Sanet - ST 1073566714
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KnygaA Course in
Modern
Macroeconomics
Pablo Kurlat
A Course in
Modern
Macroeconomics
Pablo Kurlat
Copyright
c 2020 Pablo Kurlat
https://sites.google.com/view/pkurlat
All rights reserved
ISBN: 9781073566716
For Pam, Felix and Emilia
Introduction
This book started as a collection of my teaching notes for the ECON 52 course that I taught at Stanford
University. The objective of that course, and of this book, is to introduce students to the ideas and way of
thinking of modern macroeconomics in a uni ed way that is accessible with a moderate amount of maths.
Modern macroeconomics emphasizes explicit microeconomic foundations and general equilibrium analysis,
combined with various kinds of constraints and market imperfections. When preparing the class I thought
none of the existing textbooks conveyed this in a way that I liked, so I prepared my own notes, which then grew
into this book. While mostly self-contained, the book is probably most useful to students who are familiar
with the basics of multivariable calculus and have taken a calculus-based microeconomics class.
The book is meant to be followed approximately in order. Later chapters contain many references to
material in earlier chapters. However, not everything from the early chapters is indispensable for what comes
next. Chapters 1 and 6-9 are the main core, but even within them everything that has to do with risk, search,
adjustment costs, or in nite-horizon problems can be skipped without compromising what comes later.
At the end of each chapter there is a series of exercises. Some are relatively direct applications of the
material in the chapter and others are more open-ended or explore topics related to but not directly covered
in the chapter. Several of the exercises can serve as the basis for a lecture, a class discussion, or the analysis
of a historical episode. The exercises vary in di culty but are intended to be relatively hard overall.
The list of interesting areas of macroeconomics is vast and growing, and the book does not aim to be
comprehensive. Probably the biggest omission is that it mostly deals with closed-economy issues and models,
so there is little discussion of exchange rates, capital ows or international trade. Somewhat relatedly, the
book is more US-centric than I would like. In many ways the US economy is not like that of a typical country,
but it is very well studied, so many of the ideas are discussed in terms of US evidence. The book is also biased
towards my own idiosyncratic tastes. For instance, there is more than one might expect on money supply and
demand, which is a somewhat old-fashioned topic, and on how to de ne living standards.
I would like to thank several generations of teaching assistants and students for their input. Alina Arefeva,
Juliane Begenau, Daniel Bennett, Ricardo de la O, Adem Dugalic, Guzman Gonzalez-Torres, Pavel Krivenko,
Krishna Rao, Yevgeniy Teryoshin, Daniel Layton Wright and Victoria Zuo all contributed to making this
book possible. I would also like to thank researchers who allowed me to include their ndings in the book,
in particular Daron Acemoglu, Daniel Andrei, Regis Barnichon, Robert Barro, Chad Jones, David Lagakos,
William Mann, Nathalie Moyen and Valerie Ramey. I also bene ted greatly from discussions with people who
have taught similar courses, especially Sebastian Di Tella, Pete Klenow, Monika Piazzesi and Alp Simsek.
7
On my website (https://sites.google.com/view/pkurlat) you can nd some complementary materials: data
and codes for some of the exercises, clari cations, corrections, downloadable gures, etc. If you have com-
ments, questions, suggestions, criticisms, corrections, or praise (especially praise!), you can reach me at
pkurlat@gmail.com.
8
Contents
I GDP and Living Standards 13
1 GDP 15
1.1 GDP Accounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2 Making Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Beyond GDP 31
2.1 The Human Development Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Beyond GDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
II Economic Growth 45
3 Basic Facts about Economic Growth 47
3.1 The Very Long Run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 The Kaldor Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Growth Across Countries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 The Solow Growth Model 53
4.1 Ingredients of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3 The Golden Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4 Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.5 Technological Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Theory and Evidence 75
5.1 The Kaldor Facts Again . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Putting Numbers on the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.3 The Capital Accumulation Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.4 Growth Accounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.5 TFP Dierences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
9
Contents
III Microeconomic Foundations 101
6 Consumption and Saving 103
6.1 Keynesian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2 Two Period Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3 Many periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.4 Behavioral Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7 Labor and Leisure 127
7.1 Measuring the Labor Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.2 Static Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.3 Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.4 A Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.5 Equilibrium in the Labor Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8 Investment 151
8.1 Present Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
8.2 Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.3 MPK and Investment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
9 General Equilibrium 165
9.1 Two-Period Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
9.2 First Welfare Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
9.3 In nite-Period Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
IV Money and In ation 189
10 Money 191
10.1 What is Money? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
10.2 The Supply of Money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
10.3 Changing the Supply of Money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
10.4 The Demand for Money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
11 The Price Level and In ation 205
11.1 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
11.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
11.3 Seignorage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
11.4 The Cost of In ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
10
Contents
V Business Cycles 223
12 Business Cycle Facts 225
12.1 What are Business Cycles? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
12.2 Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
12.3 Who cares? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
13 The RBC Model 241
13.1 A Two-Period Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
13.2 Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
13.3 Productivity Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
13.4 Other Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
13.5 Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
14 The New Keynesian Model 261
14.1 A Historical and Methodological Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
14.2 Monopoly Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
14.3 Sticky Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
14.4 IS-LM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
14.5 Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
14.6 Simpli ed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
14.7 Partially Sticky Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
15 Fiscal and Monetary Policy 289
15.1 Fiscal Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
15.2 Monetary Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
15.3 Monetary Policy Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
15.4 The Liquidity Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
11
PART I
GDP and Living Standards
This part of the book explores the meaning and measurement of living standards.
In Chapter 1 we look at one of the main variables that macroeconomists care about: the
Gross Domestic Product, or GDP. We go over its de nition, the accounting conventions
used to measure it and some of the conceptual issues behind the accounting conventions.
In Chapter 2 we study some of the shortcomings of GDP as a measure of living
standards, some alternatives that have been proposed, and how one can use economic
theory as a guide to improved measurements.
13
CHAPTER 1
GDP
1.1 GDP Accounting
One of the basic questions economists are interested in, when analyzing a country, is how much is produced
in that country in a given year. The basic measure of this is a country's gross domestic product (GDP). The
idea is simple: to record the value of everything that is produced in the country in a year and add it up. GDP
accounts can be constructed in three dierent (but equivalent) ways, based on measuring production, income,
or expenditure.
Production Income Expenditure
Agriculture & Mining Employee compensation Consumption
Construction Proprietor's income Investment
Manufacturing Rental income Government
Services Corporate pro ts Net exports
Government Interest income
Depreciation
1
In each of the three measures we can choose how much detail to go into. For instance, in the produc-
tion approach we don't need to lump all services together. We can instead separate healthcare, education,
entertainment, retail trade, etc., into separate accounts.
The accounting identity from the expenditure approach is sometimes written algebraically as:
Y =C +I +G+X −M (1.1.1)
where Y stands for GDP, C stands for consumption, I stands for investment, G stands for public goods and
services, X stands for exports and M stands for imports. We'll return to this equation many times.
The three measures of GDP are equal to one another. The logic is that whenever goods and services
are produced, whatever is spent on them will also constitute someone's income. A good description of how
the accounts are constructed can be found at https://www.bea.gov/resources/methodologies/measuring-the-
1
Including depreciation as a form of income doesn't seem to make much sense, but see below.
15
1.1. GDP Accounting
economy. Table 1.1 shows measures of GDP for the US for 2017 computed according to each of the three
approaches.
Table 1.1: US GDP in 2017 according to the three methods. Figures in billions of dollars. Source: BEA.
Production Income Expenditure
Agriculture 169 Employee Comp. 10,421 Consumption 13,321
Mining 269 Corporate Pro ts 1,807 Investment 3,368
Utilities 308 Proprietor's income 1,501 Govt. spending 3,374
Construction 781 Rental income 730 Exports 2,350
Manufacturing 2,180 Depreciation 3,116 Imports -2,929
Wholesale + Retail 2,261 Interest Income 768
Transport 609 Taxes 1,286
Media 1,051 Statistical discrep. -143
Finance + Insurance 1,466
Real Estate 2,591
Professional services 2,426
Educ. + Health 1,700
Arts + Entertainment 805
Other services 416
Government 2,454
Total 19,485 Total 19,485 Total 19,485
In this section we will try to understand the logic of GDP accounts through a series of examples.
Example 1.1.
Amy, who is self-employed, produces lettuce in her garden and sells it to Bob for $1. Bob eats it.
Production Income Expenditure
Agriculture 1 Proprietor's income 1 Consumption 1
The production approach measures the value of the lettuce that was produced, which is $1.
The income approach looks at how much income is derived from productive activities. In our example,
Amy obtains $1 of income from selling the lettuce. Since Amy is self-employed, we classify her income as
proprietor's income (self-employed people are sometimes called sole proprietors).
The expenditure approach looks at what the production was used for. Here the lettuce was consumed.
Value Added
Production typically takes place in several stages. Someone's output becomes somebody else's input. We want
to measure the value at the end of the production process, avoiding double counting.
16
1.1. GDP Accounting
Example 1.2.
Amy is the shareholder of a corporation that operates a fertilizer plant. The corporation hires Bob to work
in the plant and pays him a wage of $0.50. The corporation sells the fertilizer to Carol, a self-employed
farmer, for $0.80. Carol uses it to produce lettuce, which she sells to Daniel for $1. Daniel eats the
lettuce.
Production Income Expenditure
Manufacturing 0.8 Wages 0.5 Consumption 1
Agriculture 0.2 Pro ts 0.3
Prop. income 0.2
Total 1.0 Total 1.0 Total 1.0
Here it would be a mistake to add the value of the fertilizer to the value of the lettuce because the fertilizer
was used up in producing the lettuce. The value added in the production of lettuce is just the dierence
between the value of the lettuce and the value of the fertilizer. Notice that doing things this way makes total
GDP consistent across the three methods.
Forms of Investment
Investment can take dierent forms, with one thing in common: it involves producing something that will be
used for production in future periods.
Example 1.3.
1. General Electric builds an X-ray machine, which it sells to Stanford Hospital for $1,000. The cost
of producing it is made up of workers' wages of $600.
Production Income Expenditure
Manufacturing 1,000 Wages 600 Investment 1,000
Corporate pro ts 400
Total 1,000 Total 1,000 Total 1,000
2. Zoe builds a house with her bare hands and sells it to Adam for $1,000.
Production Income Expenditure
Construction 1,000 Proprietor's income 1,000 Investment 1,000
3. Dunder Mi in produces 500 tons of white paper worth $40,000 and stores them in its warehouse
while it waits for customers to buy them. The cost of producing them is made up of workers' wages
of $50,000.
17
1.1. GDP Accounting
Production Income Expenditure
Manufacturing 40,000 Wages 50,000 Investment 40,000
Corporate pro ts -10,000
Total 40,000 Total 40,000 Total 40,000
In part 1, the X-ray machine will be used to produce X-ray scans in the future. In part 2, the house will
be used to produce shelter (housing services) in future periods. Equipment (as in part 1) and structures
(as in part 2) are the largest components of investment.
Part 3 is a little bit more subtle. The paper was produced to be sold and used, not in order to be left
lying around in the warehouse. However, sometimes production and use are not synchronized. The goods
that are held in order to be used later are called inventories and include nished goods but also inputs and
half- nished products that will be part of a further productive process. Since inventories are something that
will be useful in the future, an increase in inventories is also a form of investment. In the example, we make the
interpretation that Dunder Mi in has invested in having paper available for when it manages to make sales.
When the paper is nally sold and inventories go back to zero we will record that as negative investment.
Example 1.4.
Warren invests $100,000 in shares of General Motors.
Production Income Expenditure
0 0 0
This example is a bit tricky because the word investment is used somewhat dierently in macroeconomics
than in other contexts. In the example above there is no investment in the macroeconomic sense. There is a
change in ownership but no new productive assets are created.
Durables
The distinction between consumption and investment is not always so clear. Above we saw that residential
construction is an investment because it will produce housing services in the future. By that logic, many
things could be considered investments. A refrigerator produces refrigeration services for a long time after
it's produced. Similarly for cars, electronics, clothes, etc. How does GDP accounting treat these?
Example 1.5.
1. Panasonic builds a TV (at zero cost) and sells it to Bob for $500.
Production Income Expenditure
Manufacturing 500 Corporate pro ts 500 Consumption of Durables 500
2. Bob watches the TV he bought last year.
18
1.1. GDP Accounting
Production Income Expenditure
Production 0 Income 0 Expenditure 0
3. A property developer builds a house (at zero cost) and sells is to Claire for $100,000.
Production Income Expenditure
Construction 100,000 Corporate pro ts 100,000 (Residential) Invest- 100,000
ment
4. Claire lives in the house she bought last year. In the rental market, a similar house would cost
$7,000 a year.
Production Income Expenditure
Housing services 7,000 Imputed owner-occupier 7,000 Consumption 7,000
income
Conceptually, what's going on with the TV and with the house is very similar. They are produced one year
but are enjoyed for a long time thereafter. However, GDP accounting conventions treat them dierently. For
most durable goods, we just treat them as being consumed at the moment of purchase, though sometimes we
classify consumption of durables separately from consumption of nondurables (e.g., food and entertainment)
just to emphasize that they are not quite the same. For housing, since it's such a large category and it's very
long-lived, we treat the initial construction as an investment and try to measure the ow of housing services
even when an homeowner is buying those housing services from herself.
Foreign Countries
GDP includes everything produced within the country, whether it's eventually used by residents or non-
residents. Conversely, goods produced abroad are not included in GDP even if they are consumed in the
country.
Example 1.6.
A car manufacturer buys components from Japan for $10 and uses half of those components in the
production of a car, which it sells to Andy for $20. There are no other production costs. It stores the
rest of the components. Amy, who is self-employed, produces lettuce in her garden and sells it to Franz
(a foreigner) for $2.
Production Income Expenditure
Car Manufacturing 15 Corporate pro ts 15 Consumption 20
Agriculture 2 Proprietor's income 2 Investment 5
Exports 2
Imports -10
Total 17 Total 17 Total 17
19
1.1. GDP Accounting
The Government
The government is a major producer of goods and services. Many of those services are provided directly, so
there is no real price for them. In order to add them to GDP accounts, they are valued at whatever it cost to
produce them.
Example 1.7.
1. Ms. Jody teaches Kindergarten in Lucille Nixon Elementary School in Palo Alto for the entire year
and earns $85,000.
Production Income Expenditure
Public Education 85,000 Wages 85,000 Government 85,000
2. The City of Palo Alto hires the Los Angeles Philharmonic to play a free concert in Stanford Stadium.
The musicians are paid $65,000 and renting the stadium costs $20,000. Four people show up.
Production Income Expenditure
Public Concert 85,000 Wages 65,000 Government 85,000
Rental income 20,000
Total 85,000 Total 85,000 Total 85,000
Notice that GDP is the same in both examples, even though in one case the publicly provided service is
something people actually value a lot and in the other case it's not.
Example 1.8.
Jack collects his $20,000 pension from Social Security.
Production Income Expenditure
0 0 0
Here the government is spending $20,000 but it's not in order to produce public goods and services. In
terms of GDP accounting, this is just a transfer, which has no impact on any of the accounts.
Example 1.9.
The state of California builds a high-speed train from Merced to Bakers eld. It pays workers a billion
dollars to build it with their bare hands.
Production Income Expenditure
Railway construction 1 billion Wages 1 billion Government Invest- 1 billion
ment
This is an example of public investment: something the public sector does that will be useful in the
future. In the expenditure approach, do we classify it as Government Spending or as Investment? In
20
1.1. GDP Accounting
equation (1.1.1), it's included within G, but more detailed GDP accounts include a further breakdown of
G into government investment and government consumption. The previous examples were all government
consumption. This example is government investment.
Depreciation
Machines and buildings usually deteriorate over time, a phenomenon we call depreciation. GDP is gross
domestic product because it is measured before taking into account of depreciation.
Example 1.10.
Zak's Transport Co. owns a eet of taxis. They are all new at the beginning of the year, worth a total
of $1,000. A taxi depreciates completely in 5 years. During the course of the year the company pays its
workers $200 in wages, has no other costs, and collects $500 in fares.
Production Income Expenditure
Transp. Services 500 Corporate Pro ts Consumption 500
Revenue 500
Wages -200
Depreciation -200
Total Pro t 100
Wages 200
Depreciation 200
Total 500 Total 500 Total 500
1000
Since the taxis depreciate over 5 years, an estimate of the amount of depreciation is
5 = 200. When
the company computes its pro ts, it understands that its eet of vehicles has lost value over the course of
the year, so it subtracts the amount of depreciation. In order to compute GDP we want to get back to a
before-depreciation measure, so we add back depreciation. This makes the income-based measure of GDP
consistent with the production-based measure and the expenditure-based measure.
Depreciation plays an important role in the theory of economic growth that we'll study in Chapter 4.
Non-Market Activities
A lot of economic activity does not involve market transactions and is usually not included in GDP calculations.
We already saw an exception to this: we impute the production of housing services even for people who live
in their own home without conducting a market transaction. This particular exception is made so that GDP
does not vary when housing shifts between tenant occupancy and owner occupancy. (Note that the imputed
rent of owner-occupied housing accounted is almost 8% of US GDP.) Most of the time, however, we compute
the value of an activity only if it is sold in the market.
21
1.2. Making Comparisons
Example 1.11.
1. Mary mows Andy's lawn for $25. Andy takes care of Mary's kids for $25.
Production Income Expenditure
Gardening services 25 Self-employment 50 Consumption 50
Babysitting 25
Total 50 Total 50 Total 50
2. Andy mows his own lawn. Mary takes care of her own kids.
Production Income Expenditure
Production 0 Income 0 Expenditure 0
These two examples show that, even though the economic activity is basically the same in both cases,
national accounts treat them very dierently.
1.2 Making Comparisons
One of the things we often want to do is compare GDP, either across countries or within a country across
time. To do this we have to be a bit careful with the units of measurement. When we compute GDP, we just
add the value of everything produced in the country. If it's for the US, it will be in dollars. The problem
with this measure is that the amount of goods and services you can get for one dollar is not the same in every
country or in every time period, because the prices of goods and services are dierent.
For this reason we make a distinction between nominal and real GDP:
• Nominal GDP: the total value of goods and services produced, valued at whatever price they had at the
time they were produced.
• Real GDP: the total value of goods and services produced, valued in units such that the values are
comparable across time.
Real GDP
Example 1.12.
The country of Kemalchistan uses the dinar as its currency. GDP in the years 2017 and 2018, measured
by the production method, was as follows:
22
1.2. Making Comparisons
2017 2018
Manufact. (50 balls, 10 dinar each) 500 Manufact. (50 balls, 20 dinar each) 1,000
Educ. (10 teachers, 100 dinar each) 1,000 Educ. (10 teachers, 200 dinar each) 2,000
Total 1,500 Total 3,000
GDP, measured in dinar, doubled between 2017 and 2018 but the amount of goods and services was the
same in both years. The reason GDP increased is because all prices increased. Often we are interested in a
measure that tracks the changes in the total amount of stu that is produced and doesn't rise just because
prices have changed.
In the example above, it's clear that real GDP is the same in both years and we can express it as either
1,500 dinars in 2017 prices or 3,000 dinars in 2018 prices. Either way, GDP did not grow between the two
years. But the example is special in two ways:
1. The relative quantities of the dierent goods produced don't change between the two years.
2. All the prices change by the same amount.
When these conditions fail, the way to measure real GDP in a way that's comparable across years is less
obvious.
Example 1.13.
The country of Expandia uses the dollar. GDP in the years 2017 and 2018, measured by the production
method, was as follows:
2017 2018
Agric.(10 tons of wheat, $50 each) 500 Agric. (11 tons of wheat, $60 each) 660
Manufact. (1 computer, $1,000) 1,000 Manufact. (2 computers, $600 each) 1,200
Total 1,500 Total 1,860
How much has the real output of the economy of Expandia grown? We know that agricultural output
has expanded 10% (from 10 to 11) and manufacturing output has grown 100% (from 1 to 2). How should we
compute the total growth? There is more than one way to do it.
Alternative 1: base year prices
One approach is to choose a base year and measure the value of all goods at the prices they used to have
in the base year. In the example above, if we chose 2017 as the base year, we'd have the following gures for
real GDP for the year 2018:
2018, at 2017 prices
Agriculture (11 tons of wheat, $50 each) 550
Manufacturing (2 computers, $1,000 each) 2,000
Total 2,550
23
1.2. Making Comparisons
We'd say that real GDP in 2018 was $2,550 at 2017 prices. If we want to compute the rate of growth of
GDP, we would have
2, 550
growth = − 1 = 70%
1, 500
The general formula for computing real GDP this way is:
X
Yt = pi0 qit (1.2.1)
i
where:
• Yt is real GDP in the year t.
• pi0 is the price of a certain good i in the base year (which we call year 0)
• qit is the quantity of good i produced in year t
Alternative 2: nal year prices
This is exactly the same, except that the base year is the last one we look at rather than the rst one. In the
example above, this means recomputing GDP in the year 2017 at the prices of 2018:
2017, at 2018 prices
Agriculture (10 tons of wheat at $60 each) 600
Manufacturing (1 computers at $600 each) 600
Total 1,200
The general formula (1.2.1) still applies, it's just that we have changed what year we call year 0. With 2018
as the base year, we'd say that real GDP in 2017 was $1,200 at 2018 prices, and the rate of growth of GDP
is
1, 860
growth = − 1 = 55%
1, 200
Notice that the two formulas give us a dierent answer to the question how much did the economy grow
overall between 2017 and 2018? This is often the case. Using an earlier year as the base year gives a higher
rate of growth if the sectors that are expanding most (in the example, manufacturing) are those whose relative
price is falling, and vice versa.
Alternative 3: chained prices
Neither of the above alternatives is obviously preferred, so another option is to do something in between. The
idea is to:
1. Start from some base year 0
2. Compute real growth between year 0 and year 1 in two ways: at year 0 and year 1 prices
3. Average the two growth rates in some way
24
1.2. Making Comparisons
4. Compute real GDP in year 1 by adding the average growth rate to year-0 GDP
5. Repeat for years 2, 3, 4, etc.
The term chained comes from the fact that the estimate of real GDP in any given year will be the result of
a chain of calculations linking that year to the base year. In general formulas:
P
pit−1 qit
gtI =P i −1 growth based on initial year prices
i pit−1 qit−1
P
pit qit
gtF = P i −1 growth based on nal year prices
i it qit−1
p
0.5 0.5
gt = 1 + gtI 1 + gtF −1 average growth; this is a geometric average
Yt = Yt−1 (1 + gt ) real GDP one year ahead
This will result in a measure of GDP in chained prices of the base year.
Comparisons Across Countries and PPP
Suppose we want to compare GDP across countries.
Example 1.14.
In 2018, GDP in the US and Mexico were as follows:
United States Mexico
GDP 20.5 trillion dollars 23.5 trillion pesos
Population 327 million 127 million
GDP per capita 62,700 dollars per person 185,000 pesos per person
Suppose we wanted to ask: did US residents produce more output per person than Mexican residents in
2018? The gures above don't quite give us the answer because they are in dierent units: GDP in the US is
measured in dollars while GDP in Mexico is measured in pesos. How do we convert everything to the same
units?
One approach is to look up the exchange rate between the Mexican peso and the US dollar. On average
during 2018, you could trade one dollar for about 19 Mexican pesos in foreign exchange markets; equivalently,
you could trade one Mexican peso for 0.053 US dollars. Using this exchange rate, we can restate Mexican
GDP in US dollars as:
GDP in Foreign Country = market exchange rate × GDP in Foreign Country
(in dollars, at market (dollars per unit of foreign (in foreign currency)
exchange rates) currency)
25
1.2. Making Comparisons
Using this approach, we'd conclude that Mexico's GDP in 2014 was 1.24 trillion dollars, or 9,700 dollars
per person.
One drawback of this approach is that it doesn't take into account that, even after converting currencies,
prices are dierent in dierent countries. In other words, if you take one dollar, use it to buy Mexican pesos,
go to Mexico and go shopping, the amount of stu you'd be able to aord need not be equal to the amount
of stu you'd be able to aord if you had just stayed in the US. When we see that a country has low GDP
when converted at market exchange rates, it could mean that their output is low or that prices, converted to
dollars, are low. How do we distinguish between these possibilities?
One way to do it is to change the way we assign dollar values to goods produced in foreign countries.
Instead of measuring their value in local currency and converting to dollars at the market exchange rate, we
look up an equivalent good in the US, see its US price and value the foreign goods at their US price. In
formulas:
PN
GDP in Foreign Country = i=1 pU
i
S
× qi
(in dollars, at PPP)
where N is the number of dierent goods that we are adding up, pU
i
S
is the market price of good i in the
US and qi is the quantity of good i produced in the foreign country. This is known as the Purchasing Power
Parity or PPP approach because it aims to adjust for the fact that the purchasing power of a dollar is dierent
in dierent countries. In practice, PPP calculations are harder to do than converting GDP at market exchange
rates: one needs to gure out what US good is the correct equivalent to each foreign good, which is not so
easy because the goods available in each country are dierent. For Mexico, most estimates of PPP put its per
capita GDP at around 18,000 dollars, almost twice as high as using market exchange rates, re ecting the fact
that goods tend to be cheaper than in the US.
A byproduct of computing GDP at PPP is to de ne a PPP exchange rate. This is an answer to the
following question: what would market exchange rates need to be for GDP at market exchange rates and
GDP at PPP to coincide? In formulas:
GDP in Foreign Country ≡ PPP exchange rate × GDP in Foreign Country
(in dollars, at PPP) (dollars per unit of foreign currency) (in foreign currency)
or
GDP in dollars at PPP
PPP exchange rate ≡
GDP in foreign currency
If PPP exchange rates and market exchange coincide it means that on average goods cost as much in the
foreign country as in the US. For many years, The Economist magazine has computed a simple indicator of
PPP exchange rates: instead of looking for the exchange rate that would make goods overall cost the same in
the US and in foreign countries, they focus on a single good: the Big Mac. This has the advantage of being
highly standardized across countries.
2 The Big Mac index is simply
Big Mac price in US (dollars)
Big Mac exchange rate =
Big Mac price in Foreign Country (foreign currency)
2
Though not fully: the price of Big Macs includes the location, cleanliness, etc. of the McDonald's restaurant and these vary
across countries.
26
1.2. Making Comparisons
Exercises
1.1 Accounting
How does GDP accounting record the following events? For each of them, describe how they would
be computed in GDP accounts using the income method, the production method and the expenditure
method.
(a) A car manufacturer buys components from Japan for $1 to be used in production later on and stores
them at its warehouse.
(b) A car manufacturer buys components from Japan for $1 and uses half of those components in the
production of a car, that it sells to Andy for $2. It stores the rest of the components.
(c) An army battalion is deployed to the border to repel a threatened Canadian invasion. The soldiers
earn wages of $10,000 and use ammunition that the government buys for $5,000. The ammunition
is produced using $2,000 of imported steel and 100 hours of work, for which the workers were paid
$1,000.
(d) Walmart sells 1000 bottles of Coca-Cola for $1,500. It had previously paid $1,200 for them.
(e) A shipyard builds a cruise ship. It pays wages of $200,000, interest on loans (from US residents)
of $100,000 and $300,000 for imported raw materials. The ship is sold for $1,000,000 to a cruise
company. In the same year, the cruise company has revenue for $50,000 from operating cruises,
pays wages of $20,000 to its workers and has no other expenses. Half the cruise revenue comes from
tourists who reside in the United States and half comes from tourists who reside abroad.
(f ) The government collects $1000 in income taxes from Roger.
(g) Roger earns $4000 for working as a babysitter and pays $1000 in income taxes.
1.2 Comparisons Across Time and Across Countries
Suppose these are the prices (in US dollars) and quantities of goods A and B produced in the US in 2017
and 2018:
pA qA pB qB
2017 4 5 3 3
2018 1 10 4 2
(a) What was nominal GDP in 2017?
(b) What was nominal GDP in 2018?
(c) What was real GDP in 2018 at 2017 prices (computed using xed 2017 prices)? Using this measure,
how much did GDP grow between 2017 and 2018?
(d) What was real GDP in 2017 at 2018 prices (computed using xed 2018 prices)? Using this measure,
how much did GDP grow between 2017 and 2018? What explains the dierence between the two
measures?
27
1.2. Making Comparisons
(e) How much did GDP grow between 2017 and 2018 using the chain-weighted method?
In Thailand, prices (in Thai baht) and quantities in 2017 were:
pA qA pB qB
2017 30 1 10 2
(f ) What was nominal GDP in Thailand in 2017, expressed in baht.
(g) Suppose the exchange rate in 2017 was 25 baht per dollar. What was GDP in Thailand in 2017,
expressed in US dollars at market exchange rates?
(h) What was GDP in Thailand in 2017 at PPP? What accounts for the dierence between the market
exchange rate measure and the PPP measure?
(i) What was the PPP exchange rate between the baht and the dollar?
1.3 Chained GDP
The country of Fructus produces Apples, Bananas and Cherries. Its production statistics are given below:
Apples Bananas Cherries
quantity price quantity price quantity price
2000 100 40 50 30 200 20
2001 103 39 47 35 206 20
2002 107 39 47 38 213 19
2003 109 39 45 38 215 18
2004 112 40 46 41 224 18
2005 112 39 46 39 227 17
2006 114 40 47 36 237 16
2007 115 40 46 42 249 16
2008 119 40 43 44 255 16
2009 118 42 40 46 262 15
2010 116 43 41 44 268 15
2011 118 42 40 50 280 15
2012 117 42 41 50 289 15
2013 119 43 40 53 290 14
2014 120 44 40 56 297 13
2015 125 45 41 59 308 13
2016 128 45 38 66 320 13
2017 129 46 38 61 330 13
2018 131 47 37 64 344 13
2019 136 47 37 61 353 13
2020 143 47 34 61 361 12
(you can download them as an Excel spreadsheet from the book website)
28
1.2. Making Comparisons
(a) Compute a real GDP series at year-2000 prices using base year prices and using the chained method.
(b) Plot both series and comment on any dierences.
(c) What was the average growth rate according to each method?
1.4 Drugs and Prostitution
Read the following article: http://www.economist.com/news/ nance-and-economics/21603073-italys-
inclusion-illicit-activities-its- gures-excites-much-interest-sex. What do you think? Should drug pro-
duction and prostitution be included in the calculation of GDP?
1.5 Changes in Relative Prices
We saw that using an earlier year as the base year to compute real GDP results in a high rate of growth
if the sectors that are expanding most are those whose relative price is falling. Can you think of reasons
why that should be the case (i.e., economic forces that make the same types of goods become relatively
cheaper and be produced in higher quantities)? Can you think of reasons why the opposite should be the
case (i.e., economic forces that make the same types of goods become relatively expensive and be produced
in higher quantities)?
29
CHAPTER 2
Beyond GDP
GDP is an incomplete indicator of standards of living, and people have realized this for a long time. For
instance, here is a speech by Robert Kennedy in 1968:
Our gross national product, now, is over $800 billion dollars a year, but that gross national product
if we judge the United States of America by thatthat gross national product counts air pollution
and cigarette advertising, and ambulances to clear our highways of carnage. It counts special
locks for our doors and the jails for the people who break them. It counts the destruction of the
redwood and the loss of our natural wonder in chaotic sprawl. It counts napalm and counts nuclear
warheads and armored cars for the police to ght the riots in our cities. It counts Whitman's ri e
and Speck's knife, and the television programs which glorify violence in order to sell toys to our
children. Yet the gross national product does not allow for the health of our children, the quality
of their education or the joy of their play. It does not include the beauty of our poetry or the
strength of our marriages, the intelligence of our public debate or the integrity of our public
o cials. It measures neither our wit nor our courage, neither our wisdom nor our learning, neither
our compassion nor our devotion to our country, it measures everything in short, except that which
makes life worthwhile. And it can tell us everything about America except why we are proud that
we are Americans.
There have been eorts to construct broader measures of living standards that address some of the limitations
of GDP.
2.1 The Human Development Index
The United Nations has for some time constructed a measure called the Human Development Index (HDI).
The HDI takes as a starting point that, in addition to high output, two other things contribute to Human
Development: a long life and good education. One might argue for the inclusion of all sorts of other things
in the index but the ones that are included are not unreasonable. Furthermore, some of the other things that
one might consider including (the beauty of our poetry?) are much harder to measure.
31
2.1. The Human Development Index
The HDI is constructed as follows. First, construct indices of each of the three components:
Life Expectancy − 20
Life Expectancy Index =
85 − 20
1 Avg. school yrs. 25-year-olds Expected schools yrs. 5-year-olds
Education Index = +
2 15 18
log (GNP per capita) − log (100)
Income Index =
log (75, 000) − log (100)
The logic of the index is to convert each of the three categories into a number between 0 and 1. Life expectancy
for the healthiest countries is around 80 years and for the least healthy countries is around 40 years, so the
Life Expectancy Index of dierent countries will range somewhere between 0 and 1. Similarly, children spend
somewhere between 0 and 18 years of life in school and GNP per capita ranges between about 400 to 75, 000
US dollars (at PPP).
1
The HDI is a geometric average of the three indices:
p
3
HDI = Life Expectancy Index × Education Index × Income Index
Figure 2.1.1 shows a scatterplot of the HDI against GDP per capita. Each observation represents a dierent
country. The correlation between the two measures is 0.94. With few exceptions, countries that have high
GDP per capital also have a high HDI.
Fig. 2.1.1: GDP per capita
and the HDI for 2017. Source:
World Bank for GDP per
capita, UN for HDI.
One unsatisfying aspect of the HDI is that the scaling and weighting of the various factors is somewhat
1
The HDI uses GNP rather than GDP. The main dierence between them is the income that is produced in the country but
belongs to nonresidents, like interest paid on debt owed to foreigners. The dierence is usually small. Some countries' GNP
actually exceeds 75,000 dollars, so their Income Index is greater than 1.
32
2.2. Beyond GDP
arbitrary. What exactly is the HDI measuring? Why convert variables into indices? Why equal weights on
the three factors? We turn next to a study that builds a measure of well-being that is more rmly grounded
in economic theory.
2.2 Beyond GDP
Jones and Klenow (2016) propose a measure of living standards that is similar in spirit to the HDI but quite
dierent in methodology. Again, the idea is to construct a measure that includes important aspects of quality
of life that GDP does not capture. Jones and Klenow focus on the following variables:
• consumption (as opposed to production),
• the value of leisure and nonmarket production,
• life expectancy,
• inequality.
De ning Living Standards
What do we mean when we talk about living standards? We'll de ne them as the answer to the following
question. Suppose you take a random person (Jones and Klenow, who have a sense of humor, call him Rawls)
and invite him to spend a year living as a resident in one of two countries.
2 The rules of the experiment are
that, beforehand, Rawls will not know what speci c individual within that country he will be: young or old,
rich or poor, etc.. The two options are:
• some country whose living standards we are trying to measure, let's call it Utilia,
• a country that is exactly like the US except that everyone's consumption is multiplied by some number
λ.
What number λ would make Rawls indierent between Utilia and the rescaled US? λ will be our measure of
living standards in Utilia. You might recall from microeconomics that λ is known as an equivalent variation.
Being a random resident of Utilia is equivalent, in utility terms, to being a random resident of the US and
having your consumption multiplied by λ.
In order to measure λ we have to come up with a way to decide how much people care about all the
variables, other than consumption, that are dierent in dierent countries. We are going to assume that
Rawls evaluates the choice according to the following utility function:
c1−σ
2
u (c, l, a) = E ū + − θ (1 − l) a (2.2.1)
1−σ
Utility is a function of three variables:
2
They also analyze a hypothetical scenario where Rawls will spend his whole life in one of these countries, but we'll focus on
the single-year experiment.
33
2.2. Beyond GDP
• c is the level of consumption. As we'll see below, this is a random variable. Dierent people within both
the United States and Utilia consume dierent amounts. Rawls does not know who he will be so he
does not know how much he will end up consuming. The operator E is there to indicate that we need
to compute the expected value of this function of a random variable.
• l stands for leisure: it's the fraction of time that people devote to activities that are not counted in GDP.
• a stands for alive. This is also a random variable. It takes the value 1 if Rawls turns out to be alive
and 0 if he turns out to be dead. Since a multiplies the rest of the formula, it means Rawls will only get
utility if he is alive. The chances of this happening depend on the country's life expectancy.
Parameters ū, σ and θ govern how much people care about each of the variables. In order to evaluate utility
we are going to have to:
1. Find a way to put concrete numbers for ū, σ and θ. We try to nd examples of situations where people
actually make choices in which they trade o dierent goals, showing how much they care about each of
them.
2. Find data for many countries on the distribution of consumption, on life expectancy and on leisure.
3. Obtain our measure of welfare for each country by solving for λ in the following equation:
u λcU S , lU S , aU S = u cU tilia , lU tilia , aU tilia
(2.2.2)
Let's look at each issue in turn.
Consumption
GDP is a measure of production, not consumption. But what enters the utility function (2.2.1) is consumption.
The logic is that what determines people's standards of living is how much they get to consume, not how
much they are able to produce. The output that Utilia dedicates to investment will not give Rawls any utility
during the year he stays in Utilia. This doesn't mean it's wasted: it will bene t future residents of Utilia after
Rawls has left. Similarly, the output that Utilia exports will not give Rawls any utility, and, conversely, the
consumption goods that Utilia imports will give Rawls utility.
A slightly subtler question is how to treat government consumption. As we saw in Chapter 1, government
purchases of goods and services are counted the same regardless of how much people actually value them. It
could be that on average a unit of public goods gives Rawls a lot more utility than a unit of private goods
(he really likes to feel protected by police o cers), or that it gives him less. We are going to assume that on
average public and private consumption give the same utility. In terms of the GDP accounting identity (1.1.1)
we'll assume that what goes into the utility function is C + G.3
3
To be more precise, G includes both government consumption and government investment. We'll assume that the utility
function values C plus the part of G that is government consumption.
34
2.2. Beyond GDP
The Veil of Ignorance, Inequality and Risk Aversion
Philosophers refer to a hypothetical choice that people must make before knowing what place in society they
are going to occupy as a decision made behind the veil of ignorance. The philosopher John Rawls famously
argued that the way to determine what is just is to ask what sort of society people would organize from behind
the veil of ignorance.
4
From behind the veil of ignorance, Rawls faces risk. He knows that, once he enters the experiment to spend
a year in Utilia, he might turn out to be rich and enjoy high consumption or poor and have low consumption.
How do we model his attitude towards these possibilities? Suppose there are N people in Utilia. Rawls knows
how much each inhabitant of Utilia consumes and he knows that his experience in Utilia is going to look
like one of them, but he does not know which one. Therefore, behind the veil of ignorance, he measures his
expected utility by:
N
c1−σ
X 1 n 2
u(c, l, a) = ū + − θ (1 − l) a (2.2.3)
n=1
N 1−σ
1
where cn denotes the consumption of the nth resident of Utilia. Rawls believes there is a probability
N of him
ending up in each point of the distribution, so he just computes the average utility he will get across all of
these possibilities.
5 This way of modeling attitudes towards risk is known as expected utility theory because
it says that people evaluate uncertain prospects according to the expected utility that they will obtain.
Under this theory, Rawls's attitude towards risk is related to the concavity of the function u (c, l, a). Before
we go into the maths of why that is, an important warning: the important economic idea here is that (most)
people don't like risk. Concave functions are just a mathematical representation of this idea. It's wrong to say
people dislike risk because their utility function is concave. Instead, one should say: in economic models,
we describe people's preferences with concave utility functions to capture the fact that people dislike risk. Do
not put the mathematical cart before the conceptual horse!
Now let's look at the maths of utility functions and risk aversion. Hold l and a constant for now (for
instance, because there is no uncertainty about them) and imagine that u is just a function of c. Suppose that
there are two people in Utilia, one is rich and the other is poor, with consumption crich and cpoor respectively.
Therefore, from Rawls' point of view, there are two possible states of the world: one in which he is rich and
one in which he is poor. We say that Rawls is risk averse if
crich + cpoor u (crich ) + u (cpoor )
u > (2.2.4)
2 2
What does this mean? Suppose someone oered Rawls insurance. Instead of consuming crich in one state of
crich +cpoor
the world and cpoor in the other, Rawls gets to consume the average for sure. That would give him
2
4
Rawls also proposed an answer to the question of what people would choose behind the veil of ignorance, which he called the
dierence principle: society would be organized in whatever way bene ts the least-well-o person. You'll see in Exercise 2.10
that, according to standard economic theory, the dierence principle would follow from behind-the-veil-of-ignorance choices only
if people were extremely risk averse. Interestingly, Rawls himself was adamant that the dierence principle had nothing to do
with risk aversion.
5
Implicit in formula is the assumption that all inhabitants of Utilia enjoy the same leisure. Otherwise we would have to have
ln in the formula instead of the same value l for everyone. This is obviously not correct but it's hard to obtain data on inequality
in leisure, so assuming it's the same for everyone is a simpli cation.
35
2.2. Beyond GDP
the utility in the left-hand-side of (2.2.4). Instead, in the actual world where he faces risk, his expected utility
is the right-hand-side of (2.2.4). Therefore (2.2.4) just says that Rawls, in expected-utility terms, would prefer
a world without risk. Generalizing from the example, Rawls is risk averse if, whenever c is uncertain,
u (E (c)) > E (u (c)) (2.2.5)
Inequality (2.2.5) holds whenever u is a concave function. Figure 2.2.1 illustrates this general principle.
E (c) is the midpoint between cRich and cP oor and u (E (c)) is just evaluating the function u at this point.
E (u (c)) is the midpoint between u (cRich ) and u (cP oor ). Since the function u is concave, E (u (c)) lies below
u (E (c)).
Fig. 2.2.1: Concavity of the
utility function and risk aver-
sion.
In function (2.2.1), the concavity of u is governed by parameter σ: when σ is high the function is very
concave, when σ approaches 0 the function is close to linear. If we take the derivative of utility with respect
to consumption, we get a formula for marginal utility:
6
u0 (c) = c−σ (2.2.6)
Figure 2.2.2 plots marginal utility for dierent values of σ. For all values of σ > 0, we have that marginal
utility is positive but decreasing, but it's decreasing faster for higher values of σ. This gives us another way
6 c1−σ
This assumes that a = 1, i.e. it's the marginal utility of consumption for those who are alive. Notice that if σ < 1, then
1−σ
c1−σ
is a positive number and if σ > 1, then
1−σ
is a negative number. Whether utility is positive or negative doesn't mean much
because we don't really have an interpretation of what the level of utility means: we just care about what a utility function says
about how people compare dierent alternatives. Regardless of whether σ>1 or σ < 1, marginal utility is positive. For σ = 1,
c1−σ
the function
1−σ
is not well de ned. However, its properties approach those of the function log (c). For instance, formula (2.2.6)
works ne if we just adopt the convention that when σ=1 the function becomes log (c).
36
2.2. Beyond GDP
of thinking about the relationship between risk aversion and the shape of the utility function. An individual
who faces risk will have a lot of consumption in some states of the world and less consumption in others.
If the utility function is very concave, this means that the dierence in marginal utility between high and
low consumption states of the world will be large and the individual would have a strong preference to make
consumption more even between the dierent states of the world, i.e. the individual will be very risk averse.
Fig. 2.2.2: Marginal utility
schedule for dierent values of
σ.
So far we've established that dierent values of σ can be used to represent dierent attitudes towards risk.
The next step is to decide what value of σ captures people's actual attitudes towards risk. One obvious caveat
to this analysis is that dierent people have dierent attitudes towards risk, so at best we will nd a value of
σ that roughly represents the behavior of some average person.
How can we measure people's attitudes towards risk? We need to nd environments where people are
actually trading o higher average consumption for more risk. Two situations in which people make this sort
of decision are in making nancial investments and buying insurance.
On average, risky investments like the stock market give higher returns than safer investments like US
government bonds, so someone who invested their wealth in risky investments would obtain higher average
consumption than someone who chose safer investments. However, there are states of the world where risky
investments turn out poorly and lead to very low consumption. One way to measure people's attitude towards
risk is to look at their investments: to what extent are they willing to bear risk in exchange for a higher
average consumption? Several studies have measured σ by doing exactly that. Friend and Blume (1975) were
among the rst to do so. We'll think more about risky investments in Chapter 8.
When someone buys insurance, they are moving consumption across dierent states of the world. Take the
example of re insurance. There is a state of the world in which my house burns down. If I had no insurance,
in that state of the world I would have to lower consumption in order to pay for the repairs on my house.
37
2.2. Beyond GDP
An insurance contract lets me pay the insurance company some money in the state of the world where my
house is ne in order to get the insurance company to pay me when my house burns down. Typically, buying
insurance makes my average consumption go down because on average the premium I pay is higher than the
bene ts I collect: that's how the insurance company covers administrative costs and makes pro ts. However,
if I'm risk averse I'll still be willing to buy insurance: it lets me consume more in those states of the world
where I need it the most. One way to measure people's attitudes towards risk is to measure the extent to
which they decide to buy insurance, an approach used by Szpiro (1986) among others.
There is a fair amount of disagreement about what the right value of σ is; estimates range from about 1
to about 10 (Jones and Klenow use σ=1 as a baseline, near the less-risk-averse end of the range of empirical
estimates). Part of the reason why estimates of σ dier is that people's attitudes towards risk depends to some
extent on the context where they are making this choice. Indeed, some researchers argue that this means the
expected utility model of risk attitudes is unsatisfactory.
In terms of comparing standards of living across countries, σ is what determines how much Rawls cares
about inequality. From Rawls' point of view, a very unequal country is risky. If Rawls is very risk-averse
(as represented by high σ ), a more egalitarian society may look more attractive to him, behind the veil of
ignorance, than a more unequal society that is richer on average.
The Value of Life Expectancy
Behind the veil of ignorance, Rawls does not know how old he'll be. Let's assume his age is a number randomly
drawn from 0 to 100. If his age turns out to be higher than life expectancy in the country he's going, he'll be
dead; otherwise he'll be alive. According to (2.2.1), if he turns out to be dead he'll get zero utility and if he
c1−σ 2
turns out to be alive he'll get ū + 1−σ − θ (1 − l) . In this formula, everyone who is alive gets ū in addition
to however much utility they get from c and l. Therefore the value of ū governs how attractive it is to live in
a society with high life expectancy.
How can we measure people's preferences for high life expectancy? We need to nd environments in which
people trade o years of life against higher consumption. We can nd evidence on this in how much people
are willing to pay for safety features in cars or in the extra money that people demand in exchange for doing
dangerous jobs. In these situations people have to choose between lower consumption but probably a longer
life (get a car with airbags, work as a librarian) or higher consumption but probably a shorter life (don't pay
for airbags, work as a drug dealer). By observing what choices people make at various prices we get a sense
of how they are willing to trade these o.
The Value of Leisure and Non-Market Production
There are many alternative uses of people's time. Some result in output that is counted in GDP and some
do not. But even activities that are not counted in GDP can contribute to utility. We saw some examples of
this in Chapter 1: cleaning one's own house, cooking for friends or taking care of one's own children are all
non-market activities that nevertheless produce something valuable. Moreover, people also enjoy time spent
in pure leisure activities like reading books or watching TV. In equation (2.2.1), the variable l stands for the
fraction of time that people, on average, spend on all these non-market activities, so 1−l is the fraction of
38
2.2. Beyond GDP
time spent at a job counted in GDP. The parameter θ governs how much people dislike working in the market
sector (notice that there's a negative sign) or, equivalently, how much utility they derive from the time they
spend in non-market activities.
How can we measure θ? We need to nd instances of people trading o higher consumption against more
free time. A direct source of evidence on this is in people's choices of how much to work: at what age they
enter the labor force, when they retire, how many holidays they take, how many hours per week they work,
etc.
7 Under the preferences given by (2.2.1), higher values of θ imply that people will choose to work less
and have more non-market time. Since we can measure how much time people on average spend on market
and non-market activities, we can estimate θ by asking what the value has to be to match our empirical
observations.
Data
Once we've settled on values for ū, σ and θ, we need to get actual data from Utilia to plug into formula (2.2.1).
Ideally, we would need to have data on:
1. the consumption of every individual in Utilia,
2. the average fraction of time that residents of Utilia spend working,
8
3. life expectancy in Utilia.
Life expectancy is the easiest, because most countries measure it relatively reliably. The consumption of every
individual, of course, is impossible to know. However, many (but not all) countries conduct surveys where they
ask a lot of households about their income or their consumption. These surveys are not always as accurate as
we would like, but at least they give a rough estimate of what the distribution of consumption looks like. As
to the fraction of time worked, the quality of data varies by country. Some countries have detailed time-use
surveys while others just report employment rates but not hours of work per employed person.
Results
Figure 2.2.3 shows a scatterplot of GDP per capita relative to the US on the horizontal axis and λ, obtained
using formula (2.2.2), on the vertical axis. Clearly the two measures are very highly correlated, although not
identical. This means that just looking at GDP per capita as a measure of welfare is not so bad (or it could
be that some other variable that is not included in the derivation of λ is important).
There are some interesting patterns. Some Western European countries like France look better in the
welfare measure than in GDP per capita. This is because they have higher life expectancy, more leisure, and
less inequality than the US. Rich East Asian countries like South Korea, Hong Kong, and especially Singapore
look worse in terms of welfare than in GDP per capita. This is mostly because they have low consumption
relative to GDP: they produce a lot but dedicate a large fraction to investment and net exports. This is also
true of oil-rich countries like Kuwait and Saudi Arabia (which are also very unequal). Many Sub-Saharan
7
This is valid as long as we believe that people are actually choosing how much to work. We'll return to this when we study
labor markets in Chapter 7.
8
Recall that we are assuming that everyone gets the average amount of leisure.
39
2.2. Beyond GDP
Fig. 2.2.3:Relative GDP and
welfare. Source: Jones and
Klenow (2016).
African countries look worse in welfare than in GDP per capita, especially some not-so-poor ones like South
Africa and Botswana. In large part this is due to low life expectancy (which itself is the result of the AIDS
epidemic), though inequality plays a role as well.
Exercises
2.1 Comparing the HDI and the Jones & Klenow Welfare Measure
Download the UN data that goes into building the HDI from http://hdr.undp.org/en/data and the Jones
& Klenow Data from http://web.stanford.edu/~chadj/papers.html#rawls. For each country, construct:
(a) Relative GDP:
GDP per capita
GDP per capita in the US
(b) Welfare-to-GDP:
λ
Relative GDP
(c) HDI-to-GDP:
HDI
Relative GDP
and plot a scatterplot of Welfare-to-GDP against HDI-to-GDP. What do each of these ratios measure?
Is the impression we get from the Jones & Klenow measure very dierent from the one we get from
the HDI? What countries look better under each measure and why?
40
2.2. Beyond GDP
2.2 Other Things that Matter
The welfare measure λ takes into account data on consumption levels, inequality, leisure and mortality.
(a) Name one other variable that might be an important determinant of welfare that is not included in
standard GDP calculations.
(b) What data would you need in order to gure out how much weight to give to this variable? Describe
how you would use that data to come up with the correct way to include the variable in question
in the welfare calculation. What choices that people make might reveal how much they care about
this variable? Don't worry too much about the feasibility of the data-collection procedure, but think
carefully about why the observed choices would be informative.
2.3 Healthcare
When we calculate consumption, one of the (many) categories of consumption is healthcare services.
(a) Look up how much healthcare is consumed in the United States per year for a recent year. State the
total dollar amount and also what fraction of GDP and what fraction of consumption is accounted
for by healthcare.
A good place to look for this data is the National Income and Product Accounts (NIPA) at https:
//apps.bea.gov/iTable/index_nipa.cfm. Browse around a little to get a sense of how the NIPA data
is presented, and nd the correct place to look up this particular fact.
(b) Suppose that we are calculating welfare in the style of Jones and Klenow, taking into account the
impact of life expectancy on utility. Should we therefore subtract consumption of healthcare services
from our measure of consumption? What do you think?
2.4 Construction and the Armed Forces
Suppose we compared welfare in two otherwise identical countries. Country A dedicates 50% of GDP to
maintaining a large army. Country B dedicates 50% of GDP to building houses. In which of the two
countries would welfare be higher, according to the measure used by Jones and Klenow. What do you
think of this?
2.5 Measuring the Value of Life Expectancy
Would each of the following observations be useful in determining how much to weigh life expectancy in
measured welfare? Explain.
(a) Observing how life expectancy varies across people with dierent income levels within a country.
(b) Observing how life expectancy varies across countries.
(c) Observing how much people are willing to pay for funerals.
(d) Observing the dierence in house prices in neighborhoods with dierent murder rates.
2.6 The Value of Life Expectancy
In the US, average consumption per capita is about $35,000. Life expectancy is 79 years, so in the Jones
41
2.2. Beyond GDP
and Klenow experiment, Rawls would have a 79% chance of being alive. In Bolivia, average consumption
per capita is $3,700 and life expectancy is 68 years. Assume that there is no inequality in either country
so that everyone who is alive gets the average level of consumption. Assume the following utility function:
c1−σ
u(c, a) = ū + a
1−σ
with σ = 0.5 and ū = 48.1.
(a) Suppose you made a US resident the following oer: you can buy a health plan that costs x dollars
per year and will extend your lifetime by one year. What is the price x that would make them
indierent between getting the health plan or not?
(b) How much would a Bolivian resident be willing to pay for such a plan?
(c) Write down the the special case of equation (2.2.1) that applies to this example and solve for λ for
Bolivia. Don't replace any numbers yet, leave it in terms of aU S , aBol , cU S , cBol , σ and ū.
(d) Replace the values of aU S , aBol , cU S , cBol , σ and ū to nd a value for λ. How does it compare to
cBol
cU S ? Why?
2.7 Measuring Risk Aversion
The students of Concave University are all identical in their ability and preferences, which are well de-
scribed by the function:
c1−σ
u(c) = E
1−σ
Two employers recruit CU graduates. Stabilis oers each graduate a xed salary of $50,000. Lotteris
instead oers them a base salary of $20,000 plus a bonus scheme that depends on their performance,
which is entirely driven by luck. The bonus is either $10,000 or $100,000, with equal probability. The job
itself is the same in both rms. About half the CU graduates choose to work for Stabilis and half prefer
Lotteris, and they all say they found it hard to choose because both oers were similarly attractive. What
value of σ is consistent with their decisions?
2.8 Inequality and Risk Aversion
Compare the following two countries. Both have a population of 100. Within each country, we label
individuals in order of increasing consumption. In country A, the consumption of individual j is:
cA (j) = 100 + 8j
while in country B the consumption of individual j is:
cB (j) = 200 + 4j
(a) Plot the consumption patterns of each country, with an individual's label j (which ranges from 1 to
100) on the horizontal axis and their consumption on the vertical axis.
42
2.2. Beyond GDP
(b) Compute per capita consumption in each country. [Note: if you want, you can approximate sums
with integrals]
(c) Suppose the utility function in both countries is:
c1−σ
u (c) =
1−σ
For what values of σ is expected utility higher in country A? Interpret your results.
2.9 Risk Aversion and Welfare in Western Europe
If instead σ = 1, Jones and Klenow had set a higher value of σ, what would this do to measured welfare
in Western European countries? Explain.
2.10 Risk Aversion and the Dierence Principle
Suppose a individual is trying to evaluate a society from behind the veil of ignorance. He knows that he
can either be rich or poor, with equal probability. Expected utility is
u (cRich ) + u (cP oor )
E [u (c)] =
2
where the utility function u is
c1−σ
u (c) =
1−σ
Let's now de ne an indierence curve. For any utility level Ū , the indierence curve plots all the combi-
nations of cRich and cP oor such that expected utility is Ū .
(a) Take the following equation that de nes an indierence curve:
1 c1−σ
Rich 1 c1−σ
P oor
Ū = +
21−σ 21−σ
and solve for cRich . This will tell you what value of cRich corresponds to each value of cP oor in the
indierence curve with level Ū .
(b) Plot (using Excel, Matlab or some other software that produces nice graphs) a map of indierence
curves for the following values of σ:
√
i. σ = 0.5. Plot three curves, for Ū = 2, 10 and 4 respectively
ii. σ = 2. Plot three curves, for Ū = −1, −0.4 and −0.25 respectively.
Note that when σ>1 utility values will be negative, but that's OK.
Also note that
1 cP oor 1−σ
Ū >
2 1−σ
is impossible, i.e. even if cRich = ∞ expected utility will be lower than Ū , so when you are
1
1−σ
plotting this indierence curve, whenever you set cP oor to the left of 2Ū (1 − σ) you should
set cRich = ∞.
43
2.2. Beyond GDP
iii. σ = 5. Plot three curves, for Ū = −0.25, − 544 and −4−5 respectively.
10 −10
4
iv. σ = 11. Plot three curves, for Ū = −0.1, − 10 11 and − 4 10 respectively.
In all cases, have cP oor on the horizontal axis and cRich on the vertical axis and let the range of the
horizontal axis be [0, 10] and the range of the vertical axis be [0, 10].
(c) Notice that as σ becomes large, the indierence curves start to look like right angles. Explain how
this relates to Rawls's dierence principle.
44
PART II
Economic Growth
This part of the book looks at economic growth.
In Chapter 3, we start by reviewing some facts about long-run economic growth.
In Chapter 4, we study the Solow growth model, which provides a theory of
how and why economic growth takes place.
In Chapter 5, we look at how the model can be used to make sense of the evi-
dence and how the evidence can be used to test, quantify and apply the lessons from
the model.
45
CHAPTER 3
Basic Facts about Economic Growth
3.1 The Very Long Run
Let's start by looking at the distant past. Following Maddison (2001), a number of economists have attempted
to measure GDP per capita for many countries going back hundreds of years. Of course, this is very hard
and involves quite a bit of guesswork, but we have some clues. First of all, there is some minimum level of
consumption (the subsistence level) below which people starve, so we know that in all societies that didn't
starve GDP per capita must have been at least that. Estimates of how much that is vary, but they are in
the order of about 400 dollars a year at current prices, close to what is nowadays considered extreme poverty.
Beyond this, before we had proper national accounts, we had pieces of data on things like people's average
height (from bones), the total number of livestock, crop yields or total output of speci c industries like iron
that can be used to piece together rough estimates of GDP.
The left panel of Figure 3.1.1 shows the evolution of GDP per capita in the UK in the very long run.
1
We focus on the UK because it has the best data but also because it was the rst country to show fast and
sustained economic growth, starting in the early 19th century. The rst fact that emerges from Maddison's
data is that, even before 1800, GDP per capita in the UK was well above subsistence levels, and growing
slowly. These are somewhat controversial points among economic historians, some of whom believe GDP per
capita was stagnant and closer to subsistence. The second fact that emerges is that something happened in
the 19th century that led to an acceleration in the rate of economic growth (on this there is less disagreement).
This change is known as the Industrial Revolution since one of the things that took place at the time was
a shift in production from agriculture to industry. We don't have a de nitive answer as to what caused the
the industrial revolution and why it rst took place in the UK in the 19th century, but it's a major question
among economic historians.
Other countries went through a similar process of an acceleration in the rate of economic growth. They
started from levels of GDP per capita not too far from subsistence and, at dierent initial dates, started
1
You'll notice that most of the graphs in this section are in log scale. What this does is convert proportional dierences
into absolute dierences: the vertical distance between 1, 000 and 2, 000 is the same as the vertical distance between 10, 000 and
20, 000. Hence a constant proportional rate of growth (e.g. 2% per year) looks like a straight line, and the slope of this line
indicates the rate of growth. If we plotted GDP in regular scale, then a constant rate of growth would look like an exponential
function.
47
3.2. The Kaldor Facts
Fig. 3.1.1: GDP per capita in the UK and selected countries. Source: Bolt et al. (2018).
growing. The right panel of Figure 3.1.1 shows some examples.
3.2 The Kaldor Facts
We turn now to some patterns that we observe in economies that are growing. Kaldor (1957) summarized
some of the main facts about economic growth in advanced economies. He called them remarkable historical
constancies and they became known as the Kaldor Facts. Let's have a look at some of those facts and ask
whether they are still approximately true, focusing on the US.
2
1. The rate of growth of GDP per capita is constant.
Figure 3.2.1 shows the evolution of GDP per capita in the US from 1800 to 2016. A straight line (in
log scale) seems to do quite well in describing how the US economy has grown for many decades. GDP
per capita has been growing at a rate of approximately 1.5% per year for a long time. Notice that while
1.5% doesn't seem like a lot, compounded over time it amounts to a huge increase in GDP per capita.
From 1800 to 2016, GDP per capita has become about 27 times higher.
2. The ratio of the total capital stock to GDP is constant
The capital stock is the sum of the value of all the machines, buildings, etc. that are currently available
for use in production. It's not an easy thing to measure. A standard way to do it is by keeping track
of investment and depreciation over time. Exercise 5.4 asks you to think more about this. Figure 3.2.2
shows the evolution of
K
Y = Capital
GDP
Stock over time. Consistent with the Kaldor Facts, this ratio has
2
Kaldor described them slightly dierently and in dierent order
48
3.2. The Kaldor Facts
Fig. 3.2.1: GDP per capita
in the US. Source: Bolt et al.
(2018).
remained more or less constant over time at about 3.2. This means that the total value of all the capital
that the US economy has accumulated is about the same as the economy produces in 3.2 years.
Fig. 3.2.2: Capital-to-output
ratio in the US. Source: Feen-
stra et al. (2015).
3. The shares of labor and capital income in GDP are constant
Recall from Chapter 1 the income method of measuring GDP. Let's take a simpli ed view of the types
49
3.2. The Kaldor Facts
of income and classify them into just two categories: labor income (that is earned for work done in the
current period) and capital income (that is earned by those who own some form of capital). Some forms
of income are easy to classify: workers' wages are labor income, corporate pro ts and real estate rents
are capital income. Others are a little bit trickier: is the income earned by small business owners a
reward for the work they do or for the investment they put into the business? One way of addressing
the issue is to leave proprietors' income our of the calculation entirely, which is equivalent to assuming
that the split between labor and capital income is the same in the sole proprietor sector as in the rest of
the economy. This method is not entirely satisfactory but it is often adopted. Figure 3.2.3 shows how
the labor share of GDP in the US has evolved over time. Until about 2000 or so, it seemed that this fact
continued to hold: the share of labor income in GDP was very stable at around 65%. More recently,
there has been a noticeable fall in this percentage: the share of GDP going to workers has fallen by
about 3 percentage points.
Fig. 3.2.3: The labor share of
GDP in the US. Labor income
is Compensation of Employ-
ees. Capital income is Corpo-
rate Pro ts + Rental Income
+ Interest Income + Depreci-
ation. Source: NIPA.
4. The average rate of return on capital is constant
By the rate of return on capital we mean how much income is earned by the owner of capital per unit
of capital that they own. This fact says that this has stayed constant over time. Strictly speaking, it's
not a separate fact since it's implied by facts 2 and 3. Let's see why:
3
Capital income
Return on capital ≡
Capital stock
Capital income
GDP
= Capital stock
GDP
3
This refers to the gross rate of return on capital. To get the net rate of return we should subtract depreciation from capital
income. If the depreciation rate is constant, the net return on capital will be constant too.
50
3.3. Growth Across Countries
Fact 2 says that the denominator is constant and fact 3 says the numerator is constant, so if these facts
are true then the return on capital must be constant too. Nevertheless, we state it as a separate fact
because the behavior of the rate of return on capital over time is an important aspect of many theories
of economic growth and it's useful to keep this fact in mind.
3.3 Growth Across Countries
Figure 3.3.1 shows the growth of countries of dierent initial income levels since 1960. Initially-rich countries
have growth rates that are quite similar to each other and near the middle of the range of other countries.
Among initially-poor countries there is a lot more variation. Some countries like South Korea (KOR) and
Botswana (BWA) have very rapid rates of growth, so that over time they are catching up to the living standards
in rich countries while some others like Congo (COD) and Madagascar (MDG) have very low or even negative
growth rates, meaning that they are falling further behind rich countries.
Fig. 3.3.1: Growth across
countries since 1960. Source:
Feenstra et al. (2015).
Exercises
3.1 The Past is a Foreign Country
Find the data compiled by Bolt et al. (2018) at https://www.rug.nl/ggdc/historicaldevelopment/maddison/
releases/maddison-project-database-2018. Starting from 1800, look up the US GDP per capita at intervals
of one decade. For each of these points in time, nd the country that currently has the closest GDP per
capita to the past US level. When did the US become richer than present-day India, present-day China,
and present-day Portugal?
51
3.3. Growth Across Countries
3.2 The Kaldor Facts in Other Countries
Look up GDP accounts for some countries other than the US (a good source is https://www.rug.nl/ggdc/
productivity/pwt/). Reproduce Figures 3.2.1-3.2.3 using data from those countries (try a rich country, a
middle-income country and a poor country). Do the Kaldor facts seem to hold for those countries as well?
3.3 Within-Region Convergence
Look up cross-country GDP data (a good source is https://www.rug.nl/ggdc/productivity/pwt/). For
each country, compute average GDP-per-capita growth 1960-2014. Produce a scatterplot of growth against
initial GDP per capita like Figure 3.3.1 but separately for countries in each of three regions of the world:
Europe, Latin America and Africa. Is it the case in any of these regions that initially-poor countries have
grown faster?
52
CHAPTER 4
The Solow Growth Model
Solow (1956) proposed a simple model that can help us to start to think about the process of economic growth.
4.1 Ingredients of the Model
Production Function
The rst ingredient of the model is a production function:
Y = F (K, L) (4.1.1)
Formula (4.1.1) says that the output of any productive process (denoted Y) depends on:
• K: the amount of capital (machines, buildings, etc.) dedicated to the production process.
• L: the amount of labor that is dedicated to the production process.
One way to interpret a production function is as a book of recipes: for any given combination of ingredients,
it says how much stu will be produced.
Example 4.1. One page of the recipe book says:
F 100 acres of Iowa land +1 tractor, 1, 000 hours of work
= 18, 000 bushels of corn
| {z } | {z } | {z }
K=$800,000 L=1,000 Y =$66,000
Another page of the book says:
F 1 garage in Palo Alto, 3, 000 hours of work by Stanford dropouts = 1 app
| {z } | {z } | {z }
K=$150,000 L=3000 Y =$100,000
53
4.1. Ingredients of the Model
We are going to assume that everyone in the country knows the production function; anyone can set up a
rm, hire L workers and K units of capital and obtain F (K, L) units of output. Furthermore, we are going
to make the following assumptions about the production function:
Assumption 4.1 (Constant Returns to Scale) .
F (λK, λL) = λF (K, L) for any λ>0
The standard justi cation for assuming constant returns to scale is that production processes can, at least
approximately, be replicated. If I have a factory that produces paint and I want to produce twice as much
paint, I build a replica of the original factory next to it, hire replicas of all the workers and I'm done.
Obviously, there are many objections to this argument. Maybe some factors of production (natural resources,
workers with speci c skills) are not easily replicable: this would push towards having decreasing returns to
scale. Alternatively, one could imagine that one large factory can be run more e ciently than two small ones
because not everything needs to be exactly duplicated: this would push towards having increasing returns to
scale. We are going to stick with the assumption of constant returns to scale: any productive process can be
exactly scaled up or down by increasing or decreasing the use of capital and labor in the same proportion.
We'll see that this assumption has profound implications.
1
Assumption 4.2 (Positive Marginal Product) .
FK (K, L) > 0
FL (K, L) > 0
Assumption 4.2 has a straightforward interpretation: adding additional workers or additional capital to a
productive process adds at least a little bit to total output.
Assumption 4.3 (Diminishing Marginal Product) .
FKK (K, L) < 0
FLL (K, L) < 0
Assumption 4.3 says that adding just one of the factors (workers without extra machines or machines
without extra workers) becomes less and less useful the more you do it.
Assumption 4.4 (Inada Conditions) .
1. limK→0 FK (K, L) = ∞
2. limK→∞ FK (K, L) = 0
1 ∂F (K,L) ∂ 2 F (K,L)
We adopt the following notation for partial derivatives: FK (K, L) ≡ ∂K
, FKK (K, L) ≡ ∂K 2
54
4.1. Ingredients of the Model
Assumption 4.4 is slightly more technical. It says that if there is very little capital then a little bit of
capital is extremely useful. Conversely, if there is a lot of capital then additional capital becomes almost
useless (because there are no workers to operate the additional machines). It's similar in spirit to Assumption
4.3 (diminishing marginal product) although mathematically one does not imply the other.
We'll see the role that each assumption plays later on.
One example of a production function that we'll often resort to is the so-called Cobb-Douglas production
function, shown in Figure 4.1.1:
Y = K α L1−α (4.1.2)
It's easy to verify that this satis es Assumptions 4.1-4.4. We'll see that parameter α in this formula has a
natural interpretation.
Fig. 4.1.1: The Cobb Douglas production function for α = 0.35.
Population and the Labor Supply
Assumption 4.5. The population grows at a constant, exogenous rate n: Lt+1 = (1 + n) Lt
Nowadays Assumption 4.5 is routinely made in a lot of work on economic growth but it's actually a very big
deal. Historically, the possibility that population growth might be endogenous and depend on living standards
(as is the case for wild animal populations) was a central preoccupation among economists. Exercise 4.5 asks
you to examine some of the ideas of the 19th century economist Thomas Malthus who wrote about this issue.
We are not going to model people's decisions over how much to work. We are going to assume that
everyone who is alive works, so L represents both the population and the labor force. When looking at data,
55
4.1. Ingredients of the Model
it is sometimes important to distinguish between GDP per capita and GDP per worker, but we are not going
to make this distinction for now. In Chapter 7 we'll go back to thinking about what incentives govern the
decision over how much to work.
Consumption and Investment
Assumption 4.6. The economy is closed and there is no government
Assumption 4.6 implies that in the accounting identity (1.1.1), X = M = G = 0, so we are left with
Y =C +I
This means that all output is either dedicated to consumption or to investment.
Assumption 4.7. The savings rate
Y −C
Y is equal to an exogenous constant s.
Recall that by de nition Y is both total output and total income. Therefore S ≡ Y −C represents savings:
S Y −C
all the income that people choose not to consume.
Y = Y is the savings rate: savings as a fraction of
S
income. Assumption 4.7 says that
Y = s: people save an exogenous fraction s of their total income. In
Chapter 6 we are going to think more about the incentives that shape people's decision of whether to consume
or save but for now we are going to take this decision as exogenous.
An immediate consequence of Assumptions 4.6 and 4.7 is
I = sY (4.1.3)
so a fraction s of output is dedicated to investment. There are actually two steps in getting to formula (4.1.3).
Assumption 4.6 implies that S = I: savings equal investment. This is always true in a closed economy.
2 The
second step uses Assumption 4.7: if savings are are a constant fraction of income then investment is a constant
fraction of output.
Depreciation and Capital Accumulation
Capital depreciates. Machines wear down, computers become outdated, buildings need repairs, etc. If we
want to keep track of the total capital stock it's important to keep this in mind.
2
In fact, this does not depend on having G = 0. Suppose that we have G > 0 and the government collects τ in taxes (which
may or may not be equal to G). Then private savings are S P rivate = Y − τ − C (after-tax income minus consumption) and public
savings are S P ublic = τ − G (tax revenues minus spending). Then total savings are
S = S P rivate + S public
=Y −τ −C+τ −G
=Y −C−G
=I
56
4.2. Mechanics
Assumption 4.8. The capital stock depreciates at a constant rate δ.
We are going to model depreciation in the simplest possible way: every piece of capital equipment loses a
fraction δ of its value every period. Therefore the total capital stock is going to evolve according to:
Kt+1 = (1 − δ) Kt + It (4.1.4)
Equation (4.1.4) says that if the capital stock in this period is Kt , then the capital stock in the next period
will consist of the sum of:
1. The portion of the capital stock that has not depreciated: (1 − δ) Kt and
2. The new capital that has been created through investment It
4.2 Mechanics of the Model
Per Capita Production Function
First we are going to rewrite the production function in per-capita terms. De ne
Y
y≡
L
K
k≡
L
y denotes GDP per capita and k denotes capital per worker.
3 Then using (4.1.1) we can write
F (K, L)
y=
L
K
=F ,1
L
≡ f (k) (4.2.1)
The rst step in (4.2.1) is just using the production function (4.1.1) to replace Y. Implicitly, what we are doing
is saying that all the capital and all the labor in the economy is used in one aggregate production process.
Thanks to Assumption 4.1 (constant returns to scale), it wouldn't make a dierence if we instead assume
that it's split up into many dierent production processes that are just a scaled-down version of the aggregate
economy. The second step uses Assumption 4.1 (constant returns to scale) directly: we are just multiplying by
1
λ= L . The last step is a de nition: we are de ning the per-capita production function f (k) as the output
that would be produced by one worker with k units of capital. Equation (4.2.1) says that GDP per capita is
going to the same as it would be if there was only one person and k units of capital in the economy.
3
Recall, since everybody works, per capita and per worker is the same in this model.
57
4.2. Mechanics
Dynamics of Capital per Worker
Using (4.1.4) we can derive a formula for how the amount of capital per worker k is going to evolve over time:
∆kt+1 ≡ kt+1 − kt (de nition)
Kt+1 Kt+1
= − kt (replacing kt+1 with )
Lt+1 Lt+1
(1 − δ) Kt + It
= − kt (using 4.1.4)
Lt+1
(1 − δ) Kt + sYt
= − kt (using 4.1.3)
Lt+1
(1 − δ) Kt + sYt Lt
= − kt (rearranging)
Lt Lt+1
1
= [(1 − δ) kt + syt ] − kt (using Assumption 4.5: constant n)
1+n
syt − (δ + n) kt
= (rearranging)
1+n
sf (kt ) − (δ + n) kt
= (using 4.2.1) (4.2.2)
1+n
Formula (4.2.2) has the following interpretation. The change in the stock of capital per worker k depends
I
on the balance of opposing forces. Investment adds to the capital stock, pushing it up. The term sf (kt ) = L
is investment per capita. Two forces push k down. The rst is depreciation, which directly subtracts from
the capital stock. The second is population growth. This doesn't literally subtract from the capital stock but
spreads the capital stock over a larger number of workers, so it also lowers the stock of capital per worker.
That's why δ and n appear together in formula (4.2.2). Figure 4.2.1 plots f (k), sf (k) and (δ + n) k on the
same graph.
Notice that both output per capita f (k) and investment per capita sf (k) are concave functions of k . Why
is that? Mathematically, it follows from Assumption 4.3 (diminishing marginal product): the derivative of the
production function is positive but decreasing and sf (k) is just multiplying by a constant, so it inherits the
same properties. Economically, what's going on is that the marginal product of capital is decreasing: adding
more and more machines per worker to the economy results in higher output (and therefore investment) per
worker but at a diminishing rate.
Notice also that sf (k) starts above (δ + n) k but ends below, i.e. the lines cross. That is a consequence of
Assumption 4.4 (Inada conditions). The rst part of this assumption implies that at rst sf (k) is very steep,
so it must be above (δ + n) k . The second part of the assumption says that eventually the slope of sf (k)
becomes zero. Since the slope of (δ + n) k is δ + n, this means that for su ciently high k the slope of sf (k)
is lower than δ + n, and therefore eventually sf (k) < (δ + n) k .
What does this imply? Whenever sf (k) > (δ + n) k , then equation (4.2.2) says that the capital stock
per worker is growing. Conversely, when sf (k) < (δ + n) k , equation (4.2.2) says that the capital stock per
worker is shrinking. Economically, this means that an economy with a su ciently low k will be accumulating
capital while an economy with a su ciently high k will tend to deplete its stock of capital. The reason for this
58
4.2. Mechanics
Fig. 4.2.1: The forces that
govern the evolution of the level
of capital per worker in the
Solow model.
is that the relative magnitude of two forces pushing ∆k in opposite directions changes with k. Depreciation
(and dilution via population growth) is just proportional: the more capital there is, the more it depreciates.
Investment is proportional to output, not to the capital stock. Due to the diminishing marginal product of
capital, the increase in output and therefore investment that you get out of a higher capital stock is smaller
and smaller as the capital stock increases. If Assumption 4.4 (Inada conditions) holds, eventually the extra
investment is less than the extra depreciation, so the two lines cross.
The point kss is the level of capital-per-worker such that the two forces are exactly equal. If k = kss , then
the capital stock per worker will remain constant from one period to the next. We refer to an economy where
k = kss as being in steady state. Notice that in a steady state output per worker also remains constant at its
steady state level yss = f (kss ).
For any k < kss we have that sf (k) > (δ+n)k , so k grows, and for any k > kss we have that sf (k) < (δ+n)k ,
so k shrinks. This implies that over time k moves closer and closer to the steady state.
4 Therefore over time
the economy converges to the steady state. Mathematically:
lim kt = kss
t→∞
and therefore
lim yt = f (kss )
t→∞
Therefore, over time, the rate of growth of GDP per capita will slow down to zero.
Notice one subtlety about terminology. When we say that in steady state the economy is not growing,
4
This is not quite a rigorous mathematical argument. We haven't ruled out the possibility that k could jump across kss from
one period to the next without really getting closer. But in fact it's straightforward to rule out this possibility.
59
4.2. Mechanics
what we mean is that k is not growing and y is not growing. However, L is growing. Therefore K = kL and
Y = yL are also growing. Economically, this means that both GDP and the capital stock are growing but just
enough too keep up with the growing population. GDP per capita is not growing.
In the case of a Cobb-Douglas production function, we can nd an expression for kss explicitly:
yt = ktα (dividing 4.1.2 by Lt )
sktα
− (δ + n) kt
∆kt+1 = (replacing into 4.2.2)
1+n
α
sk − (δ + n) kss
0 = ss (applying the de nition of a steady state)
1+n
1
1−α
s
kss = (rearranging)
δ+n
α
1−α
s
yss = (using the production function)
δ+n
Growth and Level Eects
Let's imagine that an economy is at its steady state and there is some change in its fundamental features. We
can ask the model what the consequences of this will be.
Suppose rst that there is an increase in the savings rate. It's often said that increasing investment (which
in a closed economy is the same as saving) is desirable. Let's see what would happen in this model.
Fig. 4.2.2: Increases in the savings rate and the rate of population growth.
Graphically, we can represent an increase in the savings rate as a proportional upward shift in the sf (k)
curve, as shown in the left panel of Figure 4.2.2. Starting from the original kss , we now have that sf (k) >
60
4.3. The Golden Rule
(δ + n) k , so the economy will begin to accumulate capital. Eventually, it will converge to a new steady state
0 0
with a higher capital stock per worker kss and higher output per worker yss .
Suppose now that there is an increase in the rate of population growth. Graphically, this is represented
by an increase in the slope of (δ + n) k , as shown in the right panel of Figure 4.2.2. Starting from the original
kss , we now have sf (k) < (δ + n) k so the stock of capital per worker will start to go down and eventually
converge to a steady state with less capital per worker and lower output per worker.
Suppose now that some new technological discovery results in a change in the production function: we
gure out a way to get more output out of the same amount of inputs. Graphically, we can represent this as
an upward shift in f (k) and therefore in sf (k), as shown in Figure 4.2.3. Starting from the original kss , we
now have that sf (k) > (δ + n) k , so the economy will begin to accumulate capital. Eventually, it will converge
to a new steady state with a higher capital stock per worker. Output per worker in the new steady state will
be higher for two reasons: k is higher (a shift along the horizontal axis) and there is a vertical shift in the
production function for any k.
Fig. 4.2.3: An upward shift
in the production function.
One thing that all these examples have in common is that none of these changes will result in sustained
long-term growth. There will be growth (or negative growth) for a while as the economy moves towards a new
steady state but this will be temporary: the long-term eect will be on the level of GDP per capita but not
on the long term growth rate of GDP per capita, which is always zero.
4.3 The Golden Rule
Suppose we were to ask the question: how much should the economy save? This is a normative question and
in order to answer it we have to have some sort of standard to make normative judgments. We'll consider
61
4.3. The Golden Rule
one possible criterion here and revisit it later in Chapter 9. The so-called Golden Rule criterion is a very
loose interpretation of the moral principle one should treat others as one would like others to treat oneself .
Applied to the question of the savings rate, it can be thought to mean that societies should save in such a way
as to maximize the level of consumption in the steady state. Whether this is a good interpretation of the moral
principle is more of a literary question than an economic one, but let's accept it for now. One justi cation for
this objective is that if you were going to be born into a society that is and will remain in steady state, the
Golden Rule society will be the one where you achieve the highest utility.
Finding the Golden Rule
If the economy is at a steady state, consumption will be
css = (1 − s) yss (4.3.1)
css depends on s in two ways. First, there is a direct eect: the more you save, the less you consume. That's
why s appears negatively in (4.3.1). Then, there is an indirect eect: the more you save, the higher the steady
state capital stock, the higher the output out of which you can consume.
We are going to restate the question of the Golden Rule a little bit. Instead of thinking about choosing s,
let's think about choosing kss . Why does this make sense? We know that changing s will change the amount
of capital the economy ends up with in steady state (that's the point of Figure 4.2.2), so we can simply think
about choosing some level of kss and then ask what s is needed to bring about this kss .
Let's start with this last step. Suppose we have decided on some level kss that we would like the economy
to have in steady state. How much does the economy need to save to make this happen?
sf (kss ) = (δ + n) kss (imposing steady state in (4.2.2))
(δ + n) kss
s= (solving for s) (4.3.2)
f (kss )
Equation (4.3.2) says that the savings rate that the economy needs to have to ensure a certain level of steady-
state capital-per-worker is equal to the ratio of depreciation-plus-population-growth (δ + n) kss to total output
f (kss ). This has a simple interpretation: the economy needs to save at a rate that is su cient to make up for
the amount of depreciation and population growth that will take place at kss .
Using (4.3.2) and (4.3.1) we can obtain an expression for css as a function of kss :
(δ + n) kss
css = 1− f (kss )
f (kss )
= f (kss ) − (δ + n) kss (4.3.3)
Equation (4.3.3) says that in steady state the economy will consume everything that is left over after making
up for depreciation and population growth. Now we want to nd the level of kss that will maximize this
62
4.4. Markets
expression. Taking rst order conditions, we obtain that the Golden Rule capital stock kgr must satisfy:
f 0 (kgr ) = δ + n (4.3.4)
The Golden Rule capital stock is such that, at the margin, the additional output you get from having more
capital exactly equals the extra investment that will be required to maintain it. Note that it's possible for
an economy to be at a steady state with kss > kgr . If this happens, this economy will have higher GDP per
capita but lower consumption per capita than a Golden Rule economy .
For the Cobb-Douglas production function, we have
α−1
αkgr =δ+n
so
1
α−1
δ+n
kgr =
α
and replacing this in (4.3.2) simpli es to
1
δ+n
α−1
(δ + n) α
s= α
δ+n
α−1
α
=α
4.4 Markets
So far we've followed a mechanical approach: factors of production get inputed in the production function
and output comes out. We haven't said anything about how this comes about: who makes what decisions and
why. Now let's imagine that decisions are made by rms and workers that interact in markets.
Factor Markets
We are going to imagine that there are perfectly competitive markets for labor and capital. The labor market
is straightforward to conceptualize (although we might not be persuaded that perfect competition is a good
assumption). There is a wage w. This means that workers get paid w goods per unit of labor that they
provide. Perfect competition means that a rm can hire as many units of labor as it wants at a wage w and
workers get paid w per unit for however many units of labor they supply (we will maintain that they supply L
inelastically). In equilibrium, w must be such that rms want to hire exactly the L units of labor that workers
supply.
The way we are going to model the market for capital is a little bit less intuitive. We are going to assume
that rms do not own the capital they use; instead they rent it from the households. This makes less of a
dierence than you might think: ultimately, the households own the rms so either directly or indirectly they
own the capital. But it's conceptually useful to make the distinction between ownership and use of capital.
Therefore we are going to assume that all capital is owned by households and rented by rms. A capital
63
4.4. Markets
rental arrangement works as follows. The rm gets the right to use one unit of capital for one period. In
K
exchange, the rm pays a rental rate r and, at the end of the period, returns the capital to its owner, with
the understanding that it will have depreciated a little bit in the meantime. Again, we are going to assume
perfect competition: rms can rent as much capital as they want at a rental rate rK and rK must be such
that they choose to rent exactly the amount K that is available in the economy.
The Firm's Decision
Anyone in the economy can set up a rm, hire labor and capital and use the production function. We'll assume
that the objective of rms is to maximize pro ts. Mathematically, this means that rm i solves the following
problem:
max F (Ki , Li ) − wLi − rK Ki
Ki ,Li
Pro ts are equal to the output the rm produces F (Ki , Li ) minus the wLi that it pays for Li units of labor
K
and the r Ki that it pays for Ki units of capital. The rst order conditions of this maximization problem
are:
FK (Ki , Li ) − rK = 0 (4.4.1)
FL (Ki , Li ) − w = 0 (4.4.2)
Fig. 4.4.1: A rm's choice of capital and labor.
Equation (4.4.1) has the following interpretation. Suppose the rm is considering whether to hire an
additional unit of capital. If it does, this will produce extra output equal to the marginal product of capital
FK (Ki , Li ) and it will cost the rm the rental rate rK . If the dierence FK (Ki , Li ) − rK were positive, it
64
4.4. Markets
would be pro table for the rm to hire more capital; if the dierence were negative, the rm would increase
pro ts by reducing the amount of capital it hires. Only if (4.4.1) holds is the rm satis ed with the amount
of capital it hires. The interpretation of equation (4.4.2) is the same but with respect to labor: only if it holds
is the rm satis ed with the number of workers it hired. Figure 4.4.1 illustrates this reasoning.
Market Clearing
In what follows, we'll rely on the following mathematical results.
Proposition 4.1 (Euler's Theorem) . The production function satis es:
FK (K, L) K + FL (K, L) L = F (K, L) (4.4.3)
Proof. Assumption 4.1 says:
F (λK, λL) = λF (K, L)
Take the derivative with respect to λ on both sides:
FK (λK, λL) K + FL (λK, λL) L = F (K, L)
Setting λ=1 gives (4.4.3)
Proposition 4.1. The production function satis es:
FK (λK, λL) = FK (K, L) for all λ>0 (4.4.4)
FL (λK, λL) = FL (K, L) for all λ>0 (4.4.5)
Proof. Assumption 4.1 says:
F (λK, λL) = λF (K, L)
Take the derivative with respect to K on both sides:
FK (λK, λL)λ = λFK (K, L)
which implies (4.4.4). Similarly, taking the derivative with respect to L on both sides leads to (4.4.5).
Now we'll use Proposition 4.1 to show that all rms in the economy will use factors in the same proportions.
65
4.4. Markets
1 1
Setting λ= Li in (4.4.4) and λ= Ki in (4.4.5) respectively we obtain:
Ki
FK (Ki , Li ) = FK ,1 (4.4.6)
Li
Li
FL (Ki , Li ) = FL 1, (4.4.7)
Ki
Equation (4.4.6) has the following interpretation. Suppose rm i Ki units of capital and Li
chooses to hire
Ki
workers. The marginal product of capital will depend on the ratio but not on the absolute values of Ki
Li
and Li . A rm with a lot of capital per worker will have a low marginal product of capital no matter how
much of each factor it has in absolute terms. The key assumption that drives this result is Assumption 4.1:
constant returns to scale. Equation (4.4.7) has the symmetric interpretation: the marginal product of labor
also depends only on the ratio of factors of production. To put it in concrete terms, suppose we are running
an orchard, which uses apple trees (a form of capital) and workers to produce apples. Equation (4.4.7) says
that the number of extra apples we'll obtain if a worker spends an extra hour picking apples depends on how
many hours per apple tree we are starting from but not on whether the farm is large or small.
The pro t-maximization conditions (4.4.1) and (4.4.2) say that each rm is equating the marginal product
to factor prices. Since they all face the same prices, they must all have the same marginal product. This in
turn implies that all rms choose the same ratio of capital to labor. The only dierence between dierent rms
is their scale, but with constant returns to scale there is no dierence between having many small rms or one
large rm that operates the same technology. Therefore we can assume without loss of generality that there
is just one representative rm that does all the production. Market clearing requires that the representative
rm hire all the available labor L and all the available capital K. Replacing Ki = K and Li = L in (4.4.1)
and (4.4.2) implies:
rK = FK (K, L) (4.4.8)
w = FL (K, L) (4.4.9)
The rental rate of capital will be equal to the marginal product of capital for a rm that hires all the capital
and all the workers in the economy; the wage will be equal to the marginal product of labor for the same rm.
Using (4.4.6) and (4.4.7), this implies that the rental rate of capital and wages and will be:
rK = FK (k, 1) (4.4.10)
1
w = FL 1, (4.4.11)
k
By Assumption 4.3, this means that, other things being equal, the rental rate of capital will be low and
wages will be high in an economy with high k. If there is a low level of capital per worker, then the marginal
product of capital will be low and so will the rental rate. Conversely, with a lot of capital to work with, the
marginal product of labor will be high, so competition between rms will drive up wages.
66
4.4. Markets
Pro ts
We can now compute the pro ts of the representative rm
Pro t =F (K, L) − wL − rK K
Proposition 4.2. The representative rm earns zero pro ts
Proof. Replace FK (K, L) and FL (K, L) in (4.4.3) using (4.4.8) and (4.4.9):
rK K + wL = F (K, L)
which gives the result
Proposition (4.2) says that all the output that is produced gets paid either to the workers or to the owners of
capital, with no pro ts left over for the owner of the rm. Even though they are trying to maximize pro ts,
the maximum level of pro ts that the rms can attain is zero.
It's very important to remember that the de nition of pro ts that we are using is dierent from the way
the term is used in accounting. Let's see an example.
Example 4.2. A corporation called Plantain Monarchy owns a retail space on the ground oor of a
building and runs a clothes shop in this retail space. In 2018, Plantain Monarchy sold clothes worth
$1,000,000, paid $300,000 in wages and spent $500,000 buying clothes from manufacturers. The rent on
a comparable retail space is $200,000 a year. What were the pro ts of Plantain Monarchy? Accountants
would measure the pro ts of Plantain Monarchy this way:
Income Statement (Accounting)
Sales 1, 000, 000
- Cost of Goods Sold 500, 000
- Expenses 300, 000
= Pro ts 200, 000
Instead, the de nition of pure economic pro ts would include the rental rate of capital as a cost.
Even though Plantain Monarchy owns the retail space and does not need to pay rent, there is still an
opportunity cost of not renting it out for $200,000. Therefore, under the de nition of pro ts that we are
using here, we have:
67
4.4. Markets
Income Statement (Economic)
Sales 1, 000, 000
- Cost of Goods Sold 500, 000
= Value Added of Plantain Monarchy 500, 000
- Wages 300, 000
- Capital Rental 200, 000
= Pro ts 0
Even though from an accounting perspective it looks like Plantain Monarchy is pro table, the ac-
counting pro ts it's earning are just the implicit rental from the capital that the rm owns.
In reality, of course, plenty of rms earn pure economic pro ts, i.e. pro ts beyond the implicit rental on
the capital they own. Many rms make losses too. There are several possible reasons why rms might earn
pro ts. One of the assumptions in our model is that there is perfect competition. If a rm has at least a little
bit of monopoly power it can earn positive economic pro ts.
5 Another assumption in the model is that there
is no risk: the output that will be produced is a perfectly predictable function of the inputs to the production
process. In reality, rms face risk. It's possible that many rms actually earn zero expected pro ts but what
we observe as nonzero pro ts is the result of either good or bad luck.
Interest Rates
Suppose that in addition to markets for hiring labor and renting capital there is a market for loans. A loan
works as follows: the lender gives x goods to the borrower in period t and the borrower pays back x (1 + rt+1 )
goods to the lender in period t + 1. rt+1 is the real interest rate on the loan. We say real interest to clarify
that it's an interest rate in terms of goods, not in terms of dollars. If the loan was described in terms of dollars,
we would need to convert dollars into goods by keeping track of how prices evolve.
6 For simplicity, we'll just
describe loans in real terms directly: as exchanges of goods in one period for goods in the next period. We'll
assume that no one ever defaults on their loans: they are always paid back. Also, we'll assume there is perfect
competition: anyone can borrow or lend as much as they want at the interest rate rt+1 . Let's gure out what
the interest rate is going to be in this economy.
Suppose someone wants to save x goods. They have two possibilities.
1. Physical investment. They convert their goods into x units of capital and rent them out in the following
period. In the following period, this will give them:
K
Rental Income rt+1 x
+ Value of depreciated capital (1 − δ) x
K
= Total 1 + rt+1 −δ x
2. Lending. They lend their x goods in the loan market and get back (1 + rt+1 ) x goods.
5
We'll think about models with monopoly power starting in Chapter 14.
6
We'll look further at the distinction between real and nominal interest rates in Chapter 11.
68
4.5. Technological Progress
They will be indierent between the two options if and only if the following condition holds:
7
K
rt+1 = rt+1 −δ (4.4.12)
K
We'll argue that condition (4.4.12) has to hold. Why? Suppose it were the case that rt+1 < rt+1 − δ.
Then one could make an in nite gain by borrowing at rate rt+1 , investing in physical capital, which earns
K
rt+1 −δ , using part of this to pay back the loans and keeping the dierence. But if borrowing to invest
is such a great deal, no one would be willing the lend and the loan market wouldn't clear. Conversely, if
K
rt+1 > rt+1 − δ then everyone would want to lend and no one would want to borrow, so again the loan market
wouldn't clear. Therefore (4.4.12) must hold. Condition (4.4.12) links the interest rate to the rental rate of
capital and therefore to the marginal product of capital. Interest rates will be high in economies where the
marginal product of capital is high (which, other things being equal, will be the case if capital per worker is
low).
Response of Prices
Equations (4.4.10), (4.4.11) and (4.4.12) give us a way to think about what happens to the level of wages
and interest rates in response to changes in the economy. Suppose there is an increase in the savings rate, as
illustrated in Figure 4.2.2. In the new steady state, there is a higher level of k, which means wages will be
higher and the rental rate of capital and therefore the interest rate will be lower. Similarly, faster population
growth will lead to a shift to a steady state will lower wages and higher interest rates.
4.5 Technological Progress
If we want to understand the growth of GDP per capita in the US over the last 250 years the model we have
studied so far doesn't have a lot of promise: it predicts that in the long run there will be no growth.
8 Now
we are going to take the same economy and see what happens when there is technological progress.
We are going to represent technological progress as a change in the production function. Figure 4.2.3 shows
what happens when there is a once-and-for-all change in the production function. Instead, we are now going
to imagine that, due to technological progress, the production function moves up a little bit every period.
We'll assume this upward shift takes a speci c form, known as labor augmenting technological progress.
This means that better technology is equivalent to having more workers. Mathematically, this means that we
will assume that the production function takes the form:
Y = F (K, AL) (4.5.1)
where the variable A represents the level of technology. (4.5.1) is a generalized version of the production
function we had considered so far. Our original production function (4.1.1) is the special case where A = 1.
7
We'll talk more about this equation in Chapter 9.
8
One could still conjecture that 250 years is not long enough to test any predictions about the long run. Strictly speaking,
we never really reach the long run. Later we'll work on putting actual numbers on our model to see how fast it approaches the
steady state. We'll see that after a few of decades we should expect almost no growth. See Exercise 5.1.
69
4.5. Technological Progress
De ne:
L̃ ≡ AL
We'll refer to L̃ as e ciency units of labor. If L workers are employed in a production process and the level
of technology is A, then the output of the production process will be the same as if L̃ = AL workers were
employed and the level of technology was 1.
Assumption 4.9. The level of technology grows at a constant, exogenous rate g : At+1 = (1 + g) At
Assumption 4.9 states that there is a stable, proportional rate of technological progress. We'll see that
under this assumption the model will be consistent with a lot of facts about economic growth, in particular
the steady rates of growth of advanced economies. On the other hand, it is rather disappointing to have to
make this assumption. Ideally, one would like to have a deeper understanding of why there is technological
progress and what determines how fast it takes place. We'll leave these important questions aside for now.
Mathematically, the model changes very little when we make Assumption 4.9. The key is to realize that
instead of stating the production function in per capita terms, as in (4.2.1), we can instead write it in per
e ciency unit of labor terms. De ne:
Yt
ỹt ≡
L̃t
Kt
k̃t ≡
L̃t
ỹ is output per e ciency unit of labor and k̃ is capital per e ciency unit of labor. These are not variables we
are actually interested in but it's a convenient way to rescale the model.
As we did before, we can write down the production function in per e ciency unit of labor terms:
F (K, AL)
ỹ =
AL
K
=F ,1
AL
= f k̃ (4.5.2)
The rst step in (4.5.2) is just applying the de nition of ỹ and using the production function (4.5.1) to replace
Y. The second step uses Assumption 4.1 (constant returns to scale) and the last step just uses the de nition
of the function f (k).
As we did before for k, we can derive an equation that describes how k̃ will evolve over time:
∆k̃t+1 ≡ k̃t+1 − k̃t (de nition)
Kt+1 Kt+1
= − k̃t (replacing k̃t+1 with )
At+1 Lt+1 At+1 Lt+1
70
4.5. Technological Progress
(1 − δ) Kt + It
= − k̃t (using 4.1.4)
At+1 Lt+1
(1 − δ) Kt + sYt
= − k̃t (using 4.5.2)
At+1 Lt+1
(1 − δ) Kt + sYt At Lt
= − k̃t (rearranging)
At Lt At+1 Lt+1
h i 1
= (1 − δ) k̃t + sỹt − k̃t (using Assumptions 4.5 and 4.9: constant n and g)
(1 + n) (1 + g)
sỹt − (δ + n + g + ng) k̃t
= (rearranging)
1 + n + g + ng
sỹt − (δ + n + g) k̃t
≈ (using that ng is small)
1+n+g
sf k̃t − (δ + n + g) k̃t
= (using 4.5.2) (4.5.3)
1+n+g
Formula (4.5.3) is a generalization of formula (4.2.2) and it has the same logic. The evolution of k̃ depends
on the balance of investment pushing it up and depreciation, population growth and technological progress
pushing it down. At rst it may seem a little bit counterintuitive that technological progress pushes k̃ down:
isn't technological progress supposed to help? The reason has to do with the way we de ne k̃ : it's capital
per e ciency unit of labor. If there is technological progress, this means that the number of e ciency units
of labor is rising, just like it would from population growth. Therefore technological progress requires that
we spread capital over more e ciency units of labor. For the purposes of calculating how k̃ evolves it doesn't
matter whether L̃ rises because L rises or because A rises. It will matter very much once we convert back the
per-e ciency-unit-of-labor measures into the per-person measures that we actually care about.
Given that formula (4.5.3) is so similar to formula (4.2.2) the dynamics that follow from it are also similar.
The economy also has a steady state. The level of capital-per-e ciency-unit-of-labor in steady state,
k̃ss , is
such that sf k̃ss = (δ + n + g) k̃ss so investment exactly balances out depreciation, population growth and
technological progress. Graphically, it can also be represented by Figure 4.2.1, except that the straight line
(δ + n) k is replaced by the slightly steeper line (δ + n + g) k . As before, k̃ increases whenever it is below k̃ss
and falls whenever it is above k̃ss .
Does this mean that technological progress makes no dierence for the predictions of the model? On
the contrary! Let's see what happens when we convert per-e ciency-unit-of-labor variables into per-person
variables. We know that in steady state, ỹ is constant. This implies that if the economy is in steady state:
ỹt+1 = ỹt (de nition of steady state)
Yt Yt+1
= (de nition of ỹ )
At Lt At+1 Lt+1
yt yt+1
= (de nition of y)
At At+1
yt+1 At+1
= (rearranging)
yt At
71
4.5. Technological Progress
yt+1
=1+g (Assumption 4.9: constant g) (4.5.4)
yt
This immediately implies the following:
Proposition 4.3.
1. In steady state, GDP per capita grows at the rate of technological progress.
2. In steady state, capital per worker grows at the rate of technological progress.
K
3. In steady state, the ratio
Y is constant.
Proof. Part 1 is just a restatement of (4.5.4). Part 2 follows from the same reasoning starting from
k̃t = k̃t+1 . Part 3 follows because:
Kt+1 k̃t+1
=
Yt+1 ỹt+1
k̃t (1 + g)
=
ỹt (1 + g)
Kt
=
Yt
Exercises
4.1 An Earthquake
Suppose an economy behaves according to the Solow growth model. It starts out at t=0 at a steady
state, with no technological progress and no population growth. Suppose an earthquake destroys half the
capital stock. As a a consequence of this, what would happen in the short run (i.e. immediately) and in
the long run (i.e. once the economy reaches steady state) to:
(a) GDP,
(b) GDP per capita,
(c) wages,
(d) interest rates.
4.2 The Black Death
In the middle of the 14th century, a plague killed about a third of the population of Europe. Assume that
the economy of Europe was well described by the Solow model. How would the following variables change
in response to the Black Death in the short run (i.e. immediately)?
(a) GDP,
72
4.5. Technological Progress
(b) GDP per capita,
(c) wages,
(d) interest rates.
4.3 Korean Uni cation
Suppose both North and South Korea have the same technology level (!?), which doesn't grow, and also
have constant population. Their respective populations and capital stocks are:
LSouth = 10
K South = 2430
LN orth = 5
K N orth = 160
The production function is:
Y = K α L1−α
with α = 0.4. The depreciation rate is δ = 0.08.
(a) Compute GDP per capita in each country.
(b) Compute wages and interest rates in each country.
K
(c) Suppose North and South Korea unify. What is the new
L in the uni ed country? What is GDP
per capita in the uni ed country?
(d) Compute wages and interest rates in the uni ed country. Interpret
(e) Will people and/or machines physically move between the North and the South? In what direction?
4.4 An AK Model
Suppose the production function takes the form:
F (K, L) = AK
(a) Which of the assumptions that we made about F does this satisfy and which does it not satisfy?
(b) Suppose every other assumption we used in the Solow model holds, and there is no technological
kt+1
progress. Find an expression for
kt (it may be useful to follow the steps that lead to expression
(4.2.2)).
(c) Will this economy grow in the long run? Explain.
4.5 A Malthusian Model
Look up Thomas Malthus and read a little about his work. We are going to formalize a simple version of
his ideas in a little model. The production function is:
Yt = At Lα
t (4.5.5)
73
4.5. Technological Progress
where At is technology, L is labor supply and α < 1.
There is no capital, no saving and no investment, so consumption is given by
Ct = Yt (4.5.6)
The population evolves according to
Ct
Lt+1 = Lt 1 + γ −c (4.5.7)
Lt
where γ and c are parameters.
(a) Malthus had in mind an agricultural economy where the total amount of land is xed. What does
that have to do with the way we wrote down the production function? What is the signi cance of
assuming α < 1?
(b) What does equation (4.5.7) mean? What is c? What is γ? Why did Malthus think that something
like equation (4.5.7) applied?
(c) Assume At = A is constant over time. Using (4.5.5), (4.5.6) and (4.5.7), nd an expression for the
percentage rate of population growth Lt+1
Lt as a function of the level of population Lt . Plot this
function.
(d) Assume At = A is constant over time. Will GDP per capita grow in the long run? Will the population
grow in the long run?
(e) Suppose there is a one-time increase in At , from A to A0 > A. What will happen to GDP per capita
and the size of the population in the short run and in the long run? Show your reasoning graphically
and/or algebraically.
(f ) Suppose now that instead of being constant, technology improves at a constant rate, so that
At+1 = At (1 + g)
Will GDP per capita grow in the long run? Will the population grow in the long run? Explain.
(g) When we have a constant rate of technological progress, how does the long-run level of consumption
per capita depend on γ? Why?
74
CHAPTER 5
Confronting Theory and Evidence
In this chapter we'll take another look at the evidence on economic growth and test some of the predictions
of the Solow model. We'll also use the model to suggest additional ways to look at the evidence.
5.1 The Kaldor Facts Again
Recall from Chapter 3 the so-called Kaldor Facts. Let's assume that advanced economies are well-described by
the steady state of the Solow model with a constant rate of technological progress. Would that be consistent
with the Kaldor Facts?
1. The rate of growth of GDP per capita is constant. In the model, this is true by assumption. Propo-
sition 4.3 says that GDP per capita grows at the rate of technological progress. Since we assume that
technological progress takes place at a constant rate, then the rate of growth of GDP per capita is
constant too. A success for the model, but not a huge one.
2. The ratio of the total capital stock to GDP is constant. This is part 3 of Proposition 4.3, so another
success for the model, this time with a result that is less obvious from the assumptions.
1
3. The shares of labor and capital income in GDP are constant. Let's see if this holds in our model. The
share of labor in GDP is:
2
wL
Labor share =
Y
1
Of course, all the results follow from the assumptions, but the usefulness of the model comes from telling us something that
follows from the assumptions is not-so-obvious ways.
2
When we write the production function with labor-augmenting technological progress as F (K, AL), it's important to distin-
∂F (K,AL)
guish between (i) (the marginal product of labor) which corresponds to taking the partial derivative with respect to
∂L
L while keeping A constant and (ii) FL (K, AL) (the derivative of F with respect to its second argument, evaluated at the point
(K, AL)). For instance, with a Cobb-Douglas production function Y = K α (AL)1−α we have
K α
∂F (K, AL)
= A(1 − α)
∂L AL
K α
FL (K, AL) = (1 − α)
AL
∂F (K,AL)
From the labor demand decisions of rms we know that w = ∂L
, i.e. wages equal to the marginal product of labor.
Instead, Proposition 4.1 is about FL .
75
5.1. The Kaldor Facts Again
∂F (K,AL)
∂L L
= (using 4.4.9)
F (K, AL)
AFL (K, AL) L
= (taking the derivative)
F (K, AL)
K
AFL AL ,1 L
= K
(using Proposition 4.1 and Assumption 4.1)
F AL , 1 AL
K
FL AL ,1
= K
(simplifying)
F AL ,1
K
Since Proposition 4.3 implies that
AL is a constant, this implies that the labor share is a constant. The
capital share is therefore also a constant because, by Proposition 4.1, they add up to 1.
We just proved that factor shares are constant if the economy is at a steady state with constant, labor-
augmenting technological progress. If we assume that the production function takes the Cobb-Douglas
form, then we can show that the labor share will be a constant even outside the steady state. Under the
Cobb Douglas production function:
1−α
F (K, AL) = K α (AL)
∂FL (K, AL)
w= = (1 − α) K α A1−α L−α
∂L
wL = (1 − α) K α A1−α L1−α
wL (1 − α) K α A1−α L1−α
= 1−α
Y K α (AL)
=1−α (5.1.1)
so the labor share is 1−α for any level of A, K and L. One reason we often assume that the production
function takes the Cobb-Douglas form is that this immediately ts the nding of constant factor shares.
Furthermore, this gives us an easy way to decide what is a reasonable value for α, since it's tied directly
to factor shares.
4. The average rate of return on capital is constant. As we saw in Chapter 3, this fact is an immediate
implication of facts 2 and 3, and therefore it also holds in the Solow model. Just to check:
rK = FK (K, AL) (using 4.4.8)
K
= FK ,1 (using Proposition 4.1)
AL
K
and since Proposition 4.3 implies that
AL is a constant, this implies that rK is a constant. rK is the
gross return on capital, meaning that it's what the owners of capital earn before depreciation. Once
we subtract depreciation, we obtain the net return on capital rK − δ , which by equation (4.4.12) must
equal the real interest rate. This will also be constant.
In terms of consistency with the Kaldor facts, the Solow model is doing quite well. Now let's see what
76
5.2. Putting Numbers on the Model
other things we can do with it.
5.2 Putting Numbers on the Model
For some purposes it is useful to put concrete numbers on models. Let's see how we can use data to guide us
towards reasonable numbers for the parameters of the Solow growth model. Let's imagine that we want our
model to approximate the behavior of the US economy.
It is standard to assume that the production function takes the Cobb-Douglas form:
3
1−α
F (K, AL) = K α (AL)
The empirical basis for this assumption comes from equation (5.1.1): if the production function takes this
form, factor shares will be constant even outside the steady state, which ts the Kaldor Facts. We just need
to choose the right number for α, and again equation (5.1.1) gives us guidance. Recall from Figure 3.2.3 that
the share of GDP that accrues to workers has averaged around 0.65 (although it's been lower recently). To
be consistent with this fact, we should have α = 0.35.
Proposition 4.3 says that GDP per capita grows at the rate of technological progress. GDP per capita in
the US has grown at approximately 1.5% per year since 1800. To be consistent with this fact, we should use
g = 0.015.
The US population has grown at a rate of approximately 1% per year since 1950. To be consistent with
this, we should use n = 0.01.
Depreciation is a little bit harder, in part because dierent kinds of capital depreciate at dierent rates,
in part because even for a speci c kind of capital it's not so easy to determine how fast in depreciates. The
Bureau of Economic Analysis uses the following values for some of the main types of capital: 0.02 for buildings,
0.15 for equipment, 0.3 for computers. We are not going to distinguish between dierent kinds of capital and
we'll just use a single number for the overall capital stock. A plausible value is δ = 0.04.
In choosing a number for s we need to use some judgment. The model we are using assumes that the
economy is closed. As we saw in Chapter 4, this implies that savings equals investment. In reality, the US
economy is not closed, and in the last couple of decades it has had higher investment than savings, as shown
in Figure 5.2.1:
4
If we want the model to replicate the investment rate that we have seen in the US in recent years, then we
should set s = 0.2 approximately; if we wanted to replicated the savings rate we would set s a little bit lower.
K
Note that these gures for s, δ , n and g are consistent with the measured
Y ratio. Recall that by equation
3
You will sometimes see this written as AK α L1−α . This doesn't make much dierence. With the Cobb-Douglas function, the
term A1−α factors out anyway so it's just changing the units in which we measure technology.
4
The counterpart to this is a trade de cit. If you go back to equation (1.1.1) from Chapter 1:
Y − C − G+ M −X = I
| {z } | {z } |{z}
Savings Trade De cit Investment
77
5.2. Putting Numbers on the Model
Fig. 5.2.1: Investment (pri-
vate plus public) and Saving as
a percentage of GDP in the US.
Source: NIPA.
(4.5.3), in a steady state we must have:
K k̃ss k̃ss
= =
Y ỹss f (k̃ss )
s
= (5.2.1)
δ+n+g
≈ 3.08
which is close to what we saw in Figure 3.2.2.
The model predicts that the interest rate should be:
5
r = FK (K, L) − δ
α−1
K
=α −δ
AL
δ+n+g
=α −δ
s
≈ 0.074
A 7.4% real interest rate is higher than has been observed in recent decades. Figure 5.2.2 shows the interest
1
5 s 1−α
There are two ways to do the second step. One is to use equation (4.5.3) to solve for kss = δ+n+g
and plug it back
FK (K,L)K
into the formula. The other is to use the previously-shown result that the capital share of GDP is equal to α, set α= Y
,
and use equation (5.2.1)
78
5.2. Putting Numbers on the Model
rate on 10-year in ation-indexed bonds (known as TIPS), which are the closest thing we have to market real
interest rate, and these have been closer to 1% or 2%. These bonds have only been around for the past couple
of decades but even before that we can reconstruct real interest rates from nominal interest rates and in ation
measures, and on average they have been signi cantly below 7.4%. Clearly the model is in con ict with the
data on this dimension. One likely source of this discrepancy is risk. In the model there is no risk, whereas
in reality most real investment involves some risk. Riskier investments tend to earn higher returns than safe
ones.
6 Maybe the more accurate comparison is between the model's prediction for interest rates and the return
on risky investments like the stock market instead of a safe asset like in ation-indexed bonds. The average
real return on the S&P500 index between 1929 and 2018 was 7.8%, much closer to the real interest rate in the
model.
Fig. 5.2.2: Interest rate
on 10-year in ation-indexed
bonds. Source: Board of Gov-
ernors of the Federal Reserve.
We saw in Chapter 4 that if an economy increases its savings rate it will reach a higher steady state level
of capital per worker. With concrete numbers for the model parameters, we can ask quantitative questions
like: how much higher? How long will it take to reach that level? Suppose the savings-and-investment rate
rises from 0.2 to 0.25, starting from a steady state. We can use equation (4.5.3) to simulate the evolution of
k̃t . Figure 5.2.3 shows the results. In the new steady state, the capital stock (per e ciency unit of labor) is
41% higher and GDP (per e ciency unit of labor) is 13% higher. The interest rate falls from 7.4% to 5.1%.
The transition to the new steady state takes time. Consumption only reaches its initial level after 17 years,
and after 30 years, GDP has only risen 9%.
6
In Chapter 8 we look at models of why that may be.
79
5.3. The Capital Accumulation Hypothesis
Fig. 5.2.3: The economy's response to a higher saving rate. The capital stock, GDP, consumption, and
investment are scaled by e ciency units of labor.
5.3 The Capital Accumulation Hypothesis
One of the major questions in all of macroeconomics is why some countries are rich and others are poor.
We can use the Solow model, together with data from dierent countries to test one possible answer to this
question.
Conjecture 5.1. Technology levels are the same across countries and the dierences in GDP per capita
K
are the result of dierences in
L.
Conjecture 5.1 is a logically possible explanation of dierences in GDP per capita across countries. Indeed,
if two countries with the same production function had dierent levels of capital per worker, they would
have dierent levels of GDP per capita. If the conjecture were true, it would have profound implications for
economic development in poor countries: if the problem is low levels of capital, then capital accumulation is
80
5.3. The Capital Accumulation Hypothesis
a good solution. However, we'll see that this conjecture is decisively rejected by the evidence. We'll address
this in dierent ways.
Convergence
K
If two countries with the same production function have dierent levels of
L , then we know that at least one
of them, and maybe both, are not in steady state. Let's compute the growth rate of a country that is not
at steady state. To avoid cluttering the algebra, let's assume that there is no technological progress and no
population growth, but the argument holds regardless. Denote the growth rate of GDP per capita by gy . Let's
compute it:
yt+1
gy ≡ −1 (by de nition)
yt
f (kt+1 ) − f (kt )
= (replacing the production function 4.2.1)
f (kt )
f 0 (kt ) [kt+1 − kt ]
≈ (this is a rst-order Taylor approximation)
f (kt )
0
f (kt ) [sf (kt ) − δkt ]
= (using 4.2.2)
f (kt )
f 0 (kt ) kt
= sf 0 (kt ) − δ (rearranging)
f (kt )
= sf 0 (kt ) − δα (letting α represent the capital share) (5.3.1)
Let's see what equation (5.3.1) means. Suppose we compare two countries with dierent levels of GDP.
The richer country, according to Conjecture 5.1, has a higher capital stock. By Assumption 4.3 (diminishing
marginal product), a higher capital stock implies a lower marginal product of capital. Therefore equation
(5.3.1) implies that the richer country will grow more slowly than the poorer country.
At some level, we sort of knew that already. By assumption, the countries are converging to the same
steady state so it makes sense that the country that is already very close would grow more slowly than the
one that is far behind. Equation (5.3.1) makes this precise: at any point in the path that leads to the steady
state, the poorer country should be growing faster than the richer country.
This is something we can test directly with cross-country data. Is it the case that initially-poorer countries
grow faster than initially-richer countries? Figure 3.3.1 from Chapter 3 gives us an answer. If Conjecture 5.1
were correct, we should see a strong negative correlation between initial GDP per capita and growth rates,
and this is not what we see.
Figure 3.3.1 is not necessarily the end of the discussion. One thing that the theory does not clearly specify
is whether a country is the right unit of observation. Maybe the lack of correlation between initial GDP and
subsequent growth is the result of a lot of weird, small, possibly irrelevant countries? An alternative way to
look at the cross-country data is to weigh each observation in proportion to the population of that country.
7
7
It's not entirely clear whether weighting by population is the right thing to do. If what we want to test is the universal validity
of Conjecture 5.1, one could make the case that a small country provides an equally valuable experiment as a large country. On
the other hand, not all important economic forces operate at the level of an entire country; perhaps each of India's 29 provinces
should be considered a separate experiment.
81
5.3. The Capital Accumulation Hypothesis
Figure 5.3.1 shows what happens if we do that.
Fig. 5.3.1: Growth across countries since 1960 and 1980, population-weighted. Source: Feenstra et al. (2015).
Weighted this way, the data do show some convergence. If we focus on the period since 1980, the pattern
is stronger. There is a simple fact behind this: China and India started this sample being very poor and
have grown very fast, especially in the last few decades. Given their large population, these two observations
largely determine the overall pattern.
Another question one can ask is whether Conjecture 5.1 might be true for speci c groups of economies,
even if it's not true for the world as a whole. For instance, dierent US states have dierent levels of GDP
per capita. Maybe for US states it's true that they have the same production function but dierent levels of
capital per worker? Capital abundance could be the main reason why Connecticut is richer than Louisiana
even if it's not the main reason the US is richer than Paraguay. Figure 5.3.2 shows what happens if we repeat
the exercise but focus, respectively, on US states and Western European countries. Here we do see strong
evidence of convergence: US states and European countries that started out poor indeed grew faster than
their richer counterparts. Note, however, that the US data in Figure 5.3.2 covers the period 1929-1988. In
more recent decades, poorer US states have not grown faster than rich ones.
Note that from a logical point of view, nding evidence consistent with a conjecture is not the same as
proving the conjecture. The evidence on convergence is consistent with a limited, intra-US or intra-Europe
version of Conjecture 5.1 but it could also be that the conjecture is wrong and these economies are converging
for other reasons (for instance, faster rates of technological progress in poor US states or European countries).
Direct Measurement
Another way to test Conjecture 5.1 is to:
82
5.3. The Capital Accumulation Hypothesis
Fig. 5.3.2: Growth across US states and Western European countries. Source: Barro and Sala-i-Martin
(1991) and Feenstra et al. (2015).
1. Directly measure the capital stock in each country
2. Assume a form for the production function (which Conjecture 5.1 assumes is the same across countries).
For instance, assume f (k) = Ak α , which we know is a decent approximation to the production function
for the United States if we set α = 0.35.
3. Predict the GDP levels that you would get by just plugging in the measured level of capital per worker
into the production function. For this, rst solve for A using data for the United States:
yU SA
A= α
kU SA
Then compute predicted GDP per capita for country j (denoted ŷj ), under Conjecture 5.1 by computing:
ŷj = Akjα
4. Compare predicted levels of GDP to measured levels of GDP
If the Conjecture 5.1 is correct, predicted and actual levels of GDP should look quite similar.
The main challenge in doing this is that measuring the capital stock in a way that is comparable across
countries is actually quite hard (Exercise 4 asks you to think about some of the di culties). So if the data
don't line up with the prediction it's possible that mismeasurement of the capital stock is part of the answer.
Figure 5.3.3 shows the comparison between predicted and actual levels of GDP, and the patterns are very
strong: in poor countries, actual GDP is consistently much lower than you would predict if you just knew
83
5.3. The Capital Accumulation Hypothesis
their capital stock and assumed the production function was the same. Furthermore, the dierence is greater
the poorer the country. If the only dierence was the capital stock, the poorest countries should have a GDP
per capita of around $10,000 instead of their actual GDP per capita, which is closer to $1,000. This is more
evidence against the common-production-function, same-technology-level assumption of Conjecture 5.1.
Fig. 5.3.3: Predicted GDP
per capita on the basis of cap-
ital stock compared to actual
GDP per capita. Source: Feen-
stra et al. (2015)
Evidence from Interest Rates and Capital Flows
Suppose we don't trust the data on capital levels across countries. Another things we can try to measure
is interest rates in dierent countries. Why would these be informative? Recall from equations (4.4.10) and
(4.4.12) in Chapter 4.4 that interest rates are equal to:
K K
r=r − δ = FK ,1 − δ
L
Other things being equal, an economy that has low levels of capital per worker will have a high marginal
product of capital, a high rental rate of capital and high interest rates.
How much higher? This depends on the exact shape of the production function. Suppose that we assume
a Cobb-Douglas production function and we compare two countries, A and B. We know that GDP per capita
in country A is x times higher than in country B but we don't know the level of their respective capital stocks
because we don't trust our measurements. If Conjecture 5.1 were true, how would the interest rates in the
two countries compare?
Using the Cobb-Douglas production function we obtain:
rK = αk α−1 (5.3.2)
84
5.3. The Capital Accumulation Hypothesis
Equation (5.3.2) tells us that if we want to predict how rental rates of capital will dier across countries we
need to nd out how k α−1 will vary across countries. Let's compute this:
yA
x= (by assumption)
yB
α
kA
= (using the Cobb-Doulgas production function)
kB
α−1
α−1 kA
x α = (rearranging) (5.3.3)
kB
(5.3.3) tells us that the comparison of k α−1 across countries (and therefore the comparison of interest rates)
is linked to the comparison of GDP per capita levels in a very speci c way. Plugging in (5.3.3) into (5.3.2) we
get:
K
rA α−1
K
=x α (5.3.4)
rB
Let's try to put some numbers on this to see what it means. Let's say we want to compare Mexico and the
US. Mexican GDP per capita is approximately 0.3 times that of the US. Setting α = 0.35, formula (5.3.4)
then implies that
K
rU S 0.35−1
K
= 0.3 0.35 ≈ 9.4
rM EXICO
so the rental rate of capital in Mexico should be more than nine times higher than in the US. In order to
account for why Mexico is so much poorer than the US under Conjecture 5.1 we must infer that the capital
stock is much lower. If capital is so scarce, then it's marginal product must be very high and so must its rental
rate.
We can link this back to interest rates by recalling that
rK = r + δ
Suppose the relevant interest rate in the US is 7.4% and the rate of depreciation is 4%, then we have that
rK = 0.114 in the US, which would mean that rK = 1.08 in Mexico and therefore we should expect r =
K
r − δ = 104% in Mexico. Table 5.1 shows the result of repeating the calculation for several countries.
85
5.4. Growth Accounting
Table 5.1: Interest rates implied by Conjecture 5.1 for dierent countries. Source for GDP data: Feenstra
et al. (2015).
Country y α−
k K
x= yU S kU S = xα r = rU Sx
α −δ
Switzerland 1.02 1.05 7.0%
USA 1 1 7.4%
Portugal 0.47 0.12 41%
Mexico 0.30 0.031 104%
China 0.25 0.019 146%
India 0.089 0.001 1,002%
Ethiopia 0.026 0.00003 9,819%
Table 5.1 shows that if Conjecture 5.1 were true, then we should observe extremely high interest rates in
poor countries, because the marginal product of capital would be extremely high.
Lucas (1990) argued that if this were true, then the incentives for investors from rich countries to invest
in poor countries would be huge. When we analyzed the Solow model we assumed that each country was a
closed economy. But no country is a fully closed economy: cross-border investment is possible, even if not
fully unrestricted. If the gures in Table 5.1 were correct, why invest in the US and earn an 7.4% return when
you can invest in Ethiopia and earn a 9, 819% return? If one believed Conjecture 5.1, it would be surprising
to observe any investment in rich countries at all.
5.4 Growth Accounting
Suppose that we observe that GDP in some country has increased and we want to understand why. Is is
because they have invested and the capital stock has increased? Has there been technological progress? Or
is it just that there are more people working? We can use a technique called growth accounting to measure
the contribution of each of these factors. The basic idea is to:
1. Measure how much capital and labor have changed.
2. Figure out how much change in output we should expect from that. This is the key step; here we rely
on combining theory and measurement.
3. Attribute all the changes in output that cannot be accounted for by changes in capital in labor to changes
in productivity.
Start from the production function:
Yt = F (Kt , Lt , At ) (5.4.1)
A couple of things to note about equation (5.4.1). First, we are including technology as a separate argument
instead of assuming it just enters as labor-augmenting. The labor-augmenting case F (K, AL) is a special case
of (5.4.1) but we want to allow technological progress to possibly take other forms. Also, we are including
time subscripts on all the variables because we want to think about how each of them changes over time.
86
5.4. Growth Accounting
Now de ne GDP growth as
Yt+1
gY ≡ −1
Yt
and compute:
F (Kt+1 , Lt+1 , At+1 )
gY = −1
F (Kt , Lt , At )
F (Kt , Lt , At ) + FK (Kt , Lt , At ) (Kt+1 − Kt ) + FL (Kt , Lt , At ) (Lt+1 − Lt ) + FA (Kt , Lt , At ) (At+1 − At )
≈ −1
F (Kt , Lt , At )
FK (Kt , Lt , At ) Kt (Kt+1 − Kt ) FL (Kt , Lt , At ) Lt (Lt+1 − Lt ) FA (Kt , Lt , At ) At (At+1 − At )
= + +
F (Kt , Lt , At ) Kt F (Kt , Lt , At ) Lt F (Kt , Lt , At ) At
| {z } | {z } | {z } | {z } | {z }
Capital Share Growth Rate of Capital Labor Share Growth Rate of Labor Solow Residual
(5.4.2)
The rst step is a rst-order Taylor approximation and the second is just rearranging.
What is equation (5.4.2) telling us? It's giving us a way to decompose the growth we observe into three
terms. The rst says how much the economy would have grown just as a result of capital accumulation, all
else equal. The second tells us how much the economy would have grown just due to changes in the size of the
labor force, all else equal. The last term measures all the growth that is not explained by changes in measured
factors of production; we collect all of this and call it a Solow residual.
What is the logic behind equation (5.4.2)? Other things being equal, more capital will increase GDP by
an amount equal to the marginal product of capital and more labor will increase GDP by an amount equal to
the marginal product of labor. So if we want to determine how much extra production is the result of changes
in capital and labor we need to know their respective marginal products. The key insight that we get from
equation (5.4.2) is that looking at each factor's share of GDP is precisely what we need to do to infer what their
respective marginal products are. Therefore the equation is telling us what things we need to measure. GDP
growth and factor shares can be read o directly from GDP accounts. The growth rate of the capital stock is
a little bit trickier because one needs to measure the original capital stock, estimate depreciation and measure
investment, but each of these things can be done, even if perhaps not perfectly. Measuring the growth rate
of labor is not just a question of measuring population growth (which is quite easy) because changes in labor
force participation, unemployment or even hours worked per employed worker can be large. With su ciently
good labor market data, this is not hard to measure either.
This leaves us with the Solow residual as the only part of equation (5.4.2) that we can't measure directly.
What do we do about it? The key is in the name: it's known as a Solow residual because you can measure
it by measuring everything else in equation (5.4.2) and then solving for what the residual needs to be for the
equation to hold, i.e.:
Solow Residual = GDP growth − Capital share × Capital Growth − Labor Share × Labor Growth
What do we do with the Solow residual? We interpret it as capturing the contribution of everything
other than accumulation of capital and increases in labor to GDP growth. It includes everything from literal
technological progress to changes in policies that lead to better (or worse) allocation of resources. It is
sometimes referred to as Total Factor Productivity or TFP.
87
5.4. Growth Accounting
Example 5.1.
In the 1960s there was a lot of worry in the West about economic growth in the USSR. There was little
reliable information about the performance of the Soviet economy but many observers had the impression
that it was growing very fast. In the context of the Cold War, many in the West were panicking about
this, for two main reasons. First, if the Soviet economy kept growing so fast, they would be able to
aord more military expenditure, changing the balance of power. Second, if they were growing so fast,
maybe they were on to something? Perhaps centralized direction of the economy was a superior technique
for economic management? One of the more famous growth-accounting exercises was undertaken to try
to answer this question. A big part of the eort was trying to reconstruct gures for GDP, the labor
force and capital accumulation, which the USSR didn't publish. After that, applying formula (5.4.2) was
relatively straightforward. Powell (1968) found that growth in the USSR between 1928 and 1966 could
be decomposed as follows:
a
GDP Growth K Share Capital Growth L Share Labor Growth Residual
5.4% 0.4 6.5% 0.6 2.8% 1.1%
This decomposition teaches several lessons. First, the Soviet economy was indeed growing very fast,
more than 2 percentage points faster than the US. O cial Soviet gures claimed even faster growth,
around 9% per year, but even 5.4% is very fast growth. Compounded over more than three decades,
this was a huge transformation. Second, this growth was not mysterious. It was in large part a result
of very high investment rates which led to fast capital accumulation and of increases in the labor force.
Investment rose from 8% of GDP at the beginning of the period to 31% of GDP towards the end. The
increase in the labor force was faster than population growth, especially due to the increased labor force
participation of women. Third, the rate of productivity growth was unremarkable, close to US levels.
Fourth, and most controversially, the fast rate of growth should not be expected to continue. Cold
War strategists should not panic. Does this conclusion follow directly from the accounting exercise? No.
But suppose that we conclude that the combination of policies pursued by the USSR boils down to: high
investment, an increase in labor force participation and mediocre TFP growth, and this is expected to
continue. High investment leads to growth for a while but eventually runs into the diminishing marginal
product of capital, as we saw in Chapter 4. Increases in labor force participation cannot continue:
eventually almost everyone is working. So unless productivity growth somehow accelerates, a simple
application of the Solow model would suggest that the USSR should be expected to slow down. To be
fair, we don't have a great understanding of what determines productivity growth, so the prediction of
a slowdown was conditional on the assumption that productivity growth would not suddenly accelerate.
The performance of the Soviet economy in the 1970s and 1980s was a great vindication of this prediction.
a Factor Shares are harder to measure in an economy that does not rely on markets, as was the case in the USSR. The
capital share of 0.4 is one point in the range of estimates that Powell (1968) considered. Also, for the period 1928-1937, the
decision of whether to measure GDP at base-year prices or nal-year prices makes a big dierence. The reason is that the
manufacturing sector was expanding more than the agricultural sector at the same time that the prices of manufactured
goods were falling. The 5.4% gure below corresponds to using nal-year prices; using base-year prices gives 6.7%.
88
5.5. TFP Dierences
5.5 Where do TFP dierences come from?
We have quite good evidence that production functions are not the same in all countries, that some countries
can simply extract more output from the same amount of capital and labor. Furthermore, the Solow growth
model predicts that growth in productivity is the only way sustain economic growth in the long run. What do
we know about what explains TFP dierences across countries? There are several conjectures, not necessarily
incompatible with each other.
Human Capital
So far we've been treating all labor as though it was the same, but dierent people's labor may contribute
dierently to total output. In fact, we have good evidence that this is the case, because dierent people get
paid very dierent wages. Economists use the term human capital to describe the productivity dierences
that are embedded in people's knowledge, skills or talent. Calling this human capital hints at the idea that
this is something that can be accumulated, for instance through education. One possible explanation for why
dierent countries have dierent productivity is that their labor is not actually the same because in some
countries labor has a lot more human capital in it.
Hall and Jones (1999) compute dierences in productivity across countries by directly measuring the
capital stock and computing a human-capital-adjusted measure of the labor supply. Their basic idea is to use
a worker's years of education as a measure of human capital. But how do you choose the units? Is a worker
with 12 years of education equivalent to 2 workers with no education? Or more? We can use evidence from
the labor market to answer this. Suppose we measure wages for workers with dierent levels of education.
Under the assumption that markets are competitive, each worker gets paid their marginal product so wages
are the correct measure of human capital. If we measure an empirical relation that says:
Wage = w (Years of Education)
then we can use the function w to convert years of education into units of human capital. Then we use
education data across countries to convert the labor force into human-capital-adjusted labor force.
Do dierences in human capital explain the cross-country dierences in productivity that we nd when we
use unadjusted labor? In part, but large dierences remain unexplained.
Figure 5.5.1 repeats the exercise from Figure 5.3.3, but using information on both human capital and
physical capital to derive the predicted level of GDP per capita. Comparing the two gures we can see that
dierences in human capital help to close some of the gap between predicted GDP per capita and actual GDP
per capita. However, most countries still have GDP per capita that is much lower than predicted. In other
words, low human and physical capital are not enough to account for the relative poverty of poor countries.
Geography
Figure 5.5.2 shows a map of the world with colors representing levels of GDP per capita.
89
5.5. TFP Dierences
Fig. 5.5.1: Predicted GDP
per capita on the basis of cap-
ital stock human capital com-
pared to actual GDP per capita.
Source: Hall and Jones (1999).
Fig. 5.5.2:GDP per capita at
PPP in 2017. Source: World
Bank.
There is a very strong pattern. Countries in colder climates tend to be much richer than countries in hotter
climates. Most of the poorest countries in the world are between the tropics. There are a number of reasons
why that might be. Sachs (2001) argued that one reason is that climate itself matters.
Crop yields are typically lower in very hot weather, higher in temperate weather and lower again in very
cold weather. This might be a xed aspect of the technology that aects productivity directly. On the other
hand, agriculture is not such a large part of the economy, especially in rich countries, so this is unlikely to
be the full explanation. In other words, the reason Sweden is rich is not that crop yields are higher than in
90
5.5. TFP Dierences
Kenya.
Another channel that links geography to productivity is disease. Many diseases, like malaria, are endemic
to hot climates. In addition to killing millions of people, these diseases may aect productivity by hindering
the physical development of children, by keeping children away from school and hurting their learning, by
keeping adults away from jobs, etc.
Institutions
The term institutions is used to refer to a whole number of political and social conditions that vary across
countries: democracy, the adherence to the rule of law, political transparency, respect for property rights, etc.
One conjecture is that these institutions could have a large eect on productivity. You'll see an example of
why that might be in Exercise 5.5.
At a basic level, we do observe a strong correlation between institutions and TFP. The same countries that
are richer are more democratic. Whether political and social institutions are the reason for their prosperity
is less clear. Perhaps it's just that as countries become richer (for other reasons), the political balance shifts
toward more democracy and openness.
One famous study (Acemoglu et al., 2001) made the case that the relationship is indeed causal, at least
in part. It focused on countries that had been at some point colonized by European powers. The study
measured the mortality rate of European settlers in these colonies and found a strong correlation between this
and present-day institutions. The measure of institutions is a risk of expropriation index developed by
the consultancy Political Risk Services that attempts to measure how likely it is that an investor's property
will be taken from them (a high score means a low chance of expropriation). The left panel of Figure 5.5.3
shows how the mortality of European settlers in colonized countries correlates with this risk-of-expropriation
measure. The correlation is strong and negative: countries where settlers had high mortality now have higher
risk of expropriation.
The authors of the study argued that this is because of the dierent types of colonization undertaken
in dierent places. In low-mortality places, Europeans had higher hopes of settling, so they brought with
them the relatively more open institutions that were developing in Europe. In high-mortality places, they had
less hope of settling so they set up what the authors call extractive institutions, which used political and
military authority to extract natural and other resources. These extractive institutions have persisted into
the post-colonial period and, the argument goes, are what explains today's low productivity. The right panel
of Figure 5.5.3 shows how how the mortality of European settlers correlates with GDP per capita. Countries
where settlers had high mortality now have lower GDP per capita.
There is some debate as to whether Acemoglu et al. (2001) have really nailed down the causal argument.
Could there be other explanations of why the places where Europeans had lower mortality in the colonial era
are wealthier today? At least two alternative explanations have been proposed. One is that the same factors
that contributed to high mortality (in particular, tropical diseases) are also directly responsible for today's
low productivity. Another is that in places of low mortality, European settlers brought other things besides
more open institutions: technology, human capital, trade links, etc.
91
5.5. TFP Dierences
Fig. 5.5.3: Mortality of colonial settlers, present-day institutions and present-day GDP per capita. Source:
Acemoglu et al. (2001).
Resource Allocation and Misallocation
Maybe poor countries get less output out of capital and labor because they don't put them to good use.
Suppose that there are many dierent rms in each country. To maximize output, each unit of capital and
each worker should be working for the rms where their marginal product is highest, which implies that the
marginal products are equated across rms. There might be several factors preventing this from happening,
and these could vary by country.
One possible factor is entry regulation. The World Bank measures all the requirements for starting a
business in dierent countries: permits, registration delays, etc. These vary greatly by country. One possibility
is that many potential businesses don't even get started because of these barriers to entry, so capital and labor
get allocated to less productive uses. Figure 5.5.4 shows the correlation between the number of days it takes
to start a new business and GDP per capita, which is negative. One possible interpretation is that the barriers
to entry for new rms, of which delay is just one example, lead to misallocation of resources and therefore
have a causal eect on productivity. Alternatively, it could be that countries that are richer for other reasons
have speedier procedures for registering new rms.
Similar eects could result from lack of credit, restrictions on foreign investment, monopoly rights or
taxation or regulation that is uneven across rms. All of these could lead capital and labor away from their
most productive uses.
One interesting piece of evidence on this comes from Bloom et al. (2013). They surveyed a sample of
textile rms near Mumbai, India, and found that they diered greatly in their productivity and management
practices. Why were the most e cient rms not expanding and taking business away from the less e cient
rms? Bloom et al. (2013) found that one of the main determinants of a rm's size is the number of male
92
5.5. TFP Dierences
Fig. 5.5.4: Time to start
a new business and GDP per
capita. Each observation rep-
resents a country. Source:
World Bank (2011).
family members the owner had. Their interpretation of this nding is that because enforcing contracts is not
easy in India, it is very hard for the owner of a rm to delegate management to outsiders who are not family
members.
8 This limits the extent to which the more productive rms can expand, so the less productive rms
end up employing an ine ciently large amount of capital and labor.
Hsieh and Klenow (2009) estimated the dispersion in the marginal product of capital and labor in manu-
facturing rms in China, India and the US. Based on these estimates, they calculated that if China and India
could reduce the degree of misallocation of capital and labor across rms to the levels observed in the US,
their TFPs could increase by 30% − 50% and 40% − 60% respectively.
Exercises
5.1 Quantifying the Solow Model
Suppose the economy is described by the Solow growth model with the parameter values we used in Section
5.2.
(a) Suppose the economy starts out with a level of GDP per e ciency unit of labor that is only 10% of
its steady state level, i.e. ỹ0 = 0.1ỹss . What is the initial capital stock k̃0 ? Use a spreadsheet to
ỹt
compute k̃t and ỹt for t= 1, 2, . . . , 100. How many years does it take for ỹss > 0.95?
(b) Suppose the economy begins in steady state but there is a fall in the rate of population growth from
n = 0.01 to n = 0. How much higher will GDP per e ciency unit of labor be in the new steady
8
Also: sexism. Only male family members were associated with rm size.
93
5.5. TFP Dierences
state? Use a spreadsheet to compute k̃t and ỹt for t = 1, 2, . . . , 100. How many years does it take for
ỹt
ỹss > 0.95?
5.2 The Speed of Convergence
Suppose the economy is described by the Solow growth model with a Cobb-Douglas production function,
no population growth and no technological progress. The economy is not in steady state. GDP capita in
period t is yt = θyss , with θ ∈ (0, 1).
kt
(a) How far away from steady state is the capital stock? Find an expression for the level of
kss that
would be consistent with yt = θyss , in terms of θ and parameters.
(b) Find an expression for kss , the steady state level of capital, in terms of parameters.
(c) Use your results from parts (a) and (b) and equation (5.3.1) to nd an expression for gy , the rate of
growth of GDP, as a function of θ and parameters.
(d) Let α = 0.35 and δ = 0.04.
Compute gy for an economy whose GDP is θ = 0.8 times its steady-state
gy
level? Compute
1−θ . What fraction of the gap between yt and yss does the economy close in one
year? Repeat this calculation for θ = 0.9 and θ = 0.99.
5.3 Causes of Growth and Growth Accounting
Suppose an economy is well described by the steady state of the Solow Growth Model with constant
technological progress and no population growth. Imagine we take data from this economy and do a
growth-accounting exercise. How much growth will we attribute to capital accumulation and how much to
technology? How does this relate to the result that says that without technological progress there would
be no growth in the steady state?
5.4 Measurement of the Capital Stock
The capital stock is not easy to measure. A common way to try to do this is to:
• guess a value for K0 at some date in the past (maybe when statistics were rst collected),
• measure investment every year thereafter (which is also not that easy), and
• use the equation
Kt+1 = (1 − δ) Kt + It (5.5.1)
to compute the capital stock for every year thereafter.
Consider an economy that is in a steady state without technological progress (and has been there for an
in nitely long time). Every year the level of investment is:
It = 0.2
and the depreciation rate is:
δ = 0.1
94
5.5. TFP Dierences
(a) What is the capital stock?
(b) The statistical o ce only started measuring investment in year 0. By then the economy was already
in steady state. They guessed (incorrectly) that the capital stock in year 0 was K0 = 1. Use equation
(5.5.1) to compute how the estimate of the capital stock changed over time after year 0 and plot your
answer. (You don't need to provide a closed form solution, a nice graph made with a spreadsheet is
enough). Compute the ratio:
KEST IM AT E
KT RU E
for the years 5, 10, and 50. (Again, you can just take the numbers from a spreadsheet).
(c) Suppose now that the statistics o ce started collecting statistics a long time ago (an in nite time
ago) so we don't have the problem of making the wrong initial guess. However, instead of using the
correct value of δ, the statistical o ce incorrectly believes that δ = 0.05. What is their estimate of
the capital stock? Why does it dier from the truth?
(d) The production function in this economy is
Y = AK α L1−α (5.5.2)
and there are accurate data on GDP and on total hours of labor supply. Using the inaccurate estimate
of K from part (c), plus accurate data on total labor and GDP, an economist is trying to measure
the economy's productivity, i.e. to solve for A in equation (5.5.2). Find an expression for
AEST IM AT E
AT RU E
as a function of
KEST IM AT E
KT RU E
Given the numbers from part (c) and using α = 0.35, how far would the economist's estimate be
from the truth?
5.5 Sources of TFP
Suppose that the true production function in Gotham is
Y = AK α L1−α (5.5.3)
Unfortunately, crime is a huge problem in Gotham, so that for each worker doing actual work rms need
to hire γ security guards just to protect their products from being stolen. The security guards will of
course describe their activity as work even though they are not actually producing anything. Use the
notation N to refer to the total labor force (including workers and guards) and denote the number of
actual production workers by L.
(a) Find an expression for total output as a function of A, K , N , γ and α.
95
5.5. TFP Dierences
(b) Write down the problem of a rm that has to choose capital and labor to maximize pro ts. Notice
that the rm will have to pay a wage to the security guards even though they will not produce
anything.
(c) If the representative rm hires all the workers and rents all the capital, what will be the wage and
the rental rate of capital? Express it as a function of A, K , N , γ and α.
(d) Suppose an economist studying Gotham is trying to estimate A using equation (5.5.3). The economist
has accurate data on K, N and Y. However, the economist doesn't really know whether workers
are involved in production or in security services: in national statistics they all look employed.
Therefore the economist will plug in the value of N instead of the value of L into the estimate of A.
What will the economist's estimate of A be? How does it compare to the true value of A?
(e) How does this relate to the ndings that link GDP levels to social and political institutions?
(f ) Suppose that, in a economy that didn't have the crime problem of Gotham, the government attempted
to create jobs by mandating that rms hire γ assistants for every production worker. The job
of assistants is to look at production workers all day long and not do anything. Using the analysis
above, what would be the eects of such a policy?
5.6 Interest Rates
Suppose we observe that Usuria (a closed economy) grows at approximately 6% a year, and we are trying
to understand why. We know the labor force has been constant.
• Conjecture 1: The economy has been at a steady-state-with-technological-progress. There has been
K
capital accumulation just to maintain
AL constant but the cause of growth has been TFP growth.
• Conjecture 2: The economy started from a very low level of capital stock (below steady state) and
has been growing because it is converging to the steady state, but TFP has been constant.
Ideally, if we wanted to distinguish between Conjecture 1 and Conjecture 2 we could do a growth-accounting
exercise. Unfortunately, Usuria does not collect reliable statistics on capital accumulation that would
enable us to do this. We do, however, have data on interest rates in Usuria. How would one use these
data to distinguish between Conjecture 1 and Conjecture 2? Be as mathematically precise as possible.
5.7 GDP Accounting, Interest Rates and Growth Accounting
Proletaria is a mythical country in Central Asia, where the currency is the ruble. The exchange rate is 10
rubles per euro.
Suppose the production function in Proletaria is given by:
Y = K α L1−α
Y
(a) Find expressions for the output-to-capital ratio
K and the marginal product of capital FK as functions
of K and L.
During 2014, the following events took place in Proletaria.
96
5.5. TFP Dierences
• Kapitas, the main manufacturer in the country, imported an industrial welder made in Germany, for
which it paid 1,000 euro.
• By the end of the year, the industrial welder was no longer new. Its estimated value in the resale
market was 960 euro.
• 100 workers worked for Kapitas the entire year making screws and nails, using the new welder. Each
of them received wages for 350 rubles.
• Each of the Kapitas workers paid 100 rubles in income taxes.
• The total output of Kapitas consisted of 1 million screws and 1 million nails. All of it was exported
to Austria for a total of 4,000 euro.
• The government of Proletaria employed one of the 100 workers as Chief of the Secret Police (in
addition to his factory job) to maintain law and order, and paid her a salary of 10,000 rubles.
• The workers ate beef imported from France, which cost a total of 2,500 euro.
(b) Construct GDP accounts (in rubles) for Proletaria by production, income and expenditure.
(c) What were the income shares of labor and capital?
(d) Suppose that we know that the capital stock in Proletaria is 100,000 rubles and that the depreciation
rate of the industrial welder is typical for this country's capital stock. What interest rate should we
expect to observe?
(e) During 2015, additional investment in Proletaria has exceeded depreciation so that the capital stock
now stands at 120,000 rubles. The labor force also grew thanks to immigration, and now consists of
105 workers instead of 100. GDP during 2015 was 55,000 rubles (prices were constant). How much of
the growth in GDP between 2014 and 2015 can be attributed to growth in Total Factor Productivity?
5.8 National Accounts and The Golden Rule
In 2018, the following events happened in Aurum, which is a closed economy that uses the Denarius
(plural: Denarii) as its currency.
• Grano, Inc. hired Cornelia to plant and harvest 500 tons of wheat on a large plot of land using a
tractor. Both the land and the tractor are owned by Grano, Inc.
It paid Cornelia 200 Denarii for her work on the harvest.
It sold the wheat to Panem, Inc. for 1,000 Denarii.
• The tractor the Cornelia used:
Was new at the beginning of the year. It was worth 2,000 Denarii.
Was no longer new at the end of the year. By then it was worth 1,800 Denarii.
• Panem, Inc. hired Gaius to grind the wheat in its mill and produce a million loaves of bread.
It paid Gaius 1,400 Denarii for his work.
It sold the bread for 4,000 Denarii to the hungry citizens, who eat it.
97
5.5. TFP Dierences
• Fabrica, Inc. hired Livia to build a combine harvester.
It paid Livia 400 Denarii for her work.
It sold the combine harvester to Grano, Inc. for 1,000 Denarii.
(a) Construct GDP accounts (in Denarii) for Aurum by production, income and expenditure.
(b) What were the income shares of labor and capital?
(c) What was the savings rate?
(d) Assume the following:
• the savings rate is constant,
• the production function has the form:
Y = AK α L1−α
and there is no technological progress,
• the depreciation rate for the tractor is typical for the capital stock as a whole,
• the population is constant.
In the long run, would the level of consumption be higher if the economy slightly increased its saving
rate?
5.9 Disease and TFP
The production function in In uenzistan is:
1−α
Y = K α (AH) (5.5.4)
where H is the total number of hours of work.
(a) Solve equation (5.5.4) for A, so that you end up with an expression for A in terms of all the other
variables and parameters.
(b) Suppose an economist wants to measure A. The economist has o cial data from the In uenzistan
Statistics O ce on:
• total hours worked by all workers,
• the total level of investment, which has been constant for many many years,
• an estimate of the average rate of depreciation of the capital stock,
• GDP and its subcomponents, measured by the income method.
How can he use this data to get an estimate of A? Don't be vague: provide exact formulas that
describe each step precisely.
(c) Suppose now that the economist has the same data as above except he doesn't know total hours
worked. Instead, the In uenzistan Statistics O ce only keeps records on the total number of employed
98
5.5. TFP Dierences
workers (call this number L), but not on how many hours each of them works. In order to get an
estimate of A, the economist assumes that the average worker works N hours per year, which is the
number of hours worked by the average worker in neighboring Healthistan. Write down an expression
for the economist's estimate of A (call this number Â).
(d) Unfortunately, the residents of In uenzistan keep getting sick, so they have to spend a signi cant
portion of their time recovering at home instead of working. As a result, they each work xN hours
Â
instead of N hours, where x < 1. Derive an expression for
A , i.e. for the ratio of the economists'
estimate of A to its true value. Interpret your nding.
99
PART III
Microeconomic Foundations
This part of the book looks at the microeconomics that is the basis of modern
macroeconomics. The Solow growth model assumes that the economy saves an
exogenous fraction of its output and, because everyone works, the size of the labor
force is exogenous. In this part of the book, we analyze the economic forces that shape
individuals' decisions to consume, save, work and invest and how they all t together.
In Chapter 6, we think about intertemporal decision making. Consumption and
savings decisions involve the relationship between the present and the future. We study
a simple two-period model of these decisions and use it to think about the evidence
on how people make these decisions. We then extend it to think about more than two
periods.
In Chapter 7, we think about the labor market. First we go over some of the
statistics that are used to measure the labor market. We then study a model of how
people divide their time between the labor market and everything else, and how labor
market equilibrium is determined. Finally, we think of reasons why the labor market
might not clear, in order to think about unemployment.
In Chapter 8, we look at investment decisions and how they relate to expecta-
tions about the future, prices of future goods, uncertainty, and asset prices.
In Chapter 9, we study general equilibrium: how all the decisions by households
and rms t together to determine economic outcomes. We also study the First
Welfare Theorem, which provides conditions under which a market economy is Pareto
E cient.
101
CHAPTER 6
Consumption and Saving
6.1 The Keynesian View of Consumption
Keynes (1936) made the following observation about consumption decisions:
The fundamental psychological law, upon which we are entitled to depend with great con dence
both a priori from our knowledge of human nature and from the detailed facts of experience, is
that men are disposed, as a rule and on the average, to increase their consumption as their income
increases but not by as much as the increase in the income.
What does this mean? Is this theory correct? Let's rst translate this theory into mathematical language and
then try to assess it. First of all, it is stating that how much people consume depends on their income, which
seems reasonable enough. Let's use C to denote consumption and Y to denote income. Keynes says that there
is a function c(·) (sometimes known as a consumption function) that relates consumption to income:
C = c (Y ) (6.1.1)
Furthermore, he is saying that consumption depends on income in a speci c way. Here Keynes' language is
more ambiguous but one possible interpretation is that he is saying that:
c0 (Y ) < 1
so when income rises by one dollar, consumption rises but by less than one dollar. The quantity c0 (Y ) is known
as the marginal propensity to consume. It measures how much of an extra dollar of income is dedicated to
consumption.
Possibly, depending on how one interprets his language, Keynes is saying something more: that if income
rises 1%, consumption rises, but by less than 1%, so the elasticity of consumption with respect to income is
positive but less than 1. Mathematically:
c0 (Y )Y ∂ log(c(Y ))
= <1
c(Y ) ∂ log(Y )
103
6.1. Keynesian
One way to test this conjecture is to take a sample of households, measure their income, measure their
∂ log(c(Y ))
consumption and see whether the best t of equation (6.1.1) has c0 (Y ) < 1 and/or ∂ log(Y ) < 1. Figure 6.1.1
shows the result of doing precisely that. The Consumer Expenditure Survey asks a sample of households to
report their income and their consumption (among other things). The gure shows scatterplots of consumption
against income for these households, both in absolute terms and in logarithmic scale, to measure c0 (Y ) and
c0 (Y )Y 1
c(Y ) respectively. The evidence seems consistent with both interpretations of Keynes's statement: both
c0 (Y )Y
the best t estimate of c0 (Y ) 0
c(Y ) are lower than 1. c (Y ) is approximately 0.25, so households whose
and
0
c (Y )Y
income is one more dollar spend an additional 25 cents.
c(Y ) is approximately 0.55, so households with 1%
higher income spend approximately 0.55% more.
Fig. 6.1.1:Evidence on the Keynesian consumption function. Each dot represents a household. Source:
Consumer Expenditure Survey, 2014.
For some time, around the mid-20th century, this type of evidence was considered quite conclusive, leading
to a rm belief in the Keynesian consumption function as a good description of consumption behavior. This
led to following kind of speculation: what is going to happen as the economy's productive capacity expands
over time? If the elasticity of consumption with respect to income is less than 1, this implies that over time,
C
as income increases, the ratio
Y will fall. Is the economy going to produce more and more goods that nobody
wants to consume? What are we going to do with all these goods?
2 Is there going to be massive unemployment
because nobody wants all the stu that we'd produce if everyone was working?
Aggregate data gives us a way to test this conjecture. Figure 6.1.2 shows the relationship between aggregate
consumption and aggregate income from national accounts. In the left panel we see the relationship in the
c0 (Y )Y
United States, where each dot represents a dierent year. The best t estimate of
c(Y ) is 0.97. There is a
1
The logarithmic scale graph only includes households with quarterly income of at least $1,000.
2
You can see echoes of this preoccupation in Orwell's famous novel 1984. It was not uncommon to interpret war, and the huge
destruction that is brings about, as a solution to the problem of over-production.
104
6.2. Two Period Model
simple explanation for this: consumption has been close to a constant fraction of GDP, approximately 65%.
c0 (Y )Y
If c(Y ) = 0.65Y then
c(Y ) = 1. The right hand panel shows the relationship across countries, where each
c0 (Y )Y
dot represents a dierent country for the year 2011. In this case, the best- t estimate of
c(Y ) is 0.85. In
both cases the estimate is much closer to 1 than in the individual household data. Overall, it does not seem
to be the case that countries consume a lower fraction of their income as they grow rich.
3
Fig. 6.1.2: Evidence on the Keynesian consumption function from aggregate data. The left panel is US
time-series evidence; the right panel is cross-country evidence. Sources: NIPA and Feenstra et al. (2015)
In the aggregate data over time we don't see the pattern that we see in the cross-sectional data. The
preoccupation about decreasing consumption rates over time seems to be unwarranted. What is going on?
Why do the two kinds of data look so dierent?
6.2 A Two-Period Model of Consumption
Let's take a step back and try to develop a theory of how households make consumption decisions and see
whether this can help us understand some of the patterns we just saw. We'll start from a very simple example
and then think about more features.
Let's imagine that this household is going to live for two periods. In period 1 they will obtain income y1
and in period 2 they will obtain income y2 . They have to decide how much they are going to consume in
period 1. The advantage of consuming is that they like to consume; the advantage of not consuming is that by
saving they can aord to consume more in period 2, which they also like. Let's assume that their preferences
3
Note the possibility of reverse causality in the cross-country data. The Solow model predicts that, other things being equal,
c0 (Y )Y
countries that choose to save more and consume less will have higher GDP. This will produce an estimate of lower than
c(Y )
1 even if the true elasticity is equal to 1.
105
6.2. Two Period Model
are described by the following utility function:
U (c1 , c2 ) = u (c1 ) + βu (c2 ) (6.2.1)
This way of thinking about the consumption decision makes it mathematically equivalent to the kind of two-
good consumption problem studied in microeconomics: here the two goods are c1 (consumption in period 1)
and c2 (consumption in period 2). In addition, for simplicity, we are assuming that the utility function is
additively separable in the two goods and that the only dierence in how much they care about each of them
is the term β. β is just some number, typically assumed to be less than 1 to represent impatience: the same
level of consumption gives the household more utility if it comes now than if it comes in the next period.
Now that we have preferences, we need to think about the household's budget. How much of each of the
two goods can the household aord? What is their relative price? Imagine that the household consumes c1 in
the rst period. This means it can save:
a = y1 − c1 (6.2.2)
The household earns interest on these savings, so by saving a, in period 2 it can aord to consume:
c2 = y2 + (1 + r) a (6.2.3)
where r is the real interest rate between periods 1 and 2. Replacing a from (6.2.2) into (6.2.3) and rearranging,
we get
1 1
c1 + c2 = y1 + y2 (6.2.4)
1+r 1+r
Equation (6.2.4) is a standard budget constraint for a two-good consumption problem.
1
The term
1+r is the price of period-2 goods in terms of period-1 goods. Why does this make sense? If you
sell one period-1 good to the market, the market is willing to give you (1 + r) period-2 goods. This is exactly
what a price means: at what rate is the market willing to exchange one good for another. High interest rates
mean that period-1 goods are expensive relative to period-2 goods: the market is willing to provide a lot of
period-2 goods in exchange for period-1 goods. Conversely, low interest rates mean that period-1 goods are
cheap.
The right hand side of (6.2.4) is the household's total budget. Why? The household's income consists
of y1 period-1 goods and y2 period-2 goods. Adding them up at their respective market prices tells us how
1
much the household can aord in total. The expression y1 + 1+r y2 is called the present value of income and
1
likewise the expression c1 + c
1+r 2 is called the present value of consumption. The idea of a present value
is to express future quantities in terms of the amount of the present goods that the are equivalent to at the
1
relevant market prices. Here the price of future goods is
1+r so we multiply by this term in order to add them
to present goods.
The term a in equations (6.2.2) and (6.2.3) represents savings, but we haven't said anything yet about
whether a needs to be a positive number. Do equations (6.2.2)-(6.2.4) also apply when a<0 ? It depends on
what we think about the household's ability to borrow. If we assume that the household can borrow as much
as it wants at the interest rate r (and must always pay back its debts), then it is OK to allow for negative
106
6.2. Two Period Model
values of a, and the budget constraint (6.2.4) still applies. a < 0 simply means that the household is borrowing
in order to pay for c1 > y 1 . For now we'll make this assumption; later on we'll think about what happens
when the household cannot borrow.
We are going to imagine that the household takes as given its current and future income y1 and y2 and
the interest rate and simply solves a standard consumer optimization problem:
4
max u (c1 ) + βu (c2 )
s.t. (6.2.5)
1 1
c1 + c2 ≤ y 1 + y2
1+r 1+r
Figure 6.2.1 shows the solution to problem (6.2.5). As is standard in microeconomics, the household will
choose the highest indierence curve it can aord, which implies that it will pick a point where the indierence
curve is tangent to the budget constraint. Notice two properties of the budget constraint. First, its slope is
− (1 + r). As usual, the slope of the budget constraint is the relative price. Higher interest rates mean a
steeper budget constraint. Second, the budget constraint goes through the point (y1 , y2 ) since the household
has the option to just consume its income each period.
Fig. 6.2.1: The consumption-
savings decision as a two-good
consumption problem.
We can also nd the solution to problem (6.2.5) from its rst order conditions. The Lagrangian is:
5
1 1
L (c1 , c2 , λ) = u (c1 ) + βu (c2 ) − λ c1 + c2 − y1 − y2
1+r 1+r
4
There is some disagreement about whether budget constraints should be written as equalities or as weak inequalities. I like
the version with weak inequality because it says that the household could, in principle, not spend all its income. Since this never
happens anyway, it's not a big deal which way we write it.
5
This problem is su ciently simple that we don't need to use a Lagrangian to solve it. We could just as easily replace
107
6.2. Two Period Model
λ is the Lagrange multiplier of the budget constraint. It has the usual interpretation of the marginal utility
of a unit of wealth. The rst order conditions of the problem are:
u0 (c1 ) − λ = 0 (6.2.6)
1
βu0 (c2 ) − λ =0 (6.2.7)
1+r
Solving equation (6.2.6) for λ, replacing in equation (6.2.7) and rearranging we obtain:
u0 (c1 ) = β (1 + r) u0 (c2 ) (6.2.8)
Equation (6.2.8) is known as an Euler equation and plays a central role in modern macroeconomics. It
describes how households trade o the present against the future, and has the following interpretation. Suppose
a household is deciding whether to allocate one unit of wealth to consumption or to save it for the future. If it
consumes it, it will obtain the marginal utility of present consumption, u0 (c1 ). This gives us the left hand side
of (6.2.8). If instead the household saves, it obtains (1 + r) units of future wealth because the market pays
interest. Each unit of future wealth gives the household the marginal utility of future consumption, which is
u0 (c2 ), multiplied by β to account for the household's impatience. This gives us the right hand side of (6.2.8).
At the margin, the household must be indierent between allocating the last unit of wealth between these two
alternatives, so (6.2.8) must hold. (6.2.8) is also the algebraic representation of the tangency condition shown
in Figure 6.2.1. The slope of the household's indierence curve is given by the marginal rate of substitution
u0 (c1 )
between period-1 consumption and period-2 consumption:
βu0 (c2 ) . The slope of the budget constraint, as we
saw, is 1 + r, so (6.2.8) says that the two are equated.
Some Examples
Figure 6.2.2 shows two possible patterns of income over time and the household's consumption decision in
each case. The left panel shows a household for whom period 1 represents their working age and period 2
represents a time in which they are planning to retire, so y2 = 0. Understanding that their income will be low
in the future, they choose a = y1 − c1 > 0 in order to be able to consume while they are retired. The right
panel represents the opposite case. Here the household has low y1 and much higher y2 , so they are optimistic
about future income compared to current income. In this example the household chooses a = y1 − c1 < 0, so
they borrow to consume more than their income.
Both examples have some features in common. In both cases expectations about the future, not just
current income, aect consumption decisions. In both cases the household is trying to even out or smooth
consumption over time, i.e. they are using borrowing and saving to prevent their consumption from moving
up and down as much as income does.
c2 = y2 + (1 + r) (y1 − c1 ) into the objective function and solve:
max u (c1 ) + βu (y2 + (1 + r) (y1 − c1 ))
c1
which gives of the same solution.
108
6.2. Two Period Model
Fig. 6.2.2: Consumption decisions in two examples.
The Eect of Interest Rates
Let's imagine that interest rates change. How do households change their consumption? The answer to this
question is going to play an important role in some of the models of the entire economy that we'll analyze
later. For now, we are going to study the question in isolation, just looking at the response of an individual
household to an exogenous change in the interest rate. For concreteness, let's imagine that the interest rate
rises.
Let's rst take a look at this question graphically. A change in interest rates can be represented by a change
in the budget constraint, as in Figure 6.2.3. The new budget constraint still crosses the point (y1 , y2 ) because
the household can aord this no matter what the interest rate is, but the slope of the budget constraint is
dierent. With higher interest rates, it becomes steeper. As with any change in prices, this can have both
income and substitution eects.
The substitution eect is straightforward: as we saw before, a higher interest rate means that present
goods have become more expensive relative to future goods. Other things being equal, this would make the
household substitute away from present goods towards future goods, i.e. save more and consume less.
The income eect is a little bit more subtle. Do higher interest rates help or hurt the household? That
depends on whether the household is borrowing or saving to begin with. If the household is saving, then higher
interest rates mean that it is earning more on its savings, which can only help them attain higher utility. This
is the case depicted in Figure 6.2.3. Instead, if the household was borrowing, then higher interest rates means
that it's paying more interest on its loans, which hurts them.
6
Graphically, it's possible to decompose income and substitution eects in the following way. First, imagine
6
There is an additional possibility, which is that the household was choosing to borrow when interest rates were low but saves
instead when the interest rate rises. In this case the income eect could go either way.
109
6.2. Two Period Model
Fig. 6.2.3: Consumption re-
sponse to higher interest rates.
changing the interest rates (and therefore the slope of the budget constraint) but adjusting the position of
the budget constraint so that the household can attain the original indierence curve and ask how much
of each good the household consumes. This is a way of isolating the substitution eect: how much the
household rebalances between present and future consumption due to the new prices while holding utility
constant. Second, move the budget constraint from the adjusted line to the actual new budget constraint.
The dierence between the household's consumption at the adjusted budget and the true new budget measures
the income eect: at the same prices, how much more or less can the household aord.
7
Let's go back to the question of how consumption reacts to a rise in the interest rate. We know that the
substitution eect would make consumption go down and the income eect could go either way. When the
income eect is negative, then both income and substitution eects go in the same direction and we know that
consumption falls when interest rates rise. When the income eect is positive, then income and substitution
eects are pushing in opposite directions and the net eect could go either way. Figures 6.2.4 shows examples
where each of these things happen. On the left panel is the saving for retirement example. Here the household
is saving so the income eect of higher interest rates is positive, and in fact stronger than the substitution
eect, so the household increases its consumption. On the left panel is the optimism example, where the
household was borrowing against its high future income. Here the income eect of higher interest rates is
negative, and reinforces the substitution eect, leading to lower consumption.
7
This way of decomposing income and substitution eects is known as the Hicks decomposition. An alternative is the Slutsky
decomposition, where the substitution eect is measured at the budget such that the original consumption plan is aordable
instead of the original utility level.
110
6.2. Two Period Model
Fig. 6.2.4: Higher interest rates in two examples.
An Explicit Example
If preferences take the CRRA form we can go beyond equation (6.2.8) and get an explicit formula for how
much the household is going to consume.
8 CRRA utility takes the form:
c1−σ
u (c) =
1−σ
so marginal utility is:
u0 (c) = c−σ
Replacing this in equation (6.2.8) gives us:
c−σ −σ
1 = β (1 + r) c2
1
⇒ c2 = [β (1 + r)] σ c1
Now replace c2 into the budget constraint (6.2.4):
1 1 1
c1 + [β (1 + r)] σ c1 = y1 + y2
1+r 1+r
and solve for c1 :
1
y1 + 1+r y2
c1 = 1 1
−1
(6.2.9)
1 + β σ (1 + r) σ
8
CRRA stands for constant relative risk aversion. We rst encountered this functional form in Chapter 2.
111
6.2. Two Period Model
Equation (6.2.9) gives us an explicit formula for how consumption depends on present income, future income
and interest rates. The numerator is the present value of income: consumption is proportional to this. The
denominator captures the eects of the household's impatience (measured by β ), the relative price of period-1
consumption (1 + r ) and the household's willingness to substitute consumption in one period for consumption
in the other (measured by σ ).
The Permanent Income Hypothesis
The model we have been analyzing gives an alternative hypothesis to Keynes's view of how households make
consumption decisions. In this model, consumption does not really depend on current income. Instead, it
depends on the total value of the household's income over time. We can see this directly in the budget
constraint (6.2.4). What determines how much consumption the household can aord is not y1 but rather
1
y1 + 1+r y2 . That same expression shows up in the numerator of equation (6.2.9).
One way of putting this into words is to say that consumption depends on permanent income, i.e. some
sort of average level of income over time. The budget constraint (6.2.4) tells us exactly what's the right way
to take the average: by weighting each period's income by the relevant price and thus computing a present
value. But this is a detail. The broader point is that what matters is average income and not any one period's
income. This way of thinking about consumption became known as the permanent income hypothesis (the
term is due to Friedman (1957)). Let's see what this means.
Suppose we ask: how much does the household's consumption increase if it nds out that its income has
increased? In this model, the answer is it depends. In particular, it depends on whether the increase in
income is perceived as temporary or permanent. Formula (6.2.9) makes this clear.
Suppose that y1 increases but the household does not change its expectation of what y2 is going to be. In
other words, this is perceived as a temporary increase, for instance because this is a worker who just received
a one-time bonus. How much is consumption going to respond?
∂c1 1
= 1
−1
<1 (6.2.10)
∂y1 1
1 + β σ (1 + r) σ
This is the marginal propensity to consume, the answer to the question: how much extra consumption does
the household choose when it gets an additional unit of (temporary) income? Equation (6.2.9) tells us that
the marginal propensity to consume is less than 1. In other words, the household will increase its consumption
by less than the increase in income, as Keynes believed.
Suppose instead that the household perceives that the increase in income is permanent, so that y2 increases
by the same amount as y1 , for instance because this is a worker who has just received a permanent raise. Let's
say the permanent raise is by some amount ∆. How much is consumption going to respond?
1
dc1 1 + 1+r
= 1
−1
(6.2.11)
d∆ 1
1 + β σ (1 + r) σ
Comparing formulas (6.2.10) and (6.2.11) immediately tells us that the marginal propensity to consume out
of permanent income is higher than the marginal propensity to consume out of temporary income. If we make
112
6.2. Two Period Model
the further simpli cation that β (1 + r) = 1, then formula (6.2.11) simpli es and we get:
9
dc1
=1
d∆
In other words, when the household perceives an increase in income as being permanent, it increases its
consumption by the same amount of the increase in income.
Reconciling with the Data
Let's go back to Figures 6.1.1 and 6.1.2. The permanent income hypothesis gives us a way to reconcile this
seemingly contradictory evidence.
If the permanent income hypothesis is correct, then the households with temporarily low income will
c
have relatively high
y because their permanent income is higher than their current income. Conversely, the
households with temporarilyhigh income will have relatively low
c
y because their permanent income is lower
than their current income.
The data in Figure 6.1.1 shows plots consumption and and total income in a single year, without making
the distinction between temporary and permanent. Total income probably re ects a mixture of of temporary
and permanent factors. Suppose we look at households who had high income this year. Some of them will
be in this group because they are always in this group (i.e. they have high permanent income). Others
will be in this group because they had a particularly good year, but their permanent income is not that
high. Conversely, if we look at households who had low income this year, this will include households with
low permanent income and households with not-so-low permanent income who had a bad year. On average,
households with higher total income are more likely to have had a better-than-usual year. According to the
c
permanent income hypothesis, these are precisely the households that should have lower
Y , resulting in the
lower-than-one slopes that we observe.
Once we add up over many people and long periods of time, then the temporary components average
out. Some people in the sample will have had a good year but others will have had a bad year, so measured
average income becomes closer to permanent average income. The permanent income hypothesis tells us that
c0 (Y )Y
consumption should be proportional to permanent income, so we expect
c(Y ) = 1, which is indeed close to
what we observe.
Taxes and Ricardian Equivalence
One of the things we are going to be interested in is how the economy reacts to changes in policies in general
and taxes in particular. One ingredient in answering that question is to gure out how household consumption
will respond to taxes.
We are going to assume that the government wants to make purchases of goods and services equal to G1
and G2 in periods 1 and 2 respectively. For now we are not going to ask why or how the government chooses
G1 and G2 , we'll just take them as given. In order to pay for this spending, the government is going to collect
9
β (1 + r) = 1 means that the household's impatience and the market interest rates exactly oset each other. If we go back to
the Euler equation (6.2.8), this implies that c1 = c2 .
113
6.2. Two Period Model
taxes τ1 and τ2 from the household in periods 1 and 2 respectively. We'll assume that these taxes are lump
sum, meaning that nothing that the household does aects how much tax it owes. The government need not
exactly balance its budget in each period. Just like the household, it can borrow and save as much as it wants
at the interest rate r, but must pay its debts.
The government budget constraint can be written as:
B = G1 − τ1 (6.2.12)
where B is the amount that the government borrows, equal to the dierence between the amount it spends in
period 1 and the taxes it collects. If B < 0, this means the government is saving. In period 2, the budget is:
τ2 = G2 + (1 + r) B (6.2.13)
The government has to collect enough taxes to pay for spending and also pay back its debt with interest.
Solving (6.2.12) for B, replacing in (6.2.13) and rearranging we get:
1 1
G1 + G2 = τ1 + τ2 (6.2.14)
1+r 1+r
Equation (6.2.14) has the same interpretation as the household's budget (6.2.4). Total government revenue
(in present value) is on the right hand side and must be equal to total government spending (in present value),
which is on the left hand side.
Now that the household has to pay taxes, its budget changes because it can only use after-tax income to
pay for consumption. Instead of (6.2.4), the household's budget is now:
1 1
c1 + c2 = y1 − τ1 + (y2 − τ2 ) (6.2.15)
1+r 1+r
Adding up (6.2.14) and (6.2.15) we get that the household's budget is:
1 1 1
c1 + c2 = y1 + y2 − G1 + G2 (6.2.16)
1+r 1+r 1+r
| {z } | {z } | {z }
Present value of consumption Present value of income Present value of government spending
The logic behind equation (6.2.16) is as follows: the budget of the household is equal to the present value
of income minus the present value of taxes. But the government budget implies that the present value of
taxes equals the present value of spending. Therefore the household's budget must equal the present value of
income minus the present value of government spending.
Equation (6.2.16) has one important implication because of what's not in it. τ1 and τ2 don't appear in the
equation. Of course, τ1 and τ2 have to be such that the government's budget constraint (6.2.14) is satis ed but
any combination of τ1 and τ2 that satis es (6.2.14) is equivalent from the point of view of the household, i.e.
the timing of taxes does not matter. This property is known as Ricardian equivalence, after David Ricardo,
a 19th century economist who rst discussed it.
How does Ricardian equivalence come about? Imagine that the government announces that it is going to
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6.2. Two Period Model
lower taxes τ1 but leave spending G1 and G2 unchanged. Upon hearing this announcement, people immediately
calculate the implications for the government budget and come to the (correct) conclusion that the government
is going to have to raise taxes τ2 in order to pay for the debts that it will incur in period 1. They realize that
their after-tax income in period 2 is going to be lower and therefore want to save now in order to pay for those
future taxes. Therefore they do not alter their consumption at all and just save all the extra after-tax income
that they get from the lower τ1 .
Note that a lot of assumptions have to be satis ed for Ricardian equivalence to hold, and we might have
good reason to doubt them. People have to be perfectly rational and understand the government's budget.
The change in taxes cannot change their expectations of future government spending. The interest rate at
which the government borrows and lends must be the same as the one faced by the households, and everyone
must be able to borrow and lend at that rate. The taxes that the government is charging must be lump-sum
so that households cannot change their tax obligations by changing their behavior. Relaxing any of those
assumptions can lead to Ricardian equivalence not holding. You'll see an example of this in Exercise 6.5.
It's also important to remember what the Ricardian equivalence result, even when it holds, does not say,
because people sometimes get this wrong. Ricardian equivalence does not say that anything the government
does is irrelevant. It also doesn't say anything about what happens if the government changes G1 or G2 . The
only thing that is irrelevant is the timing of taxes, everything else held equal.
Precautionary Savings
So far we have been assuming that the household faces no uncertainty. In particular, it knows exactly how much
income it's going to have in the future. How would the household's decisions change if it faced uncertainty?
We are going to compare two households:
• Household A. Its period-1 income is y1 and it knows that its period-2 income will be y2 .
• Household B. Its period-1 income is also y1 but is it is uncertain about its period-2 income. It can be
y2 + (with 50% probability) or y2 − (with 50% probability); is some positive number.
On average (across the possible states of the world) both households make the same lifetime income. Does
this mean they are going to make the same consumption choices? Let's see.
We have already solved household A's problem, so let's look at household B. Equation (6.2.2) still applies:
if the household consumes c1 its savings will be a = y1 − c1 . This level of savings will result in two possible
levels of period-2 consumption, depending on whether it ends up having high or low income. Its consumption
will be:
cH
2 = y2 + + (1 + r) (y1 − c1 )
if it has high income or:
cL
2 = y2 − + (1 + r) (y1 − c1 )
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6.2. Two Period Model
if it has low income. The household will therefore solve:
1 1
max u (c1 ) + β u (y2 + + (1 + r) (y1 − c1 )) + u (y2 − + (1 + r) (y1 − c1 ))
c1 2 2
As we did in Chapter 2, we are assuming that the way the household deals with uncertainty is by maximizing
average (or expected) utility.
The rst order condition for this problem is
1 0 1
u0 (c1 ) − β (1 + r) u (y2 + + (1 + r) (y1 − c1 )) + u0 (y2 − + (1 + r) (y1 − c1 )) = 0
2 2
1 1 0 L
⇒ u0 (c1 ) = β (1 + r) u0 cH
2 + u c2 (6.2.17)
2 2
Equation (6.2.17) is the generalization of equation (6.2.8) to the case where the household faces uncertainty.
The dierence comes from the fact that the household doesn't know what its marginal utility of consumption
in period 2 is going to be. Therefore it chooses savings on the basis of the expected marginal utility of
period-2 consumption. Will this uncertainty make the household save more or save less? Let's look at this
mathematically rst and then try to make sense of what it means.
Proposition 6.1. If u0 (c) is a strictly convex function, then household B saves more than household A.
Proof. Assume the contrary, i.e. cB A
1 ≥ c1 . Then:
u0 cB 0 A 0
1 ≤ u c1 (u (c) is a decreasing function)
1 1 0 L
β (1 + r) u0 cH ≤ β (1 + r) u0 cA
2 + u c2 2 (using (6.2.8) and (6.2.17))
2 2
1 0 H 1
u c2 + u0 cL ≤ u0 cA
2 2 (simplifying)
2 2
1 H 1 L 0
c + c2 > cA (u (c) is decreasing and strictly convex)
2 2 2 2
y2 + (1 + r)(y1 − cB A
1 ) > y2 + (1 + r)(y1 − c1 ) (using the budget constraints)
cB A
1 < c1 (rearranging)
which is a contradiction.
Figure 6.2.5 shows the reasoning graphically. If the household had no uncertainty, it would consume E(c) in
0
period 2 and have marginal utility u (E(c)). Introducing uncertainty means it may consume either cH or cL .
0 0 0
If u (c) is convex, as in the gure, then E[u (c)] > u (E[c]). Introducing uncertainty increases the expected
period-2 marginal utility and makes saving more attractive.
The consumption-reducing, savings-increasing eect of future uncertainty on consumption behavior is
known as precautionary savings. If the future is uncertain, households reduce their consumption as a pre-
caution. Proposition 6.1 says that if marginal utility is convex then households will have precautionary savings
behavior. Do we have any reason to believe that u0 (c) is indeed convex? Probably the best reason to believe
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6.3. Many periods
Fig. 6.2.5: Convex marginal
utility and precautionary sav-
ings.
this comes from reasoning in the opposite direction. If we believe, as many economists do, that precaution-
ary savings are an empirically important phenomenon, then a utility function with convex marginal utility
is probably the right way to represent preferences. For what it's worth, the commonly-used CRRA function
c1−σ
u (c) = 1−σ satis es this, since:
u0 (c) = c−σ
u00 (c) = −σc−(1+σ)
u000 (c) = σ (1 + σ) c−(2+σ) > 0
and u000 (c) > 0 means u0 (c) is convex.
6.3 Extension to Many Periods
For many questions, the simpli cation of only considering two periods is good enough. For others, explicitly
taking into account that there are more periods can be useful. We'll see examples of this later on. For now,
we'll just look at how to analyze mathematically a many-period household savings problem. This turns out
to be very similar to analyzing a two-period problem.
The household's preferences are:
T
X
β t u (ct ) (6.3.1)
t=0
This extends the idea of (6.2.1) to T periods. The consumption of each of the future periods aects the
household utility, but since β < 1, future periods matter less the further away they are.
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6.3. Many periods
One common assumption is to set T = ∞. The usual justi cation for this assumption is that, even though
people die, they take into account their eect of their decisions on the money they leave to their children, on
their children's children, etc.
Let's now think about the household's budget. Let at denote the level of savings that the household has
at the beginning of period t. At the beginning of the following period, the household will have savings of:
at+1 = (1 + r) at + yt − ct (6.3.2)
The idea behind (6.3.2) is to keep track of everything that adds or subtracts from the household's savings. The
household increases savings by earning interest and by getting income and reduces them by consuming. Note
that household faces an in nite number of constraints like (6.3.2), each linking the savings in two consecutive
periods.
How do we analyze a maximization problem with an in nite number of constraints? There is more than
one way. Here what we'll do is collapse them all into one by substituting one constraint into the next over
and over again. Start from the period-0 and period-1 constraints:
a1 = y0 − c0 + (1 + r) a0 (6.3.3)
a2 = y1 − c1 + (1 + r) a1 (6.3.4)
Replace a1 from (6.3.3) into (6.3.4)
2
a2 = y1 − c1 + (1 + r) (y0 − c0 ) + (1 + r) a0
Similarly, the period-2 constraint can be written as:
a3 = y2 − c2 + (1 + r) a2
2 3
= y2 − c2 + (1 + r) (y1 − c1 ) + (1 + r) (y0 − c0 ) + (1 + r) a0
Generalizing, we can write the period-T constraint as:
T T
T −t T −t
X X T +1
aT +1 = yt (1 + r) − ct (1 + r) + (1 + r) a0
t=0 t=0
or, rearranging:
T T
aT +1 X ct X yt
T
+ t = t + (1 + r) a0 (6.3.5)
(1 + r) t=0 (1 + r) t=0 (1 + r)
PT ct
Equation (6.3.5) is conceptually very similar to equation (6.2.4). On the left we have the term t=0 (1+r)t ,
PT yt
which is the present value of all the consumption over T periods and on the right we have the term t=0 (1+r)t ,
the present value of income over T periods. In equation (6.2.4) we had those same terms for the special case
of T = 2.
There is an extra (1 + r) a0 term on the right. This is the value (including interest) of any savings that
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6.3. Many periods
the household was born with. In the two-period example we were implicitly assuming that a0 = 0 so this term
didn't appear.
aT +1
Also, there is an extra term on the left. This is the present value of any savings that the household
(1+r)T
has after the nal period. If the household plans to have aT +1 > 0 then all the savings it has left over after
period T cut into what it can aord to consume. Conversely, if the household chooses aT +1 < 0 this means
that the household is planning to leave debts behind after period T. If the household could choose any value
of aT +1 it wanted, then the best plan is clear: choose aT +1 = −∞, i.e. leave in nite debts behind in order to
be able to aord in nite consumption. Clearly this is not a reasonable model. A standard assumption, which
we implicitly made in the two-period model, is that the household must pay back any debts by period T, i.e.
aT +1 ≥ 0. Imposing this leads to constraint:
T T
X ct X yt
t ≤ t + (1 + r) a0 (6.3.6)
t=0 (1 + r) t=0 (1 + r)
How about when T = ∞? In this case there is no last period where we can say: you need to pay your
debts by this date. On the other hand, there should be some limit over how much debt you can accumulate:
otherwise the household can aord vast amounts of consumption by simply running up ever-greater debts. A
standard way to model the limits on the household's debts is to impose:
aT
lim T
≥0 (6.3.7)
T →∞ (1 + r)
Constraint (6.3.7) allows the household to have large amounts of debt, as long as those debts don't grow to
in nity too fast over time. It's sometimes known as a no-Ponzi condition, i.e. it says that household cannot
run a Ponzi scheme.
10 If we impose the no-Ponzi condition, then (6.3.5) reduces to:
∞ ∞
X ct X yt
t ≤ t + (1 + r) a0 (6.3.8)
t=0 (1 + r) t=0 (1 + r)
which is just like equation (6.3.6) with T = ∞.
Putting everything together, we can write the household's maximization problem as:
T
X
β t u (ct )
t=0
s.t
T T
X ct X yt
t ≤ t + (1 + r) a0
t=0 (1 + r) t=0 (1 + r)
10
Ponzi schemes are names after Charles Ponzi. Ponzi was an Italian immigrant in Boston in the 1920s. He oered to pay 50%
interest in 45 days and attracted a lot of money from investors. He had no way to deliver those huge interest rates but as long
as more investors kept coming in su ciently fast he was able to pay old investors with the money he got from new ones. The
scheme collapsed spectacularly after a few months.
119
6.4. Behavioral Theories
Now let's set up a Lagrangian:
T
" T T
#
X
t
X ct X yt
L (c0 , c1 , . . . ) = β u (ct ) − λ t − t − (1 + r) a0
t=0 t=0 (1 + r) t=0 (1 + r)
and take rst order conditions with respect to ct for some generic periods t and t + 1:
1
β t u0 (ct ) − λ t =0
(1 + r)
1
β t+1 u0 (ct+1 ) − λ t+1 = 0
(1 + r)
Solving for λ and rearranging we get:
u0 (ct ) = β (1 + r) u0 (ct+1 ) (6.3.9)
Equation (6.3.9) is the Euler equation (6.2.8) again. Now it describes the tradeo for consuming in any
two consecutive periods instead of just periods 1 and 2, but the interpretation is the same as before. In a
sense, going through the many-period case doesn't tell us all that much that we didn't know already from the
two-period case. We'll see some of the uses of the in nite-period model later on.
6.4 Behavioral Theories
In everything we've done so far the assumption has been that consumption-savings decisions result from a
rational calculation. If you have ever met actual people, you might have doubts about this assumption. The
challenge for macroeconomic theory is that there is one way to behave rationally and many, many ways to
behave not-quite-rationally. Which of the many possible departures from perfect rationality is important
enough to take into account when we think about the macroeconomy? This is very much an open question.
One of the best-known pieces of evidence showing that something other than full rationality is at play comes
from 401(k) default options. 401(k) plans (named after the section of the tax code that governs them) are
tax-advantaged investment plans that are oered by some rms to their employees. The employee contributes
a fraction of their salary to their individual account to be withdrawn upon retirement. In the meantime, all
the returns on investment earned within the account are not taxed. Typically, employees can choose how
much of their salary to contribute to their account. If they don't make a decision of how much they want to
contribute, then their contribution is set to some default option. In the world of perfect rationality, employees
would calculate the optimal level of 401(k) savings based on preferences, interest rates, etc. and set their
contribution accordingly. The default option should have no eect on how much they contribute. In practice,
researchers have found that default options tend to have very large eects on what people end up doing. Choi
et al. (2004) studied three companies that switched the default option from contributing zero to contributing
between 2% and 3% of the employees' salary. As a result of this, the percentage of employees who saved in
401(k) funds rose by more than 40 percentage points, even though their actual options had not changed at all.
There are several types of not-quite rational models of consumption behavior. One type of model is based
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6.4. Behavioral Theories
on the idea that people don't pay attention to all the relevant factors all the time: they just follow some
approximate rule that works sort-of-OK for them, such as keep 3 months of salary in the bank account, save
$200 per month or neither save nor borrow.
11 Under this sort of model, our calculations of how households'
consumption react to interest rates might be all wrong: it's possible that households don't pay attention to
the interest rate at all.
Another class of models is based on the idea that people have poor self-control: whenever they encounter
a good that they like, they buy it (as long as they can pay for it either using their savings or by borrowing).
Under the extreme version of this model, nothing about the future aects consumption decisions because
people are just not thinking about the future.
12 A variant of this idea says that people have poor self-control
but in their calmer moments they understand this and try to arrange things to avoid falling into temptation,
for instance by limiting their access to their own savings.
In the rest of this book we are going to stick with the simple rational model, but it's useful to keep in the
back of our mind that consumption behavior could depart from rationality in all sorts of interesting ways.
Exercises
6.1 Two-Period Problem with Taxes and Initial Wealth
Suppose a household solves the following two-period consumption-savings problem with taxes:
max u (c1 ) + βu (c2 )
c1 ,a,c2
s.t.
a = a0 + y1 − τ1 − c1
c2 = y2 − τ2 + (1 + r) a
c1−σ
with u (c) = 1−σ , where: c1 is consumption at time 1, c2 is consumption at time 2, y1 is household income
at time 1, y2 is household income at time 2, τ1 are taxes at time 1, τ2 are taxes at time 2, and a0 is initial
wealth.
(a) Solve for the household's choice of c1 , c2 and a in closed form.
13
c1
(b) How does
y1 depend on y2 ? What would happen if households suddenly became optimistic about
the future?
c1
(c) How does
y1 depend on a0 ? Interpret.
c1
(d) How does
y1 depend on β? Interpret.
11
One variant of this idea, known as rational inattention, imagines that paying attention is costly and households rationally
choose what things they are going to pay attention to and which things they are going to ignore.
12
There is also a rational version of this model. It's not that people have poor self-control, it's that they have a very low β so
they strongly prefer to enjoy the present at the expense of the future. In this version, it may be rational to consume as much as
possible in the present, even in the full understanding that it will mean consuming less in the future.
13
In closed form means that you have an explicit expression for something. Suppose I ask you to determine some variable x.
Then something like x = y2 − z is a solution in closed form; something like ex + x − b = 0 also implicitly tells what x should be
but is not in closed form.
121
6.4. Behavioral Theories
∂c1
(e) Suppose y2 = τ1 = τ2 = 0 and compute
∂r . How does the answer depend on σ? Interpret the
answer. [Hint: this is a hard question, not the maths but the interpretation. Think about what σ
means for the relative importance of income and substitution eects]
6.2 Savings Rates for Dierent People
An economist has data on the occupation, this year's income (denoted y) and this year's consumption
(denoted c) of a sample of 26-year-olds. Within this sample, some are top professional athletes and others
are medical doctors in their rst year of residence. [Make whatever assumptions you think are reasonable
and, if you wish, refer back to Exercise 6.1]
y−c
(a) Suppose one computed s= y for each individual in the sample. Should we expect s to be higher
for athletes or doctors?
(b) Suppose interest rates go down. How should we expect the response of s to dier between the two
groups?
6.3 Consumption and Interest Rates
A household solves a special case of the problem in Exercise 6.1, with τ1 = τ2 = a0 = 0. Suppose that, by
coincidence, the values of β, r, y1 and y2 are such that it is optimal for the household to consume:
c1 = y 1
c2 = y 2
(a) What will happen to c1 if interest rates increase? [A graph will be helpful. Make sure you draw it
carefully, it's probably useful to make it large]
(b) Does the overall utility achieved by the household increase, decrease or stay the same?
(c) Suppose this household was the only household in the economy and Jones & Klenow, using this year's
data, applied their measure of welfare to this economy. How would their measure of welfare change
with the change in interest rates?
(d) Explain the relationship between the answer to part (b) and the answer to part (c).
6.4 Consumption and Income
Suppose there are two households in the economy. Each of them solves a special case of Exercise 6.1,
where τ1 = τ2 = a0 = 0, β = 1 and r = 0.
(a) Solve for c1 as a function of y1 and y2 .
(b) Suppose the incomes of each of the households are:
y1 y2
Household A 2 4
Household B 6 4
Compute c1 and c2 for each of the households.
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6.4. Behavioral Theories
(c) Suppose an economist is trying to decide what is a reasonable model for consumption behavior and
only has data for period 1. Is the data supportive of the Keynesian view of the consumption function?
Explain.
(d) Suppose the same economist now looks at data for period 2 in addition to data for period 1. Is the
data supportive of the Keynesian view of the consumption function? Explain.
6.5 Credit Constraints and Ricardian Equivalence
Suppose a household solves the following variant of the problem in Exercise 6.1:
max u (c1 ) + βu (c2 )
c1 ,a,c2
s.t.
a = y1 − τ1 − c1
c2 = y2 − τ2 + (1 + r) a
a ≥ −b (6.4.1)
(a) What does equation (6.4.1) mean? What does b represent?
(b) Plot the household's budget constraint. In the same graph, plot constraint (6.4.1).
(c) Solve for c1 , c2 and a.
[Hint: notice that the constraint (6.4.1) is a weak inequality, not an equality, so it may or may not
be binding. If it is not binding, then you can use the answer from Exercise 6.1. Then think about
what happens if it is indeed binding. Then gure out whether or not it will be binding]
(d) Show that, other things being equal, constraint (6.4.1) is more likely to be binding if
i. y2 − τ2 is high,
ii. y1 − τ1 is low,
iii. b is low.
Interpret each of these conditions.
(e) Suppose that the government announces a stimulus package of size ∆. This involves lowering τ1
by ∆ and increasing τ2 by ∆ (1 + r) so that the present value of taxes is unchanged. How does c1
respond to the stimulus package if we start from a situation where constraint (6.4.1) is NOT binding?
How does c1 respond to the stimulus package if we start from a situation where constraint (6.4.1) is
binding? Explain.
(f ) Suppose that instead of announcing a stimulus package, the government announces that it will allow
households to borrow ∆ from the government and repay it back (with interest) at t = 2. How do the
eects of this policy compare with the eects of the stimulus package? Explain.
6.6 A Tax on Savings
A household solves a special case of the problem from Exercise 6.1, with τ1 = τ2 = a0 = 0. Suppose now
that the government introduces a tax on interest income, so that a household that saves a (and therefore
123
6.4. Behavioral Theories
earns interest ra) will have to pay τ ra in taxes. (If the household borrows instead of saving it pays no
tax).
(a) Plot the new budget constraint.
(b) Show graphically:
i. an example where the new policy persuades households to save more,
ii. an example where the new policy persuades the household to save less,
iii. an example where the household does not change its decision in response to the new policy.
Explain.
6.7 One Rational Household
Consider an economy that is well described by the Solow growth model. The production function is:
Y = K α L1−α
The population is constant an equal to 1 and there is no technological progress. The saving rate is s and
the depreciation rate is δ.
(a) What will the capital stock be in the long run?
(b) What will be the real interest rate in the long run?
For the rest of the question, assume that s = 0.4, α = 0.35 and δ = 0.1 and the economy is initially at a
steady state
(c) If the savings rate increases from s = 0.4 to s = 0.5:
i. Will GDP increase in the long run?
ii. Will consumption increase in the long run?
(d) Suppose a single household in the economy (the Friedmans) decides that, instead of just saving an
exogenous fraction s of their income, they are going to start choosing consumption and saving to
maximize the following standard preferences:
∞
X
β t u (ct )
t=0
The Friedmans can borrow or lend at the market interest rate. Since it's just the Friedmans who start
acting this way, and they are small relative to the economy, we are going to assume that the aggregate
economy (aggregate quantities, prices, etc.) remains unchanged. Will the Friedmans' consumption
be high initially and then fall over time, be low initially and then rise over time, or remain constant?
6.8 The Marginal Propensity to Consume
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6.4. Behavioral Theories
Suppose a household has preferences given by (6.3.1) over consumption across in nite periods. Its discount
factor and the real interest rate satisfy:
1
1+r =
β
The household starts with no wealth, will receive income y0 in period t = 0 and income ȳ in every
subsequent period, from t=1 onwards.
(a) Use the Euler equation (6.3.9) and the budget constraint (6.3.8) to nd an expression for initial-period
consumption c0 .
∂c0
(b) Compute
∂y0
(c) Suppose r = 0.04. What is the marginal propensity to consume out of a purely temporary increase
in income? Describe in words what a household does with a temporary increase in income.
6.9 The Marginal Propensity to Consume in Proportional Terms
A household solves a special case of the problem in Exercise 6.1, with τ1 = τ2 = a0 = 0 and y2 = my1 ,
where m is some number.
(a) Suppose that both y1 and y2 increase by x%. By what percentage do c1 and c2 increase?
(b) Suppose y1 increases by x% but y2 remains unchanged. By what percentage does c1 increase? How
does this depend on m and r? Explain.
125
CHAPTER 7
Labor and Leisure
When we looked at the Solow model we assumed that everyone in the population was working. In this chapter
we are going to think about the labor market in a little bit more detail. We'll start by looking at some of
the ways we measure what's going on in the labor market. Then we'll think about the incentives that govern
the decision of how much to work. At rst, we'll maintain the assumption that the labor market is perfectly
competitive, so a worker can supply as much labor as they want at the equilibrium wage w. Finally we'll think
about non-perfectly-competitive models of the labor market.
7.1 Measuring the Labor Market
The usual way to measure the state of the labor market is by surveys of both individuals and employers. In
the US, the main survey of individuals is conducted by the Bureau of Labor Statistics (BLS) and is called the
Current Population Survey (CPS). Other countries conduct similar surveys although the exact questions they
ask vary slightly from one country to another.
Individuals are classi ed into three main groups:
• Employed: if they have worked (including as employees or self-employed) in the past week.
• Unemployed: if they did not work during the past week but actively looked for a job.
• Out of the labor force: if they did not work and did not look for a job in the past week.
Based on this classi cation we de ne:
Labor force ≡ Employed + Unemployed
Participation rate ≡ Labor force
Population
Employment rate
Employed
≡ Population
Unemployment rate ≡ Unemployed
Labor force
Figure 7.1.1 shows the movement over time in these quantities. The labor force participation rate increased
for most of the second half of the twentieth century, peaking around 67% in 2000, and has since fallen to about
127
7.1. Measuring the Labor Market
63%. It moves more smoothly than the employment rate, which has higher-frequency uctuations, which
correspond to movements in unemployment. The unemployment rate is quite volatile, moving up and down
between about 3% and 10%.
Fig. 7.1.1: Labor market indicators in the United States. Source: CPS.
Flows Across Employment Status
The CPS also keeps track of how people shift between employment, unemployment and out of the labor force.
At any point in time, there are large numbers of people who change status in every direction. Figure 7.1.2
shows the magnitude of these ows for the month of October, 2018.
Using the data on stocks and ows we can compute the rates at which people transition from one status to
another. The (monthly) job nding rate is de ned as the number of workers who shift from unemployment to
employment, expressed as a fraction of the pool of unemployed workers. The (monthly) job loss rate is de ned
as the number of workers who shift from employment to unemployment, expressed as a fraction of the pool
of employed workers. Figure 7.1.3 shows the evolution over time of these rates. The job loss rate oscillates
between 1% and 2% per month, while the job nding rate oscillates around 30% per month.
1
Vacancies and the Beveridge Curve
Across the market from workers looking for jobs are rms looking for workers. Starting in 2001, the BLS has
conducted a survey called the Job Openings and Labor Turnover Survey (JOLTS) which asks rms, among
other things, how many job openings (sometimes called vacancies) they currently have. For earlier periods,
1
Both of these rates are somewhat underestimated. If a worker switches status back and forth within the same month, the
monthly survey will not detect this and will record no transition. This is especially important for the job nding rate since the
denominator is smaller.
128
7.1. Measuring the Labor Market
1.3
Employed Unemployed
157 5.8
1.6
4.2 4.5 1.7 1.4
Out of the labor force
96
Fig. 7.1.2: Stocks and ows of workers across labor market status in October, 2018. Figures in millions of
workers. Source: CPS.
Fig. 7.1.3: Monthly job loss rate and job nding rate. Source: CPS.
there are measures of job openings based on sources like help wanted ads in newspapers. The vacancy rate is
de ned as the ratio of vacancies to the total labor force.
129
7.1. Measuring the Labor Market
Figure 7.1.4 shows the relationship between the vacancy rate and the unemployment rate in the US economy.
There is a strong negative relationship. High vacancy rates have tended to coincide with low unemployment
rates. This negative relationship between vacancies and unemployment is known as the Beveridge Curve.
When the ratio of vacancies to unemployed workers is high, the labor market is said to be tight.
Fig. 7.1.4: The US Beveridge
Curve, 1948-2018. Each dot
is one month. Sources: Un-
employment from CPS. Vacan-
cies from NBER Macrohistory
Database, Barnichon (2010)
and JOLTS.
What do the Measures Tell Us?
Let's start with the most widely reported statistic: the unemployment rate. A high unemployment rate is
typically viewed as a problem while a low unemployment rate is viewed as a success, and with good reason.
By de nition, people who are unemployed would like to be employed but have not been able to achieve this.
However, just looking at the unemployment rate does not give a full account of what is going on in the labor
market.
First, searching for a job is a productive use of somebody's time. We often, including in this book, treat
all workers and all jobs as being identical, but it's obvious that this is not literally true. Finding a job requires
search eort because workers are trying to nd jobs that suit them and employers are trying to nd workers
that suit them. Looking at help wanted ads, writing resumes, contacting potential employers, etc., are part
of the process of getting the right person into the right job. Unemployment is partly a re ection of the fact
that this whole process is time-consuming.
On the other hand, people are counted as unemployed only if they took active steps to try to nd a job.
There are plenty of people who would like a job but have not taken active steps within the past week to
nd one. We can see evidence for this directly from Figure 7.1.2: there is a large ow of people from out
of the labor force into employed every month: these are workers who were not actively looking for a job
but nevertheless found one and took it. One reason why people who want a job might not be looking for
130
7.2. Static Model
one is that they might believe that it's very unlikely that they will nd one. These are sometimes known as
discouraged workers. If a large fraction of the people counted as out of the labor force are discouraged
workers, then a low unemployment rate need not mean that the outcomes in the labor market are good.
An alternative approach is to avoid making distinctions between people who are actively looking for work
and those who are not. Notice that the denominator in the employment rate and the unemployment rate is
dierent. The employment rate looks at how many people are employed as a fraction of the population rather
that as a fraction of those in the labor force. This measure treats those who don't work by choice, discouraged
workers, and the unemployed in the same way. If unemployed workers become discouraged and leave the labor
force, then the unemployment rate goes down but the employment rate is unchanged.
Is a high employment rate the best indicator of good outcomes in the labor market? Not necessarily.
There are many reasons why some people choose not to work: they retire, they take care of their families, they
study full time, etc. A low employment rate could be a symptom of changes in the extent to which people are
choosing these alternative uses of their time and not necessarily a problem with the functioning of the labor
market or the overall economy.
Despite their limitations, these measures do tell us something useful about the economy. Looking at how
these variables behave will be one way to assess various theories about how the economy works.
7.2 A Static Model of the Labor Market
We are going to imagine that a worker has the preferences
U (c, l) = u (c) + v (l) (7.2.1)
where c stands for consumption and l stands for leisure. The function u (c) describes how much the worker
enjoys consumption and the function v (l) describes how much the worker enjoys dedicating time to non-market
activities (we call these leisure but they could include non-market production such as doing laundry). We are
going to imagine that both u (c) and v (l) are concave functions. This means that the household experiences
diminishing marginal utility of both leisure and consumption.
The worker has a total of one unit of time, so the amount of time he spends working is given by
L=1−l
You'll sometimes see preferences over consumption and leisure expressed in terms of dis utility from working
rather utility from leisure, with a function of the form:
U (c, L) = u (c) − z (L)
Setting z (L) = −v (1 − L) makes the two formulations exactly equivalent. We'll stick to expression (7.2.1).
The worker has to decide how much of his time to dedicate to market work and how much to dedicate to
leisure. One way to interpret this decision is literally: imagine that the worker has a job that allows him to
choose how many hours to work (for instance, the worker is an Uber driver) and think about how the worker
131
7.2. Static Model
makes this choice. More broadly, there are many decisions that involve trading o higher income against less
leisure: choosing between a full-time job and a part-time job; choosing between a high-stress, highly paid
job and a lower-paid, more relaxed job; choosing at what age to retire; choosing how many members of a
many-person household will be working in the market sector, etc. We can think about the choice of leisure
as summarizing all of these decisions.
The worker gets paid a wage w per unit of time. Below we'll think about where this wage level comes from
but for now we are just thinking about the worker's decision problem, which takes the wage as given. The
total amount the worker can spend on consumption goods is given by the budget:
c ≤ w (1 − l)
The worker solves the following problem:
max u (c) + v (l)
c,l
(7.2.2)
s.t.
c ≤ w (1 − l)
This is a two-good consumption problem. The two goods here are time and consumption goods. The only
thing to keep in mind is that the household is initially endowed with one unit of time, and it has to choose
how much of it to sell in order to buy consumption.
Figure 7.2.1 shows the solution to problem (7.2.2). The worker will choose the highest indierence curve
he can aord, which implies that he will pick a point where the indierence curve is tangent to the budget
constraint. For any wage, the budget constraint always goes through the point (1, 0): the worker can always
choose to enjoy his entire endowment of time in the form of leisure and consume zero. The slope of the budget
constraint is −w: w is the relative price of time in terms of consumption goods. When w is high, time is
expensive relative to goods, so the budget constraint becomes steeper.
We can also nd the solution to problem (7.2.2) through its rst order conditions. The Lagrangian is:
2
L (c, l, λ) = u (c) + v (l) − λ [c − w (1 − l)]
The rst order conditions are:
u0 (c) − λ = 0
v 0 (l) − λw = 0
v 0 (l)
⇒ =w (7.2.3)
u0 (c)
2
This problem is su ciently simple that we don't need to use a Lagrangian to solve it. We could just as easily replace
c = w (1 − l) into the objective function and solve:
max u (w (1 − l)) + v (l)
l
which gives us the same solution.
132
7.2. Static Model
Fig. 7.2.1: The consumption-
leisure decision as a two-good
consumption problem.
Equation (7.2.3) describes how the worker trades o dedicating time to market work or to leisure. If the
worker allocates a marginal unit of time to leisure, he simply enjoys the marginal utility of leisure v 0 (l). If
instead the worker spends that time at work, he earns w, so he is able to increase his consumption by w; this
0
gives him w times the marginal utility of consumption u (c). At the margin, the worker must be indierent
between allocating the last (in nitesimal) unit of time between these two alternatives, so (7.2.3) must hold.
(7.2.3) is also an algebraic representation of the tangency condition shown in Figure 7.2.1. The slope of the
v 0 (l)
indierence curve is given by the marginal rate of substitution between leisure and consumption:
u0 (c) . The
slope of the budget constraint is w, so (7.2.3) says that the two are equated.
The Eect of Wage Changes
Let's imagine the wage w changes. How does the worker change his choice of leisure and consumption? The
answer to this question is going to play an important role in some of the models of the entire economy that
we'll analyze later. For now, we are going to study the question in isolation, just looking at the response of
an individual worker to an exogenous change in the wage. For concreteness, let's imagine that the wage rises.
Let's take a rst look at this question graphically. A change in wages can be represented by a change in the
budget constraint, as in Figure 7.2.2. The new budget constraint still crosses the point (1, 0), but the slope
of the budget constraint is steeper. As with any change in prices, this can have both income and substitution
eects.
The substitution eect is straightforward: a higher wage means that time is more expensive. Other things
being equal, this would make the worker substitute away from leisure (which has become relatively expensive)
towards consumption (which has become relatively cheap). This makes the worker work more.
In addition, the higher wage unambiguously helps the worker: the worker is selling his time so a higher
133
7.2. Static Model
Fig. 7.2.2: Consumption and leisure response to higher wages. Income and substitution eects.
price is good for him. In other words, there is a positive income eect. For the consumption choice, the income
eect reinforces the substitution eect since both push the worker to consume more. For the leisure choice,
this goes in the opposite direction as the substitution eect: as the worker becomes richer, he wants more
of everything, including leisure. In the example depicted on the left panel of Figure 7.2.2, the substitution
eect dominates. The worker ends up at a point to the left of where he started, showing he has decided to
work more (and get less leisure) when wages rise. The right panel shows an example where the income eect
dominates so the worker decides to work less (enjoy more leisure) when the wage rises.
An Explicit Example
Suppose that the utility function takes the following form:
c1−σ
u (c) =
1−σ
θ 1+
v (l) = − (1 − l)
1+
where θ and are parameters. For this case, we can get an explicit formula for how much consumption and
leisure the worker will choose. Marginal utility of consumption and leisure are, respectively:
u0 (c) = c−σ
1
v 0 (l) = θ (1 − l)
134
7.2. Static Model
so replacing in equation (7.2.3) we get
1
θ (1 − l)
=w
c−σ
Using the budget constraint to replace c we get:
1
θ (1 − l)
−σ =w
(w (1 − l))
1 1−σ
− 1 +σ 1
1−l =θ w +σ (7.2.4)
Equation (7.2.4) gives us an explicit formula for how much the worker will choose to work depending on
the wage and the parameters in the utility function. Will this worker work more or less when the wage is
higher? Mathematically, this depends on whether the exponent on w is positive or negative. If σ<1 then
the exponent is positive and the worker will work more when the wage is higher, i.e. the substitution eect
dominates; if σ>1 then the income eect dominates and the worker works less when the wage is higher. As
we've seen before, in this particular utility function, the parameter σ tells us about curvature. In economic
terms, it tells us how fast the marginal utility of consumption goes down when consumption increases. Why is
this the relevant aspect of preferences that governs income and substitution eects? If the marginal utility of
consumption falls rapidly as consumption increases, the worker is unwilling to substitute towards even more
consumption when wages rise and he chooses to enjoy more leisure instead.
Adding Taxes and Transfers
Let's try to gure out how the worker's decisions will respond to changes in tax policy. We'll represent tax
policy in a highly simpli ed way, with just two numbers: τ and T. τ is the tax rate on the worker's income:
the government collects a fraction τ of the worker's income as taxes. T is a transfer that the worker gets from
the government. This is intended to represent the multiple income-support policies that many countries have
in place: unemployment insurance, food assistance, pensions, public healthcare, public education, etc. The
worker's budget constraint is:
3
c ≤ w (1 − l) (1 − τ ) + T
If we were looking at this from the government's perspective, we would have to think about how τ and T are
linked: the government must set τ to collect enough revenue to aord T. For now, we'll look at this from the
worker's perspective, taking τ and T as given.
Figure 7.2.3 shows how taxes and transfers aect the worker's budget constraint. There are two eects,
one from the taxes and one from the transfers. The eect of transfers is to simply shift the budget constraint
up: for any given level of leisure, the worker can consume T more. In particular, he can consume T without
3
T and τ are taken as constants for simplicity, but this is less restrictive than you might think. For instance, suppose a housing
program oers subsidized rents to low income households. The poorest households receive $3,000 per year in subsidies but this
bene t is phased out depending on the household's income until a household making $30,000 a year receives no bene t at all. We
could represent this policy as T = 3, 000 and τ = 0.1 because the loss of housing subsidies is, eectively, a tax on the household's
labor earnings. What we miss by having constant values for T and τ is that a lot of policies are non-linear in complicated ways.
Even the simple program described above has an implicit tax rate of 0.1 on incomes below $30,0000 but no taxes on marginal
income above $30,000.
135
7.2. Static Model
working at all. The eect of taxes is to lower the slope of the budget constraint: from the point of view of the
worker, the price at which he can sell time to obtain consumption is the after-tax wage w (1 − τ ).
Fig. 7.2.3: The worker's bud-
get constraint when there are
taxes and transfers.
The eect of higher transfers is a pure income eect. Prices have not changed but the worker is richer, so
he chooses higher consumption and higher leisure. This is illustrated in Figure 7.2.4.
Fig. 7.2.4: Consumption
and leisure response to higher
transfers.
136
7.3. Evidence
The eect of higher tax rates is just like the eect of lower wages. There is a substitution eect (the
after-tax price of time has gone down, so the worker chooses higher leisure) and an income eect (the worker
is poorer, so he chooses less leisure), and they push the leisure choice in opposite directions. The rst-order
condition for the household's problem becomes:
v 0 (l)
= w (1 − τ ) (7.2.5)
u0 (c)
The worker equates the marginal rate of substitution to the after-tax wage w (1 − τ ) rather than the full
wage w.
7.3 Some Evidence on Labor and Leisure Decisions
Long-Run Evidence on Income and Substitution Eects
We have seen that in general higher wages could make workers choose to work more or less. Which way does
it go in practice? International and historical data oers us the chance to see how workers' choices vary across
settings with very dierent wages.
Ramey and Francis (2009) piece together data from time-use surveys for the US for the period 1900-2006
to try to determine whether the amount of time spent on leisure has gone up or down. We know that wages
have increased a lot over the last 100 years (by a factor of 9 approximately). If the income eect dominates,
we should see that over time people are choosing more leisure; if the substitution eect dominates, we should
see that over time people are choosing less leisure. Figure 7.3.1 shows the trends in hours per week spent on
leisure, broken down by age. Overall, there seems to be a very slight upward trend in leisure, especially from
14-17 year olds and over-65-year olds.
4 This suggests that income and substitution eects almost cancel each
other out, and perhaps income eects dominate slightly.
Bick et al. (2018) do a similar measurement, but instead of looking at variation in wages and leisure over
time they look at variation in wages (or more exactly, in GDP per capita) and leisure across countries at a
given point in time. Figure 7.3.2 shows how hours worked correlate with GDP per capita across countries.
There is a negative correlation, suggesting that overall the income eect tends to dominate and people work
less as wages increase. However the relationship is not very strong, suggesting that income and substitution
eects are not so far from canceling each other out.
Comparing the US and Europe
Figure 7.3.3 shows how hours worked per employed person in the US and some European countries have
evolved over the last few decades.
5 Up until the early 1970s or so, the US and Western Europe looked very
similar but then they started to diverge so that today Europeans work less than Americans. How come?
4
The study separately measures hours spent on market work, nonmarket work, schoolwork and pure leisure. The gure
shows trends in pure leisure time.
5
The gure only looks at employed persons and says nothing about employment rates, but these tell the same story: the US
and Europe look similar until the 1970s and then employment rates are higher for the US.
137
7.3. Evidence
Fig. 7.3.1: Average hours of
leisure per week for everyone
ages 14+ in the US. Source:
Ramey and Francis (2009).
Fig. 7.3.2: Average hours of
work per week across countries.
Source: Bick et al. (2018).
One hypothesis, put forward by Prescott (2004), is that the reason is dierences in tax and social security
policy. Europe has higher tax rates and social spending than the US. These, the argument goes, discourage
Europeans from working as hard as Americans. Prescott proposes a simple version of the model in this
chapter and argues that it can explain the magnitude of the dierences between US and European labor
markets. Exercise 7.5 asks you to go through the details of Prescott's calculation and to think about the role
138
7.3. Evidence
Fig. 7.3.3: Hours worked per
year per employed person in
the US and selected European
countries. Source: OECD.
of income and substitution eects.
There is no consensus among economists about whether Prescott's hypothesis is correct. It has been
criticized from a few dierent angles. One criticism focuses on the elasticity of labor supply. Exercise 7.5
asks you to compute the elasticity of labor supply that is implicit in Prescott's calculations. This matters
because it governs how much labor supply responds to changes in incentives. Most microeconomic estimates
of this elasticity are quite a bit lower than Prescott's value, though there is some debate as to how to translate
microeconomic estimates into macroeconomic calculations. A second line of criticism focuses on timing.
Policies in the US and Europe became dierent in the 1960s and 1970s but the dierences in labor supply
continued to widen well after that, suggesting something else was going on (or that policies take a very long
time to have an eect).
Some other explanations for the US-Europe dierence have been proposed. Blanchard (2004) argues that
dierences in preferences may be a large part of the reason: maybe Europeans place a higher value on leisure
than Americans. Economists tend to be a little bit uncomfortable with explanations based on dierences in
preferences. We cannot observe preferences directly so these theories are very hard to test (but this does not
necessarily mean they are wrong). Furthermore, we need to explain why Europeans work less than Americans
now but this was not the case in the 1950s. Perhaps cultural dierences only become relevant once society
reaches a certain level of income.
Alesina et al. (2006) argue that a large part of the explanation may have to do with the dierent role of
labor unions in the US and Europe. Unions tend to be stronger in Europe and union contracts tend to specify
shorter hours and longer holidays than non-union contracts.
6 One possibility is that Europeans work less than
6
A separate, interesting, question is why union contracts look dierent than non-union contracts. A naive answer would be to
say that unions have more bargaining power with respect to employers than individual workers (which is probably true) so they
get better terms. But these better terms could be in the form of higher wages or less work. Why do unions prioritize leisure over
139
7.4. A Dynamic Model
Americans because they live in a more unionized labor market.
7.4 A Dynamic Model
So far we have looked at consumption-savings decisions and consumption-leisure decisions as separate problems.
In reality, households are making both decisions: how much to work and how much to consume. Do those
decisions interact or is it OK to look at them in isolation? Let's see how a household would behave if they
had to make both decisions at once.
The household solves the following problem:
max u (c1 ) + v (l1 ) + β [u (c2 ) + v (l2 )]
c1 ,l1 ,c2 ,l2
(7.4.1)
s.t.
1 1
c1 + c2 ≤ w1 (1 − l1 ) + w2 (1 − l2 )
1+r 1+r
The household now has four goods to choose from: consumption in each period and leisure in in each period.
As in the consumption-savings problem from Chapter 6, β represents how much it discounts the future. The
budget constraint just says that the present value of consumption must be less or equal than the present value
of income. Income in period t is given by the wage wt times the amount of time the household dedicates to
market work 1 − lt .
The Lagrangian for this problem is:
1 1
L (c1 , l1 , c2 , l2 , λ) = u (c1 ) + v (l1 ) + β [u (c2 ) + v (l2 )] − λ c1 + c2 − w1 (1 − l1 ) − w2 (1 − l2 )
1+r 1+r
and the rst-order conditions are:
u0 (c1 ) − λ = 0
v 0 (l1 ) − λw1 = 0
1
βu0 (c2 ) − λ =0
1+r
1
βv 0 (l1 ) − λw1 =0
1+r
which we can summarize as:
v 0 (lt )
= wt for t = 1, 2 (7.4.2)
u0 (ct )
u0 (c1 ) = β (1 + r) u0 (c2 ) (7.4.3)
Equation (7.4.2) is just like equation (7.2.3), except that now it applies to both periods. In each period the
consumption more than individual workers?
140
7.4. A Dynamic Model
household must be indierent at the margin between dedicating a unit of time to leisure or to market work.
Equation (7.4.3) is the Euler equation again: no matter how the household obtains its income, this equation
describes how it distributes its consumption over time.
So far it would seem that looking at the consumption-savings decision and the consumption-leisure decision
together doesn't add very much. We just get back the conclusions we got when we looked at the two problems
separately. However, looking at the decisions jointly allows us to ask some questions that were not possible
before. One that will be important later is the following. Suppose wages increase temporarily : how does the
household change its labor supply? How does the answer change if the increase is permanent?
Let's look at this mathematically rst and then think about what it means. Take equation (7.4.2) and
solve for labor supply in period 1, which we denote by L1 = 1 − l1 .
v 0 (l1 )
= w1
u0 (c1 )
⇒ v 0 (l1 ) = w1 u0 (c1 )
−1
⇒ l1 = (v 0 ) (w1 u0 (c1 ))
−1
⇒ L1 = 1 − (v 0 ) (w1 u0 (c1 )) (7.4.4)
−1 −1
where (v 0 ) denotes the inverse of v0 . Since the marginal utility of leisure is decreasing, (v 0 ) is a decreasing
function. Now compare two experiments: a temporary increase in wages (w1 rises but w2 does not) or a
permanent increase (w1 and w2 both rise). If we look at the right hand side of (7.4.4), w1 is, by assumption,
the same in both experiments. However, c1 will be dierent. As we saw in Chapter 6, consumption reacts
more strongly to a permanent increase in income than to a temporary increase in income. That means
that if we compare the two experiments, the one where the wage increase is temporary
lower c1 ,
will have:
(because the raise is temporary); therefore higher u (c1 ) (because u (c) is a decreasing function); therefore lower
0 0
(v 0 ) (w1 u0 (c1 )) (because v 0 (l) is a decreasing function); therefore lower l1 and higher L1 . Conclusion: the
−1
household's labor supply rises more in response to a temporary increase in wages than it does to a permanent
one.
What is going on in economic terms? If a wage increase is temporary, the household doesn't really feel
much richer than before, so the income eect that pushes it to increase leisure is relatively weak. This makes
the substitution eect dominate: time is temporarily very expensive so the household decides to sell more of
it and keep less of it.
This eect is at the heart of Uber's surge-pricing strategy. When Uber detects that there are many users
who want rides and not a lot of riders available, it increases fares. From the point of view of Uber drivers,
this is like a temporary increase in wages. Because the increase is temporary, Uber drivers don't really feel
much richer than before, and they react by getting in their car and oering rides to take advantage of the
surge pricing.
7 Interestingly, in a study of New York City taxi drivers, Camerer et al. (1997) argued that
they did not behave in this way at all and, if anything, reduced their working hours on days where they were
temporarily getting higher income per hour. The exact interpretation of these ndings is somewhat disputed.
7
Also, Uber users may react to surge pricing by not taking as many rides. This is just the price adjusting to equate supply and
demand! The only dierence is that, instead of anonymous market forces, here Uber is actively managing the price adjustment.
141
7.5. Equilibrium in the Labor Market
7.5 Equilibrium in the Labor Market
So far we have looked at the household's labor supply decision in isolation. Let's put this together with labor
demand to nd an equilibrium. We'll rst look at the frictionless competitive case and then we'll think about
what happens if matching workers to jobs involves a process of search. We'll limit ourselves to the one-period
case.
A Competitive Labor Market
As we saw in Chapter 4, in a competitive market rms will demand labor up to the point where the marginal
product of labor equals the wage. If the production function is F (K, L), then (taking the capital stock
K as given), the equation w = FL (K, L) gives us a labor demand curve. For each possible wage level w,
solving this equation for L tells us how much labor the representative rm is willing to hire. This demand
curve is downward sloping. This comes from the fact that the production function is assumed to be concave
in L, which is equivalent to saying that there is a diminishing marginal product of labor. For example, if
F (K, L) = K α L1−α then we have:
w = (1 − α)K α L−α
1 1
⇒ L = (1 − α) α Kw− α
On the workers' side, we can nd a labor supply curve by doing the same steps that lead to equation
(7.2.4). Replacing c from the budget constraint into the rst order condition (7.2.3) we get:
v 0 (l)
=w
u0 (w(1 − τ )(1 − l) + T )
This equation implicitly de nes a relationship between the wage w and the amount of labor the worker
supplies 1 − l, i.e. a labor supply curve. As we have seen, the slope of this relationship is ambiguous in
general, depending on the relative strengths of income and substitution eects, so the labor supply curve
could be upward or downward sloping (a downward sloping labor supply curve is sometimes called backward-
bending).
By plotting the labor demand curve and the labor supply curve on the same plot, we can determine
the equilibrium wage and the equilibrium amount of labor, as shown in Figure 7.5.1. The gure shows two
examples, one where the substitution eect dominates so the labor supply curve is upward-sloping and one
where the opposite happens. In either case, we can nd the equilibrium wage and the equilibrium amount of
labor by nding the intersection of the two curves.
We can also ask what would happen in the labor market in response to various changes. Figure 7.5.2
illustrates two possibilities. In the rst panel we have the eects of a shift in the marginal product of labor
curve FL (K, L). At any given wage level, rms want to hire more workers. This leads to an increase in wages
and (in this example, where the substitution eect dominates), to an increase in L. The second panel shows
an increase in government transfers T. As we have seen, this has an income eect, so for any given wage level,
the worker wants to work less, leading to a leftward shift in the labor supply curve. This leads to higher wages
142
7.5. Equilibrium in the Labor Market
Fig. 7.5.1: Equilibrium in the labor market.
and a fall in L.
Fig. 7.5.2: Eects of changes on the labor market.
143
7.5. Equilibrium in the Labor Market
Search
So far we have assumed that the labor market is competitive and everyone can work as much or as little as
they want at the market wage. This is not a useful way to think about unemployment because, by assumption,
there is no unemployment. For that matter, it's also not a good way to think about vacancies either because,
by assumption, rms can hire as many workers as they can and they will never have un lled vacancies. One
theory of why unemployment and un lled vacancies coexist, developed by Diamond (1982), Mortensen (1982)
and Pissarides (1985), is built on the assumption that the process of search is frictional: the workers that are
looking for jobs and the rms who need them might fail to nd each other. Let's look at how this theory
works.
To keep things simple, we are going to assume away everything that has to do with the decision of how
much to work. We normalize the size of the population to 1 and assume everyone in the population is in the
labor force. In order to actually produce output, a worker has to be matched with a rm, because rms have
access to the productive technology. If a worker is matched with a rm, we call that a job; it assigns the
worker a task in the production function, which results in y ≡ FL (K, L) additional units of output.
The process by which workers nd jobs works as follows. At the beginning of the period, 1−U workers
already have jobs, while U are still looking, so U represents the initial unemployment rate, which we take as
given. Firms that want workers advertise their vacancies. Advertising a vacancy costs χ units of output. This
represents literal advertising costs as well as the cost of things like selecting and interviewing applicants. The
number of workers who nd jobs depends on the total number of unemployed workers and the total number of
vacancies that rms create (denoted by V ), according to some function m (V, U ). This is known as a matching
function. It attempts to capture, in simpli ed form, all the process of workers searching for jobs and rms
searching for workers. We'll assume that m is increasing and concave in both U and V. This means that
more workers will nd jobs when there are more vacancies, but the marginal impact of an additional vacancy
on the number of jobs created is decreasing; similarly, more rms will be able to ll vacancies if there are
more unemployed workers looking for jobs, but the marginal impact of an additional unemployed worker is
decreasing. De ne the vacancy lling rate as:
m(V, U )
q(V, U ) =
V
The concavity of m implies that q(V, U ) is decreasing in V: the probability of lling any one vacancy falls
with the number of vacancies. Also, q(V, U ) is increasing in U.
If a worker and a rm manage to match, then they need to decide what wage the worker will be paid.
We are going to assume that they bargain over the wage in a speci c way, known as Nash bargaining. They
rst look at what's going to happen if they cannot reach an agreement. In this case, the rm will have an
un lled vacancy that will produce no output and no pro ts, while the worker will be unemployed and get
some exogenous amount b, which could represent unemployment bene ts or the value of leisure. After doing
that, they say to each other: if we do reach an agreement, we'll get a total of y instead of b, so that's y−b
better. Let's nd a deal so that we split the y−b of surplus that is generated by this job. They then set the
wage w so that the worker gets a fraction µ of the surplus and the rm gets 1 − µ. The parameter µ measures
the bargaining power of the worker. This means that the wage will be: w = b + µ(y − b) and the rm's pro ts
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7.5. Equilibrium in the Labor Market
will be y − w = (1 − µ)(y − b).
Now turn to the rm's decision to post a vacancy. A vacancy costs χ and then, with probability q(V, U ), it
results in a hire, which gives the rm a pro t of (1 − µ)(y − b). Firms will nd it pro table to create vacancies
until:
χ = q(V, U )(1 − µ)(y − b) (7.5.1)
Solving equation (7.5.1) for V tells us how many vacancies will be created, as illustrated by Figure 7.5.3. If V
is higher than the number that makes (7.5.1) hold, then the probability of lling a vacancy is too low, rms
will on average not recoup the vacancy costs and would prefer not to hire. If V is lower, then rms will on
average make pro ts beyond what is needed to recoup vacancy costs and would open more vacancies.
Fig. 7.5.3: Equilibrium
in the Diamond-Mortensen-
Pissarides search model.
Knowing V lets us gure out what the unemployment rate will be once hires have taken place. Denote
this by U 0. Since a total m (V, U ) workers nd jobs, we have:
U 0 = U − m(V, U ) (7.5.2)
Let's see what happens to the labor market when something changes. The left panel of Figure 7.5.4 shows
how the labor market reacts to an increase in the marginal product of labor y. Other things being equal,
this makes posting vacancies more pro table, so rms react by posting more vacancies until condition (7.5.1)
is restored. The right panel shows what happens if unemployment bene ts b are increased. The eects here
are more subtle. Higher unemployment bene ts increase workers' outside option if they were fail to reach an
agreement while bargaining. This improves their bargaining position, and therefore raises equilibrium wages,
lowering rm pro ts. Firms respond to this by posting fewer vacancies, restoring condition (7.5.1).
Seeing how V reacts to various changes gives us a way to think about the Beveridge Curve. Equation (7.5.2)
145
7.5. Equilibrium in the Labor Market
Fig. 7.5.4: Eects of changes in the Diamond-Mortensen-Pissarides search model.
implies a negative relationship between vacancies and after-hiring unemployment (taking initial-unemployment
as given), which is what the Beveridge Curve says. Other things being equal, anything that leads rms to
post more vacancies will result in more workers nding jobs and therefore lower unemployment. There is
some debate as to whether this is a satisfactory explanation. Shimer (2005) argued that the movements in the
vacancies-to-unemployment ratio produced by this model are too small to t the data well.
Exercises
7.1 Labor Supply
Suppose household preferences are given by:
u (c, l) = c + θ log (l)
where c is consumption, l is leisure and θ is a parameter. Households have a total of one unit of time and
can supply labor in a competitive labor market at a wage w.
(a) Find an expression for the fraction of their time that households spend in market work.
(b) If this was the right model and one looked at households in dierent countries, how would hours of
work correlate with wage levels? How does this compare to the empirical evidence?
7.2 Military Service
146
7.5. Equilibrium in the Labor Market
An economy is populated by a representative household, whose preferences are described by:
u (c) + v (l)
where c is consumption and l is leisure. The household has 1 unit of time so its budget is:
c = w (1 − l) − τ
where w is the wage rate and τ is a tax that the government collects. Notice that τ is a lump-sum tax,
not an income tax: the household must pay the same amount regardless of how much income it earns.
(a) Set up the Lagrangian for the household's optimization program and nd rst-order conditions.
(b) Use the budget constraint to replace c in the rst-order condition to obtain a single equation that
relates the household's choice of l to w and τ.
(c) Suppose that the government uses all the tax revenue to hire soldiers for the army, paying them
the prevailing wage rate. How many units of soldiers' time can the government aord? Denote this
number by m.
(d) Now suppose that instead of taxing citizens to hire soldiers, the government imposes compulsory
military service: the representative household has to dedicate m units of time to serving in the army,
unpaid. The rest of their time they can use as they please, and they pay no taxes. Set up the
household's optimization program under this policy.
(e) Show that the representative household's level of consumption, leisure, army labor and non-army
labor are the same under both policies. Explain.
7.3 Consumption Taxes and Labor Supply
Suppose a household solves the following problem:
max u (c) + v (l)
c,l
s.t. (1 + τc ) c ≤ w (1 − τ ) (1 − l)
where c is consumption, l is leisure and τ is an income tax. τc is a consumption tax, so that if the household
wishes to consume c it must pay (1 + τc ) c.
(a) Draw the household's budget constraint. What is the slope?
(b) Find the rst-order conditions for the household's consumption-leisure decision. How do the eects
of τ and τc relate to each other? Explain.
(c) If the household chooses c∗ and l∗ , how much tax revenue does the government collect?
(d) Suppose the government had originally set τc = 0 and τ >0 and now wants to enact a tax reform
that uses consumption taxes instead of income taxes. What would be the level of τc that leaves the
decisions of the household unchanged?
147
7.5. Equilibrium in the Labor Market
(e) How much revenue does the government collect under the new system? How does it compare to the
old system? Explain.
7.4 Puerto Rico
Read the following article: https://www.economist.com/united-states/2006/05/25/trouble-on-welfare-
island. Is what was going on in Puerto Rico consistent with the theories discussed in this chapter?
Discuss.
7.5 Prescott's Calculation
Suppose preferences for consumption and leisure are:
u (c, l) = log (c) + α log (l)
and households solve:
max u (c, l)
c,l
s.t. c = w (1 − τ ) (1 − l) + T
(a) Find rst-order conditions for the consumption-leisure decision.
(b) Use the budget constraint to solve for leisure l. You should get an explicit expression for l as a
function of w, τ , T and α.
(c) Suppose T = 0. How does l respond to the tax rate τ? What does this mean?
Now suppose that in both Europe and the US we have:
α = 1.54
w=1
but in the US we have:
τ = 0.34
T = 0.102
while in Europe we have:
τ = 0.53
T = 0.124
(d) Compute the amount of leisure chosen in the US and Europe. If we interpret 1 as your entire adult
lifetime, what fraction of their adult lives do people in Europe and the US work? Comment on the
respective role of taxes and transfers in this analysis using your answers to parts (b) and (c).
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7.5. Equilibrium in the Labor Market
(e) The values for τ and T above are not arbitrary. If you did the calculations correctly, you should nd
that both governments have balanced budgets (up to rounding error), i.e. they redistribute all the
tax revenue back as transfers. Check that this is the case.
(f ) Assuming the production function is Y = L = 1 − l, how much lower is GDP per capita in Europe
compared to the US?
(g) Compute the relative welfare of Europe by solving for λ in the following equation:
u (cEurope , lEurope ) = u (λcU S , lU S )
Interpret the number λ that you nd
(h) How do the answers to questions (f) and (g) compare? Why?
(i) Suppose a European policymaker sees Prescott's calculation and concludes that Europe could increase
1
its welfare by a factor of
λ by reducing its tax rate and level of transfers to US levels. Do you think
they are right? Why? Don't answer this question mechanically: think about what this calculation
does and what it leaves out.
In any calculation of this sort, an important parameter is the elasticity of labor supply. One de nition
of elasticity that is often looked at by labor economists is known as the Frisch elasticity. It is based on
the answer to the following question: suppose we increased wages but adjusted the household's income
so that consumption remained constant: how would labor supply change?
8 Let's calculate the Frisch
elasticity in Prescott's model.
(j) Use your answer to part (a) to nd an expression for labor supply (1 − l) in terms of w (1 − τ ), c and
α. Notice that now we are holding consumption constant, so the idea is that we don't replace c from
the budget constraint like we did in part (b).
∂(1−l) w(1−τ )
(k) Use your answer to part (j) to nd an expression for
∂w(1−τ ) 1−l , i.e. the elasticity of labor supply
with respect to after-tax wages, holding consumption constant.
(l) Plug in the values of α, τ , w, c and l that you found for the US case into the expression for elasticity.
What number do you get? Empirical estimates of this elasticity are usually in the range of 0.4 to 1.
How does that compare to the elasticity implied by Prescott's model? Why does that matter for our
conclusions about tax policy?
7.6 The Protestant Work Ethic
Suppose that the representative household has preferences given by:
u(c, l) = log(c) + θ log(l)
where c is consumption, l is leisure and θ is a parameter. The production function is given by:
F (K, L) = K α L1−α
8
Notice that it's not exactly the same as isolating the substitution eect as in Figure 7.2.2, because that was holding utility
constant instead of consumption; but it's related.
149
7.5. Equilibrium in the Labor Market
where K is the capital stock, which we take as given and equal to 1, L is labor and α is a parameter.
Firms hire workers in a competitive labor market where the wage is w.
(a) What will be the equilibrium wage w?
(b) How does w depend on θ? Suppose the household suddenly develops a so-called Protestant work
ethic and stops enjoying leisure so much, what will happen to real wages? Explain.
7.7 The Beveridge Curve
Assume the labor market is well described by the search model presented in Section 7.5.
(a) Suppose there is an improvement in recruiting technology, so that the matching function becomes
Am(V, U ), with A > 1. What happens to the number of vacancies? What happens to the after-hiring
unemployment rate? Would this produce something that looks like a Beveridge Curve?
(b) Suppose there is an increase in the initial (before-hiring) unemployment rate. What happens to the
number of vacancies? What happens to the after-hiring unemployment rate? Would this produce
something that looks like a Beveridge Curve?
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CHAPTER 8
Investment
When we looked at the Solow model we assumed that investment was an automatic consequence of savings. In
this chapter we are going to think about investment decisions directly. What incentives govern the decisions
to invest? Let's start with an example.
Example 8.1.
The world lasts two periods. A rm is considering building a factory in period 1 in order to produce
output in period 2. Building the factory costs 1000 (the units here are consumption goods). If the rm
builds the factory, this will result in additional revenues of 2500 and additional costs of 1400 in period 2.
After period 2, the world is over. Is building the factory a good idea?
Let's start with a naive calculation. Net of costs, in period 2 the rm will get an additional 2500 − 1400 =
1100 if it builds the factory. Since this is more than the cost of building the factory, this would seem to suggest
that building the factory is a good idea.
What's missing from this analysis? Building the factory requires giving up 1000 in period 1 in order to
obtain 1100 in period 2. But goods today and goods tomorrow are dierent goods! Just concluding that you
should build the factory is like saying that converting one large apartment into two studios is a good idea
because two is more than one. To do it right, you need to know the relative price of apartments and studios.
Likewise, in order to decide whether you should build the factory, you need to know the relative price of goods
in dierent periods.
As we saw in Chapter 6, the real interest rate is the relative price of goods in dierent periods. Let's go
over why this is the case. Imagine that anyone can borrow or lend as much as they want at the real interest
rate r, and all loans are always repaid. This means that anyone can take one good to the market in period 1
and exchange it for 1+r goods in period 2. Conversely, anyone can obtain one good in period 1 in exchange
1
for giving up 1+r goods in period 2. Hence, one good in period 2 is worth
1+r goods in period 1.
Now let's go back to the investment decision. If the rm invests, it gives up 1000 period-1 goods in exchange
1100
for 1100 period-2 goods that are worth
1+r period-1 goods. This is a good idea as long as:
1100
> 1000
1+r
151
8.1. Present Values
⇒ r < 0.1
If the real interest rate is low enough, then this investment project is worthwhile; otherwise it's not.
Notice that whether this factory is a good idea does not depend on whether the rm has 1000 to begin
with. If r < 0.1 and the rm doesn't have 1000, it is a good idea to borrow them in order to build the factory:
the net revenues from the factory will be more than enough to pay back the loan with interest. Conversely, if
r > 0.1 and the rm does have 1000 available, it is better o lending and earning interest than building the
factory. Of course, this relies on the assumption of limitless borrowing or lending at the same interest rate.
8.1 Present Values
Let's try to generalize from the example above. Most investment projects are expected to produce revenues
for more than two periods. How should one evaluate them? The key is to gure out what is the right price at
which to value goods that one will receive several periods in the future. How much is a period-t good worth?
Let's imagine that interest rates are constant at r per period. This means you can take one good in period
2
0, lend it to obtain (1 + r) in period 1, then lend (1 + r) in period 1 to obtain (1 + r) in period 2 and so on
t 1
until you are nally left with (1 + r) goods in period t. This means that one good in period t is worth
(1+r)t
goods in period 0.
Now let's evaluate some arbitrary project. We'll summarize the project in terms of the dividends it will
produce at each point in the future. We'll denote the dividend from the project in period t by dt . The total
value of the project is the sum of all of these dividends, each of them valued in terms of the period-0 goods
that they are worth. In other words:
∞
X dt
V = t (8.1.1)
t=1 (1 + r)
Formula (8.1.1) is known as a present-value formula: it tells us what is the present value of any possible
sequence of dividends.
Armed with formula (8.1.1), deciding whether an investment project is worthwhile is straightforward.
Suppose the project costs I, then the net present value of the project is de ned as:
NPV ≡ V − I
Projects are worth doing if and only if the net present value is positive. In the example above, the net present
value was:
NPV = V − I
1100
= − 1000
1+r
which was positive if r < 0.1.
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8.1. Present Values
Example 8.2.
Opening a restaurant costs $100,000 dollars in year 0. In year 1 the restaurant will not be very well
known, so it is expected to make a loss of $10,000. In year 2, the restaurant will exactly break even.
Starting in year 3, the restaurant will be a big success, earning $40,000, $50,000, $60,000 and $70,000 in
years 3 to 6 respectively. In year 7, quinoa burgers will suddenly fall out of fashion so the restaurant will
close down forever. The interest rate is 10%. Is opening the restaurant a good idea?
Let's compute the net present value:
−10, 000 0 40, 000 50, 000 60, 000 70, 000
NPV = + + + + + − 100, 000
1.1 1.12 1.13 1.14 1.15 1.16
= 31, 881
In this example, the net present value is positive so opening the restaurant is a good idea.
The Gordon Growth Formula
There is a special case in which formula (8.1.1) becomes very simple. Suppose that the dividends from a
project are expected to grow at the constant rate g forever, so that dt+1 = (1 + g) dt . This means that
t−1
period-t dividends will be dt = d1 (1 + g) . In this case, formula (8.1.1) becomes:
∞ t−1
X d1 (1 + g) t−1
V = t (replacing dt = d1 (1 + g) )
t=1 (1 + r)
∞ t
d1 X 1 + g
= (rearranging)
1 + g t=1 1 + r
1+g
d1 1+r
= (appying the formula for a geometric sum)
1 + g 1 − 1+g
1+r
d1
= (simplifying) (8.1.2)
r−g
Formula (8.1.2) is known as the Gordon growth formula. It tells us the present value of any project as a
function of the initial level of dividends, its growth rate and the interest rate. Notice that V is high when
either g is high or r is low. High r pulls V down because it makes future goods less valuable in terms of
present goods. High g osets this and pulls V up because it makes future dividends larger.
Asset Prices, Adjustment Costs and the Q theory of investment
So far we have used formula (8.1.1) to think about the value of potential projects. But one can apply the same
logic to think about productive projects that are already in place. Formula (8.1.1) is also the answer to the
following question: if the project is already in place, how much should you be willing to pay to buy it?
This is not merely a hypothetical question. There are many markets where people actively trade assets
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8.1. Present Values
that are already producing dividends. In the stock market, people trade shares in companies that are already
operating; in the commercial real estate market, people trade buildings that are already yielding rental income;
in the housing market, people trade houses that are already producing housing services; in the bond market,
people trade bonds that are already paying coupons. Formula (8.1.1) tells us what should be the price at
which people trade in these markets.
Looking at asset prices in actual markets can be extremely useful in guiding investment decisions. Suppose
that a construction company knows that it can build an o ce building for X dollars. In order to decide
whether this is a good idea, they need to decide whether the present value of the rental income they will
obtain is more than this construction cost. This can be di cult to predict. However, if there is an active
market where comparable o ce buildings change hands, they can look at what prices people are paying to
buy them. If comparable o ce building sell for more than X, then it must be that investors think the present
value of rents is more than X, which would make the project a good idea.
One version of this idea is known as the Q theory of investment. It starts by de ning an object called Q,
or sometimes Tobin's Q (after James Tobin):
Market Value
Q≡ (8.1.3)
Book Value
The book value of a company measures its accumulated investment, net of depreciation. In theory, this should
approximately measure how much you need to invest to build a company just like it. Suppose this was exactly
true and you could build an exact replica of a company by investing an amount equal to the rm's book value.
Since it's an exact replica, its present value should be equal to the original company's market value. Whenever
Q > 1, then building a replica is a good idea. Under this extreme assumption of perfect replicability, we should
see in nite investment whenever Q>1 and no investment at all whenever Q < 1.
A less extreme version of this argument involves adjustment costs. Suppose expanding a rm involves
paying an adjustment cost in addition to the cost of the additional capital. This adjustment cost can be the
cost of physically installing machines, training workers to use them, or even the cost of making the decision
to invest. Adjustment costs can be represented by a function Ψ (I, K) that says how much extra a rm needs
to pay if its current capital stock is K and it wants to invest I to expand up to K0 = K + I. This function
could take many possible shapes; we'll assume that:
ψ I2
Ψ (I, K) = (8.1.4)
2K
The parameter ψ scales adjustment costs, so higher ψ means that adjusting the capital stock is more expensive.
Furthermore, (8.1.4) implies that the marginal adjustment cost is:
∂Ψ (I, K) I
=ψ
∂I K
which is higher when investment is higher as a proportion of existing capital. In other words, proportionately
small adjustments to the rm's scale are cheap to do, but large adjustments are expensive. This is an
assumption, and it's not obvious whether it's a good one or not. Indeed, some economists have argued that
154
8.2. Risk
Ψ (·) could involve a xed adjustment cost of undertaking any non-zero level of investment. Still, we are going
to assume (8.1.4) holds and see what the implications are.
Consider the problem of a rm that needs to decide how much to invest today. Its capital stock is K and
0
its stock market value is QK . Furthermore, it knows that if increases its capital stock to K , the stock market
value will rise to QK .1
0
The rm wants to maximize its stock market value net of the cost of investment. The
problem of the rm is:
ψ I2
max Q (K + I) − I +
I 2K
The rst order condition is:
ψI
−1− +Q=0
K
I 1
⇒ = (Q − 1)
K ψ
Without adjustment costs, we had that investment would be in nite when Q>1 and zero when Q < 1. With
I
adjustment costs, we get a less extreme version of the same idea: the investment-to-capital-ratio
K depends
1
positively on Q. If adjustment costs are low (so
ψ is high), then small changes in Q will lead to a strong
I
response of
K , and vice versa.
Figure 8.1.1 shows the relationship between Q and investment for the aggregate US economy. The rela-
tionship is positive, and has become especially strong in the last couple of decades.
8.2 Risk
In everything we've done so far, we have assumed that future dividends are perfectly known. Of course,
uncertainty is a central aspect of any investment decision. How should this be taken into account?
Let's think again about a two-period problem, although the same principles apply to problems with any
horizon. Imagine a project that will pay a dividend in period 2 but the size of the dividend depends on the
state of the world. The probability of a state s is denoted by Pr (s) and the dividend that the project will
pay in state s is denoted d (s).
Example 8.3.
The project is a farm. The dividend it will pay in period 2 depends on whether it rains. The interest rate
is 10%.
1
We sometimes distinguish between average Q, which is de ned by equation (8.1.3), and marginal Q, which is de ned by:
∂ Market Value
Marginal Q≡
∂ Investment
A rational rm should make its investment decisions on the basis of marginal Q: how much does the value of the rm change
with an additional unit of investment. By assuming that the stock market value will be QK 0 for any K0 we are assuming that
marginal Q is equal to average Q.
155
8.2. Risk
Fig. 8.1.1: Tobin's Q and in-
vestment. Source Andrei et al.
(2019).
State of the world Probability Pr (s) Dividend d (s)
s = Rain 0.75 100
s = Drought 0.25 8
We are going to look at this problem from the perspective of a household who has to answer the question:
is it worth investing p in period 1 to buy a unit of this project? What is the price p that makes the household
indierent between investing in the project or not? One naive way to do this would be to take an average of
the possible dividends and discount it at the interest rate. This would give the following answer:
0.75 × 100 + 0.25 × 8 77
p= = = 70 (8.2.1)
1 + 0.1 1.1
On average, the farm will pay a dividend of 77. Discounting it at an interest rate of 10% gives a present value
of 70. However, this way of thinking about the problem is not quite right because it ignores the fact that
people are risk averse and the farm is risky. What's the right way to do it?
It turns out that the answer depends on how this particular project ts with the rest of the household's
decisions. Suppose that, if it does not invest in this project, the household will be consuming c1 in period 1
and c2 (s) in period 2 in state of the world s. Now let's ask the household how many units of the project it
wants to buy, assuming that it has to pay p for each unit. Mathematically, the household's problem is:
X
max u (c1 − xp) + β Pr (s) u (c2 (s) + xd (s)) (8.2.2)
x
s
Buying x units of the project reduces period-1 consumption by xp, bringing it down to c1 − xp. On the other
hand, owning x units of the project increases period-2 consumption by xd (s) in state of the world s, bringing
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8.2. Risk
it up to c2 (s)+xd (s). Equation (8.2.2) describes the objective function of a household that wants to maximize
expected utility.
The rst order condition for this problem is:
X
− pu0 (c1 − xp) + β Pr (s) u0 (c2 (s) + xd (s)) d (s) = 0 (8.2.3)
s
Suppose we wanted to nd the price at which the household is exactly indierent between buying a little bit
of the asset and not buying at all. We would look for the price such that x = 0 satis es equation (8.2.3).
Setting x=0 and solving for p we get:
Pr (s) u0 (c2 (s)) d (s)
P
β s
p= (8.2.4)
u0 (c1 )
Now let's assume that the choices of c1 and c2 (s) are consistent with intertemporal maximization, so that the
Euler equation holds:
2
X
u0 (c1 ) = β (1 + r) Pr (s) u0 (c2 (s))
s
Replacing u0 (c1 ) into (8.2.4) and simplifying gives:
0
P
1 s Pr (s) u (c2 (s)) d (s)
p= P 0
(8.2.5)
1+r s Pr (s) u (c2 (s))
Equation (8.2.5) contains everything you might ever want to know about nance. It says that when there
is risk, the present value of an asset comes not just from discounting average dividends but by discounting
a weighted average of dividends, where the weights are proportional to marginal utility. If u0 (c2 (s)) was the
same for all s, then formula (8.2.5) reduces to;
1 u0 (c2 ) s Pr (s) d (s)
P
0
p= (u (c2 ) factors out)
1 + r u0 (c2 ) s Pr (s)
P
P
Pr (s) d (s)
= s (probabilities add up to one)
1+r
E (d)
= (8.2.6)
1+r
which is what our naive calculation in (8.2.1) was doing. The reason this is not quite right in general is that
u0 (c2 (s)) might dier for dierent s.
Let's break down formula (8.2.5) a bit more. Recall from statistics the de nition of a covariance between
two random variables X and Y:
Cov (X, Y ) ≡ E (XY ) − E (X) E (Y )
2
This is the generalized version of the Euler equation that has expected marginal utility on the right hand side, as in equation
(6.2.17)
157
8.2. Risk
Rearranging we obtain:
E (XY ) Cov (X, Y )
= E (Y ) + (8.2.7)
E (X) E (X)
Letting X = u0 (c) and Y = d, we can use formula (8.2.7) to rewrite the right hand side of (8.2.5) and obtain:
E (d) + Cov (d, u0 (c))
p= (8.2.8)
1+r
Formula (8.2.8) shows us the consequences of using a marginal-utility-weighted average rather than a simple
average to value the project's dividends. Compared to (8.2.6) there is an additional term: Cov (d, u0 (c)). How
much a household is willing to pay for a marginal unit of a project depends on average dividends and on how
those dividends co-vary with marginal utility. What is this telling us? The marginal utility of consumption
measures how much the household values extra consumption in a particular state of the world. If dividends
co-vary positively with marginal utility, then they provide the household extra consumption exactly when the
household values it the most. This makes the asset attractive, so the household is willing to pay more than
E(d)
1+r for it. Conversely, if dividends co-vary negatively with marginal utility, then the asset gives the household
E(d)
extra consumption exactly when the household values it the least. The household will pay less than
1+r to
hold such an asset.
Let's see some examples.
Example 8.4.
The asset is a bet on a (fair) coin ip. If the coin turn out heads, it pays one dollar; if it turns up tails,
it pays zero. The interest rate is zero.
Here
E(d)
1+r = 0.5. Furthermore, Cov (d, u0 (c)) ≈ 0. Why? Because the event the coin turn up heads is
independent of the events that determine whether the household has high or low consumption (such as losing
a job, getting a promotion, etc.). Therefore the household is willing to pay p = 0.5 for this asset.
Example 8.5. The asset is a bet on a (fair) coin ip. If the coin turn out heads, it pays one million
dollars; if it turns up tails, it pays zero. The interest rate is zero.
Here
E(d)
1+r = 500, 000. However, Cov (d, u0 (c)) < 0. Why is this dierent from the previous example?
Because now, if the household buys the asset, the outcome of the coin ip is a very big deal: the household
will consume much more if the coin comes up head than if it comes up tails. Therefore marginal utility of
consumption will be lower exactly when the asset pays a high dividend. A rational, risk-averse household will
value this asset at p < 500, 000.
Example 8.6. The asset is one dollar in Facebook shares and Sheryl is a Facebook employee.
Here it's likely that Cov (d, u0 (c)) < 0. Why? The dividend from the asset is not by itself a big determinant
of Sheryl's consumption because she only owns one dollar of it. A bigger determinant is how she's doing at
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8.3. MPK and Investment
her job: whether she gets a raise, gets red, etc. However, she is more likely to get a raise and less likely to
get red if Facebook is doing well. Hence the asset tends to have higher dividends in the states of the world
E(d)
where Sheryl's consumption is high and values them less. Sheryl will be willing to pay less than
1+r for this
asset.
Example 8.7.
The asset is a one-dollar health insurance contract. A household member gets sick with probability 0.1.
If this happens, the insurance policy gives the household one dollar that can be used towards medical
expenses. The household has no other medical insurance. The interest rate is zero.
Here
E(d)
1+r = 0.1 but Cov (d, u0 (c)) > 0. Why? Absent other insurance, getting sick is expensive, so the
household has to cut back on consumption to pay medical bills if a household member gets sick. This means
that the asset pays exactly in those states of the world where marginal utility is high. Therefore a risk averse
household would be willing to pay p > 0.1 for this asset.
8.3 The Marginal Product of Capital and Aggregate Investment
So far we have been talking about the decisions to invest in individual projects. In macroeconomics we usually
care about what determines the overall level of investment. To think about that, we are going to abstract
from the features of each individual investment project and go back to assuming that every project is identical.
What determines how many investment projects take place?
We are going to imagine that the representative investment project consists simply of converting one unit
of output into a unit of capital. If this is done in period t, then in period t + 1, this unit of capital can be
K
rented out to a rm at the rental rate of capital rt+1 . As we saw in Chapter 4, with competitive markets we
K
have rt+1 = FK (Kt+1 , Lt+1 ). In addition to the rental income, the investor gets back the depreciated capital,
so in total he gets FK (Kt+1 , Lt+1 ) + 1 − δ goods at t + 1. The net present value of the project is:
1 + FK (Kt+1 , Lt+1 ) − δ
NPV = −1 (8.3.1)
1 + rt+1
The rst term is what the investor gets in period t + 1, valued in terms of period-t goods. Minus 1 is the cost
of the project in period t. The NPV of a representative investment project is a decreasing function of the
following-period capital stock Kt+1 , other things being equal. Why? Because a higher capital stock means a
lower marginal product of capital and therefore a lower rental rate of capital.
How does formula (8.3.1) help us gure out the total level of investment? The key thing to notice is that
the NPV of the representative investment project must be exactly zero. Why? If it was positive, there would
be positive-NPV projects left undone; conversely, if it was negative, it means negative-NPV projects are being
done. Neither of these possibilities is consistent with projects being carried out whenever they have positive
NPV. Setting the NPV to zero in (8.3.1) we get:
1 + FK (Kt+1 , Lt+1 ) − δ
−1=0
1 + rt+1
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8.3. MPK and Investment
⇒ FK (Kt+1 , Lt+1 ) − δ = rt+1 (8.3.2)
We have already seen equation (8.3.2) before. It's identical to equation (4.4.12) in Chapter 4. There we
were asking the question the other way around: given a level of investment, what must the interest rate be?
Here we are asking: given an interest rate, what will be the level of the capital stock? The level of the capital
stock must be such that the rental rate of capital makes the NPV of the representative investment project
equal to zero.
Formula (8.3.2) is stated in terms of the level of the capital stock. In order to know the level of investment,
we use that:
Kt+1 = (1 − δ) Kt + It
and replace Kt+1 to get:
FK ((1 − δ) Kt + It , Lt+1 ) − δ = rt+1 (8.3.3)
Figure 8.3.1 shows equation (8.3.3) graphically. The left-hand side of the equation is a downward-sloping
curve, which inherits the shape of the marginal product of capital curve. Sometimes this is known as an
investment demand schedule, meaning that it tells us how much investment would be carried out at each
possible level of interest rates. The right hand side is represented with a horizontal line since we are taking
interest rates as given.
Fig. 8.3.1: The determina-
tion of aggregate investment.
We can use equation (8.3.3), either graphically or algebraically, to ask how the level of investment responds
to dierent changes. The left panel of Figure 8.3.2 shows how investment responds to an increase in expected
productivity, represented by an upward shift in the FK curve. At the original level of investment, higher
productivity would make the representative investment project positive-NPV. This encourages additional
160
8.3. MPK and Investment
investment until the decreasing marginal product of capital ensures that NPV equals zero again. Hence,
investment rises. The right panel shows how investment responds to a rise in the interest rate. At the original
level of investment, a higher interest rate would make the representative investment project negative-NPV.
This leads to a fall in investment until the higher marginal product of capital ensures that NPV equals zero
again. Hence, investment falls.
Fig. 8.3.2: Investment response to changes in productivity and interest rates.
Exercises
8.1 Valuation
Charlie's cheese factory has a very precise business plan for 2019-2028, shown below (you can download
it as an Excel spreadsheet from the book website):
Year Pro ts Investment Dividend
2019 80 50 30
2020 77 49 28
2021 86 49 37
2022 91 47 44
2023 98 47 51
2024 98 48 50
2025 109 200 -91
2026 120 57 63
2027 119 57 62
2028 125 61 64
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8.3. MPK and Investment
(in 2025 the main storage facility will need to be replaced, hence the higher investment). From 2029
onwards, dividends will increase at a rate of 2% a year forever. The interest rate is 6%.
(a) Use the Gordon growth formula (8.1.2) to calculate the value that the factory will have in 2028 after
paying dividends (i.e. the value not including the value of the dividends it will pay in 2028).
(b) Compute the present value of the entire in nite stream of dividends that starts in 2019.
8.2 Bond Pricing
Suppose the interest rate is 4% and is expected to stay at 4% forever. On January 1, 2018, the government
issued a T -year bond, which pays coupons of 4 cents every year on January 1 (starting in 2019) and then
pays 1 dollar (in addition to the 4-cent coupon) on January 1 of the year 2018 + T .
(a) Use formula (8.1.1) to compute what the market price of the bond should be. How does it depend
on T? Explain.
(b) Suppose that after the bond has been issued, market conditions change and interest rates fall to 3%,
and are expected to remain at 3% forever. What is the market price of the bond now? How does it
depend on T? Explain.
8.3 Looking at the Stock Market
Speci c Mills is a company that has just started operations. Its business consists of buying wheat from
farmers, grinding it to produce our and selling the our. It is very small relative to both the wheat
market and the our market. Its assets consist of a single plant that cost $10 million to build. It issued 1
million shares, which currently trade in the stock market at a price of $10 per share.
(a) What is Q for Speci c Mills?
(b) Speci c Mills' share price suddenly rises to $15 per share. What is Q now?
Management is considering expanding the plant. It has calculated that in order to expand the plant to λ
times its current capacity, it is going to have to carry out additional investment that will cost
(λ − 1) + ψ(λ − 1)2
times its original investment, with ψ = 1. Assume that management trusts the stock market investors'
opinion about the pro tability of the grain-processing industry.
(c) By how much should it expand its capacity?
(d) How much will it spend in total to do so?
(e) Now suppose that instead of Speci c Mills, which is small relative to the industry it operates in, the
same situation arose for a company that was large relative to its market. How does its reaction to
its stock market price compare to that of Speci c Mills? Why?
162
8.3. MPK and Investment
8.4 The Magnitude of Adjustment Costs
The slope of the relationship illustrated in Figure 8.1.1 is approximately 0.01, so an increase in Q of 1 is
I
associated with an increase in the investment-to-capital ratio
K of 0.01.
(a) If adjustment costs are given by formula (8.1.4), what value of ψ would be consistent with this
observation?
(b) If a rm's investment is equal to 10% of its capital stock, how much does it have to spend in adjustment
costs as a fraction of its total investment? How about if a rm's investment is 20% of its capital?
8.5 A Risky Asset
The world lasts two periods. Danube.com is a large internet-based retailer. At t = 2, it will pay a dividend
of $150 per share if the economy is doing well and $50 per share if the economy is doing poorly. The
representative household will consume $40,000 if the economy is doing well and $30,000 if the economy
c1−σ
is doing poorly. Its preferences are given by E . The interest rate is 10%. Use formula (8.2.5) to
1−σ
determine at what price the representative household would be indierent with respect to buying shares
in Danube.com, for the following values of σ: 0.5,1,2 and 10. Explain.
8.6 A Risky Investment
Ingrid plans to live for two periods and only cares about consumption in the second period. She is
considering getting her MBA in the rst period. If she does it, it's going to cost her $200,000 between
tuition and foregone wages. Once she graduates, there is a 50% chance that in the second period she
will get the job she wants, which will pay her $5,000,000 and an 50% chance she'll get a regular job that
will pay her $80,000. If she doesn't get her MBA, she'll put $200,000 in the bank, where it will earn
10% interest, and then get a regular job that pays her $80,000. In all cases, since she will only live for
two periods, she will consume everything she has in period 2. Suppose Ingrid's preferences over period-2
c1−σ
consumption are given by E .
1−σ
(a) For what values of σ is it a good idea to get an MBA? Explain.
(b) Now suppose σ = 1.2. For what values of the interest rate is it a good idea to get an MBA? Explain.
8.7 Credit Constraints
Two world lasts two periods. A rm's production function is F (K, L) = At K α L1−α , where At can take
dierent values in each period. In each period, the rm can hire labor in a competitive labor market at
the same wage w, which the rm takes as given. However, there is no market for renting capital: the
rm can only use capital that it owns. If the rm owns K units of capital and decides to hire L workers,
then it earns F (K, L) − wL. The rm starts o having K1 units of capital in the rst period. There is
no depreciation. At the end of period 1, the rm can buy capital for period 2 by either using its earnings
or by borrowing. Loans must be paid back in period 2. The interest rate is r. The rm can also use its
earnings to pay dividends to its shareholders in period 1. The maximum amount that lenders are willing
to lend to the rm is b.
163
8.3. MPK and Investment
(a) Set up the problem of the rm that needs to decide how many workers to hire in each period. Note
that this problem can be solved period-by-period taking Kt as given. Find an expression for the
period-t pro ts of a rm that takes as given its capital stock Kt and chooses how much labor to hire,
i.e. for π(Kt ) ≡ maxL F (Kt , L) − wL.
(b) Set up the problem of the rm that must decide how much to pay in dividends, how much to borrow
and how much to invest.
(c) Assume b = ∞. Show that the optimal amount of investment depends on A2 but not on A1 . Explain.
(d) Assume b = 0. Show that if A1 is su ciently high, then the optimal amount of investment is the
same as in part (c). Find the minimum level of A1 such that this is the case, and denote it A∗1 . Show
that for A1 < A∗1 , investment depends on A1 . Explain.
(e) Suppose A1 < A∗1 . How does the rm react to an increase in A2 ? How does the rm react to an
increase in b? Explain.
8.8 An Earthquake
Suppose an earthquake destroys a large part of the capital stock at time t. Assume interest rates and
future labor supply are not aected by the earthquake, and there are no adjustment costs.
(a) What will happen to investment?
(b) How does Kt+1 compare with and without the earthquake?
8.9 Aggregate Investment with Risk
The world lasts two periods. The aggregate production technology for period 2 is given by:
F (K, L) = AK α L1−α
K is the aggregate capital stock in period 2 and L is the labor force, which is exogenous and normalized
to L = 1. Period 2 is the end of the world, so capital depreciates fully (δ = 1) and the representative
household will consume F (K, L) = AK α L1−α . The utility function is u(c) = log(c). A is a random
variable, which can take two possible values: AH = 1 + ε or AL = 1 − ε , with equal probability. The
interest rate between periods 1 and 2 is r.
Now consider the following investment project: investing an additional unit in period 1 to obtain an
additional unit of capital in period 2.
(a) What is the dividend produced by the marginal unit of capital? Express it as a function of the
aggregate capital stock K and realized productivity A.
(b) Suppose ε = 0. For what value of K is the net present value of additional investment exactly zero?
(c) Now suppose ε > 0. For what value of K is the net present value of additional investment exactly
zero? (Use equation (8.2.5) to guide your answer)
(d) How does K depend on ε? Explain.
164
CHAPTER 9
General Equilibrium
In Chapters 6-8 we have studied the decisions of households and rms in isolation. In this chapter we look at
how they all t together.
In microeconomics we say that there's an equilibrium in a competitive market for some good if supply
equals demand: everyone buys or sells as much as they want and the outcome is that sales equal purchases.
General equilibrium is the same idea but applied to many goods at once.
The model economy we'll be looking at will have:
• a representative household making consumption-savings-leisure decisions
• a representative rm choosing to hire labor and rent capital
• a representative investment rm carrying out investment
We'll rst look at this in a simpli ed two-period model and then we'll extend the analysis to an in nite-period
case.
9.1 General Equilibrium in a Two-Period Economy
We are going to assume that the economy lasts only for two periods. In the rst period, there is already some
level of capital K1 in place, and it's owned by the representative household. Each period, the production
function is F (K, L) and there are competitive markets for labor and renting capital. The initial capital stock
depreciates at rate δ between period 1 and period 2, and it's possible to invest in order to build more capital.
After period 2 it's the end of the world, so capital depreciates completely (δ2 = 1) and there is no more
investment.
Household, Firm and Investment Firm Problems
The representative household must choose how much to consume and also how much of its time to dedicate
to leisure and consumption. It solves a problem like the one we looked at in Section 7.4:
165
9.1. Two-Period Economy
max u (c1 ) + v (l1 ) + β [u (c2 ) + v (l2 )]
c1 ,l1 ,c2 ,l2
(9.1.1)
s.t.
1 1
w2 (1 − l2 ) + K1 1 + r1K − δ + Π
c1 + c2 ≤ w1 (1 − l1 ) +
1+r 1+r
The only thing that is dierent between this problem and the one we know from Section 7.4 is that
the household starts with some initial wealth: it owns the initial capital stock and earns a rental (net of
depreciation) for it in the rst period. Also, the household owns the rms, so if they were to make pro ts, the
present value of those pro ts, which we denote by Π, would be part of the household's budget. Π is given
Π2F
by Π= Π1F + 1+r +Π I F
, where Πt are the pro ts of productive rms in period t and Π I
are the pro ts of
investment rms. In equilibrium, it will be the case that Π = 0, so we can just ignore this part.
There is a representative rm. In each period, the rm solves the pro t maximization problem we saw in
Chapter 4:
ΠtF = max F (K, L) − wt L − rtK K
K,L
Investment consists of building period-2 capital. We are going to imagine that there is a representative
investment rm that carries out all the investment. This rm buys the (1−δ)K1 units of used period-1 capital,
adds I units of investment and obtains K2 = (1 − δ)K1 + I units of period-2 capital (here we are assuming
there are no adjustment costs like the ones we had in Section 8.1). The investment rms then rents out the
K2 units of capital at the rental rate r2K (given that the second period is the end of the world, they fully
r2K
depreciate in period 2). Converted back to present value, this means the investment rm earns
1+r K2 from
renting out the capital. The problem of the investment rm is:
r2K
Π I = max [(1 − δ)K1 + I] − [(1 − δ)K1 + I] (9.1.2)
I 1+r
| {z } | {z }
Present Value of K2 rental Cost of K2
Equilibrium De nition
De nition 9.1.
Given an initial K1 , a competitive equilibrium consists of:
1. An allocation {c1 , l1 , c2 , l2 , I, K2 , L1 , L2 }.
2. Prices w1 , w2 , r1K , r2K , r .
such that:
1. {c1 , l1 , c2 , l2 } solves the household's problem, taking prices as given.
2. Lt , Kt solve the rm's problem for t = 1, 2, taking prices as given.
166
9.1. Two-Period Economy
3. I solves the investment rm's problem, taking prices as given.
4. Markets for goods and labor in each period clear:
(a) Goods:
F (K1 , L1 ) = c1 + I
|{z} (9.1.3)
Investment
|{z}
Consumption
| {z }
GDP
F (K2 , L2 ) = c2 (9.1.4)
|{z}
Consumption
| {z }
GDP
(b) Capital:
K2 = K1 (1 − δ) + I (9.1.5)
(c) Labor:
Lt + lt = 1 (9.1.6)
A competitive equilibrium satis es two basic properties:
• Everyone is, individually, making the best choices they can. This is represented by conditions 1-3, which
say that each individual household and each individual rm makes its choices in their own best interest.
• Things add up, i.e. everyone's choices are consistent with everyone else's choices. This is represented in
condition 4. Condition 4(a) says that all the output in the economy is used for either consumption or
investment in the rst period and, since it's the end of the world, only for consumption in the second
period. (Recall that this is a closed economy with no government, so there is no other use for output).
Condition 4(b) says that the capital that rms want to use in period 2 is equal to the amount of period-1
capital that remains plus the amount that investment rms chose to build. Condition 4(c) says that the
labor that rms choose to hire (Lt ) plus the amount of time the households choose to dedicate to leisure
(lt ) add up to the entire amount of time available, which is normalized to 1.
Describing an Equilibrium
In this section we'll nd a system of equations whose solution represents the economy's competitive equilibrium.
For now we are going to leave it as a mathematical expression and the economics it contains might be a little
hard to discern. We'll use these equations to think more about economics in later chapters.
We know from Section 7.4 that the rst order conditions are (7.4.2) and (7.4.3), which we just restate here:
v 0 (lt )
= wt (9.1.7)
u0 (ct )
u0 (c1 ) = β (1 + r) u0 (c2 ) (9.1.8)
Equation (9.1.7) is the same as (7.4.2) and it describes how the household trades o leisure and consumption.
Equation (9.1.8) is the same as (6.3.9) and (7.4.3) and it describes how the household trades o present and
future consumption.
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9.2. First Welfare Theorem
As we saw in Chapter 4, the solution to the rm's problem can be summarized by the rst-order conditions:
FK (Kt , Lt ) − rtK = 0 (9.1.9)
FL (Kt , Lt ) − wt = 0 (9.1.10)
(which are the same as (4.4.1) and (4.4.2)).
The investment rm's problem (9.1.2) is linear in I. This means that unless 1 + r = r2K , investment
rms would be able to make in nite pro ts by choosing either I =∞ or I = −∞ depending on which way
the inequality goes. This would be inconsistent with capital-market clearing. Therefore it must be that in
equilibrium:
1 + r = r2K (9.1.11)
which is just equation (4.4.12) when depreciation is set to 1. Recalling the de nition of net present value from
Chapter 8, this says that in equilibrium the NPV of investment must be zero.
Replacing (9.1.10) into (9.1.7) we obtain:
v 0 (lt )
= FL (Kt , Lt ) (9.1.12)
u0 (ct )
| {z } | {z }
Marginal Rate of Substitution Marginal Rate of Transformation
Equation (9.1.12) summarizes how this economy will allocate the use of time. On the left hand side, the
v 0 (lt )
expression 0
u (ct ) describes how the representative household is willing to trade o leisure against consumption.
On the right hand side, FL (Kt , Lt ) describes how the available technology is able (at the margin) to convert
time into output.
Replacing (9.1.11) and (9.1.9) into (9.1.8) and we obtain:
u0 (c1 )
= FK (K2 , L2 ) (9.1.13)
βu0 (c2 )
| {z } | {z }
Marginal Rate of Substitution Marginal Rate of Transformation
Equation (9.1.13) summarizes how this economy will allocate output between the present and the future. On
u0 (c1 )
the left hand side, the expression
βu0 (c2 ) describes how the representative household is willing to trade o
present consumption against future consumption. On the right hand side, FK (K2 , L2 ) describes how the
available technology is able, by building a marginal unit of capital, to convert current output into future
output.
Equations (9.1.12) and (9.1.13) summarize what the general equilibrium will look like. We'll come back to
them many times.
9.2 The First Welfare Theorem
A competitive economy will allocate resources in one speci c way. What if we could choose a dierent
allocation? Would that be a good idea?
168
9.2. First Welfare Theorem
In order to answer this question it is convenient to invoke the metaphor of a social planner. The idea is
to imagine that, instead of making their decisions individually, everyone delegates decisions to a benevolent
social planner. Is it the case that the social planner would want to change the allocation of resources to
something other than the competitive equilibrium? We'll show that, under the assumptions we've made so
far, the answer is no. Even a social planner that was perfectly benevolent and had no practical, political
or cognitive di culties in choosing among all the possible allocations of resources would be satis ed with the
outcome of competitive markets.
The ctitious social planner solves the following problem:
max u (c1 ) + v (l1 ) + β [u (c2 ) + v (l2 )]
c1 ,l1 ,c2 ,l2 ,L1 ,L2 ,K2
s.t.
K2 ≤ (1 − δ) K1 + F (K1 , L1 ) − c1
(9.2.1)
c2 ≤ F (K2 , L2 )
Lt ≤ 1 − lt for t = 1, 2
K1 given
What does this optimization problem represent, in economic terms? First, the planner indeed is benevolent.
Its objective function is the same as the objective of the representative household: the planner wants to make
the representative household happy. Second, the planner is quite powerful: it can tell everyone exactly how
much to work, consume and invest without worrying about whether they will obey the instructions. In that
regard, our ctitious social planner is much more powerful than a government could possibly be. Lastly, the
planner is not all-powerful: it is constrained by the technological possibilities of the economy. The constraints,
which are identical to conditions (9.1.3)-(9.1.6), say that in order to accumulate capital it is necessary to give
up consumption and in order to produce output it is necessary to use labor, which requires giving up leisure.
These constraints are not exactly budget constraints since the planner is not buying or selling from anyone;
rather, they are technological constraints.
Let's replace the constraints in the objective function and then solve the planner's problem. Replace
c1 = (1 − δ) K1 + F (K1 , L1 ) − K2 , c2 = F (K2 , L2 ) and lt = 1 − Lt 1
into the objective to get:
max u ((1 − δ) K1 + F (K1 , L1 ) − K2 ) + v (1 − L1 ) + β [u (F (K2 , L2 )) + v (1 − L2 )]
L1 ,L2 ,K2
The rst order conditions with respect to L1 , L2 and K2 respectively are:
u0 ((1 − δ) K1 + F (K1 , L1 ) − K2 ) FL (K1 , L1 ) − v 0 (1 − L1 ) = 0
β [u0 (F (K2 , l2 )) FL (K2 , L2 ) − v 0 (1 − L1 )] = 0
−u0 ((1 − δ) K1 + F (K1 , L1 ) − K2 ) + βu0 (F (K2 , l2 )) FK (K2 , L2 ) = 0
which simplify back to (9.1.12) and (9.1.13), which are the equations that describe the competitive equilibrium.
1
Alternatively, we can set up a Lagrangian with all the constraints.
169
9.2. First Welfare Theorem
Therefore the equations that de ne the solution to the social planner's problem are the same as those which
de ne the competitive equilibrium! This is not a coincidence. The following result proves that the allocations
that result from a competitive equilibrium must achieve the optimum in the planner's problem.
Proposition 9.1 (First Welfare Theorem) . If an allocation {c1 , l1 , c2 , l2 , I, K2 , L1 , L2 } is part of a com-
petitive equilibrium then it solves the social planner's problem.
Proof. Proceed by contradiction. Suppose that an allocation {c1 , l1 , c2 , l2n
, I, K2 , L1 , L2 } with priceso
ĉ1 , ˆl1 , ĉ2 , ˆl2 , I,
ˆ K̂2 , L̂1 , L̂2
w1 , w2 , r1K , r2K , r is a competitive equilibrium but there is another allocation
that satis es the constraints on the social planner's problem but achieves strictly higher utility for the
household. This implies that:
1 1
ĉ2 > w1 1 − ˆl1 + w2 1 − ˆl2 + K1 1 + r1K − δ + Π
ĉ1 + (9.2.2)
1+r 1+r
Equation (9.2.2) says, if prices are the equilibrium prices, then the household cannot aord the consumption-
n o
leisure combination ĉ1 , ˆl1 , ĉ2 , ˆl2 . Otherwise, since it's presumed to be strictly better, the household
would have chosen it. Furthermore, the fact that in equilibrium rms and investment rms are maximiz-
ing pro ts implies that by choosing the equilibrium quantities they make at least as much pro t as they
would by choosing the alternative quantities:
Π1F = F (K1 , L1 ) − w1 L1 − r1K K1 ≥ F K1 , L̂1 − w1 L̂1 − r1K K1
Π2F = F (K2 , L2 ) − w2 L2 − r2K K2 ≥ F K̂2 , L̂1 − w2 L̂1 − r2K K̂2
K h i rK h
r2 ˆ
i
I
Π = − 1 (1 − δ)K1 + I ≥ 2
− 1 (1 − δ)K1 + Iˆ
1+r 1+r
Therefore, using the de nition:
Π2F
Π ≡ Π1F + + ΠI
1+r
to add the pro ts of production rms and investment rms, we have that:
F K̂2 , L̂1 − w2 L̂1 − r2K K̂2
r2K
h i
Π ≥ F K1 , L̂1 − w1 L̂1 − r1K K1 + + − 1 (1 − δ)K1 + Iˆ (9.2.3)
1+r 1+r
Replacing (9.2.3), L̂t ≤ 1 − ˆlt and Iˆ = K2 − (1 − δ)K1 into (9.2.2) and simplifying, we obtain:
1 F K̂2 , L̂1
ĉ1 + K2 + ĉ2 > F K1 , L̂1 + (1 − δ)K1
1+r 1+r
This implies that one of the following statements must be true. Either:
ĉ1 + K2 > F K1 , L̂1 + (1 − δ)K1
170
9.2. First Welfare Theorem
or
ĉ2 > F K̂2 , L̂1
n o
either of which contradicts the assumption that ĉ1 , ˆl1 , ĉ2 , ˆl2 , I,
ˆ K̂2 , L̂1 , L̂2 satis es the constraints on
the planner's problem.
What is the economic logic behind the First Welfare Theorem? Why are the social planner's choices the
same as those the market produces? The social planner chooses an allocation to maximize utility subject to
technological possibilities. Conversely, in a competitive economy, each household maximizes utility subject to
prices (i.e. wages, the rental rate of capital and the interest rate), as equations (9.1.7) and (9.1.8) indicate;
however, those prices in turn re ect technological possibilities, as equations (9.1.9) and (9.1.10) indicate.
Therefore the household is also, indirectly, maximizing utility subject to technological possibilities.
What we have shown is really just a special case of the First Welfare Theorem. The result is much more
general. In particular, it is still true if:
• There are many dierent goods at each date (e.g. apples, oranges, etc.) instead of just a general
consumption good.
• There is (exogenous) technological progress.
• There is uncertainty, as long as there are complete markets, i.e. markets to trade insurance against
every possible state of the world, at competitive prices.
• There are many dierent households with dierent preferences, dierent abilities and dierent wealth
instead of a representative household. However, for this case the theorem needs to be stated slightly
dierently. If an economy has dierent households, there is no unique way to de ne the social planner's
problem because the planner would have to decide how much the utility of each dierent household
matters. Hence, the more general version of the FWT says that if an allocation is part of a competitive
equilibrium then it is Pareto optimal: there is no technologically feasible way to make someone better
o without making someone else worse o.
The FWT is an extremely useful guide for thinking about public policy. In an economy where the conditions
for the theorem hold, then any policy that makes a household better o, relative to the competitive equilibrium,
necessarily makes someone else worse o. Does this mean that no policy is ever justi ed? Some people interpret
the theorem to imply just that, but this conclusion requires an extra bit of political philosophy. Depending
on one's views on the nature of justice, it is quite possible to advocate for a policy that bene ts some groups
of people at the expense of others (for instance, policies that bene t the poor at the expense of the rich or the
old at the expense of the young). What the FWT does clarify is that, if the economy is competitive, the only
possible economic justi cation for a policy is that one views the resulting redistribution as desirable.
The conditions for the FWT are quite strict and no one believes that they hold exactly in practice. Two
of the main things that would make the FWT not hold are:
2
2
Other features that would make the FWT fail include asymmetric information, incomplete markets and borrowing constraints.
171
9.3. In nite-Period Economy
• Monopoly power. Monopolists reduce quantity, relative to a competitive producer, in order to raise
prices. Therefore it is not true that they equates the value of the marginal product to the factor prices,
which is the key step that leads to equations (9.1.12) and (9.1.13). Hence, the FWT does not hold. In
Chapter 14 we study a model that features monopoly power.
• Externalities. In an economy with externalities, the private value and the social value of a good do not
coincide. If I decide to hire a (good) orchestra to play in my garden and all my neighbors enjoy it, the
social value this produces exceeds the private value that I obtain. When I decide whether or not to hire
the orchestra, I will ignore the bene t to my neighbors while a benevolent social planner would take it
into account, so I will hire an orchestra less often than what the social planner would want. Similarly, I
will over-pollute, under-invest in preventing communicable diseases, etc.
Again, the FWT helps to organize the analysis of public policy. If a policy is not about redistribution,
then it can only be justi ed economically on the basis of which failure of the FWT it's designed to address.
3
For instance, zoning regulations can (perhaps) be justi ed as a way to deal with externalities: if I could build
a tall building next to your house, I would make your garden less sunny. Antitrust policies can (perhaps) be
justi ed as a way to deal with monopoly power. It is often useful to begin thinking about a policy problem
by asking what is the failure of the FWT that the policy is designed to address.
9.3 General Equilibrium in an In nite-Period Economy
Let's now look at what happens in an economy that lasts in nite periods. We'll proceed in the same way as
before: describe the household's problem, the rm's problem, the investment decision and nally de ne and
describe an equilibrium.
The representative household solves the following problem:
∞
X
max β t [u (ct ) + v (lt )]
ct ,lt ,at+1
t=0 (9.3.1)
s.t.
at+1 = (1 + rt ) at + wt (1 − lt ) + ΠtF + ΠtI − ct (9.3.2)
a0 = K0 1 + r0K − δ given
a no-Ponzi-game condition like (6.3.7)
In each period, the household obtains utility from both consumption ct and leisure lt . Equation (9.3.2) is
the budget constraint; more precisely, there is an in nite number of budget constraints, each on them linking
one period to the following one. ΠtF and ΠtI represent the pro ts of the productive and investment rms
respectively, which will turn out to be zero.
3
There's a gray area with policies that are justi ed on the basis that people make the wrong choices. Should a benevolent
social planner want to give people what they themselves would choose or what the social planner knows is best for them?
This issue famously comes up in discussions of drug policy but also, for instance, in nancial regulation. The answer involves
a philosophical discussion that economists usually don't specialize in. Thaler and Sunstein (2008) discuss many policy issues
related to this question.
172
9.3. In nite-Period Economy
The rm's problem is unchanged: it just maximizes pro ts period-by-period. The investment rm's prob-
lem is similar to (9.1.2), except that capital does not fully depreciate in one period, so a rm that invests at
time t knows that it will get back depreciated capital at time t+1 in addition to a rental. Therefore, the
investment rm solves:
K
rt+1 +1−δ
ΠtI = max [Kt (1 − δ) + I] − [Kt (1 − δ) + I] (9.3.3)
I 1 + rt+1
De nition 9.2.
A competitive equilibrium consists of:
∞
1. An allocation {ct , lt , It , Kt+1 , Lt }t=0 .
∞
2. Prices wt , rtK , rt t=0 .
such that:
∞
1. {ct , lt }t=0 solves the household's problem, taking prices as given.
2. Lt , Kt solve the rm's problem for every t, taking prices as given.
3. It solves the investment rm's problem for every t, taking prices as given.
4. Markets for goods and labor in each period clear:
(a) Goods:
F (Kt , Lt ) = ct + It (9.3.4)
|{z} |{z}
Consumption Investment
| {z }
GDP
(b) Capital:
Kt+1 = Kt (1 − δ) + It (9.3.5)
(c) Labor:
Lt + lt = 1 (9.3.6)
As you can see, there isn't much conceptual dierence between how we de ne an equilibrium in a two-
period model and in an in nite-period model. In fact, the description of how the equilibrium behaves is also
very similar. The rst order conditions for the household's problem are:
v 0 (lt )
= wt (9.3.7)
u0 (ct )
u0 (ct ) = β (1 + rt+1 ) u0 (ct+1 ) (9.3.8)
For the rms problem:
FK (Kt , Lt ) − rtK = 0 (9.3.9)
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9.3. In nite-Period Economy
FL (Kt , Lt ) − wt = 0 (9.3.10)
And setting the NPV of investment to zero to prevent investment rms from choosing in nite investment:
K
rt+1 = rt+1 −δ (9.3.11)
which is just equation (4.4.12).
Replacing (9.3.10) into (9.3.7) we obtain:
v 0 (lt )
= FL (Kt , Lt ) (9.3.12)
u0 (ct )
| {z } | {z }
Marginal Rate of Substitution Marginal Rate of Transformation
and replacing (9.3.11) and (9.3.9) into (9.3.8) and we obtain:
u0 (ct )
= (1 + FK (Kt+1 , Lt+1 ) − δ) (9.3.13)
βu0 (ct+1 )
| {z } | {z }
Marginal Rate of Substitution Marginal Rate of Transformation
All of these equations have the same interpretation as in the two-period economy. The in nite-horizon model
will just let us ask some additional questions, such as what will the economy look like in the long run? or
how does the economy react today to news about things that will happen in the distant future?
Dynamics
When we studied the Solow model in Chapter 4, we concluded that an economy with a constant savings rate
and a xed labor supply would converge towards a steady state, where the capital stock and output were
constant and investment was just enough to make up for depreciation. Now that we have a theory of what
determines the savings rate, we can ask the same questions again: what will the economy look like in the long
run? How will it behave in the meantime?
To keep things relatively simple and focus just on the consumption / investment problem, we are going to
go back to the assumption that the labor supply is xed, so instead of equation (9.1.12) we'll just have Lt = 1.
c1−σ
Furthermore, we'll assume that the representative household has the utility function u (c) = 1−σ , which we
rst introduced in Chapter 2. With this utility function and setting Lt = 1, equation (9.3.13) reduces to:
ct+1 1
= [β (1 + FK (Kt+1 , 1) − δ)] σ (9.3.14)
ct
Equation (9.3.14) gives us a relationship between the rate of growth of consumption between t and t + 1 and
the level of the capital stock at time t + 1. If the capital stock is low, consumption should be growing over
time. What is the economic logic that it represents? Suppose, for instance, that the capital stock at t+1 is
low. This implies that the marginal product of capital ishigh, due to diminishing marginal product; therefore
the rental rate of capital will be high, by equation (9.1.9); therefore the interest rate will be high, by equation
(9.1.11); therefore current consumption is expensive relative to future consumption, as we saw in Chapter 6;
174
9.3. In nite-Period Economy
therefore the household chooses to consume more in the future than in the present, by the Euler equation
(9.1.8), which is exactly what a high rate of growth of consumption means.
In addition, we have the market clearing condition (9.1.3), which we can rewrite as:
Kt+1 = (1 − δ) Kt + F (Kt , 1) − ct (9.3.15)
This gives us a relationship between the future capital stock, the current capital stock, and consumption. The
economic logic here is simple: more consumption implies less investment and therefore a lower capital stock
in the following period.
We can represent equations (9.3.14) and (9.3.15) by means of a phase diagram. This is a graph that shows
us, for each possible level of c and k , in what direction c and k are supposed to be moving if they are to satisfy
(9.3.14) and (9.3.15). This is shown in Figure 9.3.1.
Fig. 9.3.1: Dynamics of the
Neoclassical Growth Model
The vertical line represents equation:
ct+1
=1
ct
1
⇒ [β (1 + FK (Kt+1 , 1) − δ)] σ = 1 (9.3.16)
1
⇒ FK (Kt+1 , 1) − δ = − 1 (9.3.17)
β
so it tells us what level of capital is consistent with constant consumption. At any point to the left, the lower
capital stock means a higher interest rate so consumption must be growing; at any point to the right, the
higher capital stock means a lower interest rate so consumption must be falling. Only if K solves equation
1
(9.3.17) is the interest rate exactly
β − 1, which persuades the household to keep consumption constant over
175
9.3. In nite-Period Economy
time.
The curved line represents the equation:
Kt+1 = Kt
⇒ (1 − δ) Kt + F (Kt , 1) − ct = Kt
⇒ ct = F (Kt , 1) − δKt
so it tells us, for each level of K, how much that household needs to consume in order to invest enough to
exactly make up for depreciation, thus keeping the capital stock constant. For all the points above the curve,
higher consumption implies that depreciation exceeds investment so the capital stock shrinks; for all the points
below the curve, lower consumption implies that investment exceeds depreciation and the capital stock grows.
Mathematically, (9.3.14) and (9.3.15) are two dierence equations in terms of Kt and ct . If we knew
the initial conditions K0 and c0 , we could trace out the entire path of both variables over time. The initial
condition for K0 is easy. We assumed it's exogenous so we just take as given its initial value. How about c0 ?
How much will the household consume in the initial period? Figure 9.3.2 shows, for two possible levels of the
initial capital stock (K0 and K00 ), the paths of Kt and ct that result from dierent possible values of c0 .
Fig. 9.3.2: Paths for the
economy dictated by equations
(9.3.14) and (9.3.15), starting
from dierent levels of c0 and
K0 .
In each case, the higher path has c0 too high. Starting from this level of c0 , the Euler equation (9.3.14)
dictates path of consumption that becomes ever-increasing, but there is not enough output so the economy
starts to deplete the capital stock, eventually depleting it completely. In the lower paths, c0 is too low. The
economy accumulates more and more capital over time and after some time the Euler equation starts to dictate
falling consumption. Eventually, all the output is being invested and consumption falls to zero. The only levels
of c0 that are consistent with optimality are the ones that gives rise to the middle paths. Here both ct and Kt
176
9.3. In nite-Period Economy
converge to css , Kss . Hence this economy, just like the Solow economy with an exogenous savings rate, has a
steady state.
In the theory of dierence equations, the paths that lead to the steady state are known as saddle paths
and the steady state is known as a saddle point. Starting from any K0 , there is a unique level of c0 such that
the dynamics implied by equations (9.3.14) and (9.3.15) lead towards the steady state. Furthermore, once ct
is on the saddle path, it will stay on the saddle path, so the economy will always be on this path.
The Golden Rule
When we looked at the Solow model we de ned a concept called the Golden Rule. This described the level
of capital Kgr (and the savings rate needed to attain it) such that steady state consumption is maximized.
We saw that, depending on the savings rate, an economy could end up with either more or less capital than
prescribed by the Golden Rule. Now that we have a theory of the savings rate we can ask how the model
predicts that Kss and Kgr will compare. Will the economy accumulate more or less capital than prescribed
by the Golden Rule? Recall from Chapter 4 that equation (4.3.4) says Kgr satis es:
4
FK (Kgr , 1) = δ
Instead, with endogenous savings, the steady state level of capital satis es (9.3.17). Rearranging, we have
1
FK (Kss , 1) = −1+δ
β
>δ (as long as the household is impatient so β < 1)
= FK (Kgr , 1)
⇒ Kss < Kgr
Therefore this economy will not attain the Golden Rule level of capital. It will remain below this.
What do we make of this? The Golden Rule seemed like a pretty desirable outcome, and the First
Welfare Theorem says that the equilibrium maximizes the household's utility: why doesn't the social planner
implement the Golden Rule? The answer comes from the household's impatience. The Golden Rule maximizes
consumption in the long run. An impatient household cares about the short run as well as the long run. It
would rather consume a little bit more in the present even if it means a lower level of consumption later. Note
that mathematically, Kss → Kgr if β → 1, so as households become very patient the economy indeed comes
closer to the Golden Rule. Note also that the argument does not depend on starting with a low level of capital.
If the economy were to start at Kgr , the household would choose to invest less than required to maintain the
capital stock, consuming more than cgr in the short run at the expense of lower consumption later.
Figure (9.3.3) shows how the steady state compares to the Golden Rule. The graph shows, for each
value of K, the level of consumption that is consistent with maintaining a constant capital stock equal to K,
i.e. c = F (K, 1) − δK . If the economy were to maintain Kgr , then it could sustain a level of consumption
cgr > css . However, attaining and maintaining such a high capital stock requires sacri cing too much present
4
In Chapter 4 we had population growth. Here we are setting n=0 but the argument works regardless.
177
9.3. In nite-Period Economy
consumption for future consumption and the household is better o with the equilibrium that converges to
Kss .
Fig. 9.3.3: How the steady
state compares to the Golden
Rule.
Anticipation Eects
Thinking in terms of general equilibrium can be useful for thinking about how anticipation of things that will
happen in the future can aect decisions in the present. Let's consider an example. Suppose the economy is
in steady state and suddenly everyone anticipates that a technological breakthrough will lead to a change in
the production function from F (K, 1) to AF (K, 1) starting in year T, where A > 1. Figure 9.3.4 shows what
the eect of this would be.
Once it happens, the technological improvement shifts the ct = ct+1 line to the right: higher A means that
1
it takes higher K to have AFK (K, 1) = β −1+δ (which is the condition for consumption stay constant).
In addition, the Kt = Kt+1 curve shifts up: higher productivity means it is possible to aord more and
still maintain the capital stock. Once period T arrives, we can apply the analysis we did for the constant-
technology case. From T onward, the economy must be on the new saddle path that leads to the new steady
state. But before period T, the dynamics of capital and consumption are still governed by the old technology.
As illustrated in the gure, the initial level of consumption must be such that, by the time period T arrives,
the dynamics under the old technology lead to the saddle path of the new technology.
In economic terms, what happens is that anticipation of a technological improvement leads to higher
consumption through a wealth eect. Since technology has not improved yet, higher consumption implies that
the economy is investing less than is necessary to maintain the old capital stock, so the capital stock begins to
shrink. A lower capital stock means that the marginal product of capital is higher and therefore the interest
1
rate is higher than
β − 1. This means that the household chooses a rising path for consumption. Hence, in
178
9.3. In nite-Period Economy
Fig. 9.3.4: An anticipated
technological improvement.
anticipation of the technological improvement, consumption rst jumps from css to c0 and then gradually rises
over time. When period T arrives, the capital stock has shrunk to KT and consumption has reached cT , so
0 0
the economy is exactly in the saddle path that will lead it to the new steady state Kss , css .
Exercises
9.1 The First Welfare Theorem
Consider the following policies. In each case, explain whether, in your view, the policy is justi ed and
why.
(a) Setting a maximum price that electrical utilities may charge.
(b) Mandatory vaccinations.
(c) Workplace safety standards.
(d) Rent control.
(e) A 35-hours-per-week limit on working hours.
(f ) Banning grocery stores from opening on Sundays.
(g) Banning self-service at gas stations.
9.2 Storage
Suppose the production function is the following:
F (K, L) = K
179
9.3. In nite-Period Economy
and the rate of depreciation is δ = 1.
(a) Why is this production function called storage?
(b) What will be the real interest rate and the wage in this economy?
(c) With standard preferences, will consumption increase over time, remain constant over time or decrease
over time?
9.3 Baumol's Cost Disease
The representative household consumes two dierent goods: x (clothes) and y (performances by string
quartets). Its preferences are given by:
5
1 −1
−1
1 −1
u (x, y) = α x + (1 − α) y
Each of the two goods is produced using only labor. If Lx units of labor work in producing clothes, output
of clothes is x = Ax L x ; if Ly units of labor play in string quartets, output is y = Ay Ly . The household
supplies one unit of labor inelastically and is indierent as to how much it supplies to each industry. The
price of clothes is denoted px and the price of a quartet performance is py . The wage rate is w.
(a) Use a software like Matlab or Excel to plot two sets of indierence curves for this utility function.
Set α = 0.5 in both cases, = 0.5 in one of them and =2 in the other.
(b) Set up the problem of a household that obtains income w from supplying one unit of labor inelastically
and has to decide how much to consume each of the two goods. Show that the household will consume:
−
w px
x= α
p p
−
w py
y = (1 − α)
p p
1
1−
where p ≡ αp1−
x + (1 − α) p1−
y . [This takes a bit of work. If you want, you can look up the
derivation on the book website].
(c) Set of the problem of a rm in each of the two industries that needs to decide how much labor to hire
px py
to maximize pro ts. What values of
w and w are consistent with each industry hiring a positive
but not in nite amount of labor?
py
(d) What will be the price of string quartet performances relative to clothes
px ?
(e) Suppose that over time Ax rises and Ay stays the same. What does this mean? Do you nd this
plausible? What will be the eect on the relative price of string quartets? Explain.
(f ) Use your answers to parts (b) and (d) to compute the relative quantities of each of the two goods
y
that will be consumed, i.e. solve for the equilibrium level of
x.
5
This is known as a constant elasticity of substitution utility function. The parameter represents the elasticity of substitution
between goods x and y: a higher number means that that are close to perfect substitutes. For reference, when → 1, these
preferences converge to the Cobb-Douglas case u(x, y) = xα y 1−α .
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9.3. In nite-Period Economy
(g) Use your answer to part (f) to compute the relative allocation of labor across the two industries, i.e.
Ly
solve for the equilibrium level of
Lx .
(h) Suppose that over time Ax rises and Ay stays the same. What happens to the quantity of string
quartet performances over time? How does the answer depend on ? Explain.
9.4 Pro-Worker Policies
Consider the following one-period economy. The production function is F (K, L). The representative
household owns all the capital stock K, which is exogenously given. The household's preferences are
u(c, l) where, as usual, l = 1 − L. The labor and capital markets are perfectly competitive.
∗
Denote by L the amount of labor that households supply in a competitive equilibrium with no government
intervention. Now suppose that the government dictates a new law that prohibits the household from
working more than L∗ − ε, where ε is a small positive number.
(a) Show that the policy will increase wages and lower the rental rate of capital.
(b) Show that the policy will make the representative household worse o.
(c) Now suppose that there are two representative households in the economy. Household A owns the
capital stock and does not work. Its preferences are given by u(c). Household B has preferences
u(c, l) and does not own any capital. Suppose the government enacts the same policy as before (i.e.
it prohibits household B from working more than L∗ − ε, where L∗ is the amount it works under a
no-intervention equilibrium). Show that this policy makes household A worse o and household B
better o.
9.5 Dierences in Preferences
There are two countries: Industria and Lethargia. Both countries have a population of 1 and identical
production functions:
Y = AL
where L is the total amount of time spent in market labor.
Preferences are given by:
u (c, l) = log (c) + θ log (l) (9.3.18)
where c is consumption and l is leisure. Within each country, everyone is identical but θ is dierent for
I L L I
residents of the two countries. Denote the two values by θ and θ and assume θ >θ .
Both countries run free market economies, with perfectly competitive labor markets.
(a) What is the wage in each country?
(b) Set up the problem of a representative household that has to decide how to divide its time (normalized
to a total of 1) between market work and leisure.
(c) What fraction of its time will the representative household spend on market work in each country?
(d) Which country has higher GDP?
181
9.3. In nite-Period Economy
(e) Suppose a researcher does the following:
i. assumes preferences are the same across the world, with functional form (9.3.18),
ii. estimates θ using data from Lethargia,
iii. gathers data on consumption and leisure in each country,
iv. computes λ: the relative welfare of residents of Industria, taking Lethargia as a benchmark with
λ=1 the way Jones and Klenow took the US as a benchmark.
Find an expression for λ. Which country would the researcher conclude has a higher standard of
living? Explain.
9.6 Capital Income Taxes
An economy has the production function
Y = K α L1−α
It is populated by two types of households: workers and capitalists.
Workers supply a total of L units of labor inelastically and consume all their income in every period, so
they don't really make any decisions. Their consumption in any given period is given by:
ct = wt L + Tt
where wt is the wage and Tt is a transfer they get from the government.
Capitalists have preferences given by:
∞
X
β t u (ct )
t=0
They do not work: they get their income from capital, and it is taxed at a rate τ. However, they can
choose how much to consume and how much to save in the standard way. Their budget constraint is:
Kt+1 = −ct + 1 + (1 − τ ) rtK − δ Kt
where Kt is the capital stock in period t, δ is the rate of depreciation, τ is the tax rate and rtK is the rental
rate of capital (which each capitalist household takes as given but depends on the total capital stock in
the usual way).
What does this budget constraint mean? For each unit of capital the capitalist has, it obtains a rental.
The government taxes this rental (net of depreciation) at a rate τ. Assume that the government taxes
interest income in the same way as it taxes income from renting capital, so that equation (9.1.11) holds.
(a) Set up the maximization problem of a capitalist household.
(b) From the rst order conditions, nd an after-tax version of the Euler equation. (You can skip steps
if you want)
(c) If in the long run this economy reaches a steady state in which consumption of the capitalists is
constant, what are the pre-tax and after-tax interest rates in this steady state?
182
9.3. In nite-Period Economy
(d) If the economy is in a steady state, what is the rental rate of capital? How does it depend on τ?
(e) If the economy is in a steady state, what is the capital stock?
(f ) How much revenue does the government collect from the capital-income tax in steady state?
(g) What is the level of wages in steady state?
(h) Suppose that the government uses all the revenue from the capital-income tax to nance transfers to
the workers. What level of consumption do workers attain in steady state? How does it depend on
τ? Explain.
9.7 Getting Old
An economy has two types of households: young and old. Let µ be the fraction of households that are
young. We hold the total population constant and consider the eects of higher or lower µ (which could
be the result of dierences in fertility and mortality). Each young household supplies x units of labor
inelastically, so the total amount of young labor is LY = µx; old households supply z units of labor, also
O
inelastically, so the total amount of old labor is L = (1 − µ) z . Assume x > z.
The production function is:
Y = K α L1−α
where K is the capital stock and L is the total amount of labor: L ≡ LY + LO .
(a) Holding K constant, how do GDP, wages and and the rental rate of capital depend on µ?
(b) How will the steady-state capital stock capital compare between two economies with dierent levels
of µ? Explain.
(c) Now imagine instead that the economy doesn't use capital and the production function is:
γ 1−γ
Y = LY LO
How do the wage levels of young and old workers depend on µ?
9.8 A New Technology
Consider a one-period economy where the representative worker has preferences:
log (c) + θ log (l)
where c is consumption and l is leisure. The worker must choose how to divide its time between leisure l
and market work L, so:
l+L=1
The labor market is perfectly competitive. The equilibrium wage is denoted w.
(a) Set up the problem of the representative worker, who must decide how to allocate his time. Assume
the worker has no source of income other that its earnings from labor. Solve for the household's
choice of L.
183
9.3. In nite-Period Economy
The production technology uses labor and capital to produce goods according to:
Y = K α L1−α
where K is the capital stock, which is exogenously given. The capital-rental market is also perfectly
competitive. The equilibrium rental rate of capital is denoted rK .
(b) Set up the problem of the representative rm, which must choose capital and labor inputs to maximize
pro ts. Derive rst order conditions and nd the level of w and rK such that markets clear.
(c) How do w and rK depend on K and θ? Explain.
(d) Suppose a team of scientists develops a new technology for producing goods. Using this new tech-
nology, it's possible to produce goods according to:
Y = AK
where A is a parameter. The scientists tried out the technology at a very small scale, so anything
they did was too small to aect equilibrium prices. Prove that the new technology turned out to be
pro table if and only if K > K̄ , where K̄ is some number. Find an expression for K̄ . How does K̄
depend on θ? Explain.
9.9 An Oil-Producing Economy
Suppose that the production function is given by F (K, L) = K α L1−α + e. GDP is the sum of regu-
lar output, which is produced using capital and labor, and oil extraction e, which is exogenous. The
representative household has preferences:
∞
X
β t log (ct )
t=0
and supplies L=1 units of labor inelastically. Capital depreciates at rate δ.
(a) On the same graph, draw two phase diagrams, one for e=0 and one for e = ē > 0 and label the
steady state in each case.
(b) Suppose that the economy starts at t=0 at the ē steady state but everyone suddenly realizes that
oil will run out at time T, so that:
(
ē if t<T
et =
0 if t≥T
Using the phase diagrams, plot the evolution of Kt and ct from t=0 until t=T and from t=T
onwards. Describe in words why consumption and investment evolve the way they do, and what
happens to wages and the interest rate.
9.10 Wizards and Cows
The Calevingians have two basic economic activities: gathering fruit and raising cows.
184
9.3. In nite-Period Economy
Wild fruit grows in the Western part of the kingdom. The Calevingians have not mastered the art of
growing fruit trees, so they have no control over how much fruit is available. They just visit the fruit trees
every week and pick whatever fruit is ripe. Luckily, the kingdom includes fruits with dierent seasonal
patters so there is ripe fruit more or less evenly throughout the year. Not all years are the same, however.
Some years are warm and fruit is plentiful, while others are colder and result in less fruit. The Calevingians
have complete trust in their chief wizard, who gives them a weather forecast (and thus a fruit forecast)
with a horizon of a few years.
Cows graze in pastures in the Eastern part of the kingdom. The grass on which cows feed grows just as
well in cold and warm weather. Each family owns a plot of land and keeps their own cows there. Cows
are raised primarily for meat; the Calevingians have not learned to milk them. One of the main decisions
that Calevingians need to make is how many of their cows to slaughter for meat each year and how many
(including newborn calves) to keep fattening from one year to the next. They understand that the more
cows they keep in their plot of land, the less grass each of them will have available for grazing.
(a) Write down mathematically the economic decision problem faced by a Calevingian household. Explain
the meaning of each equation you write down.
(b) Derive rst-order conditions (you may skip steps if you want).
(c) Suppose the chief wizard announces that the next few years will be cold and therefore yield less fruit.
How do households react to this announcement?
(d) Despite their rather simple economy, the Calevingians have a fairly sophisticated legal system; the
concept of a sale, a loan and an interest rate are well-established and contracts are enforced very
eectively. It's possible, for instance, to borrow in order to buy more cows or, conversely, to sell one's
cows and lend the proceeds of the sale to someone else. What happens to interest rates when the
wizard announces the upcoming cold weather?
9.11 Patience and Investment
Suppose the production function is F (K, L) = K α L1−α , and the labor supply is exogenous and equal to
L = 1. The representative household's preferences are given by:
∞
X c1−σ
t
βt
t=0
1−σ
and the depreciation rate is δ.
(a) Use equation (9.3.17) to nd an expression for the level of capital that this economy will have in the
steady state.
(b) Find an expression for the level of GDP that the economy will have in the steady state.
(c) Find an expression for the steady-state investment-to-GDP ratio. How does it depend on β? Explain.
9.12 Optimal vs Fixed Savings Rates
185
9.3. In nite-Period Economy
Suppose the economy is as in Exercise 9.11, with α = 0.4, β = 0.95, δ = 0.08 and σ = 2. The economy
starts with an initial capital stock K0 = 2. We are going to compare how this economy behaves under a
xed investment rate and under an optimally-chosen investment rate.
(a) First suppose that, like we assumed in Chapter 4, the investment rate is exogenous and constant,
and equal to the steady-state investment-to-GDP ratio that you found in Exercise 9.11. Compute
GDP, consumption, investment and the capital stock for the rst 150 years of this economy.
(b) Now suppose that consumption and investment are chosen in accordance with (9.3.13). Find the
level of initial consumption that is consistent with the system of equations (9.3.15) - (9.3.13) and the
economy converging to a steady state. (You can nd a Matlab code that will compute this at the
book website). Compute GDP, consumption, investment and the capital stock for the rst 150 years
of this economy.
(c) Plot the consumption paths of the two economies over the rst 150 years on the same graph.
(d) Replace the computed paths of consumption from parts (a) and (b) into the utility function to
compute how much utility the representative household will obtain in the rst 150 years of each
economy.
(e) By what fraction would one have to reduce consumption in the optimal-savings economy for the
household to get the same utility as in the exogenous-savings economy?
9.13 The Enclosure Acts
The Enclosure Acts were a series of laws that converted common property into private property in Great
Britain starting around the XVII century. This exercise asks you to think about their macroeconomic
eects.
Consider the following one-period economy.
• There are two sectors:
Agriculture produces output according to:
YA = N α L1−α
A
where YA is agricultural output, N is natural resources (land) and LA is labor dedicated to
agriculture.
Industry produces output according to:
YI = K α L1−α
I
where YI is industrial output, K is capital and LI is labor dedicated to industry.
For simplicity, assume that both sectors produce the same good (or that both goods are perfect
substitutes), so GDP is Y = YA + YI .
186
9.3. In nite-Period Economy
• The industrial sector operates as a competitive market, with a wage rate w and a rental rate of
K
capital r . The capital stock is exogenously given and is owned by capitalists, who do not work.
• Agriculture is organized as follows:
no one owns the land and anyone can use it,
once all the agricultural output is produced, it gets shared equally among agricultural workers.
• The amount of land available and the capital stock are exogenously given.
• Each worker supplies one unit of labor. The total number of workers is L.
• Each worker freely decides whether to work in industry or in agriculture:
if the worker chooses industry, he gets a wage w,
if the worker chooses agriculture, he gets a share of total agricultural. output
LA
(a) Find an expression for the ratio
LI that would lead to maximum GDP.
(b) If there are LA workers in agriculture, what is the income of a worker who chooses agriculture?
(c) If there are LI workers in industry, what is the income of a worker who chooses industry?
LA
(d) What is the ratio
LI that would make workers indierent between choosing agriculture and industry?
Argue that this is the allocation of labor that the economy will have. How does it compare to the
allocation that would maximize GDP? Explain.
(e) Suppose there is an Enclosure Act that does the following:
• It splits up the land equally among agricultural workers, giving each of them ownership of a
speci c piece of land.
• It establishes a competitive market for renting land and hiring agricultural workers.
After the reform:
i. Which of the sectors will increase the number of workers it employs? Explain.
ii. What will happen to GDP? Explain.
iii. What will happen to industrial wages? Explain.
iv. What will happen to the rental rate on industrial capital? Explain.
(f ) How would the answers to part (e) change if instead of being split up equally the land was given to
only some of the agricultural workers? Why?
187
PART IV
Money and In ation
This part of the book looks at the role of money in the economy.
In Chapter 10 we study what money is, how the quantity of money is deter-
mined and why people choose to hold it.
In Chapter 11 we study how the equilibrium in the money market is determined, what
determines the price level and how in ation comes about.
189
CHAPTER 10
Money
10.1 What is Money?
This question is trickier than is seems.
1 The standard answer is that money can be anything that can serves
as:
1. A store of value. One can save it (for instance, by keeping it in one's pocket) and use it later.
2. A unit of account. We can express the prices of things in terms of how much money it takes to buy
them.
3. A medium of exchange. Money changes hands when people pay for things.
Many dierent things have been used as money at various times and places: pieces of paper with the faces of
historical gures, gold and silver, cigarettes.
Why do we use money at all? The main reason is that it solves what's known as the double coincidence of
wants problem. Using money, I don't need to nd someone who has exactly what I need and wants exactly
what I have in order to trade. I can accept money in payment for the goods I sell knowing that others will
accept money in payment for the goods I want.
For something to be convenient to use as money, it typically needs to have several properties:
1. It has to be hard to counterfeit. If you want to pay for something with money, you don't want the seller
to be wondering whether you are giving them real money or fake money.
2. It has to be easy to carry, since transactions happen in many dierent places.
3. It has to be durable, otherwise it's not a very good store of value. It makes more sense to use coee
beans than strawberries as money because strawberries are likely to spoil before they can be used in the
next transaction.
4. It has to be easily divisible. For transactions to go smoothly it's important to be able to pay the exact
price without too much rounding and to be able to make change. The wonderful book by Sargent and
1
Asmundson and Oner (2012) have a good introduction to this question.
191
10.2. The Supply of Money
Velde (2014): The big problem of small change tells the history of how Europe dealt with the problem
of making change.
5. It has to be commonly accepted. Money is only useful is everyone agrees that it is indeed money and
accepts it as payment. Sometimes this acceptance is purely a social convention, sometimes it is reinforced
by laws.
The properties listed above are satis ed to dierent degrees by dierent assets. As a result, there is no unique
measure of what is the type of money that is used in any economy or how much of it there is. By convention,
several de nitions of money are typically studied, each of which draws a somewhat arbitrary line between
money and not money. Here are the main ones:
Monetary base M0 M1 M2
Physical Currency Physical Currency Physical Currency Physical Currency
Central Bank Reserves Demand deposits Demand deposits
Savings deposits
Small time deposits
Money market mutual funds
Let's start from M0. This measure counts as money only physical bills and coins. It quite clear that
physical currency meets the conditions for something to be money pretty well. Not perfectly, though: it's
possible to counterfeit and there are some places where it's not accepted as a means of payment. M1 is a
broader measure of money because it also includes demand deposits (basically checking accounts). For most
purposes, a checking account satis es the de nition of money: for most transactions, either a check or a
debit card is acceptable.
2 M2 is a still broader measure because it includes other types of deposits (savings
accounts and smaller time deposits held by individuals) as well as shares in money market mutual funds held
by individuals.
3 The assets in this broader measure of money are slightly less easy to use in transactions.
That's why these assets have an intermediate degree of moneyness and are included in the broader measures
of money but not the narrower ones.
The monetary base has this name because it's the only part of money that is under the direct control of
the government. Therefore it forms the base on which every other measure is built. To see this, it is useful
to go into the mechanics of how the quantity of money is determined.
10.2 The Supply of Money
The monetary aggregates de ned above are in part chosen by the government, which decides how much physical
currency to issue, and in part determined by what happens in the banking sector, since deposits are bank
liabilities. How does this all t together?
2
But not for every transaction. Some stands in the Palo Alto farmers' market will only take physical cash. So (I've been told)
will most drug dealers.
3
A mutual fund is an investment vehicle by which investors each own a proportion of a pool of assets. A mutual fund is called
a money market fund if it invests in very safe assets so that the total value of the pool of assets doesn't move much, making it
very similar to money as a store of value.
192
10.2. The Supply of Money
To address this, a good place to start is by looking at what a bank balance sheet looks like. Here is a
typical bank balance sheet:
Balance Sheet
Assets Liabilities
Reserves
Deposits
Bonds
Loans Other Liabilities
Other Assets Net Worth
On the left are all the bank's assets: loans, government bonds, etc. One of the main sources of income for
banks is the interest it earns on these assets (the other is fees of various kinds). One asset in particular will
be relevant to us: Central Bank reserves. The Central Bank acts as a bank for banks and reserves is just the
name given to the deposits that banks hold at the Central Bank (i.e. this is an asset for banks and a liability
for the Central Bank). Typically, these reserves earn either no interest or a very low rate.
4 Why do banks
hold them?
There are two reasons, whose relative importance has been dierent at dierent times in history. One
reason is that reserves are a way to meet unexpected withdrawals of deposits. Most bank assets are relatively
long term and hard to sell so if depositors want their money right away it's useful for the bank to have an
asset that can be converted into cash very quickly, and reserves provide this: in most countries, the Central
Bank stands ready to exchange reserves for cash whenever banks want it. Nowadays this is usually not the
main reason banks keep reserves since deposit insurance has made bank runs quite rare and there are explicit
arrangements for banks to get emergency loans to meet deposit withdrawals. Instead, the main reason banks
keep reserves is that they are required to do so by regulation. Typically, banks are required to hold a certain
minimum level of reserves, set as a percentage of the bank's deposits. The exact percentage usually depends
on the type of deposit, though the details vary a lot from one country to another.
On the right hand side are the bank's liabilities: mostly deposits but sometimes also non-deposit borrowing
such as long term bonds that the bank has issued. The bank's net worth is the dierence between its assets
and liabilities.
Let's compute the supply of money in a simpli ed example. There are three relevant entities:
• the Central Bank,
• a single commercial bank, which is meant to represent the sum of all banks in the economy,
• a single representative household.
These are their balance sheets:
4
This is changing. In recent years many central banks are paying interest on reserves. We look into this further below.
193
10.3. Changing the Supply of Money
Central Bank Private Bank Household
Assets Liabilities Assets Liabilities Assets Liabilities
Bonds: b Reserves: ρd Reserves: ρd Deposits: d Currency: c 0
Currency: c Bonds: B Deposits: d
Net Worth Loans: L Net Worth Net Worth
b − ρd − c ρd + B + L − d c+d
The household owns currency and deposits and, in this example, has no liabilities or any other assets. The
bank's assets are made up of loans, government bonds and Central Bank reserves. The reserve requirement in
this example is set at a fraction ρ of deposits and the bank is satisfying it exactly, so it has ρd in reserves. The
Central Bank owns a certain amount of government bonds b and its liabilities are the private bank's reserves
and the outstanding currency.
5
In this economy we have that, following the de nitions above:
• the monetary base is c + ρd,
• M0 is c,
• M1 is c+d (assuming the deposit is a demand deposit).
10.3 Changing the Supply of Money
In this section we'll study the traditional way in which central banks would change the supply of money. The
way central banks operate has been changing over the last decade or so. Exercise 10.3 asks you to think about
how more modern operating procedures compare with this traditional approach.
Suppose that the Central Bank wants to change the supply of M1 money. We'll come to the reasons why
the Central Bank might want to do that later on, but for now let's just accept that the Central Bank wants
to do this. The Central Bank doesn't directly control the amount of deposits, which represent the biggest
component of M1, but it aects it indirectly through open market operations. Let's see how this works.
An open market operation is a trade by the Central Bank where the Central Bank either buys bonds and
pays for them with reserves or sells bonds and accepts reserves as payment. It's called open market operation
because the Central Bank is trading just like anyone else in the open market. Let's work through how an open
market operation takes place in the example above. Suppose that the Central Bank buys government bonds
worth ∆ from the private bank and pays for them by crediting ∆ reserves to the private bank. These are new
reserves: they come into existence because the Central Bank creates them. Balance sheets are now:
5
Thinking of currency as a liability of the Central Bank is a bit counterintuitive at rst: the Central Bank doesn't really
have an obligation to pay anything to holders of currency. This wasn't always the case: it used to be that currency represented
a promise by the Central Bank to deliver gold to the holder. Then currency was in every sense a liability, and the accounting
re ected that.
194
10.3. Changing the Supply of Money
Central Bank Private Bank Household
Assets Liabilities Assets Liabilities Assets Liabilities
Reserves: Reserves: Currency: c
Deposits: d 0
ρd + ∆ ρd + ∆
Bonds: b+∆ Currency: c Bonds: B−∆
Net Worth Loans: L Net Worth Deposits: d Net Worth
b − ρd − c ρd + B + L − d c+d
But this is not the end of the story. Now the private bank has reserves of ρd + ∆ but it's only required
to have ρd. It now has excess reserves. Since reserves don't earn any interest, the bank will try to lend
out these excess reserves.
6 Let's imagine that the bank makes a loan of ∆ to the household. What exactly
happens when the bank makes a loan?
• The bank gives the borrower a check for ∆ in exchange for a promise that the borrower will pay it back
with interest later.
• The borrower deposits the check. Maybe the borrower deposits it in the same bank; maybe the borrower
deposits it in a dierent bank; maybe the borrower hands over the check to someone else (for instance,
someone who sells him a car) and then that person deposits the check in their bank. Since we are adding
up over all banks and over all households, all these variants are equivalent.
The balance sheets are now:
Central Bank Private Bank Household
Assets Liabilities Assets Liabilities Assets Liabilities
Reserves: Reserves: Deposits: Currency: c
Loans: ∆
ρd + ∆ ρd + ∆ d+∆
Bonds: b+∆ Currency: c Bonds: B−∆ Deposits:
d+∆
Loans: L+∆
Net Worth Net Worth Net Worth
b − ρd − c ρd + B + L − d c+d
Notice that even though the bank lent out the excess reserves the reserves don't actually disappear.
They are still there. It's more accurate to say: the private banks take advantage of the relaxation of the
reserve-to-deposit ratio to expand loans and deposits.
Now suppose that the borrower (or anyone that the borrower made a payment to) wants to take out a
fraction of the ∆ new deposits in cash. Call that fraction χ, so that the borrower wants to have χ∆ cash and
(1 − χ) ∆ deposits. The borrower goes to the ATM and makes a withdrawal. What exactly happens when the
borrower does this?
• The bank asks the Central Bank for cash (note that in our example the bank held zero cash to begin
with).
6
This assumes that there are lending opportunities out there where the bank will in fact earn a positive interest rate. Later
we'll think about what happens if interest rates are zero, or when the Central Bank pays interest on reserves.
195
10.3. Changing the Supply of Money
• The Central Bank prints physical currency and gives it to the bank. In return, it reduces the amount of
reserves owed to the bank.
• The bank hands over the cash to the borrower. In return, it reduces the balance on the borrower's
deposit.
The balance sheets are now:
Central Bank Private Bank Household
Assets Liabilities Assets Liabilities Assets Liabilities
Reserves: Reserves: Deposits: Currency: Loans: ∆
ρd + (1 − χ) ∆ ρd + (1 − χ) ∆ c + χ∆
Bonds: b+∆ Currency: Bonds: B−∆ d + (1 − χ) ∆ Deposits:
c + χ∆
Loans: L+∆
Net Worth Net Worth d + (1 − χ) ∆ Net Worth
b − ρd − c ρd + B + L − d c+d
Notice that nobody's net worth changes in this whole series of transactions. The participants are just
exchanging dierent types of assets and liabilities with each other.
The process is not over. The bank has ρd + (1 − χ) ∆ reserves but the reserve requirement is only
ρ (d + (1 − χ) ∆) so there are excess reserves of (1 − ρ) (1 − χ) ∆. Therefore the process repeats itself, just
2 3
scaled down by (1 − ρ) (1 − χ), and then by ((1 − ρ) (1 − χ)) , and then by ((1 − ρ) (1 − χ)) , etc. This de nes
a geometric series, so we can compute the total eect as follows:
∞
X n (1 − χ) ∆
Change in Deposits = ((1 − ρ) (1 − χ)) (1 − χ) ∆ =
n=0
| {z } ρ + χ − ρχ
First round eect
∞
X n χ∆
Change in Currency Holdings = ((1 − ρ) (1 − χ)) χ∆ =
|{z} ρ + χ − ρχ
n=0
First round eect
ρ(1 − χ)
Change in Reserves = ∆ − Change in Currency Holdings = ∆
ρ + χ − ρχ
Change in the Monetary Base = Change in Reserves + Change in Currency Holdings =∆
∆
Change in M1 = Change in Deposits + Change in Currency Holdings =
ρ + χ − χρ
At the end, we have balance sheets that look like this:
196
10.3. Changing the Supply of Money
Central Bank Private Bank Household
Assets Liabilities Assets Liabilities Assets Liabilities
Reserves: Reserves: Deposits: Currency: Loans:
ρ(1−χ) ρ(1−χ) 1−χ χ 1
ρd + ρ+χ−ρχ ∆ ρd + ρ+χ−ρχ ∆ d+ ρ+χ−χρ ∆ c+ ρ+χ−χρ ∆ ρ+χ−χρ ∆
Bonds: b+∆ Currency: Bonds: B−∆ Deposits:
χ 1−χ
c+ ρ+χ−χρ ∆ d+ ρ+χ−χρ ∆
Loans:
1
L+ ρ+χ−χρ ∆
Net Worth Net Worth Net Worth
b − ρd − c ρd + B + L − d c+d
so now the private banks are meeting the reserve requirement exactly:
ρ(1−χ)
Reserves ρd + ρ+χ−ρχ ∆
= 1−χ =ρ
Deposits d+ ρ+χ−χρ ∆
The quantity:
Change in M1 1
ω≡ = (10.3.1)
Change in the Monetary Base ρ + χ − χρ
is known as the M1 money multiplier. It's called a multiplier because whenever the Central Bank changes the
monetary base (which it controls directly), the magnitude of the change in M1 is the change in the monetary
base times the multiplier.
The money multiplier depends on two things:
• ρ: the ratio of reserves to deposits ρ. For the most part, the Central Bank can change this number by
changing the legal reserve requirement.
7
• χ : the fraction of their M1 money that people want to hold in physical cash. This is not under the
control of the Central Bank: it can vary over time depending of the evolution of payment systems or the
public's con dence in the banking system.
In most of what we'll do later, we'll just say that the Central Bank controls the money supply. What we
mean by this is that the Central Bank directly controls the monetary base. By understanding how the money
multiplier works, it can control M1 fairly accurately, albeit indirectly.
What if the Interest Rate is Zero?
The analysis above was built on the assumption that banks try to maintain reserves as low as possible. This
makes sense as long as reserves pay zero interest and other assets pay positive interest. However, if the interest
rate fell all the way down to zero (or if the Central Bank started paying market interest rates on reserves), this
logic would break down. Banks would be perfectly willing to hold reserves above the legal requirement since
the alternatives are not better. This would mean that changes in the monetary base need not lead to changes
7
As mentioned before, this assumes that banks are trying to keep the minimum possible level of reserves so that the legal
requirement is binding.
197
10.3. Changing the Supply of Money
in the M1 money supply, since all the new reserves would just sit in bank balance sheets without triggering
an expansion in loans and deposits.
This scenario has been realized in recent years, as shown in Figure 10.3.1. Nominal interest rates fell to
almost zero in late 2008. At around the same time, the Federal Reserve decided to start paying interest on
excess reserves (i.e. reserves above the legal reserve requirement). Reserves were suddenly a more attractive
asset for banks to hold, and banks started holding large amounts excess reserves. As a result, the M1 money
multiplier fell from about 2 to less than 1. The monetary base was increased almost vefold but M1 increased
much less. We'll think more about what happens when the interest rate is near zero in Chapter 15.
%
Fig. 10.3.1: Monetary Aggregates in the US when nominal interest rates reached zero. Source: Board of
Governors of the Federal Reserve System.
198
10.4. The Demand for Money
10.4 The Demand for Money
Let's focus on M1 as our de nition of money. Why do people hold money? Why is it that instead of only
holding assets that earn interest (like physical capital, government bonds, etc.) people choose to hold physical
currency and checking deposits, which earn no interest?
8 We'll propose a simple model based on the idea that
money is necessary to carry out transactions. This model is known as the Baumol-Tobin model since it was
rst analyzed by Baumol (1952) and Tobin (1956).
Let's imagine that there are two types of assets:
• M1 money, which does not pay any interest,
• assets that do pay interest, all of which pay the same nominal interest rate i.
In the course of a period (for instance, the period can be a year), a household will spend Y in real terms (Y
stands for real GDP). The price of a good is p, so in nominal terms the household will spend pY . This spending
is not all at once: it's spread evenly over the period. For instance, if the period is a year, the household spends
pY
365 each day.
Whenever the household wants to pay for something, it must use money. However, this does not mean that
the household needs to have pY dollars all at once. Whenever it wants, the household can go to the bank '
and swap some of its interest-bearing assets for money. One way of going to the bank is to go to the ATM
and get physical currency (which is money) from one's savings account (which is not money under the M1
de nition). But going to the bank need not mean literally going to a physical bank branch. Another way
of going to the bank is to go to their online brokerage account and sell some bonds (which are not money),
depositing the proceeds in a checking account (which is money under the M1 de nition). We are going to
assume that there is a xed cost F (in real terms) of going to the bank. F can literally represent ATM fees
but also the time and mental cost of dealing with the issue.
The problem of the household is to decide how many times per period it goes to the bank. The advantage
of going to the bank many times is that it allows the household to have very low levels of money, so that most
of the household's wealth is earning interest most of the time. The disadvantage is that it requires paying the
xed cost F many times. Let N denote the number of times per period that the household goes to the bank.
Figure 10.4.1 shows how the amount of money held by the household evolves over time for two values of N.
pY
Each time the household goes to the bank, it brings up its money balance to
N . Then the balance starts
to decrease gradually as the household spends the money. Eventually, when the balance reaches zero, the
household goes to the bank again to get more money. It's clear from the picture that the household will, on
average, hold less money the more often it goes to the bank. Indeed, the average money balance is simply:
pY
M= (10.4.1)
2N
How does the household choose N? Mathematically, it solves the following problem:
pY
min pF N + i (10.4.2)
N 2N
8
Some checking deposits do earn interest, but it's typically much lower than what one could earn by holding some other asset.
199
10.4. The Demand for Money
Fig. 10.4.1: Money balances
over time in the Baumol-Tobin
model.
What does this mean? The household is trying to minimize the overall cost of having money for transactions.
This cost has two parts. First, if it goes to the bank N times, it pays the cost F each time. Expressed in
pY
nominal terms, this gives us pF N . Second, if it goes to the bank N times it will on average hold
2N dollars
in money. Since this money does not earn interest, there is an opportunity cost of holding it: the foregone
pY
interest that the household could have earned if it had held less money. If the interest rate is i, then i 2N is
the foregone-interest cost of the household's money holdings.
The rst-order condition for problem (10.4.2) is:
ipY −2
pF − N =0
2
so we can solve for N to get:
r
iY
N= (10.4.3)
2F
Equation (10.4.3) tells us that the household will go more times to the bank if i is high and if F is low. What's
the economic logic of this? If i is high, then the opportunity cost of holding money is high and the household
will be willing to go to the bank many times in an eort to hold low amounts of money. On the other hand, if
F is low, going to the bank is cheap and the household will, other things being equal, be willing to go to the
bank more times.
Replacing (10.4.3) into (10.4.1) and rearranging, we get an expression for the average money balances:
r
YF
M =p
2i
200
10.4. The Demand for Money
or, dividing by the price level, for average real money balances:
r
M YF
= (10.4.4)
p 2i
Real money balances are the answer to the question: how many goods would the household be able to buy
with the amount of money it holds? Equation (10.4.4) is telling us that real money balances will be higher
when:
• Y is high. If the household wants to spend more, this will involve more payments and therefore the
household will choose to carry higher real money balances.
• i is low. i is the opportunity cost of holding money. If this is low, the household will choose to hold
higher money balances to save on trips to the bank.
• F is high. If going to the bank is costly, the household will choose to hold higher money balances to
save on trips to the bank.
Figure 10.4.2 shows the shape of the money-demand function that results from the Baumol-Tobin model.
The quantity of money that people want to hold decreases with the interest rate. A higher cost of going to
the bank shifts the entire demand schedule to the right.
Fig. 10.4.2: The money de-
mand function in the Baumol-
Tobin model.
The Baumol-Tobin model makes very speci c assumptions about how exactly households manage their
money: every trip to the bank costs the same, spending is spread out exactly over time and perfectly pre-
dictable, etc. We will sometimes want to think about the basic economic forces that the Baumol-Tobin model
captures while not expecting the exact formula (10.4.4) to hold. For this purpose, we will sometimes want to
201
10.4. The Demand for Money
q
YF
think of a generalized money-demand function mD (Y, i), increasing in Y and decreasing in i. mD (Y, i) = 2i
is just a special case of this more general formula.
There is some debate as to whether the money demand function is su ciently stable over time to be a
useful thing to look at. At times, the quantities of M0, M1 and M2 that people held have moved around quite
a bit without changes in interest rates or GDP that would account for them. However, Lucas and Nicolini
(2015) and Kurlat (2019) argue that if one constructs an appropriately-weighted composite measure that
takes into account how people substitute between dierent subcomponents of money like physical currency,
checking accounts, savings accounts, etc., the resulting money demand has a relatively stable relationship with
the interest rate, as predicted by the theory. This is illustrated in Figure 10.4.3.
Fig. 10.4.3: Money demand
in the United States. Each dot
represents one month between
1980 and 2013. Source: Kurlat
(2019)
Exercises
10.1 Central Bank Instruments
Suppose the Central Bank wants to reduce the M1 money supply but does not want to change the
monetary base. In what direction should it change reserve requirements?
10.2 Pickpockets
Suppose there is an increase in the number of pickpockets. How would that change the fraction of their
money that people want to have in cash as opposed to checking deposits? If the Central Bank keeps the
monetary base constant, what will happen to the M1 money supply?
10.3 Interest on Reserves
202
10.4. The Demand for Money
Consider an economy where households almost use no cash (so χ → 0). Furthermore, assume that the
Central Bank makes lowers the reserve requirement to almost zero (so ρ → 0).
(a) What is the value of the money multiplier as χ and ρ become small?
(b) Suppose that the Central Bank decides to pay interest on reserves at the market rate i. Does
equation (10.3.1) still hold? Why or why not?
(c) Under this policy regime, what determines the quantity of money?
10.4 ATMs
Suppose one day, suddenly and unexpectedly, ATMs are invented, which make getting cash more conve-
nient than before. What would be the eect of this invention on the demand for money?
10.5 Going to the Bank
The average person in the US has about $10,000 in M1 money and spends around $60,000 per year.
Suppose this person uses its checking account to pay for all its purchases, behaves according to the
Baumol-Tobin model and only has two investment options: its checking account (which pays zero interest)
and an investment account (which pays interest). De ne going to the bank as transferring a balance
from the investment account to the checking account.
(a) How often does the average person go to the bank?
(b) Suppose the interest rate is 2%. What must be the perceived cost of each visit to the bank for this
type of behavior to be optimal?
(c) How often would the average person go to the bank if the interest rate rose to 3%?
(d) Now suppose the checking account does pay interest, but it only pays 0.5% instead of the full 2%
one can get from the investment account. What is the opportunity cost of holding balances in the
checking account? What do we infer now about the perceived cost of going to the bank?
203
CHAPTER 11
The Price Level and In ation
11.1 Measurement
In ation is de ned as a generalized increase in the level of prices. If the prices of all goods increased by the
same percentage, then measuring in ation would be straightforward. It becomes harder when dierent prices
are changing at dierent rates, or even going in dierent directions. How do we de ne the overall level of
change?
We already encountered this issue when we discussed real and nominal GDP in Chapter 1. There the
question was how to measure the overall change in output when prices of dierent goods were changing by
dierent percentages. Here we are interested in prices for their own sake.
The basic idea is going to be to de ne what is known as a basket of goods (i.e., a list of speci c quantities
of various goods) and measure how the total price of the basket changes. We call this total price a price index.
The dierent methods of measuring the total change in prices have to do with dierent ways of choosing and
updating the basket of goods.
The GDP De ator
The GDP de ator is a price index that is a side product of the calculation of real GDP. It is de ned as
Nominal GDP
GDP de ator = × 100.
Real GDP
Look back at Example 1.13 from Chapter 1. Suppose that Expandia computes real GDP at 2017 prices, so
that real GDP in 2018 is $2,550, while nominal GDP is $1,860. Then the GDP de ator would be
$1, 860
GDP de ator2018 = × 100 ≈ 73.
$2, 550
By de nition, the GDP de ator is 100 in the base year, so in this case we would say that overall prices went
down. Notice that not all prices went down: some went up and some went down. By using the GDP de ator
as a price index we are implicitly choosing to weigh each good in proportion to its share of GDP.
205
11.1. Measurement
The Consumer Price Index
The most commonly used price index weighs the prices of dierent goods by how much they are consumed
rather than how much they are produced. The basket for the CPI is constructed by conducting a survey
asking households how much they consume of each good. The CPI is then de ned by:
P
j qj pjt
CP I = P × 100
j qj pj0
where:
• qj is the quantity of good j in the basket,
• pjt is the price of good j in period t,
• pj0 is the price that good j used to have in the base year.
Example 11.1.
The residents of Luxuria consume only three goods: Ferraris, caviar, and champagne.
Good (j ) Quantity (qj ) Price in 2017 (pj0 ) Price in 2018 (pjt )
Ferrari (units) 2 100 115
Caviar (kg) 20 4 3
Champagne (liters) 10 2 4
Total basket 1 300 330
CPI 100 110
In ation
Having measured a price index, we calculate in ation (denoted by the letter π) by applying the following
formula:
Pt
πt = −1
Pt−1
where Pt is a price index in period t. In Example 1.13, in ation (in terms of the GDP de ator) was −27%.
When in ation is negative we call it de ation: a general fall in prices. In Example 11.1, in ation (in terms of
the CPI) was 10%.
Notice that for a given country there will coexist several measures of in ation, each derived from a dierent
price index. Usually it doesn't make much dierence which price index one looks at because the production-
based basket that is used in constructing the GDP de ator and the consumption-based basket that is used in
constructing the CPI are not that dierent, at least in the US. It could make a bigger dierence in countries
that produce and consume very dierent goods. For instance, in a country that produces oil and exports most
of it, a rise in the price of oil would result in a big rise in the GDP de ator but not as much in the CPI.
206
11.1. Measurement
Figure 11.1.1 shows the evolution of CPI in ation in the US. In ation was very variable until the 1950s,
with times of over 20% in ation and over 10% de ation. Between the 1960s and the early 1980s in ation
tended to increase. Since the mid-1980s in ation has been quite low and stable.
Fig. 11.1.1: CPI in ation in
the US. Source: BLS.
Nominal and Real Interest Rates
A basic lending transaction works as follows:
• a lender gives a borrower one dollar in period t,
• the borrower pays back 1 + it+1 dollars in period t + 1.
it+1 is the interest rate between periods t and t + 1.1
We will often talk about the interest rate, although in reality there is no single interest rate for all loans.
Typically, the interest rate on government debt is the lowest rate in the country (at least in the US, where
the government is perceived as reliable) and rates paid by private borrowers are higher, which compensates
for administrative costs, the probability of default, etc.
We are often interested in expressing interest rates in terms of goods rather than in terms of dollars.
Example 11.2.
The interest rate in Usuria on a one year loan that is issued in January 2018 and will be paid back in
1
An interest rate always involves more than one period: when the loan starts and when it ends. We will adopt the convention
to label interest rates according to the period when the loan has to be paid back. Hence it+1 refers to the interest rate on loans
that are issued in period t and are due in period t + 1.
207
11.2. Equilibrium
January 2019 is 11%. Everyone expects that in ation between those dates will be 2%. Suppose someone
lends 100 dollars in January 2018. What are they giving up? What do they get in return?
January 2018 January 2019
Price index 100 102
Loan issued / repayment received (in dollars) 100 111
100 111
Loan issued / repayment received (in goods)
100 =1 102 ≈ 1.088
In the example, the 100 dollars of the original loan would be enough to buy exactly 1 consumption basket
at the time the loan is granted. By the time the loan is repaid, the 111 dollars that are paid back are not
enough to buy 1.11 consumption baskets because prices have risen in the meantime: it is only enough to buy
1.088 consumption baskets. In other words, for each good that the lender gave up at the beginning, he is
getting back 1.088 goods one year later. The 0.088 extra goods that the lender obtains are what we call a
real interest rate. We call it real because it is expressed in terms of goods as opposed to a nominal rate
that is expressed in dollars. Whenever we have referred to interest rates in Chapters 4-9 we have meant real
interest rates because we were thinking about exchanges of real goods over time. Instead, when we studied
money demand in Chapter 10 it was the nominal interest rate that mattered because that's what determines
the opportunity cost of holding money.
In general, if it+1 is the nominal interest rate, the real interest rate is de ned by the following expression:
Goods you can aord with loan repayment
1 + rt+1 =
Goods you could aord with the loan when issued
1+it+1
Pt+1 1 + it+1
= 1 = Pt+1
Pt Pt
1 + it+1
=
1 + πt+1
⇒ rt+1 ≈ it+1 − πt+1 (11.1.1)
(The last approximation is accurate when πt+1 is small.) Equation (11.1.1) is known as the Fisher equation,
named after Irving Fisher.
It is not always easy to know what the real interest rate is. There is always some uncertainty as to what
in ation is going to be. If one lends or borrows in dollars, as is usual, then until the end of the loan one is not
certain how many goods the future dollars are going to be worth. Sometimes we make the distinction between
ex-ante real interest rates (meaning the real rate that was expected at the beginning, based on expected
in ation) and ex-post real interest rates (based on what in ation turned out to be).
11.2 Equilibrium in the Money Market
In Chapter 10 we looked at the money supply and the money demand separately. An equilibrium in the money
market requires that supply equals demand: all the money that is created jointly by the Central Bank and the
private banks must be held by someone, voluntarily. We can write the money-market equilibrium condition
208
11.2. Equilibrium
as:
M S = mD (Y, i) · p (11.2.1)
The left hand side of (11.2.1) is the money supply. We are going to imagine that the Central Bank simply
chooses the money supply, by choosing the monetary base and understanding the money multiplier. The right
hand side of (11.2.1) is the money demand. This is the result of households' decisions of how much money to
hold.
How does a money market equilibrium come about? Suppose that the central bank increases MS, what
changes to induce households to increase their money holdings? The right hand side of (11.2.1) gives us a list
of the things that could possible change to restore equilibrium:
• p. The price level could rise. If the price level is higher, then the same amount of real transactions
requires more money, so households would want to hold extra money.
• i. Nominal interest rates could fall. If interest rates are lower, the opportunity cost of money is lower
and households would be willing to hold more of it.
• Y. GDP could go up. If GDP is higher, there are more real transactions to carry out, which requires
more money.
There are dierent views on which of these three variables tends to respond and why. This turns out to be an
extremely important issue. In this chapter we'll look at the so-called classical view, which postulates that
the real side of the economy is separate from anything having to do with money. Real variables like real GDP
and real interest rates are determined purely by real factors (technology, preferences, etc.) that do not change
when the money supply changes. One way of stating this view is to say that money is neutral. In everything
we have done so far we have implicitly adopted this classical view: we studied the forces that determine real
variables without any reference to the money supply. In Chapter 14 we'll think about reasons why money
might not be neutral. For now, let's see how prices and in ation behave if the classical view is correct.
An Economy in Steady State with a Constant Money Supply
Imagine rst that the economy is in a steady state where Y and r are constant and the Central Bank holds the
S
money supply M constant as well. We'll conjecture that in this economy the price level will be constant as
well, and then verify that this is consistent with an equilibrium in the money market.
2 If indeed the price level
is constant, then the nominal interest is equal to the real interest rate. Therefore, solving for p in (11.2.1), we
get:
MS
p= (11.2.2)
mD (Y, r)
which indeed is constant, because we have assumed that MS, Y and r are constant. Equation (11.2.2) tells
us that an economy where the money supply is higher will, other things being equal, have higher prices.
2
This method of guring out the equilibrium of a model is sometimes called guess and verify. Technically, we will show that
there is an equilibrium where prices are constant but not that it's the only equilibrium.
209
11.2. Equilibrium
MS
People want a certain level of real money balances given by mD (Y, r), so the price level will be such that
p
corresponds to these desired real money balances.
An Economy in Steady State with a Growing Money Supply
Maintain the assumption that Y and r are constant but now assume that the money supply grows at a constant
S
rate µ, i.e. Mt+1 = (1 + µ) MtS . µ,
We'll conjecture that in this economy the price level will also grow at rate
and then check that this is consistent with equilibrium in the money market. If:
pt+1 = (1 + µ) pt
then in ation πt+1 is:
pt+1
πt+1 ≡ −1=µ
pt
and therefore the nominal interest rate is:
it+1 = r + πt+1 = r + µ
If the money market is in equilibrium in period t, then:
MtS = mD (Y, r + µ) pt
⇒ MtS (1 + µ) = mD (Y, r + µ) pt (1 + µ)
S
⇒ Mt+1 = mD (Y, r + µ) pt+1
which implies that the money market is also in equilibrium in period t + 1. This con rms our conjecture.
Economically, what's going on is the following. Since GDP and nominal interest rates are constant, people
want to hold constant real money balances. Since the money supply is growing, prices must be growing too
in order to keep money balances constant.
Figure 11.2.1 looks at data on in ation and the growth rate of the money supply over a long period in
many countries. Comparing across countries, the data shows that in ation is almost exactly proportional to
the growth rate of the money supply.
A Growing Economy
Now suppose that the economy is in a steady-state-with-growth, with Y growing at a constant rate g and a
constant real interest rate r. The money supply grows at a constant rate µ. Let's try to nd the in ation rate
in this economy. Start from (11.2.1) and take the derivative with respect to time:
dM S ∂mD (Y, i) dY ∂mD (Y, i) di
dp
= + · p + mD (Y, i)
dt ∂Y dt ∂i dt dt
210
11.2. Equilibrium
Fig. 11.2.1: In ation and
the growth rate of the money
supply in the long run. Each
dot represents one country.
Source: World Bank.
Now divide by (11.2.1) on each side:
dM S ∂mD (Y,i)
" #
dY dp
∂mD (Y, i)
dt Y dt ∂i di
= + + dt
MS ∂Y D
m (Y, i) Y mD (Y, i) dt p
∂mD (Y,i)
" #
∂mD (Y, i)
Y ∂i di
µ= g+ +π
∂Y mD (Y, i) mD (Y, i) dt
∂mD (Y,i) Y
Let η≡ ∂Y mD (Y,i)
. η represents the elasticity of money demand with respect to GDP. It is the answer
to the question: if GDP rises x%, by what percent does the demand for real money balances increase? Assume
the function mD is such that this elasticity is constant, so:
∂mD (Y,i)
" #
∂i di
µ = ηg + +π
mD (Y, i) dt
di
If in ation is constant, then i=r+π will be constant so
dt = 0. Then the equation reduces to:
µ = ηg + π
and therefore:
π = µ − ηg (11.2.3)
so, indeed, in ation is constant. Equation (11.2.3) tells us that, other things being equal, a growing economy
will have lower in ation. Why is this? A growing economy means a growing number of transactions and
211
11.2. Equilibrium
therefore a growing demand for real money balances. This means that the economy can absorb growing
quantities of money without resulting in in ation. Why does η show up in the formula? η measures how
much the demand for money increases when the economy grows. The higher this number, the faster the
money supply can grow without leading to in ation. Note that formula (11.2.3) encompasses the steady-state-
without-growth examples as special cases.
A One-Time Increase in the Money Supply
Suppose that, starting from a steady state with a constant money supply, at time t there is a sudden, unex-
S S0
pected increase in the money supply, from M to M . After this, the money supply is expected to remain
constant at M S0 forever. What's going to happen to the price level?
MS
Before time t we had that p= mD (Y,r)
. After that, the money supply will again be constant, except that
the level will be higher. Therefore we are going to be back in a constant-money-supply steady state, where
M S0
p0 = mD (Y,r)
. The eect on prices is therefore:
p0 M S0
=
p MS
In other words, prices jump immediately to their new level, and the size of the jump is proportional to the
size of the increase in MS.
The exact causal chain that leads from an increase in the money supply to a rise in the price level is a
matter of some debate. Why does everyone immediately raise their prices? In terms of the logic of the model,
what happens is that when the Central Bank increases the money supply, everyone suddenly has more money
than they would like, so they try to reduce their money balances. But it is impossible for everyone to do this
at the same time because someone has to hold the money. Everyone immediately realizes what's going on so
money immediately loses value, which is exactly what an increase in the price level means. Note that one
condition for this reasoning to be correct is that prices must be exible, reacting immediately to changes in
the supply of money. Starting in Chapter 14, we'll think about the possibility that prices might be sticky
and react slowly to changes in the money supply. This will be a source of monetary nonneutrality, i.e. of
interaction between money and the real economy.
A Change in the Rate of Growth of the Money Supply.
Now let's do a slightly more subtle exercise. Suppose we start at a steady state with the money supply growing
at rate µ and, therefore, an in ation rate of µ. At time t there is a sudden, unexpected increase in the rate of
growth of the money supply, form µ to µ0 . After this, the rate of growth of the money supply is expected to
remain at µ0 forever. What's going to happen?
A naive guess would be to say that in ation will simply increase from µ to µ0 . This guess is not wrong, but
0
it's incomplete. If the in ation rate changes from µ to µ, then the nominal interest rate rises from i=r+µ
0
to i = r+µ. Using (11.2.1), this implies that real money balances must fall. Higher nominal interest rates
increase the opportunity cost of holding money, so people want to hold less of it. But the level of MS does
212
11.2. Equilibrium
not change at time t: it simply starts growing at a dierent rate. What makes the money market clear? The
price level must rise!
Economically, this is what's going on. People are all simultaneously trying to reduce their money balances
because the opportunity cost of holding them has gone up. Since the total (nominal) supply of money has not
changed, money loses value, which is the same thing as saying the prices rise.
Figure 11.2.2 shows how the price level evolves over time in the dierent examples above.
Fig. 11.2.2: The evolution of prices in several examples.
The Velocity of Money
The velocity of money refers to the number of times a unit of money is used per period. Let's see an
example.
213
11.2. Equilibrium
Example 11.3.
The money supply is $2. At the beginning of the period, Ann and Bob each hold one dollar. Over the
course of the year, the following things happen:
• Ann produces an apple and sells it to Bob for $1. Bob produces a banana and sells it to Ann for $1
• Ann produces asparagus and sells it to Bob for $1. Bob produces a blueberry and sells it to Ann
for $1
• Ann produces an apricot and sells it to Bob for $1. Bob produces a blackberry and sells it to Ann
for $1
In the example, nominal GDP is $6, the money supply is $2 and and each dollar changes hands 3 times, so
the velocity of money is 3. In general, we have that:
M V ≡ p Y (11.2.4)
|{z} |{z} |{z} |{z}
Money Supply Velocity Price Level Real GDP
| {z }
Nominal GDP
Equation (11.2.4), sometimes known as the quantity equation, is a de nition. It's true because this is the way
we de ne the velocity of money.
How does equation (11.2.4) relate to the money-market equilibrium condition (11.2.1)? We can use (11.2.1)
M
to replace
p in (11.2.4) and rearrange to obtain:
Y
V = (11.2.5)
mD (Y, i)
Equation (11.2.5) says that any theory of money demand, summarized by a function mD (Y, i), is also a theory
of velocity. Once we have a mD (Y, i) function, we can simply plug it into (11.2.5) to obtain velocity as a
function of Y and i.
Our theory of money demand implies that velocity is an increasing function of the nominal interest rate.
We can see this in equation (11.2.5) by noting that mD is decreasing in i, which implies V is increasing in i.
Economically, what this is saying is that if interest rates are higher, people will hold less money, so in order
to carry out the same amount of real transactions, each dollar will have to change hands more times.
Figure 11.2.3 shows how the velocity of money has evolved over time. Notice that the velocity of M1 is
higher than the velocity of M2. Recall from the de nitions of M1 and M2 that M2 includes more things than
M1. Using (11.2.4), this implies that the velocity of M2 must be lower.
One assumption that people sometimes make is that V is constant. Figure 11.2.3 shows that this is not
completely justi ed, since velocity has moved around quite a bit over time. Furthermore, a standard model
of the money demand says that velocity should not be expected to remain constant: when interest rates rise,
money demand falls and therefore velocity rises.
Nevertheless, sometimes it is useful to assume that velocity can be held constant in an other things being
214
11.3. Seignorage
Fig. 11.2.3: The velocity of
money in the US, using M1 and
M2 as de nitions of money.
Source: Federal Reserve Bank
of St. Louis.
equal sense when one considers some other change in the economic environment. If one assumes that V is
constant, then equation (11.2.4) changes from being a de nition to being a theory. In fact, it is sometimes
known as the quantity theory, because it says that that the price level will be exactly proportional to the
quantity of money.
The evidence from Figure 11.2.1 is sometimes interpreted as supportive of the quantity theory, since the
quantity theory implies that there should be an exact linear relationship between changes in the money supply
and changes in prices. Notice that the relationship between money growth and in ation becomes much closer
for countries with high in ation. Even if velocity is not exactly constant, compared to the scale of changes
in the money supply in those countries it doesn't move that much, so the quantity theory is not such a bad
approximation.
11.3 Seignorage
Nowadays, most countries tend to keep in ation quite low, though usually not at zero. We'll look at some
of the arguments in favor of positive in ation later on. Historically, one of the reasons why in ation has
sometimes been high is that governments used in ation to obtain seignorage.
The term seignorage, which derives from the French word for lord (seigneur), originally referred to the
pro t made in the production of coins. Back when coins were usually made of precious metals, the value of a
minted coin was typically above the value of the metal used to produce it. Why? Because minted coins were
better money than raw metal since they were standardized to be useful in transactions. The pro t earned by
the mint by turning metal into money was known as seignorage. Nowadays the term is used more broadly to
refer to the resources obtained thanks to the ability to create money.
215
11.3. Seignorage
When we looked at the process of money creation in Chapter 10, we didn't pay too much attention to
the Central Bank's balance sheet, but if you go back to it, you'll notice that, in the process of increasing the
monetary base by ∆, the Central Bank increased both its assets and its liabilities by ∆. This doesn't seem
like a big deal, but there is one important dierence. The assets that the Central Bank obtains (government
bonds) earn interest while the liabilities that it issues (currency and reserves) typically do not. Standard
accounting still treats them as liabilities but in a certain sense they are not. Furthermore, the Central Bank
is just a branch of the government.
3 Therefore increasing the monetary base is a way for the government to
get a loan that it will never have to repay and doesn't pay any interest. Indeed, one way to write down the
government's budget constraint is:
B
− MtB = pt Gt + (1 + it ) Bt
Bt+1 + pt τt + Mt+1 (11.3.1)
where:
• Bt+1 is nominal public debt,
• Gt is real government spending,
• τt is real government tax revenue,
• pt is the price level,
• it is the nominal interest rate,
• MtB is the monetary base.
Let's go through the terms in (11.3.1) to see what it means. The right hand side represents all the payments
the government must make, in nominal terms. pt Gt is how much the government must pay for the current
period's spending. (1 + it ) Bt is how much it must pay on the debts it had at the beginning of the period,
including the interest that accrued in the current period. The left hand side represents all the resources the
B
government can use to make its payments. pt τt is how much it raises in taxes, in nominal terms. Mt+1 − MtB
is the increase in the monetary base. This is all the payments the government can make just by virtue of
having created extra money. Bt+1 is the amount of payments the government can make by virtue of issuing
debt that will have to be repaid in the future.
We can also rearrange (11.3.1) to express it as:
B
Mt+1 + Bt+1 = pt [Gt − τt ] + (1 + it ) Bt + MtB
This formulation makes it easier to see that the monetary base is just like debt (in the sense that it enters the
government budget in the same way), except that it doesn't pay interest.
Historically, governments have used expansion of the monetary base as a way to satisfy the government
budget in various circumstances. Sometimes it's a result of di culties in collecting regular taxes, due to tax
3
This is literally true in some countries and sort-of-true in others. In the US, the Federal Reserve has a mixed governance
structure with some in uence from the private sector. However, it rebates its pro ts to the Treasury, so in that sense it's part of
the Federal Government.
216
11.4. The Cost of In ation
evasion or political indecision about what other taxes to use. Sometimes it's a result of a rapid increase in
government spending that leaves no time to increase regular taxes, as in wartime. Sometimes it's the result
of the inability to borrow, perhaps because lenders don't trust the government to pay back its debts. In other
instances it may have been to a misperception that increasing the monetary base is a way for the government
to obtain resources without really taking them away from anyone. Often it could be several of these reasons
at the same time.
One subtle question is who exactly the government is taxing when it increases the monetary base. It's
clear that the government can, at least to some extent, pay for goods and services with monetary expansion.
But nobody seems to be paying for this. How does it all add up? The answer is that anyone who holds money
is implicitly paying a tax to the government when the monetary base expands. We know that in a money
market equilibrium, an expansion of the monetary base leads (through the money multiplier) to an increase
in the money supply and then to an increase in the price level. Anyone who holds the monetary base while
it's losing value is implicitly giving up some of their wealth to the government, just as they would if they were
paying a regular tax. That's why seignorage revenue is also sometimes referred to as an in ation tax.
4
Governments usually limit how much seignorage revenue they try to raise because they want to avoid
creating high in ation. But even if they didn't care about in ation there is a limit to how much seignorage
revenue they can obtain. To obtain high revenue, they need to expand the monetary base very fast. But
fast expansion of the monetary base leads to high in ation, which leads to high nominal interest rates, which
means that the real money demand falls. Since implicitly seignorage is a tax on money holdings, this means
that the tax base shrinks. Exercise 11.6 asks you to work out the limit on seignorage revenue and compare it
to some historical experiences of very high in ation.
11.4 The Cost of In ation
In ation is generally seen as undesirable. There are several reasons for this.
One reason can be understood directly from the Baumol-Tobin model of money demand. Other things
being equal, more in ation implies higher nominal interest rates, which means that people will go to the
bank more times to avoid holding high money balances. Each of those trips to the bank has a cost of F .5
Using (10.4.3) we can compute the total cost of trips to the bank as:
Cost = N F
r
iY F
=
2
r
(r + π) Y F
= (11.4.1)
2
At the times of high in ation in Argentina in the late 1980s, my dad would literally go to the bank twice a
day, once around noon and once after work just to make sure that he had exactly enough money for the day's
4
Technically, this is not quite right. The government can obtain seignorage revenue with zero in ation if the economy is
growing. Exercise 11.3 asks you to work out how much.
5
Sometimes this is known as shoe leather cost: people wear out their shoes by walking to the bank all the time.
217
11.4. The Cost of In ation
expenses and no more. That time spent dealing with the problem of how much money to hold has a real
opportunity cost.
On the basis of a reasoning like this, Milton Friedman advocated keeping nominal interest rates at or
very near zero, a policy known as the Friedman Rule. The idea of the Friedman Rule is to eliminate the
opportunity cost of holding money. In formula (11.4.1), having i = 0 would make the cost equal to zero,
because it would mean that you don't ever need to go to the bank: since it has no opportunity cost, you can
just hold all your wealth in money. Notice that in order to have i = 0, one would need to have π = −r. Since
the real interest rate is usually positive, this means that implementing the Friedman Rule requires de ation.
The Friedman Rule is usually seen as a theoretical extreme, more valuable for the underlying logic than as a
concrete policy proposal.
Economists sometimes refer to menu costs as part of the cost of in ation. Sometimes there are real
resources that need to be dedicated to put in place a change in prices. For instance, restaurants need to print
new menus, shops need to print new signs, etc. When in ation is high this needs to be done more often, which
is a real cost. Minimizing menu costs would require keeping in ation at zero, rather than running de ation
as implied by the Friedman Rule. Even this would not eliminate menu costs: zero in ation means that the
price index would stay constant, but the prices of individual goods would still move up and down a lot, and
making those changes would incur menu costs. There is some disagreement among economists about whether
it's plausible that menu costs are large.
Another cost of in ation is that it creates uncertainty about relative prices. In order to decide what to
buy, people need to know the prices of dierent goods. But they don't look at all the prices of all the goods
at the same time. Typically, a consumer just looks at the price of a few dierent goods at a time and relies
on his knowledge of approximately how much stu you can get for a dollar to assess whether prices of the
goods he is considering are worth paying. High in ation makes it harder to keep track of how much a dollar
is worth, which makes it harder to make the right consumption decisions. It's a little bit like trying to take
measurements with a ruler that keeps changing size. Producers face the same problem: in order to decide what
price to charge for the goods they sell, they need to keep track of how much a dollar is worth, and in ation
makes this harder.
The government is not the only one to earn seignorage. Part of the money supply is made up of bank
deposits, which either pay no interest or pay less than market rates, so they are earning seignorage as well.
An additional source of costs of in ation is that it allows banks to earn more seignorage, leading to excessive
entry into the banking industry. Exercise 11.7 asks you to compute how much seignorage banks earn.
Exercises
11.1 The Elasticity of Money Demand
Recall the Baumol-Tobin model of Chapter 10. Compute η, the elasticity of money demand with respect
to GDP, in the money-demand function that arises from this model.
11.2 The Quantity Theory
The quantity theory of money assumes that money velocity is constant. Try to come up with a money-
218
11.4. The Cost of In ation
demand function that would imply that velocity is indeed constant. What does this money-demand
function say about how money holdings depend on interest rates? What does it say about how households
manage their trips to the bank?
11.3 Seignorage with Zero In ation and Growth
Suppose an economy is growing at a constant rate g, money demand comes from the Baumol-Tobin
model and the money multiplier is ω. Suppose the government wants to maintain zero in ation. How
much seignorage revenue can it obtain as a fraction of GDP? In other words, nd an expression for the
level of:
MtB − Mt−1
B
pt Yt
that is consistent with zero in ation. How does this depend on g, F and ω? Why? [Hint: use the result
from Exercise 11.1.]
11.4 Growth in the Money Supply
There are two otherwise identical economies. In both of them GDP is constant, prices are exible and
the real interest rate is constant. The rate of growth of the money supply in each of them is:
Growth rate of the money supply
Year Country A Country B
2009 3% 6%
2010 3% 5%
2011 3% 4%
2012 3% 3%
2013 3% 2%
2014 3% 1%
2015 3% 0%
How does in ation in the two countries in 2012 compare? Explain.
11.5 In ation Targeting
(a) Suppose a Central Bank has decided it wants to keep in ation at exactly 2% every year. The
economy is not growing and the real interest rate is 3%. Describe what the Central Bank must do
to the money supply to achieve its objective.
(b) Suppose the real interest rate suddenly and unexpectedly falls from 3% to 1%. How should the
Central Bank respond? Explain in detail.
11.6 Seignorage with High In ation
Suppose that money demand is given by:
−µ
mD (Yt , it+1 ) = Yt · (1 + it+1 )
219
11.4. The Cost of In ation
where µ is a parameter. This is sometimes known as the Cagan money-demand function. Equilibrium
in the money market is given by:
−µ
MtS = Yt · (1 + it+1 ) pt
The real economy is in a steady state so that:
Yt = Yss
rt+1 = rss
The government obtains seignorage revenue St (in real terms) by expanding the monetary base. St is
given by:
B
MtB − Mt−1
St = (11.4.2)
Pt
Let ω be the money multiplier, so that M S = ωM B .
Suppose that the rate of growth of the money supply is constant at rate γ, i.e.:
MtS = (1 + γ) Mt−1
S
(11.4.3)
(a) What will be the rate of in ation?
(b) What will be nominal interest rate?
(c) Find an expression for the level of real money balances. How does it depend on γ? Why?
S
(d) Solve equation (11.4.3) for Mt−1 and replace this in (11.4.2) to obtain an expression for seignorage
MtS MtS
revenue St in terms of γ and
Pt . How does St depend on γ ? How does it depend on Pt ? Why?
MtS
(e) Replace the value of
Pt that you found in part (c) into the expression for St that you found in part
(d) to obtain an expression for seignorage revenues St in terms of γ , Yss , rss , ω and µ.
(f ) Assume the following parameter values:
Yss = 1
µ=4
rss = 0.0025
ω=5
Note that µ = 4 is close to the value that has been empirically estimated in some high in ation
countries when the time period is one month. To be consistent, rss should be interpreted as a
monthly real interest rate and Yss as monthly GDP.
(g) Plot St against γ for values of γ between 0 and 0.8.
(h) What is the monthly rate of growth of the money supply that maximizes seignorage revenue? Call
this γ∗.
220
11.4. The Cost of In ation
(i) Why is seignorage revenue decreasing in γ for γ > γ∗?
(j) What monthly in ation rate does γ = γ∗ imply?
(k) What monthly nominal interest rate does γ = γ∗ imply?
(l) What yearly rate of in ation does γ = γ∗ imply?
Mt
(m) What is
Pt if γ = γ∗?
(n) What is the maximum revenue as a fraction of GDP that the government in this example can
constantly collect from seignorage?
(o) Look at the data from Sargent (1982) on the German hyperin ation of the 1920s (you can nd it
at the book website). What was the maximum monthly in ation rate that Germany experienced?
How does it compare to γ∗?
(p) Suppose that the government is increasing the money supply at a rate γ = γ∗, and in month t it
B ∗
credibly announces that (i) Mt+1 will not be equal to MtB (1 + γ ) the way it would have been if
the policy had continued as usual, but it will be some other level M̄ (maybe higher, maybe lower
than MtB (1 + γ ∗ )) (ii) from then on, it will set
B
Mt+s = M̄ for every s≥1 (i.e. the money supply
will be constant).
(q) What will be the rate of in ation from period t+1 onwards?
(r) What will be the price level Pt+1 in period t + 1? (This will depend on the government's choice of
M̄ )
Suppose the government sets M̄ at the level that will ensure that Pt+1 = Pt (i.e. the level that will
immediately stop in ation).
(s) What value of M̄ will it need to choose? How does it compare to MtB (1 + γ ∗ )?
(t) How much seignorage revenue will the government obtain in period t + 1? How does this level of
seignorage compare to what the government was getting every month before making this change?
(u) Look carefully at the timing of the end of hyperin ation in Germany? When exactly did the money
supply stop growing? When exactly did hyperin ation stop? How can we make sense of this?
11.7 Bank Seignorage
Suppose that the demand for M1 money is given by
M
pY = A−ηi where M is the quantity of M1 money,
pY is nominal GDP, i is the nominal interest rate and A and η are parameters. Households want to hold
a fraction χ of their M1 money in the form of cash and a fraction 1−χ in the form of checking accounts,
which earn no interest. Banks earn seignorage by taking checking deposits and investing in assets that
earn the nominal interest rate.
(a) Find an expression for the ratio of total seignorage earned by banks to GDP. Call this ratio s.
∂s
(b) Compute
∂i . Why does this number depend on η?
221
11.4. The Cost of In ation
(c) Look up data on M1 and its components in 2018. What is a reasonable value for χ? What was
M
the value of
pY ? If the average nominal interest was 2%, how much seignorage did banks earn as a
fraction of GDP?
(d) If η = 0.2, how much seignorage would banks earn as a fraction of GDP if the nominal interest rate
went up to 3%?
11.8 Real Interest Rates
De ne realized real interest rates rt+1 as:
rt+1 ≡ it+1 − πt+1
where it+1 is the nominal interest rate between period t and period t+1 and πt+1 is the rate of in ation
between period t and period t + 1. If there is uncertainty about what the rate of in ation is going to be,
this implies that there is uncertainty about what realized real interest rates will be. De ne the expected
real interest rate as:
E
rt+1 ≡ it+1 − E (πt+1 )
where E (πt+1 ) is the expected rate of in ation.
Suppose there are two parties to a nominal lending contract: a borrower and a lender. Who bene ts when
realized real interest rates turn out to be higher than expected real interest rates? What monetary
policy will each of them support?
11.9 Money among Prisoners of War
Read The Economic Organisation of a P.O.W. Camp, which you can nd at: https://www.jstor.org/
stable/2550133. Pick one passage out of the article and explain how it relates to the models of money
supply, money demand and in ation from Chapters 10 and 11.
222
PART V
Business Cycles
This part of the book looks at business cycles, relatively short-term movements in the
aggregate economy.
In Chapter 12 we look at patterns in business cycle data to establish some of
the basic facts of what business cycles look like. Then we turn to attempts at explain-
ing why business cycles happen and study versions of two of the leading theories: real
business cycle models and Keynesian models. We'll do them in reverse historical order.
The real business cycle model was rst proposed in the late 1970s and early 1980s,
in part as a result of some economists' dissatisfaction with earlier Keynesian models.
However, the model is a bit simpler so we do it rst, in Chapter 13, leaving the New
Keynesian model for Chapter 14.
Finally, in Chapter 15, we take a look at some of the policies that are used to
try to manage the business cycle.
223
CHAPTER 12
Facts about Business Cycles
12.1 What are Business Cycles?
There is no unique de nition of exactly what is meant by a business cycle. Here is one traditional de nition,
proposed by Burns and Mitchell (1946):
Business cycles are a type of uctuation found in the aggregate economic activity of nations that
organize their work mainly in business enterprises: a cycle consists of expansions occurring at about
the same time in many economic activities, followed by similarly general recessions, contractions,
and revivals which merge into the expansion phase of the next cycle.
Figure 12.1.1 shows the evolution of GDP in the US since 1947.
Fig. 12.1.1: Real GDP in the
US since 1947. Source: NIPA.
225
12.1. What are Business Cycles?
As we know, the general trend is upwards, but it's far from a straight line. Often the term business cycle
is used to refer to the wiggles in the trajectory of GDP and the movements in other economic variables that
accompany them. The word cycle itself is a little bit misleading, since it evokes a regular, periodic, oscillation
like that of a sine curve. The evolution of GDP is not like that, it's irregular. Sometimes GDP grows at a
fairly steady rate for a long time; other times it expands rapidly and then falls steeply.
The terms recession and expansion are often used in the context of describing business cycles. Expan-
sions are periods when GDP is growing; recessions are periods when GDP is shrinking. Sometimes a recession
is de ned more precisely as a period when GDP shrinks for two consecutive quarters, but not everyone adheres
to that de nition.
The NBER Business Cycle Dating Committee
There is a committee within the National Bureau of Economic Research dedicated to the task of declaring
when recessions and expansions begin and end. The end of an expansion/beginning of a recession is called a
peak and the end of a recession/beginning of an expansion is called a trough. They don't have an exact
rule of how they determine peaks and troughs (if they did, one wouldn't need a committee!) and look at a
broad range of indicators, not just GDP. At some level, this labeling exercise is a bit absurd: there is more to
be learned by looking at the entire data than by just having the labels recession and expansion. On the
other hand, it is sometimes useful to have a simple classi cation of which way economic activity is headed.
Figure 12.1.2 shows the same data for GDP as Figure 12.1.1 with the NBER-designated recessions shaded in
gray.
Fig. 12.1.2: Real GDP in the
US since 1947 and NBER re-
cessions. Source: NIPA and
NBER.
226
12.1. What are Business Cycles?
The Hodrick-Prescott Filter
Hodrick and Prescott (1997) propose an algorithm for distinguishing a cycle from a trend in economic
data. The idea is to separate out the long-run growth (the trend) of any economic variable, such as GDP,
from the shorter term deviations around that trend, which we will label a cycle.
Suppose we observe a variable Xt from period t = 1 until period t = T . We are going to de ne an arti cial
variable X̂t and call it the trend in Xt . We are going to want the trend to have the following properties:
1. It cannot be too far away from the actual variable Xt .
2. It has to move smoothly, i.e. the rate of growth in the trend should not change very much from one
period to the next.
Mathematically, we are going to de ne the trend as the solution to the following problem:
XT 2 T
X −1 h i2
min Xt − X̂t +λ X̂t+1 − X̂t − X̂t − X̂t−1 (12.1.1)
T
{X̂t }t=1 t=1 | {z } t=2 | {z }
Distance between trend Change in the trend's
and actual variable growth rate
where λ is a parameter. Figure 12.1.3 shows the trend in GDP using λ = 1, 600, which is a standard value for
quarterly data. Trend GDP ends up being a smoother version of actual GDP.
Fig. 12.1.3: Real GDP in the
US since 1947 and HP trend.
Once we de ne a trend, the cycle or cyclical component is simply de ned as
X̃t ≡ Xt − X̂t
227
12.1. What are Business Cycles?
i.e. as the deviation of the variable from its trend. Figure 12.1.4 shows the cyclical component of GDP
X̃t
(expressed as a percentage of trend GDP, i.e.
X̂t
), compared again with NBER-de ned recessions. The
gure shows that what the NBER committee determines is not that dierent from what HP- ltering does:
NBER-de ned recessions are periods when then cyclical component of GDP moves down.
Fig. 12.1.4: Cyclical compo-
nent of real GDP in the US
since 1947 obtained with HP
lter.
Note that dierent values of λ will result in dierent de nitions of what the trend is and therefore dierent
de nitions of what the cyclical component of GDP is. Figure 12.1.5 shows the cyclical component of GDP
for dierent values of λ. In expression (12.1.1), a high value of λ penalizes changes in the trend growth rate
very heavily. As a result, the HP lter will make the trend close to a straight line and, as a result, allow the
cyclical component of GDP to be large. Conversely, a low value of λ will result in a trend that changes quite
a bit in order to stay very close to actual GDP. As a result, the implied cyclical component will be small.
This can matter. For instance, the standard value of λ implies that after the recession of 2008-2009 GDP
returned to trend fairly rapidly and was back at trend by 2012 approximately. Mathematically, the reason is
that after several years of slow growth, the HP lter infers that the trend has slowed down, so actual GDP is
catching up to trend despite slow growth. In a sense, this level of λ imposes a limit on how much a recession
can really last. Conversely, under a higher value of λ, the procedure insists that the trend continues to grow
at close to its long-term average, so actual GDP continues to fall behind trend. We are going to use the HP
lter with the standard value of λ to systematically describe the patterns that we observe in business cycles
but it's worth bearing in mind that there is quite a bit of judgment going into how we construct the de nition
of the business cycle.
1
1
Hamilton (2018) argues that these and other problems make the HP lter completely useless.
228
12.2. Patterns
Fig. 12.1.5: Cyclical compo-
nent of real GDP in the US
since 1947 obtained with HP
lter for dierent values of λ.
12.2 Patterns in Business Cycles
The Post-War US Business Cycle
Table 12.1 shows some of the patterns displayed by the US business cycle since 1947. It shows the cyclical
patterns of several interesting macroeconomic variables. For each variable Xt , we follow these steps:
1. HP- lter the data and subtract the trend to compute the cyclical component X̃t .
2. Compute the standard deviation of X̃t .2 This gives us a sense of how far away from its trend the variable
Xt tends to be.
3. Compute the correlation between X̃t and the cyclical component of log (GDP ).3 This gives us a sense of
whether Xt tends to move together with GDP, in the opposite direction, or with an unrelated pattern.
2
PT
The standard deviation of a variable Xt T is the number of observations. X̄ = T1
is de ned as follows. t=1 Xt is the mean.
1 PT 2 p
V ar (X) = T t=1 Xt − X̄ is the variance. σX = V ar (X) is the standard deviation.
3 1 PT
The correlation between two variables X and Y is de ned as follows. Cov (X, Y ) =
T t=1 Xt − X̄ Yt − Ȳ is the
Cov(X,Y )
ρX,Y = σ σ
covariance. is the correlation. It takes values between −1 and 1. ρXY = 1 means X and Y are exactly aligned;
X Y
ρXY = −1 means X and Y are exactly aligned but move in opposite directions; ρXY = 0 means that movements in X and
movements in Y go in the same direction and the opposite direction just as much.
229
12.2. Patterns
Table 12.1: Business cycle properties of macroeconomic variables. Y, C, I, G, M, X, total hours, TFP and
real wages are measured in log scale so the units are comparable. All variables are detrended using an HP lter
with λ = 1, 600. Sources: NIPA for GDP and its components; BLS for labor market data including wages and
for in ation; Fernald (2014) for TFP; Board of Governors of the Federal Reserve System for interest rates.
Relative standard Correlation with
Variable Standard deviation
deviation GDP
GDP 1.6% 1 1
Consumption 1.2% 0.74 0.78
Durable Goods 4.7% 2.93 0.60
Non-durable Goods 1.5% 0.95 0.58
Services 0.8% 0.54 0.57
Investment 7.3% 4.59 0.83
Government spending 3.3% 2.06 0.16
Exports 5.3% 3.31 0.42
Imports 5.0% 3.12 0.72
Total hours of work 1.8% 1.13 0.85
TFP (Solow residual) 1.3% 0.78 0.80
Real wages 0.7% 0.47 0.31
Unemployment rate 0.8% −0.86
In ation 3.2% 0.26
Nominal Interest Rate 1.1% 0.38
We are going to focus on the following facts that can be gathered from Table 12.1:
1. It is typical for GDP to be about 1.6% away from trend (either above or below).
2. Total consumption is less volatile than GDP. If we break it down into categories, consumption of durables
is more volatile than GDP and consumption of nondurables and especially services less so.
3. Investment is much more volatile than GDP.
4. Consumption, investment, productivity, and hours of work are all highly positively correlated with GDP.
Unemployment is highly negatively correlated with GDP.
5. Real wages are less volatile than GDP and only weakly positively correlated.
6. In ation and nominal interest rates have weak positive correlations with GDP.
Figure 12.2.1 shows the cyclical components of GDP, total consumption, investment and total hours. They
move up and down together, with investment moving much more than GDP, consumption a little bit less and
hours about the same amount.
For now, we'll take these as the main facts about business cycles which we are tying to understand. We'll
come back to some of the other patterns documented on Table 12.1 later on.
230
12.2. Patterns
Fig. 12.2.1: Cyclical com-
ponent of real GDP, con-
sumption, investment and total
hours.
The Phillips Curve
Phillips (1958) documented a relationship between between the rate of nominal wage increases and the un-
employment rate in the UK between 1861 and 1957. He found that times of low unemployment were also
times of fast wage growth. This pattern came to be known as the Phillips Curve. There are several, slightly
dierent, versions of the Phillips Curve. They all relate some measure of nominal price changes with some
measure of economic activity such as unemployment or GDP growth. Probably the most standard version
these days has in ation on the vertical axis and unemployment on the horizontal axis. Another version has
the cyclical component of GDP on the horizontal axis instead of unemployment. Figure 12.2.2 shows how that
relationship looks like for the US.
Overall, there is a mild negative relationship between in ation and unemployment. The slope of −0.34
on the left panel means that 1 percentage point higher unemployment is associated with 0.34 percentage
points lower in ation. However, the association between the variables is quite noisy, their correlation is just
−0.36. Similarly, the slope of 0.59 on the right panel means that 1% higher GDP is associated with 0.59
percentage points higher in ation. Again, the association is quite noisy, with a correlation of just 0.29. Since
unemployment and the cyclical component of GDP are so highly correlated (as shown in Table 12.1), the two
panels tell the same story: business cycle expansions are associated with higher in ation, but only mildly.
If we break down the relationship into dierent subperiods, as shown in Figure 12.2.3, an interesting
pattern emerges. Until the mid-1960s, we observe the negative Phillips curve relationship. It somewhat noisy,
but it's clearly visible. Between the mid-1960s and the late 1970s the relationship seems to break down, and
in ation and unemployment become positively associated. Then, from the late 1970s until the mid-1980s, the
relationship is again negative, and very strong. Finally, since the mid-1980s, the relationship is negative again,
but very weak.
231
12.2. Patterns
Fig. 12.2.2: The Phillips Curve relationship in the US. Annual data for 1929-2018 on the left panel; annual
data for 1947-2018 on the right panel. Source: NIPA and BLS.
One of the features of business cycles that we going to try to understand is why the Phillips Curve
relationship sometimes seems to hold and other times seems to not hold.
The Great Depression
Much of macroeconomics was originally motivated by trying to understand the Great Depression of the 1930s.
Figure 12.2.4 shows some facts about what happened in the 1930s in the US.
Between its peak in 1929 and its trough in 1933, real GDP fell by about 27%, investment fell by more than
80%, unemployment rose from about 5% to over 20% and there was de ation, with prices falling by about
27%. Qualitatively, the Great Depression is consistent with the typical patterns of business cycles:
1. investment moved in the same direction as GDP but much more,
2. unemployment rose at the same time as GDP fell,
3. de ation coincided with high unemployment, as the Phillips Curve stipulates.
However, the magnitude of the movements was much higher than in typical business cycles: a typical recession
will have GDP a couple of percentage points below trend; in the Great Depression the fall in GDP was a
staggering 27%. How could something like this happen? What should be done to prevent it from happening
again?
At a more theoretical level, an important question is whether the Great Depression was just another
business cycle but much larger or whether it was a fundamentally dierent phenomenon.
232
12.3. Who cares?
Fig. 12.2.3: Cyclical component of unemployment and CPI in ation in the US. Source: BLS.
12.3 Who Cares about the Business Cycle?
There has been an enormous amount of research into how the business cycle works. Part of the objective of
this research is to nd ways to make the economy more stable. Lucas (1987) asked the following question:
suppose we could gure out a way to eliminate the business cycle altogether, how valuable would that be? He
proposed answering the question by doing a version of the following calculation.
Suppose the representative household in the economy has standard preferences over consumption, given
by:
∞
X c1−σ
t
βt
t=0
1−σ
233
12.3. Who cares?
Fig. 12.2.4:The US economy during the Great Depression. Sources: Johnston and Williamson (2019) and
NIPA for GDP, NIPA for investment, BLS for CPI and Smiley (1983) for unemployment.
Imagine that trend consumption is given by :
t
ĉt = c0 (1 + g)
so that it grows at a constant rate g after starting at the level c0 at some initial period 0. Actual consumption
can dier from the trend consumption because of the business cycle. We are going to have the following
representation of the business cycle. Half the time, the economy is in an expansion period so consumption is
234
12.3. Who cares?
above trend and half the time the economy is in a recession with consumption below trend:
(
ĉt (1 + f ) in an expansion
ct = (12.3.1)
ĉt (1 − f ) in a recession
where f is a parameter that measures the amplitude of the business cycle.
We can compute the utility that the representative household will experience as a function of initial
consumption c0 , the growth rate (denoted g) and the magnitude of business cycle uctuations f:
∞
X
t 1 t
1
t
W (c0 , g, f ) = β u c0 (1 + g) (1 + f ) + u c0 (1 + g) (1 − f )
t=0
2 2
∞
1 c1−σ h
1−σ 1−σ
iX h
t
i1−σ
= 0
(1 + f ) + (1 − f ) β t (1 + g)
21−σ t=0
1−σ 1−σ
1 c1−σ
0 (1 + f ) + (1 − f )
= 1−σ
21−σ 1 − β (1 + g)
In the rst line, we are applying a version of the behind-the-veil-of-ignorance argument we rst encountered
in Chapter 2. In each period, the household might nd itself either in an expansion or in a recession. The
household computes how much utility it's going to experience in each case and then takes an average.
In order to compute the value of eliminating the business cycle, we are going to solve for λ in the following
equation:
W (λc0 , g, f ) = W (c0 , g, 0) (12.3.2)
What's the logic of equation (12.3.2)? It de nes λ as the answer to the following question. Suppose someone
oered two possibilities to the representative household: either multiply its consumption by some number λ
each period (the left hand side) or keep average consumption the same but, by making f = 0, eliminate the
business cycle (the right hand side). What is the value of λ that would make the household indierent? Let's
solve:
1−σ 1−σ 1−σ
1 (λc0 ) (1 + f ) + (1 − f ) 1 c1−σ
0 2
1−σ =
2 1−σ 1 − β (1 + g) 2 1 − σ 1 − β (1 + g)1−σ
1
1 σ−1
1−σ 1−σ
λ= (1 + f ) + (1 − f ) (12.3.3)
2
Now let's plug in some actual numbers into (12.3.3). From Table 12.1 we know that the standard deviation
of the cyclical component of log (ct ) is 1.2% of its trend level, so let's set f = 0.012. As we have seen before,
there is a range of estimates for the value of σ, so we'll try a couple of dierent ones. Table 12.2 shows the
results:
4
4
As we saw in Chapter 2, for σ=1 we replace the utility function with log (c). Re-doing the steps that lead to (12.3.3) results
in:
1
λ = ((1 + f ) (1 − f ))− 2
235
12.3. Who cares?
Table 12.2: The value of eliminating the business cycle, according to the Lucas (1987) calculation.
σ 1 2 5 10
λ 1.00007 1.00014 1.00036 1.00072
The values of λ are all greater than 1. This means that the representative household views the elimination
of the business cycle as equivalent, in utility terms, as an increase in consumption. The reason is that the
household dislikes the variability in consumption that the business cycle entails, so it attains higher utility if
this variability can be eliminated.
However, the the gain is tiny. Take, for instance, σ = 2. For this value we get λ = 1.00014. This means that
the household is indierent between eliminating the business cycle and obtaining an increase in consumption
of 0.014%. To put in in dollar terms, consumption per person per year in the US is about $40,000; according
to this calculation, eliminating the business cycle would be worth just $5.80 per year to the average person.
Even for higher values of σ, which make the household dislike variability more, the numbers are still very
small. For σ = 10, eliminating the business cycle is still worth only $28.80 per year to the average person.
Reasons why the Business Cycle Could Matter More
If Lucas's calculation is correct, then the business cycle is simply not a big deal, and we should probably
devote a lot less intellectual energy to understanding it and a lot less policy eort to stabilize it. However, the
calculation leaves out a lot of things, and some have argued that taking these into account would signi cantly
alter the conclusion.
5
One reason why stabilizing the business cycle might be more valuable than Lucas's calculation suggests is
the way recessions interact with individual-level risk. Lucas's calculation assumes that there is a representative
household whose consumption moves up and down exactly with aggregate consumption. If instead a lot of
the volatility in aggregate consumption is concentrated in certain households (for instance, the ones that are
prone to losing their jobs in recessions and experiencing long spells of unemployment), then the value for these
particular households of stabilizing the business cycle would be much larger, and the overall behind-the-veil-
of-ignorance value could be higher as well. Exercise 12.3 asks you to consider this possibility.
Lucas's calculation assumes that the best one could hope for with stabilization policy is to make con-
sumption stable without changing the average level. Underlying this is a view that business cycles represent
movements up and down around some normal level. An alternative view is that the peak of the business cycle
is the normal level for the economy and business cycles are downwards deviations from this normal level.
If this view is right, then in principle the ideal policies could make the economy always remain at its peak,
raising average consumption as well as making it more stable, as illustrated in Figure 12.3.1. According to
standard preferences, this is much more valuable.
As formula (12.3.3) makes clear, the value of eliminating business cycles depends on f, i.e. on how large
these business cycles were to begin with. By historical and international standards, the postwar US economy
was relatively stable, which results in a small value of f. Lucas's conclusion that further stabilization is not
that valuable for an economy like the US does not imply that a very volatile economy would not bene t from
5
Barlevy (2005) surveys many of these arguments.
236
12.3. Who cares?
Fig. 12.3.1: Dierent views
on what stabilizing the business
cycle might be able to achieve.
policies that make it as stable as the US. Exercise 12.2 asks you to consider the value of stabilizing an economy
where recessions of the magnitude of the Great Depressions happen often.
Finally, there might be factors that are not well captured by our models (the political repercussions of high
unemployment, the stress of not knowing whether one's business will survive in a recession) that might make
business cycles more important than what is captured by equation (12.3.3).
Exercises
12.1 Business Cycles in Other Countries
(a) Look up GDP accounts for a country other than the US. Download quarterly data for GDP, con-
sumption and investment.
(b) Apply the HP lter to create trend and cyclical components. To do this, rst transform the data by
taking logarithms, i.e. compute log(Y ), log(C) and log(I). Then apply formula (12.1.1) to compute
a trend component for each log series. If you use Excel, you can nd an add-in to do this at
https://web-reg.de/webreg-hodrick-prescott- lter/; if you use Matlab, the command hp lter does
it for you. Finally, compute the cyclical component by subtracting the trend from the un ltered log
series.
(c) Compute the standard deviation of the cyclical component of GDP, consumption and investment.
(d) Compute the correlation between the cyclical components of consumption and investment with the
cyclical component of GDP.
(e) How do the patterns compare with the US? What is similar and what is dierent?
237
12.3. Who cares?
12.2 Stabilizing a Very Volatile Economy
We are going to redo Lucas's calculation for an economy that keeps undergoing the Great Depression all
the time. Look up the NIPA GDP data for 1929 (the peak before the Great Depression) and 1933 (the
trough of the Great Depression). Imagine an economy where half the time consumption is at the level
of 1929 and half the time consumption is at the level of 1933. What is the value for the representative
household in this economy of complete consumption stabilization at the average level (which, incidentally,
is pretty close to the level of consumption in 1931)? Try the following values of σ : 2, 5 and 10.
12.3 Heterogeneity and the Value of Stabilization
Imagine an economy that spends half the time in a recession and half the time in an expansion. In
this economy there are two kinds of households: stable and volatile. Stable households consume c every
period, so they are immune to the business cycle. Volatile households consume:
(
c (1 + v) in an expansion
cV olatile =
c (1 − v) in a recession
Suppose that a fraction µ of the households in the economy is volatile and a fraction 1−µ is stable.
(a) What is total consumption in the economy, in an expansion and a recession respectively?
(b) Suppose aggregate consumption is well described by equation (12.3.1). Given a value of µ, what
value of v would make aggregate consumption have uctuations of amplitude equal to f?
For the remaining questions, assume σ = 5, f = 0.012 and µ = 0.05.
(c) What is the value for a stable household of eliminating the business cycle?
(d) What is the value for a volatile household of eliminating the business cycle?
(e) What is the value of eliminating the business cycle for a household that, behind the veil of ignorance,
is not sure whether they are going to be stable or volatile?
12.4 The Value of Growth
Lucas (1988) argued:
Is there some action a government of India could take that would lead the Indian economy to
grow like Indonesia's or Egypt's?
6 If so, what exactly? If not, what is it about the nature of
India that makes it so? The consequences for human welfare involved in questions like these
are simply staggering: once one starts to think about them, it is hard to think about anything
else.
In this exercise we'll try to make sense of what Lucas had in mind.
6
This was back before the India started to grow fast. India's GDP per capita grew less than 1% per year in the 1970s, just
over 3% per year in the 1980s and 1990s and over 5% per year since 2000.
238
12.3. Who cares?
(a) Solve for λ in the following equation:
W (λc0 , g, f ) = W (c0 , g 0 , f )
with g = 0.01, g 0 = 0.02, f = 0.01 and σ = 2.
(b) What does the number you found represent?
239
CHAPTER 13
The Real Business Cycle Model
One of the leading theories of why business cycles happen is the so-called Real Business Cycle model. It has
this name because, unlike the Keynesian-inspired theories we'll look at later, money and nominal variables
play no role in it. The idea of the RBC model is to take the same model we used to think about long run
growth and apply it to questions about the short run. In particular, we are going to maintain the assumption
that all markets are competitive.
We'll proceed as follows. First, we'll write down a simpli ed version of our general equilibrium economy,
which only contains the minimal ingredients that are needed to think about short-run problems. Second, we'll
introduce shocks, exogenous changes in some aspect of the economy. We'll then work out how the endogenous
outcomes in the model economy (employment, output, etc.) react to these exogenous shocks. Finally, we'll
compare the patterns that emerge from the economy reacting to shocks to the empirical patterns we found in
Chapter 12 to assess whether we have a plausible theory of business cycles.
13.1 A Two-Period Model
We are going to look at an even-more-simpli ed version of the two-period, general-equilibrium economy we
studied in Chapter 9.
We are going to assume that the economy starts o having no capital and the production function in period
1 only uses labor:
Y1 = F1 (L)
Why make this assumption? At any point in time, the capital stock is the result of past decisions and cannot
be changed, so we keep things simple by just not having it in the model.
Conversely, we are going to assume that in the second period, the production function only uses capital,
i.e.:
Y2 = F2 (K)
Why the dierence? Remember, we are trying to understand behavior in period 1. The capital stock in period
241
13.1. A Two-Period Model
2 is the result of investment decisions taken in period 1, so it's important to think about what governs these
decisions. We don't have labor in the period-2 production function in order to have one less object to think
about.
Since this economy has competitive markets and no externalities, we know that the First Welfare Theorem
holds. Therefore, in order to nd the competitive equilibrium we can just solve the problem of a ctitious
social planner. We'll do that rst and then we'll go back to thinking about how markets attain this outcome.
The planner solves:
max u (c1 ) + v (l) + βu (c2 )
c1 ,c2 ,l,L,K
s.t.
c1 + K = Y1
(13.1.1)
Y1 = F1 (L)
c2 = F2 (K)
L=1−l
Just like we had in Chapter 9, the planner's objective function is to maximize the utility of the representative
household. The household gets utility from consumption and leisure in period 1 and from consumption in
period 2, discounted. Since we have not included period-2 labor in the model, we also don't model the
household's preferences for period-2 leisure. The rst constraint is the GDP accounting identity: in a closed
economy with no government, total output must be split between consumption and investment, and since the
economy starts with no capital, investment and the period-2 capital stock K are the same thing. The second
constraint says that output results from the production function, which uses labor. The third constraint says
that in the second period all output is consumed (since it's the end of the world, there is no point in investing
for later). The last constraint just says that the household's total time is divided between labor and leisure.
The rst order conditions for this planner's problem are familiar:
v 0 (l)
= F10 (L)
u0 (c1 )
u0 (c1 ) = βF20 (K) u0 (c2 )
These are just like equations (9.1.12) and (9.1.13) from Chapter 9. The rst describes the tradeo between
consumption and leisure in period 1 and the second describes the tradeo between consumption in period 1
and consumption in period 2. In both cases, the planner equates the household's marginal rate of substitution
to the marginal rate of transformation.
From a mathematical point of view, we now have a system of 6 equations (4 constraints and 2 rst order
conditions) and 6 unknowns (Y1 ,c1 , c2 , l , L and K ). Solving this system of equations will tell us everything
that's going to happen in our model economy. At some abstract level, we are done: we have succeeded in
reducing an economic question into a mathematical question. We could in principle just solve this system of
equations in a computer and it would tell us what's going to happen to the economy. However, we don't just
want to compute a solution, we want the model to teach us some conceptual lessons as well. For this, we are
going to have to do a bit more work.
242
13.1. A Two-Period Model
We'll start by simplifying the system of equations a little bit by substituting out c2 and l using the
constraints to get:
Y1 = F1 (L) (13.1.2)
v 0 (1 − L)
= F10 (L) (13.1.3)
u0 (c1 )
u0 (c1 ) = βF20 (K) u0 (F2 (K)) (13.1.4)
0 −1
Y1 = (u ) [βF20 (K) u0 (F2 (K))] + K (13.1.5)
Now we have a system of 4 equations in 4 unknowns: GDP Y1 , consumption c1 , investment K and total
employment L. This system of equations can be a little bit hard to interpret, so we'll start by looking at
them graphically. Notice that by setting up the equations this way, we have only two of the four endogenous
variables showing up in each equation. For instance, equation (13.1.3) tells us about the relationship between
c1 and L, taking as given the values of all exogenous parameters. This makes it possible to plot each of these
equations in a simple graph to help interpret what each of them says. Figure (13.1.1) shows each of these four
relationships.
1
The top left graph shows the production function (13.1.2). This shows a positive relationship between
employment L and total output Y1 . This is simply due to the fact that producing more output requires more
labor.
The bottom left graph shows the consumption-labor tradeo equation (13.1.3). This de nes a negative
relationship between consumption c1 and employment L. What does this mean in economic terms? The
household must, at the margin, be indierent between dedicating a unit of time to leisure or to market
production. A higher level of L means that the marginal unit of time dedicated to production: (i) produces
less output (due to diminishing marginal product of labor, i.e. lower F10 (L)) and (ii) costs more in utility
0
terms (due to diminishing marginal utility of leisure, i.e. higher v (1 − L)). Therefore the household will only
be willing to supply this extra labor if the marginal value of the output it obtains is higher. Due to diminishing
1 c1−σ
It sometimes helps a little bit to use speci c functions to get a little more sense of what's going on, so let's set u (c) = 1−σ
,
1+
(1−l)
v (l) = −θ 1+
F1 (L) = L1−α and F2 (K) = K α . This results in
Y1 = L1−α
θL
= (1 − α) L−α
c−σ
1
c−σ
1 = βαK α−1−σα
1
−σ
Y1 = βαK α−ασ−1
+K
If we set α = 0.5, θ = 1, β = 1 and σ==1 this further simpli es to:
Y1 = L0.5
1 −1.5
c1 = L
2
c1 = 2K
Y = 3K
243
13.1. A Two-Period Model
Fig. 13.1.1: Equilibrium conditions in the RBC model.
marginal utility, u0 (c1 ) can only be higher if c1 is lower. In other words, other things being equal, households
will choose to work more if they feel poorer and so place a higher value on marginal consumption. Changes
in the household's wealth will produce movements along the curve: a household that is wealthier chooses
more consumption and less labor, and vice versa. Changes in the production technology will produce shifts of
the curve: if the marginal rate of transformation between time and consumption changes, the household will
choose a dierent amount of consumption to go along with any given amount of labor.
The bottom right graph shows the consumption-investment tradeo equation (13.1.4). This de nes a
positive relationship between consumption c1 and investment K. What does this mean? The household must,
at the margin, be indierent between dedicating a unit of output to consumption or to investment. A higher
level of K means that the marginal unit of output dedicated to investment: (i) produces less period-2 output
(due to diminishing marginal product of capital, i.e. lower F20 (K)) and (ii) produces less period-2 marginal
244
13.1. A Two-Period Model
utility per unit of output (due to diminishing marginal utility of consumption, i.e. lower u0 (c2 )). Therefore
the household will only be willing to dedicate this extra output to investment if the marginal utility of present
consumption is also lower. Due to diminishing marginal utility, u0 (c1 ) can only be lower if c1 is higher. In
other words, other things being equal, households will choose to invest more only if they are also consuming
more.
Finally, the top right corner shows the relationship between investment and output implied by equation
(13.1.5). This is derived from the GDP accounting identity / market-clearing condition for period-1: Y1 =
c1 + K . We know from (13.1.4) that there is a positive relationship between consumption and investment.
Solving (13.1.4) for c1 we obtain:
−1
c1 = (u0 ) [βF20 (K) u0 (F2 (K))]
and then replacing this in the market clearing condition Y1 = c1 + K we obtain (13.1.5). This de nes a positive
relationship between GDP Y1 and investment K. If investment is higher, the positive relationship between
consumption and investment implies that consumption-plus-investment will be higher too. If households want
to invest more and consume more, this is only possible if they also produce more.
We are going to be interested in solving the system of equations (13.1.2) - (13.1.5) in order to gure out
how much output, consumption, investment and employment there's going to be in our model economy, and
how these change in response to dierent things. There is more than one way to solve these equations. Figure
13.1.2 shows a way to solve the equations graphically.
1. Start from the top-left panel which shows the production function (13.1.2). If we guess some level for
GDP Y1 , this immediately implies a level of employment L needed to produce this amount.
2. Now move to the bottom-left panel, which shows the consumption-labor tradeo (13.1.3). Given the
level of employment L, satisfying this condition implies a level of consumption c1 .
3. Now move to the bottom-right panel, which shows the consumption-investment tradeo (13.1.4). Given
the level of consumption c1 , satisfying this condition implies a level of investment K.
4. Finally circle back to see if our guess for GDP was correct by looking up in the top-right panel what level
of GDP Y1 is consistent with K. If it coincides with our original guess, then we have found a solution.
Otherwise we need to adjust our guess.
One thing to notice is that we could have gone through the graphs in a dierent order: instead of 1-2-3-4
we could have done, for instance, 1-4-3-2. The only question is whether we go back to the original guess at
the end. If the answer is yes, then all the equilibrium conditions are satis ed.
Figure 13.1.3 shows graphically what happens if the guess for Y1 is not correct. High Y1 implies high L
(through the production function). High L implies low c1 (through the labor-consumption tradeo ). Low
c1 implies low K (through the consumption-investment tradeo. Low c1 and K imply low Y1 (through the
condition Y1 = c1 + K ), so the original guess does not satisfy the equilibrium conditions.
245
13.2. Markets
Fig. 13.1.2: Graphical Solution of the RBC model.
13.2 Markets
So far we have looked at our model economy as though it was administered by a social planner. The rst
welfare theorem justi es this approach: we know that competitive equilibrium allocations will coincide with the
social planner's decisions. But we are also interested in understanding how markets bring about this outcome.
In particular, we want to know what are the prices that are associated with the competitive equilibrium
allocations.
These prices can be recovered from the rst order conditions of the representative household and the
representative rm, which we obtained in Chapter 9. Conditions (9.1.10) and (9.1.7) imply that the wage
level must be equal to the marginal product of labor, and also to the marginal rate of substitution between
consumption and leisure:
v 0 (1 − L)
w = F10 (L) = (13.2.1)
u0 (c1 )
246
13.2. Markets
Fig. 13.1.3: An incorrect guess for Y1 results in an inconsistency.
Condition (9.1.9) implies that the rental rate of capital must be equal to the marginal product of capital:
rK = F20 (K) (13.2.2)
and since period 2 is the end of the world, it's as though we had δ = 1, so conditions (9.1.11) and (9.1.13)
imply that the real interest rate is:
u0 (c1 )
1 + r = F20 (K) = (13.2.3)
βu0 (c2 )
Equations (13.2.1)-(13.2.3) allow us to think about how prices, in addition to quantities, behave in the RBC
model, which will be useful both to understand how the model works and to compare it to evidence.
247
13.3. Productivity Shocks
13.3 Productivity Shocks
A productivity shock is an unexpected change in the productive capacity of the economy, i.e. a change in
the production function. Productivity shocks are one of the main shocks that have been proposed as possible
drivers of business cycles. Let's use our model to think through how the economy would react to a productivity
shock.
Let's imagine that there is an improvement in the production function, which goes from Y1 = F1 (L) to
Y1 = AF1 (L), where A is some number greater than 1. This shock may represent a technological discovery,
an improvement in some policy, or anything that makes the economy more productive. Let's assume that this
shock is short-lived, so that there is no change in F2 .
The productivity shock aects the equilibrium conditions in two places:
Y1 = AF1 (L)
v 0 (1 − L)
= AF10 (L)
u0 (c1 )
u0 (c1 ) = βF20 (K) u0 (F2 (K))
−1
Y1 = (u0 ) [βF20 (K) u0 (F2 (K))] + K
The shock aects the production function: for the same amount of labor, the economy gets more output,
which it can distribute between consumption and investment. It also aects the labor-consumption decision
because it aects the marginal product of labor and thus the marginal rate of transformation between time
and consumption. Figure 13.3.1 shows how this aects the equilibrium.
The shock aects two of the four curves in the gure. The production function curve moves up: for any
level of L, Y1 moves up. Second, the consumption-labor curve moves to the right: dedicating one unit of time
to production produces more goods than before, so other things being equal, the household should be willing
to work more. The consumption-investment-output curves do not move, so any movement must be a shift
along these curves.
Let's trace out the eect of this shock on the equilibrium. It will be useful to do it in the order 1-4-3-2.
Start with panel 1. The most direct eect is that Y1 rises. If we trace out the rise of Y1 from panel 1 to
panel 4, we see that higher Y1 must necessarily imply higher investment K. Using panel 3, this implies higher
consumption. If we then go to panel 2, we see that there are two opposing eects on L. First, the curve shifts
to the right: holding c1 xed, the household would choose higher L. However, there is also a movement along
the curve, because c1 increases. This lowers L. In this example, the net eect is that L rises, but it's possible
to construct examples where L falls.
What's going on economically? The economy has become temporarily more productive. The total eect
of this on the economy is the result of the various ways in which the household wants to react to this. The
representative household wants to:
1. Increase consumption, because more productivity makes the household feel wealthier.
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13.3. Productivity Shocks
Fig. 13.3.1: The economy's reaction to a productivity shock.
2. Increase investment, in order to smooth out the increase in consumption between the present and the
future.
3. Work more, to take advantage of the improved technology for transforming time into consumption goods.
4. Enjoy more leisure, because more productivity makes the household feel wealthier.
The rst two forces are unambiguous: the household will react by increasing consumption and investment.
The last two forces point in opposite directions. The substitution eect pushes the household to work more
while the income eect pushes the household to work less. This is the same issue that came up in Chapter
7. If we imagine that the productivity increase is temporary, then the income eect is likely to be small and
the substitution eect dominates, persuading the household to work more. This is the case depicted in Figure
13.3.1.
The RBC model with productivity shocks therefore oers us one possible account of how and why business
cycles take place. If our model economy undergoes a productivity shock, then productivity, GDP, consumption,
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13.4. Other Shocks
employment and investment will all move in the same direction. In this account, an economic expansion is a
time when households choose to work more in response to temporarily high productivity, which, by equation
(13.2.1), translates into temporarily high wages. Conversely, a recession is a time when households choose to
work less in response to temporarily low wages.
13.4 Other Shocks
Just like we did with productivity shocks, we can also ask how the economy would react to other kinds of
shocks.
Impatience Shocks
Let's imagine that households suddenly decide they really want to consume now rather than later. We'll see
later that eects of this kind can be important in Keynesian models. We are going to model higher impatience
as a fall in β: the value that the households place on the future falls, so they really want to prioritize the
present.
In our system of equations, β enters equations (13.1.4) and (13.1.5) More impatient households place
lower value on the future so, for any given level of consumption, the level of investment they choose is lower.
Graphically, this means that the curves in panels 3 and 4 shift to the left.
The net eect is shown in Figure 13.4.1. Households consume more which, through equation (13.1.3),
means they also work less. Less work means lower total output, and since consumption has risen, investment
must fall. Overall, this does not look like the patterns in business cycles: consumption goes up while output,
employment and investment go down. Empirically, all these variables tend to move together.
Optimism
Let's imagine that households become optimistic about the future. How would this aect the economy today?
We are going to model optimism as a rise in future productivity, so that the production function for period 2
becomes:
Y2 = AF (K)
where A is a number greater than 1. For our purposes, it doesn't really matter whether the optimism is
justi ed, i.e. whether, once period 2 comes along, productivity does in fact rise. Since we are focusing on
how the economy behaves in the short run, all we are interested is how the expectation of future productivity
aects decisions in the present.
In our system of equations, A enters through the consumption-investment equation (13.1.4) and the market
clearing condition(13.1.5), which contain the term:
β AF 0 (K) u0 (AF2 (K))
| 2{z } | {z }
increases with A decreases with A
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13.4. Other Shocks
Fig. 13.4.1: The economy's reaction to an increase in impatience.
The net eect of A on the relationship between c1 and K is ambiguous since A aects the consumption-
investment decision in two opposite ways. First, higher period-2 productivity makes investment more attractive
by raising the marginal product of capital. Other things being equal, this leads to more investment (and less
consumption). But other things are not equal. For any given level of investment, higher period-2 productivity
means higher period-2 consumption, and therefore lower marginal utility of period-2 consumption. This makes
the household want to smooth consumption by raising period-1 consumption (and lowering investment). Figure
(13.4.2) shows an example where the rst eect dominates and investment rises but the net eect could go
either way.
In order to liberate resources for investment, households end up consuming less. Since households are
cutting back on period-1 consumption, their marginal utility of consumption rises, so households react by
working more i.e. they move along the curve in panel 2. Since households work more, GDP rises.
This pattern captures some of the features the we observe empirically in business cycles: output, em-
ployment and investment move in the same direction. However, it misses one very important dimension:
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13.4. Other Shocks
Fig. 13.4.2: The economy's reaction to optimism about future productivity.
consumption moves the wrong way. Since the pro-cyclicality of consumption is such a central fact of business
cycles, we must conclude that this type of shock in an RBC model cannot be the main driver of business
cycles.
Figure (13.4.2) focuses on a case where, other things being equal, higher A makes investment rise. Could
it be that making the opposite assumption xes the problem? No. In the case where the consumption-capital
shifts to the left, consumption does indeed rise. But then movement along the labor-consumption curve implies
that L falls, which means that GDP must fall, and investment falls as well. Again, we would have consumption
moving in the opposite direction as everything else.
Laziness or Taxes
Let's imagine that households become lazy. More precisely, they change their relative preference for consump-
tion and leisure. We are going to model an increased desire to enjoy leisure by saying that preferences change
252
13.4. Other Shocks
to:
u (c1 ) + θv (l) + βu (c2 )
where θ is a number greater than 1.
In our system of equations, θ enters through the labor-consumption equation (13.1.3), which becomes:
θv 0 (1 − L)
= F10 (L) (13.4.1)
u0 (c1 )
Graphically, this is a shift to the left of the labor-consumption curve in panel 2: for any given level of
consumption, households want to supply less labor.
Fig. 13.4.3: The economy's reaction to an increase in preference for leisure or an increase in income taxes.
The net eect is shown in Figure (13.4.3). Households work less, which lowers output. This fall in output
results in both a fall in consumption c1 and a fall in investment K, i.e. a movement along the consumption-
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13.5. Assessment
investment curve in panel 3.
This type of shock does produce a reaction that looks like a recession: employment, output, consumption
and investment all fall together. On the other hand, the account of recessions that this type of shock leads
to is not terribly satisfactory: recessions are what happens when everybody simultaneously decides that it's
time to get some rest.
One alternative that is mathematically equivalent to an increase in laziness is an increase in taxes. Re-
member from equation (7.2.5) in Chapter 7 what happens when you introduce a labor-income tax into a
consumption-leisure choice: the worker equates the marginal rate of substitution to the after-tax wage w (1 − τ )
rather than the pre-tax wage w. Since the representative rm is still equating the pre-tax wage to the marginal
product of labor, equation (13.1.3) becomes:
v 0 (1 − L)
= w(1 − τ ) = F10 (L) (1 − τ )
u0 (c1 )
1
v 0 (1 − L)
⇒ 1−τ 0 = F10 (L) (13.4.2)
u (c1 )
Comparing equations (13.4.1) and (13.4.2) we can see that for every laziness shock θ>1 there is a tax-rate
shock τ > 0 such that the eects of either of these shocks are equivalent.
2 While perhaps more appealing
than pure changes in preferences, a theory of business cycles based on changes in tax rates has one empirical
shortcoming: we simply don't change tax rates so often, and the changes we do make are not so strongly
correlated with business cycles.
13.5 Assessing the RBC model
Why Some Kinds of Shocks Produce Business Cycles but not Others
As pointed out by Barro and King (1984), the failure of either impatience shocks or optimism to produce
something that looks like a business cycle in an RBC model has a common source: the labor-consumption
decision. As long as nothing changes how people trade o time against consumption goods (i.e. as long as
nothing causes a shift of the labor-consumption curve in panel 2 of Figure 13.1.2) then anything that raises
consumption must lower employment. After any change that does not aect the labor consumption tradeo,
the representative household either: (i) feels richer and thus consumes more and works less or (ii) feels poorer
and thus consumes less but works more.
The reason why productivity shocks or changes in taxes could, in principle, produce a business-cycle-like
reaction is that they aect how households perceive the tradeo between dedicating time to work or to leisure.
The other shocks we looked at do not aect this margin directly and therefore cannot make consumption and
employment move in the same direction.
One maintained assumption in all of this analysis is that the labor market is competitive: households can
choose, without any restrictions, how much labor to provide each period at the market wage. Therefore this
2
To make the equivalence exact we need to assume that this is a pure tax-rate shock with no change in government spending,
so that the government gives back all the revenue it collects in the form of a lump-sum transfer.
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13.5. Assessment
model can be used to think about employment but not about un employment, since in the model no one is ever
unemployed. Later on we will consider the possibility that the labor market is not competitive, so households
are not necessarily choosing how much to work every period. This will open up other possibilities of why
employment and consumption can move together.
Quantitative Assessment
One way to assess whether the RBC is a good theory of business cycles (and, more generally, to assess models
of the macroeconomy) is the following procedure, advocated by Prescott (1986):
1. Construct a full version of the model. In this chapter we have looked at a simpli ed two-period version
of the model but in a full assessment one would use a model with an in nite horizon, labor and capital
in every period, investment, depreciation, etc. and perhaps other ingredients as well.
2. Set values for the parameters of the model, based as much as possible on micro economic data. For
instance, one would use data on household labor supply to set values for the parameters that govern
preferences for consumption and leisure.
3. Do a growth-accounting exercise like we did in Chapter 4 with data from the US economy (or some other
country) and nd the Solow residuals for each period (usually one quarter). The point of this is that
if we are trying to assess a model that says business cycles are the result of changes in productivity we
need to have a sense of how large these changes in productivity actually are. Implicitly, we are assuming
that measured Solow residuals are accurate measures of exogenous technological shocks.
4. Simulate how the model economy would respond to the types of technological shocks extracted in the
previous step.
5. Measure the behavior of the variables of interest in the model economy (GDP, employment, etc.) and
compare it to the same measurements taken from the real economy.
Applying this procedure, Kydland and Prescott (1982) and others found that the model RBC economy could
2
produce about
3 of the volatility of GDP observed in the real data. Furthermore, the model economy could
reproduce the main correlations that de ne business cycles: productivity, output, employment, consumption
and investment all move together, investment is more volatile than GDP and consumption less so. They
concluded that, while not perfect, the RBC model is a satisfactory approximation to how the real economy
behaves.
Policy Implications
If the RBC model is correct, then the implications for macroeconomic policy are profound. In the RBC
model, the First Welfare Theorem holds. This means that no social planner, no matter how unrealistically
powerful, could improve upon what the market economy is doing. In particular, nothing should be done to
prevent or stabilize business cycles. Business cycles are just the e cient way for the economy to respond to
changes in productivity. When productivity rises, it's a good time to produce goods, so households should
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13.5. Assessment
work more; when productivity falls, it's a bad time to produce goods, so households should enjoy more leisure.
Any attempt to stabilize employment or output, even if it could succeed, would reduce the welfare of the
representative household.
When it was rst proposed, this conclusion ran very counter to decades of thinking about macroeconomic
policy, which had made stabilizing the business cycle one of its priorities. At the very least, the model forces
us to ask harder questions. The model proves that observing economic uctuations does not imply that
something is wrong and needs to be xed. Economic uctuations can be perfectly consistent with a world in
which nothing is wrong. Therefore any argument for trying to stabilize the business cycle must rst make the
case of why such stabilization is desirable. We'll come to some of these arguments later on.
Furthermore, if the model is correct, conventional macroeconomic policy might not work anyway. One of
the main means by which policymakers attempt to stabilize the economy is by using monetary policy (we'll
come to some of the reasons for this later on). But the RBC is a completely real model: there is no room for
monetary policy because there is no money at all in the model. Hence, if the RBC model is right, it would
be a good idea to close central banks, or at least to drastically limit what they do. According to the model,
anything they do is irrelevant and, if it were to have an eect, it would be counterproductive.
Criticisms
One of the earliest criticisms made of the RBC model is that many economists just don't nd the mechanism
for generating recessions plausible.
3 If we interpret productivity literally as the result of technological progress,
some economists don't think it's plausible that the rate of technological progress can uctuate so much over
the course of a year or two to result in the types of business cycle movements we observe. Furthermore, the
example in Figure 13.3.1 shows what happens after an improvement in productivity. We at least know that
technology does improve over time, even if we disagree about how smooth this progress is. In order for the
model to produce a recession, there needs to be a fall in productivity. What exactly is this fall supposed to
represent? Can an economy, from one year to the next, lose the ability to successfully employ technologies
that it used the year before? Most defenders of the RBC model argue that productivity shocks should be
interpreted less literally. They argue that other things like changes in regulations or speci c problems at
individual large companies can make the economy behave as if it had experienced technological regression.
Another criticism of the model has to do with the parameter values that one needs to use in order to get
the model to work quantitatively. In particular, there is much disagreement as to the right numbers to use
to describe household's willingness to substitute between consumption and leisure, which in turn governs the
elasticity of labor supply.
4 For the RBC model to produce large uctuations in employment, the elasticity
of labor supply needs to be high. Remember, all the changes in employment in the model are the result of
households willingly changing how much they work. In order to produce the changes in employment that we
observe, households must respond strongly to changes in wages. Many economists regard the elasticity usually
used in the RBC model as contradictory with microeconomic evidence on household behavior. Much of the
debate is about the right way to derive the response of the aggregate labor supply on the basis of individual
3
Many of these criticisms can be found in Summers (1986).
4
We already encountered this issue in Exercise 7.5 when we thought about the response of labor supply to dierent tax rates
in Europe and the US.
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13.5. Assessment
labor supply.
The simulations of the RBC model often focus on how quantities move: output, employment, investment,
etc. But, using equations (13.2.1) and (13.2.3), one can also look at what the model implies for how prices
(wages and interest rates) move. One criticism of the model is that it doesn't t the behavior of prices as
well as it ts the behavior of quantities. In particular, the model implies that wages should be more variable
than we actually observe. The evidence in presented in Table 12.1 indicates that real wages do not co-move
strongly with the business cycle. But it's also possible that standard ways of measuring wage movements
underestimate how much wages actually move.
In Chapter 7 we made a distinction between being unemployed (not working but looking for work) and
being out of the labor force (not working and not looking for work). In the RBC model, there is never any
unemployment, since the labor market is competitive and everyone who wants to work nds a job. The
model can therefore provide a theory of changes in employment but not of changes in unemployment. Many
economists consider changes in un employment as a central feature of business cycles and therefore consider
the RBC unsatisfactory on those grounds.
5
Another line of criticism of the RBC model has to do with the practice of treating measured Solow residuals
are exogenous productivity. One of the reasons why this might be inaccurate is that mismeasured capacity
utilization can contaminate measurement of the Solow residual. To see why that is, imagine a restaurant at
a time when business is slow. The restaurant still has its usual level of capital (the building, the kitchen
equipment, etc.) and all its employees. However, it is producing fewer meals than usual because customers are
not showing up. If we go back to equation (5.4.2) that describes how one would construct a Solow residual,
we'll see that for this particular restaurant:
gY = Capital Share × 0 + Labor Share × 0 + Solow Residual
Since neither the labor nor the capital it employs has changed, this accounting procedure would attribute all
the change in the number of meals the restaurant produces to lower productivity. At a literal level, this is
not wrong: the restaurant is being less productive by producing fewer meals with the same amount of labor
and capital. However, this lower productivity is endogenous, it's the result of whatever it is that is causing
business to be slow so that factors of production are not being fully utilized, not of the restaurant having
become technologically worse at producing meals. Later on we'll study models where business can be slow
for the overall economy, not just a single restaurant. If these models are right, then the practice of treating
measured Solow residuals as exogenous technological changes is inaccurate.
6
5
We saw one theory of unemployment in Chapter 7, based on modeling the search process by which rms nd workers and
workers nd jobs. Inserting this type of model of unemployment into an RBC model can account for why there is unemployment
but has a hard time in getting unemployment to change very much with productivity shocks. See Shimer (2005) for a discussion
of this point.
6
Basu et al. (2006) attempt to correct for this by constructing a utilization adjusted measure of TFP.
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13.5. Assessment
Exercises
13.1 Prices in the RBC Model
Consider the two-period RBC model summarized by equations (13.1.2)-(13.1.5).
(a) Suppose there is a temporary, positive, productivity shock in period 1 only. What happens to the
wage w and the real interest rate r? Explain in words why it is that each of the prices change.
(b) Suppose there is temporary increase in households preference for leisure. What happens to the wage
w and the real interest rate r? Explain in words why it is that each of the prices change.
13.2 Stabilizing a Real Business Cycle
Suppose the economy is well described by the Real Business Cycle model. The economy has just had a
positive shock to aggregate productivity.
(a) Suppose the government wants GDP to remain the same as it would have been without the pro-
ductivity shock. Describe one possible policy instrument that could be used for this. Explain why
it works.
(b) Suppose instead that the government wants employment to remain the same as it would have been
without the productivity shock. Does the policy need to be applied more intensely or less intensely?
Why?
13.3 Good News
Consider the two-period RBC model summarized by equations (13.1.2)-(13.1.5). Suppose that the rep-
resentative household develops a very particular form of optimism about the future. It believes that the
period-2 production function is:
Y2 = F2 (K) + A
where A is some positive number.
(a) How does this type of optimism dier from the one we looked at in Section 13.4?
(b) What will happen to employment, output, consumption, investment, the wage and the real interest
rate?
(c) Does this type of optimism produce something that looks like a business cycle? Why or why not?
13.4 Government Spending
Consider an economy that is well described by the RBC model with one modi cation: there is a govern-
ment that spends G in period 1. It pays for this by collecting lump-sum taxes from the representative
household (Ricardian equivalence holds, so we don't need to specify when the government collects these
taxes). G does not enter the representative household's utility function.
(a) Where in the system of equations (13.1.2)-(13.1.5) would G show up? Explain.
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13.5. Assessment
(b) Suppose there is an increase in G. What happens to output, consumption, employment and invest-
ment? What happens to wages and interest rates? Show this graphically and explain in words what
is going on.
13.5 Fish
In the town of New Oldport in Maine, the main economic activity is shing. Crews sail out to New
Oldport Bay in the morning and return in the evening with their catch. It's a very unstable occupation
because it depends on the migration patterns of cod and bass, which are erratic. Sometimes (when a
large school of sh is passing by) sh are plentiful in the New Oldport Bay; otherwise they are scarcer.
Sonar tracking gives residents a reasonably reliable daily estimate of the number of sh in the bay. On
average, large schools of sh come by the Bay a few days a year.
(a) Write down two production functions for the New Oldport economy: one for when sh are plentiful
and one for when they are not.
(b) Write down the problem of a household that resides in New Oldport and has to decide how much
to consume and how much to work each day. Derive rst-order conditions (you may skip steps if
you want).
(c) Suppose you compare a day when sh are plentiful and one when they are not. On what day will
workers choose to work longer hours?
(d) Suppose in 2018 the migration patterns of cod and bass change such that now sh in the New
Oldport Bay are always plentiful. How will the number of hours that households choose to work on
a typical day in 2018 compare to the number of hours they worked on a plentiful- sh day in 2017?
How will they compare to the average daily number of hours in 2017?
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CHAPTER 14
The New Keynesian Model
14.1 A Historical and Methodological Note
In this chapter we'll study a simple version of the so-called New Keynesian model. For a long time, starting
around the 1930s, the ideas of Keynes (1936) were the dominant way to think about business cycles. We'll be
more precise about this, but broadly speaking the main idea in Keynes' work was that output and employment
could fall short of their normal level due to a lack of demand.
Keynes himself made his arguments in prose, rejecting the mathematical formulations that are preferred
nowadays. As a result, his writings can be a little bit hard to interpret. The so-called IS-LM model, rst
proposed by Hicks (1937), is one popular mathematical representation of Keynes' ideas. There is some debate
as to whether the IS-LM model accurately represents what Keynes really meant. The best answer is probably
who cares?: the merit of the model or lack thereof must be judged on its own and not on the basis of its
faithfulness to Keynes.
The RBC model that we looked at in the last chapter, developed around the late 1970s, was a departure
from the IS-LM model on two dimensions. First, a methodological dierence. The IS-LM model started by
describing relationships between aggregate variables (such as total investment and total consumption) without
being totally precise about how those relationships came about. In contrast, everything that happens in the
RBC model is a result of households and rms making decisions in a way that is explicitly modeled. This
type of model is sometimes known as a microfounded model, because it is built on an explicit microeconomic
model of decision-making. Arguably, such an approach makes models clearer because it allows one to keep
track of where each result comes from. It also allows one to test the model against a wider range of evidence,
testing not just the implications of the model for aggregate variables but also its implications for microeconomic
outcomes. Finally, since one of the building blocks of a microfounded model is a utility function, the model
can be used to evaluate the welfare of the household in the model in an internally consistent way.
The second dierence between the RBC model and the earlier IS-LM model is substantive. In the
Keynesian/IS-LM account, recessions happen because things go wrong with markets, which implies that
perhaps the government should do something about it. Later on we'll think more about exactly what it is
that goes wrong and what the government can do to x it. In the RBC model, instead, business cycles do not
re ect any failure of the market economy and the government should not do anything about them, even if it
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14.2. Monopoly Power
could.
The New Keynesian model that we'll study in this chapter is, like the RBC model, a microfounded model
where everything that happens results from decisions that are explicitly modeled. The New in New Keynesian
comes from this methodologically more modern way of thinking. In contrast to the RBC model, it is a model
economy where things do go wrong in speci c ways. The Keynesian in New Keynesian come from the fact
that the speci c types of market failures in the model are very similar in spirit to the earlier generation of
Keynesian models.
The starting point for our New Keynesian model will be the RBC model summarized by equations (13.1.2)-
(13.1.5). We'll then add three ingredients: monopoly power, sticky prices and the theory of money markets
we developed in Chapters 10 and 11. Adding these ingredients will lead to a model of the economy made up
of two equations that we'll label IS and LM, i.e. a version of the IS-LM model.
14.2 Monopoly Power
The Microeconomics of Monopoly Power
Let's review from microeconomics how a monopolistic rm sets its price. Suppose the rm faces a demand
function q (p), meaning that it will sell q units if it sets the price p. Let the total cost of producing q units be
c (q). Then the rm's problem is:
max pq (p) − c (q (p))
p | {z } | {z }
Total Revenue Total cost
The rst-order condition for this problem is:
pq 0 (p) + q (p) − c0 (q (p)) q 0 (p) = 0
q (p)
⇒ p+ 0 = c0 (q (p))
q (p)
| {z } | {z }
Marginal Revenue Marginal cost
The monopolist will equate marginal revenue from selling an extra unit to the marginal cost of producing it.
Note that since q 0 (p) < 0 , marginal revenue is below the price. In order to sell an extra unit the monopolist
needs to lower the price; the price cuts on all the units it was going to sell anyway subtract from what it earns
on the last unit.
We can further rearrange the rst order condition to get:
q (p)
p − c0 (q (p)) = − p
q 0 (p) p
1
= p
η
η
p = c0 (q (p)) (14.2.1)
η−1
| {z } | {z }
Marginal cost Markup
262
14.2. Monopoly Power
0
where η ≡ − qq(p)
(p)p
is, by de nition, the elasticity of demand. Since
η
η−1 > 1, formula (14.2.1) says that the
monopolist will set its price above marginal cost. The dierence between price and marginal cost is called a
markup. Formula (14.2.1) also tells us that the markup will depend on the elasticity of demand faced by
this rm. If demand is very elastic (i.e. η is very high), then the markup will be small. In the limit of η = ∞,
we are back to the case of perfect competition, where the rm sets price equal to marginal cost.
Figure (14.2.1) illustrates this principle, showing how two dierent monopolists with dierent demand
elasticities set their price. In the left panel, the monopolist faces a relatively inelastic demand. This means
that in order to sell an additional unit it needs to lower the price a lot. As a result, the marginal revenue
curve is far below the demand curve. The monopolist chooses a low quantity and a large markup. In the right
panel, the monopolist faces a very elastic demand curve, so it sets a lower markup.
Fig. 14.2.1: The monpolist's price-and-quantity decision.
Introducing Monopoly Power into the RBC Model
We are going to assume that markets are not perfectly competitive. There are many ways for markets to be
not fully competitive and we are going to model a simple one. Imagine that instead of selling their labor to
the representative rm in a competitive labor market, each individual worker operates their own small rm.
This small rm can produce output in period 1 with the same production function we had before:
Y1 = F1 (L)
Each of these small rms produces a slightly dierent good, and this dierentiation gives them some market
power. We are going to assume that all these small rms are symmetric and face identical demand curves. Let
η denote the elasticity elasticity of demand that each of them faces. Therefore we know that they are going
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14.2. Monopoly Power
η
to set price at marginal cost times a markup
η−1 . What exactly is the marginal cost of production for one
of these monopolistic producers? The marginal cost is the answer to the question: if I tell the producer to
produce one more unit of his speci c good, what do I have to give him in compensation to leave him exactly
indierent? Let's construct an answer to this question.
F10 (L) is the marginal product of labor. This means that one marginal unit of labor will produce F10 (L)
1
additional units of output. Therefore, producing one extra unit of output will require
F10 (L) additional units
of labor. Since v 0 (1 − L) is the marginal utility of leisure and supplying additional labor reduces leisure,
1 0
supplying enough labor to produce one marginal unit of output costs the household
F10 (L) v (1 − L) units of
utility. How much extra consumption would the household need to receive to be exactly compensated for the
disutility of supplying this extra labor? If u0 (c1 ) is the marginal utility of consumption and the household
1 0
needs to receive
F10 (L) v (1 − L) extra units of utility, then it requires:
v 0 (1 − L)
Real Marginal Cost = (14.2.2)
u0 (c1 ) F10 (L)
We sometimes also want to express this cost in nominal terms, i.e. how many dollars does the worker
require in order to compensate the marginal disutility cost of producing an extra unit of output. Let p be the
price of the average good produced by all the dierent small rms. Then the nominal marginal cost is:
v 0 (1 − L)
Nominal Marginal Cost =p
u0(c1 ) F10 (L)
η
How will an individual small rm set its price? It will just set a markup of
η−1 over the marginal cost. In
nominal terms, this means that rm i will set its price pi at:
v 0 (1 − L) η
pi = p (14.2.3)
u0 (c1 ) F10 (L) η − 1
But we know that all of these rms are symmetric: they all have the same production function and face
the same elasticity of demand. It stands to reason that they will all set the same price, so the average price p
will be the same as the price of any one of them: pi = p for all i. Using this in (14.2.3) implies:
v 0 (1 − L) η
1=
0 0
u (c1 ) F1 (L) η − 1
v 0 (1 − L) η−1
⇒ 0
= F10 (L) (14.2.4)
u (c1 ) η
Contrast equation (14.2.4) with condition (13.1.3) from the RBC model. The equations are the same
η−1
except for the term
η < 1. In fact, equation (14.2.4) is exactly equivalent to (13.4.2), which describes how
the economy reacts to an increase in taxes. Monopoly power ends up having the exact same eect as a tax on
1
labor income, with a tax rate of τ= η.
What's going on? Each worker is acting as a monopolist: reducing supply in order to maintain a high
price. But since they are all doing it at the same time, none of them succeeds in raising the relative price of
their own product and the aggregate eect is just that they all reduce supply relative to what would happen
264
14.3. Sticky Prices
η−1
in a competitive market. Notice that as η → ∞, the term
η converges to 1 and we are back to competitive
markets.
In terms of how the model economy reacts to various shocks, having monopoly power in the model does
not make too much dierence. We just start from a situation where the labor-consumption curve is shifted to
the left, as in Figure (13.4.3). Any further shock will have the same consequences that they do in the plain
RBC model.
One major dierence that monopoly power does make is that the First Welfare Theorem no longer holds.
In an economy where there is monopoly power, an all-powerful social planner would like to implement a
dierent outcome compared to what the market equilibrium is bringing about. In particular, a social planner
would like to undo the eects of monopoly power. The planner would like to get workers to work a bit more,
knowing that the value of the goods they'd produce exceeds the disutility that they would incur.
In discussions of economic policy it is often taken as obvious that increasing employment (creating jobs)
is desirable. The First Welfare Theorem is a useful reminder that the desirability of higher employment is not
obvious: if markets are competitive, increasing employment will lower welfare because the value of additional
consumption does not make up for the loss of leisure. Conversely, this model with monopoly power gives a
precise sense in which the commonly-held view is correct: if there is monopoly power, the value of additional
consumption does make up for the loss of leisure and higher employment is desirable.
14.3 Sticky Prices
How Monopolists Want to Adjust Prices
Suppose, that, for whatever reason, the demand curve faced by one of our monopolistic producers shifts up.
How should we expect the rm to adjust its price? We know that the rms wants to maintain a markup of
η
η−1 over marginal cost, and let's assume that the elasticity of demand η hasn't changed. Then the rm will
want to change its price in proportion to the change in its marginal cost. Figure 14.3.1 shows an example of
this, where demand rises but the elasticity doesn't change. The monopolist increases the quantity it supplies;
since marginal costs rise, then in order to maintain a constant proportional markup, the monopolist raises its
price.
Note that the key to why the monopolist raises its price is that its marginal costs are increasing. If
marginal costs were constant, the monopolist would react to the increase in demand by increasing output
but keeping the price constant. How do we know that marginal costs are indeed increasing? In our model of
owner-operated monopolistic rms, we can see this follows from equation (14.2.2). There are three eects, all
pointing in the same direction:
1. Diminishing marginal product of labor (decreasing F10 (L)). As the worker works more, it requires more
additional labor to produce each successive unit of output.
2. Diminishing marginal utility of leisure (v
0
(l) decreasing in l and therefore v 0 (1 − L) increasing in L). As
the worker works more, each additional unit of leisure that it gives up is more valuable than the previous
one because there are fewer of them left. A tired worker nds additional work especially unpleasant.
265
14.3. Sticky Prices
Fig. 14.3.1: The monpolist's
reaction to higher demand.
3. Diminishing marginal utility of consumption (decreasing u0 (c1 )). As the worker works more and earns
more income, he increases consumption; this makes him place lower value on the additional units of
consumption he'll be able to aord by working more.
To see more the forces at work more concretely, imagine that our worker- rm is a freelancer that sets his
own prices, like a wedding photographer, a plumber or a maths tutor. He reacts to higher demand by saying:
I'm getting a bit tired of working so much, so I'll raise my prices. I know this will limit how much my
business expands, which will cut into my total income, but I'm doing ne and the extra time o will make it
worthwhile. Conversely, he would react to lower demand by saying: I'm not getting enough work and I'll
have trouble paying my bills, so I'll lower my prices to get more clients and increase total revenue. It'll mean
more work than if I kept my prices as they are, but since business is slow I have a lot of free time so I don't
mind.
Introducing Price Stickiness
The key assumption in the New Keynesian model is that prices are in exible. Ultimately, it is an assumption
about the timing of decisions. We are going to assume that rms set prices before knowing exactly what's
going to happen in the macroeconomy. Once a rm has set its price, we'll assume that it's in exible: the rm
cannot change it when it faces a change in demand for its product. Sometimes this is referred to as prices
being sticky.
There are many reasons why prices might be sticky. There could be contracts such as rental contracts,
service agreements or union pay scales that prevent price changes for a speci ed period. There could be costs
to physically implementing price changes, for instance printing new menus (these types of costs are sometimes
266
14.3. Sticky Prices
known as menu costs). It could be that it takes time for rms to realize that circumstances have changed in
a way that would make them want to change prices, perhaps because they are rationally choosing not to pay
too much attention to macroeconomic news.
Price stickiness might not seem like a big deal but it has major consequences for how the economy as a
whole behaves. Let's start by seeing how a monopolist whose price is sticky reacts to a change in demand for
its product. Figure 14.3.2 shows the same increase in demand as Figure 14.3.1 for a rm whose price is sticky.
If the rm could change prices when demand increases, it would choose price pF (the F stands for exible)
F
in order to maintain a constant markup, and quantity would increase up to q . This is exactly what Figure
14.3.1 shows. If instead the price is stuck at pS (the S stands for sticky), then the producer would produce
enough to satisfy all the demand he faces at price pS , so quantity would increase all the way to qS . The fact
the prices don't react means that the quantity produced reacts more than it would under exible prices.
Fig. 14.3.2: A monopolist
with sticky prices.
Going back to the example of the freelancer, imagine he has already advertised his prices for the year and,
since a lot of people have seen them, he cannot easily change them. He then reacts to demand by simply
adjusting how hard he works: if more clients want to hire him, then he works more; if fewer clients want to
hire him, he works less and enjoys more leisure. Notice that even though higher demand makes the freelancer
tired, he does not want to turn away clients, because for each additional unit, it's still true that price is higher
than marginal cost, so he's happy to supply it.
1
1
If demand were to rise much more then there would reach a point where marginal costs are so high (when the worker is very
tired) that the worker- rm with sticky prices would want to turn away clients, but we'll assume we are not at that point. See
Exercise 14.1.
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14.4. IS-LM
14.4 Putting Everything Together
Now we are going to put all the ingredients together to work out how our model economy will behave. We
did the same thing when we studied the RBC model, but there we had the advantage that the First Welfare
Theorem applied. Therefore we could simply imagine that a social planner was choosing the allocation and
study what the planner's decisions looked like. Here the First Welfare Theorem doesn't hold so we need to be
careful about who makes each decision and how they all t together.
Household, Firm and Government Decisions
1. First of all, each individual small rm sets a price, understanding that the price will be sticky. They
would like to follow the pricing rule (14.2.3) but at the time they set the price they don't exactly know
what c1 and L are going to be, so they can't always get it exactly right. Since all rms are symmetric,
we'll assume they all set the same price and call it p1 . Later on we are going to think more carefully
about how rms choose p1 . For now we are going to take it as given because by the time interesting
things start to happen in the economy, p1 has already been chosen.
2. The government chooses a level for the money supply MS, and possibly other policies as well.
3. The household makes a consumption-saving decision. These decisions will satisfy the usual Euler equation
(6.2.8):
u0 (c1 ) = β (1 + r) u0 (c2 )
4. Each rm will make an investment decision for period 2. These will satisfy the investment-demand
condition (8.3.2). Since period 2 is the last period we set δ=1 so this reduces to:
F20 (K) = 1 + r (14.4.1)
Let:
−1
K (r) ≡ (F20 ) (1 + r) (14.4.2)
The function K (r) comes from solving (14.4.1) for K. It simply tells us the level of investment as a
function of the real interest rate. As we know from Chapter 8, this will be a decreasing function: higher
interest rates lower the NPV of investment projects so, other things being equal, fewer of them will be
undertaken.
Notice that we are describing savings and investment as separate decisions. We know from our basic
GDP accounting that in a closed economy savings and investment are equal by de nition. We'll see
below what makes this equality hold despite the decisions being taken separately.
5. The household decides how much money to hold. Its real money demand is:
mD (Y1 , i)
268
14.4. IS-LM
Notice that investment decisions are governed by the real interest rate because investment transforms
real goods today into real goods tomorrow. Instead, money demand is governed by the nominal interest
rate, because the opportunity cost of holding money depends on the nominal interest rate.
6. In period 2 the government chooses a new level for the money supply. We are not going to model the
period-2 money demand explicitly. We'll just assume that the price level in period 2, denoted p2 , depends
on the money supply in period 2 in some way. We are going to take p2 as exogenous for now, but we'll
see that expectations about p2 will play an important role. Since we have also taken p1 as exogenous,
p2
we are eectively treating in ation π≡ p1 −1 as exogenous. The reason we care about p2 and in ation
is that in the model we have both nominal and real interest rates playing a role, so we need to connect
them. Recall that by de nition r = i − π.
Market Clearing Conditions
There are several markets in the model: markets for goods in periods 1 and 2, a market for labor and a market
for money. Let's think about what it means for each of them to clear.
1. Goods in period 1. Denote GDP in period 1 by Y1 . Goods produced in period 1 can be used either for
consumption or for investment, so market-clearing requires:
Y1 = c1 + K
This is the basic GDP identity (1.1.1) for a closed economy with no government. We can also rearrange
it as:
Y1 − c1 = K
|{z}
Investment
| {z }
Saving
By imposing market clearing for goods we are making sure that saving equals investment even though
they are separate decisions.
2. Goods in period 2. Since period 2 is the last period, the household consumes all the output produced in
period 2, so:
c2 = F2 (K)
This was also true in the RBC model.
3. Labor. Here we need to remember what we assumed about how the owner-operated rms behave. We
assumed that the workers who run the rms work exactly as much as they need to meed demand for
their product. Total demand includes demand for consumption and demand for investment, so it's equal
to Y1 . The production function Y1 = F1 (L) implies that in order to produce Y1 , each worker has to
supply:
L = F1−1 (Y1 ) (14.4.3)
units of labor. Note that this is very dierent from how the quantity of labor is determined in the RBC
model. Here the worker is not optimizing how much to work. Instead, he works exactly as much as the
269
14.4. IS-LM
demand for his product requires. He does not work more because no one would want to buy the extra
units he would end up producing.
4. Money. The money-market clearing condition is just given by (14.4.9).
We are going to condense the four market-clearing conditions into just two, known as the IS and LM equations.
First, we are going to ignore the labor market clearing condition. Why? Because we know that L will just
respond to Y1 according to (14.4.3), so if we determine the level of Y1 we can always gure out L very easily.
Second, we are going to use the goods market clearing conditions for both periods, together with the Euler
equation, to derive an equation known as the IS equation. IS stands for Investment=Savings, i.e. market
clearing for period 1 goods. Finally, we are going to re-name the money market clearing condition as the LM
equation. LM stands for Liquidity=Money, where liquidity is another way of saying money demand, so
Liquidity=Money just means market clearing for money.
Both the IS equation and the LM equation are going to be relationships between between GDP Y1 and
the nominal interest i. These are two endogenous variables, and in the end we are going to nd the values of
Y1 and i that are consistent with both IS and LM. This is similar to how we think of supply and demand in
microeconomics: supply and demand each de ne a relationship between price and quantity, both of which are
endogenous variables, and the equilibrium is the price and quantity that is consistent with both supply and
demand.
The IS Relationship
Start from the Euler equation:
u0 (c1 ) = β (1 + r) u0 (c2 ) (14.4.4)
Replace c1 and c2 using the market clearing conditions for goods in each period:
u0 (Y1 − K) = β (1 + r) u0 (F2 (K))
Replace K using the investment-demand condition
u0 (Y1 − K (r)) = β (1 + r) u0 (F2 (K (r)))
Finally, replace r by i − π:
u0 (Y1 − K (i − π)) = β (1 + i − π) u0 (F2 (K (i − π))) (14.4.5)
Equation (14.4.5) is the IS equation, the rst of our two equations relating Y1 and i. Let's rst see
mathematically which way the relationship goes and then try to interpret what it means in economic terms.
Let's rewrite (14.4.5) as:
∆ = u0 (Y1 − K (i − π)) − β (1 + i − π) u0 (F2 (K (i − π))) = 0
270
14.4. IS-LM
and then take the derivative with respect to Y1 and i:
∂∆
= u00 (Y1 − K (i − π)) < 0 (14.4.6)
∂Y1 | {z }
(−)
∂∆
= − u00 (Y1 − K (i − π)) K 0 (i − π) −β u0 (F2 (K (i − π))) (14.4.7)
∂i | {z } | {z } | {z }
(−) (−) (+)
00
− β (1 + i − π) u (F2 (K (i − π))) F20 (K (i − π)) K 0 (i − π) < 0 (14.4.8)
| {z }| {z } | {z }
(−) (+) (−)
The signs of the derivatives come from the fact that:
• the marginal utility of consumption is positive: u0 (·) > 0,
• the marginal utility of consumption is decreasing: u00 (·) < 0,
• the marginal product of capital is positive: F20 (·) > 0, and
• investment is decreasing in the interest rate: K 0 (·) < 0.
Equations (14.4.6) and (14.4.8) imply that IS imposes a negative relation between Y1 and i. Increases in either
of these variables make ∆ go down, so in order to maintain ∆ = 0, if Y1 goes up then i must go down, and
vice-versa. Formally, the Implicit Function Theorem says that:
∂∆
di
= − ∂Y
∂∆
1
<0
dY ∂i
What is this telling us in economic terms? At the heart of the IS equation is the assumption the producers
respond to demand: if demand for their product goes up, they just produce more. The IS relationship follows
from answering the following question: what will aggregate demand (including demand-for-consumption and
demand-for-investment) be for each possible level of nominal interest rates?
Let's start with investment demand. Equation (14.4.2) says that investment demand will be a decreasing
function of r: at higher interest rates, fewer investment projects look attractive, so there is less investment.
Since r =i−π and we are taking π as exogenous, this means that K depends negatively on i.
Turn now to consumption demand. Let's try to gure out c1 by studying the Euler equation (14.4.4). This
tells us that c1 depends:
• negatively on r, taking c2 as given. This is the intertemporal substitution motive. If r is high, present
goods are expensive relative to future goods so the household will consume fewer of them. Since r = i−π ,
this means that c1 depends negatively on i.
• positively on c2 , taking r as given. This is the consumption-smoothing motive. If the household expects
more consumption in the future, it will smooth this out by consuming more in the present. But we know
that c2 will be equal to Y2 , which is a function of K , which depends negatively on i. Therefore indirectly
this also makes c1 depend negatively on i.
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14.4. IS-LM
This means that there are two channels by which higher interest rates induce lower consumption. First, higher
interest rates persuade households to tilt their consumption pattern away from present consumption by making
present goods expensive. Second, by lowering investment they lower expectations of future consumption, which
persuades households to lower consumption in every period.
Therefore, in our model, both investment demand and consumption demand (and therefore aggregate
demand) are decreasing in the interest rate. If we plot the IS equation, it will look like a downward-sloping
curve.
The LM Relationship
We are going to import the theory of money markets we developed in Chapters 10 and 11 into our model
of how the economy works. If you recall from Chapter 11, the so-called classical view states that money is
neutral, i.e. it has no eect on real quantities. We also saw that under this view, changes in the money supply
translate immediately into proportional changes in the price level. Therefore the classical view is incompatible
with sticky prices. If prices are sticky, money will not be neutral and we need to think about how the money
market interacts with the real economy.
The LM relationship is just a re-naming of the money market clearing condition:
M S = mD (Y1 , i) · p1 (14.4.9)
Remember, we are treating the price level p1 as exogenous because prices are sticky, and MS is exogenous
as well because it is chosen by the government. Therefore (14.4.9) also gives us a relationship between Y1
D
and i, which are linked because both are arguments of the money-demand function m (Y1 , i). Recall from
Chapter 10 that the money demand is increasing in Y1 (more transactions require more money) and decreasing
in i (higher interest rates make households hold lower money balances). Since Y1 and i move money demand
in opposite directions, maintaining money-market clearing requires that they move in the same direction.
Formally, the implicit function theorem says that:
D
∂m
dY ∂i
= − ∂m D > 0
di ∂Y 1
Therefore the money-market / LM equation imposes a positive relationship between Y1 and i.
What is this telling us in economic terms? Suppose GDP is higher. People will want to carry out a lot
of transactions, so they will want to hold a lot of money to do this. But, by assumption, the money supply
is xed, so it's impossible for all of them to hold more money at the same time. So the opportunity cost
of holding money (the nominal interest rate) rises until people are content with holding exactly MS units of
money.
How exactly does this adjustment take place? In the background, there is a market where people exchange
interest-bearing assets like bonds for non-interest-bearing money. If people want to carry out a lot of transac-
tions, they will be trying to sell bonds in exchange for money, so what happens is that the price of bonds falls.
If you recall the basic Present Value formula from Chapter 8, then for an asset like a bond that promises a
272
14.5. Shocks
xed future payment, a fall in the price is the same thing as a rise in the interest rate.
IS-LM
If we put the IS equation and LM equations together, we have:
u0 (Y1 − K (i − π)) = β (1 + i − π) u0 (F2 (K (i − π))) (14.4.10)
M S = mD (Y1 , i) · p1 (14.4.11)
By solving this pair of equations (with M S , p1 and π taken as exogenous) we can jointly gure out the level of
output and interest rates, as shown in Figure 14.4.1. The gure shows the downward-sloping IS relationship
and the upward-sloping LM relationship. The point Y1 , i is the only combination of Y1 and i that satis es
both equations. Since the model predicts that both the IS and the LM equations hold, it predicts what the
levels of GDP and interest rates are going to be.
Fig. 14.4.1: Equilibrium in
the IS-LM representation of the
New Keynesian Model.
14.5 Shocks
Let's see how the economy would respond to various shocks. We'll start by analyzing the same shocks that
we looked at in Chapter 13 and then we'll look at other possibilities.
273
14.5. Shocks
Productivity Shocks
Suppose we look at the same kind of productivity shock that we looked at in Chapter 13: the production
function goes from Y1 = F1 (L) to Y1 = AF1 (L) with A > 1. Other things being equal, how would this aect
the economy?
If you look at equations (14.4.10) and (14.4.11), you'll notice that the production function does not appear
anywhere. What's going on? In this model, output is demand-determined: We have assumed that rms
expand and contract output to meet demand, and workers work however much it takes to satisfy this demand.
An increase in productivity means that the economy can produce more output but not that it will produce
more output. Output will only increase if there is more demand, and this shock does not aect demand
directly.
Sometimes economists make the distinction between supply eects and demand eects. These terms
are often used imprecisely but in this example they are reasonably clear. Higher productivity is a supply eect:
more can be produced with the same inputs. However, with no change in demand there will be no change in
quantity.
Instead, what will happen is that employment will fall. We know this from equation (14.4.3). Since workers
are able to produce more output per hour but total demand hasn't changed, they will get their work done in
fewer hours. Gali (1999) analyzed evidence that suggested that this is indeed what happens, spurring a large
debate on whether the evidence was interpreted correctly.
Impatience Shocks
Suppose that β falls: households really want to consume now rather than later. β enters the IS equation
(14.4.10). It shifts the right-hand side down, shifting the entire curve up and to the right, as shown in Figure
14.5.1.
When people become impatient, they want to consume more now. All the purchases they make induce
producers to increase production, leading to an increase in output and employment. At the same time, the
increase in output requires more transactions, which increases the demand for money. Since we are holding
the money supply constant, this means that the interest rate rises to clear the money market: a movement
along the LM curve. In turn this higher interest rate lowers investment.
Notice how dierent this is from what happens in the RBC model. In the RBC model, an increase in
impatience makes consumption go up but employment go down, because there is no way to escape the logic
of equation (13.1.3):
v 0 (1 − L)
= F10 (L)
u0 (c1 )
which implies that if households want more consumption then, other things being equal, they want more
leisure as well. In the New Keynesian model this condition doesn't hold because workers are not choosing
how much they work in response to shocks in a utility-maximizing way: they just accommodate the level of
demand they face.
Still, as a theory of why we observe business cycles we haven't quite nailed it. Impatience shocks make
output, employment and consumption move together, but investment moves the opposite way, so they cannot
274
14.5. Shocks
Fig. 14.5.1: The eects of an
increase in impatience (fall in
β ).
be the whole story.
Optimism
Suppose households predict an increase in future productivity: the production function for period 2 goes from
Y2 = F2 (K) to Y2 = AF2 (K) with A > 1. Now the investment function (14.4.2) becomes:
−1 1+r
K (r, A) ≡ (F20 )
A
which means investment is increasing in A. The IS equation is now:
u0 (Y1 − K (i − π, A)) = β (1 + i − π) u0 (AF2 (K (i − π, A)))
It's easy to trace out that higher A lowers the right hand side and raises the left hand side, which has the
eect of shifting the IS curve to the right, as shown in Figure 14.5.2.
What's going on? There are two eects, both going in the same direction. The rst eect is through
investment. If people believe that productivity will improve, then they expect the marginal product of capital
to be higher. This means that a lot of investment projects are worth doing. All the resources needed to carry
out these investment projects have the direct eect of increasing demand for output.
The second eect is through consumption. If productivity in the future will be higher, then future output
and therefore future consumption will be higher. In addition, the fact that investment increases reinforces the
eect. Households want to smooth out this anticipated future consumption by consuming more in the present.
This adds to the increase in current demand.
275
14.5. Shocks
Fig. 14.5.2: The eects of
optimism (an increase in future
productivity).
As in the previous example, this requires an increase in the interest rate, i.e. a movement along the LM
curve, in order to clear the money market.
This oers us a possible account of business cycles that ts the basic facts: output, consumption, employ-
ment and investment all move in the same direction at the same time. Unlike the example with impatience,
a wave of optimism makes investment move together with consumption because optimism directly aects the
perceived pro tability of investment projects. Unlike optimism shocks in the RBC model, a wave of opti-
mism in the New Keynesian model gets employment to move together with consumption by getting rid of the
labor-consumption choice (13.1.3).
Interestingly, this type of eect may be close to what Keynes originally had in mind:
a large proportion of our positive activities depend on spontaneous optimism rather than mathe-
matical expectations, whether moral or hedonistic or economic. Most, probably, of our decisions to
do something positive, the full consequences of which will be drawn out over many days to come,
can only be taken as the result of animal spiritsa spontaneous urge to action rather than inaction,
and not as the outcome of a weighted average of quantitative bene ts multiplied by quantitative
probabilities.
Laziness or Taxes
Suppose, like we did in Chapter 13, that there is an increase in labor-income taxes (or equivalently, households'
preference for leisure increases), so that equation (14.2.4) becomes:
v 0 (1 − L) η−1
= F10 (L) (1 − τ )
u0 (c1 ) η
276
14.5. Shocks
Other things being equal, how would this aect the economy?
If you look at equations (14.4.10) and (14.4.11), you'll notice that neither preferences for leisure, taxes nor
the production function appear anywhere. As with productivity shocks, this would have no eect on output
or interest rates because there is no eect on demand.
2 Taxes certainly aect worker's incentives to choose
between work and leisure but we have assumed that, due to sticky prices, they are not responding to those
incentives anyway, so they have no eect.
Changes in the Money Supply
Suppose the government decides to increase the money supply MS. This increases the left-hand side of the
LM equation (14.4.11), which results in a shift to the right in the LM curve, as shown in Figure 14.5.3.
Fig. 14.5.3: The eects of an
increase in the money supply.
This change in monetary policy leads to higher output and lower interest rates. What's going on? The
money supply has increased, so something has to adjust for the money market to clear. Given the money-
demand function mD (Y1 , i) and sticky prices, two things could happen to persuade households to hold the
extra money that has been created. One possibility is that GDP rises, so that households need the extra
money to carry out extra transactions. The other possibility is that the interest rate falls, so that households
face a lower opportunity cost of holding money. The shift of the LM curve down and to the right shows the
new combinations of Y1 and i that are consistent with market clearing.
The IS curve hasn't shifted but there is a movement along the curve. A lower interest rate means that
more investment projects are worth doing, so investment rises. Furthermore, present consumption rises both
2
How come an increase in taxes has no eect on demand? Doesn't the fact that the government takes away part of people's
income reduce their consumption? In the background, what's going on is Ricardian equivalence. We have assumed that the
government increases taxes but not spending, so households rationally perceive that whatever the government is taking away it
will give back, either at the same time or in the future.
277
14.5. Shocks
to smooth out the higher future consumption and because lower rates have made present consumption cheaper.
As a result of the increase in demand, output and employment rise.
This gives us another possible source of business-cycle-like movements: changes in the money supply can
make output, employment, consumption and investment move in the same direction. It is an example of how
money is not neutral in the New Keynesian model: changes in the supply of money can have eects on real
variables. Friedman and Schwartz (1963) argued that changes in the money supply were one of the main
sources of US business cycles. This led them to the conclusion that monetary policy should aim to keep the
money supply as steady as possible to avoid causing business cycles, a point of view that came to be known
as monetarism.
Changes in Expected In ation
Suppose that people start to believe that prices are going to rise between t=1 and t = 2, perhaps because of
news about what future monetary policy is going to be like. In our model, this is captured by an increase in
π, all else being equal. π enters the IS equation (14.4.10); higher π raises the right hand side and lowers the
left hand side, which has the eect of shifting the IS curve to the right, as shown in Figure 14.5.4. Output
and the interest rate both rise.
Fig. 14.5.4: The eects of an
increase in expected future in-
ation.
More precisely, higher π shifts the IS curve up. To understand why, recall that what matters for the IS
relationship is the real interest rate r = i − π, because this is what governs both investment and consumption
decisions. What higher in ation does is change the relationship between nominal and real rates. For any given
nominal rate i, higher in ation implies a lower real rate r = i − π , so it's like shifting the axis that governs the
IS relationship. In equilibrium, even though the nominal rate goes up, the real rate goes down, so investment
and consumption rise.
278
14.6. Simpli ed
14.6 Simpli ed Versions of IS-LM
For some purposes it is useful to look at special cases of the IS and LM relationships in order to have fewer
things to think about. In this section we'll consider some examples.
Exogenous Investment
Imagine that the period-2 production function is:
F2 (K) = min AK, AK̄ (14.6.1)
Under this function, the marginal product of capital is equal to A for levels of investment up to K̄ and then
drops to zero. If A is su ciently high, this means that investment will always be equal to K̄ , no matter what
the interest rate is, so GDP and consumption in period 2 will be Ȳ2 = AK̄ . By doing this, we are eectively
treating investment as exogenous, so the only thing left to be determined is consumption. Replacing K = K̄
and c2 = Y2 into the IS equation (14.4.5), we get a closed form expression for i as a function of Y.
u0 Y1 − K̄
i= −1+π (14.6.2)
βu0 Ȳ2
This simpli ed IS relationship just captures the intertemporal substitution eect of interest rates on con-
sumption. c1 is decreasing in the interest rate, and therefore so is Y1 = c1 + K . This gives us the negative
IS relationship between interest rates and GDP. The basic economic message hasn't changed: total demand
decreases with the interest rate. We have just reduced the model to its minimal essential economic ingredients.
The Old Keynesian Model
The model we have been studying is a version of the New Keynesian model. The New comes from the fact
that it's built up from explicit microeconomic foundations. The traditional (Old) Keynesian model has many
ingredients in common. The main underlying assumptions are mostly the same (even though Old Keynesian
models sometimes didn't state them precisely), and it's possible to summarize an Old Keynesian model with
IS-LM equations, just like we did with the New Keynesian model. The main dierence between the New
and Old Keynesian models is that the Old Keynesian model is built on a less fancy theory of consumption.
For some purposes, this makes a big dierence.
Imagine that instead of assuming that consumption is a result of intertemporal optimization, we just
proposed the Keynesian consumption function that we looked at in Chapter 6: consumption depends on
current income:
c1 = c (Y1 ) (14.6.3)
where c(·) is some function. With this consumption rule, households are not looking at the future when
deciding how much to consume and they are also not looking at the interest rate.
As we saw in Chapter 6, the quantity c0 (Y1 ) is known as the marginal propensity to consume, and is the
answer to the following question: if income goes up by one dollar, by how many dollars does consumption go
279
14.7. Partially Sticky Prices
up? According to the intertemporal consumption theory we looked at in Chapter 6, consumption depends on
the present value of the household's income. The income of one individual period is only a small part of this,
so the marginal propensity to consume should be low (see Exercise 6.8). Johnson et al. (2006) and Parker et al.
(2013) looked at how households responded to temporary tax rebates implemented in 2001 and 2008. They
found that consumption reacted more than would be predicted by the pure intertemporal model. It is possible
that some households just have a simple budgeting rule that says how much they'll consume as a function of
their after-tax income, ignoring anything else. Also, some households may be borrowing-constrained and just
consume as much as they can. Either of these assumptions could justify something like (14.6.3).
If consumption follows (14.6.3), the IS equation follows directly from the period-1 goods market clearing
condition:
Y1 = c (Y1 ) + K (i − π) (14.6.4)
Qualitatively, this IS equation is not that dierent from the New Keynesian IS equation. It also relates
output Y1 and the nominal interest i. Let's check that it is indeed downward-sloping. Restate (14.6.4) as:
∆ = Y1 − c (Y1 ) − K (i − π) = 0
and then take the derivative with respect to Y1 and i:
∂∆
= 1 − c0 (Y1 ) > 0
∂Y1
∂∆
= − K 0 (i − π) > 0
∂i | {z }
(−)
so, using the Implicit Function Theorem:
∂∆
di
= − ∂Y
∂∆
1
dY1 ∂i
1 − c0 (Y1 )
= <0
K 0 (i − π)
However, the Old and New Keynesian IS relationships do have dierent implications for some important
questions. Section 15.1, for instance, studies how scal policy works dierently depending on what version of
the IS curve applies.
14.7 Partially Sticky Prices and the Phillips Curve
So far we have made the extreme assumption that all producers set their prices in advance, so prices are
perfectly sticky. Let's now consider an intermediate case where prices are somewhat sticky but not perfectly
so, and ask how the economy would behave in this case. There are two reasons to analyze this intermediate
case. First, both the assumptions of perfectly exible prices and perfectly sticky prices are extreme, so the
intermediate case is probably more empirically relevant. Second, there are some interesting eects in the
280
14.7. Partially Sticky Prices
partially-sticky-price model that don't arise in either of the extreme cases.
To keep things relatively simple, we are going to use the exogenous-investment assumption from Section
14.6, so that investment is just K̄ . It will be useful to use the market clearing condition Y1 +c1 +K to compute
how much consumption is associated with a given level of employment:
c1 (L) = F1 (L) − K̄ (14.7.1)
c1 (L) de nes a positive relationship between consumption and employment.
Suppose a fraction µ of producers have sticky prices and a fraction 1−µ have exible prices. The exact
timing is as follows:
1. First, the sticky-price producers set their price. Once they set their price, the cannot change it. Let's
call this price pS1 .
2. Second, all the macroeconomic shocks are realized and any policies that the government will enact are
put in place.
3. Finally, the exible-price producers set their price, knowing everything that happened before. Let's call
this price pF
1.
The average price of a good will be simply a price index:
3
p1 = µpS1 + (1 − µ) pF
1 (14.7.2)
As we saw in Section 14.2, a exible-price producer will want to set its price at a markup over marginal cost.
Using (14.2.3) and (14.7.1): this means:
v 0 (1 − L) η
pF
1 = p1 (14.7.3)
(c1 (L)) F10 (L) η − 1
u0
| {z } | {z }
Real Marginal Cost Markup
Replacing the pricing rule (14.7.3) into the price index (14.7.2) and solving for the price index p1 we get:
µpS1
p1 = 0 (14.7.4)
1 − (1 − µ) u0 (cv1 (L))F
(1−L) η
0 (L) η−1
1
Equation (14.7.4) implies a positive relationship between p1 and L. What's going on? Producers with
exible prices adjust their price in response to everything that happens in the economy. If economic shocks
take place that lead to higher employment, then their marginal costs of production rise. In response to this,
they set higher prices.
Now imagine that in some previous period 0 the price level was p0 . In ation between periods 0 and 1 is
3
This is a xed-weight price index. Technically, one should allow for the fact that as prices change consumer substitute between
exible-price and sticky-price producers. We are going to assume this away. For small shocks, this doesn't make much dierence.
281
14.7. Partially Sticky Prices
given by
p1 −p0
p1 4
p0 , so higher means higher in ation. Therefore equation (14.7.4) gives us a positive relation
between in ation and employment. In Chapter 12 we called this relationship a Phillips Curve. So now we
have a theoretical justi cation of what gives rise to a Phillips Curve: in a partially-sticky-price model, shocks
that make employment go up will also make prices go up.
We can also write down the Phillips Curve as a relationship between prices and output instead of employ-
ment. Invert the production function to write L(Y1 ) = F1−1 (Y1 ) and replace L(Y1 ) into (14.7.4) to obtain:
µpS1
p1 = 0 (14.7.5)
1 − (1 − µ) u0 (Yv1 −(1−L(Y 1 )) η
K̄)F 0 (L(Y1 )) η−1
1
Equation (14.7.5) says that there is a positive relationship between prices and output, an equivalent way of
expressing the Phillips Curve relationship.
Adding a Phillips Curve to an IS-LM Model
With fully sticky prices, the IS-LM model boiled down to a system of two equations in two unknowns: Y1 and
i. With partially sticky prices, we also have to solve for the period-1 price level p1 (or equivalently, period-
1 in ation). The Phillips Curve relation, which as we saw is derived from exible-price-producers' optimal
pricing, gives us an additional equation relating Y1 and p1 , so we now have a system of three equations in
three unknowns. The IS equation and the Phillips Curve are easy to see graphically because they only involve
two of the three endogenous variables: the IS equation relates Y1 to i and the Phillips Curve relates Y1 to p1 .
The LM equation is a little bit trickier because it has all three endogenous variables in it. One way to see
the LM equation graphically is to write down a modi ed version of it. Start from the LM equation (14.4.11)
and replace the term p1 with the expression that comes from the Phillips Curve (14.7.5) to obtain:
µpS1
M S = mD (Y1 , i) 0 (14.7.6)
1 − (1 − µ) u0 (Yv1 −(1−L(Y 1 )) η
K̄)F 0 (L(Y1 )) η−1 1
As with the basic LM equation (14.4.11), the right hand side of (14.7.6) is increasing in Y1 . Recall that
the right hand side of the LM equation represents the demand for nominal money balances. There are two
eects going in the same direction. First we have the basic eect we already know about: higher GDP means
households want to carry out more transactions so, other things being equal, they demand more money. In
addition, we have an eect coming from the Phillips Curve: since higher GDP is associated with higher prices,
the nominal amount of money required to carry out a given level of transactions also goes up. On the other
hand, the right hand side of (14.7.6) is decreasing in i for the usual reason that the demand for money is
decreasing in the interest rates. Overall, this implies that the modi ed LM equation (14.7.6) describes a
positive relationship between GDP and interest rates, just like the basic LM equation (14.4.11).
Figure 14.7.1 shows how the IS-LM model ts in with the Phillips Curve. The top graph shows the
simpli ed IS equation (14.6.2) and the modi ed LM equation (14.7.6); the bottom graph shows the Phillips
Curve (14.7.5). Together, the IS and LM curves determine the level of Y1 and i, and then the Phillips Curve
4
Once we are thinking of three periods: 0 (the past), 1 (the present) and 2 (the future), there are two in ation levels to keep
in mind: π1 (in ation between 0 and 1) and π2 (in ation between 1 and 2). Here we are referring to π1 .
282
14.7. Partially Sticky Prices
M
tells us what level of p1 (and therefore in ation π1 ) goes with this level of Y1 . We are holding constant
p
(the real money supply), pS1 (sticky prices) and π2 (expected in ation between periods 1 and 2).
Fig. 14.7.1: How the IS-LM curves
and the Phillips Curve t together.
Let's re-do some of the exercises we did with fully sticky prices to see whether the conclusions change once
we have partially sticky prices. Figure 14.7.2 shows the eects of a change in impatience, as captured by the
discount factor β. Inspecting equations (14.6.2), (14.7.6) and (14.7.5), the only place where β shows up is in
the IS equation, which shifts up and to the right. This is the same eect shown in Figure 14.5.1. Just like with
purely sticky prices, the result is an increase in GDP, as impatient households demand more output. With
partially sticky prices, we also have an increase in the price level, as exible-price producers raise their prices
to maintain their markups.
Figure 14.7.3 shows the eects of a productivity shock, with the production function increasing to Y1 =
AF1 (L) with A > 1. As we saw before, productivity does not enter the IS and LM equations.
5 However, it
5
Technically, productivity does enter the modi ed LM equation (14.7.6). The Phillips Curve shifts and (14.7.6) builds the
283
14.7. Partially Sticky Prices
Fig. 14.7.2: An increase in impa-
tience.
does enter the Phillips Curve:
µpS1
p1 = Y
v 0 (1−L( 1 )) η
1 − (1 − µ) u0 (Y −K̄)AF 0A L Y η−1
1 1 ( ( A ))
Higher productivity lowers marginal costs through three dierent channels. First, given a level of employment
L, the marginal product of labor AF10 (L) rises. This means less extra labor is needed to produce an extra
Y
unit of output. Second, the amount of labor needed to produce a given level of output L A falls. Since
there is a diminishing marginal product of labor, lower employment means a higher marginal product of
labor, which also implies lower marginal costs. Third, lower employment means more leisure and therefore a
Y1
v0 1 − L
lower marginal utility of leisure . Since the opportunity cost of producing output is giving up
A
Phillips Curve relationship into the LM equation, so it also shifts. This is not terribly important so in order to be able to visualize
things in a two-dimensional graph, we are going to disregard it.
284
14.7. Partially Sticky Prices
leisure, this means lower marginal costs. Therefore, the Phillips Curve shifts down: other things being equal,
exible price producers lower their prices to maintain their desired markups. As in the fully-sticky-price case,
productivity shocks have no eect on output and interest rates and they lower employment. With partially
sticky prices, we also have a fall in the price level.
Fig. 14.7.3: An increase in produc-
tivity.
Exercises
14.1 Too Many Customers
Jackson & Jackson is a shampoo manufacturer. Its engineering department estimates that if it decides
to produce q units of shampoo, the total production cost would be c(q) = aq + 2b q 2 dollars. Its marketing
department estimates that if it sets a price of p dollars per unit of shampoo, it will sell q(p) = α − βp
285
14.7. Partially Sticky Prices
α
units, where
β > a.
(a) What price should Jackson & Jackson set in order to maximize pro ts?
(b) Suppose Jackson & Jackson has chosen the pro t-maximizing price and advertised it to all its clients,
so that at this point it is impossible to change it. Soon after, it learns that the marketing department
turned out to be too pessimistic, and actual demand for shampoo is q(p) = α0 − βp, where α0 > α.
If Jackson & Jackson decides to satisfy every purchase order it receives, how many units will it end
up selling? How does this number depend on α and α0 ? Explain.
(c) For what values of α0 would it be advantageous for Jackson & Jackson to turn away some customers?
(d) Now suppose that the marketing department had the correct prediction but the engineering depart-
ment underestimated production costs, which turn out to be c(q) = a0 q + 2b q 2 , with a0 > a . For
what values of a0 would Jackson & Jackson want to turn away some customers?
14.2 Stickiness and the Phillips Curve
How does the slope of the Phillips Curve (14.7.4) depend on µ? What does this mean?
14.3 Reserve Requirements
Suppose the government decides to lower banks' reserve requirements.
(a) What will happen to the (M1) money multiplier?
(b) What will happen to the (M1) money supply?
(c) If prices are partially sticky, what will happen to GDP, interest rates and the price level?
14.4 Gold
For a long time, gold was used as money. This meant that the quantity of money depended on the
quantity of gold that had been dug up from mines. Suppose a new mine is discovered.
(a) What should we expect to happen to GDP, nominal interest rates and the price level?
(b) How does the answer depend on how exible prices are?
14.5 Future In ation and Present In ation
Suppose the economy is well described by a New Keynesian model with partially sticky prices. Let π1
refer to in ation between period 0 and period 1 and π2 refer to in ation between period 1 and period 2,
which we have taken as exogenous. Suppose there is an exogenous increase in π2 , for instance because
the government announces that it will increase the money supply in the future. What will happen to π1 ?
What are all the steps that lead to this conclusion?
14.6 The Eect of Investment
Consider the simpli ed New Keynesian model with exogenous investment from Section 14.6, with a
period-2 production function given by (14.6.1). Suppose the utility function is u(c) = log(c), the house-
hold's discount factor is β=1 and in ation is π = 0.
286
14.7. Partially Sticky Prices
(a) Derive an IS equation for this special case. This should be a relationship between Y1 and i for given
values of the exogenous parameters A and K̄ .
(b) Holding the interest rate constant, by how much does output increase if K̄ increases by one unit?
Explain in words why the answer depends on A.
Now consider a simpli ed version of the Old Keynesian model with exogenous investment. The consump-
tion function is c(Y1 ) = a + bY1 and the period-2 production function is given by (14.6.1).
(c) Use the market-clearing condition to solve for Y1 as a function of the exogenous parameters A, K̄ ,
a and b. Explain in words why it does not depend on i.
(d) By how much does output increase if K̄ increases by one unit? Explain in words why the answer
depends on b but not on A. Explain why the answer is dierent from the answer to part (b).
14.7 Blowing Stu Up
Imagine that the economy is well described by the New Keynesian model with perfectly sticky prices,
with one twist. There is already some capital in place, which can be used to produce output in period
2. Denote this capital by K0 and, for simplicity, assume that it does not depreciate. The total capital
stock that's used in production in period 2 is therefore:
K = K0 + I
where I is investment.
(a) Let I (K0 , r) be the level of investment as a function of K0 and r. How does it depend on K0 ?
(b) Write down the market-clearing condition for period-1 goods.
(c) Derive an IS equation for this economy.
(d) Suppose someone blows up part of the original capital stock, so that we start with K0 − X instead
of K0 . How does the IS curve shift in response to X? What happens to GDP and interest rates?
Explain.
14.8 A Recession
Consider the following data:
2015 2016
Capital Stock 10, 000 10, 000
Employment 1 0.81
GDP 4, 000 3, 600
Labor Income 2, 000 1, 800
Capital Income 2, 000 1, 800
Investment 800 600
Consumption 3, 200 3, 000
Price level 100 90
Nominal Interest Rate 5% 10%
287
14.7. Partially Sticky Prices
Suppose that 2015 was a totally normal year (almost identical to 2012, 2013 and 2014). This question
asks you to think about what are the possible causes of recessions and how to use the data to determine
what may have caused the 2016 recession.
(a) Name two possible causes of recessions that don't seem to t the data. For each of them, explain:
i. Why, according to some model, it's possible for this to cause a recession. You can make the
argument using equations, graphs or words, as long as you are precise.
ii. What features of the data indicate that this is not what happened.
(b) Name one possible cause of recessions that does t the data. Explain:
i. Why, according to some model, it's possible for this to cause a recession. You can make the
argument using equations, graphs or words, as long as you are precise.
ii. What features of the data indicate that this is what may be going on?
288
CHAPTER 15
Monetary and Fiscal Policy
Most policymakers around the world, especially in central banks, have some version of the New Keynesian
model in mind when they are setting policy. In this chapter we'll look at what the model tells us about the
eects of monetary and scal policy.
15.1 Fiscal Policy
So far we have modeled an economy where there is no government spending. Imagine now that there is a
government, which chooses some level of spending G. How does this change the model? What is the eect of
an increase in G?
For now let's imagine that there is only government spending in period 1 (Exercise 15.6 will ask you to
look at the eect of government spending in period 2). We are going to imagine that government spending is
nanced entirely by lump-sum taxes and that Ricardian equivalence holds, so we don't need to specify when
the government collects taxes. Furthermore, we are going to assume that government spending does not enter
the household's utility function, i.e. households do not value public goods (Exercise 15.7 will ask you to look
at what happens under dierent assumptions in this regard).
The period 1 goods market clearing condition is now:
Y1 = c1 + K + G
Solving for c1 and replacing this in the Euler equation (14.4.4) leads to the following modi ed version of the
IS relationship:
u0 (Y1 − G − K (i − π)) = β (1 + i − π) u0 (F2 (K (i − π))) (15.1.1)
An increase in government spending leads to a horizontal rightward shift of the IS curve, as show in Figure
15.1.1. Output and the interest rise.
What's going on? The government is directly demanding goods and producers are responding by producing
more goods. This is, at least in part, the logic behind the scal stimulus plans that are sometimes carried
out. The objective is to make production and employment rise by directly demanding goods and services,
with the understanding that producers will expand output to meet demand.
289
15.1. Fiscal Policy
Fig. 15.1.1: The eects of an
increase in government spend-
ing.
The Government Spending Multiplier
An increase in G leads to a rightward shift in the IS curve, resulting in higher GDP and higher interest rates.
We'll now look at two further questions about this. First, we'll ask how much does the IS curve shift. We'll see
that the New Keynesian model and the Old Keynesian model that we looked at in Section 14.6 give dierent
answers to this question. Next, we'll ask how much of the eect will be on GDP and how much on interest
rates.
Suppose that there is an increase in G. Holding everything else constant (in particular, holding i constant),
how much does the IS curve shift to the right? We'll ask this rst with the New Keynesian IS curve and then
with the Old Keynesian IS curve. Mathematically, what we'll be trying to compute is:
∂∆
∂Y1 ∂G
= − ∂∆
∂G ∂Y
1
(while holding i constant). For the New Keynesian IS curve (15.1.1) we have:
∂∆
∂Y1 ∂G
= − ∂∆
∂G ∂Y 1
−u0 (Y1 − G − K (i − π))
=− 0 =1 (15.1.2)
u (Y1 − G − K (i − π))
so the IS curve shifts to the right by the exact amount of G.
290
15.1. Fiscal Policy
Instead, by adding government spending into the Old Keynesian IS (14.6.4) we get:
Y1 = c(Y1 ) + K(i − π) + G (15.1.3)
Therefore, using the implicit function theorem, we have:
∂∆
∂Y1 ∂G
= − ∂∆
∂G ∂Y 1
1
= >1 (15.1.4)
1 − c0 (Y1 )
so the IS curve shifts more than one-for-one with G. For instance, if the marginal propensity to consume is
0.75, the IS curve will shift to the right by 4 dollars for each dollar of government spending.
What explains the dierence between (15.1.2) and (15.1.4)? Why does the Old Keynesian model predict
that the IS curve shifts more than one-for-one with changes in G, while the New Keynesian model does not?
The answer has to do with Ricardian equivalence.
Let's take the New Keynesian model rst. When the government increases spending on public goods,
there is a direct eect: producers increase output and earn additional income from selling these goods to the
government. However, they understand that, either now or in the future, the government will increase taxes
to pay for this spending. That's why we don't need to be speci c about when the government collects taxes:
the present value of these taxes will be exactly equal to the additional income generated by selling goods to
the government. Therefore the present value of after-tax income has not changed, so consumption does not
change. As a result, the IS curve shifts exactly by the amount of the increase in G.
In the Old Keynesian model, this works dierently. Households pay no attention to the fact that future
taxes will increase, so Ricardian equivalence does not hold. Therefore we do need to be speci c about when
the government collects taxes. Assuming households base consumption decisions on current after-tax income,
then they will react dierently to an increase in spending paid for by current taxes (which households pay
attention to) or by future taxes (which households ignore). Exercise 15.3 asks you to compute how dierent the
reaction will be. Let's assume that the increase in government spending is paid for entirely with future taxes.
(Equation (15.1.3) implicitly assumes this; otherwise consumption would be c(Y1 − τ1 )). Households just see
that they are earning extra income from selling goods to the government, so they go out and consume more.
How much more? That depends on the marginal propensity to consume: they will consume an extra c0 (Y1 )
per dollar of extra income. But this is not the whole story. This additional consumption will lead producers to
increase production further, and make them earn extra income, which in turn leads to extra consumption, and
so on. Mathematically, what results is a geometric series. For one dollar of additional government spending,
we get:
∂Y1 2 3
= 1 + c0 (Y1 ) + (c0 (Y1 )) + (c0 (Y1 )) + . . .
∂G
1
=
1 − c0 (Y1 )
which is exactly what (15.1.4) is saying.
291
15.1. Fiscal Policy
What we have asked so far is how far the IS curve moves to the right. The overall eect of the change in G
will depend on how the IS curve interacts with the LM curve. If the LM curve is very steep, then the interest
rate will rise a lot and GDP will increase little. Conversely, if the LM curve is relatively at, then GDP will
increase a lot with little increase in interest rates. Figure 15.1.2 illustrates these dierent cases. Exercise 15.2
asks you to think about what underlying models of money demand would produce steep or at LM curves.
Fig. 15.1.2: Crowding out.
The term crowding out is used to refer to situations where an increase in G results in higher interest rates
but not higher GDP, as shown in Figure 15.1.2 for the case of a steep LM curve. The term comes from the
idea that G crowds out other types of spending. In particular, it is sometimes said that higher government
spending may crowd out investment if it leads to a rise in interest rates.
dY1
The quantity
dG is sometimes known as the government spending multiplier. It is the answer to the
question: if government spending rises by one dollar, by how much does GDP rise? It's called a multiplier
because GDP changes by some multiple of the change in G. The size of the multiplier depends on:
1. The size of the shift in the IS curve. As we saw, in our version of the New Keynesian model, this will
be equal to 1, whereas the Old Keynesian model says this will be greater than 1.
2. The slope of the LM curve, since this determines the degree of crowding out.
The second point is a little bit subtle. The idea of the multiplier is to ask how GDP changes with an increase
in government spending holding monetary policy constant. However, holding monetary policy constant could
mean more than one thing. Figure 15.1.2 implicitly assumes that by holding monetary policy constant we
mean holding the money supply constant. However, monetary policy is often described simply in terms of
choosing an interest rate (instead of choosing a money supply that will then result in an interest rate). If by
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15.2. Monetary Policy
holding monetary policy constant we meant adjusting to money supply so that the interest rate remains
constant then that changes the multiplier. Exercise 15.1 asks you to think about this.
There is a lot of disagreement about the size of the multiplier. In particular, there is disagreement about
whether it's greater than 1, as would be implied by an Old Keynesian IS curve (combined with either a
relatively at LM curve or a monetary policy response that keeps nominal rates unchanged). Ideally, we
would like to have many experiments where G is changed randomly and measure how the economy reacts to
these. Since we don't run these sorts of macroeconomic experiments, we need to gure out the right way to
interpret the data that we do have, and there is quite a bit of disagreement on how to do that. Ramey (2011)
surveys some of the evidence on measuring the multiplier and nds that the most plausible values are between
0.8 and 1.5.
The size of the multiplier is important because, if the multiplier is large, then scal policy can be very
powerful: a relatively small change in the level of government spending can have a large eect on GDP.
15.2 Monetary Policy
Suppose that there is some shock that, other things being equal, would lead to a recession. For instance,
suppose that households become pessimistic about future productivity, so the IS curve shifts to the left. Other
things being equal, this would lead to lower GDP and a lower interest rate.
Suppose the government wants to prevent GDP from falling. One possible way to respond is to increase
the money supply. This will lead to a rightward shift in the LM curve, further lowering interest rates and
counteracting the fall in output, as shown in Figure 15.2.1.
Fig. 15.2.1: Using monetary
policy to counteract a negative
shock.
What's going on? Pessimism about the future, other things being equal, leads households to reduce
293
15.2. Monetary Policy
consumption and investment, which leads to lower output. In order to persuade households not to reduce
their spending, the government tries to engineer a fall in the interest rate, to generate movement along the
new IS curve. A lower interest rate means that more investment projects are worth doing, so investment rises,
and present goods are cheaper relative to future goods, so consumption rises. If the government gets the size
of the policy reaction exactly right, then it can oset the fall in GDP exactly, as in the example in Figure
15.2.1.
Often this type of policy is simply described as lowering the interest rate. Ultimately, by controlling the
money supply, the Central Bank can control the interest rate, so it is sometimes useful to think of the Central
Bank as just picking what interest rate it wants. Indeed, in practice that's how most central banks operate
these days. They decide on a target level for the interest rate and then conduct open market operations to
adjust the money supply however much it takes for the target they chose to actually be the equilibrium rate.
Note that the objective of policy need not be to stabilize GDP. As we saw in Chapter 13, a completely
e cient economy where the First Welfare Theorem holds may still have uctuations in GDP in response to
shocks of various kinds. One possible objective for monetary policy is to get the economy to behave the way it
would if prices weren't sticky, which would not entail complete stabilization. One challenge in implementing
this objective is that it's hard to know in real time what kinds of shocks are taking place, which makes it hard
to decide when monetary policy should refrain from attempting to stabilize GDP.
Expectations-Augmented Phillips Curve
For a long time it was believed that the Phillips Curve implied a fundamental tradeo: higher output (and
employment) was thought to necessarily go together with higher in ation. Policymakers, the argument went,
faced a choice: would they rather increase employment or lower in ation? If policymakers wanted to in-
crease employment, then the Phillips Curve implied that they had to be willing to tolerate higher in ation.
Conversely, if they wanted to combat in ation, then they had to be willing to tolerate lower employment.
No matter what choice the policymakers made, it was believed that macroeconomic policy oered the tools
to pick any point on the Phillips Curve. Figure 15.2.2 illustrates how, according to the model, monetary policy
can be used to increase output and employment, while inducing higher in ation. The gure shows an increase
in the money supply. This shifts the LM curve to the right, leading to higher GDP. Since marginal costs have
risen, exible price producers raise prices, generating in ation. Conversely, in order to reduce in ation, the
same policy can be used in reverse, which lowers in ation, employment and GDP.
The Phillips Curve de ned by equation (14.7.4) takes as given the level of sticky prices pS1 . Let's now go
back and think about how sticky-price producers set their prices. Assume that these producers are smart.
They understand that, once they have chosen a price, they will be stuck with it. So they will try to choose a
price that is approximately right on average. Speci cally, let's assume that they form an expectation of what
the exible price producers will do and then set their own price equal to that.
pS1 = E pF
1 (15.2.1)
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15.2. Monetary Policy
Fig. 15.2.2: The eects of monetary
expansion on GDP, in ation and in-
terest rates.
Now let's compute the price index. Using (14.7.2), we get:
p1 = µE pF F
1 + (1 − µ) p1 (15.2.2)
The average price is a weighted average of what sticky producers thought that exible producers would do
and what they ended up doing. The expected average price is then:
E (p1 ) = E µE pF F
1 + (1 − µ) p1
= E pF
1 (15.2.3)
so it's also equal to the expectation of what the exible-price producers will do.
295
15.2. Monetary Policy
Depending on what shocks and policies end up taking place, the actual average price might be dierent
from what the sticky price producers expected. De ne the price surprise as:
p1
≡ (15.2.4)
E (p1 )
measures the ratio between the actual average price and the expected average price. >1 means that the
price level turned out higher than expected, so there was higher-than-expected in ation. Conversely, <1
means lower-than-expected in ation. Replacing (15.2.2) and (15.2.3) into (15.2.4):
µE pF F
1 + (1 − µ) p1
= F
E p1
pF
1
= µ + (1 − µ) (15.2.5)
E pF1
Since the sticky price producers are stuck, the price surprise depends only on how the exible-price producers
deviated from what was expected.
Now turn to the exible price producers. They will set prices as a markup over marginal costs, according
to equation (14.7.3). Using (15.2.1) to replace pS1 , this reduces to:
v 0 (1 − L) η
pF F
1 = E p1
u0 (c1 (L)) F10 (L) η − 1
pF
1 v 0 (1 − L) η
⇒ = 0 (15.2.6)
E p1F u (c1 (L)) F10 (L) η − 1
Finally, replacing (15.2.6) into (15.2.5) and rearranging we get:
−µ v 0 (1 − L) η
= 0 0 (15.2.7)
1−µ u (c1 (L)) F1 (L) η − 1
What is equation (15.2.7) telling us? It says there is a positive relationship between and L, i.e. between
price surprises and employment. Once we take into account how the sticky price producers set their prices, the
model doesn't quite predict a relationship between in ation and employment. It only gives us a relationship
between higher-than-expected in ation and employment. Equation (14.7.4) gives us a Phillips Curve only
S
because we were holding p1 constant.
Rational Expectations, the Natural Rate and the Value of Commitment
Suppose it turns out that = 1, which means that the sticky price producers got their expectation of pF
1
exactly right. Then (15.2.7) reduces to:
v 0 (1 − L) η−1 0
= F1 (L) (15.2.8)
u0 (c1 (L)) η
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15.2. Monetary Policy
which is exactly condition (14.2.4) for the exible-price case. This implies that if there were never any
price surprises, the New Keynesian model would behave exactly like the RBC model.
1 Equation (15.2.8) is
sometimes described as a vertical Phillips Curve. It is vertical because in ation does not show up anywhere,
so if you were to plot it in terms of in ation against L you would nd a vertical line, with the same level of
employment for any possible level of in ation.
The level of economic activity that comes out of equation (15.2.8) is known as the natural rate (of
employment, unemployment, GDP, etc.). The natural rate is the answer to the question: if prices were
completely exible, what is the level of this variable that would prevail? One way of restating the meaning
of equation (15.2.7) is that output and employment can only deviate from their natural rate if there is higher-
that-expected or lower-than-expected in ation.
How exactly do people form their expectations? One popular hypothesis is that they form these expecta-
tions rationally.
2 What does this mean? This means that expectations are derived correctly from knowledge
of how the economy works, including how the government usually behaves, and are updated on the basis of
all available information. Does this mean that expectations are always correct, so that (15.2.8) always holds?
No. Even with rational expectations, people understand that random factors will cause in ation to dier from
what they predicted. But the predictions themselves are not systematically biased, so they are correct on
average.
If the rational expectations hypothesis is correct, higher average in ation cannot lead to higher average
employment. Rational expectations imply that producers expect the high average in ation, so it doesn't come
as a surprise and does not lead to higher output. Therefore, on average, output and employment are equal to
their natural rate, no matter what the level of in ation is. In other words, if expectations are rational, then
over longer periods there is no tradeo between in ation and employment, and the long-run Phillips Curve is
indeed vertical.
Suppose we accept the hypothesis of rational expectations. How much in ation should producers expect?
This depends on a lot of factors. One in particular is very important: how producers expect the government
to behave. Let's think a little bit about what the government might want to do and how that gets built into
expectations. Formally, what are going to do is describe a game, rst analyzed by Barro and Gordon (1983),
between the government and sticky-price producers. The government tries to pursue bene cial policies taking
as given the decisions that sticky-price producers have taken and the sticky-price producers, when setting
prices, try to anticipate what the government will do.
Let's put ourselves in the position of a government that has to decide on macroeconomic policy. Sticky-price
producers have already set their prices at pS1 = E (p1 ). The government knows that, by changing monetary
policy, it can change the level of GDP, and understands the relationship between GDP and in ation implied
by the expectations-augmented Phillips Curve (15.2.7). Assume the government is benevolent. What does the
government want to do?
∗
Let Y1 be the ideal level of GDP that the government would like to attain. What do we know about
Y1 ? First: Y1∗ is not in nity. In order to increase output, producers need to work more. At some point, the
∗
1 η−1
More precisely, like an RBC model where there is either a tax or monopoly power so that the term is present.
η
2
Another hypothesis is that they just extrapolate from recent experience: if in ation was 3% last year, they expect 3% again
this year. This is known as adaptive expectations.
297
15.2. Monetary Policy
marginal utility of leisure will be greater than the marginal utility of the consumption goods that you can
obtain by working harder. A benevolent government wants to increase GDP up to the point where equation
(13.1.3) from the RBC model holds, but no more. (Remember, in the RBC model, the First Welfare Theorem
holds)
Second: Y1∗ is greater than the natural rate of output Y1N , which is de ned by Y1N = F (L) with the level
of L that satis es (15.2.8). If we compare equations (13.1.3) and (15.2.8), we can see that the dierence comes
η−1
from the term
η , which measures monopoly power. If η → ∞, then we are back to perfect competition
and the natural rate of output coincides with what a benevolent government wants to attain. Away from that
limit, the government would like to undo the eects of monopoly power on the economy. The government
reasons: All these monopolist producers are producing a bit less than they should, reducing quantity in order
to keep their (relative) price high. Collectively, their eorts are self-defeating: they cannot all raise their price
relative to each other, and the only eect is to decrease total output. If, by raising demand, I can get them
all to increase output, this will make everyone better o .
Let's imagine that the government tries to balance two objectives: getting Y1 as close as possible to Y1∗
and keeping in ation close to zero. The government's objective is to maximize:
2
W = − (Y1 − Y1∗ ) − φπ12 (15.2.9)
Equation (15.2.9) says there are two things the government tries to avoid: GDP away from Y1∗ and in ation
away from zero. The parameter φ measures how much the government cares about each of the objectives. A
high value of φ means the government really dislikes in ation. The quadratic terms in the objective imply that
large deviations from the target are disproportionately more painful than small deviations.
3 In other words,
the marginal cost of deviating from target is increasing in the size of the deviation.
The government cannot just choose any values of Y1 and π1 it wants. If it could, the solution would be
simple: Y1 = Y1∗ and π1 = 0. Unfortunately, the government is constrained by the Phillips Curve: policies
that raise GDP will also raise in ation. Rather than work with the full-blown Phillips Curve (15.2.7), let's
propose a simpli ed version that captures the same idea:
π1 − E (π1 ) = a Y1 − Y1N
(15.2.10)
The economic content of equation (15.2.10) is just like that of equation (15.2.7): output will be above its
natural rate if and only if in ation is higher than expected. The only dierence is that we have written a
simpli ed, linear version, instead of the original (15.2.7). a is just a parameter which governs the slope of the
Phillips Curve; higher a means a steeper Phillips Curve. Note that the Central Bank takes E(π1 ) as given: by
the time the Central Bank chooses policies, sticky-price producers have already made their decisions based on
E(π1 ) and cannot change them.
3
We'll just take the fact that the function is quadratic as an assumption, although it can be justi ed as a second-order Taylor
approximation to the representative household's utility.
298
15.2. Monetary Policy
The Central Bank solves:
2
max − (Y1 − Y1∗ ) − φπ12
Y1 ,π1
s.t. (15.2.11)
N
π1 − E (π1 ) = a Y1 − Y1
Fig. 15.2.3: The Central
Bank's in ation-output trade-
o given a Phillips Curve.
Figure 15.2.3 shows the Central Bank's problem graphically. Once expectations have been set at E(π1 ),
the Phillips Curve (15.2.10) is a constraint which limits the combinations of output and in ation that are
attainable. The slope of the Phillips Curve is given by the parameter a and it goes through the point
Y1N , E(π1 ) : if in ation is equal to expectations, output will be at its natural level. The objective function
∗
(15.2.9) implies that the Central Bank's indierence curves are ellipses centered on the ideal outcome Y1 = Y1 ,
π = 0: the further away from this outcome, the worse it is for the Central Bank. The Central Bank optimizes
by choosing a point that is on the best indierence curve that is consistent with the Phillips Curve.
The Lagrangian for problem (15.2.11) is:
2
L(Y1 , π1 , λ) = (Y1 − Y1∗ ) − φπ12 − λ π1 − E (π1 ) − a Y1 − Y1N
with rst order conditions::
−2 (Y1 − Y1∗ ) + λa = 0
−2φπ1 − λ = 0
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15.2. Monetary Policy
and therefore:
−2 (Y1 − Y1∗ ) − 2φπ1 a = 0 (replacing λ = −2φπ1 )
π1 − E (π1 )
−2 + Y1N − Y1∗ − 2φπ1 a = 0 (using (15.2.10) to replace Y1 )
a
a 1
Y1∗ − Y1N +
⇒ π1 = 2
E (π1 ) (15.2.12)
1 + φa 1 + φa2
Equation (15.2.12) tells us what in ation level the government will choose as a function of:
• The distance between the target level of output and the natural level Y1∗ − Y1N . If this distance is large,
the government's desire to raise output is strong, so the government will be willing to bring about higher
in ation in order to increase output.
• Expected in ation E (π1 ). If in ation expectations are high, achieving low actual in ation means creating
a negative in ation surprise, which is costly in terms of output. Therefore the government will respond
to higher expected in ation with higher actual in ation. Notice that
∂π1 1
= <1
∂E (π1 ) 1 + φa2
so the government responds less than one-for-one to expected in ation. If the Phillips Curve is very
steep (high a) or the government strongly dislikes in ation (high φ) then it will respond little to in ation
expectations. Instead, if the Phillips Curve is at or the government doesn't mind in ation very much
it will be more responsive of in ation expectations.
Now let's impose the hypothesis of rational expectations. In this context, rational expectations means that
sticky-price producers have gured out (15.2.12). They understand what the government is trying to do and
how it trades o its dierent objectives. Therefore the rational way to set expectations is to set
E (π1 ) = π1 (15.2.13)
What are (15.2.12) and (15.2.13) telling us? The government sets in ation in response to in ation expec-
tations, but under rational expectations, expected in ation rationally anticipates what the government will
want to do. Using the rational expectations hypothesis, we can solve for what output and in ation will be.
First, replace (15.2.13) into (15.2.12) and solve for π1 :
a 1
Y1∗ − Y1N +
π1 = 2
π1
1 + φa 1 + φa2
1
Y ∗ − Y1N
⇒ π1 = (15.2.14)
φa 1
Now replace (15.2.13) into (15.2.10) and solve for Y1 :
0 = a Y1 − Y1N
⇒ Y1 = Y1N (15.2.15)
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15.2. Monetary Policy
Equations (15.2.14) and (15.2.15) tell us what the level of output and in ation will end up being. Let's
focus on (15.2.15) rst. This says that output will be exactly at the natural rate, so the government will not
have any success at all in raising it. Why is this? According to (15.2.10), the only way to raise output above
the natural rate is with higher-than-expected in ation. But under rational expectations, in ation cannot be
higher-than-expected.
4 Hence the government's attempts to raise output will be futile.
Figure 15.2.3 shows this result graphically. If expectations were set at some level like E(π1 )0 , the Phillips
Curve would be the dotted line. The Central Bank's optimal decision would be to choose output Y0 and
0
in ation π1 . But this in ation does not coincide with what the market expected, so this outcome is inconsistent
with rational expectations. The rational-expectations outcome is for the market to expect E(π1 ), which leads
N
to the solid Phillips Curve. When faced with this Phillips Curve, the Central Bank chooses Y and π1 = E(π1 ),
which con rms the market's expectation.
Let's now think about what (15.2.14) is telling us. This economy will experience positive in ation. In ation
will be higher if Y1∗ − Y1N is high, φ is low or a is low. Why is this? Each of these factors means that, taking
E (π1 ) as given, it is very attractive for the government to choose higher in ation. High Y1∗ − Y1N means that
raising GDP above the natural rate is highly desirable; low φ means that in ation is not too unpleasant; low
a (a not-too-steep Phillips Curve) means that the rise in in ation per unit of additional GDP is not large.
Due to rational expectations, the factors that make choosing higher GDP and in ation desirable are fully
anticipated, so they end up leading to in ation, but not to higher GDP.
The government in this problem faces what's known as a time-inconsistency problem. What does this
mean? Imagine that the government could announce a level of in ation before sticky producers set their
prices, and was then committed to sticking to the announcement. This would change the government's
problem entirely. If the government is committed to an in ation level, there can never be surprise in ation,
and hence GDP will be at its natural rate. Knowing that it will be committed, the government no longer has
any reason to choose a level of in ation higher than zero, so we'd end up with π1 = 0 and Y1 = Y1N . This
is a strictly better outcome than the problem with no commitment: it has the same level of GDP but lower
in ation. Notice that it's important for the government to actually commit to this. If, once expectations have
been set, the government could disregard its commitments, it would want to deviate from its announced plan
and set the in ation given by (15.2.12) instead. Time inconsistency refers to the fact that the government
would like to commit to an outcome, but then has incentives to undo this commitment.
This type of argument has been extremely in uential in the design on macroeconomic policy institutions.
In the last couple of decades many countries have introduced reforms to make their central banks more
independent of elected governments. The idea behind this is to try to isolate monetary policy from the forces
that push policymakers (for entirely benevolent reasons!) to renege on their commitment to low in ation.
Rogo (1985) argued that one way in which society could deal with the time-inconsistency problem is by
appointing a conservative central banker, i.e. Central Bank authorities who dislike in ation more than the
average person. The logic is that if the central banker's preferences have very high φ, then equation (15.2.14)
implies that in ation will be lower, and output will end up at its natural rate anyway.
4
In this model there are no shocks. In a model with shocks, rational expectations would mean that in ation cannot be higher
than expected on average; there could be shocks that lead to higher-than-expected in ation as long as there are other shocks that
lead to lower-than-expected in ation.
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15.3. Monetary Policy Regimes
15.3 Monetary Policy Regimes
As we have seen, expectations in general, and expectations about monetary policy in particular, are important
determinants of economic decisions such as setting prices. This forces us to think about monetary policy
regimes and not just isolated monetary policy decisions. A monetary policy regime is a set of norms, objectives
or procedures that govern how monetary policy is chosen. If expectations are rational, they will depend on
the type of policy regime that is in place. One possible regime, which is implicit in the Barro and Gordon
(1983) game, is for the Central Bank to have full discretion to pursue objective (15.2.9) at every point in time.
As we saw, this tends to produce more in ation than is socially desirable even if (especially if !) the Central
Bank is benevolent. Other monetary policy regimes attempt to overcome the in ation bias that comes with
discretionary policy.
Money Growth Rules
One extreme form of non-discretionary policy is for the Central Bank to commit to keeping the money supply
growing at a constant rate, for instance 2% per year, or even not growing at all. If this commitment is rm
(a big if !), then the Central Bank is not be able to choose a point in the output/in ation tradeo and thus
cannot yield to the temptation to increase GDP and in ation with monetary expansion: monetary policy is
chosen automatically by whatever the money-growth rule speci es. If expectations are rational (another big
if !), they anticipate this, so a su ciently tight money-growth rule can be eective in keeping in ation low.
One problem with a money-growth rule is that it is quite in exible. (That's the point, it's designed to be
in exible). This means that it prevents monetary policy from acting to stabilize the economy in response to
shocks. For instance, the actions illustrated in Figure 15.2.1, where the money supply is increased to bring
down interest rates and stabilize GDP, would be precluded by a strict money-growth rule. The in exibility
of a money-growth rule may lead the economy to be more volatile than it otherwise could be. Partly as a
result of this, money growth rules have fallen out of favor in recent years. Exercise 15.8 asks you to go over
an example where changes in transactions technology create volatility under a money-growth rule.
The Gold Standard and Fixed Exchange Rates
Another extreme form of a non-discretionary policy, which was important historically, is the so-called Gold
Standard. Under the simplest version of this system, the Central Bank commits to exchange currency or
Central Bank reserves for gold at a xed exchange rate. Modern versions, which are conceptually quite
similar, use a foreign currency like the US dollar or the Euro instead of gold. If the Central Bank is committed
to a xed exchange with respect to either gold or a foreign currency (again, a big if ), this removes any discretion
it might have in setting monetary policy: it must adjust the monetary base in response to any requests to
exchange gold for currency or vice versa. This prevents the Central Bank from actively choosing a point in
the in ation/output tradeo. If the regime is credible, this can be eective in keeping in ation low.
It has the disadvantage that it xates on the price of one particular good. In the gold standard, it's the
price of gold. In a xed exchange rate, it's the price of the US dollar (or more broadly, the imported goods
that can be bought for US dollars at relatively stable nominal prices). If the relative price of this good with
302
15.3. Monetary Policy Regimes
respect to other goods changes, then that will have side-eects on the economy. For instance, suppose that
jewelry becomes very fashionable, so gold becomes expensive relative to other goods. The Central Bank has
committed to keeping the nominal price of gold in terms of currency xed. This means that the nominal price
of other goods has to fall (otherwise, everyone would rush to sell their currency to the Central Bank to get
gold). The Central Bank will be forced to contract the money supply, creating de ation and a fall in GDP.
These types of side eects led Keynes to call the gold standard a barbarous relic.
In ation Targeting
In ation targeting has become popular in recent decades. Under this regime, the Central Bank is given a
formal mandate to keep in ation as close as possible to a pre-speci ed target (2% per year is a typical gure).
In the purest version, the Central Bank is supposed to ignore any other objective it might have, such as
keeping employment high, and just focus on keeping in ation as close as possible to the target. In principle,
this approach can allow the Central Bank to respond to shocks without giving it the full discretion in setting
policy that can lead to in ation bias.
Figure 15.3.1 shows how this would work, continuing the example from Figure 15.2.1. In the example,
worsening expectations about the future make the IS curve shift to the left. If the Central Bank did not react
to this (for instance, because it was following a zero-money-growth rule), then output would fall from Y1 to Y10
0
and the interest rate would fall to i. This would induce movement along the Phillips Curve, so that the price
0
level would be p1 instead of p1 . A Central Bank that followed an in ation target would react to this because
it doesn't want in ation to fall below its target. In order to keep the price level at p1 it will increase the
money supply, bringing the interest rate all the way down to i00 . This would have the double eect of keeping
the in ation rate on target and restoring output to its original level. Therefore even though the Central Bank
is instructed to only worry about in ation, by doing so it also stabilizes output and employment. Blanchard
and Gali (2007) refer to this as a divine coincidence.
Figure 15.3.2 shows how in ation targeting would be applied to the example from Figure 14.7.3. In the
example, a temporary increase in productivity leads to a shift in the Phillips Curve. If there is no reaction
from monetary policy, output does not change and employment falls. However, staying at the original output
Y1 while the Phillips Curve has shifted down would mean that the price level is p01 instead of p1 , so the Central
Bank would miss its in ation target. (In the background, what's going on is that exible-price rms set lower
prices because higher productivity means lower marginal costs). In order to hit its in ation target, the Central
Bank increases the money supply, lowering the interest rate until the increase in consumption and investment
means that output reaches Y10 , which restores the intended price level. In this example, in ation targeting
does not stabilize output. Instead, it makes output react to the shock in a way that it wouldn't under a
xed-money-growth rule. Note, however, that in the Pareto e cient RBC economy, output would react to a
productivity shock: if productivity is temporarily high it is desirable to produce more output.
Taylor Rules
A Taylor Rule, named after Taylor (1993), is similar to an in ation target but with a built-in procedure for
correcting mistakes. The Central Bank rst sets a target π∗ for the level of in ation it wants to attain. Then,
303
15.3. Monetary Policy Regimes
Fig. 15.3.1: Monetary policy under
in ation targeting.
at any point in time it sets (i.e. targets by adjusting the money supply) the nominal interest rate i according
to a formula like:
i = rN + π + a(π − π ∗ ) + b(Y − Y N ) (15.3.1)
where:
• YN is the natural level of real GDP; Y is its current level.
• π∗ is the target level of in ation; π is its current level.
• rN is the natural level of the real interest rate.
• a and b are parameters; a typical level is a = b = 0.5.
What's the idea behind a Taylor Rule? Monetary policy attempts to keep the economy at the natural level
304
15.3. Monetary Policy Regimes
Fig. 15.3.2: In ation targeting re-
sponse to a productivity shock.
of GDP YN and the in ation target π∗ . Note that the target is the natural level of GDP YN and not the
rst-best level Y ∗, which would necessitate higher-than-expected in ation. If the Central Bank is succeeding
in keeping both GDP and in ation on target, then it just sets the nominal interest rate at i = rN + π∗ : the
N N
natural real interest plus target in ation. (One practical challenge is to accurately measure Y and r in
real time). If in ation starts to deviate above the target, the Central Bank adjusts the interest rate upwards.
Crucially, since a > 0:
∂i
=1+a>1
∂π
so the nominal interest rate adjusts more than one-for-one to deviations of in ation from its target. This is
sometimes known as the Taylor principle. It means that if in ation starts to rise above its target, the Central
Bank increases the real interest rate. This implies a upward/leftward movement along the IS curve, lowering
GDP, and a downward/leftward movement along the Phillips Curve, lowering in ation. In other words, if
305
15.3. Monetary Policy Regimes
in ation starts to get out of hand, the Central Bank is willing to generate a recession in order to bring it back
in line. Conversely, if the economy starts to experience a recession, with Y below its natural level Y N, then
the Central Bank will lower the interest rate, generating in ation and a rise in GDP. The Central Bank is
willing to tolerate some in ation in order to prevent output from falling below its target.
One useful metaphor is to think of the Taylor Rule as a thermostat. A thermostat adjusts the intensity
of the boiler to keep the temperature close to a target. Under a Taylor Rule, the Central Bank adjust the
interest rate to keep both in ation and GDP close to their targets. The parameters a and b indicate how
strongly the Central Bank responds to deviations in in ation and GDP respectively. Pure in ation targeting
would correspond to a→∞ and b = 0, where the Central Bank does whatever it takes to keep in ation at its
target and does not look at GDP.
Interpreting the Historical Data
We saw in Chapter 12 that the Phillips Curve relationship sometimes seems to hold but not always. Let's
see if we can make sense of that by considering dierent possible policy regimes and dierent shocks that the
economy might experience.
Suppose rst that monetary policy does not respond to macroeconomic shocks (for instance because we
are under a xed-money-growth regime) and the main shocks have to do with shifts in either the IS or the
LM curve: changes in expectations of future productivity as in Figure 14.5.2; in the money supply as in
Figure 14.5.3; in government spending, as in Figure 15.1.1, etc. In this case expectations of in ation will be
approximately constant, so whenever in ation deviates from its usual level it comes as a surprise. Then any
shocks will lead to movements along a xed Phillips Curve. Expansionary shock will lead to higher output,
employment and in ation, and vice versa. If we look at data generated by an economy like this, we will see a
clear Phillips Curve. Arguably, this was a plausible description of the US economy until the mid-1960s.
Suppose instead that monetary policy follows in ation targeting (and follows it perfectly) and shocks
consists of some mixture of shocks that move the IS curve and productivity shocks that, as we saw in Figure
15.3.2, do not. Then we will see that in ation is almost constant no matter what the shocks are. For shocks
that move the IS curve, monetary policy will respond so that neither in ation nor GDP react at all. For
productivity shocks, monetary policy will respond as in Figure 15.3.2. In ation will not react but GDP and
employment will. If we look at data generated by an economy like this, the Phillips Curve will look almost
perfectly at. Arguably, this has been a plausible description of the US economy since the late 1980s.
5
Now suppose that the monetary policy regime is not entirely clear, so the main thing that happens is
the people keep changing their in ation expectations in a way that does not exactly correspond to rational
expectations. Suppose further that the government reacts to changing in ation expectations the way equation
(15.2.12) and Figure 15.2.3 say it will: if expected in ation rises, then the government pursues higher in ation,
but less than one-for-one, and vice versa. This means that when expected in ation E(π) rises, actual in ation
π rises but unexpected in ation falls. If we go back to the expectation-augmented Phillips Curve (15.2.7)
or its simpli ed version (15.2.10), this means that output and employment will be lower. If we look at data
generated by this economy, it will look like there's a Phillips Curve in the opposite direction than usual!
5
See Fitzgerald and Nicolini (2014) and McLeay and Tenreyro (2019) for a discussion of this point.
306
15.4. The Liquidity Trap
What's going on? If they expect high in ation, sticky-price producers raise their prices. The Central Bank
then faces an unpleasant tradeo. If it wants to maintain low overall in ation, it must persuade exible price
producers to lower their prices to balance out the increases from sticky price producers. The only way to do
so is to engineer a recession in order to lower marginal costs. Instead, if the Central Bank wants to avoid a
recession it must tolerate higher in ation. According to (15.2.12), the Central Bank chooses to compromise
and tolerates both higher in ation and lower output and employment. This outcome is sometimes known as
stag ation (for stagnation plus in ation). Conversely, if in ation expectations fall, the Central Bank will
take advantage of this to obtain both higher output and lower in ation. Arguably, this pattern is a plausible
description of what went on from the 1970s until the mid-1980s.
15.4 The Liquidity Trap
In Section 15.2 we looked at how monetary policy can be used to oset negative shocks to the economy.
Lowering the interest rate (or, more precisely, increasing the money supply so that the LM curve shifts,
leading to a lower interest rate) produces a movement along the IS curve, which can oset the eect of
negative shocks on GDP. Now we'll see that this type of policy has some limits.
The Zero Lower Bound on Nominal Interest Rates
Figure 15.4.1 shows how the LM curve shifts as the money supply increases. As we know, a higher money
supply shifts the LM curve down and to the right: if there is more money around, people will only hold it if
either they need to carry out more transactions or the opportunity cost falls. However, the LM curve never
goes below i = 0. Why not?
Fig. 15.4.1: The LM curve
and the zero-lower-bound on
interest rates.
307
15.4. The Liquidity Trap
Remember, the LM curve is just the representation of the money-market equilibrium condition. If i = 0,
there is no opportunity cost of holding money: other assets like bonds also pay zero interest. If interest rates
reach this point, then further increases in the money supply cannot lower the interest rate any further: people
are perfectly willing to hold more and more money instead of other assets. In other words, since money always
pays zero interest, it cannot be the case that other assets pay negative interest rates because people would
just hold money instead.
6
Limits on Monetary Policy
Suppose now that an economy suers a large negative shock, as shown in Figure 15.4.2. This is like the shock
we looked at in Section 15.2, just larger. In fact, the negative shock is so large that even bringing the interest
rate all the way down to zero with very expansionary monetary policy is not enough to restore output to its
previous level. This situation is known as a liquidity trap.
Fig. 15.4.2: The liquidity
trap.
It's a trap in the sense that conventional monetary policy has no power to help the economy escape. It is
sometimes said that expanding the money supply in a liquidity trap is like pushing on a string.
7
6
This argument has been tested recently. Some countries like Switzerland and Sweden have had negative nominal interest
rates. It turns out that the theoretical argument that once the interest rate becomes negative people would hold all their wealth
as physical cash in a safe deposit box in order to earn zero interest is not exactly right. Storing physical currency has its own
disadvantages: it can get stolen or lost, safe deposit boxes are costly and, unlike bank deposits, physical cash cannot be used to
make online payments. Still, it is believed to be unlikely that interest rates could be very negative for very long.
7
This metaphor is often attributed to Keynes, but it's unclear whether he is the original source.
308
15.4. The Liquidity Trap
Fiscal Policy in a Liquidity Trap
Many economists have argued that since monetary policy is ineective, when an economy is in a liquidity trap
it would be a good idea to use scal policy instead. As we saw above, an increase in G leads to a shift in the
IS curve, so in principle this can be used to oset the eects of a negative shock.
In fact, as long as the economy is in a liquidity trap, there would be no crowding out eect from higher
government spending, because the LM curve is at at zero, so the shift in the IS curve would translate one-
for-one into higher output. Part of the argument in favor of the American Recovery and Reinvestment Act of
2009 was precisely this. The economy was in a deep recession and the interest rate was already very close to
zero, so there was little scope to restore usual levels of employment and GDP using monetary policy alone.
Therefore, it was argued, an increase in government spending was the main tool of macroeconomic policy
available. Exercise 15.6 asks you to look at the importance of timing in this type of scal policy.
Forward Guidance
What else can be done when an economy is in a liquidity trap? Krugman (1998) famously argued that it
would be useful if the Central Bank could credibly promise to be irresponsible. What does this mean? A
Central Bank is often described as responsible if it is committed to pursuing low in ation without falling
into the temptation to try to push output above its natural level. However, if the economy nds itself in a
liquidity trap, it may actually be useful for the Central Bank to persuade the public that it will pursue high
in ation. Let's see why that may be.
Let's go back to the beginning our our analysis. One of the things we are holding constant is expected
in ation between periods 1 and 2, which we denoted π. Expected in ation matters because it determines
how nominal interest rates are converted to real interest rates. We haven't really said very much about where
this expected in ation comes from. Given what we saw in Chapter 11 about the relationship between money
and in ation, it seems reasonable to assume that it depends, at least in part, on expectations about future
monetary policy. Suppose that the Central Bank could make a credible announcement of what monetary
policy is going to do in the future, and thereby the Central Bank could aect π. What would the Central
Bank want to do?
As we saw in Figure 14.5.4, higher expected in ation results in an upward shift of the IS curve, because it
lowers real interest rates for any given level of nominal interest rates. The combination of promising higher
in ation with keeping nominal interest rates at zero makes the real interest rate negative. According to the
model, this can succeed in raising output when the economy is in a liquidity trap. Therefore, in a liquidity
trap it could be useful for the Central Bank to convince the public that it will not keep in ation low.
More broadly, it is increasingly recognized that communication about future policy is a very important
aspect of how central banks do their job. Central banks are increasingly choosing to provide forward guidance,
i.e. indications of what they plan to do in the future, as a way to exert in uence on the economy by changing
expectations.
309
15.4. The Liquidity Trap
Exercises
15.1 Interaction Between Fiscal and Monetary Policy
The US government has decided to go to war in order to conquer Canada an incorporate it as the 51st
US state. The Canadian government has politely agreed to carry out the war according to the following
rules:
• Each country will build a large amount of tanks and set them on re.
• The country whose tanks make the most noise will be declared the winner of the war.
• If the US wins the war, Canada will become a US state; if Canada wins the war we will leave them
alone.
• Either way, no policies will change in any of the two countries, nothing besides the tanks will be
destroyed and no one will be hurt.
In preparation for the war, the US government orders a large amount of new tanks from its suppliers
of military equipment.
Suppose throughout that the economy is well described by a New Keynesian model with partially sticky
prices.
(a) Suppose the Federal Reserve follows a constant-money-supply policy.
i. What will happen to GDP in the US?
ii. What will happen to nominal interest rates?
iii. What will happen to the price level?
(b) Suppose now that the Federal Reserve decides to adjust the money supply to keep nominal interest
rates constant.
i. What does the Federal Reserve need to do to the money supply?
ii. How does the reaction of GDP compare to part (a)?
iii. How does the reaction of the price level compare to part (a)?
(c) Suppose now that the Federal Reserve follows a strict in ation-targeting policy.
i. What does the Federal Reserve need to do to the money supply?
ii. How does the reaction of GDP compare to part (a)?
iii. How does the reaction of nominal interest rates compare to part (a)?
15.2 Money Demand and Fiscal Policy
Let's look at some special cases of money-demand functions:
• Case 1:
mD = a · Y0
310
15.4. The Liquidity Trap
• Case 2:
mD = b − x · i1
where a, b and x are positive constants.
In case 1 the money demand depends on GDP but is completely insensitive to interest rates. Case 2 is
the opposite: money demand depends on the nominal interest rate but not on GDP.
(a) Find an expression for money velocity in each of the two cases. Does the quantity theory of money
hold in each case?
(b) Draw the LM curve that results from each of these two assumptions. How does the LM curve shift
in response to an increase in the money supply?
(c) Suppose the government decides to undertake a scal expansion, i.e. increases G.
i. How does that shift the IS curve?
ii. What is the eect on interest rates and on GDP in case 1 and case 2 respectively?
iii. Suppose the scal expansion was undertaken with the objective of increasing GDP. You are
trying to judge whether the policy is working, but you don't have the latest GDP gures yet.
You do, however, have very good data on interest rates. How could you use the IS-LM model
together with data on interest rates to get a sense of the eectiveness of the policy?
15.3 Taxes and Spending in the Old Keynesian Model
Suppose household consumption behavior is well described by the Old Keynesian model, with a con-
sumption function:
c1 = a + b(Y1 − τ1 )
where a and b are constants, b < 1 and τ1 is the level of taxes collected by the government, so that Y1 − τ1
is the household's after-tax income. The level of government spending is G.
(a) Use equation (15.1.3) and the implicit function theorem to compute how much the IS curve shifts
to the right in response to an increase in government spending G and in response to a decrease in
∂Y1 ∂Y1
taxes τ1 , i.e. compute
∂G and ∂τ1 .
(b) If this model is correct, will period-1 GDP increase more in response to an increase in government
spending of 100 million dollars or a tax cut of 100 million dollars? Explain.
15.4 Patriotic Consumption
The President has just tweeted:
We need to help our economy. I'm going to ask that each of you go out and buy stu. If you
all do that, that'll get the economy going.
Suppose the typical household reacts to the announcement as follows:
I wasn't planning to do this, but the President seems like such a wise leader that I'm going to
do my patriotic duty and buy myself a lawnmower.
311
15.4. The Liquidity Trap
(a) Describe one model according to which the overall eect of the appeal to patriotic consumption is
detrimental to welfare. Describe what happens and why it's detrimental.
(b) Describe one model according to which the overall eect of the appeal to patriotic consumption has
desirable consequences. Describe what happens and why its desirable.
You can make the argument using equations, graphs or words, as long as you are precise.
15.5 Taxes and In ation
Suppose the economy is well described by a New Keynesian model with partially sticky prices. The
government wants to bring down in ation. The Central Bank, for some reason, is unable or unwilling to
change monetary policy so the government decides to try to use scal policy. The rst idea it considers
is to lower the level of government spending but it decides not to do it because this would cut into public
services that are considered too important. Two other proposals are considered:
• Proposal 1: An immediate, temporary, increase in the level of taxes, done in a lump-sum way:
everyone must pay an extra ∆ in taxes this year.
• Proposal 2: An immediate, temporary, increase in consumption taxes: everyone must pay extra
taxes in proportion to this year's consumption
Suppose that the size of the tax increase is such that the government will raise the same revenue from
both plans.
(a) If the present and future level of government spending is unchanged, what should households expect
about future taxes?
(b) Will either or the two policies be eective in lowering in ation?
i. If yes, what is the mechanism?
ii. If no, why not?
(c) Suppose a lot of households are borrowing-constrained, how does that aect the answer?
15.6 The Timing of Fiscal Policy
Equation (15.1.1) describes how to modify the New Keynesian IS equation when there is government
spending in period 1. Now imagine that there is government spending in both periods: G1 and G2 .
(a) Write down the market-clearing condition for goods in period 2.
(b) Derive an IS equation that includes both G1 and G2 .
(c) How does a change in G2 move the IS curve? Interpret what this means.
(d) Suppose that the government wants to use increased spending (higher G1 ) to raise GDP. However,
it chooses some spending projects that will take some time to get started, so it ends up increasing
G2 instead, leaving G1 unchanged. What will happen?
312
15.4. The Liquidity Trap
15.7 Government Spending in the Utility Function
Suppose that government spending enters the utility function (because households care about public
goods) and consider two possible utility functions:
u (ct , Gt , lt ) = u (ct ) + w (Gt ) + v (lt ) (15.4.1)
u (ct , Gt , lt ) = u (ct + Gt ) + v (lt ) (15.4.2)
(a) Explain in words what each of these two utility functions mean.
(b) What would the Euler equation for intertemporal choice look like under each of these utility func-
tions?
(c) Derive an IS equation in each case.
(d) Assume prices are sticky. How eective are increases in Gt in increasing output in each of the two
cases? Explain why.
15.8 ATMs
Assume that the economy is well described by the New Keynesian model with partially sticky prices.
As we did in Exercise 10.4 suppose that one day, suddenly and unexpectedly, ATMs are invented, which
make getting cash more convenient than before.
(a) If the Central Bank follows a policy of keeping the money supply constant, what will happen to
GDP, interest rates and prices?
(b) Describe in detail what the Central Bank should do if it wanted to prevent GDP from changing in
response to this change.
15.9 The Bond Market
Suppose that the Central Bank is known to use monetary policy to try to stabilize GDP. Suppose that a
new employment report is released, which suggests that employment growth is lower than people thought
it was going to be. What will happen to the price of government bonds? Explain.
15.10 Reputation
In this exercise we are going to think about how the government can build a reputation. The government
and producers play the game we looked at in Section 15.2, with two dierences:
• Instead of just playing the game once, they are going to play the same game twice.
• Producers don't actually know the value of φ (i.e. how much the government dislikes in ation).
They are going to try to gure it out by watching what the government does the rst time they
play the game.
Let's start with the rst time they play the game. Producers don't know the value of φ but they have a
guess, which we denote by φ̂. Assume that producers believe that the government's preferences are given
by φ̂ and that the government will play the rst game just like the single-game case.
313
15.4. The Liquidity Trap
(a) What level of in ation do they expect to see? Call this level of in ation πE .
(b) Now assume that (i) φ is not actually equal to φ̂ (it could be higher or lower) and (ii) the government
does indeed behave in the rst game just like in the single-game example. What level of in ation
does the government actually choose?
(c) Now suppose that in preparation for the second game, producers try to gure out the true value
of φ by looking at what the level of in ation turned out to be. Use your result from part (b) to
derive an expression for how the new guess about about φ (let's call this φ∗ ) depends on the level
∗
of in ation. Explain in words how φ depends on π and why.
(d) Now let's consider the second game. If producers' guess about φ is φ∗ , what level of in ation do
they expect?
(e) Find an expression for W (the level of welfare the government attains in the second game) as a
∗
function of φ and φ . Does the government want producers to think that φ is high or low? Why?
(f ) Now go back to thinking about the rst game. Suppose that in the rst game, the government is not
just trying to maximize W (as we assumed in part (b)), but is also trying to aect how producers
∗
will form their beliefs φ .
i. Will they choose higher or lower in ation than what you found in part (b)? Why?
ii. Will GDP be higher or lower than what you found in part (b)?
314
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Jerey D. Sachs. Tropical underdevelopment. 2001. NBER Working Paper 8119.
Thomas J Sargent. The ends of four big in ations. In In ation: Causes and eects, pages 4198. University
of Chicago Press, 1982.
Thomas J. Sargent and François R. Velde. The Big Problem of Small Change. Princeton University Press,
2014.
Robert Shimer. The cyclical behavior of equilibrium unemployment and vacancies. American Economic
Review, 95(1):2549, 2005.
Gene Smiley. Recent unemployment rate estimates for the 1920s and 1930s. Journal of Economic History, 43
(2):487493, 1983.
Robert M. Solow. A contribution to the theory of economic growth. Quarterly Journal of Economics, 70(1):
6594, 1956.
Lawrence H. Summers. Some Skeptical Observations on Real Business Cycle Theory. Federal Reserve Bank
of Minneapolis Quarterly Review, 10(4):2327, 1986.
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George G. Szpiro. Measuring risk aversion: an alternative approach. Review of Economics and Statistics, 68
(1):156159, 1986.
John B. Taylor. Discretion versus policy rules in practice. In Carnegie-Rochester Conference Series on Public
Policy, volume 39, pages 195214, 1993.
Richard H. Thaler and Cass R. Sunstein. Nudge: Improving Decisions about Health, Wealth, and Happiness.
Yale University Press, 2008.
James Tobin. The interest-elasticity of transactions demand for cash. Review of Economics and Statistics, 38
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World Bank. Doing Business 2012. 2011.
319
Kurlat has filled an essential void by producing a text, accessible to undergraduates
with a moderate amount of mathematical training, that introduces students to the
frontier of modern macroeconomics. I will be teaching from it this year and
recommend it to instructors at the intermediate or masters level and to
mathematically-inclined students who want to learn what macroeconomics is all
about.
Gabriel Chodorow-Reich, Harvard University
This outstanding book covers modern macroeconomic ideas with extreme rigor but
without heavy math and keeping the focus on real-world applications and policy
implications. Readers will find a very accessible coverage of microeconomic
foundations and a thoughtful treatment of long-run and short-run macroeconomic
models. Every instructor who teaches undergraduate macroeconomics at an
intermediate or advanced level should consider using this book.
Alp Simsek, Massachusetts Institute of Technology
A fantastic introduction to macroeconomics for advanced undergraduates. I use it as
background reading for my masters level course at the LSE on growth as well. What
makes the book stand out is a very accessible, indeed, a sparkling conversational
style combined with analytical rigour and a masterly overview of the literature. It is
rare to find a textbook that is so lucidly written and yet analytically so solid.
Maitreesh Ghatak, London School of Economics
Pablo Kurlat is Associate Professor of
Economics at the University of
Southern California. He got his Ph.D. in
Economics from MIT in 2010.
His research focuses on information
economics, monetary policy and
financial markets.
This book grew out of his teaching
notes for the Intermediate
Macroeconomics course he used to
teach at Stanford University.
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