝑣𝑣 and 𝑣𝑣′ from firms. As in the earlier
analysis of consumption, the household faces a lifetime budget constraint:
𝑤𝑤 ′ (ℎ − 𝑙𝑙 ′ ) + 𝑣𝑣 ′ − 𝑡𝑡′
𝑐𝑐′
= 𝑤𝑤 (ℎ − 𝑙𝑙 ) + 𝜈𝜈 − 𝑡𝑡 +
𝑐𝑐 +
1 + 𝑟𝑟
1 + 𝑟𝑟
Here, income 𝑦𝑦 has been replaced by the sum of wage income 𝑤𝑤𝑁𝑁 𝑠𝑠 and dividends 𝑣𝑣.
Income is no longer exogenous because the household can change 𝑦𝑦 through its choice of
labour supply 𝑁𝑁 𝑠𝑠 .
The household’s optimal plans equate marginal rates of substitution to relative prices
implied by the lifetime budget constraint for all the possible pairs of 𝑐𝑐, 𝑐𝑐′, 𝑙𝑙, and 𝑙𝑙′. Most of
these have already been analysed. First, considering current leisure 𝑙𝑙 and current
consumption 𝑐𝑐, holding fixed future 𝑐𝑐 ′ and 𝑙𝑙 ′ , the relative price is the current real wage 𝑤𝑤
and the optimality condition is 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝑐𝑐 = 𝑤𝑤, as seen in 0. This is gives the optimal labour
supply from a static point of view. The same logic applies in the future period as well,
considering 𝑙𝑙′ and 𝑐𝑐′ and holding 𝑐𝑐 and 𝑙𝑙 fixed. The relative price is the future real wage 𝑤𝑤′
and the optimality condition is 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙′ ,𝑐𝑐 ′ = 𝑤𝑤 ′ .
We have also studied the trade-off between 𝑐𝑐 and 𝑐𝑐′, holding 𝑙𝑙 and 𝑙𝑙′ fixed (which is
equivalent to fixing incomes 𝑦𝑦 and 𝑦𝑦′). The relative price is 1 + 𝑟𝑟 and the optimality
condition is 𝑀𝑀𝑀𝑀𝑆𝑆𝑐𝑐,𝑐𝑐 ′ = 1 + 𝑟𝑟. So far, the optimality conditions have been derived and
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explained earlier. The new condition we will look at comes from the trade-off between
current leisure 𝑙𝑙 and future leisure 𝑙𝑙′.
Sacrificing leisure in the current period (reducing 𝑙𝑙) increases labour supply (higher 𝑁𝑁 𝑠𝑠 ) and
raises current income by the wage 𝑤𝑤 for each hour of leisure given up. This income could be
spent now on consumption but it could also be saved. By saving the extra wage 𝑤𝑤, future
financial assets are increased by (1 + 𝑟𝑟)𝑤𝑤. This extra wealth could be spent on consumption
but it could also be used to reduce labour supply in future by replacing future wage income.
For example, earlier retirement with no loss of future consumption becomes possible if the
household has more financial assets.
Mathematically, holding 𝑐𝑐 and 𝑐𝑐′ constant, a reduction of 𝑙𝑙 by one unit means that 𝑙𝑙′ can
increase by (1 + 𝑟𝑟)𝑤𝑤/𝑤𝑤′. Hence, (1 + 𝑟𝑟)𝑤𝑤/𝑤𝑤′ is the relative price of the household’s
current time in terms of its future time and this relative price matters for how much it is
optimal to work now compared to working in the future period. The relative price increases
when the current wage 𝑤𝑤 rises relative to the future wage 𝑤𝑤′, or when the real interest rate
𝑟𝑟 increases. The household’s optimal plan must feature:
𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝑙𝑙′ = (1 + 𝑟𝑟)𝑤𝑤/𝑤𝑤′
Figure 3.27 draws indifference curves and a lifetime budget constraint in a diagram with
current leisure 𝑙𝑙 on the horizontal axis and future leisure 𝑙𝑙′ on the vertical axis. The range is
limited to leisure between 0 and the maximum available time ℎ where the household would
not participate in the labour market. All other variables 𝑐𝑐 and 𝑐𝑐′ are held fixed. The budget
constraint is downward sloping and has gradient −(1 + 𝑟𝑟)𝑤𝑤/𝑤𝑤′. The diagram shows the
case where it is feasible to have one of, but not both, 𝑙𝑙 and 𝑙𝑙′ equal to ℎ, so the household
could choose not to participate in the labour market in one of the periods. However, given
the indifference curves as shown, the household chooses to participate in both periods to
some extent.
Figure 3.26: Optimal timing of leisure/labour supply
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The optimality condition 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝑙𝑙′ = (1 + 𝑟𝑟)𝑤𝑤/𝑤𝑤′ implies the real interest rate 𝑟𝑟 changes
optimal labour supply in the same direction as the real wage 𝑤𝑤. A higher 𝑟𝑟 raises the
incentive for work because the extra income obtained can be saved to earn a higher return.
Or in other words, a higher 𝑟𝑟 increases the incentive to save and one way to do this is to
earn more and not to spend it. Since 𝑟𝑟 matters for the labour supply decision, it follows that
the labour supply curve 𝑁𝑁 𝑠𝑠 shifts when 𝑟𝑟 changes. Where labour supply increases with the
real wage 𝑤𝑤, the upward-sloping 𝑁𝑁 𝑠𝑠 curve shifts to the right when 𝑟𝑟 rises.
3.11 A dynamic macroeconomic model
Here we take all that we have learned about household and firm behaviour and put
together the pieces:
•
•
Labour demand and supply
Consumption and investment.
We will set up a dynamic macroeconomic model and see how to analyse the economy in
general equilibrium. General equilibrium means there is equilibrium in all markets
simultaneously. The three markets we consider are:
•
•
•
Goods market
Labour market
Bond market.
For now, we will work with the case of a closed economy. Section 9.1 shows how to adapt
our dynamic macroeconomic model to an open economy.
3.11.1 A representative household
We simplify matters by assuming a representative household. This means that all
households in the economy have the same preferences and the same level of wealth.
Specifically, all households are paid the same amount for work per hour, face the same
taxes and all own an equal number of shares in firms. Since the economy is closed, firms
must be owned by domestic residents.
The consequence of the representative household assumption is that all households will
ultimately make the same choices, even if they act individually. With 𝑁𝑁 households in the
economy, aggregate consumption is 𝐶𝐶 = 𝑁𝑁𝑁𝑁, where 𝑐𝑐 is an individual household’s
consumption choice. The same is true for other aggregates, which are simply scaled-up
versions of each individual’s choices and we will often ignore the distinction between
individual and aggregate variables. While we cannot use the representative-household
framework to study how inequality affects the economy, it is nonetheless a useful starting
point for many other issues.
One simplification that comes from there being a representative household in a closed
economy is that income effects from changes in the wage 𝑤𝑤 or interest rate 𝑟𝑟 can be
ignored. All else being equal, a higher wage makes households better off as workers but
worse off as owners of firms and these two effects cancel out. All else being equal, a higher
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interest rate makes households better off as savers but worse off as owners of firms with
higher borrowing costs and worse off as taxpayers if the government has higher borrowing
costs.
With no income effects coming from 𝑤𝑤 or 𝑟𝑟, there are only substitution effects to consider.
A higher wage 𝑤𝑤 implies an incentive to supply more labour, so the labour supply curve 𝑁𝑁 𝑠𝑠
is upward sloping. A higher interest rate 𝑟𝑟 increases saving, which lowers the consumption
demand 𝐶𝐶 𝑑𝑑 . The left panel of Figure 3.28 depicts a downward-sloping consumption demand
curve with 𝑟𝑟 on the vertical axis and 𝐶𝐶 on the horizontal axis. Higher 𝑟𝑟 also increases labour
supply as discussed in the analysis of the dynamics of labour supply. This means the labour
supply curve 𝑁𝑁 𝑠𝑠 shifts to the right when 𝑟𝑟 increases as shown in the right panel of the
figure.
Figure 3.27: Consumption demand and labour supply
It is important to note this logic does not mean all income effects can be neglected. Often,
the exogenous cause of a change in 𝑤𝑤 or 𝑟𝑟 will make households better off or worse off
overall. For example, higher productivity (TFP 𝑧𝑧 or 𝑧𝑧′) makes households better off, raising
either wages, profits, or some combination of the two. The argument is that conditional on
the level of TFP, households are not made better off overall by higher wages. Another
example is when the tax burden falls, which makes households better off, all else equal. It is
not always possible to infer from the direction of change in GDP whether households are
better off or not, as we will see in later examples.
Since consumption and leisure are both normal goods, anything that makes households
better off overall leads to a ‘wealth effect’ that increases consumption demand and reduces
labour supply. These wealth effects shift the 𝐶𝐶 𝑑𝑑 curve to the right and the 𝑁𝑁 𝑠𝑠 curve to the
left. If households are made worse off, the wealth effects are lower consumption demand
𝐶𝐶 𝑑𝑑 and higher labour supply 𝑁𝑁 𝑠𝑠 .
3.11.2 Firms
We assume a representative firm. All firms share the same neoclassical production function
with the same total factor productivity. Firms own the capital they use and all have the
same capital stock, so there is no need to consider a rental market for capital. Investment is
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financed by issuing bonds. Firms are owned by the representative household. There is no
need for trade in shares, so we do not consider the stock market in the dynamic
macroeconomic model.
3.11.3 Government
We will analyse the government’s fiscal policies further in Chapter 4. At this stage we simply
note that the government chooses current and future public expenditure 𝐺𝐺 and 𝐺𝐺′, which is
part of the economy’s demand for goods and services along with private expenditure 𝐶𝐶 𝑑𝑑
and 𝐼𝐼 𝑑𝑑 on consumption and investment. The government levies the same lump-sum taxes 𝑡𝑡
and 𝑡𝑡′ on all households, and finances any budget deficit by issuing bonds.
3.12 General equilibrium
We now investigate how the economy reaches general equilibrium in goods, labour and
bond markets.
The analysis starts in the labour market. There, firms’ labour demand 𝑁𝑁 𝑑𝑑 is determined by
the equation 𝑀𝑀𝑃𝑃𝑁𝑁 = 𝑤𝑤. Households’ labour supply 𝑁𝑁 𝑠𝑠 (𝑟𝑟) is determined by the static and
dynamic optimality conditions 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 = 𝑤𝑤 and 𝑀𝑀𝑅𝑅𝑆𝑆𝑙𝑙,𝑙𝑙′ = (1 + 𝑟𝑟)𝑤𝑤/𝑤𝑤′. At this stage, we
take the real interest rate 𝑟𝑟 as given, which influences the position of the labour supply
curve. The left panel of Figure 3.29 depicts the labour market diagram. The real wage
adjusts to 𝑤𝑤 ∗ to match demand and supply and clear the labour market. Intuitively, wages
affect the cost of hiring labour for firms and the incentive to work for households.
Figure 3.28: Labour-market equilibrium
The equilibrium level of employment 𝑁𝑁 ∗ in the labour market has implications for the goods
market through the amount of goods and services firms produce. The production function is
𝑌𝑌 = 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑁𝑁). Given exogenous TFP 𝑧𝑧 and a predetermined current capital stock 𝐾𝐾, firms’
output of goods 𝑌𝑌 rises or falls with employment. The right panel of the figure depicts the
production function and shows how the supply of goods 𝑌𝑌 ∗ is determined by equilibrium
employment 𝑁𝑁 ∗ .
In the goods market, firms’ supply curve for goods is derived from labour-market
equilibrium. The supply of output comes from the production function 𝑌𝑌 = 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑁𝑁 ∗ ) given
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the equilibrium level of employment 𝑁𝑁 ∗ in the labour market. The derivation of the output
supply curve 𝑌𝑌 𝑠𝑠 is shown in Figure 3.30. The right panel is the goods market diagram where
the quantity of output (real GDP 𝑌𝑌) is on the horizontal axis and the real interest rate 𝑟𝑟 is on
the vertical axis. The interest rate is relevant because 1 + 𝑟𝑟 is relative price of current goods
in terms of future goods.
A higher real interest rate 𝑟𝑟 shifts 𝑁𝑁 𝑠𝑠 (𝑟𝑟) to the right as shown in the left panel. This
increases employment 𝑁𝑁 ∗ leading to a movement up the production function in the middle
panel, which raises the supply of output 𝑌𝑌 𝑠𝑠 . This explains the upward-sloping 𝑌𝑌 𝑠𝑠 curve seen
in the right panel.
Figure 3.29: Derivation of the output supply curve
The analysis of the goods market is completed by deriving the demand curve. This sums up:
•
•
•
Consumption demand 𝐶𝐶 𝑑𝑑
Investment demand 𝐼𝐼 𝑑𝑑
Government expenditure 𝐺𝐺.
The equation describing the aggregate demand 𝑌𝑌 𝑑𝑑 for goods and services is:
𝑌𝑌 𝑑𝑑 = 𝐶𝐶 𝑑𝑑 + 𝐼𝐼 𝑑𝑑 + 𝐺𝐺
Consumption demand and investment demand depend negatively on the real interest rate
𝑟𝑟. Summing up the components of aggregate demand implies a downward-sloping demand
curve for goods as shown in Figure 3.31.
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Figure 3.30: Derivation of the output demand curve
With upward-sloping 𝑌𝑌 𝑠𝑠 and downward-sloping 𝑌𝑌 𝑑𝑑 curves, the goods market clears through
adjustment of the real interest rate 𝑟𝑟. A higher real interest rate discourages spending on
consumption and investment, which reduces aggregate demand for goods. A higher real
interest rate also encourages greater labour supply so households can save more, which
raises employment and production. Figure 3.32 shows the equilibrium real interest rate 𝑟𝑟 ∗
where the 𝑌𝑌 𝑑𝑑 and 𝑌𝑌 𝑠𝑠 curves intersect. In this model, notice that the real interest rate is
determined by competitive markets, not by the central bank’s monetary policy. This is
different in the sticky-price New Keynesian model we will study in Chapter 8.
Figure 3.31: Goods market equilibrium
The intersection of the 𝑌𝑌 𝑑𝑑 and 𝑌𝑌 𝑠𝑠 curves is not only a point of goods-market equilibrium. It
is also a labour-market equilibrium because every point on 𝑌𝑌 𝑠𝑠 represents equilibrium in the
labour market by construction. That covers two of the three markets in the dynamic
macroeconomic model. What about the remaining market, the bond market?
It might be expected the interest rate 𝑟𝑟 would adjust to achieve bond-market equilibrium
but it must be at 𝑟𝑟 = 𝑟𝑟 ∗ to clear the goods market. However, it turns out that the bond
market is in equilibrium once the labour and goods markets are both in equilibrium. This
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follows from the logic of budget constraints. Spending (demand) in markets other than the
bond market is balanced by earnings (supply) in those markets, so budget constraints then
imply a balance between spending and earning in final market, which is equivalent to
demand equals supply in that market. This result is known as Walras’ law and consequently,
we do not need to consider the bond market separately.
The complete dynamic macroeconomic model can then be summarised by the labourmarket and goods-market diagrams shown in Figure 3.33.
Box 3.6: Growth slowdowns and real interest rates
Recent decades have seen persistently low real interest rates in advanced economies.
Why might this have happened? One hypothesis is that there is greater pessimism about
the economy’s future growth potential. We will explore the connection between
expectations of the economy’s prospects and the equilibrium level of real interest rates
using the dynamic macroeconomic model.
We represent greater pessimism about future growth by lowering expectations of future
TFP 𝑧𝑧′ while leaving current TFP 𝑧𝑧 unchanged. Lower 𝑧𝑧′ than otherwise reduces the future
marginal product of capital 𝑀𝑀𝑃𝑃𝐾𝐾′ , so investment demand 𝐼𝐼 𝑑𝑑 falls for each interest rate 𝑟𝑟.
This shifts the 𝑌𝑌 𝑑𝑑 curve to the left. Lower 𝑧𝑧 ′ also reduces households’ expectations of
income in the future, which makes them worse off and means a negative wealth effect
and less demand for all normal goods. The reduction in 𝐶𝐶 𝑑𝑑 shifts the 𝑌𝑌 𝑑𝑑 curve further to
the left. The decrease in demand for leisure means an increase in labour supply 𝑁𝑁 𝑠𝑠 . This
causes rightward shift of the 𝑌𝑌 𝑠𝑠 curve. These effects are shown in Figure 3.34.
Figure 3.32: Diagrams of dynamic macroeconomic model
With a leftward shift of 𝑌𝑌 𝑑𝑑 and a rightward shift of 𝑌𝑌 𝑠𝑠 , the equilibrium real interest rate
declines unambiguously. The impact on the other variables 𝑌𝑌, 𝑁𝑁, and 𝑤𝑤 is ambiguous but
this analysis helps to explain the observation of persistently low real interest rates.
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Figure 3.33: Effects of productivity growth slowdown
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Chapter 4: Fiscal policy and credit-market
imperfections
In Chapter 3 we studied the determinants of consumption and investment expenditure,
covering the two components of private expenditure in a closed economy. The main
omission was fiscal policy, the subject of the first part of this chapter. We will explore to
what extent public expenditure and changes in taxes can be used by governments as a tool
of demand management.
Another limitation of what we covered so far in Chapter 3 is that consumption and
investment are analysed in an economy with perfect financial markets. There were no limits
on access to credit other than the ability to repay and everyone faced the same interest
rate. The second part of this chapter introduces what are known as ‘credit-market
imperfections’ and explores how these change our understanding of consumption and
investment. We will see that there are important limitations on the power of fiscal policy in
an economy with perfect financial markets but that fiscal policy becomes both more
powerful and more useful when there are credit-market imperfections.
Essential reading
•
Williamson, Chapter 10.
4.1 Taxes and the government’s budget constraint
How do the government’s tax policies affect aggregate expenditure? In particular, can tax
cuts be used as a tool of demand management to stimulate demand when needed? This is
the first aspect of fiscal policy we study, taking as given for now the government’s own
direct public expenditure.
A key point in the analysis of taxation is that changes in taxes – if not matched by changes in
government expenditure – affect the government’s budget deficit and, in turn, the level of
government debt. The need to repay government debt in the future if a default is to be
avoided has implications for the level of future taxes. These considerations lead us to the
idea that the government itself has a budget constraint.
We will begin by analysing taxes and government debt using the two-period model from
Section 3.3 that was central to our study of households’ consumption decisions. That model
has a current period and a future period.
Government expenditure, or public expenditure, refers to purchases of goods and services
by the government, irrespective of whether these are produced by private firms or are the
result of the government’s own production. This does not include transfer payments, for
example, those associated with the welfare state. Government expenditure in the current
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period is 𝐺𝐺 and 𝐺𝐺′ in the future period. These variables are in real terms, as are all variables
in this chapter.
Suppose the government collects tax revenue 𝑇𝑇 in the current period and expects to collect
𝑇𝑇 ′ in the future. Any transfer payments where the government makes payments to others
rather than receiving tax revenue are counted as negative values of tax revenue.
The government does not have to match current expenditure 𝐺𝐺 and taxes 𝑇𝑇: it can run a
budget deficit or surplus. The government’s budget deficit is defined as 𝐵𝐵 = 𝐺𝐺 − 𝑇𝑇, with a
negative value of 𝐵𝐵 denoting a surplus. If the government runs a deficit it borrows by issuing
bonds with real interest rate 𝑟𝑟. We assume the government begins from a blank slate at the
beginning of the current period with no initial debt outstanding. This means the total stock
of government debt is 𝐵𝐵 and an amount (1 + 𝑟𝑟)𝐵𝐵 including interest falls due for repayment
in the future.
In the future period, the government must repay (1 + 𝑟𝑟)𝐵𝐵 otherwise it defaults on its debt.
The analysis here assumes the government debt is in the form of inflation-indexed bonds
paying a known real interest rate 𝑟𝑟. Later in 0 where money is studied, we will see how
governments can print money and how a ‘soft’ default can occur through inflation. Here, we
can still make our equations consistent with the possibility of a ‘hard’ default by considering
a default as a special form of wealth tax applying to holdings of government bonds.
The budget constraint on the government in the future period is therefore that the budget
surplus 𝑇𝑇 ′ − 𝐺𝐺′ is sufficient to cover the repayment (1 + 𝑟𝑟)𝐵𝐵, because in a two-period
model there is no possibility of rolling over the debt any longer. In accounting for the
government’s interest payments 𝑟𝑟𝑟𝑟, it is possible to distinguish between the primary
budget surplus 𝑇𝑇 ′ − 𝐺𝐺′ that excludes the interest cost, and the budget surplus 𝑇𝑇 − 𝐺𝐺 ′ − 𝑟𝑟𝑟𝑟
that deducts the interest cost.
Dividing both sides of the budget constraint 𝑇𝑇′ − 𝐺𝐺′ = (1 + 𝑟𝑟)𝐵𝐵 by 1 + 𝑟𝑟 implies a
constraint on the amount of bonds the government can issue in the current period:
𝐵𝐵 =
𝑇𝑇 ′ − 𝐺𝐺 ′
1 + 𝑟𝑟
Substituting the budget deficit definition 𝐵𝐵 = 𝐺𝐺 − 𝑇𝑇 yields a present-value constraint:
𝑇𝑇 +
𝑇𝑇′
𝐺𝐺 ′
= 𝐺𝐺 +
1 + 𝑟𝑟
1 + 𝑟𝑟
This equation states that the government must raise tax revenue across the two periods of a
present value sufficient to cover the present value of all government expenditure across the
two periods.
We now want to consider how taxes and government debt affects private expenditure in
the economy. We start by supposing all taxes are lump-sum taxes levied on households. A
lump-sum tax is one where the amount of tax paid by a household does not depend on the
household’s behaviour. Most taxes do not have this form, for example, income taxes or
sales taxes. We consider later taxes where revenue raised depends on choices.
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We also make an assumption that the government is not redistributing when levying taxes
on households. This means that everyone shares the tax burden equally. Assume there are
𝑁𝑁 households in the economy and these households remain unchanged over the two
periods (later we will consider what happens if new generations are born). If 𝑡𝑡 and 𝑡𝑡′ are the
lump-sum amounts of tax collected from a household in the current and future periods then
an equal distribution of taxes means 𝑡𝑡 = 𝑇𝑇/𝑁𝑁 and 𝑡𝑡 ′ = 𝑇𝑇 ′ /𝑁𝑁.
What are the implications of the government’s tax policy for household budget constraints
under the assumptions made so far? We have seen that a household’s present value of
consumption must be equal to its lifetime wealth ℎ or present value of income after tax:
ℎ = 𝑦𝑦 − 𝑡𝑡 +
𝑦𝑦′
𝑡𝑡 ′
𝑦𝑦 ′ − 𝑡𝑡 ′
= �𝑦𝑦 +
� − �𝑡𝑡 +
�
1 + 𝑟𝑟
1 + 𝑟𝑟
1 + 𝑟𝑟
The second expression for ℎ above breaks down lifetime wealth into the present value of
pre-tax income and the present value of the taxes a household will face. Using government
budget constraint and the equal distribution of taxes 𝑡𝑡 = 𝑇𝑇/𝑁𝑁 and 𝑡𝑡 ′ = 𝑇𝑇 ′ /𝑁𝑁:
𝑡𝑡 +
1
𝑇𝑇 ′
1
𝐺𝐺 ′
𝑡𝑡′
= �𝑇𝑇 +
� = �𝐺𝐺 +
�
1 + 𝑟𝑟
𝑁𝑁
1 + 𝑟𝑟
1 + 𝑟𝑟 𝑁𝑁
It follows that lifetime wealth can be expressed as the present value of pre-tax income net
of a 1/𝑁𝑁th share of the present value of all government expenditure.
𝐺𝐺 ′
1
𝐺𝐺
𝐺𝐺
𝑦𝑦′
𝑁𝑁
� − �𝐺𝐺 +
� = 𝑦𝑦 − +
ℎ = �𝑦𝑦 +
1 + 𝑟𝑟
𝑁𝑁
𝑁𝑁
1 + 𝑟𝑟
1 + 𝑟𝑟
′
𝑦𝑦 ′ −
Note that the exact values of taxes 𝑡𝑡 and 𝑡𝑡′ drop out of the equation. It is therefore
sufficient to know the government’s plans for public expenditure because these determine
the present value of taxes given the government’s budget constraint. The exact combination
of 𝑡𝑡 and 𝑡𝑡′ is not relevant in calculating ℎ.
Since the position of a household’s lifetime budget constraint can be determined by
knowing ℎ and 𝑟𝑟, a corollary is that the position of the budget constraint can known simply
with reference to the present value of pre-tax income and the government’s expenditure
plans. Another way to deduce this is to note that the household’s budget constraint must
always pass through the point (𝑦𝑦 − 𝐺𝐺 ⁄𝑁𝑁 , 𝑦𝑦 ′ − 𝐺𝐺 ′ ⁄𝑁𝑁). Figure 4.1 illustrates the combined
household and government budget constraint.
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Figure 4.1: The combined household and government budget constraint
4.2 Ricardian equivalence
This section outlines a famous argument claiming the government’s tax policy has no power
to affect aggregate demand. The argument is known as ‘Ricardian equivalence’ and, if valid,
it says that if the government increases households’ current disposable income by cutting
taxes, households will not spend the extra income. The reason is that a tax cut is not a free
lunch. It increases the budget deficit and raises government debt, which leads to higher
future taxes. The government’s present-value budget constraint is central to this claim.
The government budget constraint implies a tax cut does not reduce the present value of all
taxes unless government expenditure is also reduced. Given 𝐺𝐺 and 𝐺𝐺′, reducing 𝑡𝑡 raises 𝑡𝑡′ by
an equal amount in present value. This means that changes in 𝑡𝑡 and 𝑡𝑡 ′ /(1 + 𝑟𝑟) cancel out
when calculating the impact on the present value of taxes and, consequently, households’
lifetime wealth ℎ is not affected by the tax cut.
In the consumption choice diagram in Figure 4.2, the endowment point on the household
budget constraint moves from 𝐸𝐸1 to 𝐸𝐸2 along an unchanged household budget constraint.
We have seen that the position of this budget constraint depends on pre-tax income and
the government’s spending plans but it does not depend on the exact timing of taxes. If
households can choose any point on the standard lifetime budget constraint then there is
no change to the optimal consumption plan. The tax cut does not increase consumption
demand but instead increases the desire to save. Intuitively, the tax cut now means higher
taxes in the future and households must save more now to sustain their optimal choice of
consumption in the future.
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Figure 4.2: Tax cut financed by higher budget deficit
We have shown that the tax cut does not affect households’ lifetime wealth ℎ. But might it
affect the real interest rate 𝑟𝑟, which determines the gradient of the budget constraint? In
the basic consumption choice model, the real interest rate is exogenous. But we have seen
in Chapter 3 how equilibrium interest rates can be found given exogenous incomes and
optimal consumption choices. Would that analysis of interest rates predict that issuing more
government bonds causes the bond price to fall and the interest rate to rise?
First note that the extra supply of bonds is equal to the size of tax cut. Our analysis above
shows that at any arbitrary real interest rate 𝑟𝑟, the optimal consumption plan (𝑐𝑐 ∗ , 𝑐𝑐 ′∗ ) does
not change following the tax cut, so private saving 𝑠𝑠 = 𝑦𝑦 − 𝑡𝑡 − 𝑐𝑐 also increases by an
amount equal to the tax cut. All else being equal, greater private saving means a larger
demand for bonds. With the supply and demand for bonds increasing by the same amount,
no change in 𝑟𝑟 is required for the extra issuance of bonds to be absorbed by the market.
An equivalent argument points out that with no change in 𝑐𝑐 for an arbitrary 𝑟𝑟 following the
tax cut and no change in 𝑦𝑦 or 𝐺𝐺, goods-market equilibrium 𝑐𝑐𝑐𝑐 + 𝐺𝐺 = 𝑦𝑦𝑦𝑦 still prevails at the
original interest rate 𝑟𝑟. Therefore, neither the position nor the gradient of households’
budget constraint changes after the tax cut.
Although the Ricardian equivalence argument is a useful benchmark for the analysis of fiscal
policy, the argument requires strong assumptions. These include the lump-sum nature of
taxes. When taxes are not lump sum, they can have incentive effects, even if the timing of
taxes does not change the present value of all tax revenue that needs to be collected. Lumpsum taxes matter only through an income effect but this channel is neutralised by the
government’s budget constraint. Other types of taxes also have substitution effects.
Another assumption is that the tax burden is shared equally among households, so taxes do
not redistribute between different households. We will see examples of how taxes with
distributional effects that affect households’ consumption choices. Different from the
analysis above, there are winners and losers whose consumption adjusts. Depending on the
exact behaviour of these groups, aggregate consumption can depend on tax policy. One
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particularly relevant case is where intergenerational distribution occurs. The current
generation benefits from a tax cut, but the higher taxes to repay the extra government debt
will be borne by future generations.
Finally, it is crucial to the Ricardian equivalence argument that there are no credit-market
imperfections. We will see examples where Ricardian equivalence breaks down when there
are limits on households’ borrowing or where borrowers face different interest rates from
savers.
Box 4.1: Taxes on consumption
Taxes are usually not lump-sum amounts. What difference does this make to our
analysis of fiscal policy? We will consider an example where there are proportional
taxes on consumption expenditure, for example, a sales tax or a value-added tax
(VAT).
Assume that households buying consumption goods 𝑐𝑐 in the current period face a
tax rate 𝜏𝜏, which means this consumption costs (1 + 𝜏𝜏)𝑐𝑐 inclusive of tax. Similarly,
in the future period, the cost of consumption 𝑐𝑐′ is (1 + 𝜏𝜏 ′ )𝑐𝑐′, where 𝜏𝜏 ′ is the future
tax rate.
To simplify matters in this example, we assume a representative household. This
means that all 𝑁𝑁 households have same incomes 𝑦𝑦 and 𝑦𝑦 ′ and same preferences.
Consequently, all households choose the same consumption plan (𝑐𝑐, 𝑐𝑐 ′ ) and the
government raises tax revenues 𝑇𝑇 = 𝜏𝜏𝜏𝜏𝜏𝜏 and 𝑇𝑇 ′ = 𝜏𝜏 ′ 𝑐𝑐 ′ 𝑁𝑁 in total.
In this setting, the lifetime budget constraint of a representative household is:
(1 + 𝜏𝜏)𝑐𝑐 +
(1 + 𝜏𝜏 ′ )𝑐𝑐 ′
𝑦𝑦′
= 𝑦𝑦 +
1 + 𝑟𝑟
1 + 𝑟𝑟
Instead of subtracting lump-sum amounts of tax revenue from the right-hand side,
tax rates multiplying consumption choices are added to the left-hand side. Observe
that the horizontal intercept of this budget constraint is 𝑐𝑐 =
(𝑦𝑦 + 𝑦𝑦 ′ /(1 + 𝑟𝑟))⁄(1 + 𝜏𝜏) and the vertical intercept is 𝑐𝑐 ′ =
((1 + 𝑟𝑟)𝑦𝑦 + 𝑦𝑦 ′ )⁄(1 + 𝜏𝜏 ′ ). The gradient of the budget constraint is −(1 + 𝑟𝑟)(1 +
𝜏𝜏)/(1 + 𝜏𝜏 ′ ). Notice that the tax rates 𝜏𝜏 and 𝜏𝜏 ′ affect the gradient. This is because
they can have implications for the relative cost of consumption goods at different
dates.
The government’s present-value budget constraint with public expenditure (𝐺𝐺, 𝐺𝐺 ′ )
is 𝑇𝑇 + 𝑇𝑇′⁄(1 + 𝑟𝑟) = 𝐺𝐺 + 𝐺𝐺′⁄(1 + 𝑟𝑟), or in terms of tax rates:
𝜏𝜏 ′ 𝑐𝑐 ′
1
𝐺𝐺 ′
𝜏𝜏𝜏𝜏 +
= �𝐺𝐺 +
�
1 + 𝑟𝑟 𝑁𝑁
1 + 𝑟𝑟
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How does this type of tax system affect the Ricardian equivalence argument seen
earlier? Consider a deficit-financed cut of current tax rate 𝜏𝜏 and hold constant the
public expenditure 𝐺𝐺 and 𝐺𝐺 ′ planned in the current and future time periods. Just as
before, the government budget constraint implies the future tax rate 𝜏𝜏 ′ must
adjust.
Reducing the current tax rate 𝜏𝜏 implies the household budget constraint’s
horizontal intercept (𝑦𝑦 + 𝑦𝑦 ′ /(1 + 𝑟𝑟))/(1 + 𝜏𝜏) rises and its gradient −(1 + 𝑟𝑟)(1 +
𝜏𝜏)/(1 + 𝜏𝜏 ′ ) is lower. This is illustrated in Figure 4.2. The reduction in the gradient is
similar to what happens when the interest rate 𝑟𝑟 declines and, in both cases,
current consumption becomes relatively cheaper compared to future consumption.
Figure 4.3: Failure of Ricardian equivalence
An immediate consequence of this is because the gradient of household budget
constraint has changed, it cannot be tangent to an indifference curve at the original
consumption plan. The household’s choice of consumption plan is affected by the
tax cut and so Ricardian equivalence fails. Intuitively, the tax cut has a substitution
effect working through the change in the budget constraint’s gradient that is absent
in our earlier analysis of lump-sum taxes.
Although Ricardian equivalence does not hold, accounting for the government
budget constraint is still important and has some interesting implications. Moving
the tax terms to the right-hand side, the representative household’s budget
constraint is:
𝑐𝑐 +
𝑐𝑐 ′
𝑦𝑦 ′
𝜏𝜏 ′ 𝑐𝑐 ′
= 𝑦𝑦 +
− �𝜏𝜏𝜏𝜏 +
�
1 + 𝑟𝑟
1 + 𝑟𝑟
1 + 𝑟𝑟
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Notice that the representative household’s choice of consumption plan (𝑐𝑐, 𝑐𝑐 ′ ) also
appears in the government budget constraint because it affects how much tax
revenue is collected from given tax rates. Since all households make the same
consumption choices here, the household and government budget constraints can
be combined:
𝑐𝑐 +
𝐺𝐺 ′
𝑦𝑦 ′ − 𝑁𝑁
′
𝐺𝐺
𝑐𝑐
= 𝑦𝑦 − +
𝑁𝑁
1 + 𝑟𝑟
1 + 𝑟𝑟
The tax rates 𝜏𝜏 and 𝜏𝜏 ′ have been eliminated and the equation has exactly the same
form as what was obtained with lump-sum taxes. Graphically, the equation is a
straight line with gradient −(1 + 𝑟𝑟) passing through the point (𝑦𝑦 − 𝐺𝐺/𝑁𝑁, 𝑦𝑦 ′ −
𝐺𝐺 ′ /𝑁𝑁). This is not the budget constraint faced by any individual household acting
independently of others but implies that after tax rates have been adjusted to
satisfy the government’s budget constraint, the consumption plan chosen by
households must lie on the same line as it must with a lump-sum tax system.
Considering the deficit-financed cut of current tax rate 𝜏𝜏, the budget constraint
faced by individual households becomes flatter. After taking account of the
adjustment of households’ choices and the future tax rate 𝜏𝜏 ′ needed to satisfy the
government’s budget constraint, the consumption plan must lie on combined
budget constraint and, without any changes to public expenditure 𝐺𝐺 and 𝐺𝐺 ′ , this
line does not shift. Hence, overall, the new consumption plan must be tangent to a
flatter individual budget constraint while also lying on the same downward-sloping
combined household-and-government budget constraint. Therefore, the new
consumption plan must feature higher 𝑐𝑐 and lower 𝑐𝑐 ′ as shown in Figure 4.4.
These unambiguous effects can be thought of as the substitution effect of lower
relative price of 𝑐𝑐 compared to 𝑐𝑐 ′ brought about by the tax rate changes, analogous
to the substitution effect of a lower interest rate. However, the income effects of
the tax changes are weakened by accounting for the government budget
constraint, as is the case with lump-sum taxes.
Although changing tax rates is seen to have an impact on consumption, the analysis
so far does not provide a reason for governments to vary tax rates over time. To
understand why, note that if 𝜏𝜏 = 𝜏𝜏 ′ , i.e. tax rates remain the same, the household
budget constraint gradient becomes −(1 + 𝑟𝑟). This allows households to reach the
highest indifference curve conditional on the government satisfying its budget
constraint, because the combined household-and-government budget constraint
has gradient −(1 + 𝑟𝑟). Any choice of 𝜏𝜏 ≠ 𝜏𝜏 ′ would mean the representative
household ends up on a lower indifference curve.
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Figure 4.4: Effects of timing of taxes
Intuitively, having 𝜏𝜏 = 𝜏𝜏 ′ avoids the government distorting relative prices of
consumption at different points in time. This is an argument for what is called ‘tax
smoothing’: minimising changes in tax rates. Note that the case for tax smoothing
applies more broadly than just this example and holds even when the tax system
cannot avoid all distortions. But an important caveat is that credit-market
imperfections, another reason for the failure of Ricardian equivalence, will give rise
to an argument for activist fiscal policy.
Box 4.2: The effects of a fiscal stimulus
If Ricardian equivalence holds, we have seen that governments cannot use changes
in (lump-sum) taxes to influence private expenditure. But can a government directly
boost real GDP by increasing public expenditure 𝐺𝐺? If so, by how much?
We will consider here a temporary fiscal stimulus, i.e. higher 𝐺𝐺 with no change in 𝐺𝐺 ′
planned. It is important to note that the analysis of a permanent change in 𝐺𝐺 is not
the same. The direct effect of higher government expenditure 𝐺𝐺 is to shift the 𝑌𝑌 𝑑𝑑
curve to the right because 𝐺𝐺 is one of the components of aggregate demand. But it
also raises the present value of taxes 𝑇𝑇 + 𝑇𝑇′⁄(1 + 𝑟𝑟) owing to the government’s
budget constraint.
The increase in the present value of taxes is the same amount as the increase in 𝐺𝐺
no matter whether 𝑇𝑇 or 𝑇𝑇′ rises. Consequently, we do not need to specify exactly
how the government pays for higher 𝐺𝐺. This is Ricardian equivalence in this context
– the timing of taxes does not matter – but it does not mean that 𝐺𝐺 has no effects,
only that any effect is the same whether current taxes or the budget deficit rise.
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The higher burden of taxes after the increase in 𝐺𝐺 has a negative wealth effect on
households because they are less able to afford a given amount of private
consumption and leisure. This leads them to reduce their demand for consumption
𝐶𝐶 𝑑𝑑 as it is a normal good, which has the effect of shifting the 𝑌𝑌 𝑑𝑑 curve to the left.
However, the desire for consumption smoothing means that 𝐶𝐶 𝑑𝑑 falls by less than
the tax burden and 𝐺𝐺 rise. Overall, the output demand curve 𝑌𝑌 𝑑𝑑 shifts to the right.
A way of avoiding some of the reduction in consumption is for households to work
more and earn more income. This is what the wealth effect of taxes on labour
supply represents – corresponding to a lower demand for leisure as a normal good.
Note that we are assuming taxes have a lump-sum form, so there are no
disincentive effects here. The labour supply curve 𝑁𝑁 𝑠𝑠 thus shifts to the right as
shown in Figure 4.5. Labour-market equilibrium now occurs at a higher level of
employment and the movement along the production function implies that firms
increase output of goods at the same real interest rate, shifting the output supply
curve 𝑌𝑌 𝑠𝑠 to the right. Since future consumption and leisure are normal goods,
households do not want to raise 𝑌𝑌 𝑠𝑠 by more than increase in tax burden (equal to
the increase in 𝐺𝐺) minus the reduction in consumption expenditure 𝐶𝐶 𝑑𝑑 .
Figure 4.5: Wealth effect on labour supply and output supply
With both the 𝑌𝑌 𝑑𝑑 and 𝑌𝑌 𝑠𝑠 curves shifting rightwards, GDP 𝑌𝑌 rises unambiguously as
shown in Figure 4.6. The overall shift of 𝑌𝑌 𝑑𝑑 is larger than the shift of 𝑌𝑌 𝑠𝑠 because
smoothing of consumption and leisure makes the combined wealth effect on 𝐶𝐶 𝑑𝑑
and 𝑁𝑁 𝑠𝑠 (and hence, 𝑌𝑌 𝑠𝑠 ) smaller than direct effect of 𝐺𝐺 on 𝑌𝑌 𝑑𝑑 . A larger shift of 𝑌𝑌 𝑑𝑑
than 𝑌𝑌 𝑠𝑠 implies the real interest rate 𝑟𝑟 rises unambiguously.
There are two shifts of 𝑁𝑁 𝑠𝑠 to right in the labour market diagram from Figure 4.6.
First, the wealth effect of the higher tax burden leading to a shift from 𝑁𝑁1𝑠𝑠 (𝑟𝑟1 ) to
𝑁𝑁2𝑠𝑠 (𝑟𝑟1 ). Second, the saving incentive effect of the higher real interest rate 𝑟𝑟
producing the shift from 𝑁𝑁2𝑠𝑠 (𝑟𝑟1 ) to 𝑁𝑁2𝑠𝑠 (𝑟𝑟2 ). Hence, we can conclude unambiguously
that 𝑁𝑁 rises and 𝑤𝑤 falls. Note that there is no effect on 𝑁𝑁 𝑑𝑑 – the increase in 𝐺𝐺 does
not boost productivity here, so our analysis might reach different conclusions if we
considered expenditure on infrastructure that increased total factor productivity
and hence, labour demand.
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Although we conclude that GDP 𝑌𝑌 rises overall, this analysis shows it increases by
less than 𝐺𝐺 does. This is due to 𝑌𝑌 = 𝐶𝐶 + 𝐼𝐼 + 𝐺𝐺 and both 𝐶𝐶 𝑑𝑑 and 𝐼𝐼 𝑑𝑑 falling.
Consumption falls because of the negative wealth effect and the negative
substitution effect of a higher real interest rate 𝑟𝑟. Investment falls because the
higher real interest rate raises the borrowing cost or opportunity cost of
investment. These displacement effect on private expenditure of higher public
expenditure 𝐺𝐺 are known as ‘crowding out’ effects.
Figure 4.6: Effects of temporary fiscal stimulus
In summary, our model predicts a temporary fiscal stimulus would boost real GDP
but with no ‘multiplier’ effect whereby 𝑌𝑌 rises more than 𝐺𝐺. What is missing? First,
we could imagine that some types of public expenditure are complementary with
production or consumption. Infrastructure expenditure might raise the return to
capital and boost investment by firms. It might also cause households to spend
more on certain types of goods (for example, cars or air travel) that are
complementary to the public expenditure (although it is also possible to envisage
goods and services that are substitutes for public expenditure).
More importantly, our analysis is missing the idea that deficit-financed public
expenditure raises households’ current disposable income through higher GDP. The
argument is that this encourages some households to spend more on consumption
because previously they were unable to pay for this consumption by borrowing
against future income. This supposes that there are credit-market imperfections of
the type we will study in the remainder of the chapter. We return to the analysis of
a fiscal stimulus with credit-market imperfections when studying macroeconomic
policy in Chapter 9.
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4.3 Credit-market imperfections
The two-period consumption model has households’ consumption plans limited only by a
single lifetime budget constraint:
𝑐𝑐 ′
𝑦𝑦 ′ − 𝑡𝑡′
𝑐𝑐 +
= ℎ = 𝑦𝑦 − 𝑡𝑡 +
1 + 𝑟𝑟
1 + 𝑟𝑟
This assumes households can borrow as much as they like at the same interest rate that
savers receive. This is subject only to it being possible to repay debts with future income. In
reality, however, households are subject to much tighter constraints on borrowing and face
higher interest rates when they do borrow. These additional restrictions arise from what are
known as ‘credit-market imperfections’.
We will start by considering a simple borrowing constraint. Households cannot borrow more
than some amount 𝐿𝐿. Example of borrowing constraints could be a credit limit on a credit
card or an overdraft limit. To begin with, we treat the maximum loan size 𝐿𝐿 as exogenous.
Also for now, assume that borrowers face the same interest rate 𝑟𝑟 as savers receive.
We now add the borrowing constraint to the two-period consumption choice model. With
no initial assets, borrowing means a negative level of saving 𝑠𝑠 = 𝑦𝑦 − 𝑡𝑡 − 𝑐𝑐. The borrowing
constraint is thus:
−𝑠𝑠 ≤ 𝐿𝐿
Using the definition of saving 𝑠𝑠, this constraint is equivalent to 𝑐𝑐 ≤ 𝑦𝑦 − 𝑡𝑡 + 𝐿𝐿, which places
an upper limit on current consumption 𝑐𝑐. This additional constraint truncates the lifetime
budget constraint to the right of 𝑐𝑐 = 𝑦𝑦 − 𝑡𝑡 + 𝐿𝐿 as shown in Figure 4.7. Depending on
income and preferences, the constraint may or may not be binding.
Figure 4.7: The borrowing constraint may or may not be binding
The borrowing constraint is not binding when the tangency point between an indifference
curve and the full lifetime budget constraint without the borrowing constraint is to the left
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of where the borrowing constraint is relevant, i.e. where the household wants to borrow
less than the borrowing constraint, or wants to save. The borrowing constraint is binding
when this tangency point lies to the right of where the budget constraint is truncated and,
hence, the choice of that level of borrowing is not feasible. In this case, the next best
consumption plan is the one at the ‘kink’ of the budget constraint where it is truncated by
the borrowing constraint. The household chooses to borrow the maximum amount allowed
by the constraint. In what follows, we shall assume the borrowing constraint is binding,
otherwise it is irrelevant and our analysis of consumption proceeds as it did in Chapter 3.
Binding borrowing constraints cause households’ consumption behaviour to behave in a
very different way from what we saw in Chapter 3 and Section 4.2. First, there is no
consumption smoothing following a temporary increase or decrease in disposable income.
This is illustrated in Figure 4.8. A decline in current income shifts the endowment point
horizontally to the left from 𝐸𝐸1 to 𝐸𝐸2 . Since the budget constraint is truncated at 𝑐𝑐 = 𝑦𝑦 −
𝑡𝑡 + 𝐿𝐿 because of the limit on borrowing, this truncation point also moves to the left by the
same amount.
Figure 4.8: No consumption smoothing
Starting from the borrowing constraint being binding, the household chooses the
consumption plan at the kink of the new budget constraint, which means that current
consumption falls by as much as current income. If current income were to increase, current
consumption would rise by the same amount if the borrowing constraint remains binding
(which it would as long as current income does not rise too much). Therefore, changes in
current disposable income have a one-for-one effect on current consumption if the
borrowing constraint is binding. In contrast, without a binding borrowing constraint,
households would choose to smooth consumption in response to temporary income shocks,
with 𝑐𝑐 moving less than one-for-one with a change in 𝑦𝑦.
A second difference compared to the earlier model of consumption is the failure of
Ricardian equivalence. A deficit-financed tax cut now increases current consumption
spending for those with a binding borrowing constraint as shown in Figure 4.9. The
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endowment point 𝐸𝐸 moves along the downward-sloping budget constraint line because of
the government’s own present-value budget constraint for the same reason explained in
Section 4.2. However, the limit on borrowing truncates the budget constraint horizontally at
disposable income plus maximum debt, so a tax cut that directly affects disposable income
moves the truncation point. Lower taxes today raise current disposable income, moving the
truncation point to the right along the lifetime budget constraint.
A household with a binding borrowing constraint would like to spend more today by
borrowing. If such a household receives more disposable income then they will spend all or
some of it on more current consumption, so Ricardian equivalence fails. If the borrowing
constraint remains binding after the tax cut, the household will spend all the extra
disposable income immediately.
Figure 4.9: Failure of Ricardian equivalence
Box 4.3: Bequests and intergenerational redistribution
The argument for Ricardian equivalence is that tax cuts are saved by households
because they know they will face higher taxes in future. But what if those higher
taxes are paid by future generations? Should the current generation benefiting
from a tax cut spend it and leave future generations to deal with the extra
government debt?
One reason to think this would not happen is that we observe people leaving
bequests – transfers of wealth to the next generation – which suggests there is
altruism across the generations. Hence, bequests might play the role of saving in
the standard Ricardian equivalence argument, with tax cuts being saved by the
current generation to leave larger bequests.
We can study this point further by reinterpreting the two-period consumption
choice model to consider bequests and intergenerational altruism. For simplicity,
suppose there are only two generations, a current generation of parents and a
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future generation comprised of their children. Each generation consumes in just
one period. Hence, 𝑐𝑐 denotes consumption of the current generation and 𝑦𝑦 − 𝑡𝑡
their income after tax, and 𝑐𝑐 ′ denotes consumption of the future generation and
𝑦𝑦 ′ − 𝑡𝑡 ′ their income after tax. Saving 𝑠𝑠 = 𝑦𝑦 − 𝑡𝑡 − 𝑐𝑐 now represents bequests, the
wealth of the current generation that is passed on to the future generation.
The current generation is altruistic in that they care about the next generation’s 𝑐𝑐′
as well as their own 𝑐𝑐 to some extent and indifference curves can be used to show
the current generation’s preferences over (𝑐𝑐, 𝑐𝑐 ′ ) as in the standard version of the
consumption choice model. The budget constraint 𝑐𝑐 ′ = 𝑦𝑦 ′ − 𝑡𝑡 ′ + (1 + 𝑟𝑟)𝑠𝑠 of the
future generation and 𝑠𝑠 = 𝑦𝑦 − 𝑡𝑡 − 𝑐𝑐 imply a present-value budget constraint on
the consumption levels of the two generations that is mathematically identical to
the standard lifetime budget constraint:
𝑐𝑐 +
𝑐𝑐 ′
𝑦𝑦 ′ − 𝑡𝑡 ′
= 𝑦𝑦 − 𝑡𝑡 +
1 + 𝑟𝑟
1 + 𝑟𝑟
The current generation chooses their own consumption 𝑐𝑐, which determines
bequests 𝑠𝑠, to reach their highest indifference curve subject to this present-value
budget constraint.
As in Section 4.2, the present-value budget constraint is not affected by the timing
of taxes after accounting for government’s own budget constraint. This means that
a deficit-financed tax cut (lower 𝑡𝑡, higher 𝑡𝑡′) does not affect the position of the
constraint. The current generation’s choice of consumption 𝑐𝑐 therefore remains
unchanged as shown in Figure 4.10. Since the endowment point moves to the right,
the implied bequest 𝑠𝑠 rises.
Figure 4.10: Ricardian equivalence with bequests
The present-value budget constraint does not rule out the current generation
choosing a level of its own consumption with 𝑐𝑐 > 𝑦𝑦 − 𝑡𝑡. But having 𝑐𝑐 > 𝑦𝑦 − 𝑡𝑡
means that 𝑠𝑠 < 0, a negative bequest. However, it is usually not possible
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to bequeath debt. This leads us to impose a non-negative bequest constraint 𝑠𝑠 ≥ 0,
which is equivalent to 𝑐𝑐 ≤ 𝑦𝑦 − 𝑡𝑡. Mathematically and geometrically, this has exactly
the same form as a borrowing constraint with the effect of truncating the presentvalue budget constraint at 𝑐𝑐 = 𝑦𝑦 − 𝑡𝑡.
As with a borrowing constraint, the non-negative bequest constraint may or may
not bind. All else being equal, it is more likely to bind if the current generation’s
altruism is weak, which corresponds to the current generation having steep
indifference curves where 𝑐𝑐′ is relatively unimportant compared to 𝑐𝑐. When the
constraint binds, bequests are exactly zero. A binding constraint causes Ricardian
equivalence to fail in essentially the same way as a binding borrowing constraint did
in Section 4.3. This case is illustrated in Figure 4.11.
The logic so far suggests that Ricardian equivalence should still hold when the
government practises intergenerational redistribution by cutting taxes and leaving
debt to future generations as long as people are observed to make positive
bequests. But there may be other motives for bequests beyond altruism. The
current generation might use bequests strategically to influence their children’s
behaviour with the threat of removal of a bequest – zero being the smallest
amount – providing incentives.
The key point here is that a larger planned bequest 𝑠𝑠 gives parents greater control
over their children. As an example, suppose we start from sufficiently large 𝑠𝑠 where
parents fully achieve their goal of influencing children’s behaviour, albeit at the cost
of making a bequest that is too large from a purely altruistic perspective. This case
is depicted in Figure 4.12. The indifference curves take account only of 𝑐𝑐 and
altruism over 𝑐𝑐′, not parents’ desire to influence their children’s behaviour. Hence,
the initial point does not feature a point of tangency between the indifference
curves and the present-value budget constraint because parents are choosing 𝑐𝑐 too
low and 𝑠𝑠 too high from a purely altruistic perspective.
Figure 4.11: Failure of Ricardian equivalence when bequests are zero
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Following a deficit-financed tax cut, the endowment point moves along the budget
constraint as usual. If the current generation maintained the same 𝑐𝑐 and increased
bequests 𝑠𝑠 then they would have even greater control over their children but we
started from a point where parents were already achieving their goals in that
respect. Hence, parents would respond by raising 𝑐𝑐 and leaving 𝑠𝑠 unchanged,
resulting in a failure of Ricardian equivalence even though bequests are positive.
Figure 4.12: Strategic bequests and a failure of Ricardian equivalence
4.4 Interest-rate spreads
Another type of credit-market imperfection is that borrowers face higher interest rates than
savers even when they have the ability and intention to repay a loan. Unlike earlier, there
are no additional restrictions on the quantity of borrowing here. Suppose savers receive an
interest rate 𝑟𝑟, while borrowers face an interest rate 𝑟𝑟𝑙𝑙 on loans with 𝑟𝑟𝑙𝑙 > 𝑟𝑟. The difference
between two interest rates is known as an interest rate spread.
Conditional on a non-negative amount of saving 𝑠𝑠 ≥ 0, the lifetime budget constraint can be
derived in exactly the same way as before in terms of savers’ interest rate 𝑟𝑟:
𝑐𝑐 +
𝑐𝑐 ′
𝑦𝑦 ′ − 𝑡𝑡′
= ℎ = 𝑦𝑦 − 𝑡𝑡 +
1 + 𝑟𝑟
1 + 𝑟𝑟
If the household is a borrower, so saving is negative, 𝑠𝑠 < 0, the lifetime budget constraint
can be derived in usual way, but with borrowers’ interest rate 𝑟𝑟𝑙𝑙 replacing 𝑟𝑟:
𝑐𝑐 ′
𝑦𝑦 ′ − 𝑡𝑡′
𝑐𝑐 +
= ℎ𝑙𝑙 = 𝑦𝑦 − 𝑡𝑡 +
1 + 𝑟𝑟𝑙𝑙
1 + 𝑟𝑟𝑙𝑙
These two versions of the lifetime budget constraint are straight lines with gradients −(1 +
𝑟𝑟) and −(1 + 𝑟𝑟𝑙𝑙 ) respectively. However, the first applies only to the left of the endowment
point and the second applies only to the right of 𝐸𝐸. Therefore, the gradient of the lifetime
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budget constraint increases (in absolute value) when a household switches from saving to
borrowing. This is depicted in Figure 4.13, where the lifetime budget constraint has a ‘kink’
at the endowment point.
Figure 4.13: Lifetime budget constraint with different interest rates for borrowers and savers
Compared to the case of no credit-market imperfections, someone who would have chosen
to borrow if 𝑟𝑟𝑙𝑙 = 𝑟𝑟 now chooses to borrow less or not to borrow at all if 𝑟𝑟𝑙𝑙 > 𝑟𝑟. The case
where a household decides not to borrow as a result of 𝑟𝑟𝑙𝑙 > 𝑟𝑟 is shown in Figure 4.14. If the
interest-rate spread is large enough to dissuade households from borrowing then they
choose a consumption plan at the kink of the lifetime budget constraint. Households in this
position will not smooth consumption after a temporary shock to income. The logic is similar
to the case with a quantitative limit on borrowing. This is easiest to see in the special case of
an extremely high interest-rate spread. The borrowing segment of the budget constraint
becomes so steep it is equivalent to a borrowing limit 𝐿𝐿 of zero.
Figure 4.14: An interest-rate spread causes a household to stop borrowing
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The presence of an interest-rate spread for borrower households also leads to a failure of
Ricardian equivalence. Although households who borrow face an interest rate 𝑟𝑟𝑙𝑙 > 𝑟𝑟, the
government is able to borrow at savers’ interest rate 𝑟𝑟 (savers may indeed be receiving 𝑟𝑟 as
a result of holding government bonds). The government present-value budget constraint is:
𝑡𝑡 +
1
𝐺𝐺 ′
𝑡𝑡 ′
= �𝐺𝐺 +
�
1 + 𝑟𝑟 𝑁𝑁
1 + 𝑟𝑟
By the logic of Section 4.1, the lifetime wealth ℎ of savers who receive interest rate 𝑟𝑟 is
unaffected by the timing of taxes 𝑡𝑡 and 𝑡𝑡′ conditional on the levels of public expenditure 𝐺𝐺
and 𝐺𝐺′. On the other hand, the timing of taxes affects the lifetime wealth ℎ𝑙𝑙 of borrowers,
defined as the present value of all disposable income discounted at rate 𝑟𝑟𝑙𝑙 :
𝑦𝑦 ′
1
𝐺𝐺 ′
(𝑟𝑟𝑙𝑙 − 𝑟𝑟)𝑡𝑡 ′
ℎ𝑙𝑙 = 𝑦𝑦 +
− �𝐺𝐺 +
�+
1 + 𝑟𝑟𝑙𝑙 𝑁𝑁
1 + 𝑟𝑟
(1 + 𝑟𝑟)(1 + 𝑟𝑟𝑙𝑙 )
A deficit-financed tax cut lowers 𝑡𝑡 but raises 𝑡𝑡′, which increases ℎ𝑙𝑙 when 𝑟𝑟𝑙𝑙 > 𝑟𝑟. Intuitively,
private borrowers effectively discount the future at a higher rate 𝑟𝑟𝑙𝑙 than the interest rate 𝑟𝑟
at which the government is able to borrow. Hence, shifting taxes away from the present to
the future actually increases the present value of lifetime income for borrower households.
For unchanged 𝐺𝐺 and 𝐺𝐺′, the tax cut shifts the kink point on households’ lifetime budget
constraint to the right as shown in Figure 4.15. This has no effect on the set of feasible
consumption plans for savers but expands the feasible set for borrowers. Consequently,
credit-constrained households are better off and choose to consume more. If a household is
dissuaded from borrowing by 𝑟𝑟𝑙𝑙 > 𝑟𝑟 then the household spends all of increase in disposable
income on higher current consumption 𝑐𝑐. If the household was still borrowing initially, then
the tax cut leads to an increase in both 𝑐𝑐 and 𝑐𝑐′. Either way, there is a failure of Ricardian
equivalence.
Figure 4.15: Failure of Ricardian equivalence with an interest-rate spread
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4.5 Asymmetric information
While the two credit market imperfections, namely borrowing constraints and interest rate
spreads, are realistic, we would like to understand why they arise. Moreover, what
determines the size of borrowing limits and interest rate spreads, and what might cause
them to change?
In this section and the next, we will look at two theories:
•
•
Asymmetric information, to explain interest rate spreads
Limited commitment, to explain borrowing constraints.
Asymmetric information is where one party to a transaction, a buyer or a seller, is better
informed than the other. We will apply this idea to study the market for loans with
asymmetric information. The asymmetric information is that borrowers know their
probability repayment probability better than lenders.
We assume just two types of borrowers for simplicity. ‘Good types’ are those who always
repay loans and ‘bad types’ are those who always fully default. Borrowers know their type
but lenders cannot directly distinguish between them.
In reality, lenders go to considerable lengths in screening and credit scoring applicants for
loans to resolve this informational asymmetry. However, these efforts are imperfect in that
some bad types will still slip through, in which case, the asymmetric information is the
residual uncertainty that remains about an individual’s type after screening. Moreover,
screening and credit scoring entail costs, and accounting for those costs has similar
implications for interest-rate spreads as the original informational asymmetry itself. For this
reason, we will not account for screening and credit scoring explicitly in our analysis.
Good types make up a fraction 1 − 𝑡𝑡 of all borrowers and bad types are a fraction 𝑡𝑡. The
value of 𝑡𝑡 (0 < 𝑡𝑡 < 1) is common knowledge to everyone including lenders.
While lenders cannot directly observe an individual’s type, bad types do not want to reveal
their type indirectly by asking for a different loan size compared to good types. Suppose
good types want to borrow 𝐿𝐿 when the interest rate on loans is 𝑟𝑟𝑙𝑙 . Bad types mimic them, so
all borrowers end up requesting the same loan size 𝐿𝐿.
Loans are provided by financial intermediaries such as banks who fund them exclusively by
taking safe deposits. There is no bank capital or reserves in this analysis – such
considerations are deferred until Chapter 7. A bank needs to take deposits 𝐿𝐿 per loan made
and depositors must be paid an interest rate 𝑟𝑟, the same as the interest rate on government
bonds. We suppose there is a competitive market for loans with free entry of lenders.
For each loan 𝐿𝐿, lenders must repay (1 + 𝑟𝑟)𝐿𝐿 to depositors. Since both good and bad types
obtain loans, with many borrowers, lenders can predict with confidence that a fraction 𝑡𝑡 will
default, repaying nothing. The repayment from the fraction 1 − 𝑡𝑡 of good types is (1 + 𝑟𝑟𝑙𝑙 )𝐿𝐿,
which implies that lender profits per loan are:
𝜋𝜋 = (1 − 𝑡𝑡)(1 + 𝑟𝑟𝑙𝑙 )𝐿𝐿 − (1 + 𝑟𝑟)𝐿𝐿
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Competition between lenders with free entry pushes down lending rates 𝑟𝑟𝑙𝑙 until profits are
zero. The equation for profits can be written as 𝜋𝜋 = �(1 − 𝑡𝑡)(1 + 𝑟𝑟𝑙𝑙 ) − (1 + 𝑟𝑟)�𝐿𝐿, so 𝜋𝜋 = 0
implies that (1 − 𝑡𝑡)(1 + 𝑟𝑟𝑙𝑙 ) = 1 + 𝑟𝑟. Solving this equation for the equilibrium loan interest
rate 𝑟𝑟𝑙𝑙 implies for any 𝐿𝐿:
𝑟𝑟𝑙𝑙 = 𝑟𝑟 +
𝑡𝑡(1 + 𝑟𝑟)
1 − 𝑡𝑡
This equation implies 𝑟𝑟𝑙𝑙 > 𝑟𝑟 because 𝑡𝑡 > 0. Asymmetric information therefore provides a
rationale for a positive interest-rate spread between borrowers’ and savers’ interest rates.
Intuitively, to pay savers an interest rate 𝑟𝑟, all borrowers, including the good types can only
receive loans with a higher interest rate than 𝑟𝑟 to compensate for the bad types defaulting.
The interest-rate spread 𝑟𝑟𝑙𝑙 − 𝑟𝑟 = 𝑡𝑡(1 + 𝑟𝑟)⁄(1 − 𝑡𝑡) increases with the fraction 𝑡𝑡 of bad
types.
If information were symmetric, good and bad types would face different offers from
lenders. We can think of there being two separate markets with different interest rates 𝑟𝑟𝑙𝑙 in
which 𝑡𝑡 is known to be either 0 (only good types) or 1 (only bad types). With 𝑡𝑡 = 0, good
types would be offered 𝑟𝑟𝑙𝑙 = 𝑟𝑟, the same interest rate as savers. This corresponds to the
implicit assumption of perfect financial markets in Chapter 3. With 𝑡𝑡 = 1, bad types would
face 𝑟𝑟𝑙𝑙 = ∞, meaning that financial intermediaries refuse to lend to them.
Box 4.4: Does the current profitability of firms matter for
investment?
Without credit-market imperfections, a firm’s investment demand is determined
by:
𝑀𝑀𝑃𝑃𝐾𝐾′ − 𝑑𝑑 = 𝑟𝑟
All that matters is the cost 𝑟𝑟 (or opportunity cost) of funding investment and the
future marginal product of capital 𝑀𝑀𝑃𝑃𝐾𝐾′ , which is related to a firm’s future profits.
However, empirical evidence suggests that firms’ current profitability has a positive
effect on investment decisions. We can understand this observation by introducing
credit-market imperfections into our analysis of investment, as we have done for
consumption.
Suppose that because of asymmetric information, firms can only borrow at interest
rate 𝑟𝑟𝑙𝑙 > 𝑟𝑟 to fund investment. Think of the ‘good types’ in the asymmetric
information model as firms with profitable investment opportunities who
undertake investment and will repay loans. The ‘bad types’ are unprofitable firms
that are borrowing to keep operating, in that these firms make no investments and
loans are simply used to pay managers and workers until the firms ultimately
default.
For good types, if they have sufficient current profits to fund investment from
retained earnings, the opportunity cost of investment is 𝑟𝑟, the cost of
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internal funds. But if they do not have sufficient profits, the cost of borrowing for
investment is 𝑟𝑟𝑙𝑙 > 𝑟𝑟, the cost of external funds. For firms with sufficient internal
funds, they prefer to finance investment at the lower cost 𝑟𝑟 < 𝑟𝑟𝑙𝑙 using internal
funds, in which case their level of investment is determined by the equation
𝑀𝑀𝑃𝑃𝐾𝐾′ − 𝑑𝑑 = 𝑟𝑟 as in the case of perfect financial markets.
If a firm must use external funds then investment is determined by 𝑀𝑀𝑃𝑃𝐾𝐾′ − 𝑑𝑑 = 𝑟𝑟𝑙𝑙
instead. Let 𝑥𝑥 = 𝑟𝑟𝑙𝑙 − 𝑟𝑟 denote the spread between cost of external funds
(borrowing) and internal funds (retained earnings). The equation that determines
investment is equivalent to:
𝑀𝑀𝑃𝑃𝐾𝐾′ − 𝑑𝑑 − 𝑥𝑥 = 𝑟𝑟
For a firm that was initially sufficiently profitable to finance investment from
internal funds, a decline in profits to the point where the firm must borrow to
invest has the effect of pushing up the funding cost of investment from 𝑟𝑟 to 𝑟𝑟𝑙𝑙 . This
is equivalent to going from 𝑥𝑥 = 0 to 𝑥𝑥 > 0 in the equation above (𝑥𝑥 = 0
corresponds to 𝑀𝑀𝑃𝑃𝐾𝐾′ − 𝑑𝑑 = 𝑟𝑟), implying a downward shift of the investment
demand curve. Hence, investment demand is also increasing in current profits for
some firms, all else equal.
Box 4.5: Financial crises
Our analysis of credit-market imperfections helps us understand why economic
activity falls sharply in a financial crisis.
A financial crisis features an increase in defaults on loans among other problems.
With asymmetric information, lenders might reassess the fraction 𝑡𝑡 of ‘bad types’
among borrowers. With higher 𝑡𝑡, the interest-rate spread 𝑥𝑥 = 𝑡𝑡(1 + 𝑟𝑟)/(1 − 𝑡𝑡)
between the interest rate 𝑟𝑟𝑙𝑙 paid by borrowers and the interest rate 𝑟𝑟 received by
savers rises. Hence, given 𝑟𝑟, borrowers face a higher loan interest rate 𝑟𝑟𝑙𝑙 = 𝑟𝑟 + 𝑥𝑥.
Firms that lack sufficient current profits (internal funds) and need to borrow have
their level of investment demand determined by 𝑀𝑀𝑃𝑃𝐾𝐾′ − 𝑑𝑑 − 𝑥𝑥 = 𝑟𝑟. A higher
interest-rate spread 𝑥𝑥 shifts investment demand curve downwards as shown in
Figure 4.16. Even with no change in the safe interest rate 𝑟𝑟 received by savers,
investment will be lower in a crisis.
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Figure 4.16: Effect of higher interest-rate spread on firms depending on external
funds
Some empirical support for the prediction of a negative relationship between the
interest-rate spread 𝑥𝑥 and investment 𝐼𝐼 is presented in Figure 4.17. The graph
shows time series of US investment and the spread between the interest rates on
BAA-rated and AAA-rated corporate bonds. The logic here is that lending to firms
with a AAA rating is less subject to problems of asymmetric information (‘bad types’
being included in the pool of borrowers) than lending to firms with a lower BAA
rating. The negative relationship between the spread and investment is particularly
striking during the 2008 financial crisis.
Figure 4.17: Corporate bond spreads and investment (USA)
To the extent that corporate profits decline in a financial crisis, there is also an
additional amplification effect. With lower profits, fewer firms can rely on internal
funds to finance investment, which means more firms have investment determined
by 𝑀𝑀𝑃𝑃𝐾𝐾′ − 𝑑𝑑 − 𝑥𝑥 = 𝑟𝑟 rather than 𝑀𝑀𝑃𝑃𝐾𝐾′ − 𝑑𝑑 = 𝑟𝑟, further shifting the aggregate
investment demand curve to the left.
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Increases in interest-rate spreads also affect households by steepening the budget
constraint in the region to the right of the endowment point where a household is a
borrower. This is shown in Figure 4.18. Those who were borrowing initially reduce
their borrowing and current consumption because income and substitution effects
of a higher interest rate 𝑟𝑟𝑙𝑙 are reinforcing for borrowers.
Figure 4.18: Effect of higher interest-rate spread on borrower household
As we will see, financial crises also have important consequences through the effect
of falling asset prices. This reduces the value of collateral and makes it harder to
access credit.
4.6 Limited commitment
A loan contract is an intertemporal exchange. A borrower receives funds now, agreeing to a
future repayment. But as the benefit to the borrower of this exchange is received up front
and only a cost remains in future, this means the borrower may decide to default on the
loan repayment. For lenders to agree to make a loan in first place, it is therefore important
to provide incentives to borrowers not to default.
The standard lifetime budget constraint assumes a household can borrow up to a maximum
amount (𝑦𝑦 ′ − 𝑡𝑡 ′ )/(1 + 𝑟𝑟). At this point, all future income would be needed to repay debt.
While it is feasible to repay, a household cannot make a binding and credible commitment
always to do so. Moreover, a household has a strong incentive to default if most future
income is simply used for debt service. Lenders cannot easily seize this labour income, or
force people to work to their full potential to repay debts. But if borrowers know lenders
cannot take their labour income, what stops them from defaulting?
We will see this problem is often resolved by the use of collateral, an asset pledged by a
borrower as security for a loan. For example, house is collateral for a mortgage loan and a
car for a car loan. Collateral assets can be more easily seized by lenders than income. Thus,
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the threat of losing collateral on default can provide discipline for borrowers to encourage
them to repay loans. However, the size of the threat depends on what the collateral is
worth. This value will set a limit on the maximum loan sizes lenders are willing to agree to.
We now add a housing asset to the two-period consumption model. Assume a household
initially owns a house. Housing is illiquid, though, and it is not possible to sell a fraction of
the house in the current period (and selling the whole house is inconvenient). We allow for
the house to be sold in future at an anticipated price = 𝑝𝑝′. Suppose the household has some
amount of existing debt 𝐷𝐷 (including principal and interest) owed at the beginning of the
current period, for example, an existing mortgage. The lifetime budget constraint is now:
𝑐𝑐 +
𝑦𝑦 ′ − 𝑡𝑡 ′ + 𝑝𝑝′
𝑐𝑐 ′
= 𝑦𝑦 − 𝑡𝑡 − 𝐷𝐷 +
1 + 𝑟𝑟
1 + 𝑟𝑟
This adds the future sale value of the house 𝑝𝑝′ to future income, which is equivalent to
adding the present value of the housing asset 𝑝𝑝′ /(1 + 𝑟𝑟) to lifetime wealth. Existing debt 𝐷𝐷
is subtracted from current income. This assumes that all of this debt is due for repayment or
refinancing in the current period. Long-term debt with interest rate 𝑟𝑟̅ fixed in the past would
enter the budget constraint with (1 + 𝑟𝑟̅ )𝐷𝐷 subtracted from future income 𝑦𝑦 ′ − 𝑡𝑡 ′ instead.
Suppose the household borrows a total amount 𝐿𝐿 (including rolling over or refinancing
existing debt 𝐷𝐷 due for repayment) between the current and future time periods. By
definition, 𝐿𝐿 = 𝐷𝐷 − 𝑠𝑠 = 𝑐𝑐 − (𝑦𝑦 − 𝑡𝑡 − 𝐷𝐷). If the interest rate is 𝑟𝑟 then the future repayment
due is (1 + 𝑟𝑟)𝐿𝐿. The housing asset is collateral for this loan. If the household defaults in the
future, this avoids the debt repayment (1 + 𝑟𝑟)𝐿𝐿, but leads to the loss of the house of value
𝑝𝑝′. Therefore, the household gains from default if 𝑝𝑝′ < (1 + 𝑟𝑟)𝐿𝐿, so to avoid borrowers
defaulting, lenders must ensure that the following collateral constraint holds:
𝑝𝑝′
𝐿𝐿 ≤
1 + 𝑟𝑟
The maximum loan size is thus limited by value of the collateral asset 𝑝𝑝′ /(1 + 𝑟𝑟). This
provides a rationale for a borrowing constraint of the form assumed in Section 4.3, with
current consumption limited by 𝑐𝑐 ≤ 𝑦𝑦 − 𝑡𝑡 − 𝐷𝐷 + 𝑝𝑝′ /(1 + 𝑟𝑟), which truncates the lifetime
budget constraint. Note that as long as the collateral constraint holds, borrowers are not
expected to default, so lenders anticipate being repaid in full. Therefore, there is no
interest-rate spread as seen in Section 4.5 and loans are available savers’ interest rate 𝑟𝑟.
In the absence of any collateral and faced with an inability to seize borrowers’ incomes on
default, the limited commitment model implies there would be no lending in equilibrium, so
the household would face the tightest credit constraint 𝑐𝑐 ≤ 𝑦𝑦 − 𝑡𝑡. The presence of suitable
collateral benefits households by giving access to low-cost borrowing up to some limit. This
helps to smooth consumption as shown in Figure 4.19.
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Figure 4.19: Advantage of having collateral
The major drawback of lending depending on collateral is that fluctuations in the value of
the collateral cause fluctuations in the availability of credit. As Figure 4.20 shows, a fall in
house prices leads to a tightening of the borrowing constraint and a drop in current
consumption if that constraint is binding (there is also a wealth effect on existing
homeowners). This is problematic because of the volatility of asset prices. Figure 4.21 shows
the sizeable fluctuations in US house prices over time, which can contribute to fluctuations
in consumption demand by tightening or loosening borrowing constraints.
Figure 4.20: Effects of a fall in house prices
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Figure 4.21: Fluctuations in house prices
Box 4.6: Interest rates and the value of housing collateral
Our earlier analysis of the effects of interest rates 𝑟𝑟 on consumption in Section 3.5
focused on income and substitution effects and we did not consider credit-market
imperfections there. How do borrowing constraints affect the sensitivity of
consumption to interest rates? Do they magnify or dampen the response of
consumption to changes in interest rates?
With an exogenous borrowing limit 𝐿𝐿, which implies 𝑐𝑐 ≤ 𝑦𝑦 − 𝑡𝑡 + 𝐿𝐿, current
consumption 𝑐𝑐 would not respond to changes in interest rates 𝑟𝑟 unless the
borrowing constraint is not binding or ceases to bind for higher 𝑟𝑟. However, the
limited commitment approach to understanding borrowing constraints suggests
that the maximum loan size 𝐿𝐿 is not exogenous but instead depends on the value of
collateral. Since asset prices are affected by interest rates 𝑟𝑟, hence, the collateral
constraint is too.
Consider mortgage loans secured by value of an illiquid housing asset with
anticipated future value 𝑝𝑝′. The collateral constraint on the maximum loan size is
𝐿𝐿 ≤ 𝑝𝑝′ /(1 + 𝑟𝑟), where the value of the house as collateral is 𝑝𝑝′ /(1 + 𝑟𝑟). Given
expectations of 𝑝𝑝′, a higher interest rate 𝑟𝑟 reduces the collateral value of house.
This is because interest payments rise relative to repayment of capital, and the sum
of principal and interest owed is what matters for a borrower’s incentive to default.
More generally, asset prices fall as interest rates rise because present values of
future payoffs are smaller.
The left panel of Figure 4.22 shows the effects of higher 𝑟𝑟 with a binding collateral
constraint. The higher interest rate means the collateral constraint cuts off the
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budget constraint at a point further to the left, which unambiguously reduces
current consumption 𝑐𝑐. If the collateral constraint remains binding, future
consumption 𝑐𝑐′ does not change.
Figure 4.22: Effect of higher interest rate with and without a collateral constraint
The right panel of the figure shows the case without a collateral constraint (or
equivalently, where it is not binding). We know from Section 3.5 that a higher
interest rate also unambiguously reduces current 𝑐𝑐 because income and
substitution effects are reinforcing. For future consumption 𝑐𝑐′, the two effects are
opposing, and 𝑐𝑐′ remains unchanged in the special case where income and
substitution effects are of exactly the same size. A dominant income effect and
weak substitution effect imply that 𝑐𝑐′ falls, with current 𝑐𝑐 declining by less than
when the two effects were balanced. A dominant income effect means that the
concern for consumption smoothing is more important than responding to
incentives for intertemporal substitution provided by the interest rate.
Therefore, comparing the left and right panels of the figure, in the case where
income effects are dominant and substitution effects weak, a binding collateral
constraint is likely to magnify the effect of 𝑟𝑟 on 𝑐𝑐. This is because the binding
collateral constraint blocks the attempt to smooth consumption that the household
prefers when income effects are strong.
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4.7 Overlapping generations
In Chapter 3, our general equilibrium analysis in the dynamic macroeconomic model was
restricted to an economy with a representative household. We would also like to think
about economies with differences among the population. One alternative approach focuses
on differences between different generations of households, referred to as ‘overlapping
generations’ (OLG).
Using an OLG approach to macroeconomics is useful to study policies where there is an
aspect of intergenerational redistribution, for example, a public pension system. Members
of different generations treated as distinct households, rather than implicitly one large
family. The government, however, spans the generations.
We will study a simple two-period lives OLG model of the economy. Each generation lives
for two periods, ‘youth’ and ‘old age’, and we can analyse the consumption choices of any
particular generation using our existing two-period model.
A crucial feature of the OLG model is that generations overlap. One generation is old at the
same time as subsequent generation is young. At a point in time, suppose the number of old
people in the population is 𝑁𝑁 and the number of young people is 𝑁𝑁′ (this notation is chosen
because the young of today 𝑁𝑁′ will be the old of the future). We allow for demographic
change by allowing the sizes of the generations to differ. Assume that 𝑁𝑁 ′ = (1 + 𝑛𝑛)𝑁𝑁,
where 𝑛𝑛 is the population growth rate between the generations.
We take incomes here as exogenous unlike the full dynamic macroeconomic model from
Chapter 3. The young and old each earn non-financial incomes 𝑦𝑦 and 𝑦𝑦 ′ that remain the
same between generations. It follows that GDP is 𝑌𝑌 = 𝑁𝑁 ′ 𝑦𝑦 + 𝑁𝑁𝑁𝑁′, which grows at rate 𝑛𝑛
from one generation to the next. Note that growth in total GDP here comes entirely from
population growth: income per person remains the same over time. If we think of the young
as ‘workers’ and the old as ‘retirees’ then we can assume 𝑦𝑦′ is zero, or at least very small
relative to 𝑦𝑦.
The main application of the OLG framework is to questions of fiscal policy. Nonetheless, as a
simplification, assume no public expenditure (𝐺𝐺 = 0). There can still be taxes and transfers,
and government debt, even without public expenditure.
If income 𝑦𝑦 ′ when old is low relative to income 𝑦𝑦 when young then households have an
incentive to save to smooth consumption as we have seen earlier. The economy is assumed
to have a bond market but no other types of financial markets. However, it turns out that
there is no scope for different generations to trade with each other. The young would like to
save income and lend to others but the only other generation alive at the same time is the
old. The old cannot borrow from the current young because they are not alive in the future
to repay. As generations only overlap once, there is no mutually beneficial trade in the bond
market.
The two-period lives OLG model has the simple feature that the young can only save by
holding bonds issued by the government. Suppose there is a quantity 𝑏𝑏 of government
bonds issued per young person. These bonds will mature when the current young are old. In
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equilibrium, the real interest rate 𝑟𝑟 on bonds adjusts so that saving of young 𝑠𝑠 = 𝑦𝑦 − 𝑡𝑡 − 𝑐𝑐
is equal to 𝑏𝑏.
To see how this works, start from the even simpler case of no government intervention.
There are no taxes and transfers (𝑡𝑡 = 0, 𝑡𝑡 ′ = 0) and no debt is issued (𝑏𝑏 = 0). In
equilibrium, saving must be zero, 𝑠𝑠 = 0, which implies the young must consume their
income, 𝑐𝑐 = 𝑦𝑦. As they do not acquire any assets because there are no government bonds,
the same is true when old, 𝑐𝑐 ′ = 𝑦𝑦′. In this no-intervention economy, the real interest rate 𝑟𝑟
adjusts so that 𝑐𝑐 = 𝑦𝑦 and 𝑐𝑐 ′ = 𝑦𝑦′. Diagrammatically, the gradient of the lifetime budget
constraint 𝑐𝑐 + 𝑐𝑐 ′ /(1 + 𝑟𝑟) = 𝑦𝑦 + 𝑦𝑦 ′ /(1 + 𝑟𝑟) adjusts until 𝑐𝑐 = 𝑦𝑦 and 𝑐𝑐 ′ = 𝑦𝑦′ is chosen by the
young generation of households as shown in Figure 4.23. The resulting real interest rate is
given by 1 + 𝑟𝑟 = 𝑀𝑀𝑀𝑀𝑆𝑆𝑐𝑐,𝑐𝑐 ′ at the endowment point.
Figure 4.23: No trade between the generations
Is the market equilibrium of this economy Pareto efficient? In other words, would
government intervention making one person better off require making someone else worse
off? There is none of the usual externalities or market failures that would suggest the
market equilibrium is suboptimal. However, as we will see, it may be possible for
government intervention to make all generations better off.
Suppose there is a social planner who can directly choose consumption for each generation
subject to the total supply of goods available. Can this social planner do better than the
market? Assume the social planner picks the same consumption plan (𝑐𝑐, 𝑐𝑐 ′ ) for youth and
old age of each generation. Given the numbers of young and old people, this plan entails
aggregate consumption 𝐶𝐶 = 𝑁𝑁 ′ 𝑐𝑐 + 𝑁𝑁𝑁𝑁′. The plan is feasible if 𝐶𝐶 = 𝑌𝑌:
𝑁𝑁 ′ 𝑐𝑐 + 𝑁𝑁𝑐𝑐 ′ = 𝑁𝑁 ′ 𝑦𝑦 + 𝑁𝑁𝑁𝑁′
Dividing both sides by 𝑁𝑁′ and using the demographic equation 𝑁𝑁⁄𝑁𝑁′ = 1⁄(1 + 𝑛𝑛):
𝑐𝑐 +
𝑦𝑦′
𝑐𝑐′
= 𝑦𝑦 +
1 + 𝑛𝑛
1 + 𝑛𝑛
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In the diagram with consumption plans (𝑐𝑐, 𝑐𝑐 ′ ), this is a straight line passing through the
endowment point (𝑦𝑦, 𝑦𝑦 ′ ) with gradient −(1 + 𝑛𝑛) as plotted in Figure 4.24. This line
resembles a budget constraint, except that the population growth rate, which is also the
economy’s GDP growth rate, determines the gradient of the constraint instead of an
interest rate.
Figure 4.24: Feasible consumption plans for social planner
When the no-government-intervention market equilibrium has 𝑟𝑟 < 𝑛𝑛, the social planner can
choose a combination of 𝑐𝑐 and 𝑐𝑐′ where 𝑐𝑐′ is higher than the market equilibrium and a
higher indifference curve is reached as illustrated in the figure. As 𝑐𝑐′ is higher, moving to this
plan would benefit the generation of old alive when it is implemented because they only
care about 𝑐𝑐′. But as a higher indifference curve is reached, all current and future
generations of young are also better off. This constitutes a Pareto improvement on the
market equilibrium.
In light of this analysis, we say that the economy is dynamically inefficient when 𝑟𝑟 < 𝑛𝑛, that
is, the real interest rate is below the population growth rate (which is also the GDP growth
rate here). However, when 𝑟𝑟 > 𝑛𝑛, this Pareto improvement is not available. Moving to a
feasible consumption plan on a higher indifference curve would mean reducing 𝑐𝑐′, which
would hurt the initial generation of old. The economy is said to be dynamically efficient in
the case 𝑟𝑟 > 𝑛𝑛.
Box 4.7: Pay-as-you-go pension systems
Many governments have established what are known as ‘pay-as-you-go’ pension
systems. These are public pensions that are paid not from an accumulated fund of
assets but from the current contributions or taxes of younger people. They are an
important example of a government policy that redistributes between different
generations. Here, we look at why governments use this type of pension system
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and whether it is a defect that the system is not fully funded. Unlike a typical
private pension, the system is not backed by any assets.
Suppose a government sets up a pay-as-you-go pension system. The young are
obliged to make contributions 𝑡𝑡 (a tax). The old are paid a pension 𝑝𝑝 (a transfer,
𝑡𝑡 ′ = −𝑝𝑝). The system has no assets. The payments to the old come from the
contributions collected at the same time from the young. Assuming the system
balances contributions from the 𝑁𝑁′ young and payments to the 𝑁𝑁 old, the system’s
budget constraint is:
𝑁𝑁𝑁𝑁 = 𝑁𝑁 ′ 𝑡𝑡
Dividing both sides by 𝑁𝑁′ and using the demographic equation 𝑁𝑁⁄𝑁𝑁′ = 1⁄(1 + 𝑛𝑛),
the contributions 𝑡𝑡 and pensions 𝑝𝑝 must be related as follows:
𝑡𝑡 =
𝑝𝑝
1 + 𝑛𝑛
The first generation of the old at the time the system is established necessarily gain
from it because they receive the benefit 𝑝𝑝 without having to pay the contributions
𝑡𝑡. Their consumption increases as they spend their additional pension income.
Given a real interest rate 𝑟𝑟, the lifetime budget constraint of current and
subsequent generations of the young is:
𝑐𝑐 +
𝑐𝑐′
𝑦𝑦 ′ + 𝑝𝑝
= 𝑦𝑦 − 𝑡𝑡 +
1 + 𝑟𝑟
1 + 𝑟𝑟
By substituting 𝑡𝑡 = 𝑝𝑝⁄(1 + 𝑛𝑛) from the pension-system budget constraint:
(𝑛𝑛 − 𝑟𝑟)𝑝𝑝
𝑐𝑐′
𝑝𝑝
𝑦𝑦 ′ + 𝑝𝑝
𝑦𝑦 ′
𝑐𝑐 +
= 𝑦𝑦 −
+
= 𝑦𝑦 +
+
1 + 𝑟𝑟 (1 + 𝑟𝑟)(1 + 𝑛𝑛)
1 + 𝑟𝑟
1 + 𝑛𝑛 1 + 𝑟𝑟
The final term is positive and the budget constraint shifts out if 𝑛𝑛 > 𝑟𝑟 holds. This
case is depicted in Figure 4.25.
Figure 4.25: Effect of introducing a pay-as-you-go pension system
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In equilibrium, the real interest rate 𝑟𝑟 rises as consumption falls when young and
rises when old, which means the gradient of the indifference curve is steeper. This
is why the gradient of the budget constraint increases as well as its position shifting
upwards in the diagram.
When 𝑛𝑛 > 𝑟𝑟, current and subsequent generations of young are better off as a
result of the pay-as-you-go pension system being established, in addition to the
current old. This means everyone gains from the introduction of the system. We
know this is possible because the economy’s equilibrium is dynamically inefficient if
𝑟𝑟 < 𝑛𝑛. Resources can be reallocated to make everyone better off, and the pay-asyou-go pension system is just one way of achieving this.
Intuitively, the market failure and case for government intervention is due to the
difficulty of trade between the generations. There is no pair of generations that can
make a mutually beneficial deal. Observe that the young lose by paying a tax that
goes to the old – they only gain because they in turn get a pension paid for by the
next generation. And that generation only gains because its pension is paid for by
the generation after, and so on. This pattern of trade between the generations is
difficult to arrange privately without government intervention.
If the economy is dynamically efficient (𝑟𝑟 > 𝑛𝑛) then introducing the pay-as-you-go
pension system benefits the currently old generation but makes the young and all
subsequent generations worse off. In the model, 𝑛𝑛 is the GDP growth rate, which
comes only from population growth and 𝑟𝑟 is the real interest rate on bonds. For
everyone to gain from pay-as-you-go, we need real interest rates that are lower
than the economy’s real growth rate. Importantly, this must be true not just
currently but at all future times as well. Pay-as-you-go pension systems were often
established during ‘baby booms’ when GDP growth was high but such favourable
demographics have not been maintained in many countries.
Box 4.8: Should pensions be fully funded?
A pay-as-you-go pension system pays pensions using the current contributions to
the system, not from assets held by the system. An alternative is a fully funded
pensions system. This system pays current pensions using assets acquired by
investing past pension contributions. Such a system could be private, arranged by
households or firms without any government intervention – pensions are then just
a special case of private saving. Or it could be run or regulated by the government,
with individuals being obliged to make pension contributions from their income.
This is an example of ‘forced saving’.
We study fully funded pensions using the two-period model of consumption where
the first period is ‘youth’ and the second period is ‘old age’. Suppose the
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government requires the young to pay an amount 𝑡𝑡 into their pension. These funds
are invested in assets with real return 𝑟𝑟. These could be managed privately or by
the government but we assume that the return 𝑟𝑟 is the same irrespective of who
invests the funds. Assets are worth (1 + 𝑟𝑟)𝑡𝑡 by old age and are used to pay out a
pension 𝑝𝑝 given by the constraint:
𝑝𝑝 = (1 + 𝑟𝑟)𝑡𝑡
This is the budget constraint of the pension system, assuming there are no
administrative costs or intermediary profits taken out. A household’s lifetime
budget constraint is:
𝑐𝑐 +
𝑐𝑐 ′
𝑦𝑦 ′ + 𝑝𝑝
= 𝑦𝑦 − 𝑡𝑡 +
1 + 𝑟𝑟
1 + 𝑟𝑟
By substituting the pension budget constraint 𝑝𝑝 = (1 + 𝑟𝑟)𝑡𝑡:
𝑐𝑐 ′
𝑦𝑦 ′
𝑦𝑦 ′
𝑐𝑐 +
= 𝑦𝑦 − 𝑡𝑡 +
+ 𝑡𝑡 = 𝑦𝑦 +
1 + 𝑟𝑟
1 + 𝑟𝑟
1 + 𝑟𝑟
Notice that the pension 𝑝𝑝 and the contributions 𝑡𝑡 cancel out because they are of
the same present value. The argument here is analogous to Ricardian equivalence.
Absent credit-market imperfections, setting up a fully funded pension system has
no effect on households. This is because without credit-market imperfections, the
lifetime budget constraint is the only constraint faced by households, and we have
seen that a fully funded pension system has no impact on this constraint. As shown
in Figure 4.26, the endowment point would move along an unchanged lifetime
budget constraint and there would be no change in households’ consumption plans
or welfare. For the consumption plan to remain the same, households reduce their
own private saving (or borrow) when forced to make pension contributions. Total
savings including the pension contributions are unaffected.
Figure 4.26: Establishing a fully-funded pension system
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However, households may not find it possible to borrow against future pension
income, which means they may not be able to cut private saving to compensate for
being obliged to contribute to the new pension system. Suppose we impose a
borrowing constraint that limits current consumption to 𝑐𝑐 ≤ 𝑦𝑦 − 𝑡𝑡. Without creditmarket imperfections, a fully funded pension system has no consequences for
welfare. With a binding borrowing constraint, a fully funded pension system
reduces current consumption if individuals are forced to save more than they
would have chosen to. In that case, individuals are made worse off by introducing
the pension system, as shown in Figure 4.26.
The analysis so far has failed to demonstrate a case for government intervention in
the pension system. However, if households were myopic and failed to make
adequate provision for the future then paternalistic government intervention might
improve welfare. It is also possible that when there are transaction costs in making
investments, the large scale of the government lowers costs relative to everyone
investing privately.
A further possible justification for imposing a fully funded pension system is that if
the government did not force people to save, political pressure might compel it to
provide public pensions to old people who have not saved enough. If such public
pensions are financed by taxes levied on the young, this would create a de facto
pay-as-you-go pension system. If 𝑟𝑟 > 𝑛𝑛 then the economy is dynamically efficient
and we know pay-as-you-go pensions make the young and subsequent generations
worse off. There is a need to impose a fully funded pension system in this case to
avoid political pressure to establish an inefficient pay-as-you-go system.
Box 4.9: Declining population growth rates and pay-as-you-go
pensions
Pay-as-you-go pensions were often established at times of high population growth
rates (‘baby booms’), population growth being one reason for growth in total GDP.
But fast population growth has not been sustained in many countries and this
demographic change affects what pay-as-you-go pension payments and
contributions are sustainable.
We know that the balanced budget constrained of a pay-as-you-go pension system
is 𝑡𝑡 = 𝑝𝑝⁄(1 + 𝑛𝑛), where 𝑡𝑡 is contributions made by the young, 𝑝𝑝 is pensions paid to
the old, and 𝑛𝑛 is the population growth rate. After a decline in population growth 𝑛𝑛,
the condition for dynamic efficiency 𝑟𝑟 > 𝑛𝑛 may now hold, undermining the case for
pay-as-you-go pensions. If this happens, should pay-as-you-go pensions be
reformed or abolished?
A pay-as-you-go pension system is one way that a government can affect the
allocation of consumption between young (𝑐𝑐) and old (𝑐𝑐′) generations. Absent
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other government intervention, in equilibrium we know that 𝑐𝑐 = 𝑦𝑦 − 𝑡𝑡 and 𝑐𝑐 ′ =
𝑦𝑦 ′ + 𝑝𝑝. Substituting these equations into the pension system budget constraint 𝑡𝑡 =
𝑝𝑝⁄(1 + 𝑛𝑛) leads to the fundamental resource constraint on feasible allocations of 𝑐𝑐
and 𝑐𝑐′:
𝑐𝑐 +
𝑐𝑐 ′
𝑦𝑦 ′
= 𝑦𝑦 +
1 + 𝑛𝑛
1 + 𝑛𝑛
Lower population growth 𝑛𝑛 reduces gradient of this constraint, pivoting it around
the point (𝑦𝑦, 𝑦𝑦 ′ ) as shown in Figure 4.27. The diagram is set up so that starting from
the population growth rate 𝑛𝑛1 , the economy would be dynamically inefficient in the
absence of a pay-as-you-go pension system (𝑟𝑟 < 𝑛𝑛1 if 𝑐𝑐 = 𝑦𝑦 and 𝑐𝑐 ′ = 𝑦𝑦′). Suppose
pension contributions 𝑡𝑡 have been raised by the maximum amount agreeable to all
generations in the past. This results in 𝑟𝑟 = 𝑛𝑛1 (the economy is just dynamically
efficient) at 𝑐𝑐 = 𝑦𝑦 − 𝑡𝑡1 and 𝑐𝑐 ′ = 𝑦𝑦 ′ + 𝑝𝑝1 . The indifference curve of each new young
generation is tangent to the government’s resource constraint at this point.
Figure 4.27: Lower population growth and pay-as-you-go pensions
The lower population growth rate 𝑛𝑛2 < 𝑛𝑛1 pivots the government’s constraint to
the left around the point (𝑦𝑦, 𝑦𝑦 ′ ). The original pension system 𝑡𝑡1 and 𝑝𝑝1 no longer
satisfies the constraint 𝑡𝑡 = 𝑝𝑝⁄(1 + 𝑛𝑛2 ). This means that either the pension 𝑝𝑝 is cut,
or the contributions 𝑡𝑡 are raised, or some combination of both.
Reducing pensions 𝑝𝑝 makes the current old generation worse off. The current old
clearly prefer contributions 𝑡𝑡 are raised to preserve their pension. However, if 𝑝𝑝 is
not cut then it is now the case that 𝑟𝑟 > 𝑛𝑛2 (the economy started from 𝑟𝑟 = 𝑛𝑛1 ). The
young are worse off if the pension 𝑝𝑝 is maintained at its former level. As the
economy has become dynamically efficient, current and future generations of
young prefer to cut pensions 𝑝𝑝 to some extent. We conclude that, unfortunately,
there is no reform that does not make some generation worse off.
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Note, however, that the young would generally not favour total abolition of the
pay-as-you-go pension system. Reducing contributions 𝑡𝑡 all the way to zero may
imply that 𝑟𝑟 falls below 𝑛𝑛2 , making the economy dynamically inefficient, from
which point a higher 𝑡𝑡 and 𝑝𝑝 would make both young and old better off. Therefore,
the young prefer pensions 𝑝𝑝 are cut until 𝑟𝑟 = 𝑛𝑛2 .
Box 4.10: Bubbles in financial markets
Our earlier analysis of financial markets in Chapter 3 explained asset prices in terms
of fundamentals. For example, share prices in Section 3.9 are equal to the present
discounted value of the dividends paid by firms. This approach to analysing financial
markets suggests there is no scope for ‘bubbles’ to arise.
A bubble occurs where an asset’s value exceeds the present value of the payments
received by owning the asset. This means some or all of asset’s current value comes
from expectations of its future value, not from expectations of the fundamentals
related to the payments it will make. Bubbles were not considered earlier in
Chapter 3 because rational investors would not want to pay more than the
fundamental worth of an asset. However, we will see that bubbles are possible in
our economy with overlapping generations in some circumstances.
Consider the following extreme example. In an overlapping generations economy,
suppose an old generation sets up companies and sells shares to young. If shares of
value 𝑉𝑉 are sold then each of the 𝑁𝑁 ′ young individuals must have paid an amount
𝑝𝑝 = 𝑉𝑉/𝑁𝑁′. Each of the 𝑁𝑁 old individuals receives 𝑉𝑉 ⁄𝑁𝑁 = 𝑁𝑁 ′ 𝑝𝑝⁄𝑁𝑁 = (1 + 𝑛𝑛)𝑝𝑝 using
the demographic equation 𝑁𝑁 ′ = (1 + 𝑛𝑛)𝑁𝑁. Assume these companies undertake no
investment and hence, cannot ever pay any dividends. Their shares have no
fundamental value, so there is a bubble if 𝑉𝑉 and 𝑝𝑝 have a positive equilibrium value.
Why would the young ever want to hold these shares? Since the shares never pay a
dividend, any expected return 𝑟𝑟 depends solely on capital gains:
𝑉𝑉 ′ = (1 + 𝑟𝑟)𝑉𝑉
However, we will see that capital gains sufficient to persuade the young to hold the
shares might be sustainable. Suppose the young generation pays 𝑝𝑝 each for shares.
When old, they can sell to next generation of young and receive (1 + 𝑛𝑛)𝑝𝑝′ each,
where 𝑝𝑝′ = 𝑉𝑉 ′ /𝑁𝑁 ′′ is the anticipated amount to be paid by the next young
generation. The capital gain is ((1 + 𝑛𝑛)𝑝𝑝′ − 𝑝𝑝)⁄𝑝𝑝, which is equal to (𝑉𝑉′⁄𝑉𝑉 ) − 1 and
hence, 𝑟𝑟 using 𝑝𝑝 = 𝑉𝑉/𝑁𝑁′. Any saving done by the young earns 𝑟𝑟, so their lifetime
budget constraint is the usual 𝑐𝑐 + 𝑐𝑐′⁄(1 + 𝑟𝑟) = 𝑦𝑦 + 𝑦𝑦′⁄(1 + 𝑟𝑟).
Here we ignore government debt and any taxes and transfers such as pay-as-you-go
pensions. To have the young paying 𝑝𝑝 each for the bubble asset, the per-person
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saving 𝑠𝑠 = 𝑦𝑦 − 𝑐𝑐 of the young must equal 𝑝𝑝. The level of 𝑠𝑠 must also maximise
lifetime utility given the expected asset return 𝑟𝑟. Conditional on 𝑝𝑝 and 𝑟𝑟, since 𝑟𝑟 is
solely a capital gain, the amount 𝑝𝑝′ paid by the next generation is:
𝑝𝑝′ = �
1 + 𝑟𝑟
� 𝑝𝑝
1 + 𝑛𝑛
These conditions must continue to hold for all subsequent generations as well.
Figure 4.28 shows that in dynamically inefficient economies with 𝑟𝑟 < 𝑛𝑛, a bubble is
possible with the young paying 𝑝𝑝 each for bubble asset. As long as 𝑝𝑝 is not too
large, the increase in 𝑟𝑟 needed to give the young an incentive to hold the asset
leaves 𝑟𝑟 no more than 𝑛𝑛. The bubble asset is sold on to next generation of young,
with each paying 𝑝𝑝′ but 𝑝𝑝′ is no more than 𝑝𝑝 because 𝑟𝑟 ≤ 𝑛𝑛. The bubble asset
remains affordable to future generations of young because the amount paid per
person does not grow. The total bubble asset value is 𝑉𝑉 = (1 + 𝑛𝑛)𝑝𝑝𝑝𝑝, thus 𝑉𝑉′⁄𝑉𝑉 =
(1 + 𝑛𝑛)(𝑝𝑝′ ⁄𝑝𝑝) ≤ 1 + 𝑛𝑛, so the total value grows no faster than the size of the
economy. The largest possible bubble size pushes 𝑟𝑟 up to 𝑛𝑛 exactly, in which case 𝑝𝑝
is constant and 𝑉𝑉 grows at same rate 𝑛𝑛 as the economy.
In the dynamically efficient economy (𝑟𝑟 ≥ 𝑛𝑛) depicted in Figure 4.29, a bubble
would require 𝑟𝑟 to rise above 𝑛𝑛, or even further above 𝑛𝑛. Since 𝑟𝑟 > 𝑛𝑛, the amount
𝑝𝑝′ that must be paid per person for the bubble asset by the next generation of
young is greater than the 𝑝𝑝 that the current generation of young pays. This perperson amount paid needs to keep rising, which is eventually impossible given the
incomes of the young. Such a bubble would ‘pop’ and the anticipation of this
prevents it forming if investors behave rationally. Therefore, no bubbles form in a
dynamically efficient economy.
Figure 4.28: A bubble in a dynamically inefficient economy
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Figure 4.29: No bubbles in dynamically efficient economies
Box 4.11: Does the government have a budget constraint when
interest rates are low?
In the two-period model, the government must adjust taxes or public expenditure
to satisfy its present value budget constraint if it is to avoid a hard default on its
debt. This government budget constraint is crucial to the argument for Ricardian
equivalence because it implies a tax cut now entails higher taxes in the future.
What difference does it make if there is no final period, as with the overlapping
generations economy? Can the government repay existing bonds by issuing new
bonds – rolling over its debts – and keep on doing this? Will investors be willing to
hold government bonds in this case?
Consider the following example to illustrate the argument. Suppose initially that no
government debt is outstanding, and there are no taxes, transfers, or public
expenditure plans and no public pension system is in place. The government then
makes a one-off transfer payment to the current old generation funded by issuing
bonds. However, the government does not ever plan to raise taxes to repay those
bonds. Instead, it simply keeps rolling over its debts as bonds mature in each time
period.
Suppose bonds of value 𝑏𝑏 per young person are issued. Each old person receives a
transfer 𝑁𝑁 ′ 𝑏𝑏⁄𝑁𝑁 = (1 + 𝑛𝑛)𝑏𝑏. Is this fiscal policy feasible? In other words, is there a
bond price (or equivalently, a yield 𝑟𝑟) at which the young, the only possible buyers
of the bonds, are willing to hold them?
Suppose 𝑁𝑁 ′ 𝑏𝑏 bonds are issued at yield 𝑟𝑟. This means that (1 + 𝑟𝑟)𝑁𝑁 ′ 𝑏𝑏 is due for
repayment when these bonds mature. There are no future taxes or transfer
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payments planned. With 𝑁𝑁 ′′ = (1 + 𝑛𝑛)𝑁𝑁 ′ young people in the future period, if new
bonds are issued to repay the existing debt then the following bond issuance 𝑏𝑏′ per
future young person is required:
𝑏𝑏′ =
(1 + 𝑟𝑟)𝑁𝑁 ′ 𝑏𝑏
1 + 𝑟𝑟
=�
� 𝑏𝑏
′′
𝑁𝑁
1 + 𝑛𝑛
Given bond issuance 𝑏𝑏, an equilibrium requires there is an interest rate 𝑟𝑟 at which
the saving 𝑠𝑠 = 𝑦𝑦 − 𝑐𝑐 of the young is equal to 𝑏𝑏. Given this 𝑟𝑟, next period’s bond
issuance is given by the equation above for 𝑏𝑏′ . There must then be a 𝑟𝑟′ such that
𝑠𝑠 ′ = 𝑏𝑏′, and so on for all future generations.
This analysis is mathematically identical to that for ‘bubbles’ in Box 4.10 by simply
replacing the per young person purchases 𝑝𝑝 of the bubble asset with per young
person government bond issuance 𝑏𝑏.
If the economy is dynamically efficient (𝑟𝑟 ≥ 𝑛𝑛), issuance of government bonds
requires 𝑟𝑟 > 𝑛𝑛 but then the amount of bonds outstanding per young person keeps
on growing until incomes are not high enough to keep rolling over the debt. There
would be a hard default and anticipation of this means that bonds not bought by
the young. In this case, it is not possible to run a budget deficit without raising
future taxes.
If the economy is dynamically inefficient (𝑟𝑟 < 𝑛𝑛) then it is possible to run a deficit
and never raise taxes in the future. The usual government budget constraint does
not hold in this case. But this conclusion is subject to the deficit not being so large
as to require 𝑟𝑟 > 𝑛𝑛, so there are ultimately still some constraints on fiscal policy.
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Chapter 5: Unemployment
The basic function of an economy is to allow people to turn their time as work into
production and incomes that allow them to consume goods and services. The occurrence of
unemployment, meaning those who do not have a job but who are actively seeking work,
suggests economies might be failing in this central task.
Our analysis of the labour market back in Chapter 1 had no place for unemployment. There,
the labour market cleared with the real wage adjusting to bring the demand for labour into
line with the supply of labour. Note that unemployment is defined as people who want to
work at the prevailing wage but who have not yet found a job. This is different from the
earlier notion of non-participation in the labour market, which refers to those who do not
want to work at the prevailing wage.
In this chapter, we will explore why unemployment occurs. Furthermore, we would like to
understand why the extent of unemployment varies over time and why unemployment can
differ significantly between countries. By understanding unemployment better, we will try
to answer the question of whether unemployment is a failure of the economy that can be
corrected with an appropriate policy response.
Essential reading
•
Williamson, Chapters 7 and 8.
5.1 Introduction to unemployment
Figure 5.1 below shows a time series of the unemployment rate in the USA. We see that the
unemployment has been around 5 per cent on average, although rising close to or above 10
per cent in severe recessions. As well as varying over time, average unemployment rates
differ considerably across countries as can be seen in Figure 5.2. Figure 5.3 plots cyclical
fluctuations in US unemployment and compares them to cyclical fluctuations in real GDP.
The unemployment rate is strongly countercyclical.
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Figure 5.1: The unemployment rate in the USA
Figure 5.2: Unemployment rates for various countries
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Figure 5.3: The unemployment rate over the business cycle in the USA
How can we understand the occurrence of unemployment? In our earlier supply and
demand analysis of the labour market, there was no unemployment because the real wage
𝑤𝑤 adjusts to a point where labour demand is equal to labour supply. This is depicted in
Figure 5.4, with market clearing at 𝑤𝑤 ∗∗ . At this point, not everyone need be participating the
labour market and working the maximum amount but there is no unemployment in the
sense of someone wanting to work more at wage 𝑤𝑤 ∗∗ but not being able to find extra
employment.
One approach to thinking about unemployment is to suppose there is some impediment to
wage adjustment. Assume that wages are ‘sticky’ downwards, so the labour market cannot
clear. With the real wage stuck at some level 𝑤𝑤 ∗ above 𝑤𝑤 ∗∗ , desired labour supply exceeds
desired labour demand, and unemployment 𝑈𝑈 ∗ is the gap between the two.
Figure 5.4: Sticky wages and unemployment
If wages are sticky, there can be a shortage of demand for labour and, hence, some who
want to work cannot find jobs. But in this case, why do wages not fall to clear the labour
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market in a way that a price would in any other market if there were excess supply? We
could identify wage contracts as creating some rigidity in wage adjustment, along the lines
of the nominal rigidities we will consider in Chapter 8. However, contracts can eventually be
renegotiated, so this in itself does not seem a promising explanation for why there is
positive unemployment even in the long run. We could also think of regulations such as
minimum wages as creating wage rigidity. However, unemployment exists even in countries
with no minimum wages, so this cannot be an important part of the story.
5.2 Efficiency wages
We will now analyse a possible reason why wages are sticky, namely, that firms would not
want to cut wages even if they could recruit staff at lower wages than they are currently
paying. This incentive for firms to pay high wages, even when there is unemployment, is
known as the theory of ‘efficiency wages’.
A crucial idea in understanding efficiency wages is that labour market does not operate like
a standard goods market where products with a known specification and quality can be
purchased. Hiring workers is not like this. Firms do not know in advance the exact abilities of
the workers they hire, nor how much effort they will exert on the job. In our earlier analysis
of the labour market, firms could hire 𝑁𝑁 workers with known skills and effort levels at wage
𝑤𝑤. Now, effective labour input 𝐸𝐸 depends on the wage paid because it affects incentives to
apply for jobs and put in effort on the job.
Assume each worker contributes effective labour input 𝑒𝑒(𝑤𝑤), which is a function of the real
wage 𝑤𝑤. The dependence of effective labour input on the wage paid creates an incentive for
firms to pay ‘efficiency wages’. Total effective labour input from 𝑁𝑁 workers is 𝐸𝐸 = 𝑒𝑒(𝑤𝑤)𝑁𝑁,
and the production function 𝑌𝑌 = 𝐹𝐹(𝐸𝐸) depends on effective labour input 𝐸𝐸 rather than
simply the number of workers employed. The production function 𝐹𝐹(𝐸𝐸) is increasing and
concave. The function 𝑒𝑒(𝑤𝑤) for effective labour input is assumed to be increasing in 𝑤𝑤.
One justification for this is based on a moral hazard problem. The firm needs to pay high
wages to discourage workers from shirking (being lazy) on the job. The argument is that
firms cannot directly control worker effort, although some imperfect monitoring of workers
is possible. Workers do not like putting in effort but fear being caught shirking and being
dismissed. For workers, the cost of losing a job depends on how high the current wage is
compared to wages in other jobs that a dismissed worker could get. Hence, a higher wage
provides a greater incentive not to shirk.
The implicit assumptions in this argument are that dismissal is the worst punishment
available if a worker caught shirking. Furthermore, workers cannot commit to put in effort
when hired irrespective of whether they have an incentive to honour their promise. Finally,
workers have no reputational concerns, which might arise from needing a reference from
their current employer to obtain another job. While some of these assumptions may not
hold, particularly the latter, in practice, this would only weaken the moral hazard problem,
not eliminate it completely.
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Another argument for having effective labour input as a function of wages is based on an
asymmetric information problem. Job applicants know more about their skills and abilities,
hence, potential earnings, than firms looking to recruit them. This means that good workers
will not want to apply for low-paying jobs. Even though firms cannot perfectly identify in
advance who will be a good worker, they know the wage they offer will affect the
composition of the group of applicants. Firms then need to offer high wages to attract
applications from good workers. A related argument makes the case that paying high wages
is important for retaining good workers.
As well as effective labour input 𝑒𝑒(𝑤𝑤) being increasing in wages 𝑤𝑤, it also makes sense to
assume 𝑒𝑒(𝑤𝑤) is a convex function for low 𝑤𝑤, which becomes concave for high 𝑤𝑤. Convexity
for low wages justified by there being some minimum level of wages required for people to
apply for jobs and exert effort on the job, so starting from very low 𝑤𝑤, the gradient of 𝑒𝑒(𝑤𝑤)
is initially increasing. Concavity for high wages justified by there being some maximum level
of effective labour input workers can physically put in, so the gradient of 𝑒𝑒(𝑤𝑤) ultimately
flattens out. An 𝑒𝑒(𝑤𝑤) function with these properties is shown in Figure 5.5.
Figure 5.5: Wages and effective labour input
There are other factors that affect workers’ effective labour input beyond just the wage paid
by their employers. This should also depend on market conditions. Higher wages 𝑤𝑤 ∗ in other
jobs weaken the incentives provided by the firm’s own wage 𝑤𝑤, all else being equal.
Workers would be less inclined to apply for a job paying 𝑤𝑤 or try to keep their current job if
wages 𝑤𝑤 ∗ at other firms are high. On the other hand, a higher unemployment rate 𝑢𝑢
strengthens incentives, making workers more inclined to apply for jobs or to try to keep
their current job. In general, we could write down a function 𝑒𝑒(𝑤𝑤, 𝑤𝑤 ∗ , 𝑢𝑢) that is decreasing
in 𝑤𝑤 ∗ and increasing in 𝑢𝑢, although we will often simplify matters by assuming that the
function depends on 𝑤𝑤 only.
How are wages determined when there is not a competitive market for workers who
provide a fixed and known amount of effective labour input? Firms are able to choose
wages 𝑤𝑤 to maximise profits, taking into account the relationship between effective labour
input and wages. A firm maximises profits 𝜋𝜋 = 𝐹𝐹 (𝐸𝐸 ) − 𝑤𝑤𝑤𝑤 with respect to both
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employment 𝑁𝑁 and wages 𝑤𝑤 subject to 𝐸𝐸 = 𝑒𝑒(𝑤𝑤)𝑁𝑁. Since 𝑁𝑁 = 𝐸𝐸/𝑒𝑒(𝑤𝑤) profits can be
written as:
𝜋𝜋 = 𝐹𝐹 (𝐸𝐸 ) −
𝑤𝑤
𝐸𝐸
𝑒𝑒(𝑤𝑤)
The interpretation of this equation is that given wage 𝑤𝑤, the cost of one unit of effective
labour input 𝐸𝐸 is 𝑤𝑤
� = 𝑤𝑤/𝑒𝑒(𝑤𝑤). Taking this as given for now, by adjusting size of the
workforce 𝑁𝑁, firms demand 𝐸𝐸 up to the point where:
𝜕𝜕𝜕𝜕
= 𝐹𝐹 ′ (𝐸𝐸 ) − 𝑤𝑤
� =0
𝜕𝜕𝜕𝜕
This gives rise to the equivalent of a labour demand curve, 𝐹𝐹 ′ (𝐸𝐸 ) = 𝑤𝑤
�, which says that firms
expand employment to where the marginal product of effective labour input is equal to the
cost per unit of effective labour input. Next, given demand for effective labour 𝐸𝐸, firms
would like to minimise the cost of eliciting this labour input from workers. This is because
profits 𝜋𝜋 = 𝐹𝐹 (𝐸𝐸 ) − 𝑤𝑤
�𝐸𝐸 are decreasing in 𝑤𝑤
�. The profit-maximising choice of 𝑤𝑤 equivalent to
minimising 𝑤𝑤
�, and the first-order condition for minimising 𝑤𝑤
� = 𝑤𝑤/𝑒𝑒(𝑤𝑤) is:
1
𝑤𝑤𝑒𝑒 ′ (𝑤𝑤)
𝜕𝜕𝑤𝑤
�
=
−
=0
𝑒𝑒(𝑤𝑤)2
𝜕𝜕𝜕𝜕 𝑒𝑒(𝑤𝑤)
This is equivalent to 𝑒𝑒 ′ (𝑤𝑤) = 𝑒𝑒(𝑤𝑤)⁄𝑤𝑤, that is, firms should raise wages to where the
marginal effect on 𝑒𝑒(𝑤𝑤) equals the average amount of labour input 𝑒𝑒(𝑤𝑤) per unit of wage
𝑤𝑤. Intuitively, firms have an incentive to set 𝑤𝑤 at the level 𝑤𝑤 ∗ where 𝑒𝑒(𝑤𝑤)/𝑤𝑤 is maximised,
which is equivalent to minimising 𝑤𝑤
�. Geometrically, this corresponds to where a ray from
the origin is tangent to the effective labour input function 𝑒𝑒(𝑤𝑤), as depicted in Figure 5.6.
Figure 5.6: Profit-maximising efficiency wage
The profit-maximising efficiency wage 𝑤𝑤 ∗ is where this tangency occurs, noting that the
initial convexity and subsequent concavity of 𝑒𝑒(𝑤𝑤) imply such a point exists. The inverse of
the gradient of the tangent and ray from the origin gives the effective cost 𝑤𝑤
� of a unit of
effective labour input.
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Since firms demand labour input 𝐸𝐸 up to the point where 𝐹𝐹 ′ (𝐸𝐸 ) = 𝑤𝑤
�, the implied demand
∗
for workers is 𝑁𝑁 = 𝐸𝐸/𝑒𝑒(𝑤𝑤 ). This is equivalent to finding 𝑁𝑁 where 𝑒𝑒(𝑤𝑤 ∗ )𝐹𝐹 ′ (𝑒𝑒(𝑤𝑤 ∗ )𝑁𝑁) = 𝑤𝑤 ∗ ,
which can be plotted as a downward-sloping labour demand curve with 𝑁𝑁 on the horizontal
axis. With wages set at 𝑤𝑤 ∗ , labour demand is 𝑁𝑁 ∗ , as seen in Figure 5.7. The supply of labour
𝑁𝑁 𝑠𝑠 depends on wages in usual way, but firms have no incentive to adjust the wage 𝑤𝑤 ∗ even
if there are unemployed workers when labour demand is 𝑁𝑁 ∗ . It is not in the interests of a
firm to hire at lower wages because the cost 𝑤𝑤
� of effective labour input would actually be
higher.
Figure 5.7: Efficiency wage and unemployment
Is there any reason to expect the efficiency wage 𝑤𝑤 ∗ to be above the market-clearing real
wage 𝑤𝑤 ∗∗? If not, there would be no unemployment and, since all firms would pay the same
wage 𝑤𝑤 ∗ , all else being equal, no worker would have an incentive to exert effort. This is
because workers could obtain the sae pay in another job with no risk of unemployment risk.
Taking the moral-hazard argument for efficiency wages, there needs to be unemployment in
equilibrium to provide incentives. Note that we argued earlier that 𝑒𝑒(𝑤𝑤, 𝑤𝑤 ∗ , 𝑢𝑢) should be
increasing in unemployment 𝑢𝑢.
With an efficiency wage 𝑤𝑤 ∗ above the market-clearing wage 𝑤𝑤 ∗∗ , the theory explains the
persistence of some amount of unemployment. Conditional on the level of the efficiency
wage, unemployment fluctuates with shifts of labour demand. Figure 5.8 shows the effect of
a decline in labour demand. This leads to a drop in employment but does not change wages
and, consequently, unemployment is higher. A change in one of the determinants of firms’
optimal efficiency wage would also have implications for unemployment, with a higher
efficiency wage causing an increase in unemployment, all else being equal.
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Figure 5.8: Decrease in labour demand
Box 5.1: Changes in firms’ ability to monitor workers
The moral-hazard argument for efficiency wages assumes that managers cannot
perfectly monitor workers on the job. But technological innovations may allow better
tracking of workers, which makes workers’ risk of being caught shirking greater. This
allows firms to elicit greater effort 𝑒𝑒 from workers for a given wage 𝑤𝑤, an upward
shift of the effective labour input function 𝑒𝑒(𝑤𝑤). Here we explore the labour-market
implications of this better monitoring technology.
For illustration, consider the following effort function:
𝑒𝑒(𝑤𝑤) = log
𝑤𝑤
𝑎𝑎
In this formula, 𝑎𝑎 is a positive parameter. It specifies the wage that must be exceeded
for workers to be motivated to exert effort on the job, that is, for 𝑒𝑒(𝑤𝑤) to be positive.
For 𝑤𝑤 ≤ 𝑎𝑎, the level of effort is treated as zero. For wages greater than 𝑎𝑎, the
function 𝑒𝑒(𝑤𝑤) is increasing and concave in 𝑤𝑤. Since no effort is exerted for 𝑤𝑤 below
𝑎𝑎, the function also satisfies the initial convexity requirement. A greater ease of
managers in monitoring workers can be represented by a lower value of 𝑎𝑎 because
this implies an upward shift of the effort function 𝑒𝑒(𝑤𝑤) = log 𝑤𝑤 − log 𝑎𝑎.
We know that the profit-maximising efficiency wage 𝑤𝑤 ∗ is the solution of the
equation 𝑒𝑒 ′ (𝑤𝑤 ∗ ) = 𝑒𝑒(𝑤𝑤 ∗ )/𝑤𝑤 ∗ . Geometrically, this is where a ray from the origin is
tangent to the effort function, as depicted in Figure 5.9. For the function considered
here, the marginal effect of the wage on effort is 𝑒𝑒 ′ (𝑤𝑤) = 1/𝑤𝑤. Hence, the efficiency
wage 𝑤𝑤 ∗ is found where 𝑒𝑒(𝑤𝑤 ∗ ) = 1. In this case, it is optimal to elicit a fixed amount
of effort from each worker. Specifically, the equation 𝑒𝑒(𝑤𝑤 ∗ ) = 1 implies 𝑤𝑤 ∗ =
𝑎𝑎 × exp(1). It can be seen immediately that 𝑤𝑤 ∗ and 𝑤𝑤
� = 𝑤𝑤 ∗ /𝑒𝑒(𝑤𝑤 ∗ ) fall with 𝑎𝑎 as the
monitoring technology improves. In the figure, there is a parallel upward shift of the
effort function and the optimal efficiency wage falls from 𝑤𝑤1∗ to 𝑤𝑤2∗ .
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Figure 5.9: Increase in firms' ability to monitor workers
A decline in 𝑤𝑤 ∗ implies a downward shift in the efficiency wage line in the labour-market
diagram in Figure 5.10. Since 𝑒𝑒(𝑤𝑤 ∗ ) = 1 for the particular functional form considered here,
the change in the monitoring technology does not shift the labour demand curve 𝑁𝑁 𝑑𝑑 . Since
the only change is the decline in 𝑤𝑤 ∗ , unemployment declines from 𝑈𝑈1∗ to 𝑈𝑈2∗ .
Figure 5.10: Labour-market implications of better monitoring technology
5.3 Search and matching in the labour market
A labour market that fails to clear through real wage adjustment is not essential to explain
unemployment. The search-and-matching approach to analysing the labour market offers
another way of understanding unemployment. In this approach, the labour market should
not be viewed as a centralised market that coordinates hiring of workers by firms. Instead,
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there are individual workers and individual firms who have to search to form mutually
agreeable matches.
This view of the labour market emphasises heterogeneity among potential workers and
among jobs. Workers have different skills and jobs have different requirements. This means
it is not straightforward to match up jobs with suitable workers.
Search theories of the labour market also draw a distinction between stocks and flows.
Stocks refer to the number of people with particular labour-market statuses, such as those
who currently have jobs, or those who are currently unemployed. Flows refer to transitions
between different labour-market statuses, such as those who find a new job, or those who
lose or leave a job and become unemployed. The analysis of stocks and flows is absent from
the usual supply-and-demand approach to the labour market where it is implicitly assumed
flows are so rapid that only stocks (employment and unemployment) need to be
considered.
However, flows are interesting in their own right. Even with no change in overall
employment or unemployment, there is continual ‘churn’ in the labour market with
individuals making transitions between employment and unemployment, and back again.
Owing to the differences among potential workers and jobs, it takes time for firms to find an
‘acceptable’ worker to fill a particular job and it takes time for someone searching for work
to find an ‘acceptable’ job. In other words, the search process takes time because of limited
information. Furthermore, it is often not in interests of firms to accept the first potential
employee they see, or workers the first job they are offered.
Search theory studies this process of search and matching, the rates at which the flows
occur and the unemployment that results from them. The theory explains why both
unemployment and job vacancies coexist, i.e. unfilled jobs alongside people who want to
find jobs.
5.4 A model of job search
We now look at a model that describes the process of job search. Consider an unemployed
person looking for a job. The key idea is that not all jobs are the same. The person is not
suitable for all jobs, so not all job applications will result in job offers. Moreover, of the jobs
for which the person is acceptable to employers, the job offers are not all equally attractive.
For a particular job, the wage 𝑤𝑤 is a measure of how attractive it is but the ‘wage’ can be
given a broader interpretation and can include other features of the job that affect how
happy a person would be to have that job.
The person searches for jobs and sends applications but there is uncertainty about which
job offers will be received. Assume this uncertainty is represented by a probability
distribution of 𝑤𝑤 with cumulative distribution function 𝐹𝐹(𝑤𝑤). If a job offer is received, the
function 𝐹𝐹(𝑤𝑤) gives the probability that the offer has a wage less than or equal to 𝑤𝑤.
When someone has found and accepted a job, the person receives wage 𝑤𝑤 for as long as the
job lasts. Assume that people leave jobs at an exogenous rate 𝑠𝑠 per unit of time. This implies
that jobs last for 1/𝑠𝑠 units of time on average and creates the need for some ‘churn’ in the
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labour market. The job separation rate 𝑠𝑠 represents dismissals, redundancies and workers
quitting jobs.
The value of currently having a job with wage 𝑤𝑤 is specified by the value function of
employment 𝑉𝑉𝑒𝑒 (𝑤𝑤). Value refers to the present value of all future payoffs starting from
having a job with wage 𝑤𝑤. The future payments in this present discounted sum are
discounted at some rate 𝑟𝑟. The value function 𝑉𝑉𝑒𝑒 (𝑤𝑤) is increasing in the wage 𝑤𝑤 because it
is better to start from a job with a higher wage, all else equal. An example value function is
sketched in Figure 5.11.
Figure 5.11: The value function of employment
While unemployed, a person receives 𝑏𝑏 instead of a wage. This includes unemployment
benefits paid by the government and the value of the time not spent working beyond what
is needed to search for jobs. While unemployed and searching for jobs, a person obtains job
offers at rate 𝑝𝑝 per unit of time, each of which is an independent draw from the probability
distribution of 𝑤𝑤. It takes 1/𝑝𝑝 units of time on average to get a job offer. The difficult and
time-consuming process of searching for vacancies, submitting applications and going
through the recruitment process can be represented by a low value of 𝑝𝑝.
Once a job offer with wage 𝑤𝑤 is received, the person has to decide whether to accept. If it is
accepted, the person becomes employed and receives a payoff with present value 𝑉𝑉𝑒𝑒 (𝑤𝑤).
The person could decide not to accept the offer and remain unemployed. In that case, let 𝑉𝑉𝑢𝑢
denote the present value of all expected future payoffs starting from being unemployed. It
is rational to accept a job offer 𝑤𝑤 if 𝑉𝑉𝑒𝑒 (𝑤𝑤) ≥ 𝑉𝑉𝑢𝑢 . The range of job offers that are acceptable
can be found by comparing 𝑉𝑉𝑒𝑒 (𝑤𝑤) and 𝑉𝑉𝑢𝑢 in Figure 5.12. The value 𝑉𝑉𝑢𝑢 is independent of any
particular wage 𝑤𝑤 now because it is based on the expected value of job offers that could be
obtained in the future by continuing to search.
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Figure 5.12: The reservation wage
The decision to accept or reject a job offer can be stated in terms of a reservation wage 𝑤𝑤 ∗
where 𝑉𝑉𝑒𝑒 (𝑤𝑤 ∗ ) = 𝑉𝑉𝑢𝑢 . In the diagram, the reservation wage is found where the value function
of employment 𝑉𝑉𝑒𝑒 (𝑤𝑤) intersects the value of unemployment 𝑉𝑉𝑢𝑢 . A job offer with wage 𝑤𝑤
should be accepted if 𝑤𝑤 ≥ 𝑤𝑤 ∗ and rejected if 𝑤𝑤 < 𝑤𝑤 ∗ because these ranges of 𝑤𝑤
correspond to where 𝑉𝑉𝑒𝑒 (𝑤𝑤) is respectively above and below 𝑉𝑉𝑢𝑢 .
Intuitively, a reservation wage captures the idea that people do not want to accept just any
job when better ones might be found by continuing to search. On the other hand, people do
not want to be so picky that they spend forever searching for a perfect job that they might
never find, forgoing the opportunity to earn a wage in an acceptable but not perfect job.
It is important to understand that the value of unemployment 𝑉𝑉𝑢𝑢 includes not only receiving
unemployment benefits 𝑏𝑏 but also the chance of receiving job offers while searching for
jobs, i.e. the value of search. This means 𝑉𝑉𝑢𝑢 > 𝑉𝑉𝑒𝑒 (𝑏𝑏), so the value of unemployment is more
than simply having a job that pays a wage equal to unemployment benefits 𝑏𝑏. It follows that
𝑤𝑤 ∗ > 𝑏𝑏, so the reservation wage lies above the level of unemployment benefits.
We now consider the implications of rational search behaviour for how quickly on average
people make the transition from unemployment to employment. Recall that job offers are
obtained at rate 𝑝𝑝 over time. The probability of a wage offer 𝑤𝑤 being less than the
reservation wage 𝑤𝑤 ∗ is 𝐹𝐹 (𝑤𝑤 ∗ ) because the cumulative distribution function 𝐹𝐹(𝑤𝑤 ∗ ) gives the
probability of 𝑤𝑤 ≤ 𝑤𝑤 ∗ . Therefore, the probability that an offer is accepted is 1 − 𝐹𝐹(𝑤𝑤 ∗ ). If 𝑓𝑓
denotes the average rate at people find jobs per unit of time then:
𝑓𝑓 = 𝑝𝑝(1 − 𝐹𝐹(𝑤𝑤 ∗ ))
The relationship between the job-finding rate 𝑓𝑓 and the reservation wage 𝑤𝑤 ∗ is illustrated in
Figure 5.13. The job-finding rate is a decreasing function of the reservation wage because a
higher reservation wage means people are more picky when searching for jobs and thus are
less likely to receive an acceptable offer. Mathematically, 𝐹𝐹(𝑤𝑤 ∗ ) increases with 𝑤𝑤 ∗ , so 1 −
𝐹𝐹(𝑤𝑤 ∗ ) declines with 𝑤𝑤 ∗ .
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Figure 5.13: The job-finding rate
The expected time taken to find and accept a job is 1/𝑓𝑓. This expected time is longer when
offers are harder to get (low 𝑝𝑝), or when the reservation wage 𝑤𝑤 ∗ is high, meaning that
people are more picky (𝐹𝐹(𝑤𝑤 ∗ ) is low).
5.5 Stocks and flows in the labour market
The model of job search is designed to explain the rate at which people make the transition
from unemployment to employment on average. This describes one of the crucial ‘flows’
that underlies the search-and-matching approach to the labour market. Here, we analyse
the relationship between the stocks and the flows in the labour market and use this
determine equilibrium unemployment.
The two flows we will focus on are the transitions from unemployment to unemployment
and from employment to unemployment. The rate at which the first flow occurs is the jobfinding rate 𝑓𝑓 and the rate at which the second flow occurs is the job-separation rate 𝑠𝑠.
Taking as given these flow rates (using, for example, the model of job search to understand
the job-finding rate 𝑓𝑓), what are the implications for stocks in the labour market? By this we
mean the number of people with jobs, the number unemployed and the unemployment
rate. We will apply a method known as stock-flow accounting to determine equilibrium
stocks given the flow rates 𝑠𝑠 and 𝑓𝑓.
5.51 Stock-flow accounting
At a point in time, suppose the number of people currently unemployed is 𝑈𝑈. Assume the
size of the labour force is 𝐿𝐿, which comprises those who are employed plus the unemployed
who are searching for jobs. Here, we assume the labour force is constant over time. This
ignores any changes in labour-market participation over time. The decision to participate in
the labour market was studied earlier in 0. Given 𝐿𝐿 and 𝑈𝑈, by definition, the number of
people currently in jobs is 𝐿𝐿 − 𝑈𝑈.
Given the job-separation rate 𝑠𝑠, over time there are inflows from employment to
unemployment of 𝑠𝑠(𝐿𝐿 − 𝑈𝑈). This is because inflows from the group of size 𝐿𝐿 − 𝑈𝑈 occur at
rate 𝑠𝑠 per unit of time. Similarly, given the job-finding rate 𝑓𝑓, there are outflows from
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unemployment to employment of 𝑓𝑓𝑓𝑓. Outflows from the group of size 𝑈𝑈 occur at rate 𝑓𝑓
over time.
Between two periods, stock-flow accounting says that the change in the number of people
unemployed is equal to inflows minus outflows. Letting 𝑈𝑈′ denote the number of people
unemployed in the next period, the stock-flow accounting identity is:
𝑈𝑈 ′ − 𝑈𝑈 = 𝑠𝑠 (𝐿𝐿 − 𝑈𝑈) − 𝑓𝑓𝑓𝑓
Generally, we are more interested in the unemployment rate 𝑢𝑢 = 𝑈𝑈/𝐿𝐿 rather than the
number of people unemployed. The stock-flow accounting identity can be transformed in
terms of unemployment rates by dividing both sides by 𝐿𝐿:
𝑈𝑈 ′ − 𝑈𝑈 𝑠𝑠 (𝐿𝐿 − 𝑈𝑈) 𝑓𝑓𝑈𝑈
=
−
𝐿𝐿
𝐿𝐿
𝐿𝐿
Since 𝐿𝐿′ = 𝐿𝐿, the unemployment rate is 𝑢𝑢′ = 𝑈𝑈 ′ /𝐿𝐿 next period, and thus:
𝑢𝑢′ − 𝑢𝑢 = 𝑠𝑠(1 − 𝑢𝑢) − 𝑓𝑓𝑓𝑓
This equation states that the inflow to unemployment increases the unemployment rate by
𝑠𝑠(1 − 𝑢𝑢) and the outflow decreases the unemployment rate by 𝑓𝑓𝑓𝑓. The relationship
between the two flows (relative to the size of the labour force) and the unemployment rate
is depicted in Figure 5.14. The outflow 𝑓𝑓𝑓𝑓 is increasing in the unemployment rate because a
greater number of people will leave unemployment for the same 𝑓𝑓 when there are initially
more people unemployed. The inflow 𝑠𝑠(1 − 𝑢𝑢) is decreasing in 𝑢𝑢 because higher 𝑢𝑢 means
fewer people have jobs, so fewer will leave for a given job-separation rate 𝑠𝑠.
Figure 5.14: Inflows and outflows to and from unemployment
5.52 The equilibrium unemployment rate
When the outflow from unemployment matches the inflow, the unemployment rate
remains constant over time (𝑢𝑢′ = 𝑢𝑢). This is a steady state for the unemployment rate,
which we will denote by 𝑢𝑢∗ . A steady state can be found by writing down an equation for
inflows equal to outflows:
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𝑠𝑠(1 − 𝑢𝑢∗ ) = 𝑓𝑓𝑢𝑢∗
This is a linear equation that can be rearranged to deduce 𝑠𝑠 = (𝑠𝑠 + 𝑓𝑓)𝑢𝑢∗ . Hence, there is a
unique steady state for the unemployment rate:
𝑢𝑢∗ =
𝑠𝑠
1
=
𝑠𝑠 + 𝑓𝑓 1 + 𝑓𝑓⁄𝑠𝑠
In Figure 5.14, this steady state corresponds to where the outflow and inflow lines intersect.
The outflow line passes through the origin and the inflow line intersects the horizontal axis
at 𝑢𝑢 = 1, so the steady state 𝑢𝑢∗ lies between 0 and 1. The inflows line is above outflows if
𝑢𝑢 < 𝑢𝑢∗ and outflows are above inflows if 𝑢𝑢 > 𝑢𝑢∗ . It follows that there is convergence over
time to the steady state 𝑢𝑢∗ . In this sense, we will describe the steady state 𝑢𝑢∗ as the
equilibrium unemployment rate implied by the search model. The time taken to converge to
the steady state is usually short enough (unlike the Solow model) that we can take the
steady-state unemployment rate to be the model’s prediction for unemployment.
To give an example, assume one unit of time is a month. Suppose that it takes unemployed
people two months on average to find and accept a job (1⁄𝑓𝑓 = 2), which means a jobfinding rate of 𝑓𝑓 = 1/2. Suppose that workers remain in jobs for four years (48 months) on
average (1⁄𝑠𝑠 = 48), which means a job-separation rate of 𝑠𝑠 = 1/48. Together, the ratio of
the flow rates is 𝑓𝑓⁄𝑠𝑠 = 24, and this yields the following steady-state unemployment rate:
𝑢𝑢∗ =
1
1
1
=
=
= 0.04 = 4%
1 + 𝑓𝑓⁄𝑠𝑠 1 + 24 25
5.53 Unemployment in the search-and-matching model
In summary, the search-and-matching model has an equilibrium (steady-state)
unemployment rate 𝑢𝑢∗ that depends on the inflow and outflow rates to and from
unemployment. A higher job-separation rate 𝑠𝑠 or a lower job-finding rate 𝑓𝑓 increase
unemployment.
While the job-separation rate 𝑠𝑠 is exogenous here, the job-finding rate 𝑓𝑓 was analysed in
the earlier job-search model, where 𝑓𝑓 = 𝑝𝑝(1 − 𝐹𝐹 (𝑤𝑤 ∗ )). The formula shows that a higher
reservation wage 𝑤𝑤 ∗ increases unemployment. Thus, the model can explain what is
sometimes called ‘voluntary’ unemployment that arises from unemployed people not being
willing to accept the first job they can find. As we have discussed, some degree of ‘pickiness’
is rational given the uncertainties in the search process. The formula for 𝑓𝑓 also implies a
lower rate 𝑝𝑝 of obtaining job offers increases unemployment. The model is also consistent
with some ‘involuntary’ unemployment that arises from frictions in matching potential
workers to jobs.
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Box 5.2: The generosity of the welfare state
Governments can establish labour-market institutions that provide insurance to workers
who lose a job. In our earlier job-search model, we assumed that benefits 𝑏𝑏 are paid while a
person remains unemployed. The size of the benefits 𝑏𝑏 but also duration of time for which
they can be claimed, varies considerably across countries. Here, we will focus on the
implications of differences in the size of 𝑏𝑏.
A higher value of unemployment benefits 𝑏𝑏 increases the value of unemployment 𝑉𝑉𝑢𝑢 relative
to the value function of employment 𝑉𝑉𝑒𝑒 (𝑤𝑤). Ignoring the cost to taxpayers of providing 𝑏𝑏,
which affects the value of wages after tax and ignoring the effect on 𝑉𝑉𝑒𝑒 (𝑤𝑤) that comes from
workers considering the risk of unemployment in the future, the main effect is higher 𝑉𝑉𝑢𝑢
because receiving 𝑏𝑏 is the one of the components of the value 𝑉𝑉𝑢𝑢 .
As shown in Figure 5.15, the upward shift of the 𝑉𝑉𝑢𝑢 line raises the reservation wage 𝑤𝑤 ∗ . This
means people become more picky during their job search. Intuitively, there is less incentive
quickly to take a low-paying job offer. The higher reservation wage lowers the job-finding
rate 𝑓𝑓, which reduces the outflow from unemployment and implies a higher unemployment
rate.
Figure 5.15: Increase in benefits paid to unemployed
Since more generous unemployment insurance raises the equilibrium unemployment rate,
why do governments set up welfare states that provide unemployment insurance? While
the diagram above shows that 𝑉𝑉𝑢𝑢 and the average value of 𝑉𝑉𝑒𝑒 (𝑤𝑤) above 𝑤𝑤 ∗ are higher, this
analysis does not account for cost of the insurance, i.e. the higher taxes on wages needed to
pay the unemployment benefits, which must be set against the gains.
Aside from distributional considerations, the main reason for governments to provide
unemployment insurance is because there is a missing market for people to obtain private
insurance against unemployment risk. Such insurance is usually not available because of the
problem of moral hazard. While this analysis lies outside the domain of our search model,
publicly provided unemployment insurance can be good because it substitutes for the
missing market.
Does the search model offer any guidance on the design of unemployment insurance
beyond its role in substituting for a missing market? At first glance, the model points to a
cost of unemployment insurance in making178
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jobs. The argument is that people would already optimally trade off higher wages against
longer search in the absence of insurance, so the insurance is distorting search decisions by
shifting the cost of delaying taking a job on to the taxpayers who fund the unemployment
benefits.
However, this argument may not be valid when the unemployed have little financial wealth
and there are credit-market imperfections that limit private insurance and the ability to
borrow against future income. In this case, people obtain extremely low utility while
unemployed compared to receiving even a very low wage. This leads them to accept lower
paid jobs that are worse matches compared to what they would with perfect credit markets.
Unemployment insurance might then help produce better quality matches between jobs
and workers by giving people the ability to search more thoroughly.
Box 5.3: Wage dispersion and incentives to search for jobs
In the search-and-matching approach to understanding unemployment, differences
between jobs and differences between potential workers are crucial. This is what explains
why time is needed to find suitable matches and why accepting the first job offer, or hiring
the first applicant, is not generally the best strategy for unemployed people and firms.
One reason why some jobs are more attractive than others is the dispersion of wages across
jobs. We will analyse here how the extent of wage dispersion affects job search behaviour
and the equilibrium unemployment rate. Recall that the value 𝑉𝑉𝑢𝑢 is the present value of all
current and future payoffs received conditional on being initially unemployed. This includes
receiving unemployment benefits 𝑏𝑏 in the current period but also the value of search
through expectations of the future wages received once a job is found.
In the job-search model, those searching for jobs receive an offer with probability 𝑝𝑝. An
offer is a wage 𝑤𝑤 drawn from a probability distribution with cumulative distribution function
𝐹𝐹(𝑤𝑤). It is this probability distribution that reflects the dispersion of wages across jobs. With
𝑤𝑤 ∗ denoting the reservation wage, job offers are accepted if 𝑤𝑤 ≥ 𝑤𝑤 ∗ , which has probability
1 − 𝐹𝐹(𝑤𝑤 ∗ ), and the job-finding rate that results is 𝑓𝑓 = 𝑝𝑝(1 − 𝐹𝐹 (𝑤𝑤 ∗ )). The value of being
unemployed 𝑉𝑉𝑢𝑢 includes the expected present value of 𝑉𝑉𝑒𝑒 (𝑤𝑤) conditional on 𝑤𝑤 ≥ 𝑤𝑤 ∗ with
probability 𝑓𝑓 in the next period and the present value of 𝑉𝑉𝑢𝑢 with probability 1 − 𝑓𝑓.
We now consider a more dispersed distribution of wages. Suppose the average wage that is
offered remains the same but low-wage offers are now further below the average and highwage offers are further above the average. As a simple example, suppose that job-offers
have one of three possible wages:
•
•
•
Low wage 𝑤𝑤𝐿𝐿
Medium wage 𝑤𝑤𝑀𝑀 (the average job-offer wage)
High wage 𝑤𝑤𝐻𝐻 .
There is more dispersion if 𝑤𝑤𝐿𝐿 falls and 𝑤𝑤𝐻𝐻 rises by same amount.
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Suppose the reservation wage is initially 𝑤𝑤𝑀𝑀 (𝑤𝑤 ∗ = 𝑤𝑤𝑀𝑀 ), so low-wage offers are rejected
and medium- and high-wage offers are accepted. Since job offers with 𝑤𝑤 = 𝑤𝑤𝐿𝐿 are not
accepted anyway, a lower value of 𝑤𝑤𝐿𝐿 does not worsen the expected payoff of the
unemployed. But as job offers with 𝑤𝑤 = 𝑤𝑤𝐻𝐻 are accepted, an increase in 𝑤𝑤𝐻𝐻 means 𝑉𝑉𝑒𝑒 (𝑤𝑤𝐻𝐻 )
is higher, so the expected payoff of those searching for jobs increases. Figure 5.16 illustrates
how the expected payoff from job search increases when wages are more dispersed. Note
that by drawing the value function 𝑉𝑉𝑒𝑒 (𝑤𝑤) as a straight line, we are assuming that people do
not inherently dislike uncertainty about what job offers they will receive.
Figure 5.16: More wage dispersion increases the value of search
The increase in the value of searching for jobs implies that 𝑉𝑉𝑢𝑢 rises. As seen earlier, a higher
value of 𝑉𝑉𝑢𝑢 leads to a higher reservation wage 𝑤𝑤 ∗ and that reduces the job-finding rate 𝑓𝑓.
Consequently, greater wage dispersion leads to more unemployment. This example thus
illustrates the important role of differences across jobs in explaining unemployment using
the search-and-matching approach.
5.6 Vacancies and unemployment
So far, we have studied the search problem in the labour market only from the perspective
of workers. That is, taking as given the job vacancies available, we looked at how workers
should search optimally and the unemployment rate that results from the process of search.
However, firms face a similar search problem when trying to fill vacancies, namely, the
challenge of finding a suitable person to fill a position. We need to understand the search
and matching problem from firms’ perspective to analyse job creation and wages.
Studying the search-and-matching problem for firms will also allow us to understand data
on vacancies and the vacancy rate. A vacancy is an unfilled job and the vacancy rate is the
ratio of vacancies to the sum of all filled and unfilled jobs (employment plus vacancies).
Figure 5.17 displays time series of the vacancy rate alongside the unemployment rate in the
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USA from 2000 to mid-2021. The vacancy rate is usually lower than the unemployment rate,
being typically around 2–3 per cent. Nonetheless, significant numbers of vacancies co-exist
with significant numbers of people searching for jobs. Although vacancies and
unemployment co-exist, it is also apparent there is almost always a clear negative
relationship between the two.
Figure 5.17: The vacancy rate and the unemployment rate
The negative relationship between the vacancy and unemployment rates is shown as a
scatterplot in Figure 5.18. Most of the time, the points trace out a downward-sloping curve
known as a ‘Beveridge curve’. However, this relationship displays occasional shifts, for
example, in 2009. An even bigger shift occurs with the COVID crisis in 2020 and, as of
writing, it remains to be seen whether subsequent data points will return to the original
Beveridge curve, or a Beveridge curve in a new position.
Figure 5.18: The Beveridge curve
5.6.1 An equilibrium search model of unemployment
To explain the Beveridge curve and to understand which point on the Beveridge curve the
economy will reach, we will now consider an equilibrium search model of unemployment.
This is called an equilibrium search model because it determines the number of jobs firms
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create and the level of wages, rather than taking these as given as we did earlier in the jobsearch model.
The equilibrium search model has three elements:
1. A ‘matching function’: a representation of the frictions in the search and matching
problem of workers and firms.
2. A wage-bargaining problem: there is no competitive labour market to determine
wages, so we must think of wages as coming from bargaining between workers and
firms.
3. A job-creation decision of firms: given the frictions in hiring and the wages that arise
from bargaining, how many jobs do firms want to create?
5.6.2 The matching function
The first element of the equilibrium search model is the matching function. This summarises
the process by which firms find suitable workers to fill vacancies and unemployed people
find acceptable jobs. Condensing all the details of the search done by individuals into a
single aggregate-level function is analogous to using a production function as an aggregatelevel summary of the economy’s production processes.
Mathematically, the matching function relates the ‘output’ of successful matches 𝑚𝑚 per unit
of time to the ‘inputs’ of unemployed workers 𝑢𝑢 and vacant jobs 𝑣𝑣:
𝑚𝑚 = 𝜇𝜇𝜇𝜇(𝑢𝑢, 𝑣𝑣)
For simplicity, we do not distinguish between the number of unemployed and the
unemployment rate, or the number of vacancies and the vacancy rate. In the expression
above, 𝜇𝜇 is a parameter representing matching efficiency. This is analogous to total factor
productivity in a production function because higher 𝜇𝜇 means a greater output of matches
for the same inputs of unemployed workers and firms’ vacant jobs. There are frictions in the
matching process so new matches do not generally occur so fast as to provide jobs for all
unemployed people instantaneously (𝑚𝑚 < 𝑢𝑢), or fill all vacant positions (𝑚𝑚 < 𝑣𝑣).
The matching function 𝑀𝑀(𝑢𝑢, 𝑣𝑣) is assumed to have the same properties as a neoclassical
production function:
1. Constant returns to scale: 𝑀𝑀(𝜆𝜆𝜆𝜆, 𝜆𝜆𝜆𝜆) = 𝜆𝜆𝜆𝜆(𝑢𝑢, 𝑣𝑣)
2. Positive but diminishing marginal products 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 and 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕
3. Inada conditions on marginal products.
An example is the Cobb-Douglas matching function with parameter 0 < 𝜂𝜂 < 1:
𝑚𝑚 = 𝜇𝜇𝑢𝑢𝜂𝜂 𝑣𝑣1−𝜂𝜂
In the earlier job-search model, the job-finding rate 𝑓𝑓 depends on both the frictions in
obtaining job offers and the ‘pickiness’ of workers in not accepting all offers. Here, to avoid
making the analysis too complicated, the matching function is used to represent both the
difficulty of finding jobs and the fact that some jobs are better than others. A ‘match’ means
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both a person receiving a job offer, so the worker is acceptable to the firm, and the offer
being acceptable to the worker. Therefore, the average job-finding rate 𝑓𝑓 is the number of
matches 𝑚𝑚 divided by number of unemployed 𝑢𝑢:
𝑓𝑓 =
𝑚𝑚 𝜇𝜇𝜇𝜇(𝑢𝑢, 𝑣𝑣)
=
𝑢𝑢
𝑢𝑢
5.6.3 Explaining the Beveridge curve
To explain the empirical regularity of a negative relationship between vacancies and
unemployment, we combine our earlier stock-flow accounting exercise and its implications
for the steady-state unemployment rate with the newly introduced matching function.
Taking as given the job-separation rate 𝑠𝑠 and the job-finding rate 𝑓𝑓, the steady-state
unemployment rate is:
𝑢𝑢 =
𝑠𝑠
𝑠𝑠 + 𝑓𝑓
We do not distinguish between the actual and steady-state unemployment rates 𝑢𝑢 and 𝑢𝑢∗
because we suppose convergence to the steady state is sufficiently rapid that the difference
can be ignored. We assume the job-separation rate 𝑠𝑠 is exogenous here, as we did earlier in
the job-search model. The job-finding rate is 𝑓𝑓 = 𝑚𝑚/𝑢𝑢, with new matches given by the
matching function 𝑚𝑚 = 𝜇𝜇𝜇𝜇(𝑢𝑢, 𝑣𝑣).
Multiplying both sides of the equation by 𝑠𝑠 + 𝑓𝑓 and noting 𝑓𝑓𝑓𝑓 = 𝑚𝑚, it follows that vacancies
𝑣𝑣 and unemployment 𝑢𝑢 satisfy the equation 𝑠𝑠𝑠𝑠 + 𝜇𝜇𝜇𝜇(𝑢𝑢, 𝑣𝑣) = 𝑠𝑠. The left-hand side is
increasing in both 𝑢𝑢 and 𝑣𝑣 and the right-hand side is a fixed parameter, hence, we deduce
that high vacancies 𝑣𝑣 must be associated with low unemployment 𝑢𝑢. A matching function
𝑚𝑚 = 𝜇𝜇𝜇𝜇(𝑢𝑢, 𝑣𝑣) that is increasing in 𝑢𝑢 and 𝑣𝑣 and the labour-market stock-flow accounting are
sufficient to explain the Beveridge curve. This theoretical Beveridge curve relationship is
depicted in Figure 5.19. A change in the matching function 𝑚𝑚 = 𝜇𝜇𝜇𝜇(𝑢𝑢, 𝑣𝑣) or the parameter
𝑠𝑠 will modify this relationship and cause the Beveridge curve to shift.
Figure 5.19: The model-implied Beveridge curve
The equilibrium search model is consistent with the Beveridge curve (BC). It is able to
explain why vacancies and unemployment co-exist, and why there is usually a negative
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relationship between the two. We now go further and analyse which point on the BC the
economy will reach. This is done by introducing the notion of ‘market tightness’, the ratio of
vacancies to unemployment.
5.6.4 Market tightness
While vacancies and unemployment co-exist, it is interesting to study the ratio between the
two. Market tightness 𝜃𝜃 is defined as the ratio of vacancies 𝑣𝑣 to unemployment 𝑢𝑢:
𝜃𝜃 =
𝑣𝑣
𝑢𝑢
The labour market is ‘tight’ when firms have opened many vacancies relative to the number
of unemployed people, in other words, when there are many ‘buyers’ compared to ‘sellers’
of labour.
In the Beveridge curve diagram with 𝑣𝑣 and 𝑢𝑢 on the vertical and horizontal axes, market
tightness 𝜃𝜃 is equal to the gradient of the ray from the origin to the point on the Beveridge
curve the labour market has reached. Using the constant-returns-to-scale property of the
matching function, market tightness determines the job-finding rate 𝑓𝑓:
𝑓𝑓(𝜃𝜃) =
𝑢𝑢 𝑣𝑣
𝜇𝜇𝜇𝜇(𝑢𝑢, 𝑣𝑣)
= 𝜇𝜇𝜇𝜇 � , � = 𝜇𝜇𝜇𝜇(1, 𝜃𝜃)
𝑢𝑢
𝑢𝑢 𝑢𝑢
This follows because scaling the inputs to the matching function by 1/𝑢𝑢 is equivalent to
scaling the output of matches by 1/𝑢𝑢. Since the matching function is increasing in both
inputs, it follows that the job-finding rate 𝑓𝑓 is an increasing function of market tightness 𝜃𝜃.
All else equal, unemployed people will find acceptable jobs faster in a tight labour market.
In the equilibrium search model, market tightness and wages are simultaneously
determined by wage bargaining and firms’ job-creation decisions.
5.7 Wage bargaining
When a firm finds a suitable worker to fill a vacancy, the firm and worker need to agree a
wage. Similarly, workers already matched to jobs can negotiate over wages with their
employers. However, there are multiple wages consistent with the firm being willing to
employ the worker and the worker being willing to do the job. Setting the wage determines
how much each party receives of the gains from a deal between a firm and a worker. Note
that there is no competitive market to pin down wages using the requirement that demand
equals supply. In the search model, not all ‘demands’ or ‘supplies’ of firms and workers are
satisfied.
We will analyse wages by thinking about bargaining between firms and workers. We identify
the surpluses of the two parties, i.e. how much they stand to gain from a deal (the worker
accepting or remaining in a job and the firm employing or continuing to employ the worker)
relative to no deal (the worker looking for another job and the firm looking for another
worker).
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Suppose a wage 𝑤𝑤 is agreed. A person who would remain or become unemployed if there
were no deal gains 𝑤𝑤 − 𝑏𝑏 per period from a deal, where 𝑏𝑏 is the level of unemployment
benefits. A worker thus has a surplus of 𝑤𝑤 − 𝑏𝑏 from employment. Assume each worker
produces goods of value 𝑦𝑦 per period once employed, hence, a firm gains 𝑦𝑦 − 𝑤𝑤 per period
by employing an extra person. Assume that each period of time a job vacancy is unfilled, a
firm must incur a recruitment cost 𝑐𝑐 to try to fill the position. Firms thus stand to gain an
amount 𝑦𝑦 − 𝑤𝑤 + 𝑐𝑐 𝑣𝑣 ⁄𝑢𝑢 per extra worker they hire, so a firm’s surplus is 𝑦𝑦 − 𝑤𝑤 + 𝑐𝑐𝑐𝑐.
Both surpluses must be positive for a deal to be in the interests of both parties. The total
surplus, the sum of the gains to both parties, is:
(𝑤𝑤 − 𝑏𝑏) + (𝑦𝑦 − 𝑤𝑤 + 𝑐𝑐𝑐𝑐) = 𝑦𝑦 − 𝑏𝑏 + 𝑐𝑐𝑐𝑐
Observe that the wage 𝑤𝑤 cancels out from this expression because it is a transfer between
the two parties. If the total surplus is positive then there are wages 𝑤𝑤 where both parties
would gain from a deal. The total surplus is always positive under the weak requirement
that 𝑦𝑦 > 𝑏𝑏, meaning a worker’s output exceeds the value of unemployment benefits. We
assume this in what follows.
When the total surplus is positive, there are many possible wages that give both parties a
positive surplus. The split of the surplus is resolved by assuming a specific form of bargaining
known as Nash bargaining. This has the bargaining powers 𝛾𝛾 and 1 − 𝛾𝛾 of workers and firms
given by an exogenous parameter 𝛾𝛾 between 0 and 1. The bargaining powers are the shares
of the total surplus received by each party.
The wage 𝑤𝑤 that achieves a split of the total surplus giving workers a share 𝛾𝛾 is given by:
𝑤𝑤 − 𝑏𝑏 = 𝛾𝛾 (𝑦𝑦 − 𝑏𝑏 + 𝑐𝑐𝑐𝑐)
The wage resulting from Nash bargaining is thus:
𝑤𝑤 = (1 − 𝛾𝛾)𝑏𝑏 + 𝛾𝛾𝛾𝛾 + 𝛾𝛾𝛾𝛾𝛾𝛾
This equation is referred to as the wage curve (WC). It implies a positive relationship
between market tightness 𝜃𝜃 and the bargained wage 𝑤𝑤. In Figure 5.20, the upward-sloping
wage curve is drawn in a diagram with 𝜃𝜃 on the horizontal axis and 𝑤𝑤 on the vertical axis.
Intuitively, a tighter labour market makes the cost to a firm of finding a substitute or
replacement for a worker more expensive because it will take longer to fill an open vacancy.
This puts workers in a stronger bargaining position.
It can be seen from the wage curve equation that WC shifts upwards if 𝑏𝑏 increases.
Intuitively, higher unemployment benefits raises the outside option of workers when
bargaining. An increase in productivity 𝑦𝑦 shifts WC upwards because this increases the gains
to a firm from striking a deal.
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Figure 5.20: The wage curve
5.8 Job creation
The wage curve shows that bargaining between workers and firms implies a relationship
between wages and labour-market tightness. To determine which point on wage curve is
reached, it is necessary to analyse how many job vacancies firms want to create given the
wages that will arise from bargaining.
After filling a vacancy, a worker subsequently produces 𝑦𝑦 and is paid 𝑤𝑤 and the employment
relationship comes to an end at rate 𝑠𝑠 over time, where 𝑠𝑠 is the job-separation rate. If
future profits are discounted at rate 𝑟𝑟, the expected present value of the profits once a
vacancy is filled is (𝑦𝑦 − 𝑤𝑤)/(𝑟𝑟 + 𝑠𝑠).
This expected gain from filling a vacancy needs to be compared to the expected cost.
Suppose vacancies are filled at rate 𝑞𝑞 over time, which means that the expected time taken
to fill a vacancy is 1/𝑞𝑞. The firm faces a recruitment cost 𝑐𝑐 each period the vacancy remains
unfilled, so the expected cost of filling the vacancy is 𝑐𝑐/𝑞𝑞.
The matching function 𝑚𝑚 = 𝜇𝜇𝜇𝜇(𝑢𝑢, 𝑣𝑣) implies that the vacancy-filling rate 𝑞𝑞 is a function of
market tightness 𝜃𝜃 using the constant-returns-to-scale property:
𝑞𝑞(𝜃𝜃) =
𝑚𝑚 𝜇𝜇𝜇𝜇(𝑢𝑢, 𝑣𝑣)
𝑢𝑢 𝑣𝑣
=
= 𝜇𝜇𝜇𝜇 � , � = 𝜇𝜇𝜇𝜇(𝜃𝜃 −1 , 1)
𝑣𝑣
𝑣𝑣
𝑣𝑣 𝑣𝑣
Since the matching function is increasing in both inputs, it follows that the vacancy-filling
rate 𝑞𝑞 is decreasing in market tightness 𝜃𝜃. Therefore, the expected cost 𝑐𝑐/𝑞𝑞 of filling a
vacancy is higher in a tighter market because it is harder for firms to find suitable workers
quickly.
Firms gain from creating more jobs and thus opening up additional vacancies if the expected
gain (𝑦𝑦 − 𝑤𝑤)/(𝑟𝑟 + 𝑠𝑠) exceeds the expected cost 𝑐𝑐/𝑞𝑞(𝜃𝜃). If firms open up more vacancies
then market tightness 𝜃𝜃 = 𝑣𝑣/𝑢𝑢 increases, which pushes up the expected cost 𝑐𝑐/𝑞𝑞(𝜃𝜃). This
means jobs are created up to the point where:
𝑦𝑦 − 𝑤𝑤
𝑐𝑐
=
𝑟𝑟 + 𝑠𝑠 𝑞𝑞(𝜃𝜃)
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This equation implies a negative relationship between wages 𝑤𝑤 and tightness 𝜃𝜃.
Geometrically, it is represented in Figure 5.21 by a downward-sloping job-creation (JC) curve
drawn on the same axes as the wage curve. Intuitively, lower wages lead to more job
creation and a tighter labour market, all else equal. The JC curve shifts upwards if
productivity 𝑦𝑦 increases, and downwards if recruiting costs 𝑐𝑐 rise.
Figure 5.21: The job-creation curve
The equilibrium search model is completed by putting together the upward-sloping wage
curve (WC) and the downward-sloping job-creation curve (JC) in a diagram with market
tightness 𝜃𝜃 on the horizontal axis and wages 𝑤𝑤 on the vertical axis. This is shown in the left
panel of Figure 5.22. The intersection between WC and JC determines equilibrium wages 𝑤𝑤 ∗
and labour market tightness 𝜃𝜃 ∗ . Once market tightness is known, this determines the
gradient of the ray from the origin to the Beveridge curve and, hence, the equilibrium levels
of vacancies and unemployment in the right panel of the figure.
Figure 5.22: Equilibrium wages and market tightness
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Box 5.4: Mismatch
Economies sometime faces shocks that require large reallocations of labour between
different sectors. Examples of this include the end of the construction boom in the USA after
the 2008 financial crisis and, more recently, with shifting patterns of demand for goods and
services during the COVID pandemic causing changes in the relative demands for hiring
workers in different sectors. Given the existing skills and experience of workers, these shifts
lead to greater mismatch between jobs and the unemployed. In the search-and-matching
model, we can represent the increase in mismatch by a decline in the efficiency parameter 𝜇𝜇
of the matching function 𝑚𝑚 = 𝜇𝜇𝜇𝜇(𝑢𝑢, 𝑣𝑣).
A decline in matching efficiency 𝜇𝜇 shifts the Beveridge curve relationship between
unemployment 𝑢𝑢 and vacancies 𝑣𝑣. The job-finding rate is 𝑓𝑓(𝜃𝜃) = 𝜇𝜇𝜇𝜇(1, 𝜃𝜃) conditional on
market tightness 𝜃𝜃 = 𝑣𝑣 ⁄𝑢𝑢. The unemployment rate is determined by 𝑢𝑢 = 𝑠𝑠⁄(𝑠𝑠 + 𝑓𝑓 (𝜃𝜃)),
and since 𝑓𝑓(𝜃𝜃) = 𝜇𝜇𝜇𝜇(𝑢𝑢, 𝑣𝑣)/𝑢𝑢, this is equivalent to the equation 𝜇𝜇𝜇𝜇(𝑢𝑢, 𝑣𝑣) = 𝑠𝑠(1 − 𝑢𝑢).
Therefore, a lower 𝜇𝜇 implies a higher 𝑢𝑢 for any given level of vacancies 𝑣𝑣, which means the
Beveridge curve shifts to the right as shown in Figure 5.23. Increases in mismatch thus
provide an explanation of the occasional shifts of the empirical Beveridge curve that are
observed.
An increase in mismatch also affects the job-creation curve (JC). Lower matching efficiency 𝜇𝜇
implies a lower vacancy-filling rate 𝑞𝑞 (𝜃𝜃) = 𝜇𝜇𝜇𝜇(𝑢𝑢, 𝑣𝑣)⁄𝑣𝑣 = 𝜇𝜇𝜇𝜇 (𝜃𝜃 −1 , 1) conditional on
market tightness 𝜃𝜃. This causes a downward shift of the JC curve as shown in Figure 5.23.
Intuitively, firms are less willing to create jobs if it is more difficult to recruit workers.
In equilibrium, the diagram shows wages 𝑤𝑤 and market tightness 𝜃𝜃 must fall. Lower 𝜃𝜃,
which reduces the gradient of the ray from the origin to the Beveridge curve, and the shift
of Beveridge curve to the right imply higher unemployment 𝑢𝑢, but have an ambiguous
overall effect on vacancies 𝑣𝑣.
Figure 5.23: Increase in mismatch
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Box 5.5: The bargaining power of workers
The search-and-matching approach to the labour market has wages determined not by the
usual forces of demand and supply but through bargaining between workers and firms. This
means that changes in the bargaining powers of workers and firms have implications for
unemployment and the labour market.
The wage curve (WC), representing the outcome of the bargaining process, is 𝑤𝑤 =
(1 − 𝛾𝛾)𝑏𝑏 + 𝛾𝛾𝛾𝛾 + 𝛾𝛾𝛾𝛾𝛾𝛾. In this equation, the parameter 𝛾𝛾 represents the bargaining power of
workers (0 ≤ 𝛾𝛾 ≤ 1). The interpretation of the parameter is that 𝛾𝛾 and 1 − 𝛾𝛾 are the shares
of the surplus from a successful match received by the worker and the firm respectively. A
reduction in workers’ bargaining power, lower 𝛾𝛾, implies a downward shift and a flattening
of the wage curve as shown in Figure 5.24.
Figure 5.24: Lower bargaining power of workers
The shift down of the wage curve reduces wages 𝑤𝑤 but equilibrium market tightness 𝜃𝜃 rises
moving along the job-creation curve (JC). Higher 𝜃𝜃 increases the gradient of the ray from the
origin to the Beveridge curve (BC), so this results in a movement up the Beveridge curve
with firms creating more vacancies 𝑣𝑣 and the unemployment rate 𝑢𝑢 falling.
As well as changing the average values of 𝑤𝑤, 𝜃𝜃, 𝑣𝑣, and 𝑢𝑢, the flatter wage curve also makes
market tightness 𝜃𝜃 more responsive to shifts of the job-creation curve. Movements along
the Beveridge curve would become larger, so unemployment 𝑢𝑢 becomes more volatile
whenever the job-creation curve shifts. This is because wages are tied more closely to
workers’ fixed outside option 𝑏𝑏, so are effectively ‘stickier’ and less responsive to shocks.
Box 5.6: ‘Furlough’ policies in the COVID pandemic
Following the outbreak of the coronavirus pandemic in 2020, some governments introduced
job-support (‘furlough’) schemes giving firms incentives to retain workers. Such policies
were used in UK and some other European countries. In the absence of a policy of this kind,
it was expected there would have been a huge increase in the unemployment rate, as was
seen in the USA.
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In the search-and-matching model, we could think of the disruption from the
pandemic and the ‘lockdowns’ that were imposed as causing a large temporary rise in
job-separation rate 𝑠𝑠 in the absence of a job-support scheme. One justification for
having such schemes was to avoid the costs of the large-scale rehiring of workers that
would be needed once the pandemic is over. But does the search-and-matching
model support this argument?
Without a job-support scheme in place, suppose the effect of the pandemic is a
temporary shutdown of employment in some sectors of the economy and a large rise
of the unemployment rate 𝑢𝑢. We assume there are no long-run effects: all exogenous
variables return to their former values once the pandemic over. This analysis ignores
any structural changes or mismatch that might also result from the pandemic. After
the pandemic, the job-creation curve and the wage curve are in their original
positions and the economy would go back to the same levels of wages 𝑤𝑤 ∗ and market
tightness 𝜃𝜃 ∗ . This position is depicted in Figure 5.25.
However, even if market tightness returns to the same equilibrium value 𝜃𝜃 ∗ once the
shutdown is over, the unemployment rate 𝑢𝑢 does not immediately go back to the
steady state 𝑢𝑢∗ . Time is required for the many unemployed people to find jobs again
and for firms to hire staff again. For each person out of a job during the transition
back to 𝑢𝑢∗ , an amount of production 𝑦𝑦 is lost, implying a net loss 𝑦𝑦 − 𝑏𝑏 at each point
in time the person is unemployed. For each additional vacancy 𝑣𝑣 unfilled above the
steady state 𝑣𝑣 ∗ at a point in time, extra hiring costs of 𝑐𝑐 are incurred by firms. Hence,
at a point in time where the unemployment rate is 𝑢𝑢 and vacancies are 𝑣𝑣, the loss is
(𝑢𝑢 − 𝑢𝑢∗ ) + 𝑐𝑐(𝑣𝑣 − 𝑣𝑣 ∗ ).
Having a job-support scheme would avoid some or all of these losses, so we can
assess the gains from the scheme by adding up all the losses from its absence. Note
that because 𝑣𝑣 = 𝜃𝜃 ∗ 𝑢𝑢 at all times during the recovery, the loss can be simplified to
(𝑦𝑦 − 𝑏𝑏 + 𝑐𝑐𝜃𝜃 ∗ )(𝑢𝑢 − 𝑢𝑢∗ ) at a point when the unemployment rate is 𝑢𝑢.
The dynamics of unemployment rate are given by 𝑢𝑢′ − 𝑢𝑢 = 𝑠𝑠(1 − 𝑢𝑢) − 𝑓𝑓𝑓𝑓, and the
steady state is the solution of the equation 𝑠𝑠(1 − 𝑢𝑢∗ ) = 𝑓𝑓𝑢𝑢∗ . Hence, the dynamics of
the gap between unemployment 𝑢𝑢 and its steady-state value 𝑢𝑢∗ are:
𝑢𝑢′ − 𝑢𝑢∗ = (1 − (𝑠𝑠 + 𝑓𝑓))(𝑢𝑢 − 𝑢𝑢∗ )
This says that the gap between 𝑢𝑢 and 𝑢𝑢∗ is closed at rate 𝑠𝑠 + 𝑓𝑓 per unit of time,
implying the average duration of the deviation of 𝑢𝑢 from 𝑢𝑢∗ is 1/(𝑠𝑠 + 𝑓𝑓). The
dynamics of the adjustment back to steady state are illustrated in Figure 5.25.
If 𝑢𝑢1 is the unemployment rate at the point in time when the shutdown is over, the
total costs of recovery back to steady state are approximately
(𝑦𝑦 − 𝑏𝑏 + 𝑐𝑐𝜃𝜃 ∗ )(𝑢𝑢1 − 𝑢𝑢∗ )⁄(𝑠𝑠 + 𝑓𝑓). This is what is gained by having the job-support
scheme that avoids the rise in 𝑢𝑢.
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Chapter 6: Money
Up to this point, our study of macroeconomics has focused only on real variables. In ignoring
any reference to money, we have implicitly assumed individuals do not suffer ‘money
illusion’ and that they are able to make decisions solely with reference to real values and
relative prices. But are there reasons why money matters that are missing from our earlier
analysis? In other words, why is money important for the functioning of markets?
Essential reading
•
Williamson, Chapters 12 and 18.
6.1 Why does money matter?
We will explore three reasons why money matters. The first is that money is used as a
means of payment. Trade between individuals and firms in the economy depends on having
an object such as money that serves as a medium of exchange. The second reason money
matters is known as ‘nominal rigidity’. Some market prices are quoted in units of money and
slow to adjust, or are set in contracts that are renegotiated only infrequently. A third is that
changes in the value of money affect the willingness of individuals to hold money. The
reasons that money matters in an economy are closely related to the three functions of
money: medium of exchange; unit of account; and store of value.
Money functions as a medium of exchange because money is accepted as payment for
goods and services. Direct barter exchange of different goods, or of labour for goods, is
inconvenient and difficult.
In addition, money functions as a unit of account. It is convenient to quote prices in terms of
money instead of relative prices among a huge range of different goods. Money is also the
conventional unit of account used to specify wages in employment contracts and
repayments in debt contracts.
Finally, money functions as a store of value. Money is an asset that can be used to transfer
purchasing power over time. However, all assets act as stores of value, so this function is
not particular to money. We will focus on the two special functions of money, medium of
exchange and unit of account, with most of this chapter devoted to money as a medium of
exchange.
6.1.1 Medium of exchange
The medium of exchange function of money arises from the problem of the absence of a
double coincidence of wants. Among two people, there is said to be only a single
coincidence of wants if person A wants a good person B has but person B does not want any
good that person A has. A double coincidence of wants occurs when both person A and
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person B have a good that the other person wants. In a specialised economy producing a
vast range of goods and services, a double coincidence of wants is rare.
For trade to take place in markets, it must be in interests of both parties. Since double
coincidences of wants are hard to find, barter exchange is difficult. Exchange is made easier
if one party is willing to accept money because then trade requires only a single coincidence
of wants.
6.1.2 Unit of account
The unit of account function of money results from there being too many relative prices to
quote directly among all the different goods and factors of production in an economy. It is
convenient to express all prices as amounts of money. Similarly, in contracts written to
govern long-term employment and creditor-debtor relations, it is convenient to specify
payments in terms of a conventional unit of account.
In Chapter 9, we will further argue that prices may sometimes remain fixed even when
conditions in a market or the economy change. This is because it is costly and inconvenient
to update prices continuously. The same applies to contracts that are costly to renegotiate
or to write with many contingencies in advance.
6.1.3 Different objects that serve as money
We have defined money in terms of its functions but what objects can serve as money? The
common forms of money in use today are notes and coins issued by governments or central
banks that do not have intrinsic value, i.e. they are not valued for the material from which
they are made. This type of money is known as ‘fiat’ currency. The other type of money
commonly in use today is deposits at commercial banks. These deposits are claims to fiat
currency, so this a type of ‘credit’ money. Commercial banks themselves hold fiat money as
vault cash or reserves at the central bank, which is another form of fiat money.
There are also some new or experimental forms of money that are yet not in general use
but may become more widespread in the next decade. These include cryptocurrencies,
which are a private and decentralised system of money. Central banks also have plans to set
up central-bank digital currencies, which are a centralised system of accounts where
individuals hold money directly at a central bank.
Historically, money took other forms that are now extremely rare. These include commodity
money, where coins are made from precious metals, or notes issued by the government
were redeemable for precious metals on demand. There were also private bank notes, a
form of credit money, which were claims to commodity money.
6.2 A search-theory perspective on money
To understand the medium of exchange function of money, it helps to step away from our
earlier models that assumed trade in the economy takes place in centralised markets. The
assumption was that everyone can buy or sell in markets subject only to a budget
constraint. The sequence of transactions was irrelevant – all that mattered was that each
person’s overall budget constraint was satisfied.
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However, unless there is a double coincidence of wants for all trade, this implicitly assumed
a very high degree of coordination, or that short-term credit is freely available and works
without any friction. The reason is that a budget constraint allows for purchases in a period
even though people may not have yet received payment for what they plan to sell. An
alternative approach is known as ‘search’ theory. In a search model, all trade is
decentralised and occurs in meetings between pairs of individuals rather than in centralised
markets.
6.2.1 A simple search model of money
We can illustrate the ideas of search theory in a simple model with three types of
individuals. Think of these individuals as having different occupations, so that they specialise
in producing different goods. Moreover, individuals have different needs and tastes, so their
preferences are not the same. We assume no double coincidences of wants to highlight the
usefulness of money. A specific example of three individuals is given in Figure 6.1.
In a search model there are no competitive, centralised markets. All trade must be bilateral,
meaning that it takes place between pairs of individuals. To keep the analysis simple,
assume only one indivisible unit of each of the goods and services can be produced. This
avoids the need to discuss prices at this stage because any trade that takes place must
involve one unit of goods being purchased or sold.
In the absence of any double coincidence of wants, no trade can take place between any
pairs of individuals. However, this is inefficient. If all three individuals could meet and
coordinate a three-way exchange centrally then all three could have their wants met by one
of the others.
Figure 6.1: The absence of a double coincidence of wants
6.2.1 Commodity money
In the example in Figure 6.1, everyone produces a service, which cannot be stored. Once we
allow for physical goods, it is possible that one or more such goods can become a
commodity money. A commodity money has intrinsic value because of what it is made of, so
it would have a value even if it were not used as money. But, crucially, a commodity money
is accepted for payments even by those who do not want to consume the commodity itself.
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Figure 6.2 below modifies the earlier example so that one person can produce a physical
good and one person wants to consume that good. However, there is still no double
coincidence of wants for direct barter exchange. But all trade is possible if everyone accepts
the physical good in exchange for what they produce even if they do not want to consume
it. This enables the physical good to serve as a commodity money.
For a good to serve as a commodity money, it should be:
•
•
•
•
Easily storable at low cost, potentially for long periods of time
Easily transportable
Straightforward to verify the quality of the good
Easily divisible, for when exchange is not one-for-one.
Figure 6.2: Trade with commodity money
In the past, precious metals were a common form of commodity money, which satisfy the
first two of these requirements well. The use of coinage and convertible notes or tokens
added extra convenience, helping to satisfy the third and fourth requirements.
The advantage of a system of commodity money is that the limited supply and intrinsic
value of the commodity should give confidence that money will be a good store of value, or
at least not too bad. The disadvantage is that the system ties up valuable goods as money,
which either cannot be used directly, or there must be extra production of the commodity,
which has a cost.
6.2.2 Credit money
An alternative to a system of commodity money is to use credit money. Credit money is
where privately issued IOUs circulate as money. An IOU is a debt i.e. a promise to deliver a
payment in the future.
The intended purpose of an IOU is a simple credit instrument, where, say, person A offers an
IOU to person B in exchange for something, which is accepted. Person A then later redeems
the IOU, giving person B what is owed.
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But IOUs could in principle become money if the initial holder uses it to make a payment to
someone else and then that person might pass it on to someone else as well. Consequently,
the IOU is held by a third party at redemption.
Figure 6.3 returns to the example with three individuals who produce only services. Since
services must usually be consumed at the point they are produced, this rules out the use of
commodity money. However, if everyone is willing to accept someone’s IOU, that IOU can
circulate as money. Through the use of this credit money, all three individuals are able to
purchase the services they desire.
Figure 6.3: Trade with private IOUs
6.2.3 Money and credit
The example of a private IOU circulating as money shows that there is often a connection
between money and credit. However, it would be a mistake to see money and credit as the
same thing. First, not all types of money are credit. For example, in the example with
commodity money, no-one owes anything to anyone else. As we will see, fiat money is also
not a debt that the government is obliged to repay.
Second, far from all credit ever becomes money. Very few individuals are sufficiently well
known that their IOUs could circulate as money. To use an IOU as credit money, everyone
who would subsequently hold the IOU needs to know and trust the issuer, in addition to the
first person to accept the IOU, which is all that would be required if the IOU were used as a
simple credit instrument. Only in the smallest communities are these requirements likely to
be met for circulation of individuals’ own IOUs.
Considering these difficulties, for credit to serve as money, the IOUs need to be issued by
large, well-known and trusted companies or organisations. In practice, this means banks.
Historically, bank IOUs took the form of their own issue of banknotes, which were promises
to repay deposits of commodity money. In the modern world, bank IOUs typically take the
form of deposits, which are claims to fiat money.
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If banks have created IOUs that are accepted for payments then it is easy to see how these
can be used to facilitate exchange among the three individuals in the earlier example. Thus,
trade can take place using credit money issued by banks even if the individuals in the
economy cannot persuade others to use their own IOUs as money.
A system of credit money has some important advantages. It is efficient system with a low
resource cost because it does not tie up goods with intrinsic value to be used as money.
Even if bank IOUs are claims to commodity money, banks would not need to hold 100 per
cent of deposits as vault cash with intrinsic value. Furthermore, as we will see, banks can
loan deposits to support long-term investment. Some of the return on these investments
can be paid to depositors as interest, making bank deposits a better store of value.
However, credit money also has disadvantages. The biggest of these is that default by banks
on their IOUs may cause a collapse of confidence in the monetary system. Bank runs and
bank failure disrupt trade and trigger financial crises. Defaults by banks may be due to losses
made on their loans, or even caused by a bank run itself. These problems also lead to
pressure for bailouts from the government, creating a problem of moral hazard (‘too big to
fail’).
As we have discussed, individuals’ own IOUs cannot circulate as money. However, it is
possible that some forms of credit can be substitutes for money in making payments, for
example, credit cards. An individual paying with a credit card does not need to hold money
at the time of making a purchase, hence, this payment method acts as a substitute for
money. Essentially, the financial intermediaries that issue credit cards endorse individuals’
IOUs so others can be assured these debts will be repaid. The ability to use credit in this way
reduces frictions in payment because there is less need to hold money. However, such a
system of credit-based payments has costs for financial intermediaries coming from the
need to track credit histories and collect debt repayments.
6.2.4 Fiat money
Another form of money in widespread use is fiat money. This is defined as governmentissued money of no intrinsic value. The term ‘fiat’ suggests this money has value by
government decree but that is misleading because ultimately the real value of fiat money is
determined in markets. Fiat money is not credit money because it is not a claim to anything
other than itself – it is not redeemable as an IOU is. Historically, government-issued notes
may have been claims to commodity money but this is no longer the case. In accounting
terms, fiat money is recorded as a liability of the government or central bank but it is
important to remember that is quite unlike private-sector liabilities such as bonds or loans.
The physical form of fiat money is cash, comprising notes and coins of non-precious metal.
Commercial banks hold some fiat money as vault cash but in modern monetary systems, the
reserves of commercial banks are usually held in accounts at the central bank. In this form,
fiat money is only an entry in a database recording how much each commercial bank has on
‘deposit’ in its reserve account at the central bank. While currently households and firms do
not hold reserves directly, commercial banks can convert reserves and cash one-for-one.
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Figure 6.4 below shows how trade can take place using fiat money in the earlier example
with three individuals. Initially, someone holds a unit of fiat money and everyone accepts it
for payments. The fiat money circulates among the individuals and everyone can consume
the service they desire. Note that the fiat money remains in circulation after all the
exchanges have taken place. This implicitly assumes the fiat money will go on being used for
future trade.
Fiat money shares the advantage of credit money in being an efficient, low-cost system
because the intrinsically worthless money that is used has a negligible resource cost (there
are still some costs of production for the notes and coins, and costs of handling cash for the
private sector). It is important that individuals can easily recognise units of fiat money as
genuine but this can be achieved to a sufficient degree of accuracy with appropriate antiforgery devices.
Figure 6.4: Trade with fiat money
The potential disadvantages of fiat monetary systems also stem from fiat money lacking any
intrinsic value. As currency has a much lower cost for the government to produce it than its
market value and, as there is no obligation to redeem it, there is a temptation to issue more
fiat currency to raise revenue. The abuse of this money-issuing power by governments
results in money being a poor store of value and, at worst, hyperinflation.
Furthermore, because fiat money is not redeemable for anything other than itself, its value
depends on the belief that others will continue to accept it for payments. Note that in Figure
6.4, the fiat money is never withdrawn from circulation. This means that those choosing to
accept money must always believe that others will continue to accept money in the future.
In principle, such beliefs could be subject to self-fulfilling shifts because the belief that
others will not accept fiat money justifies individuals choosing not to accept it.
However, in practice, the concern about self-fulfilling losses of confidence in fiat currency
may be mitigated by the government’s power of taxation. Governments can insist on
payment of taxes in their own currencies, which ensures there is always some demand for
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money. The payment of taxes in fiat money to the government could also be seen as a
mechanism through which fiat money is withdrawn from circulation.
Box 6.1: Cryptocurrencies
Recent years have seen a rise to prominence of cryptocurrencies, most famously
Bitcoin, but now many others too. Cryptocurrencies are a type of privately created
money in an electronic form. The operation of cryptocurrencies is a decentralised
system, unlike the centralised control of fiat money by governments.
Nonetheless, cryptocurrencies share some features of fiat money. They have no
intrinsic value, which makes them very different from commodity money.
Cryptocurrencies are also not IOUs in any sense, which means they are not credit
money.
Proponents of cryptocurrencies have put forward several advantages. First, the
blockchain technology they build on makes transactions very secure. Second,
because the supply of a particular cryptocurrency is limited by design, it is argued
there is less risk of the cryptocurrency losing value due to oversupply – in contrast
to fiat money where governments have discretion to create more – making
cryptocurrencies a better store of value. However, limited supply is necessary but
not sufficient to preserve value, which also depends on a stable or growing demand
for a currency.
Critics of cryptocurrencies argue there are serious disadvantages. First, the value of
a cryptocurrency depends on others’ beliefs about its future value, which risks
significant volatility (in theory, this is also a drawback of fiat money). Values of
cryptocurrencies have indeed been extremely volatile, making them far from a
traditional risk-free asset. Second, as with cash, the anonymity allowed by
cryptocurrencies may facilitate criminal activity, although this anonymity might also
be valuable in fostering civil liberties. Third, there is the cost of the computing
power used to maintain the distributed ledger, implying that cryptocurrencies may
entail significant resource costs, a disadvantage shared with commodity money.
How do cryptocurrencies fit into our analysis of money? In this chapter, we will
usually think of money as an asset that serves as a medium of exchange but which
is less good compared to other assets as a store of value. But cryptocurrencies are
currently little used as a traditional medium of exchange – not many purchases of
goods and services use cryptocurrency. Cryptocurrencies have had high (although
volatile) rates of return, unlike traditional forms of money. In light of these
observations, it may be better to think of people holding cryptocurrencies as a
financial asset rather than as money in the usual sense of the term. With no
dividends and all returns coming from capital gains, one approach to analysing
cryptocurrencies is as ‘bubbles’ in the overlapping generations model.
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6.3 Money and assets as stores of value
Storing value is a function of money but this function is not unique to money. All assets
must act serve as stores of value to some extent and many often do this better than money.
For example, bonds may offer a real return 𝑟𝑟 from interest payments, shares pay dividends,
property earns rents and shares and property may benefit from capital gains.
Considering money, let 𝑖𝑖𝑚𝑚 denote the return on holding money for a period in terms of units
of money itself. For example, if money is interpreted as funds in a bank account, 𝑖𝑖𝑚𝑚 is the
interest rate paid on deposits. If money is cash for which no interest is paid then 𝑖𝑖𝑚𝑚 = 0.
Note that 𝑖𝑖𝑚𝑚 is a nominal interest rate and a nominal return: it is the percentage increase in
the amount of money held simply by holding on it for some amount of time.
The terminology we will use throughout this chapter is that nominal refers to something
measured in units of money, while real refers to something measured in units of goods. If
the nominal return on money is 𝑖𝑖𝑚𝑚 , what is the implied real return? The real return on
money, and nominal assets more broadly, depends on the inflation rate in the economy.
6.3.1 Inflation
Inflation is defined as a general rise in prices quoted in terms of money. Inflation affects
how good or bad money is as a real store of value.
In previous chapters, we have measured real variables in terms of a homogeneous good or
basket of goods. Let 𝑃𝑃 denote the price of this good, or basket of goods, in terms of units of
money. If the price level in the current period is 𝑃𝑃, the notation for the price level in the
next period is 𝑃𝑃′. The rate of inflation between these time periods is denoted by 𝜋𝜋:
𝜋𝜋 =
𝑃𝑃′ − 𝑃𝑃
𝑃𝑃
Note that this definition of inflation refers to the percentage change in prices between the
current level and the level that will prevail in the future. It is also possible to measure
inflation between the past and current periods and where that inflation rate is relevant, the
notation will be adjusted to accommodate it. Note also that the future price level 𝑃𝑃′ and the
resulting inflation rate are not known in current period. Where the distinction between and
expected inflation is important, the notation 𝜋𝜋 𝑒𝑒 will be used to denote expected inflation.
Figure 6.5 shows data on inflation for the USA in the post-war period. Inflation is volatile at
the end of the 1940s but becomes very low and stable in the 1950s. The 1960s see an
increase in the inflation rate, which reaches double digits in the 1970s. Inflation is brought
under control in the 1980s and remains stable throughout the 1990s. This stability continues
into the 2000s except for the years around the 2007–8 financial crisis and its aftermath.
To calculate the real change in spending power when holding money, the value of money,
plus any interest 𝑖𝑖𝑚𝑚 that accrues, is adjusted for changes in the money prices of goods and
services. Take the amount of money 𝑃𝑃 that currently buys one unit of goods. If held simply
as money then this becomes (1 + 𝑖𝑖𝑚𝑚 )𝑃𝑃 units of money in the future period and with a price
level 𝑃𝑃′ , it would be possible to buy (1 + 𝑖𝑖𝑚𝑚 )𝑃𝑃/𝑃𝑃′ units of goods in the future. The
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definition of the real return 𝑟𝑟𝑚𝑚 on money is that holding an amount of money sufficient to
purchase a unit of goods now yields purchasing power over 1 + 𝑟𝑟𝑚𝑚 units of future goods.
The percentage real return on money 𝑟𝑟𝑚𝑚 is therefore calculated from the equation:
1 + 𝑟𝑟𝑚𝑚 =
(1 + 𝑖𝑖𝑚𝑚 )𝑃𝑃 1 + 𝑖𝑖𝑚𝑚 1 + 𝑖𝑖𝑚𝑚
=
=
𝑃𝑃′
𝑃𝑃′⁄𝑃𝑃
1 + 𝜋𝜋
Observe that (1 + 𝑟𝑟𝑚𝑚 )(1 + 𝜋𝜋) = 1 + 𝑖𝑖𝑚𝑚 implies 1 + 𝑟𝑟𝑚𝑚 + 𝜋𝜋 + 𝑟𝑟𝑚𝑚 𝜋𝜋 = 1 + 𝑖𝑖𝑚𝑚 . If 𝑟𝑟𝑚𝑚 𝜋𝜋 is small
compared to 𝑟𝑟𝑚𝑚 and 𝜋𝜋, the real return on money is approximately given by 𝑟𝑟𝑚𝑚 ≈ 𝑖𝑖𝑚𝑚 − 𝜋𝜋. In
the case where no interest is paid on money (𝑖𝑖𝑚𝑚 = 0), for example, when money is
interpreted as cash, then the real return is approximately 𝑟𝑟𝑚𝑚 ≈ −𝜋𝜋. This says that the
inflation rate is approximately the percentage loss of purchasing power of money over time.
Figure 6.5: US inflation
6.3.2 The Fisher equation
The equivalent calculation of the real return on holding nominal bonds is known as the
Fisher equation. A nominal bond is one that specifies payments in terms of units of money.
It is natural for bonds to make payments in this form following on from our earlier
discussion of money’s role as a unit of account.
We consider somewhat more general nominal bonds in Chapter 7 but here suppose that a
nominal bond makes a single payment of interest in terms of money in the next period and
this payment is certain. The interest rate specified by the bond is 𝑖𝑖, the nominal interest rate
and nominal return on holding the bond. This is known when the bond is purchased. The
substantive assumption here is that the bond payment is not indexed to inflation.
The Fisher equation gives the implied real return, referred to as the real interest rate 𝑟𝑟:
1 + 𝑟𝑟 =
(1 + 𝑖𝑖 )𝑃𝑃 1 + 𝑖𝑖
=
1 + 𝜋𝜋
𝑃𝑃′
The justification for this equation is that one unit of goods costs 𝑃𝑃 units of money in the
current period. If this money is used to buy bonds that offer a nominal interest rate 𝑖𝑖, the
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amount of money returned when the bond matures in the next period is (1 + 𝑖𝑖 )𝑃𝑃. Dividing
this by the future price level 𝑃𝑃′ gives the amount of future goods that can be purchased. The
equation then follows by noting that 𝑟𝑟 is defined so that buying nominal bonds worth a unit
of goods today gives the ability to buy 1 + 𝑟𝑟 units of goods in the future.
Rearranging the equation gives 𝑖𝑖 = 𝑟𝑟 + 𝜋𝜋 + 𝑟𝑟𝑟𝑟, so if 𝑟𝑟 and 𝜋𝜋 are small, the term 𝑟𝑟𝑟𝑟 is
negligible compared to the other terms, and it follows that 𝑟𝑟 ≈ 𝑖𝑖 − 𝜋𝜋. This equation, which
can also be written as 𝑖𝑖 ≈ 𝑟𝑟 + 𝜋𝜋, is the approximate version of the Fisher equation. It states
that the real return on bonds is the difference between the nominal interest rate and the
inflation rate. This is an approximation, and in contexts where the inflation rate can be very
high, the exact version 1 + 𝑖𝑖 = (1 + 𝑟𝑟)(1 + 𝜋𝜋) will be used.
6.3.3 Ex-ante and ex-post interest rates
We have seen that inflation affects the real return on nominal assets such as money and
nominal bonds. However, inflation 𝜋𝜋 as defined earlier is the percentage change in the price
level between the current and future time periods, which is therefore not known in
advance. This means the Fisher equation can be used with either actual inflation 𝜋𝜋 or
expected inflation 𝜋𝜋 𝑒𝑒 as appropriate.
Using the expected inflation rate 𝜋𝜋 𝑒𝑒 leads to an expected (ex-ante) real interest rate 𝑟𝑟 𝑒𝑒 ≈
𝑖𝑖 − 𝜋𝜋 𝑒𝑒 , or by giving this equation in its exact form, 1 + 𝑟𝑟 𝑒𝑒 = (1 + 𝑖𝑖)/(1 + 𝜋𝜋 𝑒𝑒 ). Using the
actual inflation leads to the actual (ex-post) real interest rate 𝑟𝑟 ≈ 𝑖𝑖 − 𝜋𝜋, or in its exact form,
1 + 𝑟𝑟 = (1 + 𝑖𝑖)/(1 + 𝜋𝜋).
Different from nominal bonds, inflation-indexed bonds have the same actual and expected
real returns. This type of bond is sometimes referred to as a real bond to distinguish it from
bonds that specify payments in terms of fixed amounts of money.
6.3.4 The opportunity cost of holding money
We have argued that money provides an important service in facilitating transactions as a
medium of exchange. Despite this advantage, money generally performs less well as a store
of value than other assets. To the extent that the return on holding money is below the
return on alternative assets, there is an opportunity cost of holding money that must be set
against its benefits in facilitating transactions. This opportunity cost is inversely related to
how well money performs as a store of value.
In what follows, we will take nominal bonds as the alternative asset to which money is
compared. The nominal interest rate on bonds is 𝑖𝑖, which is also the nominal return on
holding bonds. If money pays interest at nominal rate 𝑖𝑖𝑚𝑚 then the relative return on bonds
compared to money is the difference between the interest rates 𝑖𝑖 − 𝑖𝑖𝑚𝑚 . Note that this
relative return is the same if calculated as a comparison of real returns 𝑟𝑟 and 𝑟𝑟𝑚𝑚 because the
same inflation rate 𝜋𝜋 is subtracted from both:
𝑟𝑟 − 𝑟𝑟𝑚𝑚 = (𝑖𝑖 − 𝜋𝜋) − (𝑖𝑖𝑚𝑚 − 𝜋𝜋) = 𝑖𝑖 − 𝑖𝑖𝑚𝑚
The opportunity cost of holding money is therefore 𝑖𝑖 − 𝑖𝑖𝑚𝑚 . As we will see, the opportunity
cost is usually positive, with holders of money forgoing a generally higher return on bonds.
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If no interest is paid on money (𝑖𝑖𝑚𝑚 = 0), for example where money is physical cash, then the
opportunity cost is simply 𝑖𝑖. In this case, the level of nominal interest rates on bonds is a
measure of the opportunity cost of holding money.
6.3.5 Real and nominal interest rates
Our earlier analyses of consumption, saving, and investment in Chapter 3 shows that it is the
(expected) real interest rate 𝑟𝑟 that is important for incentives. For example, the theory of
investment links the real interest rate to the marginal product of capital net of depreciation.
We have seen there are reasons to believe that real interest rates should be positive on
average if the productivity of capital is sufficiently high, or households are impatient. But
theory does not rule out times when real interest rates are negative.
A time series of US real interest rates is shown in Figure 6.6. This shows that real interest
rates are positive but low on average (around 2 per cent). The 1980s featured much higher
real interest rates, which peaked at close to 10 per cent. Real interest rates were positive
but lower in the 1960s and the 1990s. There are also times of negative real interest rates in
the late 1940s, 1970s and, more recently, from the aftermath of the 2008 financial crisis
through to 2021.
Figure 6.6: US real and nominal interest rates
As we will see in this chapter, the nominal interest rate also matters independently of the
level of real interest rates. This is because it affects the relative returns on money and bonds
in a world where at least some forms of money pay no interest, which influences how
households allocate wealth between different assets.
Empirically, US nominal interest rates have almost always been positive, though there have
been long spells where they have been close to zero, most notably after the 2008 financial
crisis. On average, nominal interest rates are higher than real interest rates, reflecting the
positive average rate of inflation. The broad pattern for US nominal interest rates is that
they were very low in the 1940s, increased over the subsequent decades to peak close to 15
per cent in the early 1980s and then declined in through to the time of writing (2021).
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Nominal interest rates are clearly positive on average, indicating a positive opportunity cost
of holding money, particularly cash. In some countries, nominal interest rates have
occasionally turned negative. We will return to that unusual case later in this chapter.
6.4 The demand for money
To understand prices, inflation and nominal interest rates, we now set up a framework for
analysing the demand for money. The basic trade-off is that money facilitates economic
activity by acting as a medium of exchange but may not be so good as a store of value
compared to alternative assets. So, while money is useful to households and firms, they
have incentives to economise on holding money, use alternatives to money, or carry out
fewer transactions.
We suppose that real GDP 𝑌𝑌 is an indicator of the number of transactions taking place in an
economy and each transaction requires using a means of payment such as money. Here, we
take the level of real GDP 𝑌𝑌 as given, returning later to the question of whether that is
affected by money and monetary policy. If the price level is 𝑃𝑃, the money value of all
transactions in the economy is 𝑃𝑃𝑃𝑃.
To represent the role of money as a medium of exchange, we impose the following
transaction constraint in addition to the budget constraints faced by agents in the economy:
𝑀𝑀 ≥ 𝑃𝑃(𝑌𝑌 − 𝑋𝑋)
This constraint specifies a minimum level of money that households and firms must hold to
carry out transactions. Mathematically, it requires that the level of money holdings on
average during a period is sufficient to pay for transactions of real value 𝑌𝑌 − 𝑋𝑋, which
correspond to an amount of money 𝑃𝑃(𝑌𝑌 − 𝑋𝑋). The variable 𝑋𝑋 has several possible
interpretations, including efforts to economise on holding money or use alternatives to
money in carrying out transactions.
We will assume in what follows that money pays no interest. Holding money balances 𝑀𝑀 on
average during a period means forgoing interest 𝑖𝑖𝑖𝑖 that could have been earned from
holding bonds instead. If money pays interest at rate 𝑖𝑖𝑚𝑚 , all we need to do is replace the
opportunity cost 𝑖𝑖 with 𝑖𝑖 − 𝑖𝑖𝑚𝑚 throughout this chapter.
Taking as given 𝑃𝑃 and 𝑌𝑌, the minimum amount of money holdings consistent with the
transaction constraint can only be reduced and thus forgone interest saved, by increasing 𝑋𝑋.
As we now discuss, increasing 𝑋𝑋 has costs that we can compare to forgone interest to derive
households’ and firms’ demand for money.
6.4.1 Economising on holding money
The first interpretation of 𝑋𝑋 is efforts to economise on the amount of money held on
average. All 𝑃𝑃𝑃𝑃 transactions are carried out with money but agents make frequent
exchanges between bonds and money to keep their average holdings of money lower.
Consequently, as 𝑋𝑋 rises, average holdings of money 𝑀𝑀 fall further below 𝑃𝑃𝑃𝑃.
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The benefit of higher 𝑋𝑋 is a reduction in forgone interest but this uses up time or incurs
transaction costs. These costs in real terms are specified by the function 𝑍𝑍(𝑋𝑋), which is
increasing in 𝑋𝑋. We assume 𝑍𝑍(𝑋𝑋) has the properties 𝑍𝑍(0) = 0, 𝑍𝑍 ′ (𝑋𝑋) > 0, and 𝑍𝑍 ′′ (𝑋𝑋) > 0,
the third of these implying that the marginal cost 𝑍𝑍′(𝑋𝑋) is increasing in 𝑋𝑋.
Conditional on 𝑋𝑋, the lowest money holdings can be is 𝑀𝑀 = 𝑃𝑃(𝑌𝑌 − 𝑋𝑋). An increase of 𝑋𝑋 by 1
reduces the need to hold real money balances 𝑀𝑀/𝑃𝑃 by 1, which reduces the real value of
forgone interest 𝑖𝑖𝑖𝑖/𝑃𝑃 on money holdings by 𝑖𝑖. The marginal benefit of higher 𝑋𝑋 is thus
equal to 𝑖𝑖. The marginal cost of higher 𝑋𝑋 is 𝑍𝑍 ′ (𝑋𝑋), which we will denote by 𝑞𝑞 in what
follows. The optimal choice of 𝑋𝑋 ∗ is where the marginal benefit equals the marginal cost:
𝑖𝑖 = 𝑍𝑍′(𝑋𝑋 ∗ )
The marginal cost function 𝑍𝑍′(𝑋𝑋) is shown as an upward-sloping line in Figure 6.7 with 𝑋𝑋 on
the horizontal axis and the marginal cost 𝑞𝑞 on the vertical axis. The optimal value of 𝑋𝑋 ∗ for a
particular nominal interest rate 𝑖𝑖 is derived by drawing a horizontal line at 𝑞𝑞 = 𝑖𝑖 and finding
where it intersects the marginal cost function. The figure shows that a higher nominal
interest rate 𝑖𝑖 leads to an increase in the optimal 𝑋𝑋 ∗ . Intuitively, if money is a worse store of
value, meaning that the opportunity cost 𝑖𝑖 is higher, it is rational to make more efforts to
avoid holding it.
Figure 6.7: Optimal reduction in money holdings
Having derived 𝑋𝑋 ∗ , agents’ demand for money 𝑀𝑀𝑑𝑑 is the minimum amount allowed by the
transaction constraint (assuming 𝑖𝑖 > 0, so there is a positive opportunity cost):
𝑀𝑀𝑑𝑑 = 𝑃𝑃�𝑌𝑌 − 𝑋𝑋 ∗ (𝑖𝑖 )�
This a money demand function of the form 𝑀𝑀𝑑𝑑 = 𝑃𝑃𝑃𝑃(𝑌𝑌, 𝑖𝑖), where real money demand
𝐿𝐿(𝑌𝑌, 𝑖𝑖 ) = 𝑌𝑌 − 𝑋𝑋 ∗ (𝑖𝑖) is increasing in 𝑌𝑌 and decreasing in 𝑖𝑖.
The choice of how much money to hold is related to the following measure of the velocity of
money 𝑉𝑉, which is defined by 𝑀𝑀𝑉𝑉 = 𝑃𝑃𝑃𝑃. This is a measure of how fast a unit of money
circulates in a given period as it is used for multiple transactions. Since 𝑉𝑉 = 𝑃𝑃𝑃𝑃⁄𝑀𝑀𝑑𝑑 =
𝑌𝑌/(𝑌𝑌 − 𝑋𝑋 ∗ (𝑖𝑖 )), velocity is inversely related to 𝑀𝑀𝑑𝑑 and increases with the opportunity cost 𝑖𝑖
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of holding money. Intuitively, money circulates faster with people holding it for shorter
periods when money is a poor store of value.
6.4.2 Alternatives to money
A second interpretation of 𝑋𝑋 in the transaction constraint 𝑀𝑀 + 𝑃𝑃𝑃𝑃 ≥ 𝑃𝑃𝑃𝑃 is using
alternatives to money as a means of payment. We discussed in Section 6.2 how credit might
be used as a substitute for money in some situations. Suppose banks offer credit facilities,
for example, credit cards and let 𝑋𝑋 now denote the real value of transactions paid for with
credit. Think of this as short-term credit, with borrowing during a period repaid at the end of
the period. A fee 𝑞𝑞 is charged for using credit as a fraction of the amount borrowed.
In offering credit, banks face costs of screening borrowers and collection of debts. Assume
that these costs are an increasing function 𝑍𝑍(𝑋𝑋) of 𝑋𝑋. In addition, the marginal cost 𝑍𝑍 ′ (𝑋𝑋) of
extending provision of credit is increasing in the amount of borrowing 𝑋𝑋. This represents the
idea that banks would face higher costs when they expand lending to a wider group of less
credit-worthy borrowers, or extend more credit to existing borrowers. Assuming the
banking system is competitive, banks offer credit 𝑋𝑋 𝑠𝑠 up to the point where the fee charged
𝑞𝑞 is equal to the marginal cost 𝑍𝑍′(𝑋𝑋):
𝑞𝑞 = 𝑍𝑍′(𝑋𝑋 𝑠𝑠 )
This yields an upward-sloping supply curve 𝑋𝑋 𝑠𝑠 (𝑞𝑞 ) for credit facilities.
Now consider the demand for credit by households and firms as a means of payment.
Transactions can equally well be carried out using credit facilities or using money. The credit
facility fee is 𝑞𝑞 per unit of spending is the cost of using credit for payments. If money is used
instead then the cost is the opportunity cost 𝑖𝑖 of holding money. Since the two means of
payment are perfect substitutes, if 𝑞𝑞 < 𝑖𝑖 then payment with credit facilities is preferred, if
𝑞𝑞 > 𝑖𝑖 then payment using money holdings is preferred, and if 𝑞𝑞 = 𝑖𝑖 then everyone is
indifferent between the two. It follows that the demand for credit facilities 𝑋𝑋 𝑑𝑑 (𝑞𝑞) is
perfectly elastic in the range 0 ≤ 𝑋𝑋 ≤ 𝑌𝑌 with respect to the fee at 𝑞𝑞 = 𝑖𝑖.
The demand function 𝑋𝑋 𝑑𝑑 (𝑞𝑞) is plotted alongside the supply function 𝑋𝑋 𝑠𝑠 (𝑞𝑞) in Figure 6.8.
The demand curve shifts vertically if the nominal interest rate 𝑖𝑖 changes, moving upwards if
𝑖𝑖 rises. The equilibrium of the market for credit facilities is at the intersection of the demand
and supply curves. Assuming 𝑋𝑋 ∗ < 𝑌𝑌, so some amount of money is held to make payments,
the equilibrium features 𝑞𝑞 ∗ = 𝑖𝑖, so the credit fee is equal to the nominal interest rate on
bonds. An increase in 𝑖𝑖 shifts the demand function upwards, so the equilibrium 𝑋𝑋 ∗ (𝑖𝑖) rises
with 𝑖𝑖. As it does not make sense to hold more money than required to satisfy the
transaction constraint when 𝑖𝑖 > 0, the money demand function is 𝑀𝑀𝑑𝑑 = 𝑃𝑃�𝑌𝑌 − 𝑋𝑋 ∗ (𝑖𝑖 )�,
which has the same form as seen earlier with the first interpretation of 𝑋𝑋.
6.4.3 The money demand function
Considering 𝑋𝑋 as either effort to economise on holding money, or substitution towards
alternatives to money, the resulting money demand function has the form:
𝑀𝑀𝑑𝑑 = 𝑃𝑃𝑃𝑃(𝑌𝑌, 𝑖𝑖)
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The function 𝐿𝐿(𝑌𝑌, 𝑖𝑖 ) = 𝑌𝑌 − 𝑋𝑋 ∗ (𝑖𝑖 ) for holdings of real money balances 𝑀𝑀𝑑𝑑 /𝑃𝑃 increases in 𝑌𝑌
and decreases in 𝑖𝑖. Nominal money demand 𝑀𝑀𝑑𝑑 is proportional to the price level 𝑃𝑃 for given
real transactions and interest rates because higher prices scale up the need for units of
money to make payments. Money demand 𝑀𝑀𝑑𝑑 increases with 𝑌𝑌 as higher GDP means more
transactions. Money demand 𝑀𝑀𝑑𝑑 decreases with 𝑖𝑖 because a higher opportunity cost
increases incentives to reduce money holdings through various means.
Figure 6.8: The market for credit facilities
The money demand function is plotted against the price level 𝑃𝑃 in the left panel of Figure
6.9 for given values of real GDP 𝑌𝑌 and the nominal interest rate 𝑖𝑖. It is an upward-sloping
straight line because nominal money demand is proportional to the price level. The demand
function pivots to the right if 𝑌𝑌 increases or 𝑖𝑖 falls.
Figure 6.9: The demand for money
The relationship between the nominal interest rate 𝑖𝑖 and real money demand 𝑀𝑀𝑑𝑑 /𝑃𝑃 is
depicted in the right panel of Figure 6.9. The negative relationship reflects the incentive to
reduce money holdings when the opportunity cost 𝑖𝑖 is high. Mathematically, the demand
curve represents the optimality condition 𝑖𝑖 = 𝑍𝑍′(𝑋𝑋), where 𝑋𝑋 = 𝑌𝑌 − (𝑀𝑀⁄𝑃𝑃) using the
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binding transaction constraint. The demand curve shifts to the right if 𝑌𝑌 increases. In the
special case 𝑖𝑖 = 0, there is no forgone interest when holding money and the optimal value
of 𝑋𝑋 is 0. Moreover, there is no incentive to reduce money holdings until the transaction
constraint just holds. Hence, with a zero nominal interest rate, money demand is 𝑀𝑀𝑑𝑑 ≥
𝑃𝑃𝑃𝑃(𝑌𝑌, 0) = 𝑃𝑃𝑃𝑃, which corresponds to a horizontal line at 𝑖𝑖 = 0. Money demand thus
becomes perfectly interest elastic at 𝑖𝑖 = 0.
Finally, we note that when money itself pays interest at rate 𝑖𝑖𝑚𝑚 , all references to 𝑖𝑖 above in
the money demand function should be replaced by the correct opportunity cost 𝑖𝑖 − 𝑖𝑖𝑚𝑚 .
6.5 Money and economic activity
Our study of money demand revealed the ways in which it is affected by real GDP and
interest rates. But does money itself matter for real GDP? The analysis here will focus on
money’s medium of exchange function, which affects the efficiency with which markets
operate. Later in Chapter 8, money’s unit of account function becomes relevant in the
presence of nominal rigidities. In this chapter, we look at the implications of economic
activity depending on holding money for some period between selling one thing and buying
another. Money that is a poor store of value over this period acts as a tax on economic
activity, therefore discouraging production and exchange.
We illustrate this idea using the labour market as an example. Suppose the period is a
month and workers are paid a wage 𝑊𝑊 per hour of labour only at the end of the month.
Wages arrive too late to be spent directly during the same month, and suppose it is not
possible for workers to barter labour for goods, or offer IOUs for payment when they buy
goods.
Households’ labour supply condition 𝑀𝑀𝑀𝑀𝑀𝑀𝑙𝑙,𝐶𝐶 = 𝑤𝑤 derived in Chapter 1 assumed an extra
hour of labour paid money wage 𝑊𝑊 buys 𝑤𝑤 = 𝑊𝑊/𝑃𝑃 goods in the same period. But to work
more and spend more during the same month in the monetary economy described above, a
household must either forgo interest by holding on to more cash at the beginning of the
month, swap money and other assets more frequently during the month at some cost, or
pay for goods using credit as an alternative to money. All of these ways of spending more
during the month before actually receiving the wage at the end of the month entail some
cost.
We derive the amount of goods 𝑤𝑤𝑚𝑚 that can be purchased in the same month when a
household supplies an additional hour of labour paid money wage 𝑊𝑊, holding constant the
household’s future plans for consumption and labour supply. The effective real purchasing
power of a household’s wages during the month is 𝑤𝑤𝑚𝑚 , which will generally differ from the
real cost 𝑤𝑤 = 𝑊𝑊/𝑃𝑃 to firms when wages are paid at the end of the month.
Real purchases 𝑤𝑤𝑚𝑚 cost 𝑃𝑃𝑤𝑤𝑚𝑚 units of money. If the household holds extra money 𝑃𝑃𝑤𝑤𝑚𝑚
instead of bonds during the month to make the purchases then this reduces nominal wealth
by (1 + 𝑖𝑖 )𝑃𝑃𝑤𝑤𝑚𝑚 at beginning of next month. To leave future spending plans unchanged, this
needs to be replenished with the extra wages 𝑊𝑊 received at the end of the month, hence,
(1 + 𝑖𝑖 )𝑃𝑃𝑤𝑤𝑚𝑚 = 𝑊𝑊. Dividing both sides by 𝑃𝑃 implies that 𝑤𝑤𝑚𝑚 is related to 𝑤𝑤 as follows:
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𝑤𝑤𝑚𝑚 =
𝑤𝑤
1 + 𝑖𝑖
This equation says that the effective purchasing power of the wages households receive is
reduced by 𝑖𝑖 because spending more requires holding more money, which forgoes interest.
Alternatively, the household could maintain the same average money holdings during the
month and avoid forgoing interest. However, this requires swapping between money and
other assets more frequently (higher 𝑋𝑋) during the month to cover the additional spending.
But this entails transaction costs 𝑞𝑞 = 𝑍𝑍′(𝑋𝑋) per unit of extra spending. Deducting these
from the wage received implies 𝑃𝑃𝑤𝑤𝑚𝑚 = 𝑊𝑊 − 𝑞𝑞𝑞𝑞𝑤𝑤𝑚𝑚 . It follows that 𝑤𝑤𝑚𝑚 = 𝑤𝑤⁄(1 + 𝑞𝑞).
Finally, credit could be used for the extra purchases made during the month. This requires
paying a fee 𝑞𝑞𝑞𝑞𝑤𝑤𝑚𝑚 at the end of month. Deducting that from the wage implies 𝑃𝑃𝑤𝑤𝑚𝑚 = 𝑊𝑊 −
𝑞𝑞𝑞𝑞𝑤𝑤𝑚𝑚 and hence, 𝑤𝑤𝑚𝑚 = 𝑤𝑤⁄(1 + 𝑞𝑞). In Section 6.4, we saw that households’ optimal choice
of money holdings implies 𝑞𝑞 = 𝑍𝑍 ′ (𝑋𝑋) = 𝑖𝑖, so this means that 𝑤𝑤𝑚𝑚 = 𝑤𝑤/(1 + 𝑖𝑖) whichever
way of paying for current consumption that households choose.
Households’ labour supply decision in a monetary economy with the timing restriction on
receiving and spending wages equates the marginal rate of substitution 𝑀𝑀𝑀𝑀𝑀𝑀𝑙𝑙,𝐶𝐶 between
leisure and current consumption to the effective current purchasing power of the wage 𝑤𝑤𝑚𝑚 :
𝑀𝑀𝑀𝑀𝑀𝑀𝑙𝑙,𝐶𝐶 =
𝑤𝑤
1 + 𝑖𝑖
The right-hand side of the equation is lower when the opportunity cost 𝑖𝑖 rises, indicating
that money is worse as a store of value. Since working and consuming more depends on
holding money for some time, a positive opportunity cost 𝑖𝑖 works in a way similar to a
proportional tax 𝜏𝜏 on wages. We know from Section 1.5 that a proportional income tax on
wages means the households’ labour supply is determined by 𝑀𝑀𝑀𝑀𝑀𝑀𝑙𝑙,𝐶𝐶 = (1 − 𝜏𝜏)𝑤𝑤.
This logic points to one way that money matters for real GDP. If money is worse as a store of
value (a high opportunity cost 𝑖𝑖) then the implicit tax on economic activity rises. This leads
to a lower labour supply, shifting the 𝑁𝑁 𝑠𝑠 curve to the left. All else equal, there is less
employment and lower production, which causes a shift of the 𝑌𝑌 𝑠𝑠 curve to the left.
Households are worse off, which reduces consumption demand and shifts the 𝑌𝑌 𝑑𝑑 curve to
the left as well. Consequently, real GDP 𝑌𝑌 is lower. If 𝑌𝑌 𝑠𝑠 and 𝑌𝑌 𝑑𝑑 shift by the same amount,
then the real interest rate remains unchanged.
6.6 The supply of money
We now turn to thinking about the supply of money. We first consider the supply of fiat
money by a government or central bank, deferring discussion of ‘credit money’ created by
the banking system until Chapter 7. Assume all money is fiat money, for which the
government is the monopoly supplier. The quantity of money in circulation is denoted by
𝑀𝑀 𝑠𝑠 . For now, we make no distinction between cash and reserves.
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How does the central bank change the money supply? In other words, how does new money
enter circulation or existing money is removed from circulation? We will see that there are
two basic ways this can happen:
•
•
Open-market operations
Transfers.
Since fiat money is intrinsically worthless, the resource costs of creating new money are
negligible and we ignore them in our analysis.
6.6.1 Open-market operations
An open-market operation is where the central bank buys or sells assets. When the central
bank buys assets, it pays with newly created money, which increases the quantity of money
in circulation. When the central bank sells assets, it receives existing money as payment,
which is effectively removed from circulation.
The central bank can in principle buy any asset in an open market operation or sell any asset
it already holds. It usually transacts with the private sector through its dealings with
commercial banks (rather than buying bonds directly from the government). Traditionally,
open-market operations were in markets for short-term government bonds, or repos
(repurchase/resale agreements) of long-term government bonds.
These assets were chosen because they have low credit risk and a short maturity and thus
protect the central bank from capital losses that would make it harder to reverse an
expansionary open-market operation in the future. But since the 2008 financial crisis, many
central banks have also made outright purchases of long-term bonds or risky assets, for
example, quantitative easing (QE) purchases of mortgage-backed securities in the US.
6.6.2 Transfers
A transfer payment is where the central bank distributes money without acquiring any asset
in return, for example, the payment of central-bank profits to a country’s finance ministry.
These profits often arise as a normal outcome of the central bank’s operations and are
distributed to the finance ministry as the owner of the central bank.
However, in principle – putting aside legal rules – a central bank can create new money and
simply distribute it to the finance ministry or others. This could mean directly paying for
government expenditure, or giving the government money to compensate for lower tax
revenues. The case of a direct payment of new money to households is known as a
‘helicopter drop’ of money, though the same economic effect could be achieved by a
transfer to the government to fund a tax cut for households.
6.6.3 Monetary policy
The decisions the central bank makes that affect the supply of money are described as its
monetary policy. For now, we assume monetary policy is an exogenous supply of money
𝑀𝑀 𝑠𝑠 . This can be represented as a perfectly inelastic money supply curve. This supply curve
shifts if monetary policy changes. We will consider later what monetary policy should be
chosen to meet the objectives of a country’s government.
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6.7 Money and prices
This section combines the demand and supply of money to see how the level of prices, the
inflation rate and the nominal interest rate are determined. Here, we suppose that goods
prices in terms of money are fully flexible. This means that the real value of money adjusts
to be equal to the real amount of money that households and firms are willing to hold. In
Chapter 8 we consider how an economy functions differently if there are nominal rigidities,
for example ‘sticky prices’.
With flexible prices, the price level 𝑃𝑃 adjusts to ensure the money market clears. The
nominal money supply 𝑀𝑀 𝑠𝑠 is assumed to be an exogenous amount 𝑀𝑀 chosen by the central
bank. Rather than consider a completely general monetary policy, we will restrict attention
here to monetary policies where the money supply is expected to grow at some exogenous
rate 𝜇𝜇 over time:
𝑀𝑀′ = (1 + 𝜇𝜇)𝑀𝑀
The money demand function is 𝑀𝑀𝑑𝑑 = 𝑃𝑃𝑃𝑃(𝑌𝑌, 𝑖𝑖), and money-market equilibrium 𝑀𝑀𝑑𝑑 = 𝑀𝑀 𝑠𝑠
therefore requires 𝑀𝑀 = 𝑃𝑃𝑃𝑃(𝑌𝑌, 𝑖𝑖) in the current period and 𝑀𝑀′ = 𝑃𝑃′ 𝐿𝐿(𝑌𝑌 ′ , 𝑖𝑖 ′ ) in the future.
Nominal and real interest rates are linked by the Fisher equation 𝑖𝑖 = 𝑟𝑟 + 𝜋𝜋, where the
inflation rate is defined by 𝜋𝜋 = (𝑃𝑃′ − 𝑃𝑃)/𝑃𝑃. The conditions for equilibrium in the money
market now and in the future are therefore 𝑀𝑀 = 𝑃𝑃𝑃𝑃(𝑌𝑌, 𝑟𝑟 + 𝜋𝜋) and 𝑀𝑀′ = 𝑃𝑃′ 𝐿𝐿(𝑌𝑌 ′ , 𝑟𝑟 ′ + 𝜋𝜋 ′ ). A
graphical representation of the current period equilibrium is shown in Figure 6.10, where
the equilibrium price level 𝑃𝑃∗ is at the intersection of 𝑀𝑀𝑑𝑑 and 𝑀𝑀 𝑠𝑠 .
Figure 6.10: Money-market equilibrium
By dividing the future money-market equilibrium condition by the current money-market
equilibrium condition we obtain the equation:
𝑀𝑀′ 𝑃𝑃′ 𝐿𝐿(𝑌𝑌 ′ , 𝑟𝑟 ′ + 𝜋𝜋 ′ )
=
𝑀𝑀
𝑃𝑃𝑃𝑃(𝑌𝑌, 𝑟𝑟 + 𝜋𝜋)
We suppose that the types of monetary policies considered here do not affect future real
GDP 𝑌𝑌′ differently from how they affect current real GDP 𝑌𝑌 (that is, they do not change the
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future real GDP growth rate), or the current real interest rate 𝑟𝑟 relative to its future level 𝑟𝑟′,
or 𝜋𝜋 relative to 𝜋𝜋′. All else being equal, this means 𝑌𝑌 = 𝑌𝑌′, 𝑟𝑟 = 𝑟𝑟′, and 𝜋𝜋 = 𝜋𝜋′. Note we have
not ruled out that monetary policy affects the levels of 𝑌𝑌, 𝑟𝑟, or 𝜋𝜋. The equation for moneymarket equilibrium then reduces to 𝑀𝑀′⁄𝑀𝑀 = 𝑃𝑃′⁄𝑃𝑃. With definitions 𝑀𝑀′⁄𝑀𝑀 = 1 + 𝜇𝜇 and
𝑃𝑃′ /𝑃𝑃 = 1 + 𝜋𝜋, money-market equilibrium therefore implies:
𝜋𝜋 = 𝜇𝜇
The rate of inflation 𝜋𝜋 is equal to the money-supply growth rate 𝜇𝜇, which means that
inflation is determined by monetary policy through the choice of 𝜇𝜇. Intuitively, increases in
the money supply shift the 𝑀𝑀 𝑠𝑠 curve to the right, which imply that the intersection with 𝑀𝑀𝑑𝑑
occurs at a higher price level 𝑃𝑃 to leave holdings of real money balances unchanged.
Given a real interest rate 𝑟𝑟, the Fisher equation implies:
𝑖𝑖 = 𝑟𝑟 + 𝜇𝜇
A higher money-supply growth rate thus raises the nominal interest rate 𝑖𝑖 if 𝑟𝑟 remains
unchanged. This is because a higher nominal interest rate is required to cancel out the
effect of inflation and leave the real return on bonds the same.
Box 6.2: The instability of money demand
The analysis of the equilibrium inflation rate might give the impression that only the
money supply growth rate matters, a form of ‘monetarism’. This is because we have
considered only shifts of the money supply curve for a completely stable money demand
curve. However, the equilibrium of the money market can also be affected by shifts of the
money demand function and this affects the equilibrium price level for a given supply of
money. If such demand shifts occur then this leads to fluctuations in inflation even if
monetary policy keeps the money supply or money growth constant.
The money demand curve 𝑀𝑀𝑑𝑑 ⁄𝑃𝑃 = 𝐿𝐿(𝑌𝑌, 𝑖𝑖) can shift because of changes in real GDP 𝑌𝑌,
which affect the need to use money for transactions. But in addition to this, the function
𝐿𝐿(𝑌𝑌, 𝑖𝑖) itself might not be stable owing to financial innovation. New ideas or technologies
can change the costs of providing substitutes for money, for example, credit cards, or
change the costs of economising on the average amount of money held to carry out
transactions, for example, ATMs, debit cards and electronic payments. We can represent
the effects of these innovations in the model by shifts of the marginal cost function 𝑍𝑍′(𝑋𝑋).
Since the money demand is determined by the equation 𝑖𝑖 = 𝑍𝑍′(𝑌𝑌 − 𝑀𝑀𝑑𝑑 ⁄𝑃𝑃), these
changes also shift the money demand function as shown in Figure 6.11. A reduction in the
marginal cost of providing substitutes for money or economising on money holdings
increases 𝑋𝑋 and reduces money demand, causing the price level to increase for a given
money supply 𝑀𝑀 𝑠𝑠 .
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Figure 6.11: Financial innovation shifts money demand
How serious an issue is the instability of money demand? The left panel of Figure 6.12
reports a time series of the quantity of money 𝑀𝑀 𝑠𝑠 in the USA relative to nominal GDP 𝑃𝑃𝑃𝑃.
Note that this uses the M1 measure of the money supply, which is broader than the
monetary base and that we study further in Chapter 7. The measure 𝑀𝑀 𝑠𝑠 /(𝑃𝑃𝑃𝑃) is the
inverse of the velocity of money 𝑉𝑉 = (𝑃𝑃𝑃𝑃)/𝑀𝑀. In equilibrium, it is also equal to
(𝑀𝑀𝑑𝑑 ⁄𝑃𝑃)/𝑌𝑌, which is real money demand relative to real GDP 𝑌𝑌. The scaling by GDP is done
to control for changes in the demand for money owing to transactions rising with GDP.
We see that M1 as a fraction of GDP followed a stable trend prior to the 1980s but has
experienced various shifts in the 1980s, 1990s, and 2000s. The right panel of the figure is a
scatterplot of 𝑀𝑀 𝑠𝑠 /(𝑃𝑃𝑃𝑃) against the nominal interest rate 𝑖𝑖, which should show the
downward-sloping real money demand curve scaled by GDP. However, the plot indicates
this relationship has been unstable.
Figure 6.12: Demand for money according to the M1 measure in the USA
The exercise is repeated for the broader M2 measure of the US money supply in Figure
6.13. The time series in the left panel suggests the demand for M2 (relative to GDP) has
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been more stable than M1. The scatterplot in the right panel comes closer to tracing out
something that resembles a negative relationship between real money demand (scaled by
GDP) and the nominal interest rate 𝑖𝑖, though this relationship still appears to shift at some
points in time.
Overall, the evidence presented here suggests we cannot be confident that regulating the
money-supply growth rate will give tight control over inflation.
6.8 Money and public finance
Governments derive a fiscal advantage from being able to issue fiat money that is
demanded by the private sector. Unlike bonds, there is no obligation to ‘repay’ or redeem
fiat money. Furthermore, money may pay no interest (𝑖𝑖𝑚𝑚 = 0), or pay a lower rate of
interest than bonds (𝑖𝑖𝑚𝑚 < 𝑖𝑖). These fiscal gains from issuing money are often referred to as
the ‘seigniorage’ revenue of the government. They represent an implicit tax on holders of
money. This section looks at how to quantify the fiscal gains that arise from different
monetary policies.
Figure 6.13: Demand for money according to the M2 measure in the USA
6.8.1 Seigniorage: ‘printing money’
If the money supply is growing at a rate 𝜇𝜇 then an amount of new money 𝜇𝜇𝜇𝜇 is created
each time. If it were directly used to finance government expenditure, this seigniorage
revenue would be worth 𝜇𝜇𝜇𝜇⁄𝑃𝑃 in real terms. Using 𝜋𝜋 = 𝜇𝜇 that results from money-market
equilibrium in Section 6.7, the real amount of seigniorage is 𝜋𝜋𝜋𝜋⁄𝑃𝑃.
This is simplest and most direct measure of seigniorage as the fiscal advantage that comes
from ‘printing money’. But this calculation ignores the saving of regular interest payments
on past spending that has been financed in this way rather than by issuing bonds. Moreover,
most central banks are not in the business of directly financing government expenditure.
What if – the usual case – the central bank is buying assets with newly created money?
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6.8.2 Seigniorage: central-bank investment income
Assume money pays no interest and all money created by the central bank has been used to
buy nominal bonds. The central bank holds bonds of monetary value 𝑀𝑀 that matches exactly
the existing supply of money 𝑀𝑀. In this case, the central bank earns interest 𝑖𝑖𝑖𝑖 in each
period and, ignoring resource costs of creating money and any operating costs, these are
profits that can be paid out to the finance ministry. Real seigniorage revenues are 𝑖𝑖𝑖𝑖/𝑃𝑃,
which represents a flow of revenue received by the government in each period. Note that
this is different from the seigniorage measure based on the real value of the increase in the
money supply.
6.8.3 A general definition of seigniorage
Even if the central bank does not buy assets, the central-bank investment income definition
of seigniorage still accurately represents the fiscal advantage derived from steady growth in
the money supply. The government reduces the cost of financing public expenditure by
creating money rather than issuing interest-bearing bonds. The size of this advantage can be
calculated as the real quantity of money 𝑀𝑀/𝑃𝑃 in circulation multiplied by the difference in
the returns on bonds and money, which is the nominal interest rate 𝑖𝑖 when money does not
pay interest. Seigniorage then simply represents money being less good as a store of value
than bonds, which is an advantage from the perspective of the issuer of money.
With 𝑖𝑖 = 𝑟𝑟 + 𝜋𝜋, and 𝜋𝜋 = 𝜇𝜇 in equilibrium with a constant money-supply growth rate 𝜇𝜇, the
central-bank profits definition of seigniorage can be broken down into:
𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟 𝜋𝜋𝜋𝜋 𝑟𝑟𝑟𝑟 𝜇𝜇𝜇𝜇
=
+
=
+
𝑃𝑃
𝑃𝑃
𝑃𝑃
𝑃𝑃
𝑃𝑃
This is the saving of real interest payments on bonds otherwise issued plus the erosion of
existing money’s real value due to new money being created.
6.8.4 Limits on real seigniorage revenues
As seigniorage arises from money being less good a store of value than other assets, it is an
implicit tax on money. Seigniorage is closely related to the notion of forgone interest we
saw in the analysis of money demand in Section 6.4 and is essentially identical to the total
amount of forgone interest on money.
If money becomes a worse store of value because the nominal interest rate 𝑖𝑖 is higher then
real money demand 𝑀𝑀𝑑𝑑 ⁄𝑃𝑃 = 𝐿𝐿(𝑌𝑌, 𝑖𝑖) falls. Real seigniorage revenues are 𝑖𝑖𝑖𝑖⁄𝑃𝑃 = 𝑖𝑖𝑖𝑖(𝑌𝑌, 𝑖𝑖),
so there are two conflicting effects of higher 𝑖𝑖. First, the direct effect of money being worse
as a store of value. Second, the indirect effect of falling real money demand reducing the
real value of seigniorage. This means the relationship between real seigniorage revenues
and 𝑖𝑖 is not unambiguously positive. Observe that seigniorage is zero if 𝑖𝑖 = 0 and becomes
zero again for high 𝑖𝑖 if real money demand 𝐿𝐿(𝑌𝑌, 𝑖𝑖) falls towards zero sufficiently fast as 𝑖𝑖
increases. This gives rise to a Laffer curve for real seigniorage revenues as shown in Figure
6.14, indicating there are limits on the amount of real seigniorage government can obtain.
6.8.5 The inflation tax
If there is an unpredictable increase in the inflation rate 𝜋𝜋, the nominal interest rate 𝑖𝑖 on
bonds cannot rise to leave the real return 𝑟𝑟 unchanged. An inflation surprise thus reduces
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both the real value of nominal government bonds as well as existing money. This is reflected
in the ex-post real interest rate 𝑟𝑟 being less than the ex-ante real interest rate 𝑟𝑟 𝑒𝑒 .
The fiscal advantage derived from such surprise inflation is referred to here as an ‘inflation
tax’. A different term is used because the mechanism through which the inflation tax works
is distinct from the source of seigniorage revenue discussed earlier. With the definitions
adopted here, seigniorage revenues derive only from money not bonds and do not depend
on inflation being a surprise. In contrast, the inflation tax depends on inflation that was
unexpected when nominal bonds were first issued. This means that, ex post, the inflation
tax does not have any incentive effects on behaviour – like a lump-sum tax – because it is
completely unexpected.
There is no inflation tax on the real value of nominal bonds when the inflation is anticipated.
In this case, the real return is protected from expected inflation when the nominal interest
rate 𝑖𝑖 adjusts in advance. Inflation-indexed bonds are also protected against surprise
inflation and offer a guaranteed real return.
Figure 6.14: Seigniorage Laffer curve
6.8.6 The government budget constraint and Ricardian equivalence
In spite of the fiscal advantage that governments can derive from issuing money – a form of
‘soft default’ on government debt – a government ‘budget constraint’ still holds once
seigniorage and the inflation tax are counted alongside other more conventional sources of
tax revenue. Moreover, seigniorage and the inflation tax also show up in households’
budget constraints alongside explicit taxes because of they bear the losses from forgone
interest and the erosion of the real value of nominal bonds by surprise inflation.
If the economy has a representative household, it is possible to combine the household and
government budget constraints in the way seen in Section 4.1. The present value of
government expenditure ultimately determines the present value of tax revenue from all
sources, including seigniorage and the inflation tax. However, Ricardian equivalence fails
because seigniorage is effectively a tax that distorts incentives as seen in Section 6.5.
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6.9 Does monetary policy matter?
This section asks whether it matters what monetary policy is chosen by the central bank or
government. We know that monetary policy should affect nominal prices and inflation but
what effects are there, if any, on real variables such as GDP? Here, we answer this question
in the context of a model where the special feature of money is its role as a medium of
exchange. Money matters in different ways, in particular, through its unit of account
function, in the models with nominal rigidities we will see from Chapter 8.
We will consider two different changes to monetary policy:
•
•
A permanent change in the quantity of money in circulation
A permanent change in the growth rate of the supply of money in circulation.
6.9.1 A permanent change in the level of the money supply
Suppose there is an exogenous permanent change in money supply 𝑀𝑀 𝑠𝑠 = 𝑀𝑀. Since this is
exogenous, it is not a reaction to other events or shocks. We assume the change is
unexpected and that no repeat is expected in the future. This means the level of 𝑀𝑀 changes
but not its subsequent growth rate 𝜇𝜇. As the future money supply 𝑀𝑀′ changes in the same
way as the current money supply 𝑀𝑀, a zero money-supply growth rate 𝜇𝜇 = 0 is expected
subsequently because 𝑀𝑀′ = 𝑀𝑀. However, when the policy change is implemented, there is
still an unexpected change in the money supply 𝑀𝑀 relative to its past level.
Since the change in the money supply is the same in the present as in the future, the effects
on the equilibrium price levels 𝑃𝑃 and 𝑃𝑃′ are the same. This means that expected inflation
𝜋𝜋 = (𝑃𝑃′ − 𝑃𝑃)/𝑃𝑃 between now and the future period is 𝜋𝜋 = 𝜇𝜇 = 0. From the Fisher
equation we therefore conclude that 𝑖𝑖 = 𝑟𝑟 + 𝜋𝜋 = 𝑟𝑟. Although there is no further inflation
expected in the future, there can still be unexpected inflation/deflation of 𝑃𝑃 relative to the
past price level.
In what follows, we take the case of an increase in the money supply 𝑀𝑀 for illustration. We
will see that the model predicts this increase in 𝑀𝑀 has no real effects at all. This result is
shown in the supply-and-demand diagrams for the goods, labour and money markets
depicted in Figure 6.15. But what is the logic for this striking claim?
Figure 6.15: Permanent increase in money supply
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First, and most importantly, prices and wages expressed in units of money are fully flexible
here. With no impediments to price adjustment, the same real wage 𝑤𝑤 and real interest rate
𝑟𝑟 can continue to ensure supply and demand are brought into equilibrium in the labour and
goods markets. Moreover, there is no money illusion – everyone’s decisions depend on
relative prices and real variables.
Second, the policy change does not affect perceptions of how good money is as a store of
value going forwards between the current and future time periods. Since 𝑖𝑖 = 𝑟𝑟, there is no
change in the nominal interest rate 𝑖𝑖 unless 𝑟𝑟 changes. The nominal interest rate 𝑖𝑖 is a
measure of how bad money is as a store of value relative to other assets. This means no
greater tax on economic activity that depends on holding money is expected and, hence,
there is no reason for the labour and output supply curves to shift through the effect of 𝑖𝑖 on
the labour-supply condition 𝑀𝑀𝑀𝑀𝑀𝑀𝑙𝑙,𝐶𝐶 = 𝑤𝑤/(1 + 𝑖𝑖).
Third, while existing holdings of money and nominal government bonds are caught by a
surprise inflation tax that reduces the real value of households’ financial assets, the inflation
tax also allows the government to reduce other taxes and still pay for the same level of
public expenditure. These tax cuts offset the reduction in the value of financial assets and
there is no wealth effect overall on households – Ricardian equivalence holds after
accounting for the government budget constraint.
Thus, we conclude there are no reasons for any shifts of the 𝑁𝑁 𝑑𝑑 , 𝑁𝑁 𝑠𝑠 , 𝑌𝑌 𝑑𝑑 , or 𝑌𝑌 𝑠𝑠 curves.
Therefore, the equilibrium values of 𝑤𝑤 ∗ , 𝑟𝑟 ∗ , 𝑁𝑁 ∗ , and 𝑌𝑌 ∗ are unaffected. A permanent change
in the money supply has no real effects. No change in 𝑌𝑌 or 𝑖𝑖 means there is no shift of the
money demand curve 𝑀𝑀𝑑𝑑 . The rightward shift of the money supply curve 𝑀𝑀 𝑠𝑠 thus leads 𝑃𝑃 to
rise in proportion to 𝑀𝑀. Given these predictions, money is said to be ‘neutral’.
6.9.2 A permanent change in the growth rate of the money supply
Alternatively, suppose there is a permanent adjustment of the growth rate of the money
supply 𝜇𝜇. This money-supply growth rate is defined by 𝜇𝜇 = (𝑀𝑀′ − 𝑀𝑀)/𝑀𝑀, hence, the future
money supply is given by 𝑀𝑀′ = (1 + 𝜇𝜇)𝑀𝑀. The change in 𝜇𝜇 is exogenous, unexpected and no
further adjustments of 𝜇𝜇 are expected. Note that there is no change in the initial money
supply 𝑀𝑀 here.
Since the policy change affects expectations of the future money supply, inflation
expectations 𝜋𝜋 = (𝑃𝑃′ − 𝑃𝑃)/𝑃𝑃 adjust. As we have seen in Section 6.7, the effect on the
equilibrium current and future price levels 𝑃𝑃 and 𝑃𝑃′ is such that 𝜋𝜋 = 𝜇𝜇, so any changes in
money-supply growth are reflected one-for-one in changes in expected inflation.
Let us take the case of faster money growth for illustration. We will see that increasing the
money growth rate does have real effects. These are depicted in the supply-and-demand
diagrams in Figure 6.16.
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Figure 6.16: Permanent increase in money supply growth rate
The logic for the real effects is that higher money growth 𝜇𝜇 raises expectations of future
inflation 𝜋𝜋. The Fisher equation 𝑖𝑖 = 𝑟𝑟 + 𝜋𝜋 then implies the nominal interest rate 𝑖𝑖 is higher
for each value of the real interest rate 𝑟𝑟. From the labour-supply equation 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 =
𝑤𝑤/(1 + 𝑖𝑖), higher 𝑖𝑖 has a negative effect on labour supply. Intuitively, because money is a
worse store of value, the implicit tax on economic activity rises, which causes the supply of
labour to decline. Consequently, the output supply curve 𝑌𝑌 𝑠𝑠 shifts to the left and, as this
change is permanent, 𝐶𝐶 𝑑𝑑 falls in line with income, leading to a leftward shift of 𝑌𝑌 𝑑𝑑 of same
size as the shift of 𝑌𝑌 𝑠𝑠 .
If a permanent change in the money supply growth were to have no effects on any real
variables then we would say that money is ‘superneutral’. The term ‘neutrality’ used earlier
refers to there being no real effects of a permanent change to the level of 𝑀𝑀. We see in
Figure 6.16 that the model predicts money is not superneutral. Permanently faster growth
of the money supply reduces real GDP and employment because money is less good as a
store of value. This inflationary policy has a negative real effect on the economy’s supply
side.
In the money market, there is no initial change in 𝑀𝑀, so no shift of the money supply curve
𝑀𝑀 𝑠𝑠 to begin with. The money demand curve 𝑀𝑀𝑑𝑑 pivots to the left as there are fewer
transactions due to lower GDP 𝑌𝑌 and more efforts to economise on holding money or make
use of money substitutes (higher 𝑋𝑋) because of higher 𝑖𝑖. This leads real money balances
𝑀𝑀/𝑃𝑃 to fall as 𝑀𝑀𝑑𝑑 /𝑃𝑃 is lower, which causes an immediate jump up in the level of prices 𝑃𝑃.
Box 6.3: Money supply increases that the central bank announces are
temporary
We have looked at the consequences for prices, inflation and real economic variables of
permanent changes to the quantity of money or the growth rate of the money supply. But
central banks might change the money supply temporarily in some circumstances.
For example, quantitative easing (QE) might increase the money supply but it is the central
bank’s stated intention to unwind the policy in the future. QE expansions of money
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supply have turned out to be persistent in most countries, although this may not have
been expected when they were first begun. There are cases where QE has been
temporary, such as the Bank of Japan QE policy from 2001, which was largely reversed in
2006. Another example of a temporary change is the ‘de-monetization’ experiment in
India in 2016, where there was a temporary decline of the money supply.
To see what difference it makes when a money-supply change is expected to be
temporary, suppose 𝑀𝑀 𝑠𝑠 = 𝑀𝑀 is expected to change for only one time period. Throughout,
we hold the expected future money supply 𝑀𝑀′ constant. Consequently, the equilibrium
future price level 𝑃𝑃′ does not change in this example. The Fisher equation 𝑖𝑖 = 𝑟𝑟 + 𝜋𝜋 and
the definition of expected inflation 𝜋𝜋 = (𝑃𝑃′ − 𝑃𝑃)/𝑃𝑃 imply that the nominal interest rate 𝑖𝑖
is:
𝑖𝑖 = 𝑟𝑟 +
𝑃𝑃′ − 𝑃𝑃
𝑃𝑃′
= 𝑟𝑟 + − 1
𝑃𝑃
𝑃𝑃
Money-market equilibrium is the equation 𝑀𝑀 = 𝑀𝑀 𝑠𝑠 = 𝑀𝑀𝑑𝑑 = 𝑃𝑃𝑃𝑃(𝑌𝑌, 𝑖𝑖), and, hence:
𝑀𝑀 = 𝑃𝑃𝑃𝑃 �𝑌𝑌, 𝑟𝑟 +
𝑃𝑃′
− 1�
𝑃𝑃
The key point to note is that a higher price level 𝑃𝑃 lowers the nominal interest rate 𝑖𝑖 here,
so the effect of 𝑃𝑃 on money demand is magnified. We ignore here any effect of 𝑖𝑖 on 𝑌𝑌 (but
accounting for that would further boost the impact of 𝑃𝑃 on 𝑀𝑀𝑑𝑑 ). In what follows, we
assume that nominal interest rate 𝑖𝑖 remains positive throughout. Figure 6.17 shows the
relationship between 𝑀𝑀𝑑𝑑 and 𝑃𝑃 in this case for given 𝑌𝑌 and 𝑟𝑟 and the relationship in the
case where changes in the money supply are permanent, in which case money demand
𝑀𝑀𝑑𝑑 = 𝑃𝑃𝑃𝑃(𝑌𝑌, 𝑟𝑟) is proportional to 𝑃𝑃 and is thus represented by a straight line in the
diagram.
Figure 6.17: Temporary increase in money supply
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Following temporary increase in 𝑀𝑀, the money supply curve 𝑀𝑀 𝑠𝑠 shifts to the right as usual.
If the policy is expected to be reversed in future, any rise in the price level is also expected
to be reversed. Therefore, a higher price level 𝑃𝑃 would create expectations of future
deflation, reducing the nominal interest rate 𝑖𝑖 and boosting money demand. In the
diagram, 𝑀𝑀𝑑𝑑 is thus less steep than the usual 𝑃𝑃𝑃𝑃(𝑌𝑌, 𝑟𝑟) money-demand function. It follows
that the price level rises by proportionately less than 𝑀𝑀 does, in contrast to the case of a
permanent change where 𝑃𝑃 rises in proportion to 𝑀𝑀.
This different prediction compared to the case of a permanent change in 𝑀𝑀 is likely to be
quantitatively significant. If there were a 25 per cent higher money supply temporarily and
𝑃𝑃 went up by 25 per cent initially then this would require 25 per cent expected deflation
subsequently. But that cannot be an equilibrium because 𝑖𝑖 ≥ 0 implies deflation cannot
exceed the much lower equilibrium value of the real interest rate 𝑟𝑟. The price level 𝑃𝑃 must
therefore rise by far less than 25 per cent.
6.10 Optimal monetary policy and the costs of inflation
Focusing on money’s role as a medium of exchange, what should the central bank do if it
desires to make the economy run smoothly?
6.10.1 Economic efficiency
To answer this question, we need to know what the socially optimal level of economic
activity is. The marginal value of households’ time in terms of goods is the marginal rate of
substitution 𝑀𝑀𝑀𝑀𝑀𝑀𝑙𝑙,𝐶𝐶 between leisure and consumption. The economy’s ability to transform
households’ time into goods at margin, the marginal rate of transformation 𝑀𝑀𝑀𝑀𝑀𝑀𝑙𝑙,𝐶𝐶 , is given
by the marginal product of labour 𝑀𝑀𝑃𝑃𝑁𝑁 . Ignoring transaction costs, efficiency therefore
requires that 𝑀𝑀𝑀𝑀𝑁𝑁 = 𝑀𝑀𝑀𝑀𝑀𝑀𝑙𝑙,𝐶𝐶 .
In respect of transaction costs, we noted that there are costs of using substitutes for money,
or using time and effort to economise on holding money. Those resource costs are
represented by the function 𝑍𝑍(𝑋𝑋) and they constitute a social cost of carrying out
transactions. On the other hand, when holding money, the cost of forgone interest is not a
social cost because the government gains an equal amount of seigniorage revenues.
Forgone interest is simply a transfer from holders of money to issuers of money.
6.10.2 Monetary policy, efficiency, and the Friedman rule
We now consider how the choice of monetary policy affects the efficiency of the economy’s
equilibrium. The demand for labour 𝑁𝑁 𝑑𝑑 is given by 𝑀𝑀𝑀𝑀𝑁𝑁 = 𝑤𝑤 and labour supply 𝑁𝑁 𝑠𝑠 is
determined by 𝑀𝑀𝑀𝑀𝑀𝑀𝑙𝑙,𝐶𝐶 = 𝑤𝑤⁄(1 + 𝑖𝑖). This equation for 𝑁𝑁 𝑠𝑠 comes from wages being paid as
money that must be held for some period before it can be spent, as explained in Section 6.5.
When the labour market is in equilibrium (𝑁𝑁 𝑑𝑑 = 𝑁𝑁 𝑠𝑠 ), it follows that 𝑀𝑀𝑀𝑀𝑁𝑁 = 𝑤𝑤 =
(1 + 𝑖𝑖 )𝑀𝑀𝑀𝑀𝑀𝑀𝑙𝑙,𝐶𝐶 . A positive value of the nominal interest rate 𝑖𝑖 implies 𝑀𝑀𝑀𝑀𝑁𝑁 > 𝑀𝑀𝑀𝑀𝑀𝑀𝑙𝑙,𝐶𝐶 , which
corresponds to employment and output being inefficiently low relative to the optimal
outcome with 𝑀𝑀𝑃𝑃𝑁𝑁 = 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 .
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It is also possible to consider efficiency in respect of transaction costs. There is a demand
𝑋𝑋 𝑑𝑑 (𝑖𝑖 ) for substitutes for money that depends on the nominal interest rate 𝑖𝑖, which is the
opportunity cost of holding money. As explained in Section 6.4, the demand for 𝑋𝑋 is
increasing in the nominal interest rate 𝑖𝑖. The social cost of transactions is then 𝑍𝑍(𝑋𝑋 𝑑𝑑 (𝑖𝑖 )),
where forgone interest is not included directly because it is not a social cost. A positive
nominal interest rate 𝑖𝑖 implies 𝑋𝑋 𝑑𝑑 (𝑖𝑖 ) > 0 and 𝑍𝑍(𝑋𝑋 𝑑𝑑 (𝑖𝑖 )) > 0, which means that transaction
costs are inefficiently high.
Now suppose monetary policy is conducted so that 𝑖𝑖 = 0. This means there is no forgone
interest when holding money. Consequently, there is no incentive to find substitutes for
money, or incur costs in economising on holding money, so transaction costs are reduced to
zero, i.e. 𝑍𝑍(𝑋𝑋) = 0. Furthermore, there is no implicit tax on economic activity (work) that
depends on holding money because money is as good a store of value as other assets. This
means that 𝑀𝑀𝑀𝑀𝑁𝑁 = 𝑀𝑀𝑀𝑀𝑀𝑀𝑙𝑙,𝐶𝐶 , so employment and output are at their efficient levels. In
conclusion, the monetary policy 𝑖𝑖 = 0 ensures the economy’s equilibrium is efficient. This
policy of keeping the nominal interest rate at zero is known as the ‘Friedman rule’.
6.10.3 The optimal rate of inflation and the Friedman rule
To implement the Friedman rule, the central bank needs to set the money supply growth
rate 𝜇𝜇 so that 𝑖𝑖 = 0 is achieved. The analysis in Section 6.7 shows that inflation and money
supply growth are equal, that is, 𝜋𝜋 = 𝜇𝜇. By using the Fisher equation 𝑖𝑖 = 𝑟𝑟 + 𝜋𝜋, it follows
that 𝜇𝜇 = −𝑟𝑟 is required to implement the Friedman rule. A positive equilibrium real interest
rate 𝑟𝑟 (which is independent of monetary policy here) implies the required money growth
rate 𝜇𝜇 is negative. Therefore, the central bank must reduce the nominal money supply over
time to implement the Friedman rule.
Since 𝜋𝜋 = 𝜇𝜇 = −𝑟𝑟 < 0, the Friedman rule requires deflation. In other words, the inflation
rate needed for economic efficiency is negative. Higher rates of inflation 𝜋𝜋, including zero or
positive rates, imply that 𝑖𝑖 is higher, which means the economy’s equilibrium is further away
from what is efficient. The Friedman rule thus provides a way to understand the costs of
inflation but also suggests that deflation is a good thing.
6.10.4 The fiscal implications of following the Friedman rule
Implementing the Friedman rule has fiscal implications because no seigniorage revenue is
received when 𝑖𝑖 = 0. Governments therefore need to find alternative sources of tax
revenue to continue to pursue their plans for public expenditure.
Another way to think of this is that the negative inflation rate required for the Friedman rule
implies there is a positive real return on non-interest-bearing money coming from its
purchasing power growing over time. This makes it as good a store of value as bonds with
real return 𝑟𝑟. Since money offers the same return as bonds, money is effectively being
treated by the government as a debt liability that must be repaid. The deflation that
supports the Friedman rule is achieved by buying back money to reduce its supply, which
works like repaying a debt. Money can only be repurchased by the central bank selling its
assets or using tax revenue transferred from the government. Hence, other taxes must rise
if government spending is to remain unchanged.
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Note that if other taxes create distortions, for example, income tax, then it may not be
optimal to follow the Friedman rule because this would replace one distortion (positive 𝑖𝑖)
with another distortion (a higher income tax rate). We have implicitly been assuming that
the lost seigniorage revenue could be replaced by lump-sum taxes.
Box 6.4: Hyperinflations
Hyperinflation refers to an extremely high rate of inflation. The exact definition is
arbitrary, but a threshold of ≥ 50 per cent inflation per month is conventional. Such high
rates of inflation occur with extremely fast money growth rates, causing nominal interest
rates to be very high as well.
Our analysis of the non-superneutrality of money in Section 6.9 provides a reason why
such high rates of inflation have a damaging effect on the real economy. As explained in
Section 6.5, inflation works as a tax on economic activity that depends on using money.
This effect was incorporated into our model through the equation 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 = 𝑤𝑤/(1 + 𝑖𝑖) for
labour supply and the Fisher equation 𝑖𝑖 = 𝑟𝑟 + 𝜋𝜋. Section 6.10 shows that these negative
supply-side effects of inflation give rise to inefficiencies and constitute a social cost of
inflation.
While such effects are theoretically present even at single-digit rates of inflation, they are
likely to be very small in that case. The implicit tax rate on economic activity coming from
inflation is approximately equal to the nominal interest rate 𝑖𝑖 and, to be precise, this is the
nominal interest rate over the period of time people cannot avoid holding on to cash they
receive, not the annual nominal interest rate. Taking that period to be no more than a
month, single-digit annual inflation rates would not generate an implicit tax rate of more
than 1 per cent, resulting in a generally small social cost. However, in a hyperinflation with
monthly inflation rates above 50 per cent, it is easy to see that the implicit tax rate can be
very high, even if people try to shorten the period over which they hold on to cash they
receive. Hence, the social cost of inflation through this mechanism is far larger in a
hyperinflation.
If hyperinflation has such serious negative effects on economy, why do governments
choose very fast money supply growth? We have seen in Section 6.8 that governments
derive a fiscal advantage from money creation (‘seigniorage’). If there is a sudden, large
change in public expenditure needs, for example, a war, seigniorage provides a quick
source of extra revenue for governments without having to adjust explicit tax rates.
However, as inflation and the nominal interest rate rise, the demand for real money
balances declines. This implies real seigniorage revenues are limited, as shown in the
seigniorage Laffer curve in Figure 6.14.
If the government’s fiscal needs in an emergency exceed the top of the seigniorage Laffer
curve then attempts to raise further seigniorage revenues may lead to explosive rates of
money growth and accelerating inflation as real seigniorage revenues fall short
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of the government’s needs. Thus, a hyperinflation can easily spiral out of control with
severe consequences for the economy unless the government can reduce its expenditure
or find alternative sources of tax revenue.
Box 6.5: Cash and tax evasion
An important feature of money in the form of physical cash is its anonymity. Unlike most
other assets, there is no register of ownership or necessarily any record of transactions in
cash. Thus, cash is sometimes described as a ‘bearer bond’, meaning the owner is deemed
to be whoever has physical possession of the asset. The anonymity of cash makes it ideal
to evade taxes, including taxes on purchases, income, or wealth. Part of the demand for
cash therefore comes from its tax-evasion advantages relative to other assets.
Our analysis of the demand for money in Section 6.4 was based on a comparison of
benefits and costs. The marginal cost of holding higher real money balances 𝑀𝑀𝑑𝑑 /𝑃𝑃 is 𝑖𝑖, the
forgone interest on bonds. The marginal benefit of higher 𝑀𝑀𝑑𝑑 /𝑃𝑃 was 𝑞𝑞 = 𝑍𝑍 ′ (𝑋𝑋)
previously. Now, a marginal value 𝑒𝑒 of the tax-evasion advantage of money is added to this
and the overall marginal benefit of higher 𝑀𝑀𝑑𝑑 /𝑃𝑃 is 𝑞𝑞 + 𝑒𝑒.
The demand for cash is found where the marginal benefit equals the marginal cost, i.e.
where 𝑖𝑖 = 𝑞𝑞 + 𝑒𝑒 = 𝑍𝑍 ′ (𝑋𝑋) + 𝑒𝑒. This is equivalent to the equation 𝑍𝑍 ′ (𝑋𝑋) = 𝑖𝑖 − 𝑒𝑒, so the
nominal interest rate 𝑖𝑖 in the usual money demand function 𝑀𝑀𝑑𝑑 /𝑃𝑃 = 𝐿𝐿(𝑌𝑌, 𝑖𝑖) is replaced by
𝑖𝑖 − 𝑒𝑒 when cash has a tax evasion advantage. The new money demand function 𝑀𝑀𝑑𝑑 /𝑃𝑃 =
𝐿𝐿(𝑌𝑌, 𝑖𝑖 − 𝑒𝑒) is higher at each interest rate 𝑖𝑖, so there is a rightward shift of 𝑀𝑀𝑑𝑑 plotted
against 𝑖𝑖 as shown in Figure 6.18.
While individuals might gain from the use of cash for tax evasion, the marginal private
benefit 𝑒𝑒 is not a social benefit of using money. When tax evasion occurs, other taxes need
to be higher to pay for public expenditure. Furthermore, cash also facilitates criminal
activity, imposing negative externalities on others. These considerations mean it is not
desirable for monetary policy to maximise individuals’ use of cash in the way that it would
if the Friedman rule were followed.
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Figure 6.18: Money demand with tax evasion
Since the only social benefit of money is in reducing transaction costs 𝑞𝑞 = 𝑍𝑍′(𝑋𝑋), optimal
monetary policy should aim to push 𝑞𝑞 to zero, ignoring both the private cost of money in
terms of forgone interest 𝑖𝑖 (not a social cost) and the private benefit 𝑒𝑒 when money is
used for tax evasion (not a social benefit). With 𝑖𝑖 = 𝑞𝑞 + 𝑒𝑒 in equilibrium, this suggests
monetary policy should aim for 𝑖𝑖 = 𝑒𝑒 > 0, a positive nominal interest rate 𝑖𝑖 and a higher
rate of inflation 𝜋𝜋 (or less deflation) than what is implied by the Friedman rule. By
following this policy, the implicit tax on money through 𝑖𝑖 > 0 cancels out the tax evasion
advantage. Money being a worse store of value makes tax evasion harder and provides a
way to tax illegal activities.
Technological advances have made transactions by debit card and bank transfers much
cheaper and easier, even for small payments. As the payments system now enables bank
deposits to be used for almost all transactions, it has been suggested that physical cash
can and should be abolished. Since bank deposits lack the anonymity of cash, eliminating
cash would help to reduce tax evasion and criminal activity. But some argue that cash
might occasionally offer greater convenience and, more importantly, the anonymity
offered by cash could be a desirable feature in preserving individuals’ privacy and civil
liberties.
6.11 Conducting monetary policy by setting interest rates
Monetary policy has so far been described as an exogenous level, or growth rate, of the
money supply. The instrument of monetary policy was the quantity of money and the target
for monetary policy was a monetary target.
However, the ultimate objective of monetary policy is usually not the money supply itself
but control of interest rates, inflation, or other macroeconomic variables. Moreover, central
banks are often described as setting interest rates rather than setting the money supply.
And we know from Box 6.2 that a fixed money supply target does not achieve price stability
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if money demand is unstable, so a monetary target might not be desirable with these other
objectives in mind. This section explores how monetary policy works if the central bank uses
an interest rate as its operating target.
Rather than setting the money supply 𝑀𝑀 at some target level, the central bank now has a
target for the nominal interest rate 𝑖𝑖. However, market interest rates are not directly
controlled by the central bank, so it needs to vary some policy instrument under its control
to achieve its interest rate target. We assume this means the use of open-market
operations. The central bank must be willing to increase or decrease 𝑀𝑀 so that the money
market is in equilibrium at its target for the nominal interest rate 𝑖𝑖. This makes the supply of
money 𝑀𝑀 become an endogenous variable determined by the equation:
𝑀𝑀
= 𝐿𝐿(𝑌𝑌, 𝑖𝑖 )
𝑃𝑃
What are the implications of conducting monetary policy in this way? We will focus here on
the determination of prices and inflation, assuming that monetary policy does not affect
real GDP 𝑌𝑌 or the real interest rate 𝑟𝑟.
The Fisher equation is 𝑖𝑖 = 𝑟𝑟 + 𝜋𝜋 ′𝑒𝑒 , where 𝜋𝜋 ′𝑒𝑒 = (𝑃𝑃′𝑒𝑒 − 𝑃𝑃)/𝑃𝑃 explicitly denotes expected
inflation between the current and future time periods, which might not be the same as the
realised inflation rate. In equilibrium, expected inflation must be
𝜋𝜋 ′𝑒𝑒 = 𝑖𝑖 − 𝑟𝑟
Given an exogenous target for 𝑖𝑖 and an equilibrium for 𝑟𝑟 that is independent of monetary
policy, this equation determines a unique equilibrium for expected inflation 𝜋𝜋 ′𝑒𝑒 . A low
nominal rate 𝑖𝑖 is associated with low expected inflation in equilibrium. But what about the
absolute level of prices 𝑃𝑃 and the inflation rate between the past and current period?
With an endogenous money supply 𝑀𝑀, there are many possible price levels 𝑃𝑃 consistent
with 𝑀𝑀⁄𝑃𝑃 = 𝐿𝐿(𝑌𝑌, 𝑖𝑖). This is because 𝑀𝑀 adjusts to ensure this equation holds at the central
bank’s target level of 𝑖𝑖. It follows that setting a target for interest rates does not by itself
give the economy a ‘nominal anchor’ – the level of prices 𝑃𝑃 in terms of money is
indeterminate. In practice, what this means is that a fixed 𝑖𝑖 policy does not rule out
unexpected fluctuations in inflation, even though expected inflation is determinate.
A traditional monetary target does provide a nominal anchor in the sense of there being a
unique equilibrium for the levels of prices and inflation. Given a demand for real money
balances, an exogenous nominal quantity of money means there is only one possible level of
prices in equilibrium. But we have seen that a fixed money supply not desirable if real
money demand fluctuates. Moving away from targeting the money supply is desirable but
simply setting a fixed target for the nominal interest rate has pitfalls. The way forward is to
consider an alternative approach to monetary policy, the use of an interest-rate feedback
rule, the most famous example of which being the Taylor rule.
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6.12 Taylor rules and the Taylor principle
The Taylor rule is an example of what is known as an interest-rate feedback rule. Instead of
an exogenous money supply or interest rate target, a feedback rule has the central bank
actively adjust the interest rate it sets to meet an objective. Assume here that an inflation
target 𝜋𝜋 ∗ is the central bank’s only objective.
A simple version of the Taylor rule is given in the following equation:
𝑖𝑖 = 𝑟𝑟̂ + 𝜋𝜋 ∗ + 𝜙𝜙(𝜋𝜋 − 𝜋𝜋 ∗ )
This describes how the central bank sets the nominal interest rate 𝑖𝑖. According to the Taylor
rule, the level of 𝑖𝑖 should depend on the actual rate of inflation 𝜋𝜋 that occurs between the
past and current period. The coefficient 𝜙𝜙 indicates how strongly the central bank reacts to
inflation 𝜋𝜋 missing its target 𝜋𝜋 ∗ . If 𝜋𝜋 is one percentage point higher then 𝑖𝑖 is raised by 𝜙𝜙
percentage points. The special case of 𝜙𝜙 = 0 represents an exogenous interest rate target
that is not adjusted to the actual inflation rate. Finally, the term 𝑟𝑟̂ denotes the central
bank’s estimate of the equilibrium real interest rate 𝑟𝑟.
It is argued that a sufficiently strong response to inflation, as measured by the parameter 𝜙𝜙,
ensures the equilibrium inflation rate 𝜋𝜋 is determinate (and will be on target 𝜋𝜋 ∗ if the
estimate of 𝑟𝑟 is correct). To be precise, a sufficiently strong response means that 𝜙𝜙 > 1, so
the nominal interest rate reacts more than one-for-one to inflation. This is known as the
‘Taylor principle’.
To see the argument, we combine the Fisher equation 𝑖𝑖 = 𝑟𝑟 + 𝜋𝜋 ′𝑒𝑒 written explicitly in terms
of expected inflation 𝜋𝜋 ′𝑒𝑒 = (𝑃𝑃′𝑒𝑒 − 𝑃𝑃)/𝑃𝑃 between the current and future periods and the
Taylor rule 𝑖𝑖 = 𝑟𝑟̂ + 𝜋𝜋 ∗ + 𝜙𝜙(𝜋𝜋 − 𝜋𝜋 ∗ ). By eliminating 𝑖𝑖 from these equations:
𝑟𝑟 + 𝜋𝜋 ′𝑒𝑒 = 𝑟𝑟̂ + 𝜋𝜋 ∗ + 𝜙𝜙(𝜋𝜋 − 𝜋𝜋 ∗ ) = 𝑟𝑟̂ + 𝜋𝜋 + (𝜙𝜙 − 1)(𝜋𝜋 − 𝜋𝜋 ∗ )
This can be rearranged to write an equation for the expected change in the inflation rate
over time:
𝜋𝜋 ′𝑒𝑒 − 𝜋𝜋 = (𝜙𝜙 − 1)(𝜋𝜋 − 𝜋𝜋 ∗ ) − (𝑟𝑟 − 𝑟𝑟̂ )
The Taylor principle 𝜙𝜙 > 1 implies the coefficient on 𝜋𝜋 − 𝜋𝜋 ∗, the extent to which inflation 𝜋𝜋
misses its target 𝜋𝜋 ∗ , is positive. Consequently, higher inflation now would mean that
inflation is expected to rise faster in the future and lower inflation now would mean that
subsequent inflation is expected to fall faster. It follows that there is only one value of
inflation 𝜋𝜋 where subsequent inflation is not expected to keep rising or keep falling
(although this argument ignores the lower bound on 𝑖𝑖 as explained later in Box 6.6).
If 𝜋𝜋 ′𝑒𝑒 = 𝜋𝜋, so inflation is neither expected to rise or fall further in future, then:
𝜋𝜋 = 𝜋𝜋 ∗ +
𝑟𝑟 − 𝑟𝑟̂
𝜙𝜙 − 1
This is the unique stable equilibrium for inflation when 𝜙𝜙 > 1. If the Taylor principle 𝜙𝜙 > 1
is not satisfied then there are many stable paths of inflation over time that are equally
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consistent with the interest-rate rule and equilibrium in the economy. We see from the
equation that inflation is on target if the central bank’s estimate 𝑟𝑟̂ of the equilibrium real
interest rate 𝑟𝑟 is correct. Underestimating this (𝑟𝑟̂ < 𝑟𝑟) leads to inflation above the target
(𝜋𝜋 > 𝜋𝜋 ∗ ).
Our analysis of the Taylor rule suggests that central banks wanting to meet an inflation
target have two key tasks. First, to make a strong reaction to any deviation of inflation from
its target so that the Taylor principle (𝜙𝜙 > 1) is satisfied. Second, to obtain an accurate
estimate of market-clearing real interest rate 𝑟𝑟.
6.13 The liquidity trap and the zero lower bound
In this section we will study two important limitations on the power of monetary policy.
6.13.1 The zero lower bound
The zero lower bound is the claim that the nominal interest rate 𝑖𝑖 on bonds cannot be
negative, so the equilibrium of the economy always features 𝑖𝑖 ≥ 0. The logic for this comes
from the money demand trade-off analysed in Section 6.4. Holding more money has the
cost of forgoing interest when 𝑖𝑖 is positive. On the other hand, money is useful for making
payments and holding more of it avoids the costs of using substitutes for money, or of
managing to carry out transactions while holding only a small amount of money on average.
The marginal benefit of holding an extra unit of real money balances is represented by the
cost reduction 𝑍𝑍′(𝑋𝑋).
The marginal benefit 𝑍𝑍′(𝑋𝑋) cannot be negative but can be zero if 𝑋𝑋 has already reached 0. If
the nominal interest rate 𝑖𝑖 on bonds were negative, this would mean that cash is a better
store of value than bonds, while also having a non-negative benefit relative to bonds in
saving transaction costs. Hence, 𝑖𝑖 < 0 would imply money is always preferred to bonds,
which is not possible in equilibrium because then there would be no demand for bonds.
6.13.2 The liquidity trap
The liquidity trap is the idea that money and bonds become perfect substitutes at the
margin once the lower bound on nominal interest rates is reached. It implies increases in
the quantity of money might be passively absorbed with no impact on the economy.
At the zero lower bound 𝑖𝑖 = 0, no interest is forgone by holding more money. Furthermore,
once holdings of real money balances 𝑀𝑀𝑑𝑑 /𝑃𝑃 reach 𝑌𝑌 (the amount needed to make all
payments using money without incurring any transaction costs), the marginal benefit of
lowering transaction costs 𝑍𝑍(𝑋𝑋) by holding more money is zero. With a zero marginal cost
and a zero marginal benefit, households and firms are indifferent about whether they hold
higher 𝑀𝑀𝑑𝑑 /𝑃𝑃 or not. But 𝑀𝑀𝑑𝑑 /𝑃𝑃 can be larger than 𝑌𝑌 at 𝑖𝑖 = 0 because money has become as
good a store of value as bonds. Consequently, money demand 𝑀𝑀𝑑𝑑 is perfectly interest
elastic at the interest-rate lower bound 𝑖𝑖 = 0.
Using the same framework as in Box 6.3, we see that a temporary expansion of 𝑀𝑀 𝑠𝑠 has no
effect on either the nominal interest rate 𝑖𝑖 or the price level 𝑃𝑃 once the zero lower bound is
reached. In the left panel of Figure 6.19, the demand for real money balances is perfectly
interest elastic (horizonal) at 𝑖𝑖 = 0. The money market is still in equilibrium at 𝑖𝑖 = 0 after a
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shift of 𝑀𝑀 𝑠𝑠 /𝑃𝑃 to the right with higher 𝑀𝑀 𝑠𝑠 (assuming no change in 𝑃𝑃 for now). This additional
money is willingly held with no change in the interest rate. To confirm that the price level 𝑃𝑃
does not change either, the right panel of the figure shows the money demand function
𝑀𝑀𝑑𝑑 = 𝑃𝑃𝑃𝑃(𝑌𝑌, 𝑟𝑟 + (𝑃𝑃′ ⁄𝑃𝑃) − 1), which combines the Fisher equation 𝑖𝑖 = 𝑟𝑟 + 𝜋𝜋 = 𝑟𝑟 + (𝑃𝑃′ −
𝑃𝑃)/𝑃𝑃 with 𝑀𝑀𝑑𝑑 ⁄𝑃𝑃 = 𝐿𝐿(𝑌𝑌, 𝑖𝑖). Taking the future price level 𝑃𝑃′ as given (because the change in
𝑀𝑀 𝑠𝑠 is only temporary), money demand is perfectly elastic with respect to the price level 𝑃𝑃
once 𝑖𝑖 = 0 is reached. The horizontal demand curve implies that the rightward of the supply
curve 𝑀𝑀 𝑠𝑠 does not change the equilibrium price level.
Figure 6.19: Money demand at zero nominal interest rate
It is important to note this argument does not apply to permanent expansions of the money
supply 𝑀𝑀 𝑠𝑠 where the central bank can convince people that it will never reverse the policy
change. This should affect expectations of future prices 𝑃𝑃′, which would shift 𝑀𝑀𝑑𝑑 =
𝑃𝑃𝑃𝑃(𝑌𝑌, 𝑟𝑟 + (𝑃𝑃′ ⁄𝑃𝑃) − 1), resulting in a change of 𝑃𝑃 or 𝑖𝑖 or both.
Box 6.6: A deflation trap
We have seen in Section 6.12 that if a central bank aims to control inflation by setting the
nominal interest rate 𝑖𝑖 then it is important to satisfy the ‘Taylor principle’, a more than
one-for-one adjustment of 𝑖𝑖 to current inflation 𝜋𝜋. However, if 𝑖𝑖 is subject to the zero
lower bound, it may not be possible to cut 𝑖𝑖 sufficiently when inflation is significantly
below target. This means the Taylor principle cannot be satisfied for all rates of inflation 𝜋𝜋.
Therefore, a monetary policy based on setting the nominal interest rate has the risk that
the economy falls into a ‘deflation trap’ where inflation is persistently negative.
The central bank is assumed to follow the Taylor rule where possible, setting 𝑖𝑖 = 𝑟𝑟 + 𝜋𝜋 ∗ +
𝜙𝜙(𝜋𝜋 − 𝜋𝜋 ∗ ) if this results in 𝑖𝑖 ≥ 0, or 𝑖𝑖 = 0 otherwise. We assume here that the equilibrium
real interest rate 𝑟𝑟 > 0 is known to central bank. We suppose the inflation target is 𝜋𝜋 ∗ ≥
0, which means a target for price stability or a positive inflation rate. We have not yet seen
any reasons why it is desirable to have 𝜋𝜋 ∗ ≥ 0, but that will be covered in Section 9.5. The
Taylor rule has 𝜙𝜙 > 1, so the central bank aims to satisfy the Taylor principle.
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With 𝜙𝜙 > 1, the Taylor rule equation implies 𝑖𝑖 = 0 is reached for some inflation rate 𝜋𝜋
lying between −𝑟𝑟 and 𝜋𝜋 ∗ , which is where the zero lower bound becomes binding. We can
study what happens to inflation in equilibrium by combing the Fisher equation 𝑖𝑖 = 𝑟𝑟 + 𝜋𝜋 ′𝑒𝑒
in terms of expected future inflation 𝜋𝜋 ′𝑒𝑒 with the Taylor rule as was done in Section 6.12.
This leads to 𝜋𝜋 ′𝑒𝑒 − 𝜋𝜋 = (𝜙𝜙 − 1)(𝜋𝜋 − 𝜋𝜋 ∗ ) if 𝑖𝑖 ≥ 0 as seen in the earlier analysis of Taylor
rules. But it leads to 𝜋𝜋 ′𝑒𝑒 − 𝜋𝜋 = −𝑟𝑟 − 𝜋𝜋 if 𝑖𝑖 = 0 because the Fisher equation implies 𝜋𝜋 ′𝑒𝑒 =
−𝑟𝑟 at the interest-rate lower bound.
The relationship between the expected change in inflation 𝜋𝜋 ′𝑒𝑒 − 𝜋𝜋 and the current
inflation rate 𝜋𝜋 is plotted in Figure 6.20. The upward-sloping segment at the right of the
diagram where the Taylor principle is satisfied has already been discussed in Section 6.12
and the inflation target 𝜋𝜋 ∗ is a steady state. The downward-sloping segment on the left is
where the lower bound binds and the Taylor principle cannot be satisfied.
Figure 6.20: Multiple equilibria with a deflation trap
The diagram shows that there are paths of inflation falling below the target 𝜋𝜋 ∗ where 𝜋𝜋
eventually becomes negative and converges to another steady state at 𝜋𝜋 = −𝑟𝑟. This
second steady state is the ‘deflation trap’ that the Taylor rule fails to avoid.
6.14 Negative nominal interest rates
The models of money we have seen in this chapter predict that bonds cannot have a
negative nominal interest rate in equilibrium. But instances of negative nominal interest
rates have been observed, for example, in some eurozone countries from around 2016.
Does this mean we missing something important about money from our model?
6.14.1 Is there a lower bound on nominal interest rates?
The logic of our earlier argument for why 𝑖𝑖 < 0 should be impossible is that a negative
nominal return on bonds would lead wealth held in the form of bonds to be switched into
cash to receive the guaranteed zero nominal return (𝑖𝑖𝑚𝑚 = 0) on cash. Note that only cash by
its nature necessarily offers a zero nominal return; other forms of money in electronic
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accounts, such as reserves held at the central bank, could in principle have 𝑖𝑖𝑚𝑚 < 0.
However, as long as physical cash exists as a form of money, its zero nominal return is
always available to investors.
We will argue the key point missing here is that switching large amounts of wealth into
physical cash would entail security costs of keeping it safe. Unlike bonds and most assets,
there no register of ownership of physical cash – this is why cash gives anonymity – and so
holders of cash must consider the cost of keeping this physical object secure. Assume that
holding cash entails a security or storage cost that is a proportion ℎ of the amount of cash
held. This is a resource cost of holding cash, in addition to forgone interest 𝑖𝑖 (if any).
Consider again the money demand trade-off from Section 6.4. Raising 𝑋𝑋 to reduce the
amount of cash held and increase bond holdings leads to a gain of 𝑖𝑖 + ℎ per unit increase in
𝑋𝑋, saving security costs plus any forgone interest. Raising 𝑋𝑋 by one unit has a marginal cost
𝑞𝑞 = 𝑍𝑍 ′ (𝑋𝑋), so the optimal 𝑋𝑋 is found where 𝑖𝑖 + ℎ = 𝑞𝑞 = 𝑍𝑍′(𝑋𝑋), not where 𝑖𝑖 = 𝑍𝑍′(𝑋𝑋) as
before. Since 𝑖𝑖 + ℎ replaces 𝑖𝑖 but the equation is otherwise the same, it follows that the
usual money demand function 𝑀𝑀𝑑𝑑 /𝑃𝑃 = 𝐿𝐿(𝑌𝑌, 𝑖𝑖) becomes 𝑀𝑀𝑑𝑑 /𝑃𝑃 = 𝐿𝐿(𝑌𝑌, 𝑖𝑖 + ℎ). As money
demand is decreasing in 𝑖𝑖, adding the positive value of ℎ shifts 𝑀𝑀𝑑𝑑 downwards.
Geometrically, this is a parallel downward shift by an amount ℎ of the real money demand
curve plotted against 𝑖𝑖, as shown in Figure 6.21.
Figure 6.21: Demand for cash with security cost
The diagram shows it is now possible to have an equilibrium with a negative nominal
interest rate 𝑖𝑖 < 0. Bonds with a negative nominal return are willingly held because the
security costs of switching to physical cash holdings are too large. Note that bonds
themselves are not subject to the same security problems of cash because there is a register
of ownership.
Although we can now understand why nominal interest rates can be negative, it turns out
that there is still a lower bound on 𝑖𝑖, only now a negative one. As can be seen from the
diagram, it is not possible to have 𝑖𝑖 be less than −ℎ, that is, 𝑖𝑖 cannot be below the negative
of the security cost as percentage of the value of cash stored. The lower bound is therefore
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𝑖𝑖 ≥ −ℎ. In practice, ℎ should not be much more than 1 per cent for large amounts of
physical cash, which would suggest a lower bound on 𝑖𝑖 of approximately −1 per cent.
6.14.2 Costs of negative interest rates
Although our analysis has explained why a negative nominal interest rate 𝑖𝑖 is possible, it
does not explain why one would be desirable – and indeed the logic suggests that negative
nominal interest rates have social costs.
Consider what is optimal monetary policy here. There are resource costs of both holding
more cash (ℎ) and using substitutes for cash or economising on cash holdings (𝑍𝑍′(𝑋𝑋)). All
means of payment therefore have a social cost. The marginal net social cost of lower 𝑋𝑋 (or
equivalently, higher 𝑀𝑀𝑑𝑑 /𝑃𝑃) is ℎ − 𝑍𝑍 ′ (𝑋𝑋), so economic efficiency requires ℎ = 𝑍𝑍′(𝑋𝑋). Since
𝑖𝑖 + ℎ = 𝑍𝑍′(𝑋𝑋) in equilibrium, the Friedman rule 𝑖𝑖 = 0 therefore achieves efficiency. Note
that it is now efficient to have 𝑋𝑋 > 0, so cash is not used for all transactions. Having a
negative nominal interest rate 𝑖𝑖 < 0 has social costs because it leads to over-use of cash,
which wastes resources on security and storage costs, just as the earlier argument for the
Friedman rule pointed to the waste of resources in finding substitutes for cash when 𝑖𝑖 > 0.
6.14.3 Lowering the lower bound
The analysis suggests there is a negative lower bound on the nominal interest rate but one
not too far below zero in practice. What if governments want to lower interest rates further
below zero?
It is possible to envisage changes to the monetary system that would permit this. For
example, cash might have an ‘expiry date’, with an amount deducted if people need to
convert old, expired cash into new cash (what is sometimes known as a ‘Gesell tax’). If 𝜏𝜏 is
an explicit tax on holding cash, money demand 𝑀𝑀𝑑𝑑 is determined by the equation 𝑖𝑖 + 𝜏𝜏 =
𝑞𝑞 = 𝑍𝑍′(𝑋𝑋). Analogous to having a security cost ℎ, this implies the lower bound on 𝑖𝑖 is now
−𝜏𝜏, which can be lowered by raising 𝜏𝜏. Similar outcomes can be achieved by limits on
convertibility between physical cash and other forms of money such as reserves and bank
deposits. Finally, abolishing physical cash and moving to a system of purely electronic
money would remove the lower bound on nominal interest rates entirely.
None of this analysis explains why governments should do such things and points to
inefficiencies in making it harder or more costly to make use of money for transactions. In
Chapter 9, we see that there might be circumstances where macroeconomic stabilisation
policies work more effectively if the lower bound on nominal interest rates can be
circumvented.
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Chapter 7: Banking and finance
Our analysis of the supply of money in Chapter 6 assumed that all of the money supply was
fiat money created by a central bank. While a central bank is the monopoly supplier of its
currency and fiat money more broadly including reserves, this is not the only form of money
in use, nor even the most quantitatively important form of money in many countries.
This chapter investigates how the banking system affects the supply of money and how the
central bank interacts with commercial banks in setting monetary policy. The chapter also
considers the roles of banks and financial markets in the economy.
Essential reading
•
Williamson, Chapter 18.
7.1 Fractional reserve banking
In advanced economies, bank deposits are the type of money most prevalent in use for
transactions purposes. Bank deposits are a form of credit money. A deposit is a liability of
the commercial bank at which the account is held and this liability is a promise to repay fiat
money to the depositor on withdrawal. Unlike bonds or loans where there is a fixed date at
repayment falls due, many bank deposits can be transferred or withdrawn on demand. For
this reason, we refer to them as ‘demand deposits’ (held in ‘checking accounts’ or ‘current
accounts’, the terminology differing between countries). In what follows, we ignore savings
deposits (‘time deposits’) that are locked in for a definite period.
For households and firms to make use of demand deposits for payments, recipients must be
willing to treat such deposits as being as good as cash. This requires not only a right to
withdraw deposits and receive cash on demand but also the ability freely to transfer
deposits to payees by means of debit cards, bank transfers and cheques to other accounts,
including those held at other banks.
As demand deposits can be withdrawn or transferred, commercial banks themselves need
to hold fiat money. When there is a withdrawal, the bank must provide cash to the
depositor. When there is a payment made to an account at another bank, the payer’s bank
must provide an asset to payee’s bank for settlement of the transfer of funds. To avoid
credit risk, banks require settlement with a risk-free asset. The ultimate risk-free asset for
transfers of funds is fiat money itself because a unit of fiat money defines what asset a bank
deposit is a claim to.
Commercial banks hold fiat money as vault cash or reserves. Reserves are deposits at
central bank, essentially money in banks’ own accounts at the central bank. These reserves
are transferrable between banks. Moreover, the central bank is willing to exchange reserves
and cash one-for-one if requested by commercial banks.
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Although commercial banks hold fiat money as reserves (or vault cash), the amount is
normally smaller than the deposits households and firms hold with the commercial banks.
This is what is known as a fractional-reserve banking system, in that bank reserves are less
than 100 per cent of deposits. Not holding reserves equal to deposits is in the interests of
commercial banks because reserves typically offer an inferior return to other assets, for
example, loans made by the commercial banks or other financial assets.
Nonetheless, it is prudent for commercial banks to hold some reserves to cover transfers or
withdrawals of deposits. There may also be regulatory reserve requirements that impose
legal minimum levels of reserve holdings relative to deposits. Even where such reserve
requirements exist, banks usually hold some amount of excess reserves above the
minimum.
With the presence of deposits, cash and reserves, there are now several distinct measures
of the supply of money. The first is quantity of money usable by households or firms for
payments, referred to as the broad money supply.
Broad money supply = Bank deposits + Cash
This corresponds to M1 measure of money supply in the USA. A narrower notion of money
considers only the supply of fiat money, the narrow money supply.
Narrow money supply = Reserves + Cash
This measure is also known as the ‘monetary base’, and it corresponds to what is called M0
in some countries such as the UK. You may also encounter the terms ‘inside money’ and
‘outside money’. Inside money is deposits, which are created by private sector and ‘outside
money’ is fiat money, which is created by governments.
If transfers of funds deposited at commercial banks are treated as equivalent to fiat money
then households and firms can use either deposits or cash to make payments. One
difference between cash and deposits is that no interest is received by holding cash, while
deposits may pay interest at rate 𝑖𝑖𝑚𝑚 , where 𝑖𝑖𝑚𝑚 refers to the interest rate paid on ‘money’ as
distinct from bonds. Assuming no difference in convenience between cash and deposits
when making payments, there would be no demand for cash if 𝑖𝑖𝑚𝑚 > 0. While cash remains
in use for payments to varying degrees around the world, making this assumption allows us
to focus on the demand for bank deposits, which is what is new in this chapter. The demand
for cash has already been studied in Section 6.4.
If the interest rate on bonds (or savings deposits that cannot be transferred to make
payments) is 𝑖𝑖 then interest 𝑖𝑖 − 𝑖𝑖𝑚𝑚 is forgone when wealth is held as money in the form of
demand deposits. A theory of the demand for money can be developed along the same lines
as Section 6.4, with the interest-rate spread 𝑖𝑖 − 𝑖𝑖𝑚𝑚 replacing the opportunity cost 𝑖𝑖 of
holding cash. The demand for deposits by households and firms is given by the equation:
𝑀𝑀𝑑𝑑
= 𝐿𝐿(𝑌𝑌, 𝑖𝑖 − 𝑖𝑖𝑚𝑚 )
𝑃𝑃
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This equation specifies a demand for real money balances 𝑀𝑀/𝑃𝑃 that depends positively on
real income 𝑌𝑌 and negatively on 𝑖𝑖 − 𝑖𝑖𝑚𝑚 . The downward-sloping demand curve is depicted in
Figure 7.1 for given 𝑌𝑌, where the opportunity cost 𝑖𝑖 − 𝑖𝑖𝑚𝑚 is the variable on the vertical axis.
Figure 7.1: The demand for deposits
7.2 The tools of monetary policy
In this chapter we will study in more detail the tools that a central bank can use to
implement monetary policy and affect the supply of money, including indirectly the supply
of deposits created by commercial banks. Our earlier analysis in Section 6.6 supposed that
all money was fiat money and the supply of this was controlled by open-market operations.
However, open-market operations are not the only tool of monetary policy.
In general, we can identify three types of monetary policy instrument:
1. Open-market operations
2. Standing facilities
3. Reserve requirements.
7.2.1 Open-market operations
Open-market operations are purchases or sales of financial assets by the central bank. These
affect the supply of fiat money because the central bank makes and receives payments in
fiat money. Let us simplify our analysis by ignoring cash, so the supply of fiat money, the
narrow money supply or monetary base, is just supply of reserves, denoted by 𝑅𝑅.
Central-bank asset purchases raise 𝑅𝑅 as payment is added to the reserve accounts of the
commercial banks from which the financial assets are purchased. Central-bank asset sales
lower 𝑅𝑅 as commercial banks’ reserve balances are reduced when they make payments to
the central bank.
In principle, open-market operations can be conducted in a wide range of financial markets.
These could be outright purchases of long-term government bonds paid for with newly
created reserves. If the central bank continues to hold these bonds then the effect on the
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supply of reserves persists until the bonds mature. However, the central bank could sell the
bonds before maturity to reverse the increase in 𝑅𝑅. An alternative to outright purchases or
sales of assets is repo or reverse-repo operations. A ‘repo’ is a sale-and-repurchase
agreement where an asset is sold but also agreed to be bought back at a prearranged price
(the percentage increase in the repurchase price relative to the sale price being the interest
rate on the repo). Open-market operations with repos change the supply of reserves 𝑅𝑅 only
temporarily.
It is important to note that, while individual commercial banks can change the quantity of
reserves they hold by buying or selling financial assets, or lending or borrowing, the central
bank is the monopoly supplier of reserves 𝑅𝑅. Transfers of reserves or loans of reserves
among commercial banks do not change the aggregate supply of reserves.
7.2.2 Standing facilities
A feature of open-market operations is that they can be conducted at the discretion of the
central bank, meaning the central bank can choose the size of the open-market operations it
carries out. An alternative monetary policy tool where the extent of its use depends on the
actions of commercial banks is known as a standing facility. An example of a standing facility
is the ‘discount window’, an arrangement whereby the central bank agrees to lend reserves
to commercial banks at a known interest rate against collateral. The interest rate charged
on such discount loans is known as the central bank’s discount rate.
While the extent to which standing facilities are used depends on the behaviour of
commercial banks, the central bank can vary the terms on which the facilities are available,
for example, by changing the discount rate. For the discount window, or ‘borrowing facility’
as it is described by some central banks, the central bank stands ready to lend reserves to
commercial banks against appropriate collateral at discount rate 𝑖𝑖𝑏𝑏 , which can be varied by
the central bank. Note that 𝑖𝑖𝑏𝑏 is not a market-determined interest rate; it is under the direct
administrative control of the central bank.
Another standing facility is the payment of interest on commercial banks’ reserve balances,
sometimes referred to as a ‘deposit facility’. The central bank can choose to vary the
interest rate 𝑖𝑖𝑟𝑟 paid on reserve balances.
7.2.3 Reserve requirements
The final monetary policy instrument we will consider is the imposition of reserve
requirements on commercial banks, which have a direct effect on the demand for fiat
money by banks. Suppose that a commercial bank with deposits 𝑀𝑀 must hold at least an
amount 𝑞𝑞𝑞𝑞 in reserves, where 𝑞𝑞 is the required reserve ratio. The ratio 𝑞𝑞 may be 0 if no
reserve requirements are present, or close to zero where reserve requirements are small.
The case 𝑞𝑞 = 1 represents a requirement to hold 100 per cent of demand deposits as
reserves, which while far from the reality of bank regulation today, has been suggested by
some as an alternative to the fractional-reserve banking system.
If a commercial bank fails to meet its reserve requirement then it is charged interest at a
penalty rate on the size of any shortfall, or it is obliged to borrow from the central bank’s
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discount window to make good the shortfall. Here, we suppose the penalty rate is the same
as the discount rate 𝑖𝑖𝑏𝑏 . In the presence of reserve requirements, the central bank can
choose to pay interest at a different rate 𝑖𝑖𝑟𝑟𝑟𝑟 on required reserves, with 𝑖𝑖𝑟𝑟 being the interest
rate on reserves held in excess of reserve requirements.
As well as mandating reserve requirements, central banks could vary the required reserve
ratio 𝑞𝑞 but this is not often done in advanced economies.
7.3 The interbank market
Reserves are used by commercial banks to settle transfers of funds between them. Banks
can manage their holdings of reserves by borrowing or lending in the interbank markets.
Commercial banks can borrow or lend reserves among themselves at interest rate 𝑖𝑖. We will
treat this interest rate as being the same as the interest rate on nominal bonds. However,
note that interbank lending is typically of a very short maturity, for example, overnight.
Loans are often not secured by collateral, although the repo market allows for secured
lending of reserves. In practice, we will ignore these subtleties and suppose that lending in
such markets is equivalent to holding short-term bonds. The interbank and repo markets are
often referred to as the ‘money markets’ but, more precisely, they are markets for
borrowing money, specifically reserves.
7.3.1 The demand for reserves
What amount of reserves should a commercial bank hold and, hence, how much should it
borrow or lend in the interbank market to obtain its desired amount of reserves? The bank’s
depositors will make payments to accounts at other banks, which need to be settled by
transferring reserves. On the other hand, the bank will receive some reserves due to
payments from other banks going into its own depositors’ accounts. The required net
transfer of reserves resulting from the payments system is not completely predictable.
If a bank is left with more reserves relative to reserve requirements at a point when it
cannot go back to the interbank market to manage its reserve holdings then it keeps excess
reserves in its account at the central bank and receives interest 𝑖𝑖𝑟𝑟 . If it has insufficient
reserves relative to reserve requirements and borrowing from other banks is not feasible
then it must borrow the shortfall from the central bank at penalty rate 𝑖𝑖𝑏𝑏 . By holding more
reserves by borrowing from other banks when the interbank markets are open, the first of
these outcomes becomes more likely and the second less likely.
Suppose a bank borrows an additional unit of reserves. The cost of this borrowing is
interbank interest rate 𝑖𝑖. The benefit of holding more reserves depends on whether the
bank ends up with excess reserves or a shortfall of reserves. With excess reserves, the
benefit is the extra interest 𝑖𝑖𝑟𝑟 paid on its reserve balance. With a shortfall of reserves, the
benefit of an extra unit of reserves is saving the borrowing cost 𝑖𝑖𝑏𝑏 because the shortfall is
smaller. The overall expected benefit of is a weighted average of 𝑖𝑖𝑟𝑟 and 𝑖𝑖𝑏𝑏 depending on the
probabilities of the two events. When reserves are initially low, the probability of a shortfall
is high, which means more weight is put on the saving 𝑖𝑖𝑏𝑏 when calculating the expected
benefit of borrowing more in the interbank market.
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It is optimal for commercial banks to borrow reserves up to the point where the cost 𝑖𝑖
equals expected benefit. This implies the demand to borrow reserves decreases with the
cost 𝑖𝑖. With the interbank interest rate 𝑖𝑖 on the vertical axis and the quantity of reserve
holdings on the horizontal axis, there is a downward-sloping reserve demand curve as
shown in Figure 7.2.
Figure 7.2: The demand for reserves
The demand curve is bounded in a range of interbank interest rates between 𝑖𝑖𝑟𝑟 and 𝑖𝑖𝑏𝑏 . This
is because there would be an unlimited demand to borrow reserves if the cost were below
𝑖𝑖𝑟𝑟 , with banks able to profit from arbitraging the difference between 𝑖𝑖 and 𝑖𝑖𝑟𝑟 . There would
be no demand to borrow reserves if the cost were above 𝑖𝑖𝑏𝑏 because commercial banks can
always borrow from central bank instead at interest rate 𝑖𝑖𝑏𝑏 . Since the height of the demand
curve represents the expected benefit of borrowing a unit of reserves, and the probabilities
of having an excess or shortfall of reserves depend on reserve holdings relative to deposits,
the appropriate horizontal-axis variable is the quantity of reserves 𝑅𝑅 relative to amount of
deposits 𝑀𝑀, the ratio 𝑅𝑅⁄𝑀𝑀.
The reserve demand curve shifts vertically if there are changes to the central bank’s
standing-facility interest rates 𝑖𝑖𝑟𝑟 or 𝑖𝑖𝑏𝑏 . Higher reserve requirements 𝑞𝑞 shift the demand
curve to the right. Note that the interest rate on required reserves 𝑖𝑖𝑟𝑟𝑟𝑟 does not affect the
reserve demand curve here because the amount of interest paid on required reserves does
not depend on banks’ decisions that affect the extent of their holdings of excess reserves.
7.3.2 Equilibrium in the interbank market
The reserve demand curve can be combined with a reserve supply curve to understand the
determinants of the interbank interest rate and how it is affected by the various tools of
monetary policy.
The total supply of reserves available to commercial banks as a whole is determined by the
central bank. This means there is an inelastic supply of reserves 𝑅𝑅 and the position of the
reserve supply curve is affected by open-market operations. Central-bank asset purchases
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shift the reserve supply curve to the right and asset sales shift the supply curve to the left.
Taking deposits 𝑀𝑀 as given for now, the supply of 𝑅𝑅⁄𝑀𝑀 is also inelastic.
Figure 7.3 represents the interbank market equilibrium by putting together the demand and
supply curves for reserves. The equilibrium interbank interest rate 𝑖𝑖 ∗ is at the intersection of
the demand and supply curves 𝑅𝑅𝑑𝑑 and 𝑅𝑅 𝑠𝑠 .
Figure 7.3: Equilibrium in the interbank market
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Box 7.1: The ‘channel’ system of monetary policy
The traditional approach to implementing monetary policy is for the central bank to use
open-market operations. These purchases or sales of assets change the supply of
reserves, which affects equilibrium interest rates in money markets. An alternative
system works by varying the terms of the central bank’s standing facilities instead.
This system depends on there being a pair of standing facilities:
•
•
Interest 𝑖𝑖𝑟𝑟 paid on excess reserves held by commercial banks
Interest 𝑖𝑖𝑏𝑏 charged on loans of reserves to commercial banks.
It is known as the ‘channel’ or ‘corridor’ system because the interbank interest rate lies in
a channel between 𝑖𝑖𝑟𝑟 and 𝑖𝑖𝑏𝑏 as shown in Figure 7.3. The system is usually a target for the
interbank interest rate 𝑖𝑖 ∗ that lies at the centre of the channel between 𝑖𝑖𝑟𝑟 and 𝑖𝑖𝑏𝑏 .
Suppose the central bank wants to raise the interest rate 𝑖𝑖 using the channel system, for
example, increase 𝑖𝑖 by 25 basis points (0.25 percentage points). To do this, the central
bank simply raises its standing-facility interest rates 𝑖𝑖𝑟𝑟 and 𝑖𝑖𝑏𝑏 in parallel by the amount it
wants to increase 𝑖𝑖. In Figure 7.4, 𝑖𝑖𝑟𝑟 and 𝑖𝑖𝑏𝑏 are raised by 25 basis points, and this leads to
a parallel upwards shift of the reserve demand curve 𝑅𝑅𝑑𝑑 by exactly the size of the
increase in 𝑖𝑖𝑟𝑟 and 𝑖𝑖𝑏𝑏 . Intuitively, 𝑅𝑅𝑑𝑑 depends on 𝑖𝑖 relative to 𝑖𝑖𝑟𝑟 and 𝑖𝑖 relative to 𝑖𝑖𝑏𝑏 . With
no change in 𝑅𝑅 𝑠𝑠 because no open-market operation is conducted, the equilibrium 𝑖𝑖 ∗ rises
by same amount as 𝑖𝑖𝑟𝑟 and 𝑖𝑖𝑏𝑏 do.
We have seen that varying the terms of a pair of standing facilities in parallel allows very
precise adjustments of interest rates in the channel system. It is not necessary to know
anything about the exact position or shape of the reserve demand curve 𝑅𝑅𝑑𝑑 to achieve
this. Furthermore, while there is no guarantee that the equilibrium interest rate is
necessarily in the middle of the channel, a narrow corridor between 𝑖𝑖𝑟𝑟 and 𝑖𝑖𝑏𝑏 means 𝑖𝑖 ∗
cannot stray too far from its target. If the central bank wants to bring 𝑖𝑖 ∗ as close as
possible to the middle of the channel then it can use ‘fine-tuning’ open-market
operations to shift 𝑅𝑅 𝑠𝑠 so that it intersects 𝑅𝑅𝑑𝑑 in the middle of the channel.
Figure 7.4: Raising interest rates using the channel system
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In principle, the channel used in the channel system could be very narrow but that would
tend to reduce trade in the interbank market and result in frequent use by commercial
banks of central bank’s borrowing facility instead. To keep the interbank market active
and avoid potential credit risk exposure of the central bank, the channel needs to be
wide enough to encourage commercial banks to borrow or lend reserves among each
other in the interbank market.
Box 7.2: The ‘floor’ system of monetary policy
Another system of monetary policy implementation is known as the ‘floor’ system. Like
the channel system, this relies on there being a central-bank standing facility to pay
interest on excess reserves. But rather than use fine-tuning open-market operations to
keep the interbank interest rate near the centre of the channel, the floor system features
a permanently large supply of reserves.
Figure 7.5 is a representation of how the floor system operates. The reserve supply curve
𝑅𝑅 𝑠𝑠 has been shifted far to the right. The supply curve intersects the reserve demand
curve 𝑅𝑅𝑑𝑑 on the flat section of 𝑅𝑅𝑑𝑑 where 𝑖𝑖 ≈ 𝑖𝑖𝑟𝑟 , so the market interest rate is very close
to the interest rate 𝑖𝑖𝑟𝑟 paid on holdings of excess reserves. Note that the very large supply
of reserves reduces commercial banks’ incentive to trade in the interbank market
because the likelihood of being short of reserves and needing to borrow from the central
bank at rate 𝑖𝑖𝑏𝑏 is very low.
When the floor system is used, market interest rates are changed by varying the interest
rates 𝑖𝑖𝑟𝑟 and 𝑖𝑖𝑏𝑏 on the central bank’s standing facilities just as is done with the channel
system. Figure 7.6 shows the effects of increasing 𝑖𝑖𝑟𝑟 and 𝑖𝑖𝑏𝑏 by 25 basis points, which
causes a parallel upward shift of 𝑅𝑅𝑑𝑑 by the same amount and thus raises the market
interest rate 𝑖𝑖 by 25 basis points.
Figure 7.5: The floor system of monetary policy
Like the channel system, we see that the floor system allows interest rates to be
controlled very precisely. Moreover, it does so without the need for any ‘fine-tuning’
open-market operations to steer the market interest rate to the centre of a channel.
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Another advantage of the floor system is that it enables interest-rate decisions
implemented by changing 𝑖𝑖𝑟𝑟 to be separated from choices about the quantity of reserves
𝑅𝑅 𝑠𝑠 the central bank supplies – as long as 𝑅𝑅 𝑠𝑠 is above the amount where the reserve
demand curve flattens out at 𝑖𝑖 ≈ 𝑖𝑖𝑟𝑟 . This allows central banks to raise interest rates
without having to unwind their quantitative easing (QE) policies. Indeed, many central
banks found themselves using the floor system by default having chosen to expand the
supply of reserves massively with QE, but also to pay interest on reserves.
Figure 7.6: Raising interest rates using the floor system of monetary policy
As we will see in Box 7.3, the floor system has an efficiency advantage because it does
not depend on maintaining an artificial scarcity of reserves. Central banks face no
resource cost in increasing the supply of reserves but reserves have a social benefit
through their implications for commercial banks’ supply of deposits. In the floor system,
reserves are supplied up to the point where commercial banks are ‘satiated’, while in the
channel system, 𝑅𝑅 𝑠𝑠 needs to be limited to keep 𝑖𝑖 ∗ at the centre of channel. Although
decisions about the supply of reserves have no resource costs, they do have implications
for the profitability of central banks and, hence, for seigniorage revenues received by
governments.
7.4 The supply of bank deposits
The money supply 𝑀𝑀 𝑠𝑠 available to households and firms is the quantity of deposits created
by the banking system, ignoring cash here for simplicity. A bank deposit is created whenever
a commercial bank makes a loan and credits an account at the bank with the funds. By doing
this, a commercial bank earns interest at rate 𝑖𝑖 on the loan and pays interest at rate 𝑖𝑖𝑚𝑚 on
the deposit. The commercial bank therefore profits from the interest margin, or interestrate spread 𝑖𝑖 − 𝑖𝑖𝑚𝑚 . If this interest margin is positive, what stops a commercial bank from
making more loans and creating more deposits?
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7.4.1 Costs of maintaining adequate reserves
Even if a commercial bank could obtain a risk-free return 𝑖𝑖 by making a loan, creating a
larger quantity of deposits increases the risk of a shortfall of reserves when the household
or firm holding those deposits uses them to make payments. This means there are costs of
reserve management that need to be set against the interest margin 𝑖𝑖 − 𝑖𝑖𝑚𝑚 .
A shortfall of reserves leaves a commercial bank having to borrow from the central bank at a
penalty interest rate 𝑖𝑖𝑏𝑏 , while excess reserves only earn a lower interest rate 𝑖𝑖𝑟𝑟 < 𝑖𝑖𝑏𝑏 .
Creating more deposits 𝑀𝑀 𝑠𝑠 relative to reserve holdings 𝑅𝑅 thus increases the expected cost
to the commercial bank of maintaining an adequate supply of reserves. Therefore, to be
willing to supply more deposits, a higher interest margin 𝑖𝑖 − 𝑖𝑖𝑚𝑚 is needed. This implies the
ratio 𝑅𝑅/𝑀𝑀 𝑠𝑠 of reserves to deposits is negatively related to the interest-rate spread 𝑖𝑖 − 𝑖𝑖𝑚𝑚 as
depicted in Figure 7.7.
Figure 7.7: Relationship between the reserve ratio and interest-rate spread
The real quantity of deposits 𝑀𝑀 𝑠𝑠 /𝑃𝑃 supplied by commercial banks can be broken down into:
𝑀𝑀 𝑠𝑠 𝑀𝑀 𝑠𝑠 𝑅𝑅
=
×
𝑃𝑃
𝑅𝑅
𝑃𝑃
The term 𝑅𝑅/𝑃𝑃 is the real supply of reserves, with the aggregate quantity of reserves 𝑅𝑅 held
by commercial banks being determined by the reserve supply 𝑅𝑅 𝑠𝑠 chosen by the central
bank. We have seen that the costs to commercial banks of maintaining adequate reserves
imply that the ratio 𝑀𝑀 𝑠𝑠 /𝑅𝑅 increases with 𝑖𝑖 − 𝑖𝑖𝑚𝑚 . Hence, given 𝑅𝑅/𝑃𝑃, the real supply of
deposits 𝑀𝑀 𝑠𝑠 /𝑃𝑃 increases with the interest margin 𝑖𝑖 − 𝑖𝑖𝑚𝑚 .
Reserve requirements imposed on commercial banks also limit deposit creation and these
impose additional costs when interest 𝑖𝑖𝑟𝑟𝑟𝑟 paid on required reserves is below the market
interest rate 𝑖𝑖. Having 𝑖𝑖𝑟𝑟𝑟𝑟 < 𝑖𝑖 means that a larger interest margin 𝑖𝑖 − 𝑖𝑖𝑚𝑚 is required by
commercial banks when creating deposits, all else equal.
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7.4.2 Bank capital requirements
A further constraint on deposit creation comes from bank capital requirements. These
impose an upper limit on a commercial bank’s assets (for example, loans) relative to the
bank’s capital. The ‘capital’ of a commercial bank refers to the bank’s equity, which strictly
defined is funds contributed by shareholders plus undistributed bank profits. Bank capital
requirements are usually expressed as a minimum ratio of bank capital to assets and those
assets may be ‘risk weighted’, with safer assets given a lower weight.
When a commercial bank creates deposits by making loans, this adds both to the bank’s
assets and its liabilities, increasing the total size of its balance sheet. However, it does not
immediately have any impact on bank capital and thus deposit creation lowers the ratio of
capital to assets. Bank capital requirements can therefore impose limits on the quantity of
deposits created by commercial banks. Whether bank capital requirements are the binding
constraint on deposit creation or the need to maintain adequate reserves depends on the
regulatory rules in place and the system of monetary policy implementation.
The need for bank capital is due to bank assets being risky while deposits should be safe.
Capital can absorb losses up to a point without jeopardising banks’ ability to repay
depositors. As we will discuss in Section 7.12, bank capital requirements are imposed by
regulators to avoid bank failures but even in the absence of explicit rules, prudential
concerns ought to constrain banks’ own decisions about deposit creation.
7.5 Equilibrium in the banking market
The equilibrium money supply, the quantity of deposits created by the banking system, can
be found by combining the demand curve for deposits 𝑀𝑀𝑑𝑑 /𝑃𝑃 = 𝐿𝐿(𝑌𝑌, 𝑖𝑖 − 𝑖𝑖𝑚𝑚 ) from Section
7.1 with the supply curve 𝑀𝑀 𝑠𝑠 /𝑃𝑃 = (𝑀𝑀 𝑠𝑠 ⁄𝑅𝑅) × (𝑅𝑅⁄𝑃𝑃) derived in Section 7.4. We have seen
that the supply of deposits 𝑀𝑀 𝑠𝑠 /𝑃𝑃 increases with the interest margin 𝑖𝑖 − 𝑖𝑖𝑚𝑚 , all else equal.
This interest margin is also the opportunity cost to households and firms of holding money
in the form of deposits and deposit demand depends negatively on it. Therefore, given 𝑃𝑃
and 𝑌𝑌, we can draw a downward-sloping deposit demand curve and an upward-sloping
deposit supply curve and find the equilibrium in the banking market. This equilibrium is
depicted in Figure 7.8.
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Figure 7.8: Equilibrium in the banking market
The banking market can be analysed in a similar way to the money market of Chapter 6, the
only difference being that the quantity of money (deposits here) is endogenous owing to the
upward-sloping supply curve. The demand curve behaves in the same way as in Chapter 6.
The supply curve shifts to the right if there is an increase in the real supply of reserves 𝑅𝑅/𝑃𝑃.
A higher interest-rate penalty 𝑖𝑖𝑏𝑏 − 𝑖𝑖𝑟𝑟 faced by banks that have a shortfall of reserves would
increase the costs of reserve management and lower 𝑀𝑀 𝑠𝑠 /𝑅𝑅, shifting the deposit supply
curve to the left. Similarly, increasing reserve requirements 𝑞𝑞 when the interest rate paid on
required reserves is low (such as if 𝑖𝑖𝑟𝑟𝑟𝑟 = 𝑖𝑖𝑟𝑟 < 𝑖𝑖𝑏𝑏 ) shifts the deposit supply curve to the left.
An increase in binding bank capital requirements would also reduce the supply of deposits.
Box 7.3: Should central banks pay interest on reserves?
The supply of fiat money comprises cash and reserves held by commercial banks in
accounts at the central bank. By its nature, it is impractical to pay interest on physical
cash in the form of notes and coins. Traditionally, central banks did not pay interest on
reserves even though there is no technical barrier to doing so. However, in the last two
decades, more central banks have begun to pay interest on reserves, including the US
Federal Reserve.
Reserves held by commercial banks can be broken down into required reserves, i.e.
those held to satisfy regulatory reserve requirements (if any), and excess reserves,
those held beyond the minimum requirements. In principle, central banks can pay
interest at different rates on these two types of reserves, with 𝑖𝑖𝑟𝑟𝑟𝑟 being the interest
rate on required reserves and 𝑖𝑖𝑟𝑟 the interest rate on excess reserves. What reasons are
there why central banks would want to have 𝑖𝑖𝑟𝑟𝑟𝑟 and/or 𝑖𝑖𝑟𝑟 be greater than zero?
Monetary policy implementation
A first reason to pay a generally positive interest rate 𝑖𝑖𝑟𝑟 on excess reserves is if the
central bank wants to use either the ‘channel/corridor’ or ‘floor’ systems of monetary
policy implementation. We have seen in Box 7.1 and Box 7.2 that the market interest
rate 𝑖𝑖 is controlled by varying 𝑖𝑖𝑟𝑟 in those systems, so it is not possible to use them if
𝑖𝑖𝑟𝑟 = 0.
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If the central bank chooses 𝑖𝑖𝑟𝑟 = 0 then this leaves it with the ‘traditional’ approach to
implementing monetary policy. In that traditional system, open-market operations are used
to shift the reserve supply curve 𝑅𝑅 𝑠𝑠 . This changes the point of intersection with the reserve
demand curve 𝑅𝑅𝑑𝑑 in the interbank market and hence, affects the equilibrium interest rate
𝑖𝑖 ∗ as seen in Figure 7.9.
Figure 7.9: The traditional system of monetary policy implementation
However, this traditional system is subject to greater practical difficulties to the extent that
the reserve demand curve 𝑅𝑅𝑑𝑑 has an uncertain shape and position, with shocks to banks’
demand for reserves shifting 𝑅𝑅𝑑𝑑 . This means it is not certain what size of open-market
operation is needed to achieve a given change in 𝑖𝑖 ∗ . Moreover, unlike the channel system
that bounds 𝑖𝑖 ∗ in a tight range between 𝑖𝑖𝑟𝑟 and 𝑖𝑖𝑏𝑏 , only the discount-window standing
facility acts to cap 𝑖𝑖 ∗ at the interest rate 𝑖𝑖𝑏𝑏 on loans of reserves from the central bank. And
often even that was not effective because of a ‘stigma’ attached to commercial banks that
were seen to borrow from the central bank through its discount window. The traditional
system can be seen as essentially a very wide channel between 𝑖𝑖𝑟𝑟 = 0 and 𝑖𝑖𝑏𝑏 + stigma.
The cost of banking and the efficiency of money as a medium of exchange
Where money is physical cash, we saw in Section 6.10 that the Friedman rule 𝑖𝑖 = 0 yields
efficiency in the use of money as medium of exchange. Note that real money demand
𝑀𝑀𝑑𝑑 /𝑃𝑃 = 𝐿𝐿(𝑌𝑌, 𝑖𝑖) is maximised at 𝑖𝑖 = 0 because this means the opportunity cost of holding
cash is zero, which avoids taxing economic activity that depends on using money.
Where money is deposits at commercial banks that pay interest rate 𝑖𝑖𝑚𝑚 , the real demand
for money is 𝑀𝑀𝑑𝑑 /𝑃𝑃 = 𝐿𝐿(𝑌𝑌, 𝑖𝑖 − 𝑖𝑖𝑚𝑚 ), with 𝑖𝑖 − 𝑖𝑖𝑚𝑚 now being the opportunity cost of holding
money. Just as there are no resource costs of central banks creating fiat money, similarly,
there are no resource costs of commercial banks creating deposits. Hence, when money is
bank deposits, the equivalent of the Friedman rule that maximises the use of money and
avoids imposing costs on economic activity that depends on money is to have 𝑖𝑖𝑚𝑚 = 𝑖𝑖.
To achieve 𝑖𝑖𝑚𝑚 = 𝑖𝑖, the real supply of deposits 𝑀𝑀 𝑠𝑠 /𝑃𝑃 must be sufficiently large. As discussed
in Section 7.1, commercial banks need to hold reserves for the payments system to
operate, even when there are no regulatory reserve requirements. Banks’ deposit creation
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decision studied in Section 7.4 then explains why deposit supply 𝑀𝑀 𝑠𝑠 is discouraged by a low
interest rate 𝑖𝑖𝑟𝑟 paid on reserves (and required reserves 𝑖𝑖𝑟𝑟𝑟𝑟 ) relative to 𝑖𝑖 and also by a high
interest rate 𝑖𝑖𝑏𝑏 for borrowing reserves if there is a shortfall.
If interest 𝑖𝑖𝑟𝑟 is paid on reserves (and 𝑖𝑖𝑟𝑟𝑟𝑟 on required reserves where relevant) then
commercial banks increase their supply of deposits for a given interest margin 𝑖𝑖 − 𝑖𝑖𝑚𝑚 . This
shifts the 𝑀𝑀 𝑠𝑠 curve to right and lowers the spread 𝑖𝑖 − 𝑖𝑖𝑚𝑚 using the banking-market
equilibrium from Section 7.5. Therefore, by paying interest on reserves, the gap between
𝑖𝑖𝑚𝑚 and 𝑖𝑖 becomes smaller and the economy moves closer to efficiency. An expansion of the
real supply of reserves 𝑅𝑅 𝑠𝑠 /𝑃𝑃 also reduces the spread 𝑖𝑖 − 𝑖𝑖𝑚𝑚 because banks paying the
borrowing cost 𝑖𝑖𝑏𝑏 becomes less likely. Paying interest on reserves – and using a floor
system – thus supports efficiency in a similar way to the Friedman rule.
It would still be possible to achieve efficiency by directly following the original Friedman
rule 𝑖𝑖 = 0. However, achieving 𝑖𝑖𝑚𝑚 = 𝑖𝑖 has the advantage that the absolute level of interest
rates 𝑖𝑖 is not restricted, and hence, the Fisher equation 𝑖𝑖 = 𝑟𝑟 + 𝜋𝜋 does not imply that
monetary policy must generate deflation (𝜋𝜋 < 0) for money to work best as a medium of
exchange.
Seigniorage revenue and the profitability of the central bank
In a monetary system with physical cash, real seigniorage revenue is 𝑖𝑖𝑖𝑖/𝑃𝑃 as explained in
Section 6.8. Where fiat money is used as reserves 𝑅𝑅 but not as cash by households and
firms, real seigniorage revenues would be 𝑖𝑖𝑖𝑖/𝑃𝑃 if no interest is paid on reserves. However,
if interest 𝑖𝑖𝑟𝑟 is paid on all reserves (assuming 𝑖𝑖𝑟𝑟𝑟𝑟 = 𝑖𝑖𝑟𝑟 as a simplification) then seigniorage
revenues would be (𝑖𝑖 − 𝑖𝑖𝑟𝑟 )𝑅𝑅/𝑃𝑃 instead. This depends on the spread between the bond
interest rate 𝑖𝑖 and the interest rate 𝑖𝑖𝑟𝑟 paid on reserves.
We see that seigniorage revenue is reduced by paying interest on reserves (𝑖𝑖𝑟𝑟 > 0). With a
floor system (𝑖𝑖 ≈ 𝑖𝑖𝑟𝑟 ), seigniorage would be reduced to zero. Hence, one drawback of paying
interest on reserves is the loss of seigniorage that requires the government to find
alternative sources of tax revenue.
7.6 Bond maturity and the yield curve
When thinking about the role of interest rates and macroeconomics, so far we have
distinguished between nominal and real (inflation-adjusted) interest rates. We have also
made a distinction between interest rates offered to borrowers and savers when there is a
problem of asymmetric information. In addition to these considerations, interest rates also
differ by maturity. Maturity refers to the length of the period of borrowing or lending.
The relationship between interest rates and maturity is known as the term structure of
interest rates. Graphically, this is shown in the yield curve, a plot of interest rates against
maturity for different bonds. Note that the terms interest rate and yield are synonymous.
Figure 7.10 shows the yield curve for US government bonds on 1 September 2021 as an
example. Interest rates for bonds with a short maturity of one year or less were close to
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zero, while as maturity increases beyond two years, interest rates are higher, reaching
almost 2 per cent for 30-year maturity bonds.
Figure 7.10: A yield curve for US government bonds
What information is conveyed by the yield curve and the term structure of interest rates?
What difference does it make to the economy whether the yield curve is upward sloping,
downward sloping or flat?
The basics of the term structure of interest rates can be illustrated by considering just two
bonds. First, a bond that pays one unit of money next period. Second, a bond that pays one
unit of money two time periods in future. These simple bonds with one payment at maturity
(no coupon payments before maturity) are known as ‘discount’ bonds. The bonds have
maturities of one and two periods respectively. Another feature of these bonds is that they
are nominal bonds, i.e. ones that specify payments in units of money that are not indexed to
inflation.
The yield to maturity (or just yield) on a bond is defined as the discount rate that makes the
present value of the payments promised by the bond equal to the market price at which the
bond currently trades. Suppose the prices in current units of money of the one- and twoperiod bonds are 𝑉𝑉1 and 𝑉𝑉2 respectively, their respective yields 𝑖𝑖 and 𝐼𝐼 are defined by the
price-yield relationships:
𝑉𝑉1 =
1
1 + 𝑖𝑖
and
𝑉𝑉2 =
1
(1 + 𝐼𝐼 )2
By definition, there is necessarily an inverse relationship between yields and bond prices. A
high bond yield 𝑖𝑖 means a one-period bond price 𝑉𝑉1 and similarly for the two-period bond.
Suppose a saver is considering which bond to hold. What returns are obtained between the
current and future periods? Here, we look at nominal returns. For real returns, we would
also need to adjust for inflation as discussed in Chapter 6. If a saver uses a unit of money to
buy one-period bonds at price 𝑉𝑉1 then an amount 1/𝑉𝑉1 is purchased. Each bond pays off a
unit of money in the next period, so the return is (1⁄𝑉𝑉1 ) − 1. Using 𝑉𝑉1 = 1/(1 + 𝑖𝑖), the
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return is equal to the yield 𝑖𝑖. In general, returns and yields are not the same but the return
on a discount bond that is held to maturity is equal to the yield when purchased, as seen in
this example.
If instead the saver buys two-period bonds, a unit of money can purchase a quantity 1/𝑉𝑉2 of
such bonds. By the next period, the bond has not yet matured, so the only return that can
be realised immediately comes from selling it. With one period remaining until maturity,
what was originally a two-period bond is now equivalent to a one-period bond, so its price is
now 𝑉𝑉1 ′, which denotes the price of a one-period bond in the future period. Using 𝑉𝑉1 ′ =
1/(1 + 𝑖𝑖′) and 𝑉𝑉2 = 1/(1 + 𝐼𝐼 )2 , the return on holding the two-period bond for just one
period is:
Return =
𝑉𝑉1′ − 𝑉𝑉2 (1 + 𝐼𝐼 )2
1 + 𝐼𝐼
=
− 1 = 𝐼𝐼 + (𝐼𝐼 − 𝑖𝑖 ′ ) �
�
′
𝑉𝑉2
1 + 𝑖𝑖
1 + 𝑖𝑖 ′
Observe that this return is not equal to the yield 𝐼𝐼, except in the special case 𝑖𝑖 ′ = 𝐼𝐼 because
this bond has not reached maturity over the period the return is calculated.
In choosing what combination of bonds to hold, savers compare the returns they offer.
These returns depend on the yields 𝑖𝑖 and 𝐼𝐼 and the return on the two-period bond also
depends on the future yield 𝑖𝑖′, which is not known with certainty now. An important point
here is that the two-period bond’s return, if it is held for only one period, is in general riskier
than that of the one-period bond. All else equal, savers are assumed to prefer higher
expected returns and lower risk.
7.7 The expectations theory of long-term interest rates
One theory of the term structure of interest rates assumes savers care only about expected
returns on assets. In this case, savers are said to be ‘risk neutral’. Given these preferences,
savers rationally choose to hold the bond with the highest expected return. However, since
all bonds that have been issued must be willingly held by someone in equilibrium, bond
prices, or equivalently, bond yields, must adjust so that all bonds offer the same expected
return. This logic implies a connection between short-term and long-term interest rates that
is called the expectations theory of interest rates.
We will illustrate the expectations theory of interest rates with reference to the one-period
and two-period bonds introduced earlier. We refer to those bonds in what follows as ‘shortterm’ and ‘long-term’ bonds, respectively. The two bonds have yields 𝑖𝑖 and 𝐼𝐼 and, for the
one-period bond, its actual and expected return is simply equal to its yield 𝑖𝑖. The return on
the two-period bond is 𝐼𝐼 + (𝐼𝐼 − 𝑖𝑖 ′ )(1 + 𝐼𝐼)/(1 + 𝑖𝑖 ′ ), which depends on the future oneperiod bond yield denoted by 𝑖𝑖 ′ . As long as 𝑖𝑖′ and 𝐼𝐼 are not too large, we can say that
(𝐼𝐼 − 𝑖𝑖 ′ )(1 + 𝐼𝐼)/(1 + 𝑖𝑖 ′ ) ≈ 𝐼𝐼 − 𝑖𝑖′, which means the approximate return on two-period bond
is 2𝐼𝐼 − 𝑖𝑖′. The expected return is 2𝐼𝐼 − 𝑖𝑖 ′𝑒𝑒 , where 𝑖𝑖 ′𝑒𝑒 denotes the expected value of the
future one-period yield 𝑖𝑖 ′ .
For risk-neutral savers to be willing to hold either bond, the expected returns must be the
same:
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𝑖𝑖 = 2𝐼𝐼 − 𝑖𝑖 ′𝑒𝑒
Solving this equation for the long-term interest rate 𝐼𝐼 shows that
𝐼𝐼 =
𝑖𝑖 + 𝑖𝑖 ′𝑒𝑒
2
The theory therefore predicts that the long-term interest rate 𝐼𝐼 is equal to an average of the
current and expected future short-term interest rates 𝑖𝑖 and 𝑖𝑖 ′𝑒𝑒 until the maturity of the
long-term bond. This idea applies to bonds with longer than a two-period maturity, where
the long-term yield is an average on expected short-term interest rates over a longer
horizon.
Box 7.4: Forecasting from the shape of the yield curve
If the expectations theory of long-term interest rates is correct, the yield curve can be used
to forecast the future path of short-term interest rates, exploiting the information and
analysis of the participants in the bond market. We will see how to derive market
expectations of interest rates from the yield curve and why an upward-sloping yield curve
implies short-term interest rates expected to rise, and a downward-sloping yield curve
implies short-term interest rates expected to fall. This relationship between the gradient of
the yield curve and the expected direction of the future path of interest rates means that
the yield-curve gradient might be used as a leading indicator of the business cycle.
We consider here just two points on the yield curve corresponding to a short-term (oneperiod) bond with yield 𝑖𝑖 and a long-term (two-period) bond with yield 𝐼𝐼. The expectations
theory of interest rates from Section 7.7 implies that 𝐼𝐼 = (𝑖𝑖 + 𝑖𝑖 ′𝑒𝑒 )/2, where 𝑖𝑖 ′𝑒𝑒 is the
expected future short-term interest rate 𝑖𝑖′. Subtracting 𝑖𝑖 from both sides implies:
𝑖𝑖 ′𝑒𝑒 − 𝑖𝑖
= 𝐼𝐼 − 𝑖𝑖
2
This implies a positive relationship between the gradient of the yield curve as represented
by the term 𝐼𝐼 − 𝑖𝑖 and the expected change in the future short-term interest rate 𝑖𝑖 ′𝑒𝑒 − 𝑖𝑖. An
explicit formula for the expected future interest rate is 𝑖𝑖 ′𝑒𝑒 = 2𝐼𝐼 − 𝑖𝑖, which can be
calculated using the yields 𝑖𝑖 and 𝐼𝐼 currently observed.
Figure 7.11 shows the relationship between spreads of 10-year over three-month US
government bonds and the subsequent changes in three-month Treasury bill yields
averaged over the following ten years. There is indeed a positive relationship, although
weaker than the expectations theory would suggest. Note that the scatterplot makes the
relationship look better than it really is because the vertical coordinates of the data points
are calculated using overlapping 10-year spells, so each point is not an independent
observation.
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Figure 7.11: Predictions from US yield curve (1934–2011)
7.8 Risk and portfolio choice
How should savers choose between different assets if they care about the riskiness of assets
as well as their expected returns? This section explores how an optimal portfolio trading off
risk and return can be selected.
Consider a saver choosing portfolio of two assets. One is a safe asset offering a risk-free real
return 𝑟𝑟𝑓𝑓 . Another is a risky asset with an uncertain real return 𝑟𝑟. We keep the analysis
simple by supposing there are two possible outcomes for the risky asset. In the ‘bad’
outcome, the return is 𝑟𝑟1 over some period and in the ‘good’ outcome the return is 𝑟𝑟2 ,
where 𝑟𝑟1 < 𝑟𝑟2 . The outcome 𝑟𝑟1 has probability 𝑡𝑡, and 𝑟𝑟2 has probability 1 − 𝑡𝑡, where 𝑡𝑡 is a
probability between zero and one. We will assume 𝑟𝑟1 < 𝑟𝑟𝑓𝑓 < 𝑟𝑟2 , so the ‘good’ outcome for
the risky asset is better than the risk-free, while the ‘bad’ outcome is worse. Otherwise, one
asset is unambiguously better than the other and the choice of portfolio is uninteresting.
The saver chooses the fraction 𝑥𝑥 of wealth to allocate to the safe asset. Suppose that wealth
is equal to 1 for illustration. This means that purchases of the safe and risky assets are 𝑥𝑥 and
1 − 𝑥𝑥 respectively. Assume that the investment proceeds will be consumed immediately
after the returns are received. Here, there is no choice of how much to save, or for how long
to save, only which assets to hold. Consequently, in the bad (1) and good (2) outcomes, the
levels of consumption 𝑐𝑐1 and 𝑐𝑐2 are:
𝑐𝑐1 = 1 + 𝑟𝑟𝑓𝑓 𝑥𝑥 + 𝑟𝑟1 (1 − 𝑥𝑥 )
and
𝑐𝑐2 = 1 + 𝑟𝑟𝑓𝑓 𝑥𝑥 + 𝑟𝑟2 (1 − 𝑥𝑥)
Individuals dislike uncertainty about whether they will get to consume 𝑐𝑐1 or 𝑐𝑐2 . We say they
are ‘risk averse’. More precisely, this means they prefer to get expected consumption 𝑐𝑐 𝑒𝑒 =
𝑡𝑡𝑐𝑐1 + (1 − 𝑡𝑡)𝑐𝑐2 for certain (𝑐𝑐 𝑒𝑒 in both the ‘good’ and ‘bad’ scenarios for the risky asset)
than receive 𝑐𝑐1 with probability 𝑡𝑡 and 𝑐𝑐2 with probability 1 − 𝑡𝑡. We can represent these
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preferences graphically using indifference curves in a similar way to the two-period
consumption choice model.
In Figure 7.12, the horizontal and vertical coordinates give the two outcomes for
consumption 𝑐𝑐1 and 𝑐𝑐2 . Geometrically, uncertainty increases as we move away from the 45𝑜𝑜
line, the set of point where 𝑐𝑐1 = 𝑐𝑐2 , representing cases where the individual faces no
uncertainty about the amount that will be consumed. In the diagram, expected
consumption 𝑐𝑐 𝑒𝑒 = 𝑡𝑡𝑐𝑐1 + (1 − 𝑡𝑡)𝑐𝑐2 remains constant along any straight line with gradient
−𝑡𝑡/(1 − 𝑡𝑡). Indifference curves representing risk averse preferences are convex to the
origin, lying above any tangent line drawn to them. On 45𝑜𝑜 line, the tangent line to
indifference curves has gradient −𝑡𝑡/(1 − 𝑡𝑡) because this is a point where the individual
faces no risk.
Figure 7.12: Indifference curves over uncertain consumption outcomes
We would like to add to this diagram the equivalent of a ‘budget constraint’, allowing us to
use the familiar framework of constrained utility maximisation to solve the portfolio choice
problem. This can be done by eliminating 𝑥𝑥 from the equations 𝑐𝑐1 = 1 + 𝑟𝑟1 + 𝑥𝑥(𝑟𝑟𝑓𝑓 − 𝑟𝑟1 )
and 𝑐𝑐2 = 1 + 𝑟𝑟2 − 𝑥𝑥(𝑟𝑟2 − 𝑟𝑟𝑓𝑓 ), showing that different portfolios lead to different
combinations of 𝑐𝑐1 and 𝑐𝑐2 with:
𝑐𝑐2 = 1 + 𝑟𝑟2 − �
𝑟𝑟2 − 𝑟𝑟𝑓𝑓
� (𝑐𝑐1 − 1 − 𝑟𝑟1 )
𝑟𝑟𝑓𝑓 − 𝑟𝑟1
This relationship between 𝑐𝑐1 and 𝑐𝑐2 is plotted in Figure 7.13. Since 𝑟𝑟1 < 𝑟𝑟𝑓𝑓 < 𝑟𝑟2 , the
equation is a downward-sloping straight line. It can be seen the line always passes through
the point (1 + 𝑟𝑟𝑓𝑓 , 1 + 𝑟𝑟𝑓𝑓 ), which corresponds to 𝑥𝑥 = 1, a portfolio comprising only safe
assets. The gradient of the line is − (𝑟𝑟2 − 𝑟𝑟𝑓𝑓 )⁄(𝑟𝑟𝑓𝑓 − 𝑟𝑟1 ) and it also passes through the point
(1 + 𝑟𝑟1 , 1 + 𝑟𝑟2 ), which corresponds to 𝑥𝑥 = 0, that is, holding none of the safe asset in the
portfolio. Unlike budget constraints with no credit-market imperfections, we do not extend
this straight line all the way to the horizontal and vertical axes as the points to the right of
(1 + 𝑟𝑟𝑓𝑓 , 1 + 𝑟𝑟𝑓𝑓 ) correspond to 𝑥𝑥 greater than 1 and the points to the left of (1 + 𝑟𝑟1 , 1 + 𝑟𝑟2 )
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correspond to negative 𝑥𝑥. Such portfolios would mean ‘negative’ or short positions in one of
the two assets, which we rule out.
The gradient of the constraint can be expressed in terms of the expected risky asset return
𝑟𝑟 𝑒𝑒 = 𝑡𝑡𝑟𝑟1 + (1 − 𝑡𝑡)𝑟𝑟2 as follows:
𝑟𝑟2 − 𝑟𝑟𝑓𝑓
𝑟𝑟 𝑒𝑒 − 𝑟𝑟𝑓𝑓
𝑡𝑡
1
=
+
�
�
𝑟𝑟𝑓𝑓 − 𝑟𝑟1 1 − 𝑡𝑡 1 − 𝑡𝑡 𝑟𝑟𝑓𝑓 − 𝑟𝑟1
The term 𝑟𝑟 𝑒𝑒 − 𝑟𝑟𝑓𝑓 is the risk premium, the expected risky-asset return 𝑟𝑟 𝑒𝑒 minus the risk-free
asset’s certain return 𝑟𝑟𝑓𝑓 . We see that the gradient of the constraint depends on the risk
premium 𝑟𝑟 𝑒𝑒 − 𝑟𝑟𝑓𝑓 relative to the amount of risk taken in holding the risky asset. The term in
the denominator is 𝑟𝑟𝑓𝑓 − 𝑟𝑟1 , the size of the loss from holding the risky asset relative to the
safe return on the risk-free asset.
The optimal portfolio choice is found where an indifference curve is tangent to the
constraint, the straight line joining the ‘all risky asset’ portfolio to the ‘all risk-free asset’
portfolio. This is depicted in Figure 7.13. The fractions 𝑥𝑥 and 1 − 𝑥𝑥 allocated to the risk-free
and risky assets can be deduced from how far the ray to the origin from the optimal
portfolio lies between the rays to the origin from the extreme portfolios.
Figure 7.13: Optimal portfolio choice
We can see from the formula for the gradient of the constraint that a positive amount of
risky asset can be held (𝑥𝑥 < 1) only if 𝑟𝑟 𝑒𝑒 > 𝑟𝑟𝑓𝑓 . The risky asset must offer a higher expected
return 𝑟𝑟 𝑒𝑒 than the certain return 𝑟𝑟𝑓𝑓 on the safe asset to be attractive to savers who dislike
risk, creating a trade-off between risk and return. A higher expected return compensates for
taking greater risk.
What happens to the optimal portfolio if the asset returns were to change? If the risk-free
return 𝑟𝑟𝑓𝑓 rises then the gradient of the constraint declines, all else being equal. If the price
of the risky asset rises, which would lower the returns 1 + 𝑟𝑟1 , 1 + 𝑟𝑟2 , and 1 + 𝑟𝑟 𝑒𝑒
proportionately, then the gradient of the constraint also declines. A lower gradient of the
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constraint in these two cases gives rise to a substitution effect would unambiguously raise
the safe-asset portfolio share 𝑥𝑥. However, the income effect on 𝑥𝑥 is ambiguous. We usually
assume preferences where the substitution effect dominates, in which case portfolio shares
adjust in the same direction as assets’ expected returns, all else equal. An example where
the expected return on the risky asset declines is illustrated in Figure 7.14.
Figure 7.14: Lower expected return on risky asset
Box 7.5: The typical shape of the yield curve
Yield curves typically have a positive gradient, that is to say, long-maturity bonds have
higher yields (interest rates) than short-maturity bonds on average. For example, taking US
government bond yields over the period 1934–2021, three-month Treasury bills have an
average yield of 3.4%, while 10-year Treasury bonds have an average yield of 5 per cent.
What explains this phenomenon of yield curves being upward sloping on average?
We investigate the question using one- and two-period discount bonds as examples of
short- and long-maturity bonds. Yields on the short- and long-term bonds are 𝑖𝑖 and 𝐼𝐼, and
their average yields are denoted by 𝚤𝚤̅ and 𝐼𝐼 ,̅ respectively. We will argue that investors’
attitude to risk provides an explanation for the observation 𝐼𝐼 ̅ > 𝚤𝚤̅ .
Suppose investors are considering holding bonds for just one period, so they have a
relatively short horizon. As explained in Section 7.6, the return on holding the short-term
(one-period) bond is 𝑖𝑖, which is the same as its yield 𝑖𝑖 because this bond is held to maturity.
Section 7.6 also shows the return on holding the long-term (two-period) bond for only one
period is 𝐼𝐼 + (𝐼𝐼 − 𝑖𝑖 ′ )(1 + 𝐼𝐼)/(1 + 𝑖𝑖 ′ ) ≈ 2𝐼𝐼 − 𝑖𝑖 ′ , where 𝑖𝑖 ′ is the yield on a one-period bond
in the future. These bonds are nominal bonds and both returns are calculated in nominal
terms. However, given 𝑖𝑖′, the real returns would be equally affected by inflation over the
first period.
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Observe that there is an important difference in the risk characteristics of the bond returns
over the same period. The nominal return on the one-period bond is known with certainty
when purchased at yield 𝑖𝑖, while the nominal return on the two-period bond (if held for
only one period) is uncertain because 𝑖𝑖′ is not known in advance. This makes the long-term
bond riskier than the short-term bond.
If investors were risk neutral and cared only about expected returns, the average returns
must be same on both bonds. Using the formulas for the returns above, the average return
on the short-term bond is 𝚤𝚤̅ and the average return on the long-term bond is approximately
2𝐼𝐼 ̅ − 𝚤𝚤̅ since �
𝚤𝚤′ = 𝚤𝚤̅. Risk neutrality implies 2𝐼𝐼 ̅ − 𝚤𝚤̅ ≈ 𝚤𝚤̅, and hence, 𝚤𝚤̅ ≈ 𝐼𝐼,̅ which means the
yield curve would be flat on average, contrary to what is observed.
Now suppose investors are risk averse regarding the return in the first period. Since the
long-term bond has a risky return if it is sold before maturity, the portfolio choice model of
Section 7.8 explains why the long-term bond needs to offer a higher average return. This
risk premium for the long-term bond requires 2𝐼𝐼 ̅ − 𝚤𝚤̅ > 𝚤𝚤̅, hence, implying 𝐼𝐼 ̅ > 𝚤𝚤̅. Investors’
dislike of risk can thus explain why the yield curve is upward sloping on average.
7.9 The functions of banks
Although saving is related to investment in equilibrium, many savers do not directly make
loans to firms or hold shares. Instead, banks are intermediaries between those who want to
save and those who want to borrow. What is the value of this intermediation service
provided by banks?
One important aspect of this intermediation is that individuals lack knowledge about
creditworthiness of borrowers and would face large costs of monitoring them closely. Banks
as intermediaries between many individual savers and borrowers with whom they have an
established relationship and thus more knowledge about are able to reduce costs arising
from asymmetries of information. A further advantage of banks coming from their scale is
that they can diversify lending much more easily than lending done by individuals, which is
more concentrated when an individual has limited funds to deploy.
Another important aspect of the intermediation done by banks is known as maturity
transformation. This is where a bank takes deposits from savers, offering to pay interest and
give access to funds on demand but uses depositors’ funds to make loans to firms for
longer-term investment projects. In other words, the bank borrows short from depositors
and lends long. This is a valuable service because borrowers typically want long maturity
loans, while savers want quick access to their funds on demand and for those funds to be
safe.
In what follows, we will focus on the role of banks in maturity transformation. We will see
how bank deposits are a more liquid asset for savers to hold than direct equity investment
in firms, so the maturity transformation done by banks has a social value. However, this
maturity transformation also exposes banks to the risk of bank runs.
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7.9.1 The Diamond-Dybvig model
We will analyse banks using a framework known as the Diamond-Dybvig model. This model
considers an economy with a large number 𝑁𝑁 of individuals who want to save but who do
not know when in future they will need access to their savings.
The model assumes three periods, 0, 1, and 2. All individuals start with one unit of real
wealth in period 0. Some will be ‘early types’ who want to consume their wealth in period 1,
while others will be ‘late types’ who want to consume at 2. No one knows in advance
whether they will be ‘early’ or ‘late’ types but all independently have probability 𝑡𝑡 of being
an ‘early’ type. Individuals learn their types in period 1.
Let 𝑐𝑐1 denote the consumption received by an early type, and 𝑐𝑐2 the consumption of a late
type. Now consider the preferences of an individual over these consumption outcomes from
the perspective of period 0 when it is not known whether 𝑐𝑐1 or 𝑐𝑐2 will be received.
Individuals prefer more of each of 𝑐𝑐1 and 𝑐𝑐2 to less but, crucially, they are assumed to be
risk averse, i.e. they dislike uncertainty about whether they will consume 𝑐𝑐1 or 𝑐𝑐2 .
We can represent risk-averse preferences over 𝑐𝑐1 and 𝑐𝑐2 graphically using indifference
curves as we did in the portfolio choice model from Section 7.8. Here, an individual
consumes either 𝑐𝑐1 or 𝑐𝑐2 , analogous to the two outcomes in the portfolio-choice model,
although here these occur at different dates. We ignore impatience to focus on risk
aversion, so we treat the two periods symmetrically even though period 2 is further in the
future than period 1.
In a diagram with 𝑐𝑐1 and 𝑐𝑐2 on the axes, indifference curves with risk aversion are convex to
the origin. This means they lie above their tangent lines, or in other words, given outcomes
𝑐𝑐1 and 𝑐𝑐2 with probabilities 𝑡𝑡 and 1 − 𝑡𝑡, an individual prefers 𝑐𝑐 𝑒𝑒 = 𝑡𝑡𝑐𝑐1 + (1 − 𝑡𝑡)𝑐𝑐2 , the
expected level of consumption, for certain. Since we ignore impatience here, the tangent
lines to the indifference curves crossing the 45𝑜𝑜 line with no consumption uncertainty all
have gradient −𝑡𝑡/(1 − 𝑡𝑡).
We turn now to what can be done with the initial real wealth individuals have in period 0 to
allow them to consume in the future time periods 1 or 2. We assume wealth can be stored
at no cost between periods 0 and 1, and between periods 1 and 2 but no return is earned
from this storage.
Wealth can be put into a long-term investment starting in period 0. One unit invested in
period 0 yields real wealth of 1 + 𝑅𝑅 in period 2 when the investment reaches maturity,
where 𝑅𝑅 denotes the total return between periods 0 and 2. However, if the investment is
abandoned before maturity in period 1 then no return is earned, although the initial
amount invested can be recovered. This means the long-term investment is fundamentally
illiquid – only long-term investors who can tie up funds between periods 0 and 2 can earn a
return on the investment.
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7.9.2 An economy with no financial intermediaries
In the Diamond-Dybvig model, we can see the benefit of having banks by first analysing
what happens if there are no financial intermediaries. To begin with, suppose there are no
financial markets active after period 0, so individuals simply choose what to do with their
wealth in period 0 and use the proceeds to consume in periods 1 or 2.
Under the assumptions, the long-term investment is never worse than storage because the
initial investment can be recovered if necessary in period 1. Hence, to save for the future, all
individuals initiate one unit of investment in period 0, or lend funds to someone to do it for
them. If they learn they are early types in period 1, they abandon the investment and
recover their initial wealth, which permits consumption 𝑐𝑐1 = 1. If they learn they are late
types, they keep the investment going until the return is received in period 2 and enjoy
consumption 𝑐𝑐2 = 1 + 𝑅𝑅. If the investment return 𝑅𝑅 is large, this means there is a large
difference between the outcomes 𝑐𝑐1 and 𝑐𝑐2 , which is bad for those who are risk averse.
Even without financial intermediaries, perhaps having a financial market where ongoing
investments can be traded for consumption goods might help. Suppose in period 1, there is
a market where ongoing investment projects can be bought or sold at price 𝑝𝑝 in units of real
consumption goods.
Since investments can be liquidated to recover the initial unit of real wealth put in, early
types only gain from selling investments in the market if 𝑝𝑝 > 1. But investments can be
bought only using stored wealth, or the proceeds from liquidating other investments, hence,
late types buying an ongoing investment at price 𝑝𝑝 would forgo 𝑝𝑝 other investments, so they
only gain from buying if 𝑝𝑝 < 1. Since there is no price 𝑝𝑝 ≠ 1 at which both early types want
to sell and late types want to buy, the equilibrium is 𝑝𝑝 = 1 and no trade takes place in the
financial market because there are no gains from trade. Having a financial market where the
investment can be traded does not circumvent its fundamental illiquidity.
7.10 Banking as maturity transformation
In the Diamond-Dybvig model, we will see that banks can improve upon the outcome with
no financial intermediation. Banks will offer to take savers’ funds as deposits and pay them
some interest, while still giving savers the right to withdraw on demand. This makes bank
deposits more liquid than the assets held by banks, namely, the lending that finances the
illiquid long-term investments.
Banks will engage in maturity transformation, issuing short-term liabilities (deposits) to fund
long-term assets (lending). Banks help by pooling the risk of individuals needing early access
to funds because they are able to take deposits from a large number of savers.
7.10.1 Bank deposits
Instead of savers directly investing or lending to investors, a bank offers to take deposits
from them on the following terms (the deposit contract). For each unit of funds deposited in
period 0, a depositor has the right to withdraw either 𝑑𝑑1 in period 1 or 𝑑𝑑2 in period 2.
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Implicitly, 𝑑𝑑1 and 𝑑𝑑2 define the interest rates 𝑟𝑟 and 𝑟𝑟′ paid on deposits between periods 0
and 1 (with 𝑑𝑑1 = 1 + 𝑟𝑟), and periods 1 and 2 (with 𝑑𝑑2 ⁄𝑑𝑑1 = 1 + 𝑟𝑟′).
An important point is that the bank does not know either ex ante or ex post who is an early
or a late type – think of the true need for funds as being private information. Everyone who
deposits one unit of funds in period 0 has the right to withdraw an amount 𝑑𝑑1 in period 1.
Suppose a large number of individuals 𝑁𝑁 each deposits one unit of funds. Deposit taking is
the only source of funds here for banks – we ignore bank capital because there is no risk
when investments are held to maturity (𝑅𝑅 is known with certainty). A bank chooses the
fraction 𝑥𝑥 of funds that are stored (a liquid asset) and the fraction 1 − 𝑥𝑥 put into the longterm investment (an illiquid asset). The bank needs to have enough liquid assets to be able
to meet the demand for withdrawals in period 1.
Suppose for now that only early types withdraw 𝑑𝑑1 in period 1, which requires among other
things that 𝑑𝑑2 ≥ 𝑑𝑑1 (i.e. the bank never offers a negative interest rate 𝑟𝑟 ′ < 0), otherwise
everyone wants to withdraw in period 1. As each of the large number 𝑁𝑁 of depositors has
an independent chance of being an early type with probability 𝑡𝑡, the law of large numbers
implies there will be 𝑡𝑡𝑡𝑡 early types in total. Given 𝑥𝑥, the value of liquid assets in period 1 is
𝑥𝑥𝑥𝑥, so 𝑑𝑑1 and 𝑥𝑥 must satisfy:
𝑡𝑡𝑡𝑡𝑑𝑑1 ≤ 𝑥𝑥𝑥𝑥
By period 2, the (1 − 𝑥𝑥 )𝑁𝑁 funds put into long-term investments have earned a return and
now have value (1 + 𝑅𝑅)(1 − 𝑥𝑥 )𝑁𝑁. All early types have withdrawn already and only the
deposits of (1 − 𝑡𝑡)𝑁𝑁 late types remain. The amount 𝑑𝑑2 promised to those withdrawing in
period 2 must satisfy:
(1 − 𝑡𝑡)𝑁𝑁𝑑𝑑2 ≤ (1 + 𝑅𝑅)(1 − 𝑥𝑥 )𝑁𝑁
7.10.2 Competition between banks
We assume that banks are competitive, so entry of new banks or the threat of entry implies
banks will make zero economic profits from the deposit contract they offer. This means the
feasibility conditions derived above hold with equality, leaving nothing to be paid out as
bank profits in either period 1 or 2. Cancelling 𝑁𝑁 from both sides, the zero-profit conditions
are 𝑡𝑡𝑑𝑑1 = 𝑥𝑥 and (1 − 𝑡𝑡)𝑑𝑑2 = (1 + 𝑅𝑅)(1 − 𝑥𝑥). Substituting the first into the second to
eliminate 𝑥𝑥 and dividing both sides by 1 + 𝑅𝑅 leads to an overall feasibility and zero-profit
condition on the terms (𝑑𝑑1 , 𝑑𝑑2 ) of a bank’s deposit contract:
𝑡𝑡𝑑𝑑1 + (1 − 𝑡𝑡)
𝑑𝑑2
=1
1 + 𝑅𝑅
This condition is the downward-sloping straight line drawn in Figure 7.15, which shows the
combinations of 𝑑𝑑1 and 𝑑𝑑2 that are feasible given the investment returns and the fraction of
early types, and where the bank makes no profits. Points below the line correspond to
positive bank profits, while those above are not feasible because banks would make losses.
The condition was derived supposing a late type is willing to wait until period 2 to withdraw,
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which is true if 𝑑𝑑2 ≥ 𝑑𝑑1 , assuming other late types wait as well. We therefore ignore the
points below the 45∘ line in the figure.
What interest rates 𝑟𝑟 and 𝑟𝑟′ can banks offer? By definition, 𝑑𝑑1 = 1 + 𝑟𝑟 and 𝑑𝑑2 =
(1 + 𝑟𝑟 ′ )𝑑𝑑1 . One possibility is 𝑟𝑟 = 0 and 𝑟𝑟′ = 𝑅𝑅, which is equivalent to 𝑑𝑑1 = 1 and 𝑑𝑑2 = 1 +
𝑅𝑅. However, this deposit contract is not particularly interesting because it simply replicates
the outcome if there were no banks in the economy. This corresponds to the point labelled
N in Figure 7.15. Another possibility is 𝑟𝑟 = (1 − 𝑡𝑡)𝑅𝑅/(1 + 𝑡𝑡𝑡𝑡) and 𝑟𝑟′ = 0, which is
equivalent to 𝑑𝑑1 = 𝑑𝑑2 = (1 + 𝑅𝑅)/(1 + 𝑡𝑡𝑡𝑡). This deposit contract is special because it
eliminates all liquidity risk for depositors – they receive the same amount whether they
need to withdraw in period 1 or period 2. Every depositor earns interest 𝑟𝑟 < 𝑅𝑅 between
periods 0 and 1. It is the point labelled L on the 45∘ line in the figure.
Figure 7.15: Feasible deposit contracts
7.10.3 The equilibrium deposit contract offered by banks
Competition compels banks to offer in period 0 the deposit contract on the highest
indifference curve of depositors subject to the zero-profit condition. Geometrically, this
corresponds to the point of tangency (𝑑𝑑1∗ , 𝑑𝑑2∗ ) between a depositor’s indifference curve and
the zero-profit line shown in Figure 7.16. If banks did not pick this deposit contract, a new
bank would be able to enter offering depositors a contract they prefer while still making
positive profits for the entrant bank.
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Figure 7.16: Optimal deposit contract
If individuals are sufficiently risk-averse, meaning that their indifference curves have
enough curvature, then banks offer a deposit contract with 0 < 𝑟𝑟 < 𝑅𝑅 and 0 < 𝑟𝑟′ < 𝑅𝑅,
which corresponds to a point in the diagram with 𝑑𝑑1 > 1 and 𝑑𝑑2 < 1 + 𝑅𝑅, that is, one to the
right of N but to the left of L. If depositors are not very risk averse then banks might offer a
point at N or even to the left, while only if depositors have the maximum risk aversion (‘L’shaped indifference curves) will banks offer L.
With the equilibrium deposit contract for sufficiently risk-averse depositors, the withdrawals
and consumption levels of the early and late types are 𝑐𝑐1 = 𝑑𝑑1 > 1 and 𝑐𝑐2 = 𝑑𝑑2 < 1 + 𝑅𝑅.
Individuals are able to get to a higher indifference curve than without banks, which
corresponds to the point N where 𝑐𝑐1 = 1 and 𝑐𝑐2 = 1 + 𝑅𝑅.
The optimal deposit contract offered by banks helps with risk sharing. It provides insurance
to those who need access to their funds early by paying some portion of the illiquid
investment return, even though investments have not yet yielded any return. The trade-off
is that late types do not get the full investment return but risk-averse individuals value this
insurance ex ante.
The reason why financial intermediation is valued here is similar to why risk-averse people
value insurance. For example, home insurance and car insurance cover the risk of
unexpectedly needing to find funds to make repairs. Banks provides liquidity insurance –
access on demand to funds without sacrificing all investment returns. A crucial difference is
that insurance policies pay out when an objectively verifiable event occurs, for example, a
fire. However, problems of asymmetric information and moral hazard mean that some risks
are not directly insurable, for example, the risk of unemployment, which is what creates a
need for liquidity insurance instead. While banks help to provide this, we will see that the
unverifiable nature of being an ‘early type’ creates a danger of bank runs.
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7.11 Bank runs
The deposit contract offered by banks can provide valuable risk sharing for those facing
liquidity shocks. But using demand deposits to fund illiquid investments makes banks
vulnerable to runs. Bank runs are self-fulfilling losses of confidence in the ability of a bank to
honour its contract with depositors. This fragility of banks comes from a maturity mismatch
between their assets and liabilities.
7.11.1 Strategic demands for withdrawals and bank failures
In the analysis of the Diamond-Dybvig model in Section 7.10, it was assumed only the early
types would want to withdraw in period 1. Clearly, all early types do want to withdraw in
period 1 but what about the late types? The bank offers 𝑑𝑑2 > 𝑑𝑑1 if they withdraw in period
2, which appears better than getting 𝑑𝑑1 at 1 and subsequently receiving no return until
wealth is consumed in period 2. But this does not consider whether the bank is able to
honour its contract and how the behaviour of other depositors affects this.
Suppose a bank has 𝑁𝑁 depositors (where 𝑁𝑁 is large) and consider a single late-type
depositor who believes all other late types will attempt to withdraw in period 1. What
should this person do? If all late-type depositors but one try to withdraw in period 1 then
𝑁𝑁 − 1 depositors in total request withdrawals (all early types and all late types but one). The
deposit contract offers 𝑑𝑑1 to those withdrawing in period 1 and remember that withdrawal
in period 1 is not restricted to early types. An individual’s type is private information and the
virtue of the liquidity insurance provided by banks is that it was not necessary for people to
prove they need funds in order to make a withdrawal.
When those who do not need funds in period 1 request withdrawal at that date, we say a
bank is faced with a ‘run’. In period 1, a bank holds 𝑥𝑥𝑥𝑥 of depositors’ funds stored in a liquid
asset and (1 − 𝑥𝑥 )𝑁𝑁 in a long-term investment. The long-term investment can be liquidated
in period 1 but no return is earned and only the initial funds (1 − 𝑥𝑥 )𝑁𝑁 are recovered. It
follows the maximum amount recoverable in period 1 by disposing of the bank’s assets is 𝑁𝑁.
Suppose a bank faces 𝑁𝑁 − 1 requests to withdraw 𝑑𝑑1 > 1 each in period 1. In this case, we
assume the bank processes withdrawal requests from depositors in a random queuing order
(‘sequential service’). For large 𝑁𝑁, we have that (𝑁𝑁 − 1)/𝑁𝑁 ≈ 1, so (𝑁𝑁 − 1)𝑑𝑑1 > 𝑁𝑁 because
𝑑𝑑1 > 1. The bank cannot recover enough from its assets to meet all withdrawal requests.
Therefore, faced with a run of 𝑁𝑁 − 1 depositors in period 1, the bank will fail because of the
return promised to depositors (𝑑𝑑1 > 1) combined with the illiquidity of its investments.
Only some of the 𝑁𝑁 − 1 withdrawal requests are met.
When the bank fails, there is nothing left in period 2. A late-type depositor who waits until
period 2 after the bank run to make a withdrawal request gets nothing for sure. This implies
that joining the queue and participating in the bank run is better than waiting because there
is a chance to get back something in period 1 (which can be stored until consumption in
period 2).
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7.11.2 Multiple equilibria and the possibility of bank runs
We conclude that the Diamond-Dybvig model does not have a unique prediction even
though depositors and bank are individually rational in their behaviour. There is the ‘good’
equilibrium that we have seen in Section 7.10 where only early types request withdrawal in
period 1 and the bank survives and succeeds in facilitating risk sharing. However, there is
also a ‘bad’ equilibrium with a bank run where all depositors request withdrawal in period 1,
the bank fails and risk sharing breaks down.
There are multiple equilibria because bank runs are self-fulfilling. It is individually rational to
join in a run if others are doing so. Although both the good and bad equilibria are consistent
with individual rationality, collectively, many are better off and no-one is worse off in the
good equilibrium compared to the bad equilibrium.
7.11.3 The 2007 Northern Rock bank run
In 2007, with the bank Northern Rock, there was the first bank run in the UK since 1866.
After a rapid expansion, Northern Rock sought liquidity support from the Bank of England on
12 September 2007 and this became public knowledge the next day. Queues outside
branches began to form on 14 September as depositors began to panic. By 17 September,
the UK government had stepped in to guarantee all deposits.
7.11.4 The ‘shadow’ banking system and the 2008 financial crisis
The Diamond-Dybvig model focuses on runs on the retail banking system but financial
markets can sometimes face situations similar to bank runs. A version of this was seen in the
so-called ‘shadow’ banking system during the 2008 financial crisis.
Prior to the financial crisis, there was a large expansion in ‘securitisation’, which is where
mortgages and other long-term loans were originated and then sold in packages to other
financial institutions such as investment banks. This ‘shadow’ banking system financed
purchases of securitised assets through short-term borrowing in the money markets, which
required frequent rollovers. In 2008, investment banks such as Bear Stearns and Lehman
Brothers struggled to roll over short-term financing. The belief that these banks and other
institutions could not repay short-term debts through the sale of illiquid assets caused new
financing to dry up, creating a self-fulfilling freeze in the money markets.
7.12 Deposit insurance and bank regulation
We have seen in Section 7.10 that banks provide a valuable service but the Diamond-Dybvig
model also highlights their inherent fragility, as indicated by the analysis of bank runs in
Section 7.11. Can government intervention prevent bank runs, leaving only the ‘good
equilibrium’ where banks are successful? Some possible policy interventions are:
•
•
•
•
A system of deposit insurance
The central bank acting as ‘lender of last resort’
Imposing bank capital requirements
Imposing reserve requirements.
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7.12.1 Deposit insurance
Suppose the government guarantees that it will compensate depositors for losses arising
from bank failures, a system of deposit insurance. This is ultimately backed by the
government’s tax revenues if needed. If deposit insurance is complete and credible then the
late types in the Diamond-Dybvig model need never request withdrawal in period 1 because
they are guaranteed to receive 𝑑𝑑2 > 𝑑𝑑1 by waiting until period 2. Therefore, no bank runs
occur and, absent any other problems, no banks will fail and the government would not
actually ever need to pay out on its guarantee to depositors.
This system of deposit insurance appears to provide a ‘free lunch’ of eliminating bank runs
at no cost. However, it comes with the drawback of removing the incentive for depositors to
avoid banks that take too much risk. To the extent that banks are ‘too big to fail’
(systemically important), there is also an implicit government guarantee to other bank
creditors. This leads banks to take excessive risks because these guarantees remove the
discipline otherwise provided by creditors. Note that this line of argument goes beyond the
basic Diamond-Dybvig model, which has no investment risk. It suggests deposit insurance
may indeed avoid self-fulfilling panics but there will still be bank failures owing to losses on
excessively risky investments.
Remember that the bank run in the Diamond-Dybvig model is due to an illiquidity problem.
A bank can repay all late types the promised amount in period 2 but cannot meet
unexpected and collectively irrational demands for withdrawals in period 1. This is different
from an insolvency problem where a bank has suffered losses on its investments that leave
it unable to honour its promises to creditors even without any panic. A system of deposit
insurance protects depositors against bank failures for both reasons but we only want this
protection for the problem of illiquidity – insurance that also operates in the case of
insolvency creates a problem of moral hazard. A serious challenge for bank regulation is that
it is hard to distinguish between illiquidity and insolvency in real time.
7.12.2 The central bank as ‘lender of last resort’
Another way to avoid bank runs is for the central bank to act as ‘lender of last resort’. This
means the central bank operates a discount window or borrowing facility that provides
liquidity to commercial banks by making loans against illiquid assets pledged as collateral.
The central bank usually lends through these facilities at a penalty rate but this is still more
favourable to a bank facing a bank run than attempting to obtain liquidity on commercial
terms and can enable the bank to survive. While ‘lender of last resort’ can stop runs, it also
leads to a moral-hazard problem because of the difficultly of distinguishing illiquidity and
insolvency in real time.
7.12.3 Bank capital requirements
Given the moral-hazard problems of policies designed to avoid bank runs, other bank
regulation is important in reducing the risk of bank insolvency and the cost of bailouts. Bank
capital requirements are a widely used regulatory tool to achieve this (‘bank capital’
narrowly defined refers to a bank’s equity). Capital requirements specify a minimum ratio of
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bank equity to total bank assets (sometimes weighted by assets’ riskiness). Funds provided
by a bank’s shareholders and retained profits not paid out to shareholders can absorb some
losses on a bank’s investments without jeopardising depositors’ funds.
7.12.4 Reserve requirements
Another regulatory tool is reserve requirements. These restrict banks to hold a minimum
fraction of deposits as reserves. The advantage of this is that reserves are the most liquid
type of asset, which helps ensure banks are able to meet demands for withdrawals
whenever they occur.
In the Diamond-Dybvig model, the liquid asset is ‘storage’, which earns no return. We saw in
Section 7.10 that banks would already choose to hold some liquid assets, so reserve
requirements may not even be binding. Moreover, reserve requirements would not
eliminate the risk of the bank runs studied in Section 7.11, only making their consequences
less severe. The only case where having reserve requirements eliminates bank runs is when
those reserve requirements are 100 per cent of deposits.
In terms of the model from Section 7.4, imposing a higher reserve-to-deposit ratio than
banks would otherwise choose reduces the supply of deposits and increases the interest
margin 𝑖𝑖 − 𝑖𝑖𝑚𝑚 in equilibrium, which is inefficient, although this problem can be mitigated if
sufficient interest is paid on the required reserves themselves.
Box 7.6: The 100 per cent reserve requirements
One radical proposal for making the banking system safer is imposing a 100 per cent
reserve requirement on banks. This policy is also known as ‘narrow banking’. As deposits
are fully backed by reserves, banks are always able to satisfy any requests for withdrawals.
Therefore, a bank cannot fail as a result of a bank run, which gives the banking system
much greater resilience.
However, 100 per cent reserve requirements mean that banks cannot make loans financed
by taking deposits, which reduces bank lending if deposits cannot easily be replaced by
other unregulated bank liabilities (bonds or equity). This means banks cannot perform
maturity transformation, which we saw in Section 7.10 has a social value in the DiamondDybvig model.
We now consider how to incorporate reserve requirements into the Diamond-Dybvig
model, which were not explicitly considered in our earlier analysis. There, the liquid asset is
‘storage’, which earns no return and investments in the illiquid asset are recoverable,
although only by sacrificing all of the investment return. The liquidation value of a bank’s
assets in period 1 is thus equal to 1 per depositor. We interpret a minimum reserve-deposit
ratio 𝑞𝑞 as a minimum value of the ratio of the asset liquidation value 1 to the amount 𝑑𝑑1 in
a depositor’s account in period 1, that is, 1⁄𝑑𝑑1 ≥ 𝑞𝑞. This is equivalent to 𝑑𝑑1 ≤ 1⁄𝑞𝑞.
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We know from Section 7.10 that banks have an incentive to offer 𝑑𝑑1 > 1. If 𝑞𝑞 is low, so 1/𝑞𝑞
is high, the reserve requirement is not binding. Reserve requirements ultimately become
binding for higher values of 𝑞𝑞 closer to 1. Once this point is reached, the deposit contract is
limited to 𝑑𝑑1 = 1⁄𝑞𝑞, which is less than 𝑑𝑑1 would otherwise be.
Reserve requirements with 𝑞𝑞 < 1 still result in 𝑑𝑑1 > 1. This means a large enough bank run
in period 1 still causes a bank to fail because 𝑑𝑑1 exceeds the liquidation value of all assets,
although a greater number of requests for withdrawals could be satisfied during a run.
Therefore, less than 100 per cent reserve requirements do not rule out self-fulfilling bank
runs in the Diamond-Dybvig model.
The 100 per cent reserve requirements (𝑞𝑞 = 1) imply that banks must offer 𝑑𝑑1 = 1. In this
case, they can always meet all requests for withdrawals, so self-fulfilling runs cannot occur,
eliminating the bad outcome of the Diamond-Dybvig model. But as shown in Figure 7.17,
the 100% reserve requirements prevent maturity transformation, so the good outcome is
worse.
Figure 7.17: The 100 per cent reserve requirements in the Diamond-Dybvig model
More generally, beyond the Diamond-Dybvig model, reserve requirements reduce the
supply of bank deposits when interest is not paid on required reserves, as explained in
Section 7.4. This reduction in 𝑀𝑀 𝑠𝑠 raises the gap between 𝑖𝑖 and 𝑖𝑖𝑚𝑚 . Having 100 per cent
reserve requirements mean that banks supply deposits up to the point where 𝑖𝑖𝑚𝑚 = 𝑖𝑖𝑟𝑟𝑟𝑟 ,
where 𝑖𝑖𝑟𝑟𝑟𝑟 is interest paid on required reserves. To avoid a reduction in the supply of
deposits 𝑀𝑀 𝑠𝑠 , the central bank must pay interest on required reserves 𝑖𝑖𝑟𝑟𝑟𝑟 and increase the
supply of reserves to the banking system. Maintaining an efficient supply of money to the
economy after imposing 100 per cent reserve requirements thus entails a fiscal cost.
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Box 7.7: Central-bank digital currency
Central-bank digital currency (CBDC) refers to an electronic form of fiat money held directly
by non-banks. CBDC would enable households and firms to keep money directly in accounts
at the central bank, similar to the way commercial banks hold reserves in accounts at the
central bank. As of 2021, apart from some small-scale experiments, CBDCs are not currently
in use – but this may soon change.
Unlike deposits at a commercial bank, there cannot be a ‘run’ on central banks offering
CBDC accounts. This is because fiat money not redeemable for anything but itself. Thus,
CBDC would offer the convenience of electronic payments without the risks associated with
commercial-bank deposits. Moreover, it would offer greater security than the physical cash
that is currently the only way non-banks can hold fiat money, removing the anonymity that
is a feature of cash.
Central banks can pay interest on CBDCs in the same way they can pay interest on reserves.
If they were to do so at a sufficiently high rate, this might lead people to switch from
commercial-bank deposits to CBDCs. That would present a similar challenge to commercial
banks – how to fund bank lending – as is present with 100 per cent reserve requirements.
In some ways, a CBDC has similarities to the proposal for 100 per cent reserve
requirements, which is also designed to make the monetary system free of ‘runs’. The
difference is that with CBDC, there would be a choice of whether to hold the CBDC or a
commercial-bank deposit backed by less than 100 per cent reserves, rather than forcing
everyone to hold fully reserve-backed deposits or cash.
CBDC may ultimately be intended as a replacement for physical cash. As discussed in
Section 6.14, this would open the door to significantly negative nominal interest rates.
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Chapter 8: Business cycles
This chapter investigates the causes of booms and busts – cyclical fluctuations in economic
activity. We will also explore whether policy interventions, such as monetary or fiscal policy,
should be used to tame business cycles. Economists have not reached a consensus on the
causes of business cycles, so we will present a number of different theories. We begin by
looking at an economy with nominal rigidity where markets fail to clear because prices are
slow to adjust. When an economy with nominal rigidity is hit by shocks, GDP tends to
fluctuate excessively because markets do not function efficiently. There is scope for central
banks or governments to intervene to deliver better economic outcomes.
We will also examine theories of business cycles that do not depend on nominal rigidity. We
cover real business cycle theory, which argues that the business cycle is simply the
economy’s efficient response to supply shocks. An alternative approach, the coordination
failure model, suggests business cycles are driven by self-fulfilling changes in optimism or
pessimism, rather than fundamental shocks.
Essential reading
•
Williamson, Chapters 13 and 14.
8.1 Nominal rigidity
Most models we have used so far feature ‘market clearing’, with the relevant prices in each
market of the economy adjusting so that desired demand and supply are equalised.
However, there is an important set of ideas about how the economy works that suggests
prices change only slowly to achieve market clearing.
In this chapter, we will explore the implications of prices quoted in units of money
remaining fixed even when supply or demand conditions change. The term ‘nominal rigidity’
is used to refer to any type of prices quoted in money not adjusting as required to clear
markets. Why the focus on prices quoted in units of money? For prices to remain fixed, they
must already have been set and specified in some particular units. We discussed in Chapter
6 the convenience advantage of quoting all prices in terms of money – what we called
money’s ‘unit of account’ function.
One common form of nominal rigidity is the ‘stickiness’ of retail prices faced by consumers,
although there are other nominal rigidities too such as sticky wages and sticky producer
prices. Why would goods prices be sticky? One basic explanation points to physical costs of
adjustment, for example, the costs of printing new price labels. Costs of this type are known
as ‘menu costs’. However, technology and online retailing have greatly reduced such costs.
More broadly, we can also envisage costs of making pricing decisions as a cost of
adjustment, for example, the managerial time and resources needed to select a new price.
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Leaving a price unchanged saves costs because no new price needs to be determined. Firms’
relationships with their customers might also create barriers to price adjustment. For
example, the coordination problem of no firm wanting to go first with a price increase and
antagonise its customers.
Data on observations of individual prices also provides some support for the relevance of
nominal rigidity. The price data shown in Figure 8.1 suggests individual goods prices quoted
in units of money change infrequently, even during times when the economy experiences
shocks.
Figure 8.1: Evidence on sticky prices
Models of the economy where there is a failure of market clearing because of nominal
rigidity are often labelled ‘Keynesian’ models. We will study extensively what is called a
‘new Keynesian’ model. This model is ‘new’ in the sense that it has many of the features of
modern macroeconomic models seen in earlier chapters but in combination with older
Keynesian ideas about the failure of markets to clear.
The only nominal rigidity in the new Keynesian model is stickiness of goods prices. Having
price stickiness in the model implicitly assumes prices are set by firms, not markets. For this
reason, the model also features an imperfectly competitive goods market.
It is important to note that the reasons for nominal rigidity are unlikely to prevent eventual
price adjustment in the long run. Prices will ultimately adjust to shocks, so markets still clear
in the long run. Consequently, the model will make a distinction between the short run
where prices are sticky and the long run where prices are flexible. The transition from the
short run to the long run is studied in Chapter 9.
There is also an important difference compared to our earlier analysis of the failure of
labour-market clearing owing to efficiency wages in Section 5.2. There, firms have incentives
to pay efficiency wages at all times, so the rigidity of wages is not temporary and does not
disappear in the long run. Moreover, efficiency wages impart rigidity to real wages, which is
conceptually distinct from nominal rigidity.
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8.2 The new Keynesian model
The new Keynesian model is essentially the dynamic macroeconomic model introduced in
Chapter 3 with nominal rigidity added to it. The goods market has sticky prices and is
treated as an imperfectly competitive market for consistency. This is in contrast to the
perfect competition and fully flexible prices of the earlier dynamic macroeconomic model.
We start with a simple version of the model where all prices are completely fixed and are
expected to remain so in the near term. Later in Chapter 9, we will add partial price
adjustment to analyse inflation. We can illustrate the main consequences of nominal rigidity
by only having sticky prices in the goods market. Nominal wages are fully flexible and we can
treat the labour market as being perfectly competitive. However, it is possible to combine
our earlier analysis of efficiency wages with the new Keynesian model and this case is also
considered later.
Goods prices being sticky has consequences for our analysis of firms’ labour demand and
the implied level of output supply in the goods market. Recall that with perfect competition,
firms can sell as much output as they like at the market price, which adjusts to clear the
goods market. Each extra unit of labour allows a firm to produce 𝑀𝑀𝑃𝑃𝑁𝑁 units of output, the
marginal product of labour, so a profit-maximising firm hires labour at real wage 𝑤𝑤 up to the
point where 𝑀𝑀𝑃𝑃𝑁𝑁 = 𝑤𝑤. This is the labour demand curve with perfect competition.
With nominal rigidity, if an imperfectly competitive firm does not change the fixed price 𝑃𝑃� of
the good it is selling then it cannot choose how much it sells. If demand falls, shifting the
demand curve for the firm’s product leftwards, it cannot sell as much as before at price 𝑃𝑃�,
and as a result hires less labour. Even if the marginal product of labour exceeds the real
wage, the firm does not hire more labour because it cannot sell the extra output that could
be produced with additional employment. If demand rises, a firm can now sell more at the
same price 𝑃𝑃�, so it hires more labour to meet the additional demand. Strictly speaking, this
is true only as long as 𝑤𝑤 ≤ 𝑀𝑀𝑃𝑃𝑁𝑁 , otherwise the firm would not want to sell more, although
as we will see, cases where this condition fails to hold are not likely to be relevant in
practice.
This logic tells us that the labour demand curve is no longer given by the marginal product of
labour but is instead perfectly wage inelastic as depicted in the right panel of Figure 8.2. The
vertical labour demand curve shifts with the aggregate demand for goods, which we
suppose affects the demand for each individual good. We truncate the vertical labour
demand curve where it goes above 𝑀𝑀𝑃𝑃𝑁𝑁 .
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Figure 8.2: Labour demand with sticky prices
The wage-inelastic labour demand curve 𝑁𝑁 𝑑𝑑 (𝑌𝑌) is determined using the production function
𝑌𝑌 = 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑁𝑁) to find what level of employment 𝑁𝑁 is needed to produce output 𝑌𝑌 sufficient
to meet demand, taking as given the stock of capital 𝐾𝐾 and TFP 𝑧𝑧. This is shown in the left
panel of Figure 8.2. An increase in 𝑌𝑌 shifts labour demand to the right.
How do we analyse outcomes in the labour market when goods prices are sticky? Since
labour demand is wage inelastic, the outcome for employment 𝑁𝑁 is directly determined by
the position of 𝑁𝑁 𝑑𝑑 (𝑌𝑌). For wages 𝑤𝑤, we need to be more specific about how the supply side
of the labour market works.
Let us first suppose that wages are fully flexible. In that case, wages are determined by a
standard upward-sloping labour supply curve 𝑁𝑁 𝑠𝑠 (𝑟𝑟) derived from households’ optimality
conditions 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 = 𝑤𝑤 and 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝑙𝑙′ = (1 + 𝑟𝑟)𝑤𝑤/𝑤𝑤′ as explained in Chapters 1 and 3. With
a representative household, Chapter 3 explains why 𝑁𝑁 𝑠𝑠 (𝑟𝑟) is unambiguously upward sloping
owing to the substitution effect of wages and shifts to the right when the real interest rate 𝑟𝑟
increases (intertemporal substitution). Although we consider a monetary economy, we
ignore here the effect of money’s medium of exchange function on 𝑁𝑁 𝑠𝑠 , that is to say, we
neglect any effects coming from money being less good as a store of value and acting as a
tax on economic activity (see Section 6.5).
With flexible wages, the real wage 𝑤𝑤 adjusts so that the labour market clears with 𝑁𝑁 𝑑𝑑 (𝑌𝑌) =
𝑁𝑁 𝑠𝑠 (𝑟𝑟) as shown in the left panel of Figure 8.3. We will see that employment fluctuates with
aggregate demand, although if there are no impediments to wage adjustment then there is
no unemployment or fluctuations in unemployment. If we want to study unemployment
over the business cycle, we can combine our earlier analysis of efficiency wages from
Section 5.2 with the new Keynesian model. In that case, the real wage is determined by
firms’ desire to pay an efficiency wage and remains constant. Employment is found on the
inelastic labour demand curve but desired labour supply can be higher, so unemployment
exists. Given desired labour supply, unemployment changes in the opposite direction to
changes in employment. This case is depicted in the right panel of Figure 8.3.
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Figure 8.3: The labour market with sticky prices
In an economy with flexible prices, the output supply curve in the goods market represent
the production of goods and services by firms given employment at the labour-market
equilibrium. However, with sticky prices, the demand for labour depends on the aggregate
demand for goods. This means there is no independent decision made by firms about how
much to sell. The supply of goods passively accommodates changes in demand and thus
there is no output supply curve relevant for determining outcomes in the goods market as
long as goods prices remain sticky. One caveat to this logic is that demand must not be so
large that firms do not want to meet it because wages are too high. We require that 𝑤𝑤 ≤
𝑀𝑀𝑃𝑃𝑁𝑁 , which in the goods market diagram is equivalent to remaining on the left of the
hypothetical output supply curve with perfect competition and flexible prices (𝑌𝑌� 𝑠𝑠 ), where
the real wage would be exactly equal to the marginal product of labour. The hypothetical
supply curve is depicted as a dashed line in Figure 8.4. This condition is not likely to be of
concern in most applications of the model.
Figure 8.4: The goods market with sticky prices
The demand curve in the goods market simply represents the same aggregate demand for
goods and services that was found in the earlier dynamic macroeconomic model from
Section 3.12. The 𝑌𝑌 𝑑𝑑 curve represents the equation 𝑌𝑌 𝑑𝑑 = 𝐶𝐶 𝑑𝑑 + 𝐼𝐼 𝑑𝑑 + 𝐺𝐺 as before and is
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shown in Figure 8.4. But since prices are sticky, the economy does not have to be at the
intersection of the 𝑌𝑌 𝑑𝑑 and 𝑌𝑌 𝑠𝑠 curves. Many points on the 𝑌𝑌 𝑑𝑑 curve can be a goods-market
equilibrium in the short run. Note that in Keynesian models, the 𝑌𝑌 𝑑𝑑 curve was traditionally
known as the 𝐼𝐼𝐼𝐼 (investment = saving) curve. The equation for investment 𝐼𝐼 being equal to
national saving 𝑌𝑌 − 𝐶𝐶 − 𝐺𝐺 is equivalent to 𝑌𝑌 = 𝐶𝐶 + 𝐼𝐼 + 𝐺𝐺.
The point on the output demand curve the economy reaches and the outcome for real GDP
𝑌𝑌 are determined by the level of interest rates. With complete stickiness of prices there is
zero inflation (𝜋𝜋 = 0) and, hence, the Fisher equation 𝑖𝑖 = 𝑟𝑟 + 𝜋𝜋 𝑒𝑒 implies that 𝑟𝑟 = 𝑖𝑖, so the
real interest rate 𝑟𝑟 is the same as the nominal interest rate 𝑖𝑖. As discussed in Chapters 6 and
7, there are various instruments of monetary policy that can control the nominal interest
rate 𝑖𝑖. With sticky prices, monetary policy also effectively sets the real interest rate 𝑟𝑟, which
selects a point on the 𝑌𝑌 𝑑𝑑 curve.
In the goods market diagram, we represent the central bank’s choice of nominal interest
rate 𝑖𝑖 (and, hence, 𝑟𝑟) with a line that we will label 𝑀𝑀𝑀𝑀 (money and monetary policy). The
𝑀𝑀𝑀𝑀 line is usually assumed to be flat or upward-sloping, which is to say that the central
bank either sets some particular interest rate, in which case 𝑀𝑀𝑀𝑀 is horizontal, or we think of
the central bank as systematically adjusting interest rates up and down with GDP. The
intersection between the 𝑌𝑌 𝑑𝑑 (𝐼𝐼𝐼𝐼) curve and the 𝑀𝑀𝑀𝑀 line determines the real interest rate 𝑟𝑟
and output 𝑌𝑌 as seen in Figure 8.4.
8.3 The real effects of monetary policy
What happens when the central bank changes its monetary policy according to the new
Keynesian model? Does monetary policy have real effects?
Suppose the central bank cuts the nominal interest rate 𝑖𝑖. If prices are completely sticky,
inflation remains zero and, consequently, the real interest rate falls. Supposing the stance of
monetary policy is represented by a horizontal 𝑀𝑀𝑀𝑀 line, the interest rate cut shifts the 𝑀𝑀𝑀𝑀
line downwards. The economy moves along the output demand curve as shown in Figure
8.5, with the lower real interest rate 𝑟𝑟 stimulating consumption and investment demand.
This works through a lower cost of borrowing for firms and a substitution effect on
households’ consumption expenditure plans. GDP 𝑌𝑌 rises with the resulting increase in
aggregate demand.
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Figure 8.5: Cutting interest rates
In the labour market, the increase in aggregate demand for goods shifts the labour demand
𝑁𝑁 𝑑𝑑 (𝑌𝑌) curve to the right. This is because firms selling goods at a fixed price want to hire
more workers when they are able sell more output. Employment rises as a result.
When wages are flexible, the real wage 𝑤𝑤 increases because of a movement along the
upward-sloping labour supply curve 𝑁𝑁 𝑠𝑠 (𝑟𝑟), which also shifts to the left when 𝑟𝑟 declines. If
real wages are rigid owing to efficiency wages then higher employment and lower desired
labour supply imply a decline in unemployment. These two cases are depicted in Figure 8.6.
Box 8.1: The Volcker disinflation
The new Keynesian model predicts changes in interest rates by central banks have
real effects. Specifically, higher interest rates reduce demand and real GDP. The
most striking evidence for this is seen following substantial increase in interest
rates in USA in the early 1980s. Paul Volcker became Chairman of the US Federal
Reserve in 1979 at a time when inflation had reached double-digit levels and there
was pressure to bring inflation back down.
Figure 8.6: Labour market with interest-rate cut
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As seen in Figure 8.7, the shift in the Federal Reserve’s monetary policy stance
brought about by Volcker saw interest rates rise from below 10 per cent to peak at
15 per cent in 1981. Inflation started to fall and was below 5 per cent by 1983.
While we cannot yet analyse inflation using our basic new Keynesian model with
completely sticky prices, we will study inflation with partial price adjustment in
Chapter 9. For now, we focus on the effects of real interest rates on aggregate
demand and GDP. Volcker’s tightening of monetary policy led the real interest to
rise from negative levels to more than 5 per cent by 1981. This was followed by a
sharp fall in real GDP during the 1981–82 recession.
Figure 8.7: The Volcker disinflation
8.4 Business cycles due to demand shocks
The new Keynesian model provides a framework for understanding business-cycle
fluctuations. Let us analyse how the economy would respond to an unexpected fall in
aggregate demand.
What are the possible sources of such a negative demand shock? One possibility is a decline
in consumption demand 𝐶𝐶 𝑑𝑑 or investment demand 𝐼𝐼 𝑑𝑑 owing to lower confidence about the
future. If expectations of the future are formed rationally, this would be triggered by some
specific bad news about the economy’s future fundamentals, for example, less optimistic
expectations of future TFP 𝑧𝑧′.
Another source of demand shocks is a worsening of credit-market imperfections. As
discussed in Chapter 4, borrowers would face a higher interest rate 𝑟𝑟𝑙𝑙 even if there is no
change in the risk-free interest rate 𝑟𝑟 received by savers. Further possibilities are shifts in
preferences toward saving more for the future, or an increase in uncertainty about the
future that triggers greater saving owing to concern about future risks, which would both
reduce consumption demand 𝐶𝐶 𝑑𝑑 . Fiscal austerity, where the government reduces its
expenditure 𝐺𝐺 is another possibility.
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Consider a decline in the expected value of future TFP 𝑧𝑧 ′ as a specific example of a demand
shock. This leads to lower 𝐶𝐶 𝑑𝑑 and 𝐼𝐼 𝑑𝑑 and shifts the 𝑌𝑌 𝑑𝑑 curve to the left. Assuming no
automatic or discretionary monetary policy response, that is to say, a horizontal 𝑀𝑀𝑀𝑀 line in
same position, the real interest rate 𝑟𝑟 remains the same and GDP 𝑌𝑌 declines as shown in
Figure 8.8.
Figure 8.8: Effects of a negative demand shock
In the labour market, lower aggregate demand 𝑌𝑌 reduces labour demand, with 𝑁𝑁 𝑑𝑑 (𝑌𝑌)
shifting to the left and resulting in lower employment 𝑁𝑁. Owing to the negative wealth
effect of lower 𝑧𝑧′, the labour supply curve 𝑁𝑁 𝑠𝑠 (𝑟𝑟) shifts rightwards. With flexible wages, the
real wage 𝑤𝑤 would decline, while with a rigid efficiency wage, higher unemployment would
result. It can also be seen from the production function that the decline in output and
employment would raise average labour productivity 𝑌𝑌/𝑁𝑁, which is the gradient of the ray
from the origin to the production function.
Box 8.2: Can the new Keynesian model match the business-cycle
stylised facts?
It is desirable that any theory of the business cycle is consistent with empirical
evidence on the behaviour of fluctuations of various macroeconomic variables. We
can document a set of business-cycle ‘stylised facts’ using methods described in
Section 3.1. Variables are detrended and their percentage deviations from trend
can be compared to those of real GDP. We have already looked at the behaviour of
fluctuations in consumption and investment in Chapter 3 and unemployment in
Chapter 5. Both consumption and investment are procyclical, meaning that their
deviations from trend are positively correlated with deviations of real GDP from its
trend, while unemployment is countercyclical, i.e. negatively correlated with real
GDP. Consumption is less volatile than GDP – its percentage fluctuations are smaller
than those of real GDP – while investment is more volatile than GDP.
We can also look at a broader range of macroeconomic variables. Figure 8.9 shows
detrended employment, which is procyclical, generally less volatile than
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We can also look at a broader range of macroeconomic variables. Figure 8.9 shows
detrended employment, which is procyclical, generally less volatile than GDP and
slightly lagging. Figure 8.10 shows real wages, for which it is harder to discern a
clear pattern but which is overall weakly procyclical and less volatile than GDP.
Figure 8.11 displays the data for average labour productivity, which is procyclical
and less volatile than GDP. The procyclicality of average labour productivity reflects
the fact that employment typically moves by less in percentage terms than GDP.
Figure 8.9: Fluctuations in employment over the business cycle (USA)
Figure 8.10: Fluctuations in real wages over the business cycle (USA)
Fluctuations of the real interest rate are shown alongside fluctuations of GDP in
Figure 8.12. The cyclicality of the real interest rate appears to change over time,
generally being countercyclical prior to the 1990s and procyclical afterwards. Taking
an overview of the whole period covered by the data, the real interest rate is
weakly countercyclical. Finally, fluctuations of inflation are shown in Figure 8.13.
Here again the relationship with GDP fluctuations appears to have changed over
time. There is a strong countercyclical relationship in the 1970s but at other times
inflation appears procyclical. Overall, we conclude that inflation is weakly
procyclical.
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Figure 8.11: Fluctuations of average labour productivity over the business cycle (USA)
Figure 8.12: Fluctuations of real interest rates over the business cycle (USA)
How do the predictions of the new Keynesian model compare to this evidence on
business-cycle fluctuations? We will consider two different types of shock studied
earlier. First, we will consider a demand shock coming from news that changes
confidence about the future, or a change in the extent of credit-market imperfections.
Second, we will look at a shift in the stance of monetary policy.
Figure 8.13: Fluctuations of inflation over the business cycle (USA)
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Consider the negative demand shock considered earlier that was caused by lower
confidence about the future. A lower expected value of 𝑧𝑧′ shifts the 𝑌𝑌 𝑑𝑑 curve to the
left due to declines in 𝐶𝐶 𝑑𝑑 and 𝐼𝐼 𝑑𝑑 . We assume a horizontal 𝑀𝑀𝑀𝑀 line in an unchanged
position, indicating a passive stance of monetary policy throughout.
Referring to Figure 8.8, the model predicts that GDP 𝑌𝑌 falls and consumption 𝐶𝐶 and
investment 𝐼𝐼 are lower. Prices and inflation do not change because all prices fixed. The
real and nominal interest rates 𝑟𝑟 and 𝑖𝑖 are unchanged because of the passive
monetary policy and the absence of any change in inflation. The production function
implies employment 𝑁𝑁 declines and the leftward shift of 𝑁𝑁 𝑑𝑑 and rightward shift of 𝑁𝑁 𝑠𝑠
in the labour market result in a lower real wage 𝑤𝑤. If we assumed firms are paying
efficiency wages instead then the real wage would remain constant and
unemployment would rise owing to the direction of the shifts of 𝑁𝑁 𝑑𝑑 and 𝑁𝑁 𝑠𝑠 .
Using the production function diagram, we can also see the model’s prediction for the
response of average labour productivity 𝐴𝐴𝐴𝐴𝐴𝐴 = 𝑌𝑌/𝑁𝑁. Average labour productivity is
given the gradient of the ray from the origin to the relevant point on the production
function. Figure 8.14 shows that since the production function has a concave shape
and does not shift with a demand shock, a decline in employment and output raises
the gradient of this ray. Intuitively, the shape of the production function comes from
diminishing returns to labour, so a decline in employment raises labour productivity.
In summary, the new Keynesian model with demand shocks predicts that consumption
and investment are procyclical (they both move in the same direction as real GDP),
employment is procyclical and average labour productivity is countercyclical. Apart
from productivity, these predictions are in line with the business-cycle stylised facts.
With a competitive labour market, the model predicts a strongly procyclical real wage
contrary to the empirical evidence but this cyclicality of wages would be weakened by
integrating the model of efficiency wages into the analysis of the labour market.
Adding efficiency wages also allows the new Keynesian model to match the
countercyclicality of unemployment. These predictions of the model and the
corresponding stylised facts from the data are summarised in Table 8.1. Note that the
model makes similar predictions for other forms of demand shock such as a worsening
of credit-market imperfections that raises interest-rate spreads.
Figure 8.14: Prediction of countercyclical average labour productivity
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As the stance of monetary policy remains completely passive following the shock by
assumption, the model predicts an acyclical real interest rate. Having an upwardsloping 𝑀𝑀𝑀𝑀 line, as discussed in Box 8.4, would result in the model predicting a
procyclical real interest rate. While the overall pattern in the data is weak
countercyclicality, there are periods where real interest rates appear procyclical, so
the model’s predictions are not too far from the empirical evidence. With completely
sticky prices, the model predicts that inflation is acyclical. We will see in Chapter 9 that
adding partial price adjustment means that inflation would be procyclical in an
economy with demand shocks, which helps to match the data.
Table 8.1: Predictions of new Keynesian model with demand shock to confidence about
the future
Variable
Model
Data
Consumption
Procyclical
Procyclical
Investment
Procyclical
Procyclical
Real interest
rate
Acyclical
(Procyclical with upwardsloping MM line)
Countercyclical (weakly)
Employment
Procyclical
Procyclical
Real wage
Procyclical
(Acyclical with efficiency wage)
Procyclical (weakly)
Unemployment
Acyclical
(Countercyclical with efficiency
wage)
Countercyclical
Average labour
productivity
Countercyclical
Procyclical
Acyclical
Procyclical
Inflation
It is also possible to consider business cycles triggered by shifts in monetary policy, for
example, the Volcker disinflation discussed in Box 8.1. Suppose the central bank shifts
the stance of monetary policy towards higher interest rates, shifting the 𝑀𝑀𝑀𝑀 line
upwards. This is the opposite of the case depicted in Figures 8.5 and 8.6 from Section
8.3.
The model predicts that consumption 𝐶𝐶 and investment 𝐼𝐼 fall, moving up the 𝑌𝑌 𝑑𝑑 curve
in the goods-market diagram. Real GDP 𝑌𝑌 falls and the real interest rate 𝑟𝑟 rises. It can
be seen from the production function and labour-market diagrams that employment
𝑁𝑁 and real wages 𝑤𝑤 are lower, with 𝑁𝑁 𝑑𝑑 shifting to the left and 𝑁𝑁 𝑠𝑠 shifting to the right.
With efficiency wages, unemployment 𝑈𝑈 rises and the real wage 𝑤𝑤 is constant. Prices
and inflation do not change because all prices are fixed.
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Table 8.2 summarises the predictions of the model and compares them to the
business-cycle stylised facts. Consumption and investment are procyclical, matching
the data. The real interest rate is countercyclical, which fits the overall pattern weakly
present in the data. Employment is procyclical, matching the data. Real wages are
strongly procyclical but that prediction can be tempered by efficiency wages, which
also allows the model to generate the countercyclical unemployment seen in the data.
Inflation is acyclical but partial price adjustment would change that prediction to
procyclicality. As with demand shocks, the model predicts countercyclical average
labour productivity, contrary to the empirical pattern.
In summary, the new Keynesian model is broadly consistent with most of the businesscycle stylised facts when the business cycle is caused by demand shocks, including
shifts in monetary policy. The only major failing is in accounting for the procyclicality
of productivity. One potential reconciliation with the productivity data is discussed in
Box 8.3.
Table 8.2: Predictions of the new Keynesian model with a monetary policy shock
Variable
Model
Data
Consumption
Procyclical
Procyclical
Investment
Procyclical
Procyclical
Real interest
rate
Countercyclical
Countercyclical (weakly)
Employment
Procyclical
Procyclical
Real wage
Procyclical
(Acyclical with efficiency wage)
Procyclical (weakly)
Unemployment
Acyclical
(Countercyclical with efficiency
wage)
Countercyclical
Average labour
productivity
Countercyclical
Procyclical
Acyclical
Procyclical
Inflation
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Box 8.3: Labour hoarding
Empirical evidence indicates average labour productivity (𝐴𝐴𝐴𝐴𝐴𝐴 = 𝑌𝑌⁄𝑁𝑁) is a
procyclical variable, moving in the same direction as real GDP 𝑌𝑌. However, the new
Keynesian model with demand shocks predicts average labour productivity is
countercyclical.
The reason for the model’s prediction of countercyclical productivity is that the
neoclassical production function 𝑌𝑌 = 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑁𝑁) features a diminishing marginal
product of labour, while demand shocks do not shift the production function by
affecting current TFP 𝑧𝑧. As the total stock of capital 𝐾𝐾 changes relatively little over
business cycle, when 𝑁𝑁 and 𝑌𝑌 fall, the marginal product of labour 𝑀𝑀𝑃𝑃𝑁𝑁 rises. This
implies average labour productivity 𝑌𝑌/𝑁𝑁 rises as employment falls.
The empirical evidence suggests a relationship in the opposite direction. However,
data on employment might not measure accurately the true labour input 𝑁𝑁 going
into the production function 𝑌𝑌 = 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑁𝑁). Data on employment is a headcount
of firms’ employees but does not capture how intensively employees are working.
There is data on total hours worked but, for many jobs, this simply measures
contractual hours of work, which does not fully measure the intensity of work. A
true measure of labour input 𝑁𝑁 would account for the intensity of work.
The claim that true labour input 𝑁𝑁 might fall substantially in a recession while firms
retain most of their staff raises the question of why firms do not lay-off workers
when fewer are needed. One argument for this ‘labour hoarding’ is that a recession
is expected to be temporary and firms want to avoid incurring the costs of hiring
again in the future recovery (the costly process of search and matching in hiring
workers was discussed in Chapter 5).
The occurrence of labour hoarding helps explain why measured average labour
productivity is procyclical. As shown in Figure 8.15, if measured labour input falls by
much less than actual labour input, the drop in output will be associated with a fall
in measured productivity (calculated using data on actual output and measured
employment). This measurement problem suggests that the observed procyclicality
of productivity may not be inconsistent with the new Keynesian model of the
business cycle with demand shocks.
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Figure 8.15: Procyclical measured productivity with labour hoarding
An issue similar to labour hoarding arises when measuring total factor productivity
(TFP). This is done by calculating a ‘Solow residual’, that is, the change in GDP not
explained by changes in inputs of labour and capital. However, estimate of the
capital stock do not fully capture changes in the usage of capital by firms because
utilisation of capital might vary over time. This is analogous to the change in labour
utilisation by firms when there is labour hoarding and suggests there is bias
towards detecting procyclicality in TFP.
8.5 The natural rate of interest
Although the objective of the new Keynesian model is to understand the functioning of an
economy with sticky prices, analysing the hypothetical case of fully flexible prices even in
the short run is nonetheless useful. This helps us understand the different predictions the
model makes for the short run and the long run. It also provides guidance on how monetary
policy should be conducted.
8.5.1 Imperfect competition and the output supply curve
We have assumed the goods market is imperfectly competitive to allow for sticky prices. We
now consider how imperfectly competitive firms would set prices if they were always free to
adjust them. In doing this, we make use of a model of monopolistic competition from
microeconomics. Each firm faces a downward-sloping demand curve for its product because
goods produced by different firms are imperfect substitutes. Conditional on aggregate
demand, a firm can only sell more of its product by charging a lower price, unlike perfect
competition where firms are able to sell any amount at the prevailing market price.
Profit maximisation by imperfectly competitive firms implies they will exploit market power
to sell at a price above marginal cost. This is because firms with market power face a
downward-sloping demand curve for their product, so they can charge a higher price by
choosing to sell less.
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Given a production function 𝑌𝑌 = 𝑧𝑧𝑧𝑧(𝐾𝐾, 𝑁𝑁) with a particular capital stock 𝐾𝐾 and level of TFP
𝑧𝑧, the decision to supply goods 𝑌𝑌 is equivalent to a decision to hire labour 𝑁𝑁. For an
imperfectly competitive firm, the effect on real revenue of hiring an extra worker is less
than the physical marginal product of labour because the price of its product relative to
other goods needs to be lowered to sell the extra output. The marginal gain in real revenue
from hiring an extra unit of labour is measured by the marginal revenue product of labour
𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 , which is below the marginal product of labour 𝑀𝑀𝑃𝑃𝑁𝑁 . The relationship between the
two is 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 = (1 − 𝜖𝜖 −1 )𝑀𝑀𝑃𝑃𝑁𝑁 , where 𝜖𝜖 is the price elasticity of the demand curve for a
firm’s product.
In choosing how much labour to hire and how much output to produce, each firm compares
the real cost of hiring a worker, the real wage 𝑤𝑤, to the marginal benefit 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 . Hence,
firms’ demand for labour is given by 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 = 𝑤𝑤 instead of 𝑀𝑀𝑃𝑃𝑁𝑁 = 𝑤𝑤 with perfect
competition. Workers being paid their marginal revenue product is equivalent to firms
pricing (𝜖𝜖 − 1)−1 per cent above their marginal cost 𝑤𝑤/𝑀𝑀𝑃𝑃𝑁𝑁 of producing a unit of output.
Assuming each individual firm faces a demand curve for its product with a constant price
elasticity 𝜖𝜖 > 1, the marginal revenue product curve 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 is simply a scaling down of the
marginal product curve 𝑀𝑀𝑃𝑃𝑁𝑁 as shown in Figure 8.16. Labour demand thus behaves in the
same way as with perfect competition, just at a lower level, all else being equal. An output
supply curve 𝑌𝑌 𝑠𝑠 can then be derived exactly as earlier in the dynamic macroeconomic model
from Section 3.12. The only difference is that 𝑌𝑌 𝑠𝑠 is lower for any given 𝑟𝑟 because
imperfectly competitive firms restrict supply to raise profits. For comparison, what the
output supply curve would look like with perfect competition is shown labelled as 𝑌𝑌� 𝑠𝑠 in the
figure.
8.5.2 Market clearing in the absence of nominal rigidities
With flexible prices and wages in the goods and labour markets, the intersection of the
demand and supply curves determines equilibrium in all markets. As shown in Figure 8.16,
there is a market-clearing real interest rate 𝑟𝑟 ∗ . This interest rate 𝑟𝑟 ∗ is known as the ‘natural
rate of interest’. It is the hypothetical real interest rate that would prevail if there were no
nominal rigidities in the economy. The ‘natural’ terminology is also applied to other
variables. The ‘natural level of output’ is the market-clearing level of real GDP 𝑌𝑌 ∗ in the
absence of any nominal rigidities. By incorporating efficiency wages into the analysis of the
labour market, there would be a ‘natural rate of unemployment’, i.e. the unemployment
rate occurring with no nominal rigidity (note that efficiency wages are not a nominal rigidity
– they explain why firms do not want to adjust real wages).
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Figure 8.16: Labour and goods markets with flexible prices
8.5.3 The long run and the short run
We now return to the actual assumption made in the new Keynesian model that goods
prices are sticky. The model implies we should think differently about how real GDP is
determined in the short run and the long run.
We take ‘long run’ to mean a situation where current market conditions have been correctly
foreseen and are not expected to change. Even if firms have sticky prices 𝑃𝑃�, all prices are set
appropriately for the current state of the economy. In this case, the new Keynesian model
predicts the real interest rate and output coincide with their ‘natural’ levels. Moreover, as
long as prices do not remain sticky forever, even if shocks do occur, the new Keynesian
model predicts all variables will tend to their respective natural levels in the long run absent
any further changes or shocks to the economy.
The ’short run’ is the time horizon in which market conditions can deviate from what was
expected when prices were originally set in the past. Shocks result in the economy
fluctuating around its natural level of output. Note that it is possible to have GDP above or
below its natural level 𝑌𝑌 ∗ . As long as 𝑌𝑌 and 𝑟𝑟 lie to the left of the hypothetical perfectcompetition output supply curve 𝑌𝑌� 𝑠𝑠 (which is true for 𝑟𝑟 ∗ and 𝑌𝑌 ∗ ) then 𝑤𝑤 < 𝑀𝑀𝑃𝑃𝑁𝑁 holds and
firms would willingly sell more if given the chance – and must sell less if demand falls.
When the economy experiences a shock and the actual level of real GDP 𝑌𝑌 deviates from its
natural level 𝑌𝑌 ∗ , we say there is an ‘output gap’ between 𝑌𝑌 and 𝑌𝑌 ∗ . We will see that there is
a case for the central bank or government to use demand-management policies to try to
close the output gap, moving GDP 𝑌𝑌 towards 𝑌𝑌 ∗.
8.6 Optimal stabilisation policy
With sticky prices and imperfect competition, the equilibrium of the economy is not
efficient. Following a shock to the economy, it is possible to obtain a better outcome for
households through a macroeconomic policy intervention. This is an improvement on
waiting for prices to adjust.
When the economy has a representative household, efficiency can be judged easily by
comparing the marginal product of labour 𝑀𝑀𝑃𝑃𝑁𝑁 to households’ marginal rate of substitution
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between leisure and consumption 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 . The 𝑀𝑀𝑃𝑃𝑁𝑁 is what can be produced if people were
able to work more and the 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 is what value (in terms of goods) people put on their
time. The economy has inefficiently low employment and production if 𝑀𝑀𝑃𝑃𝑁𝑁 > 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶
because the value of people’s time is less than the value they put on the extra goods that
can be produced and consumed if there were more economic activity.
How efficiently the economy is operating can be judged from the goods market diagram by
comparing the outcome for 𝑌𝑌 and 𝑟𝑟 to the hypothetical perfectly competitive output supply
curve 𝑌𝑌� 𝑠𝑠 , on which 𝑀𝑀𝑃𝑃𝑁𝑁 = 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 . All points to the left of 𝑌𝑌� 𝑠𝑠 have 𝑀𝑀𝑃𝑃𝑁𝑁 > 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 ,
meaning that output 𝑌𝑌 is too low. Inefficiency is thus measured by how far the economy is
to the left of the 𝑌𝑌� 𝑠𝑠 curve.
Why is the market equilibrium in the new Keynesian model generally not efficient? To
simplify the analysis, we ignore some other reasons for inefficiency we have studied
elsewhere that are not central to the new Keynesian model. First, wages are flexible, so the
labour-market equilibrium is always on the labour supply curve. This ignores the persistent
unemployment that results from firms’ incentives to pay ‘efficiency wages’ as seen in
Section 5.2. Second, we ignore the implications for labour supply of needing to use money
as a medium of exchange that were studied in Section 6.5. This neglects any inefficiencies
resulting from a failure of monetary policy to follow the ‘Friedman rule’ as discussed in
Section 6.10. The consequence of these simplifications is that real wages 𝑤𝑤 are always equal
to the marginal rate of substitution 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 between leisure and consumption.
There are two distinct reasons why output is inefficiently low in the New Keynesian model.
First, the natural level of output 𝑌𝑌 ∗ is already too low because imperfect competition gives
firms an incentive to reduce production. Even without nominal rigidity, imperfect
competition would result in 𝑤𝑤 = 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 < 𝑀𝑀𝑃𝑃𝑁𝑁 at 𝑌𝑌 = 𝑌𝑌 ∗ . Second, when prices are sticky, a
negative demand shock pushes GDP 𝑌𝑌 below 𝑌𝑌 ∗ and, as diminishing returns to labour then
implies 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 and 𝑀𝑀𝑃𝑃𝑁𝑁 rise while 𝑤𝑤 = 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 falls, we have 𝑤𝑤 < 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 and the economy
is even further away from what is efficient.
How can economic policy achieve a better outcome? Let us consider the example studied in
Section 8.4 of a negative demand shock due to a decline in expected future TFP 𝑧𝑧′. As shown
in Figure 8.17, the 𝑌𝑌 𝑑𝑑 curve shifts to the left and output 𝑌𝑌 falls below 𝑌𝑌 ∗ if the 𝑀𝑀𝑀𝑀 line
remains in its original position. Now, instead of leaving monetary policy unchanged, the
central bank lowers the nominal and real interest rates 𝑖𝑖 and 𝑟𝑟. Reducing 𝑟𝑟 moves the
economy along the 𝑌𝑌 𝑑𝑑 curve, raising GDP 𝑌𝑌. Since 𝑀𝑀𝑃𝑃𝑁𝑁 > 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 , the representative
household gains from this policy intervention. Fiscal policy could also be used to provide a
stimulus to demand. An increase in public expenditure 𝐺𝐺 would raise 𝑌𝑌, although its effects
are not exactly equivalent because the composition of aggregate demand 𝐶𝐶 + 𝐼𝐼 + 𝐺𝐺 would
be different compared to the case where monetary policy is used.
What is the best monetary policy to follow? To close the output gap between 𝑌𝑌 and 𝑌𝑌 ∗
exactly, the central bank should set the nominal interest rate 𝑖𝑖 (and, hence, 𝑟𝑟) equal to the
natural rate of interest 𝑟𝑟 ∗ . This moves the 𝑀𝑀𝑀𝑀 line to where it intersects the 𝑌𝑌 𝑑𝑑 curve in the
same place as the output supply curve 𝑌𝑌 𝑠𝑠 as shown in Figure 8.17. Such a monetary policy
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achieves the same economic outcome as if prices were flexible. The policy intervention thus
neutralises the negative consequences of slow price adjustment following a shock. The
central bank adjusting interest rates compensates for the slow pace of price changes.
Figure 8.17: Optimal stabilization policy
To implement this optimal monetary policy, the central bank needs to know the natural rate
of interest 𝑟𝑟 ∗ . This is a practical problem because 𝑟𝑟 ∗ is not directly observable and needs to
be estimated. A more fundamental challenge is that it must be feasible to reduce the
nominal interest rate 𝑖𝑖 if 𝑟𝑟 ∗ falls. As explained in Section 6.13, nominal interest rates are
subject to a lower bound, so the required interest rate cut might not be possible if the lower
bound on 𝑖𝑖 is reached. Chapter 9 discusses alternative policies that could be used if the
lower bound is binding.
The stabilisation policy described here aims to close the output gap between actual real
GDP 𝑌𝑌 and its natural level 𝑌𝑌 ∗ . But that does not mean the policy should aim for GDP to be
stable if 𝑌𝑌 ∗ itself varies over time. Furthermore, it might be wondered why the policy
intervention should stop when 𝑌𝑌 reaches 𝑌𝑌 ∗ . That addresses only one of the two sources of
inefficiency in the new Keynesian model – remember output at 𝑌𝑌 ∗ is still inefficiently low.
Should the central bank continue to push output above 𝑌𝑌 ∗ to where 𝑌𝑌 𝑑𝑑 intersects 𝑌𝑌� 𝑠𝑠 and
𝑀𝑀𝑃𝑃𝑁𝑁 = 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 hold? While at first glance this appears desirable, we will argue in Chapter 9
that such a policy would be unsustainable and give rise to negative side effects.
Box 8.4: Modelling monetary policy using Taylor rules and LM
curves
In the new Keynesian model so far, we have represented monetary policy using a
horizontal 𝑀𝑀𝑀𝑀 line. This shifts vertically if the central bank changes the nominal
and real interest rate. But the shape of the 𝑀𝑀𝑀𝑀 line is not an inherent feature of
the new Keynesian model, it depends on the most appropriate way to describe the
conduct of monetary policy.
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More generally, we consider the 𝑀𝑀𝑀𝑀 (‘money and monetary policy’) line to be a
representation of how the money market and the monetary policy regime imply a
link between interest rates and GDP. We can draw 𝑀𝑀𝑀𝑀 lines for monetary policies
that target the money supply, or an interest rate feedback rule such as a Taylor
rule.
8.6.1 Money supply targets, the LM curve and the IS-LM model
Suppose the central bank’s monetary policy is a target for the money supply 𝑀𝑀 𝑠𝑠 . Assume
the money-supply target 𝑀𝑀 𝑠𝑠 = 𝑀𝑀∗ is exogenous. For this monetary policy regime, the
interest rate 𝑖𝑖 is endogenous. It is still the case that real and nominal interest rates are same
because prices are sticky, hence, 𝑟𝑟 = 𝑖𝑖. With a target for the money supply, the interest
rates 𝑖𝑖 and 𝑟𝑟 and GDP 𝑌𝑌 are determined jointly in the goods and money markets. The
equivalent of the 𝑀𝑀𝑀𝑀 line in this case represents money-market equilibrium and it is often
labelled as the ‘𝐿𝐿𝐿𝐿’ curve for this particular monetary policy.
The 𝑀𝑀𝑀𝑀 line/𝐿𝐿𝐿𝐿 curve for a money-supply target is upward sloping, as shown in the goods
market diagram in Figure 8.18. The 𝐿𝐿𝐿𝐿 curve is upward sloping because higher output
increases the real demand for money for transactions, as shown in the left panel of the
figure representing the money market. With a fixed nominal money supply 𝑀𝑀∗ and a sticky
price level 𝑃𝑃�, there is an inelastic supply of real money balances 𝑀𝑀∗ /𝑃𝑃� and the nominal
interest rate 𝑖𝑖 rises to restore equilibrium in the money market. Money-market equilibrium
thus requires higher real interest rates 𝑟𝑟 when real GDP 𝑌𝑌 is higher, explaining the upwardsloping 𝐿𝐿𝐿𝐿 curve.
Figure 8.18: The LM curve with a money supply target
Combining the 𝐿𝐿𝐿𝐿 curve (𝑀𝑀𝑀𝑀 line) with the output demand curve 𝑌𝑌 𝑑𝑑 in the goods market
leads to the IS-LM model, which is a special case of our new Keynesian model. What is called
the 𝐼𝐼𝐼𝐼 curve in that model is simply another label for what we call output demand 𝑌𝑌 𝑑𝑑 .
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A change in the money-supply target causes the 𝐿𝐿𝐿𝐿 curve to shift, which has real effects on
the economy. Increasing the money supply 𝑀𝑀∗ implies the supply of real money balances
𝑀𝑀∗ /𝑃𝑃� is larger (the price level remaining constant at 𝑃𝑃�). Given the real demand for money
at a particular level of real GDP 𝑌𝑌, the intersection of money supply and demand occurs at a
lower nominal interest rate 𝑖𝑖 and, hence, also 𝑟𝑟. Since the 𝐿𝐿𝐿𝐿 curve represents
combinations of 𝑌𝑌 and 𝑟𝑟 where the money market is in equilibrium, the 𝐿𝐿𝐿𝐿 curve must shift
downwards and real GDP increases.
8.6.2 The Taylor rule
Another example of a monetary policy is to have the central bank adjust the nominal
interest rate 𝑖𝑖 systematically in response to inflation 𝜋𝜋 and output 𝑌𝑌, for example, by
following a Taylor rule. We have seen an example of a Taylor rule in Section 6.12 but that
focused only on the response of 𝑖𝑖 to inflation 𝜋𝜋. In the basic new Keynesian model, prices
are completely fixed, so there is no inflation and response of 𝑖𝑖 to 𝜋𝜋 not relevant here.
In response to changes in real GDP, the Taylor rule calls for a higher interest rate 𝑖𝑖 in a boom
and a lower 𝑖𝑖 in a recession. With real and nominal interest rates being equal, 𝑟𝑟 = 𝑖𝑖, the
positive response of 𝑖𝑖 to 𝑌𝑌 can be represented by an upward-sloping 𝑀𝑀𝑀𝑀 line as depicted in
Figure 8.19.
Figure 8.19: Using a Taylor rule
Why should the central bank want to choose interest rates that are positively related to 𝑌𝑌?
One argument is that this helps to stabilise an economy that is hit by demand shocks,
avoiding large output gaps between actual GDP 𝑌𝑌 and the natural level of output 𝑌𝑌 ∗ .
Furthermore, if 𝑌𝑌 ∗ is known or estimated, the interest-rate rule can be refined to react to
the gap 𝑌𝑌 − 𝑌𝑌 ∗ , or an estimate of this output gap.
We know from Section 8.6 that the optimal monetary policy is for the central bank to set
𝑖𝑖 = 𝑟𝑟 ∗ , where 𝑟𝑟 ∗ is the natural rate of interest. However, the central bank may not have
perfect information about 𝑟𝑟 ∗ . In that case, Figure 8.20 shows having a positive response of 𝑖𝑖
to 𝑌𝑌 can yield a better outcome for the economy than having monetary policy keep 𝑖𝑖
constant.
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Figure 8.20: Demand shocks when a Taylor rule is used
8.7 Real business cycle theory
In the new Keynesian model, the business cycle is the economy’s inefficient response to
shocks, usually demand shocks, owing to the failure of prices to adjust. This justifies policy
interventions to temper the business cycle. An alternative approach argues that business
cycles are simply the economy’s efficient response to variations in its ability to produce
goods due to supply shocks. Policy intervention is futile or counterproductive in this view.
This way of understanding fluctuations in the economy is known as real business cycle (RBC)
theory.
How does such a theory of the business cycle work? The RBC model is essentially just the
dynamic macroeconomic model developed earlier in Chapter 3. The core of the model
features flexible prices and perfectly competitive markets studied in general equilibrium as
we did in Section 3.12.
For completeness, we add a money market alongside the labour and goods markets studied
in the dynamic macroeconomic model. Money demand and supply come from our analysis
in Chapter 6 and we assume the central bank chooses exogenous path of the money supply.
This emphasises the medium of exchange function of money but has no special role for the
unit of account function owing to nominal rigidities unlike the earlier new Keynesian model.
For simplicity, we ignore the effect of money’s medium of exchange function on the labour
supply curve, or we assume the central bank is following the Friedman rule. This means that
the labour supply curve 𝑁𝑁 𝑠𝑠 (𝑟𝑟) derives from the household optimality conditions 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 =
𝑤𝑤 and 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝑙𝑙′ = (1 + 𝑟𝑟)𝑤𝑤/𝑤𝑤′ discussed in Sections 1.4 and 3.10.
The RBC approach to understanding business cycles looks at how the equilibrium of the
economy in the goods, labour and money markets shown in Figure 8.21 is affected by supply
shocks.
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8.8 Business cycles due to supply shocks
RBC theory identifies exogenous shocks to total factor productivity (TFP) as the source of
the business cycle: supply shocks (or ‘technology’ shocks). In the study of economic growth,
technological progress was seen as driving permanent increases in TFP. However, in RBC
theory, supply shocks are increases or decreases in TFP that are eventually reversed.
Figure 8.21: The RBC model
8.8.1 Supply shocks
Supply shocks in the RBC model are usually assumed to be deviations from the trend in TFP
growth that are expected to persist beyond current time period to some extent but which
are not permanent. The empirical counterpart to TFP is the Solow residual, representing
changes in the level of real GDP that cannot be explained by changes in inputs of capital and
labour. The deviations from trend of the Solow residual in the USA are shown in Figure 8.22
alongside the deviations from trend of real GDP. We see that movements in the Solow
residual display a clear positive correlation with real GDP, indicating the Solow residual is
procyclical. While it is less volatile than GDP, we do see transitory fluctuations that could be
a cause of business cycles. As discussed in Box 8.3, it is possible some of this procyclicality
could be the result of measurement error in accounting for factor inputs.
Figure 8.22: Fluctuations of the Solow residual (USA)
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Assuming the evidence from the Solow residual correctly indicates that transitory supply
shocks are hitting the economy, what might such shocks represent? One possibility is an
uneven pace of technological progress, where TFP might rise by more than usual in some
years, or by less in others. However, the Solow residual often falls sufficiently far below its
trend that the implied level of TFP actually declines in absolute terms. That is hard to
understand if TFP is representing technology because we would not expect that to go
backwards.
However, there are a number of other possible sources of transitory changes in TFP:
•
•
•
•
Fluctuations in energy costs, which affect overall production costs
Supply disruptions, for example due to natural disasters, wars, pandemics
Changes in regulations that affect firms’ productivity
Weather (in an agricultural economy).
8.8.2 The predictions of the RBC model
Representing a supply shock as a transitory change in TFP 𝑧𝑧, the effects of a negative shock
to 𝑧𝑧 are shown in Figure 8.23. The production function moves down, shifting the 𝑌𝑌 𝑠𝑠 curve to
the left. Lower 𝑧𝑧 implies lower 𝑀𝑀𝑃𝑃𝑁𝑁 , which shifts the 𝑁𝑁 𝑑𝑑 curve to the left and results in the
𝑌𝑌 𝑠𝑠 curve moving further to the left.
To the extent that the shock persists for some time, expectations of future TFP 𝑧𝑧′ decline,
which implies lower 𝑀𝑀𝑃𝑃𝐾𝐾′ and shifts the 𝐼𝐼 𝑑𝑑 and 𝑌𝑌 𝑑𝑑 curves to the left. There is lower 𝐶𝐶 𝑑𝑑 and
higher 𝑁𝑁 𝑠𝑠 owing to the negative wealth effect of lower 𝑧𝑧 (and 𝑧𝑧′), although there is
consumption smoothing because the shock is not permanent. These wealth effects imply a
leftward shift of 𝑌𝑌 𝑑𝑑 and a rightward shift of 𝑌𝑌 𝑠𝑠 . The wealth effect on labour supply is smaller
than the impact of 𝑧𝑧 on 𝑌𝑌 𝑠𝑠 both directly and through 𝑀𝑀𝑃𝑃𝑁𝑁 . Hence, the overall effects are
that the 𝑌𝑌 𝑑𝑑 and 𝑌𝑌 𝑠𝑠 curves shift to the left, so real GDP 𝑌𝑌 falls.
Figure 8.23: A negative supply shock in the RBC model
In the goods market, the real interest rate 𝑟𝑟 rises if 𝑌𝑌 𝑠𝑠 shifts more than 𝑌𝑌 𝑑𝑑 . The 𝑌𝑌 𝑑𝑑 shift
becomes larger if the drop in TFP lasts longer because that leads to a greater impact on
consumption and investment demand (less consumption smoothing and a greater impact on
the expected future marginal product of capital). The effect on employment 𝑁𝑁 in the labour
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market is ambiguous because 𝑁𝑁 𝑑𝑑 falls but 𝑁𝑁 𝑠𝑠 can rise. The wealth effect on 𝑁𝑁 𝑠𝑠 is smaller
when TFP is expected to recover quickly. If 𝑁𝑁 𝑑𝑑 shifts to the left and 𝑁𝑁 𝑠𝑠 shifts to the right,
the real wage 𝑤𝑤 increases. A smaller wealth effect on 𝑁𝑁 𝑠𝑠 means that 𝑤𝑤 rises by less. Finally,
in the money market, 𝑀𝑀𝑑𝑑 falls with lower 𝑌𝑌, which leads to a higher price level 𝑃𝑃.
8.8.3 Stabilisation policy?
When a recession occurs because of lower TFP 𝑧𝑧, this clearly makes households worse off.
While the recession is bad, in the RBC model, it does not follow that the government should
intervene. The model predicts that policy intervention, even if it succeeds in raising GDP,
makes households worse off. First, with flexible prices, there is limited scope to raise GDP
with monetary policy – it is not possible to improve on following the Friedman rule. Second,
while increasing public expenditure 𝐺𝐺 raises GDP 𝑌𝑌 as we saw in Box 4.2, this is inefficient
because it leads people to work more when productivity is low.
Box 8.5: Sources of supply shocks in the RBC model
Considering a transitory positive supply shock, we have seen that the RBC model
makes the following predictions:
•
•
•
•
•
•
GDP 𝑌𝑌 rises as the 𝑌𝑌 𝑑𝑑 and 𝑌𝑌 𝑠𝑠 curves shift to the right
The real interest rate 𝑟𝑟 falls as 𝑌𝑌 𝑠𝑠 shifts by more than 𝑌𝑌 𝑑𝑑
Consumption 𝐶𝐶 rises because of the wealth effect and lower 𝑟𝑟
Investment 𝐼𝐼 rises because of higher expectations of 𝑀𝑀𝑃𝑃𝐾𝐾′ and lower 𝑟𝑟
Real wage 𝑤𝑤 rises as 𝑁𝑁 𝑑𝑑 shifts to the right and 𝑁𝑁 𝑠𝑠 shifts to the left
Employment 𝑁𝑁 rises if 𝑁𝑁 𝑑𝑑 shifts more than 𝑁𝑁 𝑠𝑠 , which occurs if the wealth
effect on 𝑁𝑁 𝑠𝑠 is weak, as would be the case for a transitory shock.
Finally, Figure 8.24 shows that average labour productivity 𝑌𝑌/𝑁𝑁 can rise with GDP
because 𝑧𝑧 increases. This is in contrast to the prediction for (correctly measured)
average labour productivity in a model with demand shocks and a neoclassical
production function.
Figure 8.24: Average labour productivity in the RBC model
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The RBC model generates business-cycle fluctuations due to occurrence of
transitory supply shocks. Its predictions are summarised in Table 8.3 alongside the
empirical evidence on the business-cycle stylised facts. We see that the fluctuations
of macroeconomic variables implied by the model are consistent with businesscycle stylised facts.
Table 8.3: Predictions of the RBC model with transitory supply shocks
Variable
Model
Data
Consumption
Procyclical
Procyclical
Investment
Procyclical
Procyclical
Real interest
rate
Countercyclical
Countercyclical (weakly)
Employment
Procyclical
Procyclical
Real wage
Procyclical
Countercyclical (weakly)
Average labour
productivity
Procyclical
Procyclical (weakly)
In spite of this success, the source of transitory supply shocks is not obvious in
many business-cycle episodes. We could consider instead supply shocks with
permanent effects that are more easily interpreted as being due to the uneven
pace of technological progress. However, it is much harder to make the RBC model
consistent with the business-cycle stylised facts when supply shocks are highly
persistent.
A permanent productivity shock changes current and future TFPs 𝑧𝑧 and 𝑧𝑧′ by a
similar amount. Considering the case of permanent positive shock, the RBC model
predicts the following in comparison to a transitory shock:
•
•
•
Wealth effects are larger, hence, 𝑁𝑁 𝑠𝑠 shifts further to the left, and 𝐶𝐶 𝑑𝑑 rises
by more, shifting 𝑌𝑌 𝑑𝑑 further to the right
There is a greater incentive for firms to increase investment, and the larger
increase in 𝐼𝐼 𝑑𝑑 means 𝑌𝑌 𝑑𝑑 shifts further to the right
The effect on 𝑁𝑁 𝑑𝑑 is same for a permanent and transitory shock to 𝑧𝑧.
Since 𝑌𝑌 𝑑𝑑 and 𝑌𝑌 𝑠𝑠 shift in the same direction, the RBC model can still generate
fluctuations in GDP 𝑌𝑌 with permanent shocks to TFP. However, as Figure 8.25
shows, the model will struggle to generate predictions consistent with the stylised
facts. The main problem is that the stronger wealth effect on labour supply 𝑁𝑁 𝑠𝑠
means that employment 𝑁𝑁 might fall when 𝑌𝑌 rises. Employment then becomes
countercyclical, which is clearly contrary to the data. Furthermore, the stronger
shift of 𝑁𝑁 𝑠𝑠 in the opposite direction to 𝑁𝑁 𝑑𝑑 implies much larger fluctuations in the
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real wage, which now becomes very procyclical, contrary to the data. These
predictions are consistent with very long-run trends where hours worked have
fallen even though productivity and wages have risen permanently.
8.9 Coordination failure model
An alternative approach to understanding business cycles emphasises what is called
‘coordination failure’. Business cycles are caused by self-fulfilling changes in beliefs – even
though everyone is rational – because the economy has multiple equilibria. This means that
the requirements for agents to make decisions rationally and for markets to clear do not pin
down unique levels of GDP, employment, wages, and interest rates. Business cycles result
from shifts between optimism and pessimism, even if economy’s fundamentals remain
unchanged. Although an equilibrium with high GDP is preferred by everyone, there is a
difficulty of coordinating expectations on this best outcome.
Our earlier dynamic macroeconomic model has a unique equilibrium, so there are no
business cycles without the occurrence of exogenous shocks. This is true for both new
Keynesian and RBC models of the economy. In the coordination failure model, all the
assumptions of the dynamic macroeconomic model are maintained, except that the
(aggregate) production function has increasing returns to scale. This leads to strategic
complementarities in firms’ employment decisions and implies there can be multiple
equilibria.
Figure 8.25: A permanent productivity shock in the RBC model
8.9.1 Labour productivity spillover across firms
At the level of an individual firm, the production function 𝑌𝑌𝑖𝑖 = 𝑧𝑧(𝑁𝑁)𝐹𝐹(𝐾𝐾𝑖𝑖 , 𝑁𝑁𝑖𝑖 ) is assumed to
have the neoclassical properties, in particular, a diminishing marginal product of labour 𝑁𝑁𝑖𝑖 .
Firms are competitive, so their labour demand curve is given by marginal product of labour.
The new feature of the model is that each firm’s TFP level 𝑧𝑧(𝑁𝑁) is positively related to
aggregate employment 𝑁𝑁 but each firm takes 𝑁𝑁 as given when choosing its own 𝑁𝑁𝑖𝑖 . Each
firm benefits from higher employment and output at other firms but this effect is not
internalised. This ‘spillover’ or externality is a source of market failure.
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Why would such positive labour productivity spillover occur? Advocates of coordinationfailure models point to several possibilities. First, there could be complementarities in
employment across firms, for example, if more people are employed writing computer
software, this makes computer hardware more useful, so workers producing the hardware
are now making something of greater value. Another example is that delivery and logistics
are more efficient when such companies have more orders to fulfil.
A second possibility is what are known as ‘thick market’ effects. This is where an increased
amount of activity in a market provides some direct benefit to market participants. For
example, it is harder to attract customers to a shopping mall when not many other shops in
the mall are open. The coordination failure model explores the implications of such thickmarket effects and complementarities. Of course, a priori, it is also possible to envisage
negative spillovers across firms, so the validity of the assumption is debatable.
The spillover effect has important consequences for firms’ demand for labour. If an
individual firm increases its employment 𝑁𝑁𝑖𝑖 , then its marginal product of labour declines, so
the firm-level labour demand curve is downward-sloping as usual. However, if all firms are
increasing employment 𝑁𝑁𝑖𝑖 together, then aggregate employment 𝑁𝑁 rises and 𝑧𝑧(𝑁𝑁)
increases. The rise in 𝑧𝑧(𝑁𝑁) boosts the marginal product of labour in each firm, offsetting the
decline due to higher 𝑁𝑁𝑖𝑖 . This spillover effect might outweigh the declining firm-level
marginal product of labour, so marginal product of labour increases with aggregate
employment 𝑁𝑁. In that case, the aggregate-level labour demand curve becomes upwardsloping. The firm-level and aggregate-level labour demand curves are plotted in Figure 8.26.
Figure 8.26: Aggregate labour demand in the coordination failure model
In what follows, we assume the positive spillover effect from firms’ employment is
sufficiently strong to make the aggregate labour demand curve upward sloping. Weaker
spillovers would mean 𝑁𝑁 𝑑𝑑 remains downward sloping, only becoming flatter. A strong
enough spillover to make 𝑁𝑁 𝑑𝑑 upward sloping implies increasing returns to labour at the
aggregate level, and results in an aggregate production function with a convex shape as
shown in Figure 8.27. Note that we assume a spillover effect – an externality – rather than
assume firms directly have an increasing-returns production function to maintain the
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framework of perfect competition. Increasing returns at the firm level requires a model with
imperfectly competitive firms, similar to that used earlier in the new Keynesian model with
sticky prices.
8.9.2 Implications for the output supply curve
The coordination failure model not only needs an aggregate labour demand curve that is
upward sloping. In addition, 𝑁𝑁 𝑑𝑑 must be steeper than labour supply 𝑁𝑁 𝑠𝑠 (𝑟𝑟), as depicted in
Figure 8.28. In the dynamic macroeconomic model studied in Section 3.12, the supply of
output by firms is derived from the equilibrium level of employment in the labour market. In
that model, the output supply curve 𝑌𝑌 𝑠𝑠 is upward sloping, meaning that the supply of goods
is positively related to the real interest rate 𝑟𝑟. Intuitively, a higher real interest rate is
needed to induce more supply by increasing the desire to save through earning more by
supplying more labour.
Figure 8.27: Aggregate production function in the coordination failure model
The relationship between 𝑟𝑟 and the supply of goods can be reversed with a strong enough
spillover effect in the coordination failure model. High output and high employment
generate a strong productivity-boosting spillover, which means firms are willing to choose
high employment even when a low interest rate 𝑟𝑟 reduces workers’ desire to save by
earning more. As shown in Figure 8.29, the coordination failure model features a
downward-sloping output supply curve 𝑌𝑌 𝑠𝑠 . This profoundly changes the predictions of the
model compared to the standard dynamic macroeconomic model.
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Figure 8.28: Labour market equilibrium in the coordination failure model
Figure 8.29: The output supply curve in the coordination failure model
8.10 Multiple equilibria and business cycles
We now investigate how business cycles can occur in the coordination failure model, even
when there are no exogenous shocks to the economy’s fundamentals. The key feature of
the model is that its output supply curve is downward sloping.
Goods-market equilibrium occurs where the 𝑌𝑌 𝑠𝑠 curve intersects the usual downwardsloping output demand curve 𝑌𝑌 𝑑𝑑 . The derivation of the output demand curve from Section
3.12 is unchanged here because the spillover effect in the coordination failure model works
through the supply side of the economy.
With output demand and supply curves both being downward sloping, there may not be a
unique equilibrium in the goods market because 𝑌𝑌 𝑑𝑑 and 𝑌𝑌 𝑠𝑠 can cross more than once. We
will focus on a case where 𝑌𝑌 𝑑𝑑 and 𝑌𝑌 𝑠𝑠 intersect twice, though this is only one of many
possibilities. This case is illustrated in Figure 8.30.
When there are multiple intersections of output demand and supply curves, the economy
has multiple equilibria. Each equilibrium is consistent with utility maximisation by
households and profit maximisation by firms, and market clearing (in both goods and labour
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markets because every point on the output supply curve represents a point of labourmarket equilibrium given the way the 𝑌𝑌 𝑠𝑠 curve is constructed). Without more assumptions,
it is not possible to say which equilibrium will be the outcome. Usually, we derive the
predictions of a model from looking at what happens in equilibrium but that is not sufficient
when there are multiple equilibria.
Figure 8.30: Two equilibria in the coordination failure model
What explains why the coordination failure model has multiple equilibria? This feature is
due to the presence of what is known as ‘strategic complementarity’. Strategic
complementarity refers to a situation where one person’s desire to perform an action
increases when others are also performing the same action. In the model, an individual
firm’s marginal product of labour rises when other firms are expanding employment and,
hence, it becomes profitable for firms to expand employment when other firms are doing
so. This creates strategic complementarity in firms’ employment decisions and sufficiently
strong strategic complementarity leads to multiple equilibria because different choices
made by individuals are mutually reinforcing.
While both equilibria in Figure 8.30 are fully consistent with rational optimisation by
individual households and firms, nonetheless, households are generally not indifferent
between them. The high-output equilibrium is good. It features high consumption, and
wages and productivity are high. Leisure is low but that choice makes sense because
productivity is high. The low-output equilibrium is bad. Consumption is low, and wages and
productivity are low, only mitigated by high leisure but that is chosen because productivity
is low. The inefficiency of the low-output equilibrium is due to the productivity spillover
effect that individuals fail to internalise.
Although everyone prefers the equilibrium with high output, there can be a coordination
problem in reaching it. At a low level of output, it makes sense for all firms in the economy
collectively to switch to the high-output equilibrium. But at the low-output equilibrium, it is
individually rational for each firm to choose low output. Therefore, the economy could get
stuck at the equilibrium with low GDP for some time, even though everyone would gain by
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coordinating on the high-GDP equilibrium. A recession can thus result from a coordination
failure.
What ultimately determines which of the two equilibria prevails? If everyone believes the
high-output equilibrium will prevail, then it will, while if everyone believes the low-output
equilibrium will prevail, then it will instead. Thus, the outcome depends on whether people
are optimistic or pessimistic about the economy’s prospects. However, both of these are
consistent with rational expectations. There is an independent role for beliefs, which are not
uniquely determined by requirement of rationality.
If the optimism or pessimism of households and firms determines which equilibrium
prevails, what explains how optimistic people should be? Again, the model provides no
direct answer. In principle, any extraneous factor could shift the economy from pessimism
to optimism and thus cause business cycles. Such extraneous factors unrelated to the
economy’s fundamentals are referred to by the term ‘sunspots’. For example, there could
be a media report on an event that triggers a wave of pessimism far beyond the importance
of the event to the economy’s fundamentals.
Business cycles in the coordination failure model can thus occur due to exogenous shifts in
optimism and pessimism, which are consistent with rational expectations because of
multiple equilibria. These induce movements of the economy between the low- and highoutput equilibria shown in Figure 8.30. If the economy goes into a recession then the real
interest rate 𝑟𝑟 rises as 𝑌𝑌 falls, moving along both 𝑌𝑌 𝑑𝑑 and 𝑌𝑌 𝑠𝑠 curves. As seen in Figure 8.31,
employment 𝑁𝑁 falls with 𝑌𝑌 when there is a movement along the aggregate production
function. Owing to there being increasing returns to labour, average labour productivity
𝑌𝑌/𝑁𝑁 falls with 𝑌𝑌. Moving along the aggregate labour demand curve 𝑁𝑁 𝑑𝑑 , the real wage 𝑤𝑤
falls with 𝑌𝑌 and 𝑁𝑁.,
Figure 8.31: Business cycles in the coordination failure model
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Box 8.6: The strength of strategic complementarities
The coordination failure model of business cycles depends on strategic
complementarity in firms’ employment decisions. This strategic complementarity
comes from a positive spillover from aggregate employment to each individual
firm’s productivity. But how strong does this spillover need to be for the model
successfully to generate business cycles?
It is clear from the workings of the model that explaining business cycles as waves
of optimism and pessimism requires multiple equilibria, which depends on the
output supply curve 𝑌𝑌 𝑠𝑠 being downward sloping. For this to happen, the spillover
must be strong enough that there are increasing returns to labour at the level of
the aggregate economy, i.e. an upward-sloping aggregate labour demand 𝑁𝑁 𝑑𝑑 . Just
having a positive spillover might not be enough to offset the usual diminishing
returns to labour, as shown in Figure 8.32.
Figure 8.32: Spillover effect too weak to generate increasing returns
Even if the aggregate 𝑁𝑁 𝑑𝑑 curve is upward sloping, this is not enough. If labour
supply is insufficiently wage elastic, i.e. 𝑁𝑁 𝑠𝑠 (𝑟𝑟) is too steep, then there is a unique
equilibrium even if there are increasing returns to labour in aggregate. This case is
depicted in Figure 8.33.
Figure 8.33: Labour supply insufficiently wage elastic
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Chapter 9: Inflation, expectations and
macroeconomic policy
This chapter considers the link between inflation and business-cycle fluctuations when
prices are neither completely sticky nor completely flexible. We will also explore the
important role of expectations in analysing macroeconomic policy and consider the
challenges faced by the central bank when monetary policy is constrained by a lower bound
on interest rates.
Essential reading
•
Williamson, Chapter 15.
9.1 Inflation and the Phillips curve
This section introduces a model that links inflation to the business-cycle fluctuations studied
in Chapter 8. The basic new Keynesian model from the previous chapter assumes all goods
prices are completely rigid, so that model says nothing about how inflation is determined.
On the other hand, full price flexibility implies the economy’s real interest rate is at its
natural rate and its real GDP reaches its natural level. In that case, Chapter 6 showed how
inflation depends on monetary policy and other real and financial variables but there was
only a very limited effect of monetary policy on real variables for moderate inflation rates
because of the absence of nominal rigidity.
We now consider a model with partial price adjustment to bridge the two extremes above.
This model also implies a close link between inflation and the state of the real economy, the
Phillips curve.
9.1.1 Firms’ incentives to adjust prices
As explained in Section 8.5, with monopolistic competition, each firm wants to set a price
where 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 = 𝑤𝑤, i.e. the marginal revenue product of labour is equal to the real wage. If
𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 < 𝑤𝑤, the price charged by a firm is too low and it would want to raise its price and
sell less. If 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 > 𝑤𝑤, the firm’s price is too high, and it would want to lower the price and
sell more.
To understand why prices are not always set so that 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 = 𝑤𝑤, we assume there are costs
of price adjustment. These were discussed in Section 8.1 and include ‘menu costs’ and the
managerial costs of making pricing decisions. Firms compare these costs with the benefits of
price adjustment when deciding whether to set a new price. The gains a firm would make by
adjusting the price of its product and the magnitude of the desired price adjustment both
increase with the size of the gap between 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 and 𝑤𝑤. We will not set up a precise
comparison of the costs and benefits of price adjustment here but, in the background, this
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trade-off is why firms will not change their prices all the time, nor leave their prices constant
forever.
9.1.2 Price changes and economic activity
We now link the incentives to adjust prices to economic activity as measured by real GDP 𝑌𝑌.
When GDP 𝑌𝑌 is at its natural level 𝑌𝑌 ∗ , firms employ workers up to the point 𝑁𝑁 ∗ where
𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 = 𝑤𝑤, as shown in Figure 9.1. Firms would have no desire to change prices in this
case. If 𝑌𝑌 is below 𝑌𝑌 ∗ , employment 𝑁𝑁 is below 𝑁𝑁 ∗ and the diagram shows that 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 > 𝑤𝑤,
so a price cut is desired. If 𝑌𝑌 is above 𝑌𝑌 ∗ , then 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 < 𝑤𝑤, so a price increase is desired.
This is because the marginal revenue product 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 falls with employment (𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 is
downward sloping), while the real wage 𝑤𝑤 rises (𝑁𝑁 𝑠𝑠 is upward sloping).
Moreover, a larger gap between 𝑌𝑌 and 𝑌𝑌 ∗ means that firms’ desired price change is larger
and more firms will prefer to adjust prices after taking account of the costs of doing so.
Intuitively, as output and employment increase, the marginal product of labour 𝑀𝑀𝑃𝑃𝑁𝑁
declines (as does 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 ), which increases firms’ marginal cost of production. The wage 𝑤𝑤
must also increase to raise the supply of labour, which additionally adds to the cost of
production. It follows that the direction and size of price changes depends on the ‘output
gap’ between actual GDP 𝑌𝑌 and its natural level 𝑌𝑌 ∗ .
Figure 9.1: Firms' incentives to change prices
Inflation is defined as the rate of change of the price level 𝑃𝑃 over time. The notation we will
use in this chapter is that 𝜋𝜋 is the inflation rate between the past and current time periods
(note this is different from our earlier notation in Chapter 6) and 𝜋𝜋′ is the inflation rate
between the current and future time periods, i.e. 𝜋𝜋 = (𝑃𝑃 − 𝑃𝑃�)/𝑃𝑃� and 𝜋𝜋 ′ = (𝑃𝑃′ − 𝑃𝑃)/𝑃𝑃,
where 𝑃𝑃� is the past level of prices. Mechanically, inflation 𝜋𝜋 is positive when firms are
increasing prices on average, or equivalently, when newly set prices are higher than the
average of past prices 𝑃𝑃�. By the same logic, future inflation 𝜋𝜋′ is expected to be positive
when firms will set higher prices than the current average 𝑃𝑃 in the future.
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9.1.3 Expectations
When there are costs of making price changes and firms do not expect to be adjusting prices
continually, they also need to consider future conditions when setting current prices. If it is
desirable to set higher prices in future time periods then expected future inflation 𝜋𝜋 ′𝑒𝑒 will
be positive. Hence, any firm adjusting its prices in current period will choose a larger price
increase when 𝜋𝜋 ′𝑒𝑒 is higher, so higher expected future inflation 𝜋𝜋 ′𝑒𝑒 leads to more inflation 𝜋𝜋
in the current time period. The effect of 𝜋𝜋 ′𝑒𝑒 on 𝜋𝜋 is less than one-for-one because less
weight is given to future conditions than current conditions when setting prices now.
9.1.4 The Phillips curve
We have seen that our model of partial price adjustment predicts inflation 𝜋𝜋 is positively
related to the output gap between 𝑌𝑌 and 𝑌𝑌 ∗ because higher 𝑌𝑌 raises 𝑤𝑤 and lowers 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 ,
leading firms to have a greater desire to make larger price increases. Inflation is also
positively related to expected future inflation 𝜋𝜋 ′𝑒𝑒 , capturing firms’ desire to raise prices preemptively in response to expected future economic conditions.
We refer to the relationship between inflation 𝜋𝜋 and the output gap 𝑌𝑌 − 𝑌𝑌 ∗ as the ‘Phillips
curve’. This is the upward-sloping line or curve 𝑃𝑃𝑃𝑃 in Figure 9.2 with 𝜋𝜋 on the vertical axis
and real GDP 𝑌𝑌 on the horizontal axis. If no future inflation is expected (𝜋𝜋 ′𝑒𝑒 = 0), the Phillips
curve passes through the point with 𝜋𝜋 = 0 when 𝑌𝑌 = 𝑌𝑌 ∗ because there is no inflationary
pressure when output is at its natural level and no future inflation is expected. The Phillips
curve shifts to the right if the natural level of output 𝑌𝑌 ∗ rises and shifts upwards if expected
future inflation 𝜋𝜋 ′𝑒𝑒 is higher.
Figure 9.2: The Phillips curve
The gradient of the Phillips curve indicates how much inflation 𝜋𝜋 rises when output 𝑌𝑌
increases. The Phillips curve would be flatter if there is less inflationary pressure because of
more nominal rigidity, for example, fewer firms being willing to change price because the
costs of price adjustment are larger.
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The Phillips curve is also flatter if there is greater ‘real rigidity’. Real rigidity refers to firms’
desired prices being less sensitive to what happens to real GDP 𝑌𝑌. For example, if the
marginal product of labour curve 𝑀𝑀𝑃𝑃𝑁𝑁 is flatter (and hence, also 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 ) because returns to
labour diminish less rapidly, then firms’ cost of production rises by less with output,
reducing inflationary pressure. This also occurs if the labour supply curve 𝑁𝑁 𝑠𝑠 is flatter, so
wages need to rise by less to induce an increase in labour supply. Another source of real
rigidity is the presence of efficiency wage concerns that mean real wages are less sensitive
to economic conditions, for example, when a particular constant real wage level maximises
the amount of effective labour input per unit of wages paid.
9.1.5 Inflation and unemployment
Traditionally, the Phillips curve was viewed as a negative relationship between the inflation
rate and the unemployment rate. By adding efficiency wages to our analysis of the labour
market, we immediately obtain this negative inflation-unemployment relationship from the
‘Phillips curve’ in terms of inflation and the output gap explained earlier.
The logic is that, for a given natural rate of unemployment and natural level of output 𝑌𝑌 ∗ , an
increase in 𝑌𝑌 raises employment and reduces unemployment relative to the natural rate of
unemployment. Hence, if unemployment below its natural rate then output 𝑌𝑌 is above 𝑌𝑌 ∗ ,
which increases inflationary pressure, while if unemployment above its natural rate then 𝑌𝑌
is below 𝑌𝑌 ∗, which decreases inflationary pressure. The downward-sloping Phillips curve in
terms of inflation and unemployment passes through the natural rate of unemployment at
𝜋𝜋 = 0 when 𝜋𝜋 ′𝑒𝑒 = 0. This Phillips curve shifts with changes in inflation expectations or
changes in the natural rate of unemployment.
9.2 Expectations and aggregate demand
We have seen that expectations of future inflation are an important feature of the Phillips
curve that explains current inflationary pressure. Expectations about the future are also
relevant to the level of aggregate demand for goods and services, which we represented
using the output demand curve 𝑌𝑌 𝑑𝑑 derived in Chapter 3.
First, inflation expectations affect the real interest rate that results from the nominal
interest rate set by the central bank. Inflation can therefore affect incentives to save or
borrow.
Second, expectations of the future state of the economy influence current consumption and
investment demand. This is because households have a desire to smooth consumption in
response to expected changes in future income, and investment demand depends on
expectations of the amount of future employment of labour.
9.2.1 Inflation expectations and real interest rates
In Chapter 3, we saw that both consumption and investment demand depend on the
expected real interest rate between the current and future time periods. Here, with our
notation that distinguishes between actual and expected inflation, we denote the ex-ante
real interest rate by 𝑟𝑟 𝑒𝑒 . The Fisher equation from Section 6.3 implies:
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𝑟𝑟 𝑒𝑒 = 𝑖𝑖 − 𝜋𝜋 ′𝑒𝑒
Given a nominal interest rate 𝑖𝑖, higher expected future inflation 𝜋𝜋 ′𝑒𝑒 reduces the expected
real interest rate 𝑟𝑟 𝑒𝑒 . This means there is less incentive to save and a greater incentive to
borrow. With these substitution effects, and ignoring income effects with a representative
household as explained in Section 3.11, higher 𝜋𝜋 ′𝑒𝑒 leads to higher 𝐶𝐶 𝑑𝑑 . It also raises
investment demand 𝐼𝐼 𝑑𝑑 according to the model from Section 3.8. Output demand 𝑌𝑌 𝑑𝑑
therefore depends positively on expected future inflation 𝜋𝜋 ′𝑒𝑒 .
9.2.2 Expectations of the economy’s future GDP
The consumption choice model studied in Section 3.3 shows that households’ optimal
consumption plan satisfies 𝑀𝑀𝑀𝑀𝑆𝑆𝐶𝐶,𝐶𝐶 ′ = 1 + 𝑟𝑟 𝑒𝑒 , where 𝑀𝑀𝑀𝑀𝑆𝑆𝐶𝐶,𝐶𝐶 ′ is the marginal rate of
substitution between current and future consumption. If consumption in the future 𝐶𝐶′ is
expected to be higher then 𝑀𝑀𝑀𝑀𝑆𝑆𝐶𝐶,𝐶𝐶 ′ rises (see Figure 3.9). Hence, for the same expected
real interest rate 𝑟𝑟 𝑒𝑒 , current consumption 𝐶𝐶 must increase to satisfy 𝑀𝑀𝑀𝑀𝑆𝑆𝐶𝐶,𝐶𝐶 ′ = 1 + 𝑟𝑟 𝑒𝑒 . This
reflects that both 𝐶𝐶 and 𝐶𝐶 ′ are normal goods, so for a given relative price of current and
future consumption as determined by the real interest rate, households want both 𝐶𝐶 and 𝐶𝐶′
to rise or fall together. This is the desire for consumption smoothing discussed in Chapter 3.
Therefore, the expectation of higher consumption in future raises consumption demand 𝐶𝐶 𝑑𝑑
in the current period.
The model of investment in Section 3.8 implies that firms invest in capital up to the point
where 𝑀𝑀𝑃𝑃𝐾𝐾′ − 𝑑𝑑 = 𝑟𝑟 𝑒𝑒 . If expectations of future employment 𝑁𝑁 ′ increase, this raises the
expected future marginal product of capital 𝑀𝑀𝑃𝑃𝐾𝐾′ . Note that any neoclassical production
function with capital and labour has the feature that a greater employment of labour
increases the marginal product of capital. Therefore, for a given real interest rate, the
expectation of higher future employment raises current investment demand 𝐼𝐼 𝑑𝑑 .
In summary, we conclude that the output demand curve 𝑌𝑌 𝑑𝑑 shifts to the right if
expectations of future inflation 𝜋𝜋 ′𝑒𝑒 increase, or there are higher expectations of future
consumption or future employment. To simplify matters, we suppose that future
consumption and employment are both positively related to future real GDP, so 𝑌𝑌 𝑑𝑑 shifts to
the right if expectations of future GDP 𝑌𝑌 ′ increase.
9.3 Aggregate demand with market imperfections
In analysing the effects of macroeconomic policy on aggregate demand, we will see that it
can be important to incorporate some of the insights of our study of credit-market
imperfections in Chapter 4 and imperfectly competitive markets in Chapter 8. For example,
it is sometimes claimed that increasing aggregate demand through extra government
expenditure 𝐺𝐺 gives rise to a positive feedback loop – a ‘multiplier’. The argument is that
higher demand raises output and income, and consumption then rises with income, further
increasing demand and so on.
However, this multiplier did not feature in our analysis of the output demand curve 𝑌𝑌 𝑑𝑑 from
the dynamic macroeconomic model of Chapter 3. There we emphasised that higher
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government expenditure 𝐺𝐺 increases the tax burden, reducing the amount of private
consumption 𝐶𝐶 that is affordable. Income 𝑌𝑌 did increase but, as seen in Box 4.2, that was
the result of the decision of households to supply more labour when faced with a higher tax
burden. Higher GDP 𝑌𝑌 caused by higher 𝐺𝐺 did not in itself make households better off.
9.3.1 Consumption and aggregate demand
In the new Keynesian model from Chapter 8, the presence of imperfect competition means
the level of GDP is not efficient. As 𝑀𝑀𝑃𝑃𝑁𝑁 > 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 , households are made better off when
higher aggregate demand for goods increases employment. Higher income itself makes
households better off because the extra ability to consume is worth more to households
than the extra time spent working. In addition, real wages 𝑤𝑤 can rise with aggregate
demand and employment in the new Keynesian model because 𝑤𝑤 begins below the
marginal product of labour 𝑀𝑀𝑃𝑃𝑁𝑁 , unlike in the standard dynamic macroeconomic model.
The consequence of the market imperfections is that consumption demand 𝐶𝐶 𝑑𝑑 now
depends directly on aggregate demand and income 𝑌𝑌 in addition to other factors – it is still
necessary to consider the effect of higher government expenditure 𝐺𝐺 on the tax burden.
The sensitivity of 𝐶𝐶 𝑑𝑑 to higher 𝑌𝑌 is labelled the marginal propensity to consume (MPC):
𝜕𝜕𝐶𝐶 𝑑𝑑
𝑀𝑀𝑀𝑀𝑀𝑀 =
𝜕𝜕𝜕𝜕
The aggregate demand for output is 𝑌𝑌 𝑑𝑑 = 𝐶𝐶 𝑑𝑑 + 𝐼𝐼 𝑑𝑑 + 𝐺𝐺. Consumption demand 𝐶𝐶 𝑑𝑑 is now a
function of 𝑌𝑌, and in equilibrium, income is equal to aggregate demand (𝑌𝑌 = 𝑌𝑌 𝑑𝑑 ). Figure 9.3
plots the expenditure function 𝑌𝑌 𝑑𝑑 against aggregate output and income 𝑌𝑌.
Figure 9.3: Output demand with multiplier
The gradient of the expenditure 𝑌𝑌 𝑑𝑑 as a function of 𝑌𝑌 is given by marginal propensity to
consume 𝜕𝜕𝐶𝐶 𝑑𝑑 ⁄𝜕𝜕𝜕𝜕. In equilibrium, 𝑌𝑌 = 𝑌𝑌 𝑑𝑑 , so the level of output demand must lie on the
45∘ line in the diagram. Any factors affecting aggregate expenditure 𝑌𝑌 𝑑𝑑 apart from 𝑌𝑌 itself
cause shifts of the expenditure function.
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If aggregate expenditure 𝑌𝑌 𝑑𝑑 were to increase, for example, owing to higher 𝐺𝐺, the
expenditure function shifts vertically upwards by the same amount. By finding the new
point of intersection with the 45∘ line, we see that the overall effect on 𝑌𝑌 is larger than the
size of the shift of the 𝑌𝑌 𝑑𝑑 function. A marginal propensity to consume between 0 and 1
(making the expenditure function upward sloping but less steep than the 45∘ line) amplifies
the effects of changes in output demand 𝑌𝑌 𝑑𝑑 because of a positive feedback loop working
through income and consumption.
The consequences of this for the output demand curve 𝑌𝑌 𝑑𝑑 (a relationship between real GDP
𝑌𝑌 and the real interest rate 𝑟𝑟) derived in Section 3.12 is that the 𝑌𝑌 𝑑𝑑 curve becomes flatter,
i.e. more sensitive to 𝑟𝑟 and shifts become larger than they would be in the standard
dynamic macroeconomic model of Chapter 3.
9.3.2 Consumption and aggregate demand with credit-market
imperfections
Even in the new Keynesian model, the multiplier effect described above is usually not strong
enough to offset the direct effect of the higher tax burden on consumption when public
expenditure is increased. However, a stronger multiplier is found with the credit-market
imperfections that were studied in Chapter 4. We assume some households face a binding
borrowing limit. They cannot generate enough income in the current period to pay for their
desired level of consumption and they cannot borrow against future income. It is important
to appreciate that not all households in the economy will be in this position – some will be
savers – so we are moving away from the usual assumption of a representative household.
As explained in Section 4.3, households who are credit-constrained have a marginal
propensity to consume of 1 from disposable income (assuming all households are able to
increase the amount they work when aggregate output 𝑌𝑌 rises). Moreover, their current
consumption does not respond to future taxes, only to current taxes through their impact
on disposable income. In contrast, households who are not credit-constrained have a much
smaller response of consumption to 𝑌𝑌 and adjust consumption in response to the present
value of all current and future taxes.
The economy’s overall marginal propensity to consume is a weighted average of the 𝑀𝑀𝑀𝑀𝑀𝑀 =
1 for credit-constrained households and the low 𝑀𝑀𝑀𝑀𝑀𝑀 of unconstrained households, the
weights depending on how much of aggregate consumption spending comes from the two
groups of households. This overall 𝑀𝑀𝑀𝑀𝑀𝑀 determines the gradient of the expenditure
function seen in Figure 9.3 and the size of the multiplier effect. If there are sufficiently many
constrained households, we will see that the overall 𝑀𝑀𝑀𝑀𝑀𝑀 can be high enough to offset the
negative impact on current consumption of the higher future tax burden when the
government increases public expenditure.
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Box 9.1: Multiplier and crowding-out effects of fiscal policy
We now re-examine the effects of a fiscal stimulus. There are two important
changes compared to our earlier analysis in Box 4.2. First, some households are
credit constrained. Second, prices are sticky (completely sticky here for simplicity)
and we suppose monetary policy is accommodative in the sense that the central
bank holds the real interest rate 𝑟𝑟 constant throughout.
Consider an increase in government expenditure 𝐺𝐺 that is financed by running a
larger budget deficit, so there is no rise in current taxes 𝑇𝑇, only future taxes. The
wealth effect from the higher tax burden reduces consumption demand 𝐶𝐶 𝑑𝑑 but
only for those households who are not credit constrained.
Credit-constrained households consume their disposable income and so have a
marginal propensity to consume of one and their current consumption does not
respond to the higher future taxes. This means that 𝐶𝐶 𝑑𝑑 depends on disposable
income 𝑌𝑌 − 𝑇𝑇, which rises with 𝑌𝑌. Ignoring the effects of the tax burden on
unconstrained households, the increase in 𝐺𝐺 raises 𝑌𝑌 𝑑𝑑 more than one-for-one
because of the multiplier effect working through the consumption of the
constrained households.
Compared to Box 4.2, the presence of some credit-constrained households means
there is less crowding out of consumption from the higher tax burden and new
multiplier effect on consumption. With no constrained households, we know the
rightward shift of the output demand curve 𝑌𝑌 𝑑𝑑 is smaller than the increase in 𝐺𝐺.
But as the fraction of credit-constrained households rises, the shift of 𝑌𝑌 𝑑𝑑 becomes
larger and can exceed than the increase in 𝐺𝐺.
The other difference compared to the analysis in Box 4.2 is that sticky prices imply
the real interest rate 𝑟𝑟 is determined by the intersection of the 𝑌𝑌 𝑑𝑑 curve and the
𝑀𝑀𝑀𝑀 line, not 𝑌𝑌 𝑑𝑑 and the output supply curve 𝑌𝑌 𝑠𝑠 . This is shown in Figure 9.4. If
monetary policy is accommodative with a horizontal 𝑀𝑀𝑀𝑀 line remaining in the
same position then the real interest rate 𝑟𝑟 does not rise, in contrast to what
happens in an economy with flexible prices.
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Figure 9.4: Effects of a fiscal stimulus with sticky prices
The constant 𝑟𝑟 reduces crowding out of private consumption and investment expenditure
when 𝐺𝐺 rises. The increase in real GDP 𝑌𝑌 is equal to the size of the horizontal shift of 𝑌𝑌 𝑑𝑑 and
we have seen that this can be larger than the increase in 𝐺𝐺 when there are sufficiently many
credit-constrained households. It is therefore possible that ‘multiplier’ effects dominate
‘crowding-out’ effects and GDP rises by more than a deficit-financed increase in government
expenditure.
Box 9.2: Asset prices and the financial accelerator
Shocks to aggregate demand can also be amplified through financial markets. For
example, supposing a decline in GDP leads to a reduction in asset prices, this
tightens credit constraints if those assets are used as collateral for borrowing. The
greater difficulty of borrowing this causes then further reduces aggregate demand
and GDP. A feedback loop of this type is known as a ‘financial accelerator’.
We can illustrate the financial accelerator using the limited-commitment model of
borrowing constraints and house prices from Section 4.6. We add to the earlier
model a reason why lower aggregate demand and GDP reduces house prices, for
example, borrowing constraints linked to the incomes of those buying houses, or
complementarities between housing and consumption expenditure on durable
goods. We assume the collateral constraint in the limited-commitment model is
binding, which implies the consumption of existing homeowners is linked to house
prices.
Suppose a negative shock to the economy reduces GDP. This leads to lower house
prices and causes a reduction in consumption of credit-constrained homeowners,
which in turn reduces aggregate demand and GDP, and depresses house prices
further.
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Box 9.3: The 2008 financial crisis
What are the links between the 2008 financial crisis in the USA and the severity of
the ‘great recession’ that followed? Our analysis of credit-market imperfections
from Chapter 4 allows us to identify a number of channels through which a financial
shock can have a large impact on aggregate demand and GDP.
First, falling house prices reduce the value of collateral available to support
borrowing. Using the limited-commitment model from Section 4.6, this reduces the
consumption demand 𝐶𝐶 𝑑𝑑 of credit-constrained households, causing 𝑌𝑌 𝑑𝑑 to shift to
the left.
A financial crisis also leads to expectations of more defaults on debts, which raises
interest-rate spreads through the asymmetric information mechanism explained in
Section 4.5. The consequences of the higher interest rates faced by borrowers are
analysed in Box 4.5 and result in lower 𝐶𝐶 𝑑𝑑 for borrower households and lower 𝐼𝐼 𝑑𝑑
for firms without sufficient internal funds to finance investment. These effects lead
to a further shift of 𝑌𝑌 𝑑𝑑 to the left.
Furthermore, falling asset prices and defaults result in losses for banks, which
reduces bank capital. As explained in Chapter 7, this means that banks may need to
restrict lending and deposit creation to satisfy bank capital requirements. Overall,
these factors imply a large leftward shift of the output demand curve 𝑌𝑌 𝑑𝑑 as shown
in Figure 9.5.
Figure 9.5: A financial shock
Finally, the Federal Reserve was unable fully to offset the large leftward shift of 𝑌𝑌 𝑑𝑑
by reducing interest rates and shifting down the 𝑀𝑀𝑀𝑀 line. This is because of the
lower bound on the nominal interest rate, an issue we will discuss further in Section
9.7. Assuming the nominal interest rate 𝑖𝑖 cannot fall below zero, the lowest
possible real interest rate was −𝜋𝜋 ′𝑒𝑒 , the negative of expected future inflation, and
inflation expectations were relatively low during the 2000s.
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9.4 Inflation, aggregate demand and monetary policy
The Phillips curve derived in Section 9.1 implies a link between economic activity and
inflation. We will now consider which point on the Phillips curve the economy will reach,
which depends on the output demand curve and monetary policy. There is an interaction
with monetary policy because the central bank may adjust interest rates in response to
changes in inflation, for example, if it uses an interest-rate feedback rule such as the Taylor
rule studied earlier in Section 6.12. We will summarise the 𝑌𝑌 𝑑𝑑 curve and the stance of
monetary policy with a single curve that can be drawn in the same diagram as the Phillips
curve and the intersection between the two determines inflation 𝜋𝜋 and real GDP 𝑌𝑌.
Suppose the central bank sets the nominal interest rate 𝑖𝑖 and increases 𝑖𝑖 in response to
higher inflation 𝜋𝜋. The ‘Taylor principle’ from Section 6.12 suggests 𝑖𝑖 should rise more than
one-for-one with 𝜋𝜋. The central bank also increases 𝑖𝑖 in response to GDP 𝑌𝑌 being higher
than 𝑌𝑌 ∗ , using the natural level of output 𝑌𝑌 ∗ as the notion of ‘potential output’. In Section
8.2, in the diagram with the 𝑌𝑌 𝑑𝑑 and 𝑀𝑀𝑀𝑀 curves (Figure 8.4), the 𝑀𝑀𝑀𝑀 curve is now upwardsloping and shifts upwards when inflation 𝜋𝜋 increases.
Higher inflation 𝜋𝜋 thus causes an upward shift of the 𝑀𝑀𝑀𝑀 curve. The higher real interest rate
induced by monetary policy causes a reduction in aggregate demand, a movement up the
𝑌𝑌 𝑑𝑑 curve and 𝑌𝑌 is lower. Taking as given expectations of 𝜋𝜋 ′ and 𝑌𝑌 ′ , this implies a negative
demand-side relationship between inflation 𝜋𝜋 and real GDP 𝑌𝑌. We label this the 𝑌𝑌 𝑑𝑑 − 𝑀𝑀𝑀𝑀
line, coming from the combination of the output demand curve and the stance of monetary
policy. Putting this together with the upward-sloping Phillips curve that represents the
supply side of the economy, the intersection between 𝑌𝑌 𝑑𝑑 − 𝑀𝑀𝑀𝑀 and 𝑃𝑃𝑃𝑃 determines
inflation 𝜋𝜋 and real GDP 𝑌𝑌. This is illustrated in Figure 9.6.
Figure 9.6: The Phillips curve, output demand, and monetary policy
The 𝑌𝑌 𝑑𝑑 − 𝑀𝑀𝑀𝑀 curve shifts to the right for the same reasons that cause the output demand
curve 𝑌𝑌 𝑑𝑑 to shift to the right. It shifts to the left if there is an exogenous increase in the
tightness of monetary policy. Using the links between expectations of the future and
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aggregate demand studied in Section 9.2, it also shifts to the right with an increase in
expected future inflation 𝜋𝜋 ′ or expected future GDP 𝑌𝑌 ′ .
9.5 The costs of inflation
Our discussions of macroeconomic policy have focused on obtaining a desirable outcome
for real GDP 𝑌𝑌, for example, by closing the output gap between 𝑌𝑌 and 𝑌𝑌 ∗ . But there are also
costs of inflation that make control of inflation a legitimate objective of policy in its own
right. We now consider these costs of inflation.
Some costs of high inflation, arising from money being a poor store of value, have already
been analysed in Section 6.10. However, the nominal rigidities and the unit of account
function of money studied in Chapter 8 and here give rise to further costs of inflation. They
also suggest the optimal inflation rate might be different from what was found in Chapter 6.
9.5.1 Money being a poor store of value
As explained in Section 6.10, for a given real interest rate on bonds, higher inflation makes
money a worse store of value. This leads to time and resources being wasted in trying to
economise on holding money or creating substitutes for money. It also reduces production
because money being a poor store of value is an implicit tax on economic activity.
Hence, money’s medium of exchange function suggests there are costs of inflation, or to be
precise, costs of anticipated inflation. The earlier analysis of the Friedman rule indicates
these costs are eliminated only when there is deflation at a rate equal to the real interest
rate (𝜋𝜋 ′𝑒𝑒 = −𝑟𝑟 𝑒𝑒 < 0). This suggests the optimal inflation rate is negative.
9.5.2 Menu costs and relative-price distortions
The nominal rigidities in the new Keynesian model imply there are additional costs of
inflation. Costs incurred by firms in adjusting prices, for example menu costs and managerial
time, are higher as inflation – or deflation – increases. Since price adjustments are not
perfectly synchronised across different firms, positive or negative inflation rates also affect
the relative prices of different goods, causing misallocation of spending.
Costs of these kinds increase as inflation 𝜋𝜋 rises above or falls below zero and occur for both
anticipated and unanticipated inflation. These considerations suggest aiming for a zero rate
of inflation.
9.5.3 Inflation and redistribution
Another potential problem of unanticipated inflation is the redistribution between creditors
and debtors it causes. To see this, suppose savers hold nominal bonds issued by borrowers.
The ex-ante Fisher equation 𝑖𝑖 = 𝑟𝑟 𝑒𝑒 + 𝜋𝜋 ′𝑒𝑒 implies that higher expected inflation 𝜋𝜋 ′𝑒𝑒 can
result in higher 𝑖𝑖, leaving the expected real return 𝑟𝑟 𝑒𝑒 on nominal bonds unchanged. The expost Fisher equation is 𝑟𝑟 = 𝑖𝑖 − 𝜋𝜋′, which indicates the real return on bonds depends on the
realised inflation rate 𝜋𝜋 ′ but the nominal interest rate 𝑖𝑖 cannot adjust if this inflation is
unexpected. This logic suggests inflation should be as predictable as possible but does not
provide specific guidance on what the optimal inflation rate is.
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Box 9.4: Inflation targeting
In light of the costs of inflation, one widely used monetary policy strategy is to
make controlling inflation the primary goal of monetary policy. Inflation targeting is
where the central bank uses its policy instruments to try to achieve inflation 𝜋𝜋 =
𝜋𝜋 ∗ , where 𝜋𝜋 ∗ is the target rate of inflation. Inflation-targeting central banks
typically have targets for 2 per cent or 3 per cent inflation with some margin for
error.
The benefits of price stability in the new Keynesian model
Although real-world central banks typically have positive inflation targets, the
nominal rigidities of the standard new Keynesian model suggest a zero inflation
target 𝜋𝜋 ∗ = 0 is best (we will consider later a reason for targeting a positive
inflation rate). The costs of inflation linked to nominal rigidity are minimised by
ensuring inflation 𝜋𝜋 is kept close to zero. But what about the consequences for the
stability of real variables such as GDP of this exclusive focus on inflation?
Assume the inflation target 𝜋𝜋 = 𝜋𝜋 ∗ = 0 is achieved (and this is expected to
continue in the future as well). The new Keynesian model with partial price
adjustment implies there is no inflation or deflation only if firms are on average
happy with the existing prices they have previously set. With imperfect
competition, these firms have no desire to raise or lower prices when their
marginal revenue product of labour 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 is equal to the real wage 𝑤𝑤. But if
𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 = 𝑤𝑤 then employment must be such that the resulting supply of output is
on 𝑌𝑌 𝑠𝑠 curve because 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 = 𝑤𝑤 is what occurs when prices are fully flexible.
This argument indicates that aiming for price stability (zero inflation) in the new
Keynesian model should result in real GDP 𝑌𝑌 being at the intersection of the 𝑌𝑌 𝑑𝑑 and
𝑌𝑌 𝑠𝑠 curves, hence, equal to the natural level of output 𝑌𝑌 ∗ . This means there would
be no output gap and the real interest rate would be at the natural rate of interest
𝑟𝑟 ∗ . The pursuit of price stability thus results in real economic outcomes that are the
same as if prices were completely flexible.
With 𝜋𝜋 ′𝑒𝑒 = 0, monetary policy needs to set a nominal interest rate 𝑖𝑖 = 𝑟𝑟 = 𝑟𝑟 ∗ to
achieve the zero inflation target. Interestingly, this is exactly the same monetary
policy as the optimal stabilisation policy from Section 8.6 where there was no need
for any concern about inflation because prices were completely sticky. Here, we
have added an inflation objective alongside the desire to close the output gap, but
the same monetary policy is able to achieve both objectives.
Flexible or strict inflation targeting?
Should central banks interpret an inflation target strictly to the exclusion of other
objectives? Or should meeting the inflation target be ‘flexible’, allowing also for
stabilisation of fluctuations in real GDP and employment?
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In the standard new Keynesian model, we have seen that aiming for 𝜋𝜋 = 0 strictly
means accepting real GDP 𝑌𝑌 = 𝑌𝑌 ∗ , which closes the output gap. This suggests a
‘strict’ inflation target is not necessarily bad. One concern might be that real GDP
𝑌𝑌 ∗ is inefficiently low because 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 = 𝑤𝑤 ∗ = 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 implies 𝑀𝑀𝑃𝑃𝑁𝑁 > 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶
(recalling that 𝑀𝑀𝑃𝑃𝑁𝑁 > 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 ). But as we will see in Box 9.5, aiming for 𝑌𝑌
systematically above 𝑌𝑌 ∗ risks an inflation bias.
Putting aside concerns that output 𝑌𝑌 ∗ is too low on average, what about
fluctuations in 𝑌𝑌 ∗ and, hence, in actual real GDP 𝑌𝑌 = 𝑌𝑌 ∗ when 𝜋𝜋 = 0? Will a strict
inflation target cause real GDP to fluctuate too much? Let us consider this point
after when efficiency wages are added to the new Keynesian model. This means the
level of employment 𝑁𝑁 ∗ associated with the natural level of output 𝑌𝑌 ∗ is found
where the efficiency wage 𝑤𝑤 ∗ equals 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 , rather than where 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 intersects
� is where 𝑀𝑀𝑃𝑃𝑁𝑁 =
the labour supply curve 𝑁𝑁 𝑠𝑠 . The efficient level of employment 𝑁𝑁
𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 , which lies on the 𝑁𝑁 𝑠𝑠 curve as shown in Figure 9.7.
Figure 9.7: Excessive real GDP fluctuations with a strict inflation target
Now consider a negative supply shock that shifts 𝑀𝑀𝑃𝑃𝑁𝑁 and 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 to the left. As
seen in the figure, this shock causes employment 𝑁𝑁 ∗ and output 𝑌𝑌 ∗ to fall too much
� ). Rigidities such as efficiency wages can therefore
compared to what is efficient (𝑁𝑁
make the levels of employment and output too volatile if the central bank follows a
strict inflation target resulting in 𝑁𝑁 = 𝑁𝑁 ∗ and 𝑌𝑌 = 𝑌𝑌 ∗ .
In these circumstances, a flexible inflation target generally performs better than a
strict inflation target. Following a temporary negative supply shock, allowing
inflation 𝜋𝜋 to rise means 𝑁𝑁 will be above 𝑁𝑁 ∗ (this is a movement along the Phillips
curve). This is a better outcome because employment drops too much if 𝜋𝜋 = 0,
although there is a trade-off between the cost of the positive inflation rate and the
better outcome for real GDP.
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9.6 Time inconsistency
We have seen that macroeconomic policies such as monetary and fiscal policy can have a
direct impact on the economy. But since expectations of the future matter for current
economic outcomes, expectations of future policies can also affect the economy today.
Policymakers will be able to achieve more if they are able to influence expectations as well
as take direct action. However, announcements of future policies might not be credible
because of the problem of ‘time inconsistency’.
A policy is said to be time inconsistent if the policymaker gains in the current period when
the announcement is believed but does not gain by implementing the policy when the time
comes in future – even if nothing fundamental has changed. A time-consistent policy is one
where there is no incentive ex post to follow a different policy from the one it was optimal
to announce earlier. Without the ability to commit to future actions, people have no
incentive to believe a policymaker will follow a time-inconsistent policy because the
announcement lacks credibility.
What makes some policies time inconsistent? Since expectations of the future influence the
economy in the present, the policymaker can achieve more in the current period if
announcements of future policy are believed and change expectations today. But that
benefit lies in the past when the time comes in the future actually to implement the
announcement, so the policy that is now optimal to the policymaker is different.
For example, consider the announcement that tax rates on capital or capital income will be
low. If this is believed, it encourages investment and raises the capital stock. Since building
up new capital takes some time, it is expected future tax rates that matters for investment
decisions. But once capital is accumulated, the government has an incentive to implement a
‘one-off’ capital levy to raise tax revenue.
Another example is announcing a target for a low rate of inflation. If this is believed, it
reduces the nominal interest rate and money becomes seen as a better store of value,
which encourages economic activity. Note that it is expected future inflation that matters
for the interest rate on nominal bonds and decisions about holding money. However, ex
post, the government has an incentive to tolerate a ‘one-off’ burst of inflation to reduce the
real value of government liabilities through an inflation tax.
Box 9.5: The inflation bias problem
In the new Keynesian model, the central bank is able to raise real GDP 𝑌𝑌 by lowering the real
interest rate 𝑟𝑟. We saw in Section 8.6 and Box 9.4 how this makes it possible for the central
bank to pursue stabilisation policy that aims to close the output gap between real GDP 𝑌𝑌 and
the natural level of output 𝑌𝑌 ∗ , as well as achieve price stability.
However, the natural level of output 𝑌𝑌 ∗ is itself inefficiently low, with 𝑀𝑀𝑃𝑃𝑁𝑁 > 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 even at
𝑌𝑌 = 𝑌𝑌 ∗ because 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 = 𝑤𝑤 = 𝑀𝑀𝑀𝑀𝑆𝑆𝑙𝑙,𝐶𝐶 and 𝑀𝑀𝑃𝑃𝑁𝑁 > 𝑀𝑀𝑀𝑀𝑃𝑃𝑁𝑁 . This inefficiency of 𝑌𝑌 ∗ arises
owing to distortions present in the economy in addition to the problem of nominal rigidity
that the stabilisation policy and inflation targeting are able to mitigate. The basic source of
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distortions is the imperfect competition in the goods market that leads firms to produce
too little as a way of boosting profits. But other distortions such as ‘efficiency wages’ or
taxes that reduce the incentive to work also worsen the inefficiency of the natural level of
output.
In light of GDP being inefficiently low, should monetary policy aim to provide additional
stimulus even when 𝑌𝑌 = 𝑌𝑌 ∗ ? Suppose monetary policy reduces the real interest rate
below the natural interest rate 𝑟𝑟 ∗ to push real GDP above its natural level 𝑌𝑌 ∗ . Taking as
given inflation expectations 𝜋𝜋 ′𝑒𝑒 , this policy moves the economy along a Phillips curve
with output 𝑌𝑌 and inflation 𝜋𝜋 both rising. This seems to improve the outcome for real
GDP, albeit at the cost of some inflation.
However, this policy is not a response to a shock but to economic activity being judged
systematically too low. Hence, the higher inflation that results from it can be anticipated.
This causes inflation expectations to rise, which shifts the Phillips curve upwards. The
upward shift of the Phillips curve worsens the combinations of inflation and real GDP that
can be attained by the central bank. Now, an even higher rate of inflation is required to
achieve a given target for the level of real GDP, and when that is anticipated, this leads to
even higher expected inflation. Eventually, inflation expectations stop rising when further
inflation would be too costly for the central bank to tolerate.
These adverse shifts of the Phillips curve mean that there is higher inflation with only a
limited (or no) gain in terms of real GDP. Monetary policy therefore suffers from an
‘inflation bias’. The freedom to use monetary policy to aim for the efficient level of real
GDP leads to higher inflation, but fails to achieve its original goal. It results in higher costs
of inflation, without gaining much or anything by way of a better outcome for real GDP.
If the central bank were to announce it would target a lower rate of inflation and inflation
expectations fell, the Phillips curve would be in a more favourable position. But then
there would be a temptation to pursue an expansionary monetary policy to raise 𝑌𝑌 above
𝑌𝑌 ∗ . This points to the time inconsistency of announcing a goal of lower inflation.
Owing to the time inconsistency problem, an announcement that the central bank will
pursue low inflation is not credible. But there would be gains from being able to reduce
inflation expectations and obtain a Phillips curve in a more favourable position if the
central bank were able to commit itself to a low-inflation monetary policy.
What institutional mechanisms might enable the central bank to do this? First, giving the
central bank independence might insulate it from political pressure to aim for real GDP 𝑌𝑌
above 𝑌𝑌 ∗. Second, establishing a framework for monetary policy where the central bank
is given the primary task of controlling inflation and is judged by the public on how well it
performs in this might shift the focus of monetary policy away from the level of real GDP.
An independent central bank given the job of meeting an inflation target is the current
consensus on how best to achieve this. The central bank is usually allowed to interpret its
inflation target ‘flexibly’ and consider business-cycle fluctuations in real GDP in its policy
deliberations but to focus only on inflation in the long run.
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9.7 Unconventional monetary policy at the interest-rate lower
bound
An important obstacle to using monetary policy to stabilise the economy in the way
described in Section 8.6 is the interest-rate lower bound. It was argued the central bank
could close the output gap and obtain 𝑌𝑌 = 𝑌𝑌 ∗ by setting the nominal interest rate 𝑖𝑖 so that
the implied real interest rate 𝑟𝑟 equals natural rate of interest 𝑟𝑟 ∗ . But as Section 6.13
explains, there is a limit on how far the nominal interest rate 𝑖𝑖 can be reduced by the central
bank. This limit used to be seen as zero but is now thought to be slightly negative (see
Section 6.14 for one reason why).
The ex-ante Fisher equation implies the real interest rate is 𝑟𝑟 = 𝑖𝑖 − 𝜋𝜋 ′𝑒𝑒 . Taking as given
inflation expectations 𝜋𝜋 ′𝑒𝑒 , if there is a lower bound 𝑖𝑖 ≥ 0 on the nominal interest rate
(taken to be zero here) then the real interest rate is subject to the lower bound 𝑟𝑟 ≥ −𝜋𝜋 ′𝑒𝑒 .
The real interest rate cannot fall below the negative of the rate of inflation that is expected.
The problem is that a large shock to the economy may reduce the natural rate of interest 𝑟𝑟 ∗
below −𝜋𝜋 ′𝑒𝑒 , meaning that it is not feasible to get to 𝑟𝑟 = 𝑟𝑟 ∗ by lowering 𝑖𝑖. This challenge for
monetary policy is illustrated in Figure 9.8.
Figure 9.8: Interest-rate lower bound and the limits to stabilisation policy
The inability to use conventional monetary policy to stabilise the economy once the
interest-rate lower bound is reached has led central banks to use or consider using
unconventional monetary policies instead, such as:
•
•
•
Quantitative easing
Forward guidance
Negative interest rates.
9.7.1 Quantitative easing
If the central bank cannot reduce the nominal interest rate 𝑖𝑖, can it stimulate the economy
by increasing the money supply instead? Policies of this type are known as quantitative
easing (QE).
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Suppose the central bank creates more money and purchases short-term nominal
government bonds. Since the interest-rate lower bound has been reached, these short-term
nominal government bonds have yield 𝑖𝑖 = 0. But as Section 6.13 has explained, at the
interest-rate lower bound, the demand for money becomes perfectly interest elastic
because money and risk-free nominal bonds are perfect substitutes at the margin – the
‘liquidity trap’. As Figure 9.9 shows, this additional money would be passively absorbed into
money demand but there would be no change in interest rates, prices, or real GDP.
Can quantitative easing be adapted so that expansions of the money supply do have an
effect on the economy? What needs to be done differently?
Figure 9.9: The liquidity trap and the limits to stabilisation policy
It is implicit in the example from Figure 9.9 that the expansion of the money supply is only
temporary and would be reversed once the interest-rate lower bound ceases to bind and QE
is no longer needed. As explained in Box 6.3, the effects of a permanent expansion of the
money supply would be quite different, causing an increase in prices and inflation, and
leading here to a movement along the Phillips curve with higher economic activity. Hence, a
commitment to maintain a monetary expansion even in normal times in the future might in
principle be a more effective form of QE, although it may suffer from a time-inconsistency
problem and lack credibility if people think the central bank will change course in the future.
Another proposal is a ‘helicopter drop’ of money, where a monetary expansion is
transferred to the government and given away in the form of lower taxes or increased
transfer payments for households. In principle, this is very similar to the permanent
expansion of the money supply described above – which would deliver a fiscal gain to the
government from reducing the real value of existing money and nominal debt, and
ultimately show up as lower taxes. But one important difference is that it would be more
difficult to reverse because the central bank has given the money away and does not hold
any additional assets that could be used to buy back the money. While this policy might be
more effective, it is widely seen as setting a dangerous precedent and blurring the lines
between monetary and fiscal policy.
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Rather than claim monetary expansions will be permanent, real-world central banks’ QE has
instead attempted to get around the liquidity trap problem by making purchases of longterm government bonds with a positive yield, or private-sector assets such as corporate
bonds or mortgage-backed securities. These assets share the feature that they are risky,
unlike short-term government bonds. Private-sector assets are subject to default risk and
even long-term government bonds are risky for those not holding them until maturity (see
Box 7.5). These risky assets are not perfect substitutes for money, unlike short-term
government bonds at interest-rate lower bound.
We can use the model of portfolio choice from Section 7.8 to analyse the effects of
purchases of risky assets by the central bank. Central-bank purchases of risky assets require
in equilibrium that private investors hold a smaller fraction of risky assets and a larger
fraction of risk-free assets (the money the central bank creates) in their portfolios. As shown
in Figure 9.10, a lower risk premium is needed for this portfolio to be chosen, so centralbank purchases result in a decline in risk premiums. This is known as the ‘portfolio balance
effect’. This then helps an economy at the lower bound by reducing the cost of credit for
risky borrowers, or those borrowing over long periods.
Figure 9.10: The portfolio balance effect
9.7.2 Forward guidance
The term ‘forward guidance’ refers to an announcement made by the central bank about
the future path of interest rates. When the current nominal interest rate is at its lower
bound, the central bank is still able to give an indication of what it will do with future
interest rates, at least to the extent that future interest rates are not already themselves
expected to be at the lower bound. For example, there could be an announcement that
interest rates will remain ‘lower for longer’, i.e. the central bank will not raise interest rates
as quickly as is currently expected.
If interest rates can be lowered in future periods then this would be expected to raise future
output 𝑌𝑌 ′ and future inflation 𝜋𝜋 ′ . Higher expectations of 𝜋𝜋 ′ and 𝑌𝑌 ′ both increase output
demand in the current period through the channels explained in Section 9.2, even though
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there is no immediate change in interest rates. Thus, forward guidance might work through
manipulating expectations of the future in a way that improves current economic outcomes.
However, the policy is not a ‘free lunch’ because future outcomes are worse if future
interest rates are set at an inappropriate level for the conditions then prevailing.
This also points to a potential problem with the credibility of forward guidance. Once the
future period is reached, the central bank may not want to keep interest rates too low. The
benefit of announcing this was better economic outcomes in the past but that is now a
bygone, while the cost is worse current economic outcomes. This time-inconsistency
problem can undermine the credibility of a forward guidance announcement. For people to
believe the announcement, the central bank needs to make a binding commitment but lacks
any straightforward way to tie its hands.
Box 9.6: Inflation targeting and the interest-rate lower bound problem
If the lower bound on nominal interest rates jeopardises macroeconomic stability and
unconventional monetary policies are seen as ineffective or costly are there any reforms
to economic policy that would mitigate the lower bound problem? For example, should
the existing framework of inflation targeting that is used in many countries be changed
or abandoned? Below we explore a number of alternatives.
Raising the inflation target
Suppose a higher target 𝜋𝜋 ∗ for inflation is chosen. The Fisher equation with the real
interest rate at its natural rate 𝑟𝑟 ∗ in the long run implies 𝑖𝑖 = 𝑟𝑟 ∗ + 𝜋𝜋 ∗ . A higher inflation
target 𝜋𝜋 ∗ thus means a higher nominal interest rate 𝑖𝑖 on average, which gives a larger
cushion to adjust 𝑖𝑖 downwards. With 𝜋𝜋 ′𝑒𝑒 ≈ 𝜋𝜋 ∗ and a lower bound of zero on the nominal
interest rate 𝑖𝑖, the real interest rate can be reduced to −𝜋𝜋 ∗ , which is lower when 𝜋𝜋 ∗ is
higher.
Therefore, if inflation targets were raised, monetary policy can shift the 𝑀𝑀𝑀𝑀 line further
downwards. This gives the central bank a greater ability to offset larger negative demand
shocks. However, there are costs of having higher 𝜋𝜋 on average, as discussed in Section
9.5.
Average inflation targeting
It would also be possible to replace the standard form of inflation targeting with what is
called ‘average inflation targeting’. This has the central bank aim for a target based on
the inflation rate averaged over a number of years. Average inflation targeting was
adopted by the US Federal Reserve in August 2020.
Consider an example of this policy where the target is for inflation averaged over two
periods. If this is credible, people expect that an average of current inflation 𝜋𝜋 and future
inflation 𝜋𝜋 ′ will remain stable even if 𝜋𝜋 does not. Suppose 𝜋𝜋 falls because of a negative
demand shock that the central bank cannot offset because of the interest-rate lower
bound. Under this policy, there is then an increase in expectations of 𝜋𝜋 ′ – assuming
monetary policy is not also constrained in the future period. As explained in Section 9.2,
higher 𝜋𝜋 ′𝑒𝑒 reduces the real interest rate and boosts demand in the current period.
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Under this policy, there is then an increase in expectations of 𝜋𝜋 ′ – assuming monetary
policy is not also constrained in the future period. As explained in Section 9.2, higher 𝜋𝜋 ′𝑒𝑒
reduces the real interest rate and boosts demand in the current period.
Price-level targeting or nominal GDP targeting
Following a similar logic to average inflation targeting, targets for the level of prices 𝑃𝑃
have also been suggested. Price-level targeting is equivalent to a target for the inflation
rate averaged over a long period. Lower inflation now causes expected future inflation to
rise so that the future price level can return to its target level or path.
Another alternative policy with some similar features is a target for the level of nominal
GDP. If credible, low inflation now means expectation of higher future inflation or higher
future real GDP, and expectations of higher 𝜋𝜋 ′ or 𝑌𝑌 ′ boost current demand.
Box 9.7: Forward guidance and confidence
Central banks have increasingly used ‘forward guidance’ when nominal interest rates are
at their lower bound. With forward guidance, the central bank provides information
about future path of interest rates. But how is that information interpreted by people in
the economy?
In Section 9.7 we described one interpretation, where people believe the central bank
will keep interest rates lower in future when it is not constrained by the lower bound
problem and that interest rates will be lower than future economic conditions warrant.
This future stimulus raises expectations of 𝜋𝜋 ′ and 𝑌𝑌 ′ , which boosts the current level of
demand. In this case, forward guidance is interpreted as a commitment to keep
monetary policy excessively loose in the future.
But another way people might interpret the forward guidance announcement is that the
central bank is more pessimistic about the economy. Hearing the forward guidance
announcement, they think the central bank must believe prospects for the economy
have become weaker if it is predicting future interest rates will be lower than before. If
the private sector revises its own beliefs after hearing the announcement then
confidence about the future declines. This leads to lower expectations of 𝜋𝜋 ′ and 𝑌𝑌 ′ ,
which reduce demand through the channels explained in Section 9.2, the opposite of
what the central bank intended if it was trying to make a commitment.
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Box 9.8: Negative interest rate policies
We saw in Section 6.14 that it is possible for a central bank to reduce the nominal
interest rate 𝑖𝑖 below zero. But there is still a lower bound −ℎ while physical cash
remains available, where ℎ is the proportional holding cost of cash taking account
of the need for secure storage. With interest rate lower bound 𝑖𝑖 ≥ −ℎ, the central
bank can lower the real interest rate 𝑟𝑟 = 𝑖𝑖 − 𝜋𝜋 ′𝑒𝑒 to −𝜋𝜋 ′𝑒𝑒 − ℎ at most. As shown in
Figure 9.11, this gives further scope for monetary stimulus, which can be effective
through the usual channels.
Figure 9.11: Negative nominal interest rates
By setting negative nominal interest rates, the central bank can shift the 𝑀𝑀𝑀𝑀 line
further downwards along the 𝑌𝑌 𝑑𝑑 curve, which allows it to stabilise the economy for
a larger range of negative demand shocks.
However, there may be costs of this negative interest rate policy. First, there can be
an inefficiently large use of cash to avoid negative rates, which wastes resources on
security and storage costs (see Section 6.14). Negative interest rates also mean
banks cannot break even without negative interest rates charged on deposits. But if
households and firms were to switch from using bank deposits to using cash as
money then this would eliminate the benefits of financial intermediation provided
by banks (see Section 7.10).
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Chapter 10: Open-economy macroeconomics
This chapter explores how an economy’s interactions with the rest of the world influence
GDP, its pattern of trade, exchange rates, and welfare.
Essential reading
•
Williamson, Chapters 16 and 17.
10.1 International trade in goods and assets
Globalisation has increased the scope for countries to trade with one another and exploit
gains from trade. We usually think of gains from trade as coming from exchanging different
types of goods. For example, Saudi Arabia has a comparative advantage in extracting oil and
China has a comparative advantage in manufactured goods. But in thinking about trade and
macroeconomics, another type of trade known as ‘intertemporal’ trade is important.
In this type of trade, countries exchange goods at different points in time. We do not usually
observe such intertemporal exchanges of goods taking place direct but this type of exchange
is mediated whenever assets are traded between countries. For example, if a country has a
low income but expects a higher income in the future, firms and households gain by
borrowing from the rest of the world to invest and smooth consumption. This works by
selling bonds to foreigners now and repaying debt in future. By selling bonds, the country
imports goods to invest and consume, and exports more in future to repay its debts.
Similarly, a country expecting low growth or a decline in income might gain by lending to the
rest of the world. It buys bonds from foreigners now and spends the payoff from owing
those bonds later. It exports goods now to acquire assets and is able to import more when it
is repaid. In these examples, countries effectively have comparative advantages in
production at different points in time.
To analyse international trade, recall the terminology of the balance of payments. The
balance of payments 𝐵𝐵𝐵𝐵 is broken down into two accounts, the current account 𝐶𝐶𝐶𝐶 and the
financial account 𝐹𝐹𝐹𝐹:
𝐵𝐵𝐵𝐵 = 𝐶𝐶𝐶𝐶 + 𝐹𝐹𝐹𝐹
The current account is the sum of net exports and net foreign income:
𝐶𝐶𝐶𝐶 = 𝑁𝑁𝑁𝑁 + 𝑁𝑁𝑁𝑁𝑁𝑁
Net exports 𝑁𝑁𝑁𝑁 refers to the value of exports minus value of imports. Net foreign income
N𝐹𝐹𝐹𝐹 is income earned from ownership of foreign assets minus foreign claims to income
earned in domestic economy.
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The terminology related to the financial account (𝐹𝐹𝐹𝐹) has changed over time. You may see it
referred to as the capital account (𝐾𝐾𝐾𝐾) in some textbooks. The financial account comprises
foreign net purchases of domestic assets minus domestic net purchases of foreign assets.
Any purchases or sales of official foreign-exchange reserves are included in 𝐹𝐹𝐹𝐹 in this
course, so the financial account captures all trade in assets, both by the private sector and
the government.
The balance of payments identity is 𝐵𝐵𝐵𝐵 = 0, so the financial and current accounts must in
equilibrium be related according to:
𝐹𝐹𝐹𝐹 = −𝐶𝐶𝐶𝐶
Next, let us see how international trade affects the goods market and the relationship
between saving and investment. Net exports 𝑁𝑁𝑁𝑁 are a component of aggregate
expenditure, so the breakdown of GDP by expenditure is now:
𝑌𝑌 = 𝐶𝐶 + 𝐼𝐼 + 𝐺𝐺 + 𝑁𝑁𝑁𝑁
An open economy also has a separate notion of gross national product (GNP) as well as GDP.
GNP is equal to GDP plus net foreign income:
𝐺𝐺𝐺𝐺𝐺𝐺 = 𝑌𝑌 + 𝑁𝑁𝑁𝑁𝑁𝑁
National saving 𝑆𝑆 is the sum of private and public saving, 𝑆𝑆 = 𝑆𝑆 𝑝𝑝 + 𝑆𝑆 𝑔𝑔 , where private saving
is 𝑆𝑆 𝑝𝑝 = (𝑌𝑌 + 𝐼𝐼𝐼𝐼𝐼𝐼 + 𝑁𝑁𝑁𝑁𝑁𝑁 − 𝑇𝑇) − 𝐶𝐶. This is disposable income minus consumption 𝐶𝐶, where
the private sector receives domestic income 𝑌𝑌, interest on government debt 𝐼𝐼𝐼𝐼𝐼𝐼, net
foreign income 𝑁𝑁𝑁𝑁𝑁𝑁, and pays taxes net of transfers 𝑇𝑇. Public saving is 𝑆𝑆 𝑔𝑔 = 𝑇𝑇 − 𝐼𝐼𝐼𝐼𝐼𝐼 − 𝐺𝐺,
with the budget deficit being −𝑆𝑆 𝑔𝑔 . National saving is therefore 𝑆𝑆 = 𝑌𝑌 + 𝑁𝑁𝑁𝑁𝑁𝑁 − 𝐶𝐶 − 𝐺𝐺 =
𝐺𝐺𝐺𝐺𝐺𝐺 − 𝐶𝐶 − 𝐺𝐺, and the national accounts and balance-of-payments identities imply:
𝑆𝑆 = 𝐼𝐼 + 𝐶𝐶𝐶𝐶
This means that a country can save either through investment in the domestic capital stock,
or through running a current account surplus (𝐶𝐶𝐶𝐶 > 0). Equivalently, a financial account
deficit (𝐹𝐹𝐹𝐹 < 0) implies lending to the rest of the world. A country can borrow by running a
current account deficit (𝐶𝐶𝐶𝐶 < 0), noting that a financial account surplus (𝐹𝐹𝐹𝐹 > 0) implies
borrowing from the rest of the world.
Saving, investment, and the current account are flow variables. Net foreign assets 𝑁𝑁𝑁𝑁𝑁𝑁 is
net stock of international savings, defined as the value of foreign assets owned by domestic
residents minus value of domestic assets owned by foreign residents. The accounting
equation for the link between current and future net foreign assets denoted by 𝑁𝑁𝑁𝑁𝑁𝑁′ is:
𝑁𝑁𝑁𝑁𝐴𝐴′ = 𝑁𝑁𝑁𝑁𝑁𝑁 − 𝐹𝐹𝐹𝐹 + Net capital gains
For the purposes of this course, we will ignore changes in asset valuations. Hence, the
current account gives the change in net foreign assets:
𝑁𝑁𝑁𝑁𝐴𝐴′ − 𝑁𝑁𝑁𝑁𝑁𝑁 = 𝐶𝐶𝐶𝐶
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We will focus on international trade done by a small open economy (SOE). An SOE is not
large enough to affect what happens in other countries or in world markets. This makes the
country a price taker in competitive world markets. The country can be affected by what
happens in the rest of the world but there is no feedback from it to the world. This
assumption is appropriate for some economies, such as Singapore, although not for others.
However, many of the lessons for SOEs carry over to the analysis of large open economies.
10.2 Gains from trade in assets
We will start by demonstrating how there can be gains from trade even if countries do not
have comparative advantages in producing different types of goods. To that end, suppose
there is a single homogeneous good everywhere in the world and all variables are measured
in real terms in units of this good.
There is a single type of asset traded internationally, namely, a bond with real interest rate
𝑟𝑟. Hence, all assets around the world have the same real return 𝑟𝑟. If a country’s net foreign
assets 𝑁𝑁𝑁𝑁𝑁𝑁 are held in this bond then its net foreign income is 𝑁𝑁𝑁𝑁𝑁𝑁 = 𝑟𝑟 × 𝑁𝑁𝑁𝑁𝑁𝑁. Initially, we
rule out any investments in physical capital (𝐼𝐼 = 0), which has the consequence that
national saving (or borrowing) is possible only through the current account because 𝑆𝑆 = 𝐶𝐶𝐶𝐶.
Our analysis is done in a two-period set-up like the consumption choice model from Chapter
3. Since the second period is final, no one wants to hold assets after (𝑁𝑁𝑁𝑁𝐴𝐴′′ = 0), which
requires 𝑁𝑁𝑁𝑁𝐴𝐴′ + 𝐶𝐶𝐴𝐴′ = 0. We start from a blank slate in the first period by considering a
country with no initial assets or debts internationally, that is, 𝑁𝑁𝑁𝑁𝑁𝑁 = 0. This means the
accumulation of future net foreign assets is solely due to the current account surplus in the
current period, 𝑁𝑁𝑁𝑁𝐴𝐴′ = 𝐶𝐶𝐶𝐶. These observations imply the country’s international budget
constraint is:
𝐶𝐶𝐶𝐶 + 𝐶𝐶𝐴𝐴′ = 0
This states that present and future current account balances must offset each other. A
country that accumulates assets today by saving will spend those savings in the future. A
country that borrows today must run a future current account surplus to repay its debts.
This budget constraint takes a more familiar form in terms of net exports. Note that 𝐶𝐶𝐶𝐶 =
𝑁𝑁𝑁𝑁 because 𝑁𝑁𝑁𝑁𝑁𝑁 = 𝑟𝑟 × 𝑁𝑁𝑁𝑁𝑁𝑁 = 0, and 𝐶𝐶𝐴𝐴′ = 𝑁𝑁𝑋𝑋 ′ + 𝑁𝑁𝑁𝑁𝐼𝐼 ′ = 𝑁𝑁𝑋𝑋 ′ + 𝑟𝑟𝑟𝑟𝑟𝑟 because 𝑁𝑁𝑁𝑁𝐴𝐴′ =
𝑁𝑁𝑁𝑁. Hence, (1 + 𝑟𝑟)𝑁𝑁𝑁𝑁 + 𝑁𝑁𝑋𝑋 ′ = 0, and dividing both sides by 1 + 𝑟𝑟 implies:
𝑁𝑁𝑁𝑁 +
𝑁𝑁𝑁𝑁′
=0
1 + 𝑟𝑟
This says that the present discounted value of current and future levels of net exports must
be zero. With no investment in physical capital, net exports are equal to 𝑁𝑁𝑁𝑁 = 𝑌𝑌 − 𝐶𝐶 − 𝐺𝐺
and so the country’s international budget constraint becomes the usual consolidated
household-and-government budget constraint in terms of aggregate consumption and
income:
𝐶𝐶 +
𝐶𝐶′
𝑌𝑌′
𝐺𝐺′
= �𝑌𝑌 +
� − �𝐺𝐺 +
�
1 + 𝑟𝑟
1 + 𝑟𝑟
1 + 𝑟𝑟
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Underlying this combined constraint is a representative household choosing a consumption
plan (𝐶𝐶, 𝐶𝐶 ′ ) to get onto highest indifference curve subject to budget constraint:
𝐶𝐶 +
𝐶𝐶′
𝑌𝑌 ′ − 𝑇𝑇 ′
= 𝑌𝑌 − 𝑇𝑇 +
1 + 𝑟𝑟
1 + 𝑟𝑟
Current and future income (𝑌𝑌, 𝑌𝑌 ′ ) are treated as exogenous here. We make the usual
assumptions on household preferences such as a desire for smooth consumption.
The government exogenously sets fiscal policy (𝐺𝐺, 𝐺𝐺 ′ ) and (𝑇𝑇, 𝑇𝑇 ′ ) subject to its budget
constraint:
𝐺𝐺 +
𝐺𝐺′
𝑇𝑇′
= 𝑇𝑇 +
1 + 𝑟𝑟
1 + 𝑟𝑟
To understand the gains from international trade, we first look at what happens in ‘autarky’
where there is no possibility of trade. In autarky, net exports and the current account are
necessarily zero, 𝑁𝑁𝑁𝑁 = 𝐶𝐶𝐶𝐶 = 0. With 𝑁𝑁𝑁𝑁 = 0 and 𝑁𝑁𝑋𝑋 ′ = 0, in equilibrium the economy
must have 𝐶𝐶 = 𝑌𝑌 − 𝐺𝐺 and 𝐶𝐶 ′ = 𝑌𝑌 ′ − 𝐺𝐺′. The economy’s real interest rate must therefore
adjust so that the chosen consumption plan (𝐶𝐶, 𝐶𝐶 ′ ) is consistent with these requirements,
just as seen earlier in Section 3.6. The real interest rate in this case is called the autarky real
interest rate 𝑟𝑟𝑎𝑎 .
Figure 10.1: Autarky real interest rate
Now consider a country open to international trade. We suppose that capital mobility is
perfect. Perfect capital mobility means there are no restrictions on capital flows. Domestic
residents can freely buy or sell foreign assets and foreigners can freely buy or sell domestic
assets. As domestic and foreign bonds are equivalent, the domestic real interest rate 𝑟𝑟 must
equal the foreign real interest rate 𝑟𝑟 ∗ and a small open economy is not big enough to
influence 𝑟𝑟 ∗ . This means the economy is effectively a participant in a perfectly competitive
world market for bonds with 𝑟𝑟 = 𝑟𝑟 ∗ .
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Since the international budget constraint always passes through the point (𝑌𝑌 − 𝐺𝐺, 𝑌𝑌 ′ − 𝐺𝐺 ′ )
that is the autarky equilibrium irrespective of the level of 𝑟𝑟 ∗ , it follows that whenever 𝑟𝑟 ∗ ≠
𝑟𝑟𝑎𝑎 , households can get on to a higher indifference curve through trade than they can reach
in autarky. This is because the indifference curve at the autarky point must cut the budget
constraint because the indifference curve has gradient −(1 + 𝑟𝑟𝑎𝑎 ) there, while the budget
constraint has gradient −(1 + 𝑟𝑟 ∗ ) everywhere.
Figure 10.2: Gains from trade
This account of the gains from international trade also points towards some potential
determinants of the current account. By making an analogy with an individual’s life cycle
and consumption choice, we can see that the stage of development of a country can be one
long-run factor that might affect its current account.
For example, a country with high GDP due to extraction of an exhaustible resource is in a
similar position to a middle-aged person planning for retirement. It makes sense to save to
smooth consumption, which for the country as a whole means running a 𝐶𝐶𝐶𝐶 surplus. This
helps to explain the large current-account surpluses of oil exporters such as Saudi Arabia.
Figure 10.3: Long-run explanations for the current account
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On the other hand, a developing country is in a similar position to a young person who
expects a much higher income in the future. It makes sense to borrow to smooth
consumption, which for the country as a whole means running a 𝐶𝐶𝐶𝐶 deficit. However, there
are many examples where developing countries are seen not to behave in this way, running
large current-account surpluses instead. This is related to the puzzle of why capital does not
flow from rich countries to poorer countries.
Although we have seen that there are gains from international trade, changes in the world
interest rate can increase or decrease the size of the gains from trade, although these
always remain positive relative to autarky. The argument is equivalent to how individual
savers and borrowers are affected by changes in the real interest rate. Recalling the
argument from Section 3.5, a higher interest rate implies a substitution effect reducing
current consumption and increasing 𝐶𝐶𝐶𝐶 for both. The higher interest rate makes a country
with a current-account surplus (a saver) better off but a country with a current-account
deficit (a borrower) worse off. The overall effect on the current account is ambiguous when
it is initially in surplus but, if it is initially in deficit, then the deficit should fall.
Figure 10.4: Increase in world interest rate
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Intuitively,
income risk
shocksharing
leads a country to lower its international
Box 10.1:a negative
International
lending or increase its international borrowing. Adjustment of the current account
Suppose a country faces a temporary negative shock to real GDP 𝑌𝑌. In the twothus helps households faced with income shocks to smooth consumption. The
period model, current income falls but expected future income remains the same.
model predicts that the current account falls with a negative income shock, so it is
Figure 10.5 shows the leftward shift of the budget constraint passing through (𝑌𝑌 −
predicted
to be a procyclical variable. Figure 10.6 shows data on the cyclicality of
𝐺𝐺, 𝑌𝑌 ′ − 𝐺𝐺 ′ ) as 𝑌𝑌 falls. The optimal choice of current consumption 𝐶𝐶 ∗ declines but by
the current account in the USA. Empirically, the US current account is actually
less than 𝑌𝑌 does. This reflects the desire of households to smooth consumption by
countercyclical, although
the USA is far from a small open economy and many
choosing a decline in 𝐶𝐶 ′∗ as well to permit a smaller fall in 𝐶𝐶. It follows that the
countries’ own business-cycle fluctuations correlate with what happens in the USA.
current account 𝐶𝐶𝐶𝐶 = (𝑌𝑌 − 𝐺𝐺) − 𝐶𝐶 declines after the income shock leaving the
If global business cycles are correlated across countries but more volatile outside
economy with a smaller 𝐶𝐶𝐶𝐶 surplus or a larger 𝐶𝐶𝐶𝐶 deficit.
the USA, it is less surprising that the USA borrows more from the rest of the world
in booms and lends to the rest of the world (or borrows less) in recessions.
Figure 10.5: Negative shock to current income
Figure 10.6: Cyclicality of the US current account
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Box 10.2: The ‘twin deficits’
How might fiscal policy affect the current account? Can a government budget
deficit cause a current account deficit – the so-called ‘twin deficits’ phenomenon?
In the USA in the 1980s, a rising government budget deficit occurred alongside a
rising current account deficit. We will explore this connection and consider whether
the link between the two deficits depends on whether the budget deficit is due to
higher public spending or tax cuts.
Suppose there is a temporary increase in government expenditure 𝐺𝐺, with 𝐺𝐺 ′
remaining constant. This raises the tax burden faced by households, shifting the
present-value budget constraint to the left in Figure 10.7. Households choose to
reduce 𝐶𝐶 ∗ but, with consumption smoothing, households also reduce 𝐶𝐶 ′∗ , which
means that 𝐶𝐶 falls by less than 𝐺𝐺 rises. This implies that the current account 𝐶𝐶𝐶𝐶 =
𝑆𝑆 = (𝑌𝑌 − 𝐺𝐺 ) − 𝐶𝐶 declines. The increase in government expenditure thus causes a
larger current account deficit.
Figure 10.7: Temporary increase in public expenditure
However, this prediction for the current account is not tightly linked to the budget
deficit. We obtain the same prediction for 𝐶𝐶𝐶𝐶 irrespective of whether current taxes
𝑇𝑇 rise to pay for higher 𝐺𝐺, in which case there is no change in the budget deficit, or
future taxes 𝑇𝑇′ increase and the budget deficit rises. National saving 𝑆𝑆 = 𝑌𝑌 − 𝐶𝐶 − 𝐺𝐺
falls by the same amount in both these cases.
This is because Ricardian equivalence implies that timing of taxes does not affect
households’ consumption choices taking as given the government’s spending plans
(𝐺𝐺, 𝐺𝐺 ′ ). Households’ budget constraint passes through the point (𝑌𝑌 − 𝐺𝐺, 𝑌𝑌 ′ − 𝐺𝐺 ′ ),
which determines its position. This means private saving 𝑆𝑆 𝑝𝑝 will adjust to offset any
changes in public saving 𝑆𝑆 𝑔𝑔 due to taxes and, consequently, national saving 𝑆𝑆 =
𝑆𝑆 𝑝𝑝 + 𝑆𝑆 𝑔𝑔 is unaffected by the timing of taxes. The same is therefore true of the
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current account. There will be no international financing of a budget deficit
resulting from a tax cut because domestic households will save the whole tax cut.
Allowing for the credit-market imperfections of Chapter 4 leads to Ricardian
equivalence failing and implies that 𝐶𝐶 rises after a tax cut. In this case, national
saving and the current account would fall following a tax cut.
10.3 Sovereign default
The earlier assumption of perfect capital mobility meant that investors were not concerned
with which country’s bonds they hold – all bonds were equivalent. But how can lenders to a
foreign government compel that government to repay its debts? The problem of contract
enforcement is more difficult than when creditor and debtor are in the same legal
jurisdiction. In this section, we study international borrowing subject to limited
commitment: default can be chosen rationally by a government in some circumstances.
In this analysis, to simplify the relationship between households and the government within
a country, we assume that private and public spending are perfect substitutes, so
households have preferences over total spending 𝐶𝐶 + 𝐺𝐺 instead of only 𝐶𝐶 and we
consolidate the nation’s private and public debts. The argument for the latter is that private
debt can often end up as public debt, for example, bailouts during a financial crisis.
Considering the usual two periods, let 𝐵𝐵 denote the nation’s consolidated private and public
debts to rest of world, including principal and interest, at the beginning of the current
period. Unlike earlier versions of the two-period model, the country does not start from
blank slate, with 𝐵𝐵 being determined by decisions made in the past.
Denoting total international debts (principal and interest) by 𝐵𝐵′ at the beginning of the
future period, issuance of bonds during the current period at interest rate 𝑟𝑟 is 𝐵𝐵′ /(1 + 𝑟𝑟).
The budget equation for current debt issuance is:
𝐶𝐶 + 𝐺𝐺 = 𝑌𝑌 − 𝐵𝐵 +
𝐵𝐵′
1 + 𝑟𝑟
If debts are repaid in full, future private and public spending is limited to income less total
debt in the future period because no further bond issuance is possible at that point:
𝐶𝐶 ′ + 𝐺𝐺 ′ = 𝑌𝑌 ′ − 𝐵𝐵′
Dividing both sides of this equation by 1 + 𝑟𝑟 and substituting in place of 𝐵𝐵′ /(1 + 𝑟𝑟) above
leads to the present-value budget constraint:
𝐶𝐶 ′ + 𝐺𝐺 ′
𝑌𝑌 ′
𝐶𝐶 + 𝐺𝐺 +
= 𝑌𝑌 − 𝐵𝐵 +
1 + 𝑟𝑟
1 + 𝑟𝑟
To analyse household and government choices, the two-period model diagram now has 𝐶𝐶 +
𝐺𝐺 and 𝐶𝐶 ′ + 𝐺𝐺′ on the horizontal and vertical axes. Indifference curves have the usual convex
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shape. The present-value budget constraint passes through the point (𝑌𝑌 − 𝐵𝐵, 𝑌𝑌 ′ ), where
initial debts 𝐵𝐵 are subtracted from current income 𝑌𝑌.
Now suppose that a nation has the option of (full) default in either the current or future
periods. If a default occurs in the future period then the country wipes out debt 𝐵𝐵′ but
incurs a penalty 𝑣𝑣, the effect of which is measured as equivalent to a reduction in output 𝑌𝑌′.
Total spending is then 𝐶𝐶 ′ + 𝐺𝐺 ′ = 𝑌𝑌 ′ − 𝑣𝑣. The penalty 𝑣𝑣 can either be direct losses or
disruption from international creditors trying to enforce their claims, or a proxy in a twoperiod model for the loss from being excluded from world credit markets further in future.
The nation can also default in the current period. In this case, the country wipes out initial
debts 𝐵𝐵 but suffers exclusion from world credit markets in the future period as well as the
same future-period penalty 𝑣𝑣 described above. The default has the effect of returning the
country to autarky, with total expenditure limited by income net of penalties in each time
period, that is, 𝐶𝐶 + 𝐺𝐺 = 𝑌𝑌 and 𝐶𝐶 ′ + 𝐺𝐺 ′ = 𝑌𝑌 ′ − 𝑣𝑣.
Default is a rational choice for the nation if it allows the representative household to reach a
higher indifference curve. In making the default decision, suppose that the government acts
in the interests of the representative household.
Since a nation can choose to default, lenders will only make or roll over current loans if no
default is expected to occur in the future. If there is no default in the second period, the
representative household benefits from expenditure 𝐶𝐶 ′ + 𝐺𝐺 ′ = 𝑌𝑌 ′ − 𝐵𝐵′, while a default
allows it to benefit from 𝐶𝐶 ′ + 𝐺𝐺 ′ = 𝑌𝑌 ′ − 𝑣𝑣. Comparing the two, it follows that a futureperiod default is rational if 𝐵𝐵′ > 𝑣𝑣. Consequently, rational lenders make loans in the current
period only if 𝐵𝐵′ ≤ 𝑣𝑣. With this restriction, the current-period budget equation implies:
𝐶𝐶 + 𝐺𝐺 ≤ 𝑌𝑌 − 𝐵𝐵 +
𝑣𝑣
1 + 𝑟𝑟
This is referred to as a limited-commitment constraint on borrowing. It takes the form of a
borrowing constraint analogous to those studied in Chapter 4. The present-value budget
constraint is truncated at 𝐶𝐶 + 𝐺𝐺 = 𝑌𝑌 − 𝐵𝐵 + 𝑣𝑣/(1 + 𝑟𝑟), where the amount that expenditure
can exceed income is 𝑣𝑣 ⁄(1 + 𝑟𝑟) − 𝐵𝐵. Subject to the limited-commitment constraint, if
lending occurs in the current period, debts will be repaid in the future if nothing
unexpectedly changes. When the limited-commitment constraint binds, 𝐶𝐶 ′ + 𝐺𝐺 ′ = 𝑌𝑌 ′ − 𝑣𝑣
because 𝐵𝐵′ = 𝑣𝑣 makes the nation indifferent between defaulting or not.
Since the possibility of default explains why a country faces a borrowing constraint, this
analysis can be used to understand why countries might fail to use international financial
markets to smooth consumption by as much as our earlier analysis would suggest.
Now consider the default decision in the current period, taking as given past lending
decisions 𝐵𝐵 (which might turn out ex post to have been unwise if default is rational). If a
default occurs, the country has no access to international financial markets and only one
expenditure point (𝑌𝑌, 𝑌𝑌′ − 𝑣𝑣) is feasible in autarky. Supposing 𝐵𝐵 ≤ 𝑣𝑣/(1 + 𝑟𝑟), the default
outcome lies to the left of the point where the limited-commitment constraint binds. It
follows that default is not chosen in this case.
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Figure 10.8: Case where no default is chosen
What happens if 𝐵𝐵 > 𝑣𝑣/(1 + 𝑟𝑟)? In this case, the autarky point A lies to the right of the
point B where the limited-commitment constraint binds, so it might be in the nation’s
interest to default. If limited commitment constraint would bind in the absence of default, it
can be seen that default is definitely preferable when 𝐵𝐵 > 𝑣𝑣 ⁄(1 + 𝑟𝑟). If the limitedcommitment constraint is not binding with the country not wanting to borrow the
maximum amount, then there is the value of being able to access world capital markets to
consider as well as the present value of the penalty 𝑣𝑣.
Figure 10.9: Case where default is chosen
This analysis points to factors that make default more likely. First, it is immediate that a
greater debt burden 𝐵𝐵 increases the likelihood of default, all else being equal. Default is also
more likely when the losses or penalties 𝑣𝑣 from default are small. For example, bailouts or
debt forgiveness might allow a nation to return to credit markets after a short spell of
exclusion.
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For a country not already borrowing the maximum amount consistent with the limitedcommitment constraint, a negative shock to income 𝑌𝑌 makes default more likely. However,
𝑌𝑌 becomes irrelevant to the default decision once the maximum borrowing amount is
reached because there is no option value of being able to borrow more to smooth
consumption.
A final crucial factor in the default decision is the interest rate 𝑟𝑟 faced by a country. If this is
high, default is more likely because a greater sacrifice of expenditure is necessary to roll
over existing debts at a higher interest rate. Countries with large debt burdens might not
have an incentive to default as long as interest rates remain low.
The link between interest rates and default decisions points to the possibility of self-fulfilling
expectations of default. Suppose international lenders believe there is a positive probability
of a country defaulting. If 𝑟𝑟 ∗ is the interest rate on a risk-free asset (a bond issued by a
country that will not default), lenders are only willing to lend to the country with default risk
if they receive a risk premium. This risk premium is a spread between 𝑟𝑟 and 𝑟𝑟 ∗ that depends
on the perceived probability of default.
Self-fulfilling defaults could occur because believing the probability of default is high drives
up 𝑟𝑟, making it more likely the condition for default will be satisfied. On the other hand, if
there is a low default probability, 𝑟𝑟 is low and default is not chosen, assuming other
fundamentals do not point to default. This logic shows that lack of commitment can lead to
self-fulfilling defaults in some circumstances.
10.4 Open-economy real dynamic model
The simple examples with exogenous income and fiscal policy illustrate the gains from
international trade in assets and how the current account responds to shocks. These
findings apply more broadly in economies where GDP is endogenous, for example, where
firms hire labour supplied by households and invest in capital as we saw in the dynamic
macroeconomic model from Chapter 3. By adding international trade to the dynamic
macroeconomic model, we can also investigate whether GDP responds differently to shocks
in an open economy.
The closed-economy dynamic macroeconomic model developed in Section 3.12 explains
real GDP 𝑌𝑌 and the real interest rate 𝑟𝑟 by finding the intersection of the output demand 𝑌𝑌 𝑑𝑑
and output supply 𝑌𝑌 𝑠𝑠 curves. For comparison, we denote by 𝑌𝑌𝑎𝑎 and 𝑟𝑟𝑎𝑎 the predictions of the
model if the economy were in autarky, i.e. cut off from international trade. Output demand
in a closed economy is given by 𝑌𝑌𝑎𝑎𝑑𝑑 = 𝐶𝐶 𝑑𝑑 + 𝐼𝐼 𝑑𝑑 + 𝐺𝐺. Unlike the new Keynesian model from
Chapter 8, we assume flexible goods prices here, which means we are examining a purely
‘real’ model with no monetary issues. We consider a model with sticky prices later in the
chapter.
10.4.1 Balance-of-payments equilibrium, capital flows and net exports
There are two important differences when we analyse an open economy using our dynamic
macroeconomic model. First, net exports 𝑁𝑁𝑁𝑁 are added as an additional component of
output demand:
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𝑌𝑌 𝑑𝑑 = 𝐶𝐶 𝑑𝑑 + 𝐼𝐼 𝑑𝑑 + 𝐺𝐺 + 𝑁𝑁𝑁𝑁
Output demand is 𝑌𝑌 𝑑𝑑 = 𝑌𝑌𝑎𝑎𝑑𝑑 + 𝑁𝑁𝑁𝑁, the sum of domestic demand 𝑌𝑌𝑎𝑎𝑑𝑑 , referred to as
absorption and net exports 𝑁𝑁𝑁𝑁. All else being equal, changes in net exports cause shifts of
the 𝑌𝑌 𝑑𝑑 curve. The second important difference is that perfect mobility of capital in an open
economy aligns the domestic real interest rate 𝑟𝑟 with the foreign interest rate 𝑟𝑟 ∗ . This is a
necessary condition for balance-of-payments equilibrium 𝐵𝐵𝐵𝐵 = 𝐶𝐶𝐶𝐶 + 𝐹𝐹𝐹𝐹 = 0. However, if
𝑟𝑟 = 𝑟𝑟 ∗ for a small open economy that takes 𝑟𝑟 ∗ as given, what ensures the goods market is in
equilibrium if the domestic real interest rate 𝑟𝑟 cannot adjust to where the 𝑌𝑌 𝑑𝑑 and 𝑌𝑌 𝑠𝑠 curves
intersect? The mechanism is now that capital flows occur until 𝑌𝑌 𝑑𝑑 and 𝑌𝑌 𝑠𝑠 cross at 𝑟𝑟 = 𝑟𝑟 ∗.
Adjustment of 𝐹𝐹𝐹𝐹 implies changes in the current account 𝐶𝐶𝐶𝐶 = −𝐹𝐹𝐹𝐹 and, hence, net
exports, which shift the 𝑌𝑌 𝑑𝑑 curve.
Consider the example in Figure 10.10. The hypothetical closed-economy output demand
curve 𝑌𝑌𝑎𝑎𝑑𝑑 intersects 𝑌𝑌 𝑠𝑠 at 𝑟𝑟 = 𝑟𝑟𝑎𝑎 , which is above the foreign real interest rate 𝑟𝑟 ∗ . In an open
economy with perfect capital mobility, 𝑟𝑟 would fall to 𝑟𝑟 ∗ on the horizontal 𝐵𝐵𝐵𝐵 line
representing balance-of-payments equilibrium. At 𝑟𝑟 = 𝑟𝑟 ∗ , the output demand curve 𝑌𝑌𝑎𝑎𝑑𝑑 is to
the right of the output supply curve 𝑌𝑌 𝑠𝑠 . With domestic demand exceeding domestic
production, imports rise, reducing net exports 𝑁𝑁𝑁𝑁, which shifts the open-economy output
demand curve 𝑌𝑌 𝑑𝑑 to the left until it intersects 𝑌𝑌 𝑠𝑠 at 𝑟𝑟 = 𝑟𝑟 ∗ . The fall in the current account
𝐶𝐶𝐶𝐶 is matched by capital inflows that raise 𝐹𝐹𝐹𝐹.
Figure 10.10: Goods market in an open economy
In what follows, we simplify matters by assuming 𝑁𝑁𝑁𝑁𝑁𝑁 = 0, which means that the current
account 𝐶𝐶𝐶𝐶 is the same as net exports 𝑁𝑁𝑁𝑁. We will also consider an economy starting from
a current account close to balance (𝐶𝐶𝐶𝐶 is close to zero). This means that households hold
assets equal to all domestic assets and the economy is neither a net saver or borrower. This
allows us to continue to ignore income effects arising from changes in real wages 𝑤𝑤 and the
real interest rate 𝑟𝑟 (see the discussion in Section 3.11). The consequence of these
simplifications is that the labour demand 𝑁𝑁 𝑑𝑑 and labour supply 𝑁𝑁 𝑠𝑠 curves, and the output
supply curve 𝑌𝑌 𝑠𝑠 , continue to behave as they do in a closed economy. The only change in an
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open economy is the presence of net exports as a component of 𝑌𝑌 𝑑𝑑 . Hence, the adjustment
to balance-of-payments equilibrium (𝑟𝑟 = 𝑟𝑟 ∗ ) in an open economy occurs solely through
shifts of the output demand curve as seen in Figure 10.10.
10.4.2 Examples
Let us consider two types of shock to illustrate how the open-economy real dynamic model
works. We proceed by finding the effects of a shock in a closed economy and then ask what
adjustment of net exports (and, hence, what shift of 𝑌𝑌 𝑑𝑑 ) is required to achieve balance-ofpayments equilibrium in an open economy.
First, consider a temporary negative supply shock (a decline in current TFP 𝑧𝑧) of the kind we
used to explain business cycles using the RBC approach from Section 8.8. In a closed
economy, this shock shifts both the output supply 𝑌𝑌 𝑠𝑠 and output demand 𝑌𝑌 𝑑𝑑 curves to the
left but the leftward shift of 𝑌𝑌 𝑑𝑑 is smaller than that of 𝑌𝑌 𝑠𝑠 owing to the desire for
consumption smoothing. This led the equilibrium real interest rate to rise in a closed
economy. In an open economy, the real interest rate cannot rise above 𝑟𝑟 ∗ and balance-ofpayments equilibrium (𝑟𝑟 = 𝑟𝑟 ∗ ) is restored by net exports 𝑁𝑁𝑁𝑁 declining instead, shifting 𝑌𝑌 𝑑𝑑
further to the left to match the shift of 𝑌𝑌 𝑠𝑠 . This analysis is illustrated in Figure 10.11. The
negative TFP shock results in a larger decline in output in an open economy as households
smooth consumption by importing more and borrowing from the rest of the world.
Figure 10.11: Temporary TFP shock in an open economy
The second example is a temporary fiscal stimulus, specifically, a temporary increase in
government spending 𝐺𝐺. In a closed economy, this policy change was analysed in Box 4.2.
Both the output demand 𝑌𝑌 𝑑𝑑 and output supply 𝑌𝑌 𝑠𝑠 curves shift to the right but 𝑌𝑌 𝑑𝑑 shifts by
more, leading to an increase in 𝑟𝑟 in a closed economy. In an open economy, balance-ofpayments equilibrium is restored by a decline of 𝑁𝑁𝑁𝑁, shifting 𝑌𝑌 𝑑𝑑 to the left until its overall
shift matches that of 𝑌𝑌 𝑠𝑠 . This is shown in Figure 10.12. The fiscal stimulus raises real GDP by
a smaller amount in an open economy because some of the additional expenditure is spent
on imports rather than buying more goods produced domestically.
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Figure 10.12: Temporary fiscal stimulus in an open economy
Box 10.3: Global imbalances
In the 1990s and 2000s, large current-account imbalances relative to GDP open up
in several countries. These became known as ‘global imbalances’, the most
noteworthy of which were the large current-account deficits of the USA and the
large current-account surpluses of China. Here, we use the international real
dynamic model to investigate some possible causes of such imbalances.
First, suppose that good news about future productivity growth creates
expectations of a high return on investments in country. We represent this in the
model by higher 𝑧𝑧′, which raises 𝑀𝑀𝑃𝑃𝐾𝐾′ . This might fit the case of the US economy in
the 1990s, which had a spell of high productivity growth and experienced an
investment boom. In a closed economy, this would shift the output demand curve
to the right (increasing 𝐼𝐼 𝑑𝑑 ) and the output supply curve to the left (when the shock
affects future 𝑧𝑧′ but not current 𝑧𝑧). The closed-economy effects are essentially the
opposite of those studied in Box 3.6, with the real interest rate rising
unambiguously.
In an open economy, adjustment to balance-of-payments equilibrium requires a
leftward shift of 𝑌𝑌 𝑑𝑑 to intersect 𝑌𝑌 𝑠𝑠 at 𝑟𝑟 = 𝑟𝑟 ∗ and, hence, a decline in net exports.
The economy runs a current-account deficit as borrowing from the rest of world
finances the investment boom. This case is illustrated in Figure 10.13.
Our second example considers a country with underdeveloped financial markets.
We interpret this as a larger spread between the safe real interest rate 𝑟𝑟 and the
interest rate 𝑟𝑟𝑙𝑙 on domestic lending. As we saw in Section 4.5, a larger spread can
reflect a more severe asymmetric-information problem in underdeveloped financial
markets. The effect of a larger spread between 𝑟𝑟 and 𝑟𝑟𝑙𝑙 is that the output demand
curve is further to the left than otherwise (for example, through lower 𝐼𝐼 𝑑𝑑 )
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because 𝑟𝑟𝑙𝑙 is higher for each real interest rate 𝑟𝑟 on the vertical axis. In a closed
economy, this would lead to a lower domestic real interest rate 𝑟𝑟 in equilibrium.
Figure 10.13: Country with a high return on capital
In an open economy, balance-of-payments equilibrium is restored by a rightward
shift of 𝑌𝑌 𝑑𝑑 with net exports rising as illustrated in Figure 10.14. The economy runs a
current account surplus with lending to the rest of the world substituting for
domestic lending. This is because domestic lending is hampered by more severe
asymmetric information problems compared to foreign financial markets. This case
arguably fits the experience of some developing economies that have run
persistently large current-account surpluses.
Figure 10.14: Country with limited financial development
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Box 10.4: Capital controls
We have assumed throughout this chapter that capital mobility is perfect, which
effectively makes the financial account 𝐹𝐹𝐹𝐹 of the balance of payments extremely
sensitive to the domestic-foreign interest rate gap 𝑟𝑟 − 𝑟𝑟 ∗ . This leads to 𝑟𝑟 = 𝑟𝑟 ∗ in
balance-of-payments equilibrium.
One reason for less-than-perfect capital mobility is that there are restrictions in
place that prevent some capital flows, such as limits on foreign purchases of
domestic assets or domestic purchases of foreign assets. These are known as
capital controls and have the effect of making 𝐹𝐹𝐹𝐹 less sensitive to 𝑟𝑟 − 𝑟𝑟 ∗ .
To see the effects of capital controls, we consider the extreme case where all
capital flows are blocked. With a zero financial account (𝐹𝐹𝐹𝐹 = 0), balance-ofpayments equilibrium then requires 𝐶𝐶𝐶𝐶 = 0. Hence, in the open-economy real
dynamic model, the shifts of 𝑌𝑌 𝑑𝑑 through net exports 𝑁𝑁𝑁𝑁 that ensured 𝑟𝑟 = 𝑟𝑟 ∗ are
prevented by the capital controls. This most extreme form of capital controls thus
leaves the economy at the same outcomes as if it were in autarky. It can be
analysed as if it were a closed economy with the domestic real interest rate
determined by the intersection of 𝑌𝑌 𝑑𝑑 and 𝑌𝑌 𝑠𝑠 with no adjustment of 𝑁𝑁𝑁𝑁. More
generally, capital controls that are less strict than this result in outcomes
somewhere between the autarky and perfect capital mobility extremes.
Figure 10.15: Negative supply shock with capital controls
Taking the example in Section 10.4 of a negative supply shock, Figure 10.15 shows
that GDP 𝑌𝑌 falls by less with full capital controls and the real interest rate 𝑟𝑟 rises.
Although GDP falls by less, households are better off with perfect capital mobility
because they can smooth consumption with cheaper international borrowing at
interest rate 𝑟𝑟 ∗ . The high interest rate 𝑟𝑟 is a reflection of the difficulties of
producing domestically after the negative supply shock.
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10.5 The terms of trade
Our analysis in this chapter assumed all goods are homogeneous to emphasise the gains
available from purely intertemporal trade between countries. It was also possible to
develop a model of the determinants of net exports even though exports and imports were
the same type of good. This is because the current account is the opposite of the financial
account in balance-of-payments equilibrium, so international saving or borrowing shows up
as trade surpluses or deficits. Nonetheless, trade in different types of goods is obviously a
feature of the real world and it is important to account for this some contexts, particularly
when we consider international competitiveness later in the chapter.
We now allow for different types of goods to be produced in the domestic and foreign
economies. We denote the relative price of imports in terms of domestically produced
goods (some of which are exported) by 𝑞𝑞. Supposing all domestically produced goods are
the same, this is a measure of the terms of trade, the relative price of imports to exports. As
defined, higher 𝑞𝑞 means imports become more expensive, which improves the
competitiveness of exports.
The quantity (or volume) of exports is denoted by 𝑋𝑋 and the quantity of imports by 𝑍𝑍. The
value of net exports in terms of domestic goods is:
𝑁𝑁𝑁𝑁 = 𝑋𝑋 − 𝑞𝑞𝑞𝑞
The spending patterns of domestic and foreign consumers on different goods depend on
relative price of imports 𝑞𝑞. The quantity of exports 𝑋𝑋 demanded increases with 𝑞𝑞 as
competitiveness improves and the quantity of imports 𝑍𝑍 demanded decreases with 𝑞𝑞 as
domestically produced goods become more competitive. We assume the value of net
exports 𝑁𝑁𝑁𝑁 = 𝑋𝑋 − 𝑞𝑞𝑞𝑞 rises with 𝑞𝑞 overall, that is, the volume effects of 𝑞𝑞 on 𝑋𝑋 and 𝑍𝑍
dominate import value effect (the direct negative effect of 𝑞𝑞 in the equation for 𝑁𝑁𝑁𝑁). This
occurs when the Marshall-Lerner condition is satisfied: the sum of the absolute value of the
elasticities of 𝑋𝑋 and 𝑍𝑍 with respect to 𝑞𝑞 is greater than 1. The relationship between net
exports and the term of trade is depicted in Figure 10.16.
Figure 10.16: Net exports and the terms of trade
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Accounting for domestic and foreign output being different types of goods does not
fundamentally affect the determination of net exports 𝑁𝑁𝑁𝑁 in the two-period model of the
current account from Section 10.2. Given the equilibrium value of 𝑁𝑁𝑁𝑁, the terms of trade 𝑞𝑞
adjusts to the level that is consistent with it according to Figure 10.16. For example, the
temporary increase in government spending 𝐺𝐺 from Box 10.2 to leads to a fall in net exports
𝑁𝑁𝑁𝑁 and now to a lower terms of trade 𝑞𝑞, which represents a real appreciation of the terms
of trade, a loss of competitiveness of domestic goods.
In the full real dynamic model, the output demand curve 𝑌𝑌 𝑑𝑑 continues to shift with 𝑁𝑁𝑁𝑁
following shocks to the economy. However, a higher value of 𝑁𝑁𝑁𝑁, associated with a
rightward shift of 𝑌𝑌 𝑑𝑑 requires higher 𝑞𝑞. This real depreciation of the terms of trade reduces
the purchasing power of the real wage 𝑤𝑤 (which is in terms of domestic goods but
households also want to buy imports), implying a leftward shift of labour supply 𝑁𝑁 𝑠𝑠 and
causing a leftward shift of the output supply curve 𝑌𝑌 𝑠𝑠 . Qualitatively, this does not change
the direction of our earlier results but implies smaller adjustments of 𝑁𝑁𝑁𝑁 and GDP 𝑌𝑌 are
needed to restore balance-of-payments equilibrium 𝑟𝑟 = 𝑟𝑟 ∗ . Hence, we shall usually ignore
this effect in practice.
10.6 Exchange rates
Our analysis of the current account and GDP in an open economy has so far made no
reference to exchange rates or monetary issues more generally (the terms of trade in
Section 10.5 is a relative price of two goods, a real variable). An exchange rate is a relative
price of different currencies in the foreign-exchange market. Exchange rates specify the
international values of currencies, so in order to study them, we must reintroduce money
into our analysis. We will consider both a model of the ‘long run’ with flexible prices based
on the treatment of money in Chapter 6 and a ‘short run’ model with sticky prices based on
the new Keynesian approach to macroeconomics from Chapter 8.
Suppose there are two currencies in the world, one for the domestic economy and one for
the foreign economy. The price of domestic goods in terms of domestic currency is 𝑃𝑃 and
the price of foreign goods in terms of foreign currency is 𝑃𝑃∗ . The exchange rate 𝑒𝑒 between
the domestic and foreign currencies, or to be precise, the nominal exchange rate is defined
as the domestic-currency price of a unit of foreign currency. Note that a higher value of 𝑒𝑒 is
a depreciation of the domestic currency with this definition.
Given the nominal exchange rate 𝑒𝑒 and the prices 𝑃𝑃 and 𝑃𝑃∗ we can calculate the implied real
exchange rate, denoted by 𝑞𝑞. This is defined as the price of a unit of foreign goods in terms
of domestic goods. The real exchange rate 𝑞𝑞 is calculated by comparing the domestic price
𝑃𝑃 of domestic goods to the price of foreign goods 𝑃𝑃∗ converted into domestic currency 𝑒𝑒𝑃𝑃 ∗:
𝑞𝑞 =
𝑒𝑒𝑃𝑃∗
𝑃𝑃
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10.6.1 Purchasing power parity
In the long run with flexible prices, one theory of exchange rates builds on the ‘law of one
price’. This states that identical goods should sell at the same price in different locations
after adjusting for prices being quoted in different currencies.
We develop this theory going back to the case where the goods bought and sold in each
country around the world are homogeneous. If a unit of goods can be bought in foreign
markets at price 𝑃𝑃∗ , this can be imported at cost 𝑒𝑒𝑃𝑃 ∗ in terms of domestic currency.
Therefore, the flexible domestic price 𝑃𝑃 should adjust so that it and exchange rate 𝑒𝑒 satisfy
𝑃𝑃 = 𝑒𝑒𝑃𝑃∗ , implying a real exchange rate of 𝑞𝑞 = 1, so the price of goods is the same in both
countries after adjusting for different currencies. In this case, we say that purchasing power
parity (PPP) holds. Note that the argument treats all goods as tradable with no shipping
costs or tariffs, and assumes perfectly competitive markets.
10.6.2 Equilibrium exchange rates in the long run
We now analyse the equilibrium exchange rate with flexible prices, assuming that PPP holds.
The assumption of flexible prices and PPP makes this case more plausible for the long run
than the short run.
We treat the real equilibrium of economy as independent of monetary policy because prices
are flexible. This is exactly (or approximately) justifiable in some circumstances as discussed
in Chapter 6. For a small open economy with perfect capital mobility, 𝑟𝑟 = 𝑟𝑟 ∗ and real GDP 𝑌𝑌
is found using the real dynamic model from Section 10.4, which is at the point on the 𝑌𝑌 𝑠𝑠
curve where 𝑟𝑟 = 𝑟𝑟 ∗ (with 𝑁𝑁𝑁𝑁 adjusting so 𝑌𝑌 𝑑𝑑 intersects 𝑌𝑌 𝑠𝑠 at that point).
For monetary variables, prices and exchange rates must PPP, so 𝑃𝑃 = 𝑒𝑒𝑃𝑃∗ and the foreign
price level 𝑃𝑃∗ is exogenous in a small open economy. There is also the Fisher equation for
the nominal interest rate 𝑖𝑖 = 𝑟𝑟 ∗ + 𝜋𝜋, where 𝜋𝜋 is the domestic inflation rate and 𝑟𝑟 ∗ is the
foreign and domestic real interest rate.
Since the nominal exchange rate 𝑒𝑒 gives the value of one currency in terms of another, we
need to describe monetary policy to determine the equilibrium exchange rate. We start
with the case where there is monetary policy autonomy, meaning that the domestic central
bank is not required to intervene in the foreign-exchange market. We say there is a ‘floating’
or ‘flexible’ exchange rate. In contrast, in a fixed-exchange rate system, we will see that
domestic monetary policy lacks autonomy.
Monetary policy is specified as an exogenous path of the money supply 𝑀𝑀 𝑠𝑠 = 𝑀𝑀. We can
analyse the determinants of the price level 𝑃𝑃 and inflation rate 𝜋𝜋 using the approach from
Chapter 6. Given the equilibrium price level 𝑃𝑃 resulting from domestic monetary policy, and
the foreign price level 𝑃𝑃∗ , the exchange rate 𝑒𝑒 is found from the PPP equation as follows:
𝑒𝑒 =
𝑃𝑃
𝑃𝑃∗
Another way to understand this is to consider that the exchange rate 𝑒𝑒 adjusts to clear the
money market, with PPP then determining the price level 𝑃𝑃 = 𝑒𝑒𝑃𝑃∗ . Substituting 𝑃𝑃 = 𝑒𝑒𝑃𝑃∗
and 𝑖𝑖 = 𝑟𝑟 ∗ + 𝜋𝜋 into the money demand curve 𝑀𝑀𝑑𝑑 = 𝑃𝑃𝑃𝑃(𝑌𝑌, 𝑖𝑖) derived in Section 6.4, the
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money demand function in terms of the exchange rate is 𝑀𝑀𝑑𝑑 = 𝑒𝑒𝑃𝑃∗ 𝐿𝐿(𝑌𝑌, 𝑟𝑟 ∗ + 𝜋𝜋). Given 𝑌𝑌,
𝑟𝑟 ∗ and 𝜋𝜋, this rises proportionally with 𝑒𝑒. Higher 𝑒𝑒 means a depreciation of the domestic
currency that raises domestic prices according to PPP, which increases the need for money
to carry out a given real quantity of transactions.
Figure 10.17 plots the money demand curve with the nominal exchange rate 𝑒𝑒 on the
vertical axis. The demand curve is upward sloping, analogous to the upward-sloping money
demand function in terms of the price level 𝑃𝑃 that were seen in Section 6.4. The money
supply curve is vertical because the quantity of money 𝑀𝑀 𝑠𝑠 is exogenous. The equilibrium
nominal exchange rate 𝑒𝑒 is found at the point of intersection between 𝑀𝑀𝑑𝑑 and 𝑀𝑀 𝑠𝑠 .
Figure 10.17: Equilibrium exchange rate
10.7 Exchange-rate regimes
A government can choose to let the exchange rate of its currency float – a ‘flexible’
exchange-rate regime – or intervene to fix the exchange rate. There are also intermediate
exchange-rate regimes between these extremes.
Under a flexible-exchange rate regime, the market determines the exchange rate 𝑒𝑒, which
will depend on the stance of monetary policy among other things. With a fixed-exchange
rate regime, there is a target value 𝑒𝑒̅ of the exchange rate and the government or central
bank intervenes in the foreign-exchange market to ensure 𝑒𝑒 = 𝑒𝑒̅.
A fixed exchange-rate regime depends on the government or central bank maintaining
sufficient foreign-exchange reserves (foreign-currency assets) to support the value of the
domestic currency if needed and being willing to accumulate more foreign-exchange
reserves if necessary. Foreign-exchange market interventions lead to the domestic money
supply 𝑀𝑀 𝑠𝑠 being endogenous.
10.7.1 Intervention in the foreign-exchange market
Suppose foreign-exchange market intervention is done by the central bank. The central
bank’s liabilities are domestic currency in circulation – the supply of fiat money 𝑀𝑀 𝑠𝑠 – and it
holds the following types of assets:
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•
•
Domestic government bonds (acquired through open-market operations)
Foreign-exchange reserves (acquired through foreign-exchange market
intervention).
A sale of foreign currency by the central bank to purchase domestic currency shrinks the
stock of foreign-exchange reserves and the domestic money supply 𝑀𝑀 𝑠𝑠 . The central bank
must have acquired sufficient foreign-currency assets in the past to do this. A sale of
domestic currency by the central bank to purchase foreign currency expands 𝑀𝑀 𝑠𝑠 and the
stock of foreign-exchange reserves. In principle, there is no limit to the extent of such sales
of domestic currency.
It is possible for the central bank to ‘sterilise’ its foreign-exchange market interventions
through open-market operations in the domestic bond market that leave 𝑀𝑀 𝑠𝑠 unchanged, for
example, buying government bonds with newly created money after shrinking the money
supply after selling foreign-exchange reserves. If this does not happen, the intervention is
said to be ‘unsterilised’ and results in a change in the domestic money supply 𝑀𝑀 𝑠𝑠 .
10.7.2 A shock to foreign prices
We illustrate how foreign-exchange market intervention is necessary to support a fixed
exchange rate following a shock. For example, suppose there is a rise in prices 𝑃𝑃∗ in foreign
markets. The equation 𝑒𝑒 = 𝑃𝑃/𝑃𝑃∗ indicates this implies a fall in the exchange rate 𝑒𝑒
consistent with PPP, so there is pressure for the domestic currency to appreciate. A sale of
domestic currency is necessary to maintain the fixed exchange rate 𝑒𝑒 = 𝑒𝑒̅.
In the money-market diagram shown in Figure 10.18, a rise in 𝑃𝑃∗ pivots the money demand
curve 𝑀𝑀𝑑𝑑 rightwards. An increase of the money supply 𝑀𝑀 𝑠𝑠 is needed to keep the
equilibrium exchange rate at 𝑒𝑒 = 𝑒𝑒̅, which corresponds to a purchase of foreign-currency
assets by the central bank by selling domestic currency. Observe that sterilisation of the
foreign-exchange intervention would not be effective – only an unsterilised intervention
shifts 𝑀𝑀 𝑠𝑠 and achieves 𝑒𝑒 = 𝑒𝑒̅. With the exchange rate 𝑒𝑒 = 𝑒𝑒̅ fixed, higher 𝑃𝑃 ∗ implies a higher
domestic price level 𝑃𝑃 = 𝑒𝑒𝑃𝑃∗ . The fixed exchange rate causes inflation in foreign markets to
be ‘imported’ to the domestic economy.
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Figure 10.18: Foreign exchange intervention with rise in foreign price level
We now look at what difference it makes if the country has a flexible exchange rate. As
before, the increase in foreign prices 𝑃𝑃∗ implies fall in exchange rate 𝑒𝑒 = 𝑃𝑃/𝑃𝑃∗ consistent
with PPP, leading to pressure for an appreciation of the domestic currency. This is shown in
Figure 10.19, where the rise of 𝑃𝑃∗ pivots 𝑀𝑀𝑑𝑑 rightwards. With domestic monetary policy
unchanged, i.e. with no shift of the money supply curve 𝑀𝑀 𝑠𝑠 , the exchange rate falls from 𝑒𝑒1
to 𝑒𝑒2 . There is an appreciation of the domestic currency and no foreign-exchange
intervention occurs to prevent this. Observe that the nominal exchange rate 𝑒𝑒 adjusts so
that 𝑃𝑃 = 𝑒𝑒𝑃𝑃 ∗ is unaffected (money-market equilibrium 𝑀𝑀 𝑠𝑠 = 𝑀𝑀𝑑𝑑 = 𝑃𝑃𝑃𝑃(𝑌𝑌, 𝑟𝑟 ∗ + 𝜋𝜋) is
reached at the same price level 𝑃𝑃). Inflation in foreign markets does not cause a rise in
domestic prices 𝑃𝑃 under a flexible exchange-rate regime.
Figure 10.19: Flexible exchange rate with rise in foreign price level
10.7.3 Monetary policy autonomy with a flexible exchange rate
With a flexible exchange rate policy, the equilibrium value of the exchange rate 𝑒𝑒 is left to
float, leaving the country free to choose its own monetary policy. For example, suppose the
central bank chooses to make a one-off permanent increase of the money supply 𝑀𝑀 𝑠𝑠 . The
effects of this are shown in Figure 10.20.
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Figure 10.20: Exchange rate with increase in the money supply
The money supply curve 𝑀𝑀 𝑠𝑠 shifts to the right, and the equilibrium exchange rate 𝑒𝑒 rises in
proportion to the increase in 𝑀𝑀 𝑠𝑠 , implying a depreciation of the domestic currency.
Purchasing power parity 𝑃𝑃 = 𝑒𝑒𝑃𝑃∗ implies the price level 𝑃𝑃 also rises in proportion to 𝑀𝑀 𝑠𝑠 . As
no further inflation expected, there is no shift of 𝑀𝑀𝑑𝑑 through higher 𝑖𝑖 = 𝑟𝑟 ∗ + 𝜋𝜋 and there
are no real effects because prices are fully flexible. This is the open-economy equivalent of
the money neutrality result from Section 6.9. Monetary policy autonomy is still valuable to
the extent that it gives monetary policy control over domestic prices and the inflation rate.
10.8 Open-economy sticky-price model
Nominal exchange rate fluctuations change international relative prices when the prices of
goods are sticky in units of money. This means that exchange rates have implications for
competitiveness, and it is important to account for this effect when studying an open
economy in the short run. We do this by adapting the new Keynesian model from Chapter 8
to an open economy.
10.8.1 Competitiveness and output demand
The domestic and foreign economies produce different types of goods. The prices of these
goods in terms of a country’s own currency are 𝑃𝑃 and 𝑃𝑃∗ . The nominal exchange rate of the
two currencies is 𝑒𝑒, defined as the domestic currency price of foreign currency. We suppose
there is no ‘pricing to market’ – no price discrimination between the domestic and foreign
markets. Domestic goods are sold at price 𝑃𝑃/𝑒𝑒 in the foreign market and foreign goods are
sold at price 𝑒𝑒𝑃𝑃 ∗ in the domestic market. The relative price of foreign goods to domestic
goods, the terms of trade 𝑞𝑞, is:
𝑃𝑃∗
𝑒𝑒𝑃𝑃∗
𝑞𝑞 =
=
𝑃𝑃⁄𝑒𝑒
𝑃𝑃
Since the domestic and foreign goods are different products, this does not have to equal 1
even though the law of one price holds for individual goods. Higher 𝑞𝑞 means a real
depreciation of the terms of trade, increasing the competitiveness of domestically produced
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goods. We assume that net exports 𝑁𝑁𝑁𝑁 are positively related to competitiveness 𝑞𝑞 as
explained in Section 10.5.
Both goods prices 𝑃𝑃 and 𝑃𝑃∗ are sticky in this model for the reasons discussed in Section 8.1.
As explained in the new Keynesian model from Section 8.2, real GDP 𝑌𝑌 becomes demand
determined, i.e. the output supply curve 𝑌𝑌 𝑠𝑠 is not relevant in the short run. Unlike the real
dynamic model in Section 10.4, only the output demand curve 𝑌𝑌 𝑑𝑑 is relevant here.
Even though prices of goods are sticky in the currency of the country in which they are
produced, competitiveness 𝑞𝑞 can change because of fluctuations in the nominal exchange
rate 𝑒𝑒. A depreciation of domestic currency, higher 𝑒𝑒, increases competitiveness, higher 𝑞𝑞,
which raises net exports 𝑁𝑁𝑁𝑁. This shifts output demand 𝑌𝑌 𝑑𝑑 = 𝐶𝐶 + 𝐼𝐼 + 𝐺𝐺 + 𝑁𝑁𝑁𝑁 to the right.
Domestic demand in 𝑌𝑌 𝑑𝑑 depends on the real interest rate 𝑟𝑟 as usual. If prices are expected
to remain constant, the Fisher equation implies the real interest rate 𝑟𝑟 equals the nominal
interest rate 𝑖𝑖 controlled by monetary policy. Although producer prices are sticky, consumer
prices can change with the exchange rate but we assume the share of spending on imports
is sufficiently small that this effect can be ignored when thinking about expected inflation.
10.8.2 Balance-of-payments equilibrium and uncovered interest parity
Assume investors are risk neutral, meaning they care only about the expected returns on
assets. For both domestic and foreign nominal bonds to be willingly held, both must offer
the same expected return. With perfect capital mobility, this means that balance-ofpayments equilibrium requires expected returns on domestic and foreign assets are equal.
Each unit of domestic currency invested in a domestic bond pays off 1 + 𝑖𝑖 units of domestic
currency in the future, where 𝑖𝑖 is nominal interest rate. To invest in a foreign bond with
nominal interest rate 𝑖𝑖 ∗ , a domestic investor converts domestic currency into foreign
currency, receiving 1/𝑒𝑒 per unit of domestic currency, which pays off (1 + 𝑖𝑖 ∗ )/𝑒𝑒 units of
foreign currency in the future when invested in the foreign bond. This is expected to be
worth (1 + 𝑖𝑖 ∗ )𝑒𝑒 ′ /𝑒𝑒 units of domestic currency in future, where 𝑒𝑒′ is the expected future
nominal exchange rate. The two bonds have the same expected return expressed in the
same currency when:
1 + 𝑖𝑖 = (1 + 𝑖𝑖 ∗ )
𝑒𝑒′
𝑒𝑒
Note that 𝑖𝑖 cannot be directly compared to 𝑖𝑖 ∗ because these are nominal returns in different
currencies. Let ∆ = (𝑒𝑒 ′ − 𝑒𝑒)/𝑒𝑒 denote the expected change in the nominal exchange rate 𝑒𝑒,
where a positive value of ∆ means an expected depreciation of the domestic currency. The
condition for domestic and foreign bonds to offer the same expected return is:
1 + 𝑖𝑖 = (1 + 𝑖𝑖 ∗ )(1 + ∆)
This is known as the uncovered interest parity (UIP) condition.
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Assuming ∆ is not too large relative to 𝑖𝑖 ∗ , the UIP equation becomes 𝑖𝑖 ≈ 𝑖𝑖 ∗ + ∆. Ignoring the
approximation, with perfect capital mobility and risk-neutral investors, balance-of-payments
equilibrium requires that the following uncovered interest parity condition holds:
𝑖𝑖 = 𝑖𝑖 ∗ + ∆
This says that the domestic nominal interest rate 𝑖𝑖 must be equal to the foreign nominal
interest rate 𝑖𝑖 ∗ plus any expected depreciation ∆ of the domestic currency relative to the
foreign currency. Given 𝑖𝑖 ∗ , the domestic interest rate 𝑖𝑖 must rise to compensate investors
for a positive expected depreciation ∆ of the domestic currency.
Assuming consumer-price inflation is approximately zero so 𝑟𝑟 ≈ 𝑖𝑖, we can state the UIP
condition in terms of domestic and foreign real interest rates 𝑟𝑟 and 𝑟𝑟 ∗ :
𝑟𝑟 = 𝑟𝑟 ∗ + ∆
10.8.3 Monetary policy and the exchange-rate regime
The equilibrium of the economy is found using a goods market diagram with 𝑟𝑟 and 𝑌𝑌 on the
axes. There is an output demand curve 𝑌𝑌 𝑑𝑑 representing aggregate demand, which includes
net exports. The position of 𝑌𝑌 𝑑𝑑 depends on the nominal exchange rate because a change in
𝑒𝑒 affects competitiveness 𝑞𝑞 and net exports 𝑁𝑁𝑁𝑁. Higher 𝑒𝑒, a depreciation of the domestic
currency, implies a rightward shift of 𝑌𝑌 𝑑𝑑 . Balance-of-payments equilibrium is represented by
a horizontal 𝐵𝐵𝐵𝐵 line drawn at 𝑟𝑟 = 𝑟𝑟 ∗ + ∆, which is the UIP condition. If no change in the
nominal exchange rate is expected (Δ = 0) then the 𝐵𝐵𝐵𝐵 line is 𝑟𝑟 = 𝑟𝑟 ∗ . Finally, as in the new
Keynesian model from Section 8.2, there is an 𝑀𝑀𝑀𝑀 line representing the central bank’s
monetary policy stance.
With a flexible exchange rate, the central bank has monetary policy autonomy, meaning it
can choose the position and shape of the 𝑀𝑀𝑀𝑀 line, which describes how it sets the domestic
nominal (and real) interest rate. In a closed economy, the outcome for real GDP is found
where 𝑌𝑌 𝑑𝑑 and 𝑀𝑀𝑀𝑀 intersect but, in an open economy, the outcome must also be consistent
with balance-of-payments equilibrium as represented by the 𝐵𝐵𝐵𝐵 line. This is reached
through adjustment of the exchange rate 𝑒𝑒.
In Figure 10.21, suppose domestic output demand 𝑌𝑌𝑎𝑎𝑑𝑑 = 𝐶𝐶 + 𝐼𝐼 + 𝐺𝐺 intersects 𝑀𝑀𝑀𝑀 at 𝑟𝑟𝑎𝑎 ,
which is above the 𝐵𝐵𝐵𝐵 line. The exchange rate appreciates (𝑒𝑒 falls), with lower
competitiveness reducing net exports and shifting 𝑌𝑌 𝑑𝑑 to the left until it intersects 𝑀𝑀𝑀𝑀 on
the 𝐵𝐵𝐵𝐵 line. This process determines equilibrium real GDP 𝑌𝑌0 and the exchange rate 𝑒𝑒0 .
Note that the 𝑀𝑀𝑀𝑀 line is drawn as upward sloping. With perfect capital mobility, an 𝑀𝑀𝑀𝑀
line that is horizontal or close to horizontal might require huge movements in the exchange
rate and real GDP to restore balance-of-payments equilibrium. For this reason, we assume
monetary policy is willing to adjust the nominal interest rate if the changes in real GDP
brought about by exchange-rate movements were too large.
If the government has a fixed exchange rate policy then the central bank must use monetary
policy to support the target 𝑒𝑒 = 𝑒𝑒̅ for the exchange rate. If the exchange rate is kept fixed at
𝑒𝑒̅ then there is no shift of the output demand curve 𝑌𝑌 𝑑𝑑 because of 𝑒𝑒. Moreover, if the
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exchange rate is expected to remain fixed in the future then there is no expected
depreciation or appreciation (∆ = 0) and the 𝐵𝐵𝐵𝐵 line is at 𝑟𝑟 = 𝑟𝑟 ∗. Given the requirement of
balance-of-payments equilibrium, Figure 10.22 shows that keeping the exchange rate fixed
requires the central bank to adjust its stance of monetary policy so that the 𝑀𝑀𝑀𝑀 line
intersects 𝑌𝑌 𝑑𝑑 on the 𝐵𝐵𝐵𝐵 line.
Figure 10.21: Open-economy sticky-price model with flexible exchange rate
With a fixed exchange rate, having the 𝑀𝑀𝑀𝑀 line shift to intersect 𝐵𝐵𝐵𝐵 and 𝑌𝑌 𝑑𝑑 is equivalent to
having a horizontal 𝑀𝑀𝑀𝑀 line in the same position as the 𝐵𝐵𝐵𝐵 line. Intuitively, this represents
the subordination of monetary policy to defending the fixed exchange rate, with the central
bank being forced set the same interest rate 𝑟𝑟 ∗ as the foreign economy.
Figure 10.22: Open-economy sticky-price model with fixed exchange rate
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Box 10.5: The trilemma
There is an important limitation on the government’s policy choices in an open
economy known as the ‘trilemma’. This says it is only possible to have two of the
following three:
1. A fixed exchange rate: a constant nominal exchange rate 𝑒𝑒 = 𝑒𝑒̅
2. Monetary policy autonomy: an independent choice of the money supply or
interest rate to pursue domestic macroeconomic objectives
3. Free capital flows: perfect capital mobility, where the UIP condition 𝑖𝑖 = 𝑖𝑖 ∗ +
∆ holds.
With the second and third of these policies, Figure 10.23 shows how a shift in the
monetary policy stance to meet some domestic objective requires that the
exchange rate 𝑒𝑒 must adjust so 𝑌𝑌 𝑑𝑑 intersects 𝑀𝑀𝑀𝑀 and 𝐵𝐵𝐵𝐵, with free capital flows
tying down the position of the 𝐵𝐵𝐵𝐵 line if no change in the exchange rate is
expected. It is therefore not possible to have the first policy of keeping the
exchange rate fixed as well.
Figure 10.23: Monetary autonomy and free capital flows
Choosing the first and third policies of a fixed exchange rate 𝑒𝑒 = 𝑒𝑒̅ and free capital
flows, the position of the output demand curve is tied down at 𝑌𝑌 𝑑𝑑 (𝑒𝑒̅). We have
seen that the stance of monetary policy must adjust so that the 𝑀𝑀𝑀𝑀 line intersects
𝐵𝐵𝐵𝐵 and 𝑌𝑌 𝑑𝑑 , as shown in Figure 10.24. This means it is not possible to have the
second policy choice, an independent monetary policy, that is, the freedom to
choose the position of the 𝑀𝑀𝑀𝑀 line.
Making the first and second policy choices of a fixed exchange rate and monetary
policy autonomy, Figure 10.25 shows that it is not possible simultaneously to be on
the perfect capital mobility 𝐵𝐵𝐵𝐵 line in general. In the absence of large private
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capital flows, a sterilised foreign-exchange intervention could succeed in keeping
the exchange rate fixed at 𝑒𝑒 = 𝑒𝑒̅ given the level of 𝑁𝑁𝑁𝑁, while monetary policy
autonomy is preserved because the intervention is sterilised and does not affect
the money supply. However, capital controls would be needed to prevent a large
adjustment of the financial account 𝐹𝐹𝐹𝐹 to the interest rate differential 𝑟𝑟 − 𝑟𝑟 ∗ , so
the policy choice of free capital flows would have to be abandoned.
Figure 10.24: Fixed exchange rate and free capital flows
Figure 10.25: Monetary autonomy and fixed exchange rate
If the government wants to maintain a fixed exchange rate and allow capital
mobility, are there policies that can substitute for an independent monetary policy?
Figure 10.26 shows how fiscal policy, for example, an increase in government
expenditure 𝐺𝐺, can shift the output demand curve 𝑌𝑌 𝑑𝑑 for a given exchange rate 𝑒𝑒̅.
This affects real GDP because the stance of monetary policy must adjust to ensure
the 𝑀𝑀𝑀𝑀 line intersects both the 𝑌𝑌 𝑑𝑑 curve
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and the 𝐵𝐵𝐵𝐵 line (which can be shown as the 𝑀𝑀𝑀𝑀 line overlapping the 𝐵𝐵𝐵𝐵 line).
Thus, fiscal policy becomes highly effective under a fixed exchange rate regime.
Note that fiscal policy would not be effective under a flexible exchange rate regime
unless accommodated by a change in monetary policy.
Figure 10.26: Fiscal policy with fixed exchange rate and free capital flows
Box 10.6: Currency crises
Currency crises occur when a fixed exchange rate policy is put under pressure by
large capital flows and collapses. There are various reasons why currency crises can
occur and here we look at two examples.
Unsustainable macroeconomic policies
Suppose the government has a fixed exchange-rate policy but is also increasing the
money supply, for example, to obtain seigniorage revenues. The central bank
continues to buy government bonds, increasing 𝑀𝑀 𝑠𝑠 . We analyse these policies in an
economy with flexible prices using the model from Section 10.6.
The left panel of Figure 10.27 shows the rightward shift of the money supply curve
𝑀𝑀 𝑠𝑠 curve is inconsistent with an equilibrium exchange rate at the target 𝑒𝑒 = 𝑒𝑒̅.
There is pressure for 𝑒𝑒 to rise, i.e. for a depreciation of the domestic currency. The
central bank intervenes by selling foreign-exchange reserves and buying back
domestic currency, which reducing 𝑀𝑀 𝑠𝑠 and shifts the money supply curve back to
the left. This maintains the fixed exchange rate but the central bank is left holding
fewer foreign-exchange reserves but more domestic government bonds. If this
continues, eventually, foreign-exchange reserves will be depleted.
With limited foreign-exchange reserves and further expansions of the money
supply, a crisis eventually occurs. Once reserves are depleted, the fixed exchange
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rate cannot be defended any longer and the exchange rate will float. With the
money supply continuing to increase, the exchange rate will depreciate over time,
with PPP implying this will lead to ongoing inflation. A higher inflation rate 𝜋𝜋 raises
𝑖𝑖 = 𝑟𝑟 ∗ + 𝜋𝜋 and reduces money demand as shown in the right panel of the figure. At
the time of the collapse of the fixed exchange rate, there is sharp reduction in
demand for domestic currency, triggering a large terminal drop in the central
bank’s foreign-exchange reserves when the crisis occurs.
Figure 10.27: Foreign-exchange interventions to defend fixed exchange rate
In this example, the government is following macroeconomic policies that are
ultimately inconsistent with a fixed exchange rate. A currency crisis is inevitable due
to bad policy.
Self-fulfilling currency crises
Under some circumstances, currency crises can have self-fulfilling features. This
means that a fixed exchange rate collapses owing to this being expected to happen
but, if this is not expected to happen, there is no fundamental reason for the fixed
exchange rate to fail. We explore this possibility using the model with sticky prices
from Section 10.8.
Suppose a currency crisis is expected to happen, implying a positive depreciation
∆> 0 of the domestic currency is expected. As shown in Figure 10.28, this shifts the
𝐵𝐵𝐵𝐵 line upwards from 𝑟𝑟 ∗ to 𝑟𝑟 ∗ + ∆, so a higher interest rate is needed to avoid
capital outflows and maintain balance-of-payments equilibrium. In order to defend
the fixed exchange rate 𝑒𝑒 = 𝑒𝑒̅, the central bank must tighten monetary policy,
shifting the 𝑀𝑀𝑀𝑀 line upwards and moving along the 𝑌𝑌 𝑑𝑑 (𝑒𝑒̅) curve. Thus, maintaining
a fixed exchange rate when a currency crisis is expected requires a higher real
interest rate 𝑟𝑟, which results in lower real GDP 𝑌𝑌.
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Figure 10.28: Expected collapse of fixed exchange rate
If the government is not willing to sacrifice real GDP and employment to defend the
fixed exchange rate then it fails to shift the 𝑀𝑀𝑀𝑀 line, which would cause the
exchange rate to depreciate to restore balance-of-payments equilibrium. However,
this justifies the initial expectation that the fixed exchange rate will be abandoned,
so the shift in expectations of the exchange rate is not irrational. This shows the
collapse of the fixed exchange rate regime can be a self-fulfilling prophecy.
However, if no collapse is expected then the 𝐵𝐵𝐵𝐵 line does not shift, the central
bank does not need to change monetary policy, and neither GDP nor the exchange
rate change, confirming the belief of no collapse is not irrational, so there are
multiple equilibria in this example.
We see that currency crisis can be self-fulfilling when the government will not
prioritise the defence of a fixed exchange rate if that jeopardises other objectives
such as GDP.
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