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Intermediate Macroeconomics
Dirk Krueger1
December 21, 2001
1
I would like to thank Charles Jones, Felix Kubler, Beatrix Pall and Tom Sargent
for stimulating discussions about teaching modern macro. All remaining errors are
mine.
ii
Contents
1 Introduction
1.1 The Scope of Macroeconomics . . . . . . . . .
1.2 US Macroeconomic Data: A Helicopter Tour
1.2.1 Real GDP . . . . . . . . . . . . . . . .
1.2.2 Digression: The Rest of the Course . .
1.2.3 Other Macroeconomic Aggregates . .
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1
1
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2 National Income and Product Accounting (NIPA)
2.1 Gross Domestic Product (GDP) . . . . . . . . . . . .
2.1.1 Computing GDP through Production . . . .
2.1.2 Computing GDP through Spending . . . . .
2.1.3 Computing GDP through Income . . . . . . .
2.2 Price Indices . . . . . . . . . . . . . . . . . . . . . .
2.3 From Nominal to Real GDP . . . . . . . . . . . . . .
2.4 Measuring In‡ation . . . . . . . . . . . . . . . . . . .
2.5 Measuring Unemployment . . . . . . . . . . . . . . .
2.6 Measuring Transactions with the Rest of the World .
2.7 Appendix A: More on Growth Rates . . . . . . . . .
2.8 Appendix B: Chain-Weighted GDP . . . . . . . . . .
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3 Economic Growth
3.1 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . .
3.1.1 Discrete vs. Continuous Time . . . . . . . . . . . . . . . .
3.1.2 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Some Useful Facts about Logs . . . . . . . . . . . . . . .
3.1.4 Growth Rates (once again) . . . . . . . . . . . . . . . . .
3.1.5 Growth Rates of Functions . . . . . . . . . . . . . . . . .
3.1.6 Simple Di¤erential Equations and Constant Growth Rates
3.2 Growth and Development Facts . . . . . . . . . . . . . . . . . . .
3.3 The Solow Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Setup of the Basic Model and Model Assumptions . . . .
3.3.3 Analysis of the Model . . . . . . . . . . . . . . . . . . . .
3.3.4 Introducing Growth . . . . . . . . . . . . . . . . . . . . .
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CONTENTS
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4 Business Cycle Fluctuations
4.1 Potential GDP and Aggregate Demand . . . . . . . . . . . . . .
4.2 The IS-LM Framework . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 The Balance of Income and Spending: Keynesian Cross
and Multiplier . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Investment, the Interest Rate and the IS Curve . . . . . .
4.2.3 The Demand for Money and the LM-Curve . . . . . . . .
4.2.4 Combination of IS-Curve and LM-Curve: Short-Run Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.5 Monetary and Fiscal Policy in the IS-LM Framework . . .
4.3 The Aggregate Demand Curve . . . . . . . . . . . . . . . . . . .
4.4 Unemployment . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Concepts and Facts . . . . . . . . . . . . . . . . . . . . .
4.4.2 Some Theory and the Natural Rate of Unemployment . .
4.4.3 Unemployment and the Business Cycle . . . . . . . . . . .
4.5 The Price Adjustment Process . . . . . . . . . . . . . . . . . . .
4.5.1 Aggregate Demand, Potential GDP and the Price Adjustment Process . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.2 Monetary Policy . . . . . . . . . . . . . . . . . . . . . . .
4.5.3 Fiscal Policy . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Stabilization Policy . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 Aggregate Demand Shocks and Their Stabilization . . . .
4.6.2 Price Shocks and Their Stabilization . . . . . . . . . . . .
4.7 Real Business Cycle Theory . . . . . . . . . . . . . . . . . . . . .
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3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.3.5 Analysis of the Extended Model . . . . . . .
3.3.6 Evaluation of the Solow Model . . . . . . . .
The Convergence Discussion . . . . . . . . . . . . . .
Growth Accounting and the Productivity Slowdown
Ideas as Engine of Growth . . . . . . . . . . . . . . .
3.6.1 Technology . . . . . . . . . . . . . . . . . . .
3.6.2 Ideas . . . . . . . . . . . . . . . . . . . . . . .
3.6.3 Data on Ideas . . . . . . . . . . . . . . . . . .
Infrastructure . . . . . . . . . . . . . . . . . . . . . .
3.7.1 Cost of Investment . . . . . . . . . . . . . . .
3.7.2 Bene…ts of Investment . . . . . . . . . . . . .
Endogenous Growth Models . . . . . . . . . . . . . .
Neutrality of Money . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . .
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5 Microeconomic Foundations of Macroeconomics
151
5.1 Consumption Demand . . . . . . . . . . . . . . . . . . . . . . . . 151
5.1.1 Data on Consumption . . . . . . . . . . . . . . . . . . . . 151
5.1.2 The Keynesian Aggregate Consumption Function and the
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.1.3 The Life Cycle/Permanent Income Model of Consumption 156
CONTENTS
5.2
v
Investment Demand . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.2.1 Facts about Investment . . . . . . . . . . . . . . . . . . . 170
5.2.2 The Theory of Investment . . . . . . . . . . . . . . . . . . 173
6 Trade, Exchange Rates & International Financial Markets
6.1 Terms of Trade, the Nominal and the Real Exchange Rate . .
6.2 E¤ects of the Real Exchange Rate on the Trade Balance . . .
6.3 Determinants of the Real Exchange Rate . . . . . . . . . . . .
6.3.1 Purchasing Power Parity . . . . . . . . . . . . . . . . .
6.3.2 Real Exchange Rates and Interest Rates . . . . . . . .
6.4 The International Financial System . . . . . . . . . . . . . . .
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7 Fiscal and Monetary Policy in Practice
7.1 Fiscal Policy . . . . . . . . . . . . . . .
7.1.1 Data on Fiscal Policy . . . . . .
7.1.2 A Few Theoretical Remarks . . .
7.2 Monetary Policy . . . . . . . . . . . . .
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vi
CONTENTS
Chapter 1
Introduction
1.1
The Scope of Macroeconomics
Macroeconomics wants to explain the evolution of the main economic aggregates
over time. We are interested in why total production (real GDP) grows over time
on average and why it shows sizeable ‡uctuations around its long-run growth
trend. We want to understand what causes unemployment and in‡ation, how
interest rates behave and what causes a trade de…cit.
In contrast to microeconomics, where the object of interest is a single …rm
or household, in macroeconomics we study the whole economy. Our reasoning, however, will be based on the insights that microeconomic theory provides
(therefore the prerequisite requirements for this course).
Why should we care about macroeconomics. I could think of three good
reasons
1. It a¤ects us on a day-to-day basis: A rise in the interest rate makes loans
for cars more expensive, raises the interest rate that you pay on a mortgage
and (usually) has a negative e¤ect on stock prices. A decline in production
leads to people being laid o¤ -and that could be a member of your family.
High in‡ation wipes out part of the value of your savings. The list goes
on and on....
2. A good understanding of macroeconomics is essential for policy makers.
Politicians can change …scal policy (how much the government spends
and how much it taxes you) and central bankers (Alan Greenspan and his
Federal Reserve Board) can change monetary policy (how much currency
to issue and how high to set the Federal Funds Rate -an important interest
rate). As we will see later …scal and monetary policy can have good and
bad e¤ects on the economy. It is crucial that policy makers and central
bankers understand macroeconomic data and macroeconomic theory to
make an informed decision about when and to what extent to change
monetary and …scal policy.
1
CHAPTER 1. INTRODUCTION
2
3. A good understanding is important for us as good citizens because it helps
us to understand and critic what politicians, central bankers and the press
tell us about the economy and what should be done to improve it.
But let’s …rst look at some data to see what it is that we’re talking about,
or, to speak with Sherlock Holmes
Data! Data! Data! I can’t make bricks without clay.
1.2
1.2.1
US Macroeconomic Data: A Helicopter Tour
Real GDP
When economists say that the US economy grew 2% last year they usually mean:
real Gross Domestic Product (GDP) was 2% higher in 2000 than in 1999. Let
us …rst de…ne what nominal GDP is.
De…nition 1 Nominal GDP is the total value of goods and services produced
in an economy during a particular time period.
Note that when talking about GDP we have to specify the GDP of what
economy (e.g. the US) for what time period (e.g. a year, say 2000) we mean.
Nominal GDP is measured in dollars. Since prices tend to increase over time
(ask your parents how much college tuition cost 30 years ago), so will nominal
GDP. To measure the economic activity of a country we are really interested
in how many real goods and services were produced in the economy. This is
measured by real GDP.
Real GDP =
Nominal GDP
Price Level
We will discuss how to compute the “Price Level” in the next section. Finally,
a growth rate of a variable is computed as follows. Let Yt denote real GDP in
period t (i.e. Y2000 is real GDP for the year 2000). Then the growth rate of real
GDP from period t ¡ 1 to period t is computed as
gY (t ¡ 1; t) =
Yt ¡ Yt¡1
Yt¡1
As an example, suppose real GDP in 1988 equals $ 585 and $ 605 in 1989, then
the growth rate of real GDP between 1988 and 1989 would equal
gy (1988; 1989) =
$605 ¡ $585
= 0:034 = 3:4%
$585
This is the number that people mean when they say that the economy grew by
3:4% in 1989.
1.2. US MACROECONOMIC DATA: A HELICOPTER TOUR
3
Let’s look at some data for real GDP. The solid line in Figure 1 shows the
evolution of real GDP for the US economy from 1967 to 2001.1 We have two
principal observations
1. Real GDP grows over time. If GDP would have grown at 2.75%, then the
graph of real GDP would have looked like the dotted line. The dotted line
is called “Trend”, because it shows how real GDP evolved on average.
2. Actual real GDP exhibits -occasionally sizeable- deviations from its long
term growth trend. These ‡uctuations are called business cycles.
Figure 2 shows these ‡uctuations in more detail. The dotted line at 0 corresponds to the trend. When the solid line takes the value -0.061 as in 1983, this
means that actual real GDP was 6.1% below the trend.
Periods in which real GDP actually declines are called recessions, and, if
these declines are extremely severe, depressions.2 From 1967 until 2001 the US
experienced 5 recessions.3 Note that, although recessions are recurrent events,
the exact timing of a recession is extremely hard to forecast.
1.2.2
Digression: The Rest of the Course
At this stage let’s have a short preview of the course. The two main sections,
Sections 3 and 4 deal exactly with the two observations we made about Figure
1:
In Section 3 we will study why, on average, the economy grows over time.
This area of study is called growth theory and we will discuss the neoclassical
growth model. As a sneak preview, the economy grows over time because:
1 The data have quarterly frequency, i.e. there one observation for real GDP for each quarter. The …rst observation is the GDP for the …rst three months of 1967, the last observation
is the GDP for April to June 2001. The data are then converted to yearly numbers (basically by multiplying them by 4). If you are interested in the actual data, on the WWW
go to http://www.economagic.com/em-cgi/data.exe/fedstl/gdp96+1#DataWhat is actually
plotted is the natural logarithm of real GDP, for the following reason. If GDP grows at a
constant rate g; then the log of GDP is a straight line with slope g: By plotting the log of
GDP we can draw the long-term growth trend as a straight line (rather than an exponential
function). This technique is used quite often by economists. Hall and Taylor plot GDP instead of log GDP, but use a logarithmic scale on the y-axis on p. 6 (observe that the distance
between 3500 and 4000 is bigger than between 6000 and 6500 on the y-axis; this is what a
log-scale does). Both tricks are equivalent.
2 The US economy as well as other economies in the world experienced a depression, the
so-called great depression, from 1929 to 1932.
3 One de…nition of a recession is “a decline in two subsequent quarters of real GDP”. If
you are interested in more detailed information about the timing and length of expansions
and recessions, visit the webpage of the National Bureau of Economic Research (NBER) at
http://www.nber.org/cycles.html. Note that, according to the o¢cial de…nition of a recession,
the U.S. economy is not currently in a recession, as real GDP growth has not been negative
in the …rst two quarters of 2001.
CHAPTER 1. INTRODUCTION
4
Real GDP in the United States 1967-2001
9.2
Log of real GDP
9
8.8
Trend
8.6
GDP
8.4
8.2
8
1970
1975
1980
1985
Year
1990
1995
Figure 1.1:
1. the population grows. A higher population means that a bigger labor force
is available for the production of goods and services.
2. more capital is accumulated. Over time, more and more machines and
other equipment are used in the production process
3. there is technological progress (e.g. the development of faster and faster
computer chips) makes capital and labor more productive in the production process.
In Section 4 we will study why there are business cycles, i.e. why the economy
‡uctuates around its long-term growth trend. In contrast to growth theory,
where the level of agreement between economists is fairly high, in business cycle
theory there is substantial disagreement about why business cycles exist and
what the government can do about them. Again a brief sneak preview:
2000
Percentage Deviation of Real GDP from Trend
1.2. US MACROECONOMIC DATA: A HELICOPTER TOUR
5
Expansions and Recessions
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
1970-71
recession
1974-75
recession
1970
1975
1990-91
recession
1980-82
back-to-back recessions
1980
1985
1990
1995
Year
Figure 1.2:
1. in this course we mostly will follow Hall and Taylor (and many others)
and assume that in the short run wages and/or prices are “sticky”, i.e. not
‡exible to adjust immediately to shocks hitting the economy. Potential
shocks could come from the private sector of the economy (a certain drop
of households’ willingness to buy cars), from world markets (remember
the oil price shocks in 1973 and 1980) or from changes in monetary and
…scal policy. The results are business cycles.
2. an alternative view holds that business cycles originate from “technology
shocks” (e.g. in certain years we have bad weather and that makes production, in particular agricultural production, more di¢cult). Prices and
wages are fully ‡exible even in the short run. People respond optimally
and work more when the conditions are such that they are productive (in
years of good technology shocks) and less when they are not so productive.
2000
CHAPTER 1. INTRODUCTION
6
Hence in good years workers supply a lot of labor and production (real
GDP) is high, in years with bad technology workers supply little labor
and real GDP is low. This view has become known as “Real Business
Cycle Theory” (“Real” because the shocks underlying business cycles are
technology shocks).4
The dispute between these two schools is not only theoretical. Based on
theory, economists from both camps have di¤erent views about economic policy. In RBC-theory business cycles arise because households react optimally
to technology shocks. Hence there is no role for government policy to improve
matters. If business cycles come about because prices and wages can’t adjust in
the short run (as in the …rst view), there may be a role for an active monetary
and …scal policy to reduce the economic ‡uctuations.
Common among both schools is that they both use models -abstract simple
descriptions of the economy, either with equations or graphs- to explain business cycles and to argue for or against a certain policy. We will follow this
methodological approach.
1.2.3
Other Macroeconomic Aggregates
Why are business cycles bad? Because if real production declines, workers get
laid o¤ and the unemployment rate increases. We should expect that the
unemployment rate follows the path of real output rather closely. Let us …rst
de…ne the unemployment rate.
De…nition 2 The labor force is the number of people, 16 or older, that are
either employed or unemployed but actively looking for a job. The unemployment
rate is given by
Unemployment Rate =
number of unemployed people
labor force
In Figure 3 we plot the unemployment rate for the US from 1967 to 2001.5
We see that in recessions the unemployment rate increases, whereas in expansion
it decreases. A variable that shows such a behavior is called “countercyclical”:
it is high when real GDP is low (relative to trend) and it is low when real GDP
is high. Also note that currently unemployment is at its lowest level since 1970.
Another important macroeconomic variable is the in‡ation rate. It measures the growth rate of the price of a particular basket of goods and services.6
4 The founders of RBC-theory are Finn Kydland from Carnegie Mellon University and Ed
Prescott from the University of Minnesota -incidentally my Ph.D. thesis advisor.
5 The unemployment rate is measured by the Bureau of Labor Statistics (BLS). Go to their
homepage at http://stats.bls.gov/top20.html if you want to have a look at the original data.
6 There are several measures of the in‡ation rate. They are distinguished by what goods
and services are included in the basket of goods whose price is measured. The two most
important indexes for in‡ation are the Consumer Price Index (CPI) and the GDP de‡ator.
Both will be discussed in the next section.
1.2. US MACROECONOMIC DATA: A HELICOPTER TOUR
Unemployment Rate for the US 1967-2001
12
1980-82
back-to-back recessions
11
1974-75
recession
10
Unemployment Rate
7
9
1990-91
recession
8
7
1970-71
recession
6
5
4
3
2
1970
1975
1980
1985
Year
1990
1995
Figure 1.3:
Let Pt be the price level in period t: Then the in‡ation rate between periods
t ¡ 1 and t is given by
¼t = gP (t ¡ 1; t) =
Pt ¡ Pt¡1
Pt¡1
Figure 4 shows the in‡ation rate for the US economy from 1967 to 2001.
We see that in‡ation rates were higher and more volatile in the 70’s and early
80’s than in the 90’s. Combining …gure 2 and 4 it is not apparent whether the
in‡ation rate is procyclical or countercyclical.
Interest Rates are important macroeconomic variables because they determine how costly it is to take out a loan to buy a car, a house, stocks, or, for
…rms, to …nance new equipment. How are interest rates computed. Suppose in
2000
CHAPTER 1. INTRODUCTION
8
Inflation Rate for the US 1967-2001
16
14
Inflation Rate
12
10
8
6
4
2
0
1970
1975
1980
1985
Year
1990
1995
Figure 1.4:
period t ¡ 1 you borrow the amount $Bt¡1 . The loan speci…es that in period
t you have to repay $Bt : In general $Bt will be bigger than Bt¡1 (since you
have to repay Bt¡1 ; the so-called principal, and the interest on the loan): The
nominal interest rate on the loan from period t ¡ 1 to period t, it ; is computed
as
Bt ¡ Bt¡1
it =
Bt¡1
This is called a nominal interest rate because it does not take into account
in‡ation. The real interest rate rt is de…ned as the di¤erence between the
nominal interest rate and the in‡ation rate:
rt = it ¡ ¼t
Note that nominal interest rates historically tend to rise with in‡ation: lenders
demand a higher nominal interest rate in times of high in‡ation as compensation
2000
1.2. US MACROECONOMIC DATA: A HELICOPTER TOUR
9
for the loss of purchasing power of their money, due to high in‡ation.
Example: In the year 2000 you borrow $15; 000 to buy a new car and the
bank asks you to repay $16; 500 exactly one year later. Then the yearly nominal
interest rate from 2000 to 2001 is
i2001 =
$16; 500 ¡ $15; 000
= 0:1 = 10%
$15; 000
Now suppose the in‡ation rate is 3% in 2001. Then the real interest rate equals
10% ¡ 3% = 7%
Note that whenever stating an interest rate, it is crucial to state the length
of the period with respect to which it applies, i.e. whether it is a yearly, a
quarterly, a monthly or a daily interest rate.
In Figure 5 the nominal interest rate for the US economy from 1967 to 2001
is plotted.7 Comparing Figure 2 and Figure 5 indicates that interest rates tend
to be procyclical: they increase during expansions and fall during recessions.
Now we have a rough idea about how the most important macroeconomic
variables evolved over the last 30 years. Now we turn to a discussion how these
variables are actually measured in the data.
7 There are many di¤erent interest rates.
rate that banks charge each other for loans
So this is a daily interest rate. This daily
yearly interest rate by “multiplying” the daily
http://www.stls.frb.org/fred/data/irates.html.
The Federal Funds rate is the interest
from one evening to the next morning.
interest rate has been converted into a
rate by 365. For the original data go to
CHAPTER 1. INTRODUCTION
10
Federal Funds Interest Rate 1967-2001
20
Nominal Interest Rate in %
18
16
14
12
10
8
6
4
2
0
1970
1975
1980
1985
Year
Figure 1.5:
1990
1995
2000
Chapter 2
National Income and
Product Accounting
(NIPA)
In this section we look in detail at how the macroeconomic aggregates whose
behavior over the last thirty years we studied in the last section are de…ned and
measured in the data. We will start with gross domestic product (GDP).
2.1
Gross Domestic Product (GDP)
We de…ned nominal and real GDP in the last section. Now will we discuss how
we measure these entities in the data. Nominal GDP can be measured in three
di¤erent ways which all lead to the same result:1
1. We can measure nominal GDP by adding together the value of production
in all di¤erent industries in the economy.
2. We can measure nominal GDP by adding together the spending on goods
and services of the di¤erent sectors of the economy (households, …rms, the
government and foreigners).
3. We can measure nominal GDP by adding together all the income that is
generated from the production process: wages, salaries and pro…ts.
In fact, the Bureau of Economic Analysis (BEA), the US government agency
that is responsible for measuring GDP, does calculate GDP in these three different ways and makes sure that the three numbers they get coincide (as they
should according to accounting principles).
1 The fact that the total value of production always equals the total value of spending and
always equals the total income is called an identity, it is inevitably true as a consequence of
accounting principles.
11
12CHAPTER 2. NATIONAL INCOME AND PRODUCT ACCOUNTING (NIPA)
2.1.1
Computing GDP through Production
We want to calculate nominal GDP by adding together the value of production
for all di¤erent industries in the economy, agriculture, mining, construction,
manufacturing etc. Can we just add together all those industries’ sales? Consider the following example: US steel produces a ton of steel and sells it to
GM for $1500. GM then uses this steel to build a car that it sells for $10,000.
Assume for the moment that a car can be produced only with steel and labor.
Should the contribution to GDP be the whole $11,500, the sum of total sales?
No, since the steel has been counted double; once when it was sold from US
Steel to GM and once when it, as a part of the car, was sold by GM. But it
was only produced once, so we should only count it once. This is achieved by
the concept of value added. It basically measures how much a …rm, in its
production process, added to the value of the intermediate goods it purchased
from its suppliers. Roughly, value added of a …rm equals its revenues from sales
minus the purchases of intermediate goods -goods that the …rm bought from
other …rms and used to produce its own products.
For the example then, the contribution should be only $1500 (from the sale
of steel to GM, the value added of US Steel) plus $8500 (the value added of
GM, equal to the total sale of $10,000 minus the purchase of the intermediate
good steel for $1,500).
So when we measure nominal GDP through production, we sum up the
value added of all industries in the economy, because the value added (and
not the sales) are the correct contributions of the industries to production. Table
1 shows the contribution of di¤erent industries to nominal GDP for 1999. The
numbers in column 2 are in billions of dollars.2
Table 1
Industries
Total Nom. GDP
Agriculture, Forestry, Fishing
Mining
Construction
Manufacturing
Transportation, Publ. Utilities
Wholesale Trade
Retail Trade
Finance, Insurance, Real Estate
Services
Government
Statistical Discrepancy
Value Added
9,299.2
125.4
111.8
416.4
1,500.8
779,6
643.3
856.4
1,792.1
1,986.9
1,158.4
-71.9
in % of Tot. Nom. GDP
100.0%
1.3%
1.2%
4.5%
16.1%
8.4%
6.9%
9.2%
19.3%
21.4%
12.5%
-0.8%
Note that total nominal GDP in 1999 was $US 9,299.2 billion, or $US
9,299,200,000,000. To make this number a little less intimidating, economists
2 All
data in this section come from the Economic Report of the President (2001).
2.1. GROSS DOMESTIC PRODUCT (GDP)
13
often report GDP per capita. On average in 1999 the population of the US was
275,372,000. Hence GDP per capita in 1999 amounted to $33,769.59. In 1999
every person in the US, from the newborns to the old, produced on average
about $34,000 worth of goods and services.
2.1.2
Computing GDP through Spending
Nominal GDP can also be computed by summing up the total spending on
goods and services by the di¤erent sectors of the economy. Formally, let
C
I
G
X
M
Y
=
=
=
=
=
=
Consumption
(Gross) Investment
Government Purchases
Exports
Imports
Nominal GDP
Then
Y = C + I + G + (X ¡ M )
Let us turn to a brief description of the components of GDP:
² Consumption (C) is de…ned as spending of households on all goods, such
as durable goods (cars, TV’s, Furniture), nondurable goods (food, clothing, gasoline) and services (massages, …nancial services, education, health
care). The only form of household spending that is not included in consumption is spending on new houses.3 Spending on new houses is included
in …xed investment, to which we turn next.
² Gross Investment (I) is de…ned as the sum of all spending of …rms on plant,
equipment and inventories, and the spending of households on new houses.
It is broken down into three categories: residential …xed investment
(the spending of households on the construction of new houses), nonresidential …xed investment (the spending of …rms on buildings and
equipment for business use) and inventory investment (the change in
inventories of …rms). To make the concept of investment clearer, we have
to take a little digression about stocks and ‡ows.
A stock is a quantity measured at a given point in time. A ‡ow is a quantity
measured per unit of time. As an example consider …lling a bathtub with water.
The amount of water in the tub is a stock -we say that the bathtub contains
3 What about purchases of old houses? Note that no production has occured (since the
house was already built before). Hence this transaction does not enter this years’ GDP. Of
course, when the then new house was …rst bought by its …rst owner it entered GDP in the
particular year.
14CHAPTER 2. NATIONAL INCOME AND PRODUCT ACCOUNTING (NIPA)
50 gallon of water. The amount of water ‡owing out of the faucet is a ‡ow
-we say that 2 gallon of water per minute ‡ow into the tub. Note that we
measure the stock by gallon, the ‡ow by gallon per minute. Often stocks and
‡ows are related. In our example the stock of water in the tub equals the
accumulated ‡ow of water out of the faucet. The same is true with investment
and the capital stock. The capital stock of an economy is the typical economic
example of a stock, whereas investment, like GDP and its other components
consumption, government purchases etc. are ‡ow variables.4 The capital stock
is the total amount of physical capital in the economy, including all buildings and
equipment. Part of the capital stock wears out every period in the production
process, a process called depreciation (which is again a ‡ow variable). We
have the following relationship between the capital stock, gross investment and
depreciation:
Capital Stock at end of this period = Capital Stock at end of last period
+Gross Investment in this period
¡Depreciation in this period
We de…ne net investment as
Net Investment = Gross Investment ¡ Depreciation
and therefore
Net Investment = Capital Stock at end of this period
¡Capital Stock at end of last period
Note that what enters nominal GDP is gross, not net investment, but that net
investment in this period equals the change of the capital stock from the end of
last to this period.
What residential and nonresidential …xed investment are and why they are
included in nominal GDP is rather obvious. So let’s spend some time to understand inventory investment. Suppose in 1999 Ford produces a car that you
purchase in 1999. Then your spending on the car enters GDP as consumption
under C: But now suppose Ford produces the car and puts it in its stock for
sale in 2000. Since the car is not sold yet, it doesn’t enter GDP as consumption
in 1999. But Ford’s production activity is the same, no matter whether the
car was sold or not in 1999, so the contribution to GDP should be the same.
The key is inventory investment: By producing now and putting the car in its
stock, Ford increased its inventory by one car, and the statisticians count this
as investment in inventories. By the same token as before
Inventory Investment = Stock of Inventories at end of this year
¡Stock of Inventories at the end of last year
4 Remember the de…nition of nominal GDP: it is the total value of goods and services
produced in an economy during a particular time period, i.e measured in units per time
period.
2.1. GROSS DOMESTIC PRODUCT (GDP)
15
Sometimes the variable …nal sales is reported in the news. (Nominal) …nal
sales equal nominal GDP minus inventory investment.
² Government spending (G) is the sum of federal, state and local government
purchases of goods and services. Note that government spending does not
equal total government outlays: transfer payments to households (such
as welfare, social security or unemployment bene…t payments) or interest
payments on public debt are part of government outlays, but not included
in government spending G:
² As an open economy, the US trades goods and services with the rest of
the world. Exports (E) are deliveries of US goods and services to the rest
of the world, imports (M) are deliveries of goods and services from other
countries of the world to the US. Why are imports subtracted from exports
when computing GDP. Suppose Boeing buys 4 jet engines from the British
company Rolls Royce, puts them into a Boeing 747 and sells the aircraft
to the French airline Air France. What has been produced in the US was
the plane, excluding the engines. So we count the plane as exports out
of the US, the engines as import into the US and the net contribution to
GDP is (X ¡M ), that is, exports minus imports. The quantity (X ¡M ) is
also referred to as net exports or the trade balance. We say that a country
(such as Germany) has a trade surplus if exports exceed imports, i.e. if
X ¡ M > 0. A country has a trade de…cit if X ¡ M < 0; which was the
case for the US in recent years.
In Table 2 you can see the composition of nominal GDP for 1997, broken
down to the di¤erent spending categories discussed above. Again the numbers
are in billion US dollars.
Table 2
Total Nom. GDP
Consumption
Durable Goods
Nondurable Goods
Services
Gross Investment
Nonresidential
Residential
Changes in Inventory
Government Purchases
Federal Government
State and Local Government
Net Exports
Exports
Imports
Final Sales
in billion $
9,299.2
6,268.7
761.3
1,845.5
3,661.9
1,650.1
1,203.1
403.8
43.3
1,634.4
586.6
1,065.8
-254.0
990.2
1,244.2
9,255.9
in % of Tot. Nom. GDP
100.0%
67.4%
8.2%
19.8%
39.3%
17.7%
12.9%
4.3%
0.5%
17.6%
6.3%
11.5%
-2.7%
10.6%
13.4%
99.5%
16CHAPTER 2. NATIONAL INCOME AND PRODUCT ACCOUNTING (NIPA)
2.1.3
Computing GDP through Income
The production of goods and services generates income, either in the form of
wages and salaries for workers, or in the form of pro…ts for individuals running
a business. This fact provides a third way of computing nominal GDP. The
broadest measure of the total incomes of all Americans is called national income. It is closely related, but not equal to nominal GDP. Remember that US
GDP is the value of goods and services produced in the US. Some people in
this country are not Americans, so, although they contribute to US GDP, their
income is not part of national income. On the other hand there are Americans
who produce goods and services abroad, so they don’t contribute to US GDP,
but their income is part of national income. When we add to GDP factor
income from the rest of the world (income of Americans not earned in
America) and subtract factor income to the rest of the world (income
of Non-Americans earned in the US, like my salary) we arrive at Gross National Product (GNP). GNP is the value of all goods and services produced
by Americans, whereas GDP is the value of all goods and services produced in
America. There are other parts of GNP that are not part of national income.
First we have to subtract depreciation, Since depreciation of capital is a cost
of producing the output of the economy, subtracting depreciation shows the
net result of economic activity. GNP minus depreciation equals Net National
Product (NNP). From NNP we subtract sales and excise taxes to obtain
national income.5 This is due to the fact that NNP is measured in terms
of the prices that …rms receive for their products, but only that part of these
prices which does not go to the government becomes income of households. So
the connection between GDP and national income is given by (in brackets the
numbers for the US in 1999, in billion $US).
Gross Domestic Product (9,299.2)
+Factor Income from abroad (305.9)
¡Factor Income to abroad (316.9)
= Gross National Product (9,288.2)
¡Depreciation (1,161.0)
= Net National Product (8,127.1)
¡Sales and Excise Taxes (718.1)
¡Other Adjustments6 (-3.8)
= National Income (7,469.7)
National Income is divided into …ve components, depending on the way the
income is earned:
5 Other minor corrections of NNP to obtain national income are the following. To NNP
we add net subsidies of the government to government businesses, and we substract business
transfers (gifts of businesses) and statistical discrepancy. These adjustments are of minor
importance.
2.1. GROSS DOMESTIC PRODUCT (GDP)
17
1. Compensation of Employees: wages, salaries and fringe bene…ts earned by
workers
2. Proprietors’ Income: income of noncorporate business, such as small farms
and law partnerships
3. Rental Income: income that landlords receive from renting, including the
“imputed” rent that homeowners pay themselves, less expenses on the
house, such as depreciation
4. Corporate Pro…ts: income of corporations after payments to their workers
and creditors
5. Net interest: interest paid by domestic businesses plus interest earned
from foreigners
Commonly the …rst component is called labor income, components 2 to 5
together are called capital income.7 The labor share is de…ned as the fraction
of national income that goes to labor income, the capital share is de…ned as
the fraction of national income that goes to capital income. Formally
Labor Share =
Capital Share =
Labor Income
National Income
Capital Income
National Income
Obviously, since national income equals labor income plus capital income, the
labor share and the capital share sum to 1. In Table 3 you can …nd national
income and its component for the US in 1999
Table 3
National Income
Compensation of Employees
Proprietors’ Income
Rental Income
Corporate Pro…ts
Net Interest
Billion $US
7,469.7
5,299.8
663.5
143.4
856.0
507.1
% of National Income
100.0%
71.0%
8.9%
1.9%
11.5%
6.8%
We see that for 1999 the labor share equals 71% and the capital share equals
29%.
Finally, let us relate national income to two other, commonly used income
concepts that may coincide more with your common understanding about what
the income of a household (or in our case the income of all households) is. A
series of adjustments takes us from national income to personal income, the
7 There is some ambiguity about counting proprietors’ income as capital income, since
arguably the labor of the farmer is one of the most important inputs to the farms’ production
of agricultural products.
18CHAPTER 2. NATIONAL INCOME AND PRODUCT ACCOUNTING (NIPA)
income that households and noncorporate businesses receive. First we have to
reduce national income by that fraction of corporate pro…ts that are not paid out
in the form of dividends. This entity is called retained earnings. Second we
have to subtract contributions for social insurance (the amount paid to the
government for social security and medicare). Third, we want to include interest
payments that households receive, rather than interest payments that businesses
pay. This is accomplished by reducing national income by net interest paid by
businesses and adding personal interest income. Finally we add to national
income transfers from the government and businesses to households, such
as social security bene…ts and pensions paid by …rms to their retired employees.
The relation between national income and personal income is then given by (in
brackets again the numbers for 1999 in billion $US)
National Income (7,469.7)
¡Retained Earnings (485.7)
¡Contributions for Social Insurance (662.1)
¡Net Interest (507.1)
+Personal Interest Income (963.7)
+Government and Business Transfers (1,016.2)
= Personal Income (7,789.6)
Finally, we arrive at Disposable Personal Income (the income that households and noncorporate businesses can spend, after having satis…ed their tax
obligations) by subtracting from personal income personal tax and nontax
payments (such as parking tickets) to the government:
Personal Income (7,789.6)
¡Personal Tax and Nontax Payments (1,152.0)
= Disposable Personal Income (6,637.6)
This concludes the discussion of how nominal GDP is measured. As you see
from the numbers for 1999 (and as you will see in the problem sets) all three
methods indeed lead to the same result.
One last, but very important fact follows from the equivalence of GDP measured by spending and measured by income. For simplicity let us consider an
economy without government and international trade.8 Saving (S) is de…ned as
income minus consumption, or
S =Y ¡C
But from the spending side of GDP we know that
Y =C +I
8 Hall and Taylor show the argument that will follow for the general case with government
and international trade. The reader is refered to the book for details.
2.2. PRICE INDICES
19
(remember that we assumed that G = X = M = 0). Substituting for Y in the
…rst equation we get
S
= Y ¡C
= C+I ¡C
= I
Hence saving equals investment. This is again an accounting identity, it is always
true. Note that this identity of saving and investment also holds for the general
case with government and foreign trade, with saving and investment rede…ned
to account for the presence of the government and other countries. It is a crucial
identity that we will use over and over again in growth theory and business cycle
theory.
2.2
Price Indices
To compute real GDP we divide nominal GDP by the “Price Level”. To compute
the in‡ation rate we need price levels in two di¤erent periods. In this section
we discuss how we measure the “Price Level”. In general economists measure
the price level by a price index. A price index is a ratio between the price of a
particular basket of goods in period t and the price of the same basket in a base
period, say period 0: There are two important questions involved in constructing
a price index: a) what period to chose as base period b) what basket of goods
to chose.
Let’s consider a very simple economy in which people just produce and
buy two goods, say hamburgers and coke. We denote by ht the amount of
hamburgers consumed (and produced) in period t; and by ct the amount of coke
consumed in period t: Also let Pht be the price of one hamburger in period t and
pct the price of one bottle of coke in period t: Let (h0 ; c0 ; ph0 ; pc0 ) denote the
same variables at period 0: Now let’s ask ourselves how one would measure the
price level in period t as compared to period 0; which we will take as our base
period? One option is to compare how expensive the basket of goods consumed
in period 0 are in period t: The result is
Lt =
pht h0 + pct c0
ph0 h0 + pc0 c0
Such a price index is called a Laspeyres price index. If, on the other hand, we
take as our basket the goods purchased in period t; then we have
P at =
pht ht + pct ct
ph0 ht + pc0 ct
Such a price index is called a Paasche price index. It turns out that all price
indices actually used in practice to compute real GDP or the in‡ation rate are
either Laspeyres or Paasche price indices. Before turning to this point, a brief
20CHAPTER 2. NATIONAL INCOME AND PRODUCT ACCOUNTING (NIPA)
comment about these indices. Unfortunately both have their problems.9 The
problem with the Laspeyres price index is that it tends to overstate in‡ation
by assuming that households buy the same basket of goods in period t as in
period 0: But as prices change from period 0 to period t; consumers tend to
substitute goods that have become relatively more expensive from period 0 to
period t with goods that have become relatively less expensive. By holding
the basket of goods …xed at the basket bought in period 0; the Laspeyres price
index ignores this substitution e¤ect, which tends to lead to an overstatement
of in‡ation. Now let’s look at the Paasche price index. Consider the following
scenario: suppose a virus is detected in all coke bottles in the country at period
t; so that at period t no coke is produced (and the price of the few bottles in the
stores from last year sky-rockets). And suppose that the price for hamburgers
stays constant between period 0 and t: What would the Paasche index say about
the price level in period as opposed to period 0: Since the Paasche index uses
the basket of goods in period t; and since no coke is produced in period t; the
price change for coke does not have any e¤ect, and the Paasche index would
be at P at = 1 (under the assumption that hamburger prices have remained
constant). But would we really think that the situation just described is one in
which prices have remained constant, as the Paasche index indicates. In general,
because of this problem the Paasche index tends to understate in‡ation. But
now let’s leave the general theory of price indices and talk about real GDP and
in‡ation
2.3
From Nominal to Real GDP
Real GDP is the meant to measure the total production of goods and services
in physical units. But how does one add 10 cars, twelve haircuts and a cruise
missile together to one number? What statisticians do in practice to determine
real GDP is the following: they pick a base year, say 1996. The contribution
of computers to real GDP in 2000 is then computed as follows: take the dollar
amount spent on computers in 2000 and divide by the price of computers in
2000 relative to 1996 (i.e. divide by the price in 2000 and multiply by the price
in 1996). The result the total value of computers sold in 2000 in prices of 1996.
Summing up all goods and commodities, evaluated at their 1996 prices, yields
real GDP. Note that for the base year nominal and real GDP always coincide.
The ratio between nominal and real GDP turns out to be a price index, the
so-called GDP-de‡ator:
GDP de‡ator =
Nominal GDP
Real GDP
9 In fact, the problem of how to construct an ideal price index is a deep methological
problem, know as the index number problem. It has not been, and in fact can’t be fully
resolved. Also it is hard to say which of the two indices discussed is superior.
2.4. MEASURING INFLATION
21
To see why this is, suppose again that our economy produces only hamburgers
and coke. Nominal GDP in 2000 would be given by
Nominal GDP = h2000 ph2000 + c2000 pc2000
Real GDP would be given by (assuming 1996 is the base year)
Real GDP = h2000 ph1996 + c2000 pc1996
From the previous formula we get
GDP de‡ator =
h2000 ph2000 + c2000 pc2000
h2000 ph1996 + c2000 pc1996
This should look familiar to you; in fact the GDP de‡ator is a Paasche price
index; compare this formula to the one for a Paasche price index in the previous
section.
2.4
Measuring In‡ation
Remember that the in‡ation rate from period t ¡ 1 to period t was de…ned as
¼t =
Pt ¡ Pt¡1
Pt¡1
where Pt is the price level in period t: One possibility to compute the in‡ation
rate is to take as the price level the GDP de‡ator from the previous section.
The basket of goods on which the in‡ation rate is then based corresponds to
the current composition of GDP. More often an in‡ation rate is reported that
uses a di¤erent basket of goods and services.10
Mostly when the in‡ation rate is reported in the news, it is based on the
Consumer Price Index (CPI), which the Bureau of Labor Statistics determines
every month. The news release of this monthly number is followed with wide
interest for the following reasons. The Federal Reserve Bank, who is responsible
for monetary policy, bases its decision on the development of the in‡ation rate,
as its major objective is to achieve “price stability”. A higher than expected
in‡ation rate causes the FED to increase interest rates, which usually a¤ect
the stock market adversely. Knowing this in advance, the stock market tends
to react negatively to higher than expected in‡ation and positively to lower
than expected in‡ation. It is also important because many contracts include socalled COLA’s, cost-of-living adjustments that specify that payments increase
proportionally to the CPI. This is the case for social security bene…ts, for example. So the CPI is likely the most-watched macroeconomic variable. How is
it computed?
10 When we are concerned about how the purchasing power of a typical household has
changed over time, a basket of goods that includes cruise missiles, oil platforms and the like
(as for the GDP de‡ator) may not be very informative.
22CHAPTER 2. NATIONAL INCOME AND PRODUCT ACCOUNTING (NIPA)
Basically, the BLS determines a basket of goods and services that a typical
American household buys in a typical month of the base year. This basket
includes 4 loafs of bread, a case of beer, 1/60 of a car, 4 haircuts and so forth.
The BLS then determines how much this basket cost in a typical month of the
base year, and how much it cost in a typical month this year. The CPI for this
month equals the ratio between the price of the basket in this year and the price
in the base year. Again suppose that the BLS decided that the correct basket
was composed only of hamburgers and coke (and the base year is 1996), then
the CPI for 2000 is given by
CPI =
h1996 ph2000 + c1996 pc2000
h1996 ph1996 + c1996 pc1996
Again note that this is exactly a Laspeyres price index from the previous section.
The in‡ation rate is then computed using as price level in period t; Pt the CPI
for period t:
There is a recent political discussion about whether the CPI overstates in‡ation. One problem that we already discussed in the previous section is that
people may substitute away from goods that have become relatively more expensive. A second problem is the introduction of new goods. Since new goods are
not included in the base year basket, they have no e¤ect on the CPI. Arguably,
however, the introduction of new goods makes consumers better o¤. A third
problem is unmeasured changes in quality. Suppose a good gets better without
this improvement being re‡ected in the price (maybe because the improvement
is hard to measure), then the CPI remains unchanged although it should have
fallen. This problem is not only academic. Because of the COLA’s, government
outlays depend signi…cantly on how in‡ation is measured. Suppose the CPI
overstates true in‡ation by one percentage point (this is the magnitude that
some economists believe is realistic), then the government in 1997 paid about
$10 billion too much for social security bene…ts, quite a signi…cant number.
2.5
Measuring Unemployment
Remember our de…nition of the unemployment rate as the ratio between the
number of unemployed people and the labor force. In practice about 100,000
adults in each month are interviewed and asked about whether they are employed, and, if not, are asked if they are actively looking for a job (i.e. if they
are in the labor force).11 . The number of people that are unemployed and the
number of people in the labor force are counted and the ratio computed, which
gives the unemployment rate for that month.
11 Asking everybody in the US would be quite expensive, and a sample of 100,000 gives a
quite accurate description of the entire population.
2.6. MEASURING TRANSACTIONS WITH THE REST OF THE WORLD23
2.6
Measuring Transactions with the Rest of the
World
We already de…ned what the trade balance is: it is the total value of exports
minus the total value of imports of the US with all its trading partners. A
closely related concept is the current account balance. The current account
balance equals the trade balance plus net unilateral transfers
Current Account Balance = Trade Balance + Net Unilateral Transfers
Unilateral transfers that the US pays to countries abroad include aid to poor
countries, interest payments to foreigners for US government debt, and grants
to foreign researchers or institutions. Net unilateral transfers equal transfers of
the sort just described received by the US, minus transfers paid out by the US.
Usually net unilateral transfers are negative for the US, but small in size (they
amounted to about 0.5% of GDP in 1999). So for all practical purposes we
can use the trade balance and the current account balance interchangeably. We
say that the US has a current account de…cit if the current account balance is
negative and a current account surplus if the current account balance is positive.
Note that the current account balance is a ‡ow (since exports and imports are
‡ows).
The current account balance keeps track of import and export ‡ows between
countries. The capital account balance keeps track of borrowing and lending
of the US with abroad. It equals to the change of the net wealth position
of the US. The US owes money to foreign countries, in the form of government
debt held by foreigners, loans that foreign banks made to US companies and in
the form of shares that foreigners hold in US companies. Foreign countries owe
money to the US for exactly the same reason The net wealth position of the
US is the di¤erence between what the US is owed and what it owes to foreign
countries. Note that the net wealth position is a stock, but that the capital
account balance, as the change in the net wealth position, is a ‡ow:
Capital Account Balance this year = Net wealth position at end of this year
¡Net wealth postion at end of last year
Compare this to the relationship between the capital stock and investment from
above: it is exactly the same principle. Note that a negative capital account
balance means that the net wealth position of the US has decreased: in net
terms, capital has ‡own out of the US. The reverse is true if the capital account
balance is positive: capital ‡ew into the US.
The current account and the capital account balance are intimately related:
they are always equal to each other. This is another example of an accounting
identity.
Current Account Balance this year = Capital Account Balance this year
24CHAPTER 2. NATIONAL INCOME AND PRODUCT ACCOUNTING (NIPA)
The reason for this is simple: if the US imports more than it exports, it has to
borrow from the rest of the world to pay for the imports. But this change in
the net asset position is exactly what the capital account balance captures. In
the next …gure we plot the trade balance for the US for the last 30 years.
Trade Balance for the US 1967-2001 (in Constant Prices)
0
-50
Trade Balance
-100
-150
-200
-250
-300
-350
-400
1970
1975
1980
1985
Year
1990
1995
Figure 2.1:
One can see that the trade balance was mostly negative during this period,
and has been particularly negative during the expansion of the 90’s. One consequence of this …gure and the accounting identity is that the net wealth position
of the US has declined over the years. Since 1989 the US, traditionally a net
lender to the world, has become a net borrower: the net wealth position of the
US has become negative in 1989.
A last variable that is of strong importance when discussing international
trade are exchange rates. The exchange rate of the dollar with the yen measures
how many yen somebody has to pay to buy one dollar (currently about 119). The
exchange rate of the dollar with the Euro measures how many euro somebody
has to spend in order to buy 1 dollar (currently about 1.1). The exchange
2000
2.7. APPENDIX A: MORE ON GROWTH RATES
25
rates are important for the following reasons: suppose the exchange rate of the
dollar with the yen increases (i.e. dollar become more expensive to buy for
Japanese households). That means it becomes more expensive for Japanese to
buy American products. Reversely if the exchange rate declines. Hence there
tends to be a close relation between exchange rates and imports and exports (and
hence the trade balance). A strong dollar (Euro are cheap, dollars expensive)
tends to increase the trade de…cit, a weak dollar tends to decrease it.
2.7
Appendix A: More on Growth Rates
Remember that the growth rate of a variable Y (say nominal GDP) from period
t ¡ 1 to t is given by
gY (t ¡ 1; t) =
Yt ¡ Yt¡1
Yt¡1
(2.1)
Similarly the growth rate between period t ¡ 5 and period t is given by
gY (t ¡ 5; t) =
Yt ¡ Yt¡5
Yt¡5
Now suppose that GDP equals $1000 in 1992. From 1992 to 1993 it grows at a
growth rate of 2%. From 1993 to 1994 it grows at a rate of 4%, from 1994 to
1995 at 7%, from 1995 to 1996 at 1% and from 1996 to 1997 at 3%. How do
we …gure out how big GDP was in 1997? We can use the formula in (2:1): Note
that
Yt ¡ Yt¡1
Yt¡1
= Yt ¡ Yt¡1
= Yt
= Yt
gY (t ¡ 1; t) =
gY (t ¡ 1; t) ¤ Yt¡1
gY (t ¡ 1; t) ¤ Yt¡1 + Yt¡1
(1 + gY (t ¡ 1; t))Yt¡1
Hence GDP in period t equals GDP in period t ¡ 1; multiplied by 1 plus the
growth rate. For the example:
Y1993
Y1994
Y1995
Y1996
Y1997
=
=
=
=
=
=
(1 + gY (1992; 1993)) ¤ Y1992
(1 + 0:02) ¤ $1000 = $1020
(1 + 0:04) ¤ $1020 = $1060:80
(1 + 0:07) ¤ $1060:80 = $1135:06
(1 + 0:01) ¤ $1135:06 = $1146:41
(1 + 0:03) ¤ $1146:41 = $1180:80
and the growth rate from 1992 to 1997 is given by
gY (1992; 1997) =
$1180:80 ¡ $1000
= 18:08%
$1000
26CHAPTER 2. NATIONAL INCOME AND PRODUCT ACCOUNTING (NIPA)
Particularly interesting is the case where a variable grows at a constant rate,
say g, over time. Suppose at period 0 GDP equals some number Y0 and GDP
grows at a constant rate of g% a year. Then in period t GDP equals
Yt = (1 + g)t Y0
(2.2)
For example, if Jesus would have put 1 dollar in the bank at year 0AC and the
bank would have paid a constant interest rate of, say, 1.5%, then in 1999 he
would have had a fortune of
Y1999
= (1:015)1999 ¤ $1
= $8; 425; 941; 823
which is almost the US GDP for this year. Sometime it is interesting to do the
reverse calculation. Suppose you know GDP at time 0 and at time t and want
to know at what constant rate GDP must have grown to reach Yt ; starting from
Y0 in t years. We can use the formula (2:2) to solve for g
= (1 + g)t Y0
Yt
(1 + g)t =
Y0
µ ¶ 1t
Yt
(1 + g) =
Y0
µ ¶ 1t
Yt
¡1
g =
Y0
Yt
As an example: Suppose we know that in the year 1900 a country has GDP of
$1,000 and in 1999 it has GDP of $15,000. Suppose we assume that the GDP
of this country has grown over these years at a constant rate g: How big must
this growth rate be? If we take 1900 as period 0; then 1999 is period t = 99:
We apply the formula to get
g
=
µ
µ
Yt
Y0
¶ 1t
¡1
¶1
$15; 000 99
¡1
$1; 000
= 0:028 = 2:8%
=
Finally, we might be interested in the following question: Suppose we know the
GDP of a country in period 0 and its growth rate g and we want to know how
many time periods it takes for GDP in this country to double (to triple and so
forth). Again we can use the formula, but this time we solve for t :
Yt
(1 + g)t
= (1 + g)t Y0
Yt
=
Y0
(2.3)
2.8. APPENDIX B: CHAIN-WEIGHTED GDP
27
Now we need a little mathematical fact about logarithms: if a and b are arbitrary
positive numbers, then
¡ ¢
log ab = b ¤ log(a)
Using this fact and taking (natural) logarithms on both sides of equation (2:3)
yields
µ ¶
¢
¡
Yt
log (1 + g)t = log
Y0
µ ¶
Yt
t ¤ log(1 + g) = log
Y0
³ ´
log YY0t
t =
log(1 + g)
Now suppose we want to …nd the number of years it takes for GDP to double,
i.e. the t such that Yt = 2 ¤ Y0 or YY0t = 2: We get
t=
log(2)
log(1 + g)
So once we know the growth rate of our country, we can answer our question.
For example with a growth rate of g = 1% it takes about 70 years, with a growth
rate of g = 2% it takes about 35 years, with a growth rate of g = 5% it takes
about 14 years and so forth.
2.8
Appendix B: Chain-Weighted GDP
In this appendix we discuss a recent development in the computation of real
GDP and the GDP de‡ator. The Bureau of Labor Statistics used to compute
real GDP and the GDP de‡ator in exactly the fashion described in the main
text. In 1996 it also introduced the Fisher indices to compute real GDP (it still
reports two measures of real GDP, the old and the revised numbers). What is
the problem with the old method?
With the old method one would pick a base year, say 1992. The contribution
of computers to real GDP in 1999 is them computed as follows: take the dollar
amount spent on computers in 1999 and divide by the price of computers in
1999 relative to 1992 (i.e. divide by the price in 1999 and multiply by the
price in 1992). The result the total value of computers sold in 1999 in prices of
1992. Summing up all goods and commodities, evaluated at their 1992 prices,
yields real GDP. Note that for the base year nominal and real GDP always
coincide. The problem is that goods whose prices have fallen a lot between
this year and the base year (like computers) receive more and more weight in
computing real GDP. I will use the same example as Hall and Taylor (p. 33),
but will deviate once I describe the reforms the BEA has undertaken. Suppose
a country produces only two goods, computers and hamburgers. The next table
describes the spending on both goods as well as their prices
28CHAPTER 2. NATIONAL INCOME AND PRODUCT ACCOUNTING (NIPA)
Table 4
Year
1992
1994
1996
1998
Spending in Current $
Computers (1) Hamburgers (2)
100
106
105
98
103
104
99
100
Prices in Current $
Computers (3) Hamburgers (4)
1.00
1.00
0.80
1.05
0.60
1.10
0.40
1.15
This is how real GDP and the GDP de‡ator are computed using the old
method. The …rst step is to determine real quantities for the years 1994, 1996
and 1998 (again, this is equivalent to valuing 1994, 1996 and 1998 quantities in
1992 prices). Note that, since we have chosen 1992 as our base year, in 1992
nominal and real quantities coincide (as we have normalized all prices in 1992 to
1). This is done by dividing the spending numbers for computers in column (1)
by the current prices for computers in column (3), and likewise for hamburgers
by dividing the numbers in column (2) by current prices in column (4). The
results are found in the …rst two columns of the next table.
Table 5
Year
1992
1993
1994
1995
Real Quantities
Computers (5) Hamburgers (6)
100.0
106.0
131.3
93.3
171.7
94.5
247.5
87.0
Real GDP
(7)=(5) + (6)
206.0
224.6
266.2
334.5
GDP De‡ator
((1)+(2))/(7)
1.000
0.904
0.778
0.595
Next we determine real GDP by summing up all real quantities, in this case
only computers and hamburgers. This is done by summing the …rst two columns
and yields the third column. Finally we compute the GDP de‡ator by diving
nominal GDP by real GDP in the di¤erent years. Nominal GDP is given by the
sum of columns (1) and (2), real GDP is given by the column labeled (7). It
yields the last column of Table 5.
The problem with the old method is evident: although in 1998 people spent
more on hamburgers than computers, the weight that computers receive in real
GDP is about three times that for hamburgers. Also, the choice of the base year
is quite important, and changes in the base year (which are done about every
5 to 7 years) can lead to serious revisions of growth rates of real GDP and the
GDP de‡ator.
The BEA reform addressed both problems. The …rst change was to introduce
chain-weighted indices. Instead of computing variables in comparison to a …xed
base year, variables computed in 1993 are based on 1992, variables in 1994 are
based on 1993 and so forth. Before they were all based on the base year, 1992.
Growth rates between 1992 and 1995 are then found by “chaining” the growth
rates for single years together (as described in the previous appendix). The
2.8. APPENDIX B: CHAIN-WEIGHTED GDP
29
second change was to allow weights for real GDP to take into account relative
price changes. I will now describe how the new method computes real GDP and
the de‡ator mechanically.12
We …rst have to introduce two quantity indices (which are very similar to
the price indices discussed before). Let
pct
ct
pht
ht
=
=
=
=
Price of a computer in period t
Number of computers bought in period t
Price of hamburgers in period t
Number of hamburgers bought in period t
Let (pc0 ; c0 ; ph0 ; h0 ) be the corresponding value for period 0: We de…ne the
Laspeyres quantity index as
LQt =
ht ph0 + ct pc0
h0 ph0 + c0 pc0
Note that here we keep prices …xed at period 0 prices and vary the quantities, whereas with the Laspeyres price index we kept quantities …xed at period
0 quantities and varied the prices. Similarly we de…ne the Paasche Quantity
index as
P aQt =
ht pht + ct pct
h0 pht + c0 pct
The new measure for real GDP, in, say 1993, is the real GDP in 1992 times the
square-root of the product of Laspeyres and Paasche quantity index between
1992 and 1993. Formally
p
real GDP in 1993 = real GDP in 1992 ¤ LQ1993 ¤ P aQ1993
where period 0 corresponds to 1992.
Let us compute real GDP for 1993, using this new method. The only thing
we need are the ingredients for our quantity indices and last periods GDP. We
have prices already given in columns (3) and (4), and quantities in (5) and (6),
as well as 1992 real GDP from summing (1) and (2) for 1992. Nothing more is
required. The Laspeyres quantity index is
LQ1993
=
=
=
h1993 ph1992 + c1993 pc1992
h1992 ph1992 + c1992 pc1992
93:3 ¤ 1 + 131:3 ¤ 1
106 ¤ 1 + 100 ¤ 1
224:6
= 1:090
206
12 This discussion is somewhat technical and di¤ers from Hall and Taylor. For further
reference, the original article from the BEA describing the procedure is by Steven Landefeld
and Robert Parker and can by found in the Survey of Current Business, May 1997, pp 58-68.
30CHAPTER 2. NATIONAL INCOME AND PRODUCT ACCOUNTING (NIPA)
The Paasche quantity index is
LQ1993
=
=
=
h1993 ph1993 + c1993 pc1993
h1992 ph1993 + c1992 pc1993
93:3 ¤ 1:05 + 131:3 ¤ 0:8
106 ¤ 1:05 + 100 ¤ 0:8
203
= 1:061
191:3
Hence real GDP in 1993 equals
p
real GDP in 1993 = real GDP in 1992 ¤ LQ1993 ¤ P aQ1993
p
= 206 ¤ 1:090 ¤ 1:061
= 221:5
The GDP de‡ator is computed as before by dividing nominal by real GDP. Let
us do one more year, since it shows the “chain” character of the new method.
For 1994 we now take 1993 as period 0: Again we have all the ingredients ready,
since we just computed real GDP for 1993. In particular
p
real GDP in 1994 = real GDP in 1993 LQ1994 ¤ P aQ1994
and we compute the Laspeyres quantity index as
LQ1994
=
=
=
h1994 ph1993 + c1994 pc1993
h1993 ph1993 + c1993 pc1993
94:5 ¤ 1:05 + 171:7 ¤ 0:8
93:3 ¤ 1:05 + 131:3 ¤ 0:8
236:6
= 1:166
203
and the Paasche quantity index as
LQ1994
=
=
=
h1994 ph1994 + c1994 pc1994
h1993 ph1994 + c1993 pc1994
94:5 ¤ 1:1 + 171:7 ¤ 0:6
93:3 ¤ 1:1 + 131:3 ¤ 0:6
207
= 1:141
181:4
We get
p
real GDP in 1994 = real GDP in 1993 LQ1994 ¤ P aQ1994
p
= 221:5 ¤ 1:166 ¤ 1:141
= 255:5
The …nal results of this exercise are given in Table 6
2.8. APPENDIX B: CHAIN-WEIGHTED GDP
31
Table 6
Year
1992
1993
1994
1995
Real GDP
206.0
221.5
255.5
294.2
GDP De‡.
1.000
0.916
0.810
0.676
Gr.R. GDP
In‡. R.
Gr.R. GDP (old)
In‡. R. (old)
7.5%
15.3%
15.1%
-8.4%
-11.6%
-16.5%
9.0%
18.5%
25.7%
-9.6%
-13.9%
-23.5%
Let us get some feeling for the results. The whole objective of computing
real GDP and the GDP de‡ator was to decompose nominal GDP into a price
component and a quantity component since we are interested about how real
economic activity in an economy evolves over time. The old method of computing GDP gives too much weight to commodities whose prices have fallen
rapidly, in our example computers. Hence the old method overstates by how
much the real component of GDP increased and understates by how much the
price component increased (in this example it overstates by how much it declined). Comparing growth rates of real GDP for both methods and in‡ation
rates for both methods we see that the new methods shows lower growth rates
of real GDP and higher in‡ation rates.13 This is exactly the problem of the old
method: it understates the importance of the price decline in computers.
Finally, the di¤erence between both methods can be sizeable, not only in
our cooked-up example. Growth of real GDP, using these two methods, seem
to di¤er by as much as 0.5 to 1% yearly. Given that policies are based on real
GDP growth numbers this is not to be underestimated in its importance.
13 Note that, as discussed before, the GDP de‡ator, computed using the old method, is a
Paasche price index, and that, as discussed in the main text, Paasche price indices tend to
understate in‡ation.
32CHAPTER 2. NATIONAL INCOME AND PRODUCT ACCOUNTING (NIPA)
Chapter 3
Economic Growth
3.1
3.1.1
Mathematical Preliminaries
Discrete vs. Continuous Time
So far in this course we have dealt with time as a discrete variable. Time could
take the values t = 0; t = 1; t = 1995 and so on, but no values in between.
For the purposes of growth theory it is often convenient to think of time as a
continuous variable, so that t = 0:3; t = 1995:25 etc are possible. When time is
continuous, we write our economic variables of interest, like GDP, population,
in‡ation rate, as functions of time. Let us look at an example.
Suppose that the population in a particular country is a function of time:
N(t) gives the population of a particular country at date t; where t can take
any value (not just integer values). So N (1995) is the population of the country
on January 1, 1995, N (1995:5) is the population on July 1, 1995 and so on.
3.1.2
Derivatives
The derivative of a function N; denoted by N 0 or dN
dt measures by how much the
population changes when the date changes by a very small bit (an instantaneous
change). If the independent variable of a function N is time (as in our example),
then it has become customary to denote the derivative of the function N by N_ .
_
Hence N 0 ; dN
dt and N all denote the same thing, namely the derivative of the
function N with respect to time. Note that when the population increases over
dN
time, then dN
dt > 0 and when it decreases, then dt < 0:
The derivative of a function with respect to time expresses the instantaneous
change of the function. It is closely related to the change of the function over a
discrete time span. Let N (1996) be the population of our country on January
1, 1996 and N(1997) be the population on January 1, 1997. Then N (1997) ¡
N(1996) is the change in the population in the time interval between January
1, 1996 and January 1, 1997. Here the time interval is one year. If we let the
time interval get shorter and shorter, the change of the variable during that
33
CHAPTER 3. ECONOMIC GROWTH
34
time interval approaches the derivative of the function. Formally, let ¢t denote
the length of the time interval, then the derivative of N with respect to time t
is de…ned as
N (t) ¡ N(t ¡ ¢t)
dN (t)
´ N 0 (t) ´ N_ (t) = lim
¢t!0
dt
¢t
There are a few basic rules to take derivatives:
1. If N (t) = tn with n a positive integer, then
N_ (t) = ntn¡1
2. If N (t) = ex ; then
N_ (t) = ex
3. If N (t) = log(t); then
1
_
N(t)
=
t
4. If N (t) = g(h(t)); with g; h functions, then
_
N_ (t) = g 0 (h(t)) ¤ h(t)
Note that whenever we use the log in this course, we mean the log with basis
e; or the natural logarithm (Sometime the symbol ln is used for the natural log,
but we will always use log to denote the natural logarithm). Examples
If N (t) = t5 then N_ (t) = 5t4
6x2
3
If N (t) = log(2x3 ) then N_ (t) = 3 =
2x
x
Also note that a very important consequence of the forth rule (the so-called
chain rule) is the following. Suppose we want to …nd the time derivative of
log (N (t)) : Then we use as our function g the log; and as our function h the
function N to get
³
´ d log(N(t))
_
1
N(t)
_
log(N_ (t)) ´
=
¤ N(t)
=
dt
N (t)
N(t)
3.1.3
Some Useful Facts about Logs
Here are some rules for the natural logarithm
log(x ¤ y)
µ ¶
x
log
y
log(xa )
log(ex )
elog(x)
= log(x) + log(y)
= log(x) ¡ log(y)
= a ¤ log(x)
= x
= x
3.1. MATHEMATICAL PRELIMINARIES
3.1.4
35
Growth Rates (once again)
Remember how growth rates were de…ned in the case where time is discrete
gN (t ¡ 1; t) =
Nt ¡ Nt¡1
Nt¡1
In continuous time growth rates are de…ned analogously. Noting that, as the
time interval between t¡1 and t converges to 0; the di¤erence Nt ¡Nt¡1 (divided
_
by the time interval) converges to N(t)
and Nt¡1 gets closer and closer to Nt :
This motivates the fact that in continuous time we de…ne the growth rate of a
variable N at time t as
gN (t) =
N_ (t)
N (t)
; i.e. we can compute the growth
Note the important fact that gN (t) = d log(N(t))
dt
rate of a variable by taking the time derivative of the log of this variable. This
fact turns out to be very useful.
3.1.5
Growth Rates of Functions
The preceding fact, plus the rules for logarithms, can be used to compute growth
rates of functions. Suppose we have a variable k(t) that is de…ned to be the
ratio of two other variables K(t) and L(t); i.e.
k(t) =
K(t)
L(t)
In our application we will denote by k(t) as capital per worker, by K(t) the
aggregate capital stock and by L(t) the number of workers at time t: Suppose
we know the growth rate of K(t) and L(t) and want to …nd the growth rate of
k(t): We do the following. First we take logs on both sides (and use the rules
for logs)
log(k(t)) = log(K(t)) ¡ log(L(t))
Now we di¤erentiate both sides with respect to time to get
d log(K(t)) d log(L(t))
¡
dt
dt
_
_
K(t)
L(t)
=
¡
k(t)
K(t) L(t)
gk (t) = gK (t) ¡ gL (t)
d log((k(t))
dt
_
k(t)
=
Hence the growth rate of the ratio K(t)
L(t) equals the di¤erence of the growth rates.
Also, if we want the ratio to remain constant over time (i.e. gk (t) = 0), this
requires that both K(t) and L(t) must grow at the same rate, i.e. gK (t) = gL (t):
CHAPTER 3. ECONOMIC GROWTH
36
Suppose that total output at period t; Y (t) depends on the total capital
stock K(t) and total number of workers L(t) used in the production process in
the following form
Y (t) = K(t)® L(t)1¡®
with ® a …xed constant between 0 and 1: This particular relationship between
output and capital and labor input is called Cobb-Douglas production function
and we will use it extensively later. Suppose we know the growth rates of capital
K(t) and labor L(t) and we want to …nd the growth rate of output. Again we
can use the trick of …rst taking logs and then di¤erentiate with respect to time.
log(Y (t))
d log(Y (t))
dt
Y_ (t)
Y (t)
gY (t)
= ® ¤ log(K(t)) + (1 ¡ ®) ¤ log(L(t))
d log(K(t))
d log(L(t))
= ®¤
+ (1 ¡ ®) ¤
dt
dt
_
_
K(t)
L(t)
= ®¤
+ (1 ¡ ®) ¤
K(t)
L(t)
= ® ¤ gK (t) + (1 ¡ ®) ¤ gL (t)
Hence the growth rate of output equals the weighted sum of the growth rates of
inputs, with the weight being equal to the (share) parameter ® in the production
function.
3.1.6
Simple Di¤erential Equations and Constant Growth
Rates
Suppose a variable,1 say output Y grows at a constant rate gY (t) from date 0
to date T and suppose we know output at period 0; Y (0): What is output at
period T ? In discrete time the answer was
YT = (1 + g)T Y0
Now we want to derive a similar formula for continuous time. We start with
the de…nition of a growth rate in continuous time (and use the fact that this
growth rate is constant from t = 0 to t = T )
g=
Y_ (t)
Y (t)
Integrating both sides with respect to time t; from 0 to T; yields2
1 This
section assumes familiarity with the theory of integration. Readers without this
knowledge may skip to the …nal formulas.
2 Keep
Y_ (t)
Y (t)
in mind that the time derivative of log(Y (t)) equals
equals log(Y (t)):
Y_ (t)
;
Y (t)
so the anti-derivative of
3.2. GROWTH AND DEVELOPMENT FACTS
Z
T
Y_ (t)
dt
0 Y (t)
= log(Y (T )) ¡ log(Y (0))
µ
¶
Y (T )
= log
Y (0)
Y (T )
=
Y (0)
gdt =
0
gT
gT
egT
Z
37
T
Y (T ) = egT ¤ Y (0)
(3.1)
Hence if output at time 0 equals Y (0) and grows at constant rate g; then at time
T output equals egT Y (0): Note that with formula (3.1) we can ask exactly the
same questions (and use exactly the same manipulations) in continuous time as
in Appendix 1 of Chapter 2 with discrete time.
We should note two things: …rst, by taking logs in the formula we get
log(Y (T )) = log(Y (0)) + gT
Hence if output (or any other variable) grows at a constant rate g; then plotting
the log of output gives a straight line with intercept log(Y (0)) and slope g:
Therefore economists often plot the log of a variable (rather than the variable
itself), because this way it is easy to see whether (and at what rate) the variable
grows over time. See Figures 7 and 8 for the e¤ect.
Second, the formulas for discrete and continuous time yield roughly the same
result (you should work out some examples with your pocket calculator). The
two formulas would in fact be identical if eg = (1 + g): That this equality is
approximately true can be seen from the Taylor series expansion of eg around
g=0
eg
= e0 + (g ¡ 0)e0 +
= 1+g+
(g ¡ 0)2 0 (g ¡ 0)3 0
e +
e +:::
2
6
g2 g3
+
+ :::
2
6
¼ 1+g
if g is not too large
3.2
Growth and Development Facts
The economist Niclas Kaldor pointed out the following stylized growth facts
(empirical regularities of the growth process) for the US and for most other
industrialized countries (look back at the …gures in the last section):
1. Output (real GDP) per worker y = YL and capital per worker k =
over time at relatively constant and positive rate.
K
L
grow
CHAPTER 3. ECONOMIC GROWTH
38
Exponentially Growing Variable
1600
1400
1200
Y(t)
1000
800
600
400
200
0
0
0.5
1
1.5
2
2.5
Time
3
3.5
4
4.5
Figure 3.1:
2. They grow at similar rates, so that the ratio between capital and output,
K
Y is relatively constant over time
3. The real return to capital r (and the real interest rate r ¡ ±) is relatively
constant over time
4. The capital and labor shares are roughly constant over time. The capital
share ® is the fraction of GDP that is devoted to interest payments on
capital, ® = rK
Y : The labor share 1 ¡ ® is the fraction of GDP that is
devoted to the payments to labor inputs; i.e. to wages and salaries and
other compensations: 1 ¡ ® = wL
Y : Here w is the real wage.
These stylized facts motivated the development of the neoclassical growth
model, the so-called Solow model, to be discussed below. The Solow model has
spectacular success in explaining the stylized growth facts by Kaldor. Note that
the growth facts pertain to data for a single country over a (long) period of
time. Such a data set is called a time series.
5
3.2. GROWTH AND DEVELOPMENT FACTS
39
Exponentially Growing Variable, Log Scale
8
7
log(Y(t))
6
5
4
3
2
0
0.5
1
1.5
2
2.5
Time
3
3.5
4
4.5
Figure 3.2:
In addition to the growth facts we will be concerned with how income (per
worker) levels and growth rates vary across countries in di¤erent stages of their
development process. The true test of the Solow model is to what extent it can
explain di¤erences in income levels and growth rates across countries, the so
called development facts. As we will see, the verdict here is mixed.
Now we summarize the most important facts from the Summers and Heston’s
panel data set. This data set follows about 100 countries for 30 years and
has data on income (production) levels and growth rates as well as population
and labor force data. In what follows we focus on the variable income per
worker. This is due to two considerations: a) our theory (the Solow model)
will make predictions about exactly this variable b) although other variables
are also important determinants for the standard of living in a country, income
per worker (or income per capita) may be the most important variable (for
the economist anyway) and other determinants of well-being tend to be highly
positively correlated with income per worker.
5
40
CHAPTER 3. ECONOMIC GROWTH
Before looking at the data we have to think about an important measurement
issue. Income is measured as GDP, and GDP of a particular country is measured
in the currency of that particular country. In order to compare income between
countries we have to convert the income measures into a common unit. One
option would be exchange rates. These, however, tend to be rather volatile and
reactive to events on world …nancial markets. Economists that study growth and
development tend to use a di¤erent procedure to measure the value of currencies
against each other. They ask how many dollars it costs to buy a middle class car
in the US, and how many yen the same type of car costs in Japan. Suppose the
numbers are $15,000 and 2,000,000 yen. Then the exchange rate, based on cars
would be $0.75 per 100 Yen. By extending this procedure to a lot of di¤erent
products and taking a weighted average one constructs an exchange rate that
measures the relative purchasing power of two currencies. This exchange rate is
called the PPP-based exchange rate, where PPP stands for Purchasing Power
Parity. All income numbers used by Summers and Heston (and used in these
notes) are converted to $US via PPP-based exchange rates.
Here are the most important facts from the Summers and Heston data set:
1. Enormous variation of per capita income across countries: the poorest
countries have about 5% of per capita GDP of US per capita GDP. This
fact is about dispersion in income levels. When we look at Figure 9, we
see that out of the 104 countries in the data set, 37 in 1990 and 38 in
1960 had per worker incomes of less than 10% of the US level. The richest
countries in 1990, in terms of per worker income, are Luxembourg, the US,
Canada and Switzerland with over $30,000, the poorest countries, without
exceptions, are in Africa. Mali, Uganda, Chad, Central African Republic,
Burundi, Burkina Faso all have income per worker of less than $1000.
Jones’ Figure 1.2. shows that not only are most countries extremely poor
compared to the US, but most of the world’s population is poor relative
to the US.
2. Enormous variation in growth rates of per worker income. This is a fact
about changes of levels in per capita income. Figure 10 shows the distribution of average yearly growth rates from 1960 to 1990. The majority of
countries grew at average rates of between 1% and 3% (these are growth
rates for real GDP per worker ). Note that some countries posted average growth rates in excess of 6% (Singapore, Hong Kong, Japan, Taiwan,
South Korea) whereas other countries actually shrunk, i.e. had negative growth rates (Venezuela, Nicaragua, Guyana, Zambia, Benin, Ghana,
Mauretania, Madagascar, Mozambique, Malawi, Uganda, Mali). We will
sometimes call the …rst group growth miracles, the second group growth
disasters. Note that not only did the disasters’ relative position worsen,
but that these countries experienced absolute declines in living standards.
The US, in terms of its growth experience in the last 30 years, was in the
middle of the pack with a growth rate of real per worker GDP of 1.4%
between 1960 and 1990.
3.2. GROWTH AND DEVELOPMENT FACTS
41
Distribution of Relative Per Worker Income
40
1960
1990
35
Number of Countries
30
25
20
15
10
5
0
0
0.2
0.4
0.6
0.8
1
Income Per Worker Relative to US
1.2
Figure 3.3:
3. Growth rate determine economic fate of a country over longer periods of
time. How long does it take for a country to double its per capita GDP
if it grows at average rate of g% per year. A good rule of thumb: 70=g
years (this rule of thumb is due to Nobel Price winner Robert E. Lucas
(1988)). Growth rates are not constant over time for a given country.
This can easily be demonstrated for the US. GDP per worker in 1990 was
$36,810. If GDP would always have grown at 1.4% , then for the US
GDP per worker would have been about $9,000 in 1900, $2,300 in 1800,
$570 in 1700, $140 in 1600, $35 in 1500 and so forth. Economic historians
(and common sense) tells us that nobody can survive on $35 per year
(estimates are that about $300 are necessary as minimum income level
for survival). This indicates that the US (or any other country) cannot
have experienced sustained positive growth for the last millennium or so.
In fact, prior to the era of modern economic growth, which started in
England in the late 18-th century, per worker income levels have been
1.4
CHAPTER 3. ECONOMIC GROWTH
42
Distribution of Average Growth Rates (Real GDP) Between 1960 and 1990
25
Number of Countries
20
15
10
5
0
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
Average Growth Rate
0.04
0.05
Figure 3.4:
almost constant at subsistence levels. This can be seen from Figure 11,
which compiles data from various historical sources. The start of modern
economic growth is sometimes referred to as the Industrial Revolution.
It is the single most signi…cant economic event in history and has, like
no other event, changed the economic circumstances in which we live.
Hence modern economic growth is a quite recent phenomenon, and so
far has occurred only in Western Europe and its o¤springs (US, Canada,
Australia and New Zealand) as well as recently in East Asia.
4. Countries change their relative position in the (international) income distribution. Growth disasters fall, growth miracles rise, in the relative crosscountry income distribution. A classical example of a growth disaster is
Argentina. At the turn of the century Argentina had a per-worker income
that was comparable to that in the US. In 1990 the per-worker income
of Argentina was only on a level of one third of the US, due to a healthy
growth experience of the US and a disastrous growth performance of Ar-
0.06
3.3. THE SOLOW MODEL
43
GDP pe r Capita (in 1985 US $): W e ste rn Europe
a nd its Offsprings
16000
14000
12000
10000
GDP per Capita
8000
6000
4000
2000
89
73
19
50
19
13
19
70
19
20
18
10
18
00
16
14
0
00
10
50
0
0
Tim e
Figure 3.5:
gentina. Countries that dramatically moved up in the relative income
distribution include Italy, Spain, Hong Kong, Japan, Taiwan and South
Korea, countries that moved down are New Zealand, Venezuela, Iran,
Nicaragua, Peru and Trinidad&Tobago.
In the next sections we have two tasks: to construct a model, the Solow
model, that a) can successfully explain the stylized growth facts b) investigate
to which extent the Solow model can explain the development facts.
3.3
The Solow Model
We look for a model that explains the stylized growth facts from above. In 1956
Robert Solow from MIT developed such a model, the Solow growth model in
his paper “A Contribution to the Theory of Economic Growth”. This brought
him the Nobel Price in 1987.
3.3.1
Models
Before discussing the Solow model, let’s brie‡y make clear what a successful
model is.
What is a model? It is a mathematical description of the economy. Why
do we need a model? The world is too complex to describe it in every detail.
A model abstracts from details to describe clearly the main forces driving the
economy. What makes a model successful? When it is simple but e¤ective in
describing and predicting how the (economic) world works. Note: A model
CHAPTER 3. ECONOMIC GROWTH
44
relies on simplifying assumptions. These assumptions drive the conclusions of
the model. When analyzing a model it is therefore crucial to clearly spell out
the assumptions underlying the model.
3.3.2
Setup of the Basic Model and Model Assumptions
The basic assumptions of the Solow model are that there is a single good produced in our economy and that there is no international trade, i.e. the Solow
model is a model of a closed economy Also there is no government. It is also
assumed that all factors of production (labor, capital) are fully employed in the
production process. The model consists of two basic equations, the neoclassical
aggregate production function and a capital accumulation equation.
1. neoclassical aggregate production function
Y (t) = F (K(t); L(t))
where Y (t) is total output produced in our economy at date t: Output
is produced using the two inputs capital K(t) and labor services L(t):
Assumptions on F :
² Constant returns to scale: doubling both inputs will result in doubled
output. Mathematically: for all constants c > 0
F (cK(t); cL(t)) = cF (K(t); L(t))
² Positive, but decreasing marginal products: holding one input …xed,
by increasing the other input we increase output, but at decreasing
rate. Mathematically
@F
@K
@F
@L
@2 F
<0
@K 2
@2F
> 0;
<0
@L2
> 0;
An important example for F is the Cobb-Douglas production function
Y (t) = F (K(t); L(t)) = K(t)® L(t)1¡®
(3.2)
where ® is a …xed parameter between 0 and 1: You should verify that
the Cobb-Douglas production function satis…es the two assumptions
made on F above. Our stylized growth facts dealt with output per
(t)
and capital per worker k(t) = K(t)
worker y(t) = YL(t)
L(t) . Dividing both
sides of equation (3:2) by the number of workers L(t) yields
µ
¶a µ
¶1¡®
K(t)® L(t)1¡®
K(t)® L(t)1¡®
K(t)
L(t)
=
y(t) =
=
= k(t)®
L(t)
L(t)® L(t)1¡®
L(t)
L(t)
3.3. THE SOLOW MODEL
45
The fact that we can write output per worker as a function of capital
per worker alone is due to the …rst assumption. The fact that there
are decreasing returns to capital per worker (an increase in capital
per worker increases output per worker at a decreasing rate) is due
to the second assumption. In summary, the aggregate production
function, written in per-worker terms for the Cobb-Douglas case, is
given by
y(t) = k(t)®
(3.3)
2. capital accumulation equation
_
K(t)
= sY (t) ¡ ±K(t)
(3.4)
_
The change of the capital stock in period t, K(t)
is given by the total
amount of investment in period t; sY (t) minus the depreciation of the
old capital stock ±K(t): Here s is the fraction of total output (income) in
period t that is saved, i.e. not consumed. If s = 0:2; then 20% of the total
output in period t is saved by the households in the economy. Similarly
± is the fraction of the capital stock at period t that wears out in the
production process. The important assumptions implicit in equation (3:4)
are
² Households save a constant fraction s of output (income), regardless
of the level of output. This is a strong assumption about the behavior
of households (and much theoretical work has been done to relax this
assumption). s is an important parameter of the model. Note that
the fact that total saving of households sY (t) equals total investment
is not an assumption, but follows from the accounting identity that
saving equals investment.
² A constant fraction ± of capital depreciates in each period. Rather
than a behavioral assumption (as the …rst one), this is an assumption
about technology: the production process is such that a constant
fraction of capital wears out in each period.
Since equation (3:3) is in per-worker terms, we look for a representation of
equation (3:4) in per-worker terms. The last assumptions that we make is that
the labor force participation rate is constant and that the populations grows
exponentially at a growth rate of n: Then the number of workers grows at rate
n; i.e.
L(t) = ent L(0)
(3.5)
Note that it follows from equation (3:5) that (remember that a dot over a
variable denotes the derivative of that variable with respect to time)
_
L(t)
nent L(0)
= nt
=n
L(t)
e L(0)
(3.6)
CHAPTER 3. ECONOMIC GROWTH
46
Now we can divide both sides of equation (3:4) by L(t) to obtain
_
K(t)
= sy(t) ¡ ±k(t)
L(t)
(3.7)
The right hand side of equation (3:7) is already in per-worker form, but the left
hand side requires more work. But
_
_
_
K(t)
K(t)
K(t)
K(t)
=
=
k(t)
(3.8)
L(t)
K(t) L(t)
K(t)
Remember that
_
_
_
_
K(t)
L(t)
K(t)
k(t)
=
¡
=
¡n
k(t)
K(t) L(t)
K(t)
Hence
_
_
k(t)
K(t)
=
+n
K(t)
k(t)
Combining equations (3:8) and (3:9) we get
Ã
!
_
_
_
K(t)
K(t)
k(t)
_ + nk(t)
=
k(t) =
+ n k(t) = k(t)
L(t)
K(t)
k(t)
(3.9)
(3.10)
Finally, we use (3:10) in (3:7) to obtain
_ + nk(t) = sy(t) ¡ ±k(t)
k(t)
or
_
k(t)
= sy(t) ¡ (± + n)k(t)
(3.11)
This is the capital accumulation equation in per-worker terms
3.3.3
Analysis of the Model
The Solow growth model characterizes output per capita and capital per capita
by the two basic equations
y(t) = k(t)®
_
k(t)
= sy(t) ¡ (± + n)k(t)
(3.12)
Substituting the …rst into the second we obtain a di¤erential equation in k; the
per-worker capital stock:
_
k(t)
= sk(t)® ¡ (± + n)k(t)
(3.13)
We will proceed by analyzing this di¤erential equation. Note that once we know
the behavior of k(t) over time, then from (3:12) we know the behavior of y(t):
Together with the knowledge of the initial number of workers L(0) and with help
of equation (3:5) we know the behavior of K(t) = k(t)L(t) and Y (t) = y(t)L(t):
Note that the values of these variables depend on the parameters s; ± and n: We
will demonstrate this below with some numerical examples.
But now let us proceed with the analysis of (3:13):
3.3. THE SOLOW MODEL
47
Graphical Analysis
Our di¤erential equation (3:13) describes how the capital stock per worker in
our model evolves over time. For example, we can analyze what happens with
the capital stock if we start at an arbitrary initial level k(0): We can also analyze
how capital per worker and hence output per worker di¤er in two economies that
di¤er in their savings or population rates.
_
Remember that k(t)
is the change in per-worker capital stock. This change
at period t is given by the di¤erence between investment (=saving) per worker
sy(t) = sk(t)® and e¤ective depreciation (± + n)k(t):3 In Figure 12 we draw
graphs of sy(t) = sk(t)® and (± + n)k(t) as functions of k(t): As a function of
k(t); the graph of (± + n)k(t) is a straight line with slope (± + n) that starts at
0: The graph of sk(t)® also starts at 0; and is very steep for small values of k(t)
and very ‡at for large values of k(t): This is a consequence of our assumptions
on the production function. Remember that the derivative of a function gives
the slope of the function. The derivative of sk(t)® with respect to k(t) is given
by
®s
k(t)1¡®
As long as 0 < ® < 1 this derivative approaches in…nity as k(t) approaches 0
and it approaches 0 as k(t) gets larger and larger.
_
The change in k(t); k(t)
is given by the di¤erence between the two graphs,
sk(t)® and (± + n)k(t): Suppose our economy starts at k(0): Since at k(0) we
_
is positive and the capital stock per
have that sk(t)® exceeds (± + n)k(t); k(t)
worker increases. This is indicated by the arrows on the x-axis. In fact, the
process of increasing k(t) continues as long as sk(t)® is bigger than(± + n)k(t):
Over time, the capital stock per worker converges to k¤ ; the capital stock at
_
= 0; i.e.
which sk(t)® = (± + n)k(t): At k¤ we have the situation in which k(t)
the capital stock per worker does not change anymore. Such a point at which
_
k(t)
= 0 is called a steady state: once the economy reaches this point, it stays
there forever. Given the properties of the production function there is a unique
positive steady state capital stock per worker in the Solow model and from any
positive initial capital stock k(0) the economy converges to this steady state over
time (we demonstrated this for k(0) < k¤ ; you should convince yourself that this
also happens if k(0) > k¤ ). Therefore this steady state is called (locally) stable:
starting close to k¤ brings the economy to k¤ over time. We make several other
observations: …rst, there is another (trivial) steady state k¤ = 0: If the economy
starts with k(0) = 0; it stays there forever. Second, once we have determined
the behavior of k(t); since y(t) = k(t)® we know the behavior of output per
worker, and also the behavior of consumption per worker c(t) = (1 ¡ s)k(t)®
over time. The behavior of total consumption, output and the capital stock
follows from the fact that the number of workers grow at constant rate n:
3 Note that k is per-worker capital. As population increases at rate n; this reduces capital
per worker (for a given capital stock). This e¤ect acts in exactly the same fashion as physical
depreciation.
CHAPTER 3. ECONOMIC GROWTH
48
(n+δ)k(t)
sy(t)
k(0)
k*
k(t)
Figure 3.6:
Steady State Analysis
We can solve for the steady state analytically. Remember that a steady state is
_
= 0: We
a situation in which per capita capital is constant over time, i.e. k(t)
¤
¤
denote steady state capital per worker by k : Obviously k solves the equation
0 = s (k¤ )® ¡ (n + ±)k¤
or
¤
k =
µ
s
n+±
1
¶ 1¡®
The steady state output per worker is then given by
®
µ
¶ 1¡®
s
¤
y =
n+±
(3.14)
3.3. THE SOLOW MODEL
49
Hence the steady state of an economy depends positively on the saving rate s
and negatively on the population growth rate n of the economy (and on the
technological parameters ±; ®). An increase in the saving rate and a decrease in
the population growth rate increases per worker capital and output. This type of
analysis -how does the steady state change with a change in model parametersis called comparative statics.
(n+δ)k(t)
s’y(t)
sy(t)
k(0)=k*
k’*
k(t)
Figure 3.7:
Let us now demonstrate the dynamic response of the economy to a change
in the saving rate from s to s0 : Suppose the economy initially is in the steady
state with the old saving rate, i.e. k(0) = k¤ : Now (for some model-exogenous
reason) the households in our economy start saving more, so that the saving
rate increases from s to s0 : As shown in Figure 13, such a change does not
a¤ect the (n + ±)k(t) line, but it tilts the sk(t)® line outwards around zero to
s0 k(t)® = s0 y(t): For the old steady state capital stock k(0) = k¤ ; now with the
_
> 0 and
new saving rate s0 we have that s0 k(0)® > (n + ±)k(0): Hence k(0)
CHAPTER 3. ECONOMIC GROWTH
50
the capital stock per worker starts growing. It continues to grow until it hits
the new steady state k0¤ > k¤ ; where it stays forever, unless new changes in
the saving rate or the depreciation rate happen. The process of the economy
moving from one steady state to the new steady state is called transition path
or transition dynamics. The same analysis can be done with a change in the
population growth rate, which is left as an exercise.
Evaluating the Basic Model
The simple Solow model gives a simple answer to the question why some countries have such a high level of output per worker and other have such a low level
of output per worker (i.e. why some countries are so rich whereas others are so
poor). Assuming that all countries have reached their respective steady states,
the Solow model predicts that countries with high saving (investment) rates s
and low population growth rates n have high per-worker output. We can use
the Summers-Heston data set to see whether this prediction of the model can
be found in the data. This is a …rst test of the model.
In Figure 14 we plot GDP per worker in 1990 against the average investment
rate (the fraction of GDP used for investment, equal to the saving rate s in our
model) between 1980-90. Each dot is one country (try to guess where the US
-or your country of birth- is located in this plot and then look at Jones’ Figure
2.6 if you want). We see the positive correlation between GDP per worker and
the investment rate: countries with higher investment rates in the data tend to
have higher GDP per worker, as predicted by the Solow model. This can be
viewed as a …rst success of the Solow model.
Figure 15 plots GDP per worker in 1990 against the average population
growth rate between 1980-90. Again each dot represents one country. As predicted by the Solow model there is a negative correlation between population
growth rates and per worker GDP. Again the data support this prediction of
the Solow model.
3.3.4
Introducing Growth
We wrote down the Solow growth model to explain the stylized facts of Kaldor,
in particular the facts that income and capital per worker grow at equal constant
and positive rates. So what about growth in the simple Solow model? We saw
that in the model capital and output per capita converged to their steady state
levels and then stayed there forever (remember that a steady state was de…ned
as a situation in which the per worker capital stock does not change anymore).
Hence in this version of the model there is no long-run growth of capital per
worker or output per worker. Output and the capital stock grow, but only at
the rate of population growth n: Fortunately this failure of the model is easy
3.3. THE SOLOW MODEL
51
GDP per Worker 1990 as Function of Investment Rate
4.5
GDP per Worker 1990 in $10,000
4
3.5
3
2.5
2
1.5
1
0.5
0
-0.5
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Average Investment Share of Output 1980-90
Figure 3.8:
to correct as we will see in a second. But …rst let’s look at growth along the
transition path to the steady state.
Dividing both sides of (3:13) by k(t) we get
gk (t) ´
_
k(t)
= sk(t)®¡1 ¡ (n + ±)
k(t)
(3.15)
Since ® < 1; ® ¡ 1 < 0 and therefore the bigger the per worker capital stock
k(t); the smaller is the growth rate of the per worker capital stock gk (t): At
the steady state k¤ the growth rate is zero (you should verify this by plugging
the formula for k¤ into (3:15) for k(t)). If the economy starts at k(0) < k¤ ;
the growth rate of k is positive. Over time capital per worker gets bigger and
bigger and the growth rate declines (but is still positive). Eventually k reaches
the steady state k¤ and growth stops as the growth rate falls to zero. Hence
there is growth along the transition path in the simple Solow model, but no
0.45
CHAPTER 3. ECONOMIC GROWTH
52
GDP per Worker 1990 as Function of Population Growth Rate
4.5
4
GDP per Worker 1990 in $10,000
3.5
3
2.5
2
1.5
1
0.5
0
-0.5
0
0.01
0.02
0.03
0.04
Average Population Growth Rate 1980-90
Figure 3.9:
sustained growth over time. The behavior of output per worker parallels that
of k; as y(t) = k(t)® : On the other hand, if the economy starts at k(0) > k¤ ;
then the growth rate is negative and the capital stock per worker declines to k¤
over time. The growth rate becomes less and less negative and …nally arrives at
0 in the steady state.
This discussion is neatly summarized in Figure 16, where on the x-axis we
have k(t) and we plot the curve sk(t)®¡1 and the line n+±: The vertical distance
_
_
k(t)
k(t)
: We see that k(t)
> 0
between the two lines represents the growth rate k(t)
whenever k(t) < k¤ and
_
k(t)
k(t)
< 0 whenever k(t) > k¤
Now let us return to the question of how to generate sustained growth in
the Solow model. The answer that Solow gave was to introduce technological
progress in the aggregate production function. Aggregate output is now given
0.05
3.3. THE SOLOW MODEL
53
α-1
sk(t)
(n+δ)
k(0)
k*
k(t)
Figure 3.10:
by
Y (t) = K(t)® (A(t)L(t))1¡®
where A(t) is the level of technology at date t: The capital accumulation equation remains unchanged. When the level of technology multiplies labor input
L(t) as above, technological progress is said to be labor-augmenting (or Harrod neutral): a higher level of technology A(t) makes a given number of workers
more productive in that the same number of workers can now produce more output. For concreteness we interpret A(t) to be the stock of ideas or knowledge
that an economy at time t has access to.
We assume that the level of technology grows at a constant rate g > 0 over
time, i.e.
_
A(t)
=g
A(t)
CHAPTER 3. ECONOMIC GROWTH
54
This is a crucial (maybe the most crucial) assumption of the model. Note that
we do not explain why the level of technology grows over time, that is we take
growth of technology as exogenously given, as manna from heaven, so to speak.
Therefore the Solow model is often referred to as an exogenous growth model. In
later sections we will look at the so-called endogenous growth theory, at models
that try to explain why we have technological progress.
Now let’s proceed with the analysis of the Solow model with technological
progress. Because of technological progress it is clear that the economy will not
any longer have a steady state in which output and capital per worker are constant. But it turns out that it possesses a very similar property. By a balanced
growth path we de…ne a situation in which output, capital and consumption per
worker grow at constant rates (which need not be the same). Note that a steady
state is just a special case of a balanced growth path in which all variables grow
at constant rate 0:
We …rst want to …nd out at what growth rate output per worker and capital per worker grow in a balanced growth path. Remember that the capital
accumulation equation is given by
_
K(t)
= sY (t) ¡ ±K(t)
Dividing both sides by K(t) yields
gK (t) ´
_
Y (t)
K(t)
=s
¡±
K(t)
K(t)
Remember that
gk (t) ´
_
_
K(t)
k(t)
=
¡n
k(t)
K(t)
Hence
gk (t) ´
_
Y (t)
k(t)
=s
¡ (n + ±)
k(t)
K(t)
In a balanced growth path by de…nition gk (t) is constant. From the equation
Y (t)
above this requires that K(t)
is constant over time, i.e. that output Y (t) and
the capital stock K(t) must grow at the same rate. It then follows that output
per worker y(t) and capital per worker k(t) must grow at the same rate in a
balanced growth path, i.e. gk = gy .4
The next question is at what common rate do y and k grow? Dividing the
aggregate production function by the labor force L(t) we get
y(t) =
=
K(t)® (A(t)L(t))1¡®
Y (t)
=
L(t)
L(t)
K(t)® (A(t)L(t))1¡®
L(t)®
L(t)1¡®
= k(t)® A(t)1¡®
4 This follows from the fact that g = g ¡n and g = g ¡n: From g
y
k
K
Y
K = gY then gk = gy
immediately follows.
3.3. THE SOLOW MODEL
55
In order to get growth rates we take logs and then di¤erentiate with respect to
time
d log(k(t))
d log(A(t))
d log(y(t))
= ®
+ (1 ¡ ®)
dt
dt
dt
gy (t) = ®gk (t) + (1 ¡ ®)gA (t)
Now we use the result that k and y grow at the same rate in a balanced growth
path and that A grows at constant rate g by assumption. We then have
gy
(1 ¡ ®)gk
gk
= gk = ®gk + (1 ¡ ®)gA
= (1 ¡ ®)g
= gy = g
Hence along a balanced growth path capital per worker and output per worker
(and consumption per worker) all grow at the same rate g; the growth rate of
technological progress. Also, if there is no technological progress, then g = 0
and there is no sustained growth in the economy (and we are back in the simple
Solow model). Therefore the engine of growth in per capita output in this model
is technological progress.
3.3.5
Analysis of the Extended Model
The previous discussion indicates that along a balanced growth path our variables of interest, y(t) and k(t) grow at constant rate g; the rate of technological
progress. To analyze the new model graphically it is convenient to work with
variables that are constant in the long run. Since y(t) and k(t) and A(t) grow
at the same rate g; we de…ne new variables
y~(t) =
~
k(t)
=
Y (t)
y(t)
=
A(t)
A(t)L(t)
K(t)
k(t)
=
A(t)
A(t)L(t)
Note that in a balanced growth path y~ and k~ are constant. We will call k~
the technology-adjusted per worker capital stock and y~ the technology-adjusted
output per worker. It turns out that once we look at the variables y~ and k~ the
analysis from the previous section goes through almost unchanged.
First look at the aggregate production function
y~(t) ´
=
Y (t)
A(t)L(t)
K(t)® (A(t)L(t))1¡®
A(t)L(t)
K(t)® (A(t)L(t))1¡®
(A(t)L(t))® (A(t)L(t))1¡®
~ ®
= k(t)
=
CHAPTER 3. ECONOMIC GROWTH
56
which is exactly the same as before, once we made the change in variables. Now
let’s look at the capital accumulation equation
_
K(t)
= sY (t) ¡ ±K(t)
Dividing both sides by A(t)L(t) we obtain
_
K(t)
A(t)L(t)
_
K(t)
A(t)L(t)
=
sY (t)
K(t)
¡±
A(t)L(t)
A(t)L(t)
~
= s~
y(t) ¡ ± k(t)
The right hand side of this equation is already in a form that we like, the left
hand side requires more work, basically the same work as in the last section:
_
_
_
K(t)
K(t)
K(t)
K(t)
~
=
=
k(t)
A(t)L(t)
K(t) A(t)L(t)
K(t)
(3.16)
Also
gk~ (t) ´
=
Hence
_
K(t)
K(t)
=
:
~
k(t)
~
k(t)
:
~
k(t)
= gK (t) ¡ gA (t) ¡ gL (t)
~
k(t)
_
K(t)
¡g¡n
K(t)
+ g + n: Substituting into equation (3:16) we get
0
1
:
:
~
_
B k(t)
C
K(t)
~ = k(t)
~
C k(t)
~
=B
+
g
+
n
+ (g + n)k(t)
A
~
A(t)L(t) @ k(t)
and hence the capital accumulation equation becomes
:
~
~
= s~
y(t) ¡ (n + g + ±)k(t)
k(t)
Summarizing, with our new variables k~ and y~ our two equations of the Solow
model become
~ ®
y~(t) = k(t)
:
~
k(t)
~
= s~
y (t) ¡ (n + g + ±)k(t)
3.3. THE SOLOW MODEL
57
~
which can be combined to the di¤erential equation in k(t):
:
~
~ ® ¡ (n + g + ±)k(t)
~
k(t) = sk(t)
(3.17)
Note how similar equation (3:17) is to equation (3:13) once we make our change
of variables. In particular, we can analyze (3:17) graphically in exactly the same
way as we did with (3:13):
Graphical Analysis
We can draw the Solow diagram for the model with technological progress.
~
In Figure 17 we let k(t)
be the variable on the x-axis and we plot the curves
~ on the y-axis. The …rst curve looks exactly as before,
~ ® and (n + g + ±)k(t)
sk(t)
the other is again a straight line, but now with slope (n + g + ±) instead with
~ ® and (n + g + ±)k(t)
~ gives the change in
(n + ±): The di¤erence between sk(t)
:
~
:
the technology adjusted capital stock per worker k(t)
~
~
~ ® exSuppose our economy starts at k(0):
Since at k(0)
we have that sk(t)
:
~
~
is positive and the technology-adjusted capital stock per
ceeds (±+n)k(t);
k(t)
worker increases. This is indicated by the arrows on the x-axis. In fact, the pro~
~ continues as long as sk(t)
~ ® is bigger than (± + n + g)k(t):
cess of increasing k(t)
¤
~
Over time, the capital stock per worker converges to k ; the capital stock at
:
®
¤
~
~
~
~
which sk(t) = (± +n)k(t)s: At k we have the situation in which k(t) = 0; i.e.
the technology adjusted capital stock per worker does not change anymore. As
~
before there is only one such k~¤ and from all positive starting values of k(0)
we
always go to k~¤ : Hence in the long run in our economy converges to the balanced
~
grow path in which capital per worker k(t) = k(t)A(t)
and output per worker
y(t) = y~(t)A(t) grow at the constant rate g: Total output Y (t) = y(t)L(t) and
the capital stock K(t) = k(t)L(t) grow at rates g + n:
Balanced Growth Path Analysis
As before we can solve for the balanced growth path analytically. We know that
:
¤
~
~
at k we have k(t) = 0; i.e. k~¤ solves
®
0 = sk~¤ ¡ (n + g + ±)k~¤
CHAPTER 3. ECONOMIC GROWTH
58
~
(n+g+δ) k(t)
~ α
sk(t)
.
~
k(0)
~
k(0)
~
k*
~
k(t)
Figure 3.11:
Hence
k~¤ =
µ
s
n+g+±
1
¶ 1¡®
µ
s
n+g+±
®
¶ 1¡®
and therefore
¤
y~ =
3.3. THE SOLOW MODEL
59
It follows that along a balanced growth path
k(t) = A(t)
µ
µ
s
n+g+±
1
¶ 1¡®
®
¶ 1¡®
s
y(t) = A(t)
n+g+±
1
µ
¶ 1¡®
s
K(t) = L(t)A(t)
n+g+±
®
µ
¶ 1¡®
s
Y (t) = L(t)A(t)
n+g+±
We can now do comparative statics as before. Suppose that in period t = T
all of a sudden the saving (investment) rate s in our economy increases to s0 .
The economic intuition of what happens is simple: people save more and hence
there are more funds for investment into capital. Therefore the capital stock
per worker will increase over time, hence output per worker will increase over
time. Let’s look at what exactly happens a bit more carefully.
From the formulas above we note the following: …rst of all, since the growth
rate of the economy is given by g; the rate of technological progress, the growth
rate of the economy is not a¤ected by the increase of the saving rate. From the
formulas above, however, we see that y~¤ increases to y~¤0 . Hence the levels of
output and capital per worker, y(t) and k(t); are higher in the new balanced
growth path, and so are the levels of total output and capital, Y (t) and K(t):
We will see this graphically from several …gures. In Figure 18 we show how a
change in the saving rate from s to s0 a¤ects the technology adjusted capital
stock per worker k~ over time.
The story is exactly the same as in Figure 13 for the simple Solow model.
Suppose the economy initially is in the steady state with the old saving rate,
~ ) = k~¤ : Now (for some model-exogenous reason) the households in our
i.e. k(T
economy start saving more, so that the saving rate increases from s to s0 : As
~ line, but
shown in Figure 18, such a change does not a¤ect the (n + g + ±)k(t)
~ ® = s0 y~(t): For the old
~ ® line outwards around zero to s0 k(t)
it tilts the sk(t)
~ ) = k~¤ ; now with the new saving rate s0 we have
steady state capital stock k(T
:
0~
®
~
~
that s k(T ) > (n + g + ±)k(T ): Hence k(T ) > 0 and the technology-adjusted
capital stock per worker starts growing. It continues to grow until it hits the
new steady state k~0¤ > k~¤ ; where it stays forever. What happens to economic
growth over time? We already concluded that in the long run an increase in the
saving rate has no e¤ect on the growth rate of this economy. But what along
CHAPTER 3. ECONOMIC GROWTH
60
~
(n+g+δ) k(t)
~ α
s’k(t)
.
~
k(T)
~
k*
~ α
sk(t)
~
k*’
~
k(t)
Figure 3.12:
the transition path? Recall our basic equation for the extended Solow model
:
~
~ ® ¡ (n + g + ±)k(t)
~
k(t) = sk(t)
~ ) = k~¤
Along the initial balanced growth path, for k(T
:
~ )=0
~ )® ¡ (n + g + ±)k(T
~
k(T ) = sk(T
3.3. THE SOLOW MODEL
61
~ ) = k~¤ we have
But now the saving rate increases from s to s0 > s; so at k(T
:
~ )>0
~ )® ¡ (n + g + ±)k(T
~ ) = s0 k(T
k(T
~
Hence the growth rate of k;
:
~
k(t)
~
k(t)
is positive. Along the entire transition path
:
~ ® curve lies above the (n + g + ±)k(t)
~ line.
~
remains positive, as the s0 k(t)
k(t)
~ increases
The di¤erence between these two lines gets smaller and smaller as k(t)
:
~
to the new steady state. Therefore k(t)
gets smaller and smaller (but remains
~ increases along the transition path. Therefore the growth rate
positive) and k(t)
of k~ behaves as follows: it is equal to zero before time T (as we are in a balanced
growth path), then at time T; jumps up to a positive number, and then over
time declines (but remains positive) to the old, zero growth rate. Figure 19
shows this discussion graphically.
Remember that we chose the ~variables only for convenience. What we are
really interested in is the behavior of capital per worker and output per worker.
Let us …rst look at the growth rates of these variables as the saving rate changes
y
K
k
Y
=A
=A
: Therefore
and y~ = AL
from s to s0 : Remember that k~ = AL
~
k(t) = k(t)A(t)
Taking logs and di¤erentiate with respect to time we …nd the growth rate of
capital per worker as
:
~
_k(t)
k(t)
=
+g
~
k(t)
k(t)
Therefore the growth rate of capital per worker reacts to an increase in the saving
rate in exactly the same way as the growth rate of k~ (just shifted upwards by
g): before time T capital per worker grows at rate g; at time T the growth rate
jumps above g and comes back to g over time. We show this in Figure 20.
It is equally straightforward to determine the behavior of the growth rate of
output per capita over time. The production function is given by
~ ®
y~(t) = k(t)
and therefore
:
y~(t)
=®
y~(t)
:
~
k(t)
~
k(t)
CHAPTER 3. ECONOMIC GROWTH
62
.
~ ~
k(t)/ k(t)
0
T
time t
Figure 3.13:
Also, since y(t) = y~(t)A(t) we have
y(t)
_
y(t)
:
y~(t)
y~(t)
=
=
:
y~(t)
+g
y~(t)
y(t)
_
¡g
y(t)
and hence
y(t)
_
=®
y(t)
:
~
k(t)
+g
~
k(t)
3.3. THE SOLOW MODEL
63
.
k(t)/k(t)
g
T
time t
Figure 3.14:
Therefore the growth rate of output also behaves similar to the growth rate
~ Before time T output per worker grows at constant rate g as we are in
of k:
the balanced growth path. At time T; the growth rate jumps up (but only by
~ and then comes back to its balanced
a fraction ® than the jump for k or k)
growth path level of g: We demonstrate this in Figure 21.
So far we only talked about growth rates. But what happens to the level of
per capita output? We know that in the old balanced growth path and in the
new balanced growth path the level of per capita income grows at constant rate
g: Along the transition path the growth rate is higher than g, i.e. output per
worker temporarily grows at a faster pace. In Figure 22 we draw the behavior
of the level of output per capita. Instead of y we plot log(y) (remember why it
is easier to plot the log of a variable that grows at a constant rate over time).
Note that the picture makes clear that there are no long run e¤ects on the
CHAPTER 3. ECONOMIC GROWTH
64
.
y(t)/y(t)
g
T
time t
Figure 3.15:
growth rate of output per capita from an increase in the saving (investment)
rate: in the long run output grows at rate g and only increases in the rate of
technological progress a¤ect the long-term growth rate of output per capita.
Therefore any policy that helps to raise the saving rate s is unsuccessful in
increasing long term growth rates of real per worker GDP (if, of course, we
believe in the Solow model). On the other hand, an increase in the saving rate
has a level e¤ect: it permanently increases the “plateau” on which output per
capita grows, as shown in Figure 22. This ends our discussion of how a change in
the saving rate a¤ects growth rates and level of output (and income) per capita
in the extended Solow model. The same techniques and graphs can be applied
when analyzing changes in the population growth rate n, the depreciation rate
± or the rate of technological progress g.
3.3. THE SOLOW MODEL
65
log(y)
level
effect
slope g
slope g
T
time t
Figure 3.16:
3.3.6
Evaluation of the Solow Model
So is the extended Solow model a success? Let us start with the growth facts.
In the Solow model, in the long run (along a balanced growth path) output
per worker and capital per worker grow at the same positive rate g; the rate of
technological progress. Hence the ratio between the aggregate capital stock and
output K
Y is constant. Therefore the …rst two stylized facts can be explained by
the Solow model. However, they are only explained when we introduce technological progress at rate g > 0: Why this progress happens is left unexplained.
In the next sections we will look at models that explicitly try to explain technological progress.
What about the other two stylized growth facts, the facts that the real
interest rate and the capital and labor shares are constant over time? It turns
out that the Solow model has the property that the real interest rate and the
capital and labor share are constant along a balanced growth path. To see this
CHAPTER 3. ECONOMIC GROWTH
66
we have to take a little detour. So far we didn’t talk about who produces the
output in our economy. So let us introduce …rms. Firms produce output by
hiring L(t) numbers of workers, which are paid a wage w(t) and by renting
capital K(t) from households who own the capital stock. Per unit of capital the
…rms have to pay rent r(t): The real interest rate equals r(t) ¡ ±: households
receive rent r(t) for one unit of capital, but a fraction ± of the capital that
they lend to …rms they don’t get back because it wears out in the production
process. Therefore the e¤ective return on lending out capital (which is the real
interest rate) equals r(t) ¡ ± We assume that …rms are price takers, hence take
the price for output p(t) and the prices for their inputs, w(t) and r(t); as given.
We normalize5 the price of output to p(t) = 1: A …rm then solves the problem
to maximize their pro…ts, which are given by the di¤erence between the sales
of their output and the payments for their inputs. They do so by choosing how
many workers L(t) to hire and how much capital K(t) to rent.
max K(t)® (A(t)L(t))1¡® ¡ w(t)L(t) ¡ r(t)K(t)
K(t);L(t)
Remember from calculus that we maximize a function by taking …rst order
conditions and setting them to 0: Taking …rst order conditions with respect to
K(t) yields
®K(t)®¡1 (A(t)L(t))1¡® = r(t)
or
®
µ
K(t)
A(t)L(t)
~ =
But remember our variable k(t)
¶®¡1
K(t)
A(t)L(t) :
= r(t)
Hence
~ ®¡1
r(t) = ®k(t)
Along the balanced growth path k~ is constant, and it follows that the rental
price for capital, r(t) is constant along the balanced growth path in the Solow
model. From this it follows that the real interest rate is constant along the
balanced growth path. What about the capital share? Total income in this
1¡®
: Per unit of capital, the amount r(t)
economy is Y (t) = K(t)® (A(t)L(t))
is earned as capital income (rent). Hence total capital income equals r(t)K(t)
and the capital share (the fraction of income that goes to capital) equals
r(t)K(t)
Y (t)
=
®K(t)®¡1 (A(t)L(t))1¡® K(t)
K(t)® (A(t)L(t))1¡®
®K(t)® (A(t)L(t))1¡®
=
K(t)® (A(t)L(t))1¡®
= ®
5 As in microeconomics, as long as there is no money in the economy we can pick one good
to be the numeraire and normalize its price to 1: In the presence of money, money is usually
taken to be the numeraire, and the degree of freedom to normalize another price to 1 is gone.
3.3. THE SOLOW MODEL
67
Hence the capital share in the Solow model equals ® (and therefore the labor
share equals 1 ¡ ®): So ® is not only a technical parameter in the production
function, but turns out to be equal to the capital share. Note that this is true
not only along a balanced growth path, but is true at all times in the Solow
model.6 This motivates economists to pick values for ® of around 13 in numerical
exercises. Finally it is easy to show that along the balanced growth path wages
also grow at rate g; the rate of technological progress (by looking at the …rst
order condition of the …rm with respect to L(t)).
We conclude that the extended Solow model is successful in explaining all
four stylized growth facts of Kaldor. What about the development facts from
the Summers-Heston data set?
1. Enormous di¤erences in income levels across countries: the Solow model
can explain di¤erences in levels by pointing to di¤erences in population
growth rates and di¤erences in saving (investment) rates. But can it
explain the magnitude of these di¤erences? We will come back to this
point, but the verdict here will be negative
2. Enormous variation in growth rates per worker. There are two answers
the Solow model can give. According to the model, two countries can permanently grow at di¤erent rates only if they have di¤erences in the rate
of technological progress g (otherwise they eventually grow at the same
rate). Given that technology, at least most of it, is based on knowledge
that freely moves across countries this would be a rather unsatisfactory
answer. The Solow model can do better than this, by appealing to transition dynamics. Remember that along the transition to a balanced growth
path the growth rate changes over time. So di¤erences in growth rates
across countries could be due to the fact that some countries are closer to
the balanced growth path than others (but eventually they will all grow
at the same rate). Germany and Japan, for example, lost most of their
capital stock during World War II. So by starting far below the balanced
growth path these countries are predicted by the model to grow faster
than countries that did not have their productive capacity destroyed during WWII.
3. Transition dynamics can also explain why the growth rate of a country is
not constant over time.
4. Changes in the relative position of a country can be explained by the Solow
model by appealing to the same features as in point 1: Countries whose
population growth rate declines or saving rates move up, relative to other
countries, should move up in the international income distribution.
6 It is a consequence of the assumption of price taking behavior and a constant returns to
scale production function of Cobb-Douglas type. Also remember from your micro class that
with constant returns to scale all …rms earn zero pro…ts and the number of …rms operating
is undetermined -and we might as well assume that there is a single …rm producing all the
output.
68
CHAPTER 3. ECONOMIC GROWTH
So in principle the Solow model can capture most of the stylized facts that
we set out in the beginning, at least qualitatively. It does so, however, by
appealing to technological progress that is left unexplained in the model. After
taking some further looks at the data we will pick this point up again.
3.4
The Convergence Discussion
We have seen that one of the most puzzling, and probably the most troublesome
facts coming from the Summers-Heston cross country data set is the enormous
disparity in incomes per worker across countries. Development economists (and
not only those) naturally ask the question of whether these di¤erences are permanent or whether we should expect that eventually the poor countries catch
up to the rich countries, a phenomenon that economists term “convergence”.
Among others, economic historians Aleksander Gerschenkron (1952) and Moses
Abramovitz (1986) have advanced the hypothesis that poor countries should
grow faster, under the appropriate assumptions, than rich countries. We term
this hypothesis the “convergence hypothesis”.
Note that the question of convergence is intimately related to the observation
of variation of growth rates across countries: a country can only catch up to
another (group of) country if it grows at a faster pace. So convergence requires
poorer countries to grow faster than richer countries. In this section we ask two
questions: a) do we see convergence in the data b) what does the Solow model
have to say about convergence.
The main prediction of the convergence hypothesis is that poor countries
grow faster than rich countries. We can test the convergence hypothesis by
looking at whether this prediction is born out in the data. This is typically
done by looking at a plot of the following sort: on the x-axis we have a variable
that indicates how rich a country initially is, typically the level of GDP per
worker or GDP per capita for the …rst year for which we have data. On the
y-axis we have the growth rate of a country from the initial to the …nal period.
Plotting lots of di¤erent countries we would expect a negative correlation between the initial level of GDP per worker and the growth rate if the convergence
hypothesis is true: rich countries grow slower than poor countries, according to
the convergence hypothesis. Let look at such plots.
In Figure 23 we use data for a long time horizon for 16 now industrialized
countries. Clearly the level of GDP per capita in 1885 is negatively correlated
with the growth rate of GDP per capita over the last 100 years across countries.
So this …gure lends support to the convergence hypothesis. We get the same
qualitative picture when we use more recent data for 22 industrialized countries:
the level of GDP per worker in 1960 is negatively correlated with the growth
rate between 1960 and 1990 across this group of countries, as Figure 24 shows.
This result, however, may be due to the way we selected countries: the very
fact that these countries are now industrialized countries means that they must
3.4. THE CONVERGENCE DISCUSSION
69
Growth Rate of Per Capita GDP, 1885-1994
Growth Rate Versus Initial Per Capita GDP
3
JPN
2.5
NOR
FIN
ITL
2
CAN
DNK
GER
SWE AUT
FRA
USA
BEL
1.5
NLD
GBR
AUS
NZL
1
0
1000
2000
3000
4000
5000
Per Capita GDP, 1885
Figure 3.17:
have caught up with the leading country (otherwise they wouldn’t be called
industrialized countries).
When we do the same plot for the whole sample of 104 countries (not just
industrialized countries) Figure 25 doesn’t seem to support the convergence
hypothesis: for the whole sample initial levels of GDP per worker are pretty
much uncorrelated with consequent growth rates. In particular, it doesn’t seem
to be the case that most of the very poor countries, in particular in Africa, are
catching up with the rest of the world, at least not until 1990 (or until 2000 for
that matter).
What does the Solow growth model have to say about convergence. Let us
distinguish two situations
CHAPTER 3. ECONOMIC GROWTH
70
Growth Rate of Per Capita GDP, 1960-1990
Growth Rate Versus Initial Per Capita GDP
JPN
5
POR
4
GRC
ESP
3
ITL
IRL
TUR
AUT
2
FRA
BEL
FIN
NOR
GER
GBR
DNK
NLD
SWE
CAN
CHE
AUS
USA
1
NZL
0
0
0.5
1
1.5
2
2.5
Per Worker GDP, 1960
Figure 3.18:
1. Suppose all countries have the same savings rates s; same population
growth rates n and the same growth rate of technological progress (because
there is free transfer of knowledge across borders). That is, all countries
have the same balanced growth path. Then the Solow model predicts
two things: a) eventually all countries reach the balanced growth path,
all countries will have the same growth rate and the same level of per
worker GDP b) countries that start with capital per worker further below
the balanced growth path (i.e. are initially poorer) grow faster along the
transition path than do countries that are initially richer. Remember Figures 16 and 17. So the Solow model predicts convergence among countries
with similar saving rates, depreciation rates and population growth rates,
convergence to the same balanced growth path. Such convergence is also
called absolute convergence, because eventually these countries will have
the same level of income per capita. Figures 23 and 24 show convergence
4
x 10
3.4. THE CONVERGENCE DISCUSSION
71
Growth Rate of Per Capita GDP, 1960-1990
Growth Rate Versus Initial Per Capita GDP
6
4
2
0
-2
KOR
HKG
OAN SGP
JPN
SYC CYP
LSO THA PRT
GRC
ESP
MYS
ITA
IDN
JOR SYR
TUR
IRL
EGY
ISRAUTFINFRABEL
YUG
ECU
CHN
BOL PRY
BRA
GER
LUX
NAM
CAN
GIN
COL
DZA
NOR NLD
CMR
TUNGAB
ISL
PNG MUS MEX
GBR
BGDCSK PAN
CHE
DNK
ZAF FJI
AUS
DOM
HND
LKA
SWE
NGA
GTM CIV
CRI
PHL COM
GNB
SLV
CHL
MAR
URY
COGJAM
IND CIV
NZL
SEN
CAF
ZMBZWE
IRN
PER
TTO
KEN
BEN
GMB
TGO
GHA
MOZ TCD
VEN
RWANIC
MLI
MRT
UGA
CAF
MDG
MLI
MWI
BFA
BDI LSO
BFA
GUY
MOZ
PAK
USA
-4
0
0.5
1
1.5
2
2.5
Per Worker GDP, 1960
Figure 3.19:
among industrialized countries. To the extent that the industrialized countries in Figures 23 and 24 have similar characteristics (similar s; n; ±; ®; g)
this is exactly what the Solow model would predict.
2. So does Figure 25 constitute the big failure of the Solow model? After
all, for the big sample of countries it didn’t seem to be the case that poor
countries grow faster than rich countries. But isn’t that what the Solow
model predicts? Not exactly: the Solow model predicts that countries
that are further away from their balanced growth path grow faster than
countries that are closer to their balanced growth path (always assuming
that the rate of technological progress is the same across countries). This
is called conditional convergence. The “conditional” means that we have
to look at the individual countries’ steady states to determine how fast a
country should grow. So the fact that poor African countries grow slowly
4
x 10
72
CHAPTER 3. ECONOMIC GROWTH
even though they are poor may be, according to the conditional convergence hypothesis, due to the fact that they have a low balanced growth
path and are already close to it, whereas some richer countries grow fast
since they have a high balanced growth path and are still far from reaching
it. To test the conditional convergence hypothesis economists basically do
the following: they compute the steady state output per worker7 that a
country should possess in a given initial period, say 1960, given n; s; ±; ®
measured for this country’s data. Then they measure the actual GDP per
worker in this period and build the di¤erence. This di¤erence indicates
how far away this particular country is away from its balanced growth
path. This variable, the di¤erence between hypothetical steady state and
actual GDP per worker is then plotted against the growth rate of GDP
per worker. If the hypothesis of conditional convergence were true, these
two variables should be negatively correlated across countries: countries
that are further away from their balanced growth path should grow faster.
Jones’ Figure 3.8 provides such a plot. In contrast to Figure 25 (or his
Figure 3.6) we see that, once we condition on country-speci…c balanced
growth paths, poor (relative to their BGP) countries tend to grow faster
than rich countries. So again, the Solow model is quite successful.
A few words of caution about the success of the Solow model. Most of the
arguments presented in this section rely on transition dynamics: countries are
not in the balanced growth path and hence can grow at rates di¤erent from g;
the rate of technological progress. There are obvious and frequent reasons why
countries may be thrown out of their balanced growth paths: wars, famines,
political instability, you name it. The Solow model is obviously silent about
why these events come about. Also, the model doesn’t answer the important
question of what it is about special countries that makes them have low saving
rates, low population growth rates and hence lower balanced growth paths. It
also does not speak to the question where technological progress, the source of
growth in the model, comes from. Finally, so far it only provides qualitatively
the right answers. But if we take reasonable numbers for s; n; ±; ® in di¤erent
countries, does the model provide reasonable numbers for the size of dispersion
in per worker output across the world. In other words: are the s; n; ±; ® in the
data really so much di¤erent for the US and Ethiopia as to give rise to 40 times
higher output per worker in the US as in Ethiopia?8
7 Which is proportional to the balanced growth path output per worker (just multiply it
by the constant A(1960)):
8 The answer to this question is highly disputed, but I doubt it. For those interested I have
further references on this issue.
3.5. GROWTH ACCOUNTING AND THE PRODUCTIVITY SLOWDOWN73
3.5
Growth Accounting and the Productivity Slowdown
The aggregate production function posits that the output Y (t) of an economy
is produced by the two factors of production: capital K(t); labor L(t); in combination with the available technology A(t): We can follows Solow (1957) and
perform some simple accounting to break down the growth rate of output into
the growth rate of capital input, the growth rate of labor input and the growth
rate of technological progress.
We rewrite the aggregate production function as
Y (t) = B(t)K(t)® L(t)1¡®
The factor B(t) captures the level of technology and equals A(t)1¡® from before.
B(t) is called total factor productivity, and a production function in which
technological progress enters the way as shown is said to have Hicks-neutral
technological progress.9 Doing our usual trick of …rst taking logs with respect
to time and then di¤erentiating with respect to time we get
gY (t) = gB (t) + ®gK (t) + (1 ¡ ®)gL (t)
or, if we work with A(t) instead of B(t)
gY (t) = (1 ¡ ®)gA (t) + ®gK (t) + (1 ¡ ®)gL (t)
The growth rate of B(t); gB (t) is called total factor productivity (TFP) growth
or multifactor productivity growth. We can use these formulas to perform our
basic accounting exercise for a particular country: …rst we have to take a stand
on what ® is. Since it turns out to be the capital share, an ® = 13 is quite
popular among economists. Next we measure the growth rate of real GDP, gY
the growth rate of the aggregate capital stock gK and the growth rate of labor
input gL from the data.10 We then use the formula above to compute gB as the
residual
gB (t) = gY (t) ¡ ®gK (t) ¡ (1 ¡ ®)gL (t)
Computed this way, gB is also called the Solow residual, it is that part of output
growth that cannot be explained by the growth in inputs capital and labor.11
9 There
are several reasons of why we make the change from A(t); multiplying labor, to
B(t); multiplying K(t)® L(t)1¡® : First, the growth rate of B is a widely used productivity
measure by economists. Second, Solow did it this way (which shows that economists cannot
be consistent with their notation). Third, in the Cobb-Douglas case both ways are equivalent,
but for more general production functions this is not true anymore.
10 Labor input is usually measured by the total number of manhours worked in the economy
in a given period. This is a more precise measure of labor input than the number of workers
as the number of hours a worker works per year may change over time.
11 In some sense it measures our ignorance in explaining growth. In the light of our previous
discussion, the Solow residual may (should!) measure technological progress.
CHAPTER 3. ECONOMIC GROWTH
74
Before actually carrying out the accounting exercise one word of caution
is in order. We will only measure TFP growth correctly if we measure the
growth in output and in labor and capital inputs correctly. Measuring gY and
gL is relatively straightforward, but measuring the growth rate of the capital
stock may be tricky. An example: suppose the capital stock of an economy
consists only of 10 486-processor computers and now the economy invests in a
new Pentium 2, which is double as fast as the 486’ers. Did the capital stock go
up by 10% (as the number of computers went up by 10%) or did it go up by 20%
(as the computing power went up by 20%)? In practice a lot of assumptions
and simpli…cations are needed when measuring the growth rate of the capital
stock and this variable is probably one of the most poorly measured economic
variables. The consequences of this problem for measuring TFP growth are
enormous: suppose we measure gK as 3% but it was in fact 6%: Then we
attribute ® ¤ (6% ¡ 3%) = 1% of output growth to productivity growth when
it was in fact due to growth in capital input. Computing productivity as a
residual leads to mismeasurement of productivity whenever inputs or output
are not measured correctly.
But now let’s go ahead and perform the accounting exercise for US data
from 1960 to 1990. In Table 7 we report averages of growth rates for output
and factor inputs for several subperiods. We assume that ® = 13 : In parenthesis
is the percent that capital, labor and TFP growth contribute to GDP growth
Table 7
Period
1960 ¡ 90
1960 ¡ 70
1970 ¡ 80
1980 ¡ 90
GDP gY
3:1
4:0
2:7
2:6
Capital ®gK
0:9 (28%)
0:8 (20%)
0:9 (35%)
0:9 (34%)
Labor (1 ¡ ®)gL
1:2 (38%)
1:2 (30%)
1:5 (56%)
0:7 (26%)
TFP (gB )
1:1 (34%)
1:9 (50%)
0:2 (8%)
1:1 (41%)
GDP p. worker gy
1:4
2:2
0:4
1:5
We see that real GDP grew strongest in the 60’s, at 4% a year, and at
about 2 12 % since then. Approximately 1 percentage point of this growth is due
to accumulation of physical capital. Between 0:7 and 1:5 percentage points
is due to growth of labor input. We see the dramatic decline of total factor
productivity in the 70’s: from 1:9% in the 60’s to just about 0: This productivity
slowdown is one of the most studied and least understood phenomena of recent
economic history; it is an international phenomenon in that a lot of countries
experienced a productivity slowdown at approximately the same time. The 80’s
showed somewhat of a recovery of TFP growth to 1:1%; and the latest numbers
indicate that for the last four years TFP growth was again at the speed of the
60’s.
Remember that GDP per worker is de…ned as y = YL : Sometimes this variable
is also referred to as labor productivity, as the ratio of output to labor input.
3.5. GROWTH ACCOUNTING AND THE PRODUCTIVITY SLOWDOWN75
We immediately have that gy = gY ¡ gL ; hence from the formulas above
gy
gy
= gB + ®gk
= (1 ¡ ®)gA + ®gk
and we see the direct impact of TFP growth on per worker income growth (or
labor productivity). As predicted by this equation, the productivity slowdown
of the 70’s led to a sharp decline of income per worker in that period, with the
growth rate of per worker income recovering in the 80’s (and even more so in
the late 90’s).
What are possible reasons for the productivity slowdown? As mentioned it
is still somewhat of a puzzle. Here are some explanations
1. Sharp increases in the price of oil which made companies use inferior technology that didn’t require oil. Problem: oil prices (adjusted for in‡ation)
are lower in the late 80’s than in the 60’s.
2. Structural changes: as the economy produced more and more services and
less and less manufacturing goods the high productivity sectors (manufacturing) become less important than the low productivity sectors (services).
3. Slowdown in resources spent on R&D in the late 60’s.
4. TFP was abnormally high in the 50’s and 60’s since all the new technologies developed for the war became available for private business sector use.
So the 70’s and 80’s are the “normal” situation.
5. Information technology (IT) revolution in the 70’s. Computers swept into
business o¢ces and for the last 10 to 15 years a lot of time was spent
learning how to use them (instead of producing output). Hence the productivity slowdown. Once the new technology is …gured out, TFP should
boom again.
Probably the truth is that all these factors contributed to the slowdown,
although I personally …nd the last explanation very intriguing, in particular
given that TFP has been extraordinarily high in the last …ve years, possibly
showing the e¤ects of investment in IT started in the 70’s and 80’s.
We can use the same framework for the analysis of the growth process in
other countries. In particular, what determinants are responsible for the growth
miracles in East Asia, the Singapores, Japans, Koreas, Hong Kongs and Taiwans? There exists a somewhat heated discussion about this issue, with one
group of economists attributing most of the fantastically high growth rates from
the 60’s to the mid 90’s to TFP growth, whereas others attribute most of it to
the fast accumulation of physical (and human) capital. In Table 8 we show
results from growth accounting for the Asian growth miracles, and, as a comparison, data for some industrialized and some Latin American countries. The
calculations are done with country-speci…c ®’s, where the ® for a particular
country is matched to that country’s average capital share during the relevant
time period.
CHAPTER 3. ECONOMIC GROWTH
76
Table 8
Country
Germany
Italy
UK
Argentina
Brazil
Chile
Mexico
Japan
Hong Kong
Singapore
South Korea
Taiwan
Time Per.
1960 - 90
1960 - 90
1960 - 90
1940 - 80
1940 - 80
1940 - 80
1940 - 80
1960 - 90
1966 - 90
1966 - 90
1966 - 90
1966 - 90
GDP gY
3:2
4:1
2:5
3:6
6:4
3:8
6:3
6:8
7:3
8:5
10:3
9:1
Cap. Sh. ®
0:4
0:38
0:39
0:54
0:45
0:52
0:63
0:42
0:37
0:53
0:32
0:29
Cap. ®gK
1:9(59%)
2:0(49%)
1:3(52%)
1:6(43%)
3:3(51%)
1:3(34%)
2:6(41%)
3:9(57%)
3:1(42%)
6:2(73%)
4:8(46%)
3:7(40%)
Labor (1 ¡ ®)gL
¡0:3(¡8%)
0:1(3%)
¡0:1(¡4%)
1:0(26%)
1:3(20%)
1:0(26%)
1:5(23%)
1:0(14%)
2:0(28%)
2:7(31%)
4:4(42%)
3:6(40%)
Although there is always the issue of mismeasurement (which is very important in these exercises) it does not appear to be the case that the bulk of East
Asia’s growth miracle is due to particularly strong TFP growth. Fast capital
accumulation (a high growth rate of the capital stock) seems to be at least as
important.
3.6
Ideas as Engine of Growth
We saw that the Solow model was very successful in explaining Kaldor’s growth
facts and fairly successful in explaining the stylized development facts that we
found from the Summers-Heston cross country data set. However, I stressed several time that the source of growth in the Solow model is technological progress
and that this technological progress is an assumption of the model. Why there
_
A(t)
= gA > 0) is not explained within the
is positive technological progress (i.e. A(t)
model. In this section we will informally discuss the main ingredients of growth
models that eliminate this shortcoming by explicitly explaining why technology
grows at a constant positive rate. In the second part of this section we will
brie‡y describe how we can measure technological progress directly from the
data.12
3.6.1
Technology
Let us …rst de…ne precisely what we mean by technology. Technology is the way
inputs to the production process (in our case labor and capital) are transformed
12 Note that we tried to measure technological progress in the last section. There technological progress or TFP was not measured directly, it was de…ned as the residual of output
growth and growth of inputs labor and capital.
TFP (gB )
1:6(49%)
2:0(48%)
1:3(52%)
1:1(31%)
1:9(29%)
1:5(40%)
2:3(36%)
0:2(29%)
2:2(30%)
¡0:4(¡4%)
1:2(12%)
1:8(20%)
3.6. IDEAS AS ENGINE OF GROWTH
77
into output. In our example without technological progress we had
Y (t) = K(t)® L(t)1¡®
In this case technology is completely described by the parameter ® (and the fact
that capital and labor input enter multiplicatively in the production function).
Not that for this case technology does not change over time. The amount of
inputs K(t) and L(t) may vary over time, and hence the amount of output
produced may change over time, but given inputs the way output is produced
does not change over time (one easy way to see this is to realize that the only
place t enters in the production function is in K(t) and L(t)).
The situation is di¤erent when the production function is given by
1¡®
Y (t) = K(t)® (A(t)L(t))
A(t) is an index of technology that the economy has access to in period t: If
A(t) changes over time, then technology changes over time. Suppose in period
T A(T ) is twice as big as A(t) in period t: Then, even if the economy uses the
same amount of labor and capital in both periods, in period T the economy
produces 21¡® times the output in period t. An easy way to see that in this
case technology is not constant is to realize that t enters not only in K(t)
and L(t); but also in A(t): Increases in the technology index A(t) are called
technological progress. When new ideas are created, new knowledge is added
to the existing stock of knowledge and more output can be produced with a
given amount of labor and capital. New ideas can come in the form of new
procedures to put more and more tansistors onto a computer chip of given size
(Moore’s Law states that the number of transistors that can be packed onto
a given chip doubles roughly every 18 months), the development of new drugs
against diseases, a new strategy to run chain stores etc. The important insight
of economists that worked in the area of growth and ideas was not so much that
new ideas can induce economic growth, but rather that ideas do not usually
simply fall from heaven, but are the result of costly e¤ort to discover new ideas.
Firms, governments and individuals spent time and money on activities that are
designed to generate new ideas that then bene…t economic growth. Our next
task is to investigate why (and under what circumstances) resources are spent
on the development of new ideas.
3.6.2
Ideas
A key feature of ideas are that they are nonrivalrous goods. If one person
uses calculus, another person is not precluded from using exactly the same
idea. This makes ideas very di¤erent from most goods. If I consume a pizza,
you cannot consume the same pizza. Pizza, as most consumption goods are
rivalrous, but ideas are not. A nonrivalrous good is a good whose use by one
person does not preclude the use of this same good by another person. An
important consequence of this fact is that usually nonrivalrous goods only have
to be produced once: once an idea has been developed it is there for use, and
78
CHAPTER 3. ECONOMIC GROWTH
it needs not be discovered again. This fact will turn out to have important
consequences.
Another key feature of ideas are they are, at least partially, excludable. A
good that is excludable is a good for which the owner of the good can charge
another person a fee for the use of it. A good can very well be excludable but
nonrivalrous: think of computer software. The fact that I use Windows NT
does not preclude you from using it, but for sure Microsoft tries to make sure
that they can charge a fee for the use of Windows NT. The legal system of most
countries has provisions that make sure that developers of new ideas have the
right to charge users of these ideas a price by providing copyright and patent
laws.
Dividing goods along the two dimensions of nonrivalry and nonexcludability
we can distinguish four groups of goods
1. Rivalrous goods that are excludable: almost all private consumption goods,
such as food, apparel and consumer durables fall into this group.
2. Rivalrous goods that have a low degree of excludability: an example is the
…sh in international waters. When the …sh is caught by American …shermen, Japanese …shermen are precluded from catching and selling them.
Hence these …sh are rivalrous goods. But American …sherman have no
possibility to exclude Japanese …shermen from …shing in international waters. Rivalrous, nonexcludable goods often su¤er from the tragedy of the
commons. The classic textbook example stems from middle age England.
English towns had plots of land, called commons, where all peasants of the
town were allowed to graze their cattle free of charge. Since no farmer was
excluded, but there was only a …xed amount of grass available, the grass
in the commons falls under this category. What happened was that, since
grazing an additional cow yielded bene…ts for a farmer and the cost was
shared among all farmers (less grass available), the commons were completely overgrazing and became useless. A similar development threatens
to happen with the stock of …sh in international waters. To avoid the
tragedy of the commons usually government intervention or private agreements to avoid overgrazing or -…shing are needed.
3. Nonrivalrous goods that are excludable: examples include the computer
code for software programs or blueprints for the production of machines,
cameras, lasers etc. Most of what we call ideas in this section would fall
under this point.
4. Nonrivalrous and nonexcludable goods: these goods are often called public
goods because they are mostly produced, or at least provided, by the
government. The prime example is national defense: the fact that the U.S.
government protects you from an aggression of some other country does
not preclude me from being protected; also usually nobody is excluded
from this good national defense (the times of outlaws are gone). Some
ideas fall under this point, too. Basic scienti…c research is such an example.
3.6. IDEAS AS ENGINE OF GROWTH
79
It is obviously nonrivalrous and I can hardly exclude you from learning
about the Solow model (even if I tried very hard so far). As we will see,
the fact that a lot of basic research is done in public or publicly funded
institutions is no accident, but follows from basic economic principles.
The distinction into excludable/nonexcludable and rivalrous/nonrivalrous
goods is not only academic. It has a huge impact on the economics of ideas.
Consider nonrivalrousness …rst. Since an idea is a nonrivalrous good, it can be
used by many people without precluding other people from using it. This just
means that the cost of providing the good to one more consumer, the marginal
cost of this good, is constant at zero (or at least very low, if the idea has to be
put on some physical object, like a ‡oppy disk). But developing the idea at …rst
may involve substantial resources, i.e. high start-up or …xed costs. Hence the
production process for ideas is usually characterized by substantial …xed costs
and low marginal costs.13
Now comes in the issue of excludability. Suppose a …rm can’t exclude another
…rm from adapting and also selling the idea (or the good based on the idea).
Competition would then drive down the price of the good to marginal cost
(remember your micro class). But because of the original …xed cost the …rm
that invented the idea would lose money by developing and then selling the idea
at marginal cost. So would any …rm ever develop an idea if it can’t exclude it
from competitors? Most likely not. Hence for the development of new ideas
by private companies it is crucial that ideas are excludable. Therefore the
existence of intellectual property rights like patent or copyright laws are crucial
for the private development of new ideas, and hence for the engine of growth.
It is also not surprising that ideas that can’t be made excludable by these laws
(or for which society decides that these ideas are so desirable that everybody
should have unlimited access to them) are usually developed by publicly …nanced
institutions.
In fact, some economic historians have made the point that this force is
so strong that it explains part of the industrial revolution. Remember that
sustained economic growth is a very recent phenomenon. Before the middle
of the 18-th century, economic growth was an unknown phenomenon. Then,
so the hypothesis of economic historian and Nobel price winner Oliver North,
institutions developed that protected intellectual property rights. Only after
this had happened could private …rms and individuals be sure that their investment into developing new ideas would be rewarded by warranting patents that
then could be sold for fees covering the initial …xed cost of development. The
number of new ideas developed increased, sustained technological progress occurred and the world, for the …rst time, experienced sustained economic growth.
The initial period of economic growth in the late 18-th and early 19-th century
is called the industrial revolution; its timing coincides pretty closely with the
drafting of the US constitution, the French revolution and following Declaration
13 This cost structure for the production of ideas is closely linked to the fact that the production process for ideas is usually characterized by increasing returns to scale. See Jones for
details.
CHAPTER 3. ECONOMIC GROWTH
80
of the Rights of Man and of the Citizen, and the publishing of the …rst book on
economics stressing private property rights, self-interest and private markets,
Adam Smith’s “An Inquiry into the Nature and The Causes of the Wealth of
Nations”.
3.6.3
Data on Ideas
How can we measure technological progress directly, i.e. not just as Solow residual in our accounting exercises? To the extent that we attribute technological
progress to the evolution of new ideas, this translates into the question of how we
can measure the amount of new ideas being produced. There are two ways: we
can try to count the number of new ideas directly or (since this may be easier)
we can try to measure the amount of resources that are spent in producing new
ideas. If more resources mean more ideas, this gives us indirect evidence about
the number of new ideas that should have been produced during a particular
time period.
How can we measure the number of ideas? One close proxy may be the
number of patents issued. Jones provides data for the number of patents the
have been issued in the US, from 1900 to 1991. The data show the following
general features:
² the number of patents issued has increased substantially: in 1900 roughly
25,000 patents were issued in the US, in 1990 the number was 96,000
² more and more patents issued in the US are issued to foreign individuals
or foreign …rms. The number of patents issued to US …rms or individuals
has been roughly constant at 40,000 per year between 1915 and 1991.
Obviously these data don’t tell us anything about the importance of each
patent. The patent for the light bulb is supposedly hundred times more important than the patent for the self-rotating hamster cage. The data do not re‡ect
this di¤erence of importance and therefore obviously give only a limited account
of how the level of ideas has evolved over time. Ignoring this caveat it seems
that the level of technology, as measured by the stock of ideas, has increased
rapidly in the US over the last century.
Jones also provides data on resources devoted to the development of new
ideas. His Figure 4.6 shows how the number of researchers engaged in research
and development (R&D) evolved in the US and in other industrialized countries
over the last 40 years. Not only did the absolute level increase substantially
(from around 200,000 to about 1,000,000 between 1950 and 1990 for the US),
but also the fraction of the labor force involved in R&D increased from about
0.25% in 1950 to about 0.75% in 1990. This also indicates, to the extent that
more researchers develop more ideas, that the number of ideas and hence the
level of technology has increased rapidly over the last 40 years or so.
So what have we accomplished in this section? We …rst de…ned what exactly
we mean by technology. We then associated improvements in technology with
the discovery of new ideas. We then argued that by its very nature ideas are
3.7. INFRASTRUCTURE
81
nonrivalrous goods and discussed what this implies for the cost structure of
producing goods based on new ideas. We then argued why it is important for
the development of new ideas that ideas are, or are made, excludable goods
and …nally we presented some data showing that indeed the number of ideas
has rapidly increased over the time horizon we have reliable data for. By doing
all this we have provided an explanation for sustained technological progress
that was the underlying force of economic growth in the Solow model. Our
discussion was purely verbal in nature. In the mid 80’s and early 90’s models
have been developed that formalized our reasoning, in particular by Stanford’s
Paul Romer (1990). Jones’ Chapters 5 and 6 discuss these models in detail and
the interested reader is invited to consult the book.
3.7
Infrastructure
In the last section we looked at how we can justify one assumption of the Solow
model, namely that the level of technology grows over time. Now we want to
look at another assumption, namely the assumption that each country saves
and invests a certain fraction of output (and consumes the rest). Why is this
important? Remember that the Solow model explains di¤erences in income
levels across countries by di¤erences in saving or investment rates. The question
is then: why do some countries save and invest such a high fraction of income,
whereas other countries don’t. Our answer will roughly be that some countries
have political institutions that make investing more pro…table than others. For
the purpose of this section we will interpret s as the investment rate rather than
the saving rate (in the Solow model both are equivalent). We will also interpret
the capital stock as including not only physical capital, but also human capital,
the skill and education that the labor force has acquired. So by investment we
will mean investment in physical capital (building new factories and the like) as
well as investment in human capital such as schooling.14
Each investment has costs and bene…ts: a …rm that contemplates building
a new factory weighs the cost of construction against the bene…ts of being able
to produce and sell products with the new factory; in your decision to invest in
your Stanford education you weigh the cost (tuition, opportunity cost of your
time) against the bene…ts (better pay and more interesting work in the future).
The reason why some countries invest more than others is then due to the fact
that either the costs of investment are lower or the bene…ts are higher (or both)
in these countries. So let’s have a look at the determinants of costs and bene…ts
of investment.
14 It is quite easy to introduce human capital into the standard Solow model, and Chapter
3 of Jones does exactly that. I skipped it because I think it does not add much to the basic
insights that can be gained from the basic Solow model.
CHAPTER 3. ECONOMIC GROWTH
82
3.7.1
Cost of Investment
The cost of investment may not only involve the resources to come up with a
new business idea and the purchase of buildings and equipment, but also the
cost of obtaining all legal permissions. That this may involve signi…cant costs
(in particular time spent) is demonstrated in the famous book by Hernando de
Soto “The Other Path” (1989). De Soto started a small business in Lima, with
the purpose of measuring the cost of setting up a small business, in particular those costs due to bureaucracy and compliance with regulations. He was
confronted with several o¢cial requirements such as obtaining a zoning certi…cate and registering with tax authorities. Meeting these requirements took an
equivalent of 289 working days and required two bribes. Overall, only the cost
of meeting these o¢cial requirements amounted to about 32 monthly minimum
wages, i.e. for the same money one worker of this company could have been paid
for almost three years. Similar stories can be told for a lot of countries and they
may provide part of the explanation for why the investment share of output is
relatively low. They almost always involve a de…cient or corrupt bureaucracy
that impedes pro…table investment activities.
3.7.2
Bene…ts of Investment
What determines the pro…tability of an investment project, over and above its
costs and the inherent quality of its idea. We will follow Jones and single out
several factors
1. The size of the market. The larger the pool of potential buyers of a
products (or skill of a person), the larger are the potential bene…ts from
an investment. Suppose Netscape’s only market would be the Bay Area.
My educated guess is that its stock price wouldn’t be where it is now
(better: where it was six months ago). But with potential buyers of
Netscape all over the world, the bene…ts for the founders of Netscape are
potentially huge. The size of the potential market for a product does
depend crucially on political decisions within the country. Countries like
the US are very open to international trade, and since the US allows foreign
…rms to sell their products in the US, usually US …rms are allowed to sell
in foreign countries. A country that decides to remain relatively closed
to international trade restricts the market for their …rms to the domestic
market and therefore reduces potential bene…ts from setting up a new or
expanding an existing business.
2. The extent to which the bene…ts from the investment accrue to the investor. Suppose the investment project actually earns some money. The
extent to which institutions in the country guarantee that the pro…t remains with the owner is an important determinant of the decision to invest.
Reasons of why pro…ts are diverted from the owner range from high taxes
to theft, corruption, the need to bribe government o¢cials or the payment of protection fees to the Ma…a or Ma…a-like organizations. These
3.7. INFRASTRUCTURE
83
features not only tax the investment project, but also may lead to ine¢cient production just to avoid the diversion of pro…ts. It also may channel
investment into unproductive sources, such as protection against crime, so
as to avoid extortion of pro…ts. To what extent pro…ts are diverted from
private investors is largely determined by the government. Hence, roughly,
countries with policies and institutions that favor investment bene…ts being diverted from investors make investments less bene…cial and hence will
have a lower fraction of output being invested.
3. Rapid changes in the economic environment in which …rms and individuals
operate may increase uncertainty of investors. Who would invest in a
country for which there is a reasonable chance that tomorrow a new radical
government will take power that nationalizes all private …rms?
This list of potential determinants of the costs and bene…ts of investment
projects is probably not complete. The next task is to determine whether, in
the data, these determinants actually have an in‡uence on the share of output
that is invested in these countries. Jones provides some …gures that shed light
on this question (see his …gures 7.1 and 7.2).
In his …gure 7.1. he plots the share of output that is invested (on the y-axis)
against a variable that is a weighted average of two variables: one that measures
the degree of openness, the other that broadly measures to what extent the
government of a particular country tries to stop diversion of pro…ts from private
investment. From the …gure one can clearly see that countries that are more
open to international trade and stop diversion of private pro…ts more e¤ectively
have a higher investment share of output. Figure 7.2 does the same, but focuses
on investment in human rather than physical capital. The measure of human
capital accumulation (plotted on the y-axis) is the average years of schooling in
a particular country, the variable on the x-axis is the same as before. Again we
see that the more open and more e¤ective in stopping diversion a country is,
the more on average do its citizens invest in school education.
So again, what have we accomplished? We explained income di¤erences of
countries, following Solow, by di¤erences in investment rates. In this section
we have discussed why investment rates are higher in some countries than in
others. The basic answer was: some countries have institutions (bureaucracy,
policies and politicians) that favor investment to a greater extent. But now the
question arises: why do some countries have better institutions (better at least
for encouraging investment and therefore economic growth) than others? It is
not that economists are completely clueless about this question15 , but instead
of speculating at this point it’s time to punt and leave this to the political
scientists.
15 There is some exciting work done in the area of political economy, which tries to explain
economic policies and institutions as the outcome of explicit or implicit voting procedures.
CHAPTER 3. ECONOMIC GROWTH
84
3.8
Endogenous Growth Models
In this section I brie‡y want to expose you to a rather di¤erent class of growth
models. The Solow model (and all its extensions) are called exogenous growth
models, as the engine of growth, technological progress, is itself exogenous to
the model. The models that explain technological progress (I alluded to them
when talking about ideas, but didn’t expose you to the formal models) are
sometimes called endogenous growth models since they explain technological
progress within the model. Most of these models share one very important
feature with the Solow model: a change in the saving (investment) rate has
e¤ects on income levels, but not on growth rates. Therefore policies that increase
the saving rate have only level, but no growth rate e¤ects.
I will now present a simple model in which policies that a¤ect the saving rate
will a¤ect the growth rate of the economy. This model (and the class of models
based on it) were the …rst type of endogenous growth models. As we will see
they do not at all rely on technological progress to generate growth: in these
models sustained growth is possible even without any technological progress.
The model consists of two equations as before, a production function and a
capital accumulation equation. Already written in per worker terms, the capital
accumulation is identical to the Solow model
_
k(t)
= sy(t) ¡ (n + ±)k(t)
The only di¤erence is the production function, which takes the form
y(t) = Ak(t)
where A is a technology parameter that does not change over time (we have no
technological progress). The only (very crucial) di¤erence to the Solow model
is that k(t) doesn’t have an exponent ® (or you may set ® equal to 1 in the
Solow model). Because of the form of the production function these types of
models are also referred to as Ak-models. The important economic assumption
that di¤erentiates it from the Solow model deals with the marginal product of
capital: in the Solow model the e¤ect on output per worker from one unit more
of capital per worker was given by
®
dy(t)
=
dk(t)
k(t)1¡®
and now it is given by
dy(t)
=A
dk(t)
The key di¤erence is that in the Solow model the marginal product of capital
was decreasing when k(t) was increasing. The additional e¤ect of one unit more
capital gets smaller and smaller, and this is the reason for the economy to …nally
come to rest at the steady state. In this model the marginal product of capital
3.8. ENDOGENOUS GROWTH MODELS
85
is constant, independently of the level of k(t): This will cause the model to not
have a steady state!!
But let us proceed. We can substitute the production function into the
capital accumulation equation to obtain
_
k(t)
= sAk(t) ¡ (n + ±)k(t)
and now can draw a diagram similar to the Solow diagram. First we make the
assumption that sA > n + ±: We then plot the sAk(t) curve and the (n + ±)k(t):
Both are straight lines, one with slope sA the other with slope n + ±: Under our
assumption the …rst curve is steeper than the second curve. Figure 26 shows
both curves.
sAk(t)
(n+δ)k(t)
.
k(t)
k(0)
k(t)
Figure 3.20:
Suppose the initial capital stock per worker is k(0): At k(0), since the sAk(t)
_ is positive and the capital
curve lies above the (n+±)k(t) curve, we have that k(t)
86
CHAPTER 3. ECONOMIC GROWTH
stock is growing. The important fact in this model is that, under the assumption
that sA > n + ±; this is the case for all levels of the capital stock per worker
k(t); so the capital stock per worker continues to grow forever. And this is true
without may technological progress, it just comes from the fact that capital
has a high marginal product that does not decline over time as the economy
accumulates more and more.
The growth rate of capital per worker is given by
_
k(t)
= sA ¡ (n + ±)
k(t)
which also equals to the growth rate of output. Two important facts: the growth
rate of the economy is constant and positive (always, not only in a balanced
growth path) and the growth rate is increasing in the saving rate. In other
words, in this model a country with a higher saving rate has a permanently
higher growth rate, not only a higher income level!!! It also follows that all
policies that increase the saving rate increase the growth rate. Hence such a
policy has growth rate and not only level e¤ects.
Note that this model can provide an alternative explanation for di¤erences in
growth rates across countries than the Solow model: di¤erences in growth rates
are due to di¤erences in saving rates, according to this model. What do the data
have to say about this prediction? Figure 27 plots the average investment rate
between 1980 and 1990 against the average growth rate of real GDP between
1960 and 1990 for the di¤erent countries in the Summers-Heston data set.
We see that investment and growth rates tend to be positively correlated,
as predicted by the Ak-model, but only weakly so. I would interpret Figure
27 carefully, as weak, but not convincing support of the Ak model. Jones (in
his section 8.4) quotes other empirical results that question the Ak-model. In
particular, the rate of investment into human capital has increased rapidly in
the US over the last 100 years (as measured by the years of schooling the average
American received). The Ak model would predict a strong increase in the growth
rate of the economy over the last hundred years. The data, however, indicate
that growth rates at the turn of the century for the US have been as high as
they are now, contradicting the prediction of the model.16
3.9
Neutrality of Money
Before summarizing our main results from the study of the theory and data of
economic growth there is one important point to be made about all of growth
theory in the way we discussed it. All the variables we looked at, real GDP, real
GDP per worker, the capital stock, the real interest rates were real variables.
In the whole last chapter not once did we talk about money, nominal interest
rates, in‡ation and the like. This is due to what is called the classical dichotomy
16 Other studies have tried to investigate whether the form of the production function (constant returns to scale with respect to capital alone) can be backed up by data. The data seems
not to be supportive of this assumption.
3.9. NEUTRALITY OF MONEY
87
Growth Rates and Investment Rates
Average Growth Rate 1960-1990
0.06
0.04
0.02
0
-0.02
-0.04
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Average Investment Rate 1980-1990
Figure 3.21:
or the neutrality of money. It is fair to say that today most economists believe
that in the long run (the time horizon for growth theory) money has no e¤ect on
real variables. A doubling in the money supply by the Federal Reserve Bank in
the long run just leads to a doubling of the price level, but leaves real variables
una¤ected. It is this neutrality of money in the long run that allows us to do
growth theory and never talk about money, since growth theory is the theory
of output and output growth determination in the long run.
We will see that when we talk about economic ‡uctuations, or the determination of output in the short run, the classical dichotomy will not always hold.
In the short run money potentially matters in determining real output, and with
it monetary policy becomes an interesting topic to study.
0.45
CHAPTER 3. ECONOMIC GROWTH
88
3.10
Summary
Let us sum up what we learned in the last chapter. From Kaldor’s stylized
growth facts we learned that over the long run capital and output per worker
grow at roughly equal and constant positive rates. From the Summers-Heston
data set we saw that there are enormous di¤erences in per-worker income levels
across countries and that countries also vary widely with respect to growth rates
of per worker GDP.
We then constructed the Solow model with technological progress. In a
balanced growth path the Solow model reproduces Kaldor’s stylized growth
facts. In particular the sustained growth of per worker GDP is explained by
technological progress, which itself has the origin in the discovery of more and
more ideas as engine of growth.
The di¤erences in income levels can be explained within the Solow model
with di¤erences in saving (investment) rates, whose di¤erences in turn can be
explained by di¤erences in institutions across countries that a¤ect the pro…tability of investment projects. A question mark remains whether the Solow model,
although capable of explaining the direction of income di¤erences, can explain
the magnitude of income di¤erences across countries.
For di¤erences in growth rates the Solow model points to transition dynamics
and the principle of conditional convergence: countries that are far away from
their balanced growth path should grow faster than those close to their balanced
growth path. The data show some support for the conditional convergence
hypothesis.
Finally we looked at a model in which changes in saving rates have e¤ects
not only on income levels (as in the Solow model), but e¤ects on growth rates
of income. These Ak-type models were found to be somewhat de…cient when
confronting their predictions with the data.
Overall I think it is fair to say that the Solow model (and its extensions) has
been an great success in addressing most of the puzzles in the data on economic
growth and development. This may explain that there is substantial agreement
among economists about what to study and teach in the area of growth theory.
As we will see in a bit, the same cannot be said for the study of economic
‡uctuation, for business cycle theory.
Chapter 4
Business Cycle Fluctuations
The modern world regards business cycles much as the ancient
Egyptians regarded the over‡owing of the Nile. The phenomenon
recurs at intervals, it is of great importance to everyone, and natural
causes of it are not in sight. (John Bates Clark, 1898)
4.1
Potential GDP and Aggregate Demand
Remember the …gure that plots real GDP per capita over the last 30 years.
For your bene…t it is reproduced here as Figure 28. Real GDP (and also real
GDP per capita) on average grows at a positive rate. We constructed the
Solow growth model to explain this fact of sustained econonomic growth. In the
Solow model all factors of production (labor and capital) were fully employed
to produce output (real GDP) according to the aggregate production
1¡®
Yt = Kt® (At Lt )
I switched back to discrete time as for the rest of the course we will work in
discrete time. The amount of output that can be produced at time t according
to the Solow model is called potential GDP or trend GDP: it is the level
of real GDP that can be produced in the economy if all factors of production
are fully employed, and it corresponds to the line labeled “Trend” in Figure 28.
When it is necessary to distinguish potential GDP from actual GDP we use Ytp
as a symbol for potential GDP and Yt for actual GDP.
At this point a word of caution: when I say that all factors of production
are fully employed I do not mean that the unemployment rate is zero and …rms
operate at 100% capacity. People voluntarily quit jobs and it takes time until
they start a new job; this relocation process generates a positive unemployment
rate even when economist speak of a situation of full employment. Losely, for
our purposes full employment means that all factors of production are used as
in “normal” times. The unemployment rate in normal times is often referred to
89
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
90
as the Natural Rate of Unemployment (and we will come back to it when
we discuss unemployment).
So now we know what factors determine potential GDP or trend GDP (Y p ),
the amount of output that all …rms together can produce. In this part of the
course we want to explain why actual GDP (Y ), the amount of output that
actually is produced, can temporarily and cyclically deviate from potential GDP,
i.e. we want to explain why there can be business cycles and what causes them.
Or, to use Figure 28, we want to explain the ‡uctuations of the solid line around
the dotted line.
Real GDP in the United States 1967-2001
9.2
Log of real GDP
9
8.8
Trend
8.6
GDP
8.4
8.2
8
1970
1975
1980
1985
Year
1990
1995
Figure 4.1:
There are several competing business cycle theories. We will …rst explore a
theory that is based on the idea the prices are not fully ‡exible in the short run.1
1 This class of theories that relies on sticky prices or wages is usually referred to as Keynesian
or New Keynesian business cycle theory. We will later take a look at a business cycle theory
that is neoclassical in spirit, i.e. has fully ‡exible prices and wages, the so-called Real Business
2000
4.1. POTENTIAL GDP AND AGGREGATE DEMAND
91
The idea goes like this: The economy (i.e. all its …rms together) can supply total
output equal to potential output.2 The total demand for output, called Aggregate Demand is the sum of demands by all households, …rms, the government
and foreign countries for domestic output. If prices were completely ‡exible
in the short run, then they would adjust instantenously to equate aggregate
demand to potential output, just as in you’ve learned in microeconomics. The
key of Keynesian business cycle theory is the assumption that in the short run
prices are not ‡exible, they are …xed (or sticky). We assume that at these …xed
prices …rms are ready to supply whatever output is demanded. In other words,
prices are assumed to be sticky in the short run, but production is assumed to
be able to adjust very rapidly to aggregate demand. New Kenesian business
cycle theory works hard to provide explanations for why prices are sticky in
the short run; please refer to Hall and Taylor’s Chapter 15 for further (quite
interesting) details. To repeat: in the short run aggregate demand determines
realized GDP; realized GDP may be smaller, may equal or may be bigger than
potential GDP.
How reasonable is this assumption? For the US …rms on average work at a
level of capital utilitization of about 80%, i.e usually only 45 of all the available
machines are actually used (or used in as many shifts as they could). Firms usually are also able to adjust labor input to changing aggregate demand by hiring
new workers (although this may be di¢cult in a tight labor market like the one
we have now) or inducing workers to work overtime, etc. So the assumption that
…rms can adjust production instantenously to demand seems quite reasonable.
The assumption of sticky prices in the short run seems harder to defend, and
the interested reader is referred to Chapter 15 in Hall and Taylor.
Business cycle ‡uctuations then come about by ‡uctuations in aggregate
demand: recessions are periods in which aggregate demand falls below potential output at the …xed price level whereas booms are times in which aggregate
demand is above potential GDP at a …xed price level. The situation is exempli…ed in Figure 29. The aggregate demand curve is downward sloping since
at lower prices consumers and …rms demand more goods and services (we will
return to this point later). Potential output is independent of the price level; it
is determined purely by the availability of production factors in the long run.
Suppose the price level is …xed in the short run at P1 . At this price level aggregate demand is lower than potential output, part of the available inputs labor
and capital are left idle (more than in normal times) and the economy is in a
recession. In the long run prices are assumed to be ‡exible so that in the long
run the price level adjusts to P0 and realized GDP equals potential GDP.
So in order to develop a uni…ed macroeconomic theory of growth and business
cycle our task is to provide a theory of aggregate demand. Growth theory
provided the explanation for the growth of potential GDP, and since business
cycle ‡uctuations are explained as short-run deviations of aggregate demand
from potential GDP, with sticky prices being responsible for these deviations
Cycle theory (RBC-theory).
2 Some economists, e.g. Gregory Mankiw in his Macroeconomics textbook, refer to potential
output as (long-run) aggregate supply.
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
92
Price Level (P)
Potential Output
Gap between
Potential
Output and
Aggregate
Demand
P
1
P
0
Aggregate
Demand
Output (Y)
Figure 4.2:
being sustained, we need to explain aggregate demand. We will proceed in two
steps:
1. We will determine what aggregate demand is for a given …xed price level P
(i.e. we will determine single points on the aggregate demand curve). In
doing so we will develop the famous IS-LM-model, due to Sir John Hicks,
who formalized the ideas of John Maynard Keynes.
2. We will investigate how changes in the price level change aggregate demand (i.e. we will trace out the entire aggregate demand curve).
4.2. THE IS-LM FRAMEWORK
4.2
4.2.1
93
The IS-LM Framework
The Balance of Income and Spending: Keynesian
Cross and Multiplier
Now we take the price level in the economy as …xed. Therefore we don’t (as
of now) have to distinguish nominal from real GDP, but for concreteness let
Y be real GDP. Multiplying Y by the …xed price level P gives nominal GDP.
Remember that actual GDP is determined by aggregate demand in the short
run. We start our analysis of aggregate demand (which equals realized GDP)
by remembering that GDP equals total income and equals total spending in
the economy (remember that this is an accounting identity, i.e. is always true).
From the spending side we have
Y = C + I + G + (X ¡ M )
(4.1)
On the other hand, how much consumers spend on consumption goods and
import goods depends on their income Yh , so C and potentially M are functions
of income, i.e. C = C(Yh ) and M = M (Yh ). But by our identity spending Y
always has to equal income Yh , and both equal GDP. So (given a price level
P ) realized GDP is that level of income Yh for which total income equals total
spending Y , i.e. that level of Y that solves
Y = C(Y ) + I + G + (X ¡ M (Y ))
The situation in which Y = Yh is called Spending Balance by Hall and Taylor.
We will now start to model each component of spending.
The Aggregate Consumption Function
We start with C; the consumption expenditures of private households. So for
now we assume that investment I; government spending G and net exports
(X ¡M) are just some constant numbers, and in particular do not depend on the
level of income in the economy. We posit a very simple theory of consumption
in this section: we assume that
C = a + bYd
(4.2)
where a and b are …xed positive constants and Yd is (personal) disposable income
of private households. Remember that disposable income is (roughly) total
income (GDP) less taxes, i.e. Yd = Yh ¡ T; where T are total taxes. Several
things should be noted:
1. The equation in (4:2) is called the aggregate consumption function and
gives total consumption as a function of current disposable income. It is a
whole contingency plan: if disposable income is 200; then total consumption equals a + b ¤ 200; if disposable income is 500; then total consumption
equals a + b ¤ 500 and so forth.
94
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
2. As every model, the aggregate consumption function is a very simple approximation of actual consumption. It is likely that actual consumption
depends on other variables besides current disposable income, as future
income expectations, current wealth, interest rates and the like. But it
is exactly its simplicity that makes the aggregate consumption function
tractable.
3. As the name says, the aggregate consumption function models aggregate,
or total consumption. It is not meant to model any speci…c household,
but all households together. Obviously, if all individual households have
individual consumption functions of the form (4:2); then the aggregate
consumption function also has this form. But it may be true that even
if individual consumption functions look di¤erent, summing them all up
gives an aggregate consumption function of the form above.3
4. The constant b is call the marginal prospensity to consume. Note
dC
= b; i.e. b is equal to the extra amount that households consume
that dY
d
if their disposable income increases by $1: For example, if b = 0:8; then an
extra dollar of disposable income makes household spend 80 cents more on
consumption. This explains the name marginal prospensity to consume: it
is the response of consumption to a marginal (small) increase in disposable
income. It is assumed that b < 1: The constant a is sometimes called
autonomous consumption, it is that part of consumption that does not
depend on income.
Let us plot the aggregate consumption function in Figure 30. On the x-axis
we have disposable income Yd and on the y-axis we have aggregate consumption.
Given the form of the aggregate consumption function, consumption is a linear
function (straight line) with intercept a and slope b: The actual constants chosen
are a = 220; b = 0:9: This is about what ones gets if one …ts US data on
consumption and disposable income for the last 30 years.4
The Keynesian Cross
Let us make another simplifying assumption and assume that income is taxed
at a constant marginal tax rate ¿ : out of each dollar of total income a fraction
¿ has to be paid in taxes. For example if ¿ = 0:2 then for every dollar of income
the household has to pay 20 cents in taxes. With this assumption the relation
between total income Yh and total disposable income Yd is given by
Yd = (1 ¡ ¿ )Yh
3 The question under which conditions individual consumption functions give an aggregate
consumption function of a particular form is actually a deep theoretical question. Aggregation
theory deals with these issues that are well beyond the scope of this course.
4 The procedure used is OLS (ordinary least squares) estimation. You will (or have learned)
this procedure in great detail in your econometrics classes.
4.2. THE IS-LM FRAMEWORK
95
Aggregate
Consumption
(in billion $)
2,000
C=a+bY
d
1,000
Slope b
1,000
2,000
3,000
Disposable
Income (Y )
d
(in billion $)
Figure 4.3:
Substituting this relationship into the aggregate consumption function yields
C = a + b(1 ¡ ¿ )Yh
(4.3)
Our very simple model consists of two equations, (4:3) and (4:1); which
determine the two endogenous variables C and Y: We now determine income
and spending where we have spending balance. Remember that total spending
was given by
Y = C + I + G + (X ¡ M )
We assumed that I; G; (X ¡ M ) are some …xed, exogenously given numbers and
substitute the aggregate consumption function to get
Y = a + b(1 ¡ ¿ )Yh + I + G + (X ¡ M )
(4.4)
96
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
Now we use the fact that at the point of spending balance we have that spending
equals income, or Y = Yh : Imposing this condition in (4:4) yields
Y = a + b(1 ¡ ¿)Y + I + G + (X ¡ M )
We can now solve for income (spending)
Y (1 ¡ b(1 ¡ ¿ )) = a + I + G + (X ¡ M)
a + I + G + (X ¡ M)
Y = Yh =
1 ¡ b(1 ¡ ¿)
The value for aggregate consumption at the point of spending balance is obtained by plugging in for Yh in (4:3): This yields
C =a+
b(1 ¡ ¿ )
(a + I + G + (X ¡ M ))
1 ¡ b(1 ¡ ¿)
We can also solve for the point of balanced spending graphically. In Figure 31 we draw the famous Keynesian Cross diagram that determines income
(spending) in the balanced spending situation.5
On the x-axis we have total income Yh and on the y-axis we have total
spending Y: Income and spending are equal at the point of spending balance,
so this point has to be somewhere on the 45-degree line (since the 45-degree
line is the collection of all points at which Yh = Y ): But which point? This
is determined by the total spending equation. Plotting this equation we note
that is a straight line with intercept a + I + G + (X ¡ M ) (all the components
of spending that do not depend on income) and slope b(1 ¡ ¿) < 1: Hence the
line starts above zero and has smaller slope than the 45-degree line. Therefore
it necessarily intersects the 45-degree line once and only once. At this point
income coincides with spending and aggregate consumption is described by the
aggregate consumption function: as we found algebraically, at this point
Y = Yh =
a + I + G + (X ¡ M)
1 ¡ b(1 ¡ ¿)
You may ask yourself: haven’t we said that the fact that income equals
spending is an identity, i.e. always true. So what is the signi…cance of the
spending balance income? Remember that the aggregate consumption function
is a whole contingency plan: for each possible perceived income it gives the
amount consumed. Spending balance is the point at which total income is exactly at its right level so that consumption spending plus all the other spending
components, which are treated as exogenous at this point exactly equals that
income. In other words it is that income for which consumers can actually afford to spend what they want to spend according to the consumption function,
because total spending generates exactly that income.
5 John Maynard Keynes was the founder of macroeconomics and was the …rst to discuss
the aggregate consumption function.
4.2. THE IS-LM FRAMEWORK
97
Total
Spending Y
45-degree line: Y=Y
h
Slope 1
Spending
Y=a+(1-τ)bY +I+G+(X-M)
h
Slope (1-τ)b
a+I+G+(X-M)
Y=Y
h
Total Income Y
h
Figure 4.4:
This also makes clear why we always have to be at spending balance: suppose
in Figure 32 the economy is at a point where income Yh = Y1 : In this situation
total spending is higher than total income: consumers spend too much relative
to their income, a situation that is not sustainable, since consumers would realize
this imbalance. The same is true for a point like Y2 ; where income is too high
relative to what consumers want to spend. Therefore the economy always has
to be in spending balance where income equals spending.
To make this point more rigorous one has to specify an adjustment process
that takes the economy from points like Y1 or Y2 to spending balance. It is
quite straightforward to do this, but we have to introduce time (and therefore
dynamics) in our analysis. Assume that the consumption function takes the
form
Ct = a + b(1 ¡ ¿ )Yh;t¡1
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
98
Total
Spending Y
45-degree line: Y=Y
h
Slope 1
Spending
Y=a+(1-τ)bY +I+G+(X-M)
h
Slope (1-τ)b
a+I+G+(X-M)
Y
1
Y=Y
h
Y
2
Total Income Y
h
Figure 4.5:
i.e. consumption this period depends on income last period. On the other hand
…rms produce whatever the sectors in the economy want to spend, so
Yt = Ct + It + Gt + (Xt ¡ Mt )
For the moment let us assume that It = I; Gt = G; (Xt ¡ Mt ) = (X ¡ M ) ; i.e.
all components apart from consumption are constant over time and exogenously
given. Plugging the consumption function into the spending equation yields
Yt = a + b(1 ¡ ¿ )Yh;t¡1 + I + G + (X ¡ M)
From our identity Yh;t¡1 = Yt¡1 : and therefore
Yt = a + b(1 ¡ ¿ )Yt¡1 + I + G + (X ¡ M )
(4.5)
This is a linear di¤erence equation that gives spending (income) this period as
a function of income (spending) last period. It is very similar in spirit to our
4.2. THE IS-LM FRAMEWORK
99
basic di¤erential equation in the Solow model, just in discrete time. Let us
graphically analyze this di¤erence equation.
Total
Spending Y
45-degree line: Y=Y
t h,t
Spending
Y=a+(1-τ)bY
+I+G+(X-M)
t
h,t-1
Y
2
Y
1
a+I+G+(X-M)
Y
h,0
Y Y Y=Y
h
h,1 h,2
Total Income Y
h
Figure 4.6:
In Figure 33 we have on the x-axis income of households at period Yh and
on the y-axis we have total spending Y: We plot two relationships, our identity
Yt = Yh;t and the equation for total spending
Yt = a + b(1 ¡ ¿)Yh;t¡1 + I + G + (X ¡ M )
Now suppose we start with total income Yh;0 : In period 1 aggregate consumption
is given by C1 = a + b(1 ¡ ¿)Yh;0 and aggregate spending is given by
Y1 = a + b(1 ¡ ¿ )Yh;0 + I + G + (X ¡ M)
Graphically we get this point by starting from Yh;0 , going to the spending line
and from there to the y-axis, as indicated by the arrows. But from our identity
100
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
Yh;1 = Y1 i.e. income in period 1 equals spending in period 1 since all spending
generates income. Graphically we …nd Yh;1 by starting at Y1 on the y-axis, going
to the 45-degree line (where Y = Yh ) and then down to the x-axis. Now we have
income in period 1: We can now repeat the same logic to …nd Y2 ; Yh;2 ; Y3 and so
forth. As the …gure indicates over time income and spending go the point where
Y = Yh and then stay there forever. This point is our steady state in which
income and spending does not change anymore. We can solve for this point, call
it Y ¤ ; analytically. At this point Y does not change anymore, so Yt¡1 ¡ Yt = 0
(this is the analog to k_ = 0 in the Solow model), so Yt¡1 = Yt = Y ¤ : Using this
in (4:5) we have
Y ¤ = a + b(1 ¡ ¿)Y ¤ + I + G + (X ¡ M )
or
Y¤ =
a + I + G + (X ¡ M)
1 ¡ b(1 ¡ ¿)
which is exactly our income at spending balance. So the dynamic model provides
the foundation for assuming that we are always in spending balance: if we start
with income below, then spending of the economy in period 1 is above income
in period 0, …rms produce to satisfy the demand and generate income in period
1 which is higher than in period 0, this leads to further spending and income
increases until the economy hits Y ¤ : Note that the previous analysis crucially
depends on the assumption that b < 1; (or better, (1 ¡ ¿ )b < 1). Repeat the
analysis with (1 ¡ ¿)b > 1 and you will see that, unless we start at Y ¤ ; we will
never get there and hence the dynamic model is not adequate for providing an
underpinning for the assumption that we always are in spending balance.6
From now on we will assume that the adjustment process to Y ¤ is rapid
enough so that we, without losing anything substantial, can assume that we will
always be at spending balance. For this we should interpret the time periods
as short, maybe a month or so. We will not consider the adjustment process
explicitly in our further analysis.
The Multiplier
We now know how the level of income (and spending) is determined, given
exogenously given values for I; G and (X ¡ M): The next question is: what
happens to income and spending if there is an exogenous change in investment,
government spending or net exports? So suppose that the government decides
to increase government spending, say because the Reagan administration fears a
nuclear attack by the Russians and decides that one should have SDI to protect
6 Note
that, since the di¤erence equation (4:5) is linear, we can actually solve it analytically.
Doing so yields
Yt = Y ¤ + (Y0 ¡ Y ¤ ) (b(1 ¡ ¿ ))t
Obviously Yt goes to Y ¤ as t becomes large as long as b(1 ¡ ¿ ) < 1:
4.2. THE IS-LM FRAMEWORK
101
its citizens. For concreteness, suppose that G increases by $50 billion to G0 .
Let ¢G = G0 ¡ G denote the change in government spending ¢Y the resulting
change in income (and spending). Let us …rst analyze the situation graphically.
Total
Spending Y
45-degree line: Y=Y
h
New Spending
Y=a+(1-τ)bY +I+G’+(X-M)
h
Spending
Y=a+(1-τ)bY +I+G+(X-M)
h
a+I+G’+(X-M)
ΔG
a+I+G+(X-M)
Y’*
Y*
ΔY
Total Income Y
h
Figure 4.7:
From Figure 34 we see that income (and spending) increase, due to the
increase in government spending, from Y ¤ to Y 0¤ : We can actually use our
model for the adjustment process to describe how the economy moves from Y ¤
to Y 0¤ over time. For now we are interested in the size of the change in income
¢Y: From the picture we see that ¢Y > ¢G; i.e. income and spending go up
by more than the initial increase in government spending. We will now show
that this not an accident of the picture, but will be true in general. From our
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
102
algebraic solution we have
Y¤
=
Y 0¤
=
a + I + G + (X ¡ M)
1 ¡ b(1 ¡ ¿)
a + I + G0 + (X ¡ M)
1 ¡ b(1 ¡ ¿ )
Hence
¢Y
= Y 0¤ ¡ Y ¤
a + I + G0 + (X ¡ M ) a + I + G + (X ¡ M )
¡
=
1 ¡ b(1 ¡ ¿)
1 ¡ b(1 ¡ ¿ )
G0 ¡ G
=
1 ¡ b(1 ¡ ¿ )
¢G
=
1 ¡ b(1 ¡ ¿ )
Since 0 < (1 ¡ ¿ )b < 1; we have that 1 ¡ b(1 ¡ ¿) < 1 and therefore ¢Y > ¢G:
Suppose that the marginal prospensity to consume equals b = 0:9 and the tax
rate equals ¿ = 0:2: Then
¢Y
=
=
¢G
1 ¡ 0:9(1 ¡ 0:2)
¢G
= 3:57 ¤ ¢G
0:28
So if government spending goes up by $50 billion, total income and spending
1
(GDP) in the economy goes up by $178.5 billion. The term 1¡b(1¡¿
) is called
the government spending multiplicator: it tells us by how much GDP goes up if
government spending goes up by $1. Similarly we can derive the investment and
the export multiplicator, where instead of an increase in government spending
we consider an exogenous increase (or fall) in investment or exports. These
1
multipliers turn out to both equal 1¡b(1¡¿
) ; i.e. are equal to the government
spending multiplicator.
What is the economics behind these results. This is most clearly demonstrated by using the adjustment process explicitly. Remember that the two
equations were
Yt
Yh;t
= a + b(1 ¡ ¿ )Yh;t¡1 + I + G + (X ¡ M)
= Yt
Now we start at Yh;0 = Y ¤ ; i.e. the old steady state corresponding to government
spending G: Now Reagan and his SDI come along and government spending
increases by ¢G to G0 : Then total spending increases from Y1 = Y ¤ to Y1 = Y ¤ +
¢G: Firms supply the desired new additional products, here the SDI system.
The additional production generates additional income, so income increases from
4.2. THE IS-LM FRAMEWORK
103
Yh;0 = Y ¤ to Yh;1 = Y ¤ + ¢G: But this is not the end of the story. Although
G does not increase further, total spending does: since income is now higher by
¢G and consumers consume a fraction b(1 ¡ ¿ ) out of every additional dollar
of income, consumption spending in the second round increases by b(1 ¡ ¿ )¢G:
Hence
Y2
= Y1 + b(1 ¡ ¿)¢G
= Y ¤ + ¢G + b(1 ¡ ¿ )¢G
= Y ¤ + (1 + b(1 ¡ ¿ )) ¢G
Again …rms stand by to produce the additional goods demanded and additional
income of size b(1 ¡ ¿ )¢G is generated: Yh;2 = Y2 : And again a fraction b(1 ¡ ¿ )
of this additional income is used for additional consumption, so that
Y3
= Y2 + b(1 ¡ ¿) ¤ b(1 ¡ ¿ )¢G
= Y2 + (b(1 ¡ ¿ ))2 ¢G
´
³
= Y ¤ + 1 + b(1 ¡ ¿) + (b(1 ¡ ¿ ))2 ¢G
This process of additional income generation and additional spending continues
ad in…nitum, until Y 0¤ is reached: additional spending generates additional
income from the production process; this additional income leads to further
additional spending and so forth. Note, however, that the income and spending
increments become smaller and smaller over time (and eventually become so
small that they are negligible); eventually we get arbitrarily close to Y 0¤ : This
adjustment process is demonstrated in Figure 35.
In Table 9 we summarize all the e¤ects of the change in government spending
from G to G0 : Again we assume b = 0:9 and ¿ = 0:2: For concreteness we assume
that Y ¤ = $1; 000 billion
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
104
Total
Spending Y
45-degree line: Y=Y
h
New Spending
Y=a+(1-τ)bY +I+G’+(X-M)
h
Spending
Y=a+(1-τ)bY +I+G+(X-M)
h
a+I+G’+(X-M)
(1-τ)bΔG
ΔG
a+I+G+(X-M)
ΔG
Y’*
Y*
ΔY
Total Income Y
h
Figure 4.8:
Table 9
Time t
0
1
2
3
..
.
t large
Additional Spending
0
¢G = 50
b(1 ¡ ¿ )¢G =
0:9(1 ¡ 0:2)50 = 36
(b(1 ¡ ¿))2 ¢G =
(0:9(1 ¡ 0:2))2 50 = 26
..
.
(b(1 ¡ ¿ ))t ¢G ¼ 0
Total Change in Income Yt ¡ Y0
0
¢G = 50
(1 + b(1 ¡ ¿ ))¢G =
(1 + 0:9(1 ¡ 0:2))50 = 86
(1 + b(1 ¡ ¿) + (b(1 ¡ ¿ ))2 )¢G =
(1 + 0:72 + (0:72)2 )50 = 112
..
.
(1 + b(1 ¡ ¿) + ¢ ¢ ¢ + (b(1 ¡ ¿))t )¢G ¼
1
1¡b(1¡¿ ) ¢G = 178:5
Yt = Yh;t
1; 000
1; 050
1; 086
1; 112
..
.
Y 0¤ = 1; 178:5
4.2. THE IS-LM FRAMEWORK
105
As argued above, in the …rst round income and spending increase exactly
by the amount of additional government spending. Additional income triggers
additional consumption spending, $36 billion in the second round, $26 billion in
the third round and so forth. Summing up all these e¤ects yields a total increase
in income and spending of $178:5 or exactly 3:57 times the initial increase in
government spending. Remember that this was exactly what we got using our
government spending multiplier. This, again is no accident: mathematically
this comes from the fact that the sum of all income increases in all rounds, if
we allow in…nitely many rounds
2
t
(1 + b(1 ¡ ¿ ) + (b(1 ¡ ¿ )) + ¢ ¢ ¢ + (b(1 ¡ ¿ )) + ¢ ¢ ¢ )¢G
1
¢G
=
1 ¡ b(1 ¡ ¿ )
equals exactly the multiplier. So again the dynamic analysis provides the justi…cation for our shortcut results.7
In all of our exercise we ignored the fact that we government has to somehow
…nance the additional government spending. The SDI project was …nanced
by issuing more government debt (which is being repaid at the moment). If
instead the increase in government spending is …nanced by increasing taxes, the
multiplier analysis is changed and the multiplier is much smaller. In fact I may
ask you in a homework to derive the famous Haavelmo multiplier, the multiplier
that results from a tax-…nanced increase in government spending.8
It is an easy modi…cation to analyze what happens if not only the amount of
consumption goods that are purchased domestically depends on current income,
but also the imported consumption goods. Suppose that imports are given by
the function
M = mYh
where m is the marginal prospensity to import. Now our key equations for
spending balance become
Y
Yh
= a + b(1 ¡ ¿ )Yh + I + G + X ¡ mYh
= Y
Doing exactly the same analysis as before this yields as income (spending) level
7 Shortcut as we ignore the adjustment process to spending balance. The formula comes
from the mathematical fact that, for any number c strictly between 0 and 1 we have
=
1 + c + c2 + c3 + ¢ ¢ ¢
1
1¡c
The expression 1 + c + c2 + c3 + ¢ ¢ ¢ is called a geometric sum (since the terms in the sum
decline geometrically to zero).
8 Named after Swedish economist and Nobel price winner (in 1989) Trygve Haavelmo the
result is that a tax-…nanced increase in government spending increases income 1 for 1, i.e. the
multiplier is exactly 1.
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
106
in spending balance
Y¤ =
a+I +G+X
1 ¡ b(1 ¡ ¿) + m
1
and a government spending (investment, export) multiplicator of 1¡b(1¡¿
)+m :
Note that the multiplicator is smaller now because the fraction m of additional
income generated by ¢G is not spent on domestic consumption and therefore
does not additional income domestically (but rather in the country from where
the additional import goods come). Further details will be investigated in some
homework problems.
4.2.2
Investment, the Interest Rate and the IS Curve
In the previous subsection we …xed the price level and investment I; government spending G and exports X at some exogenously given numbers. Now we
look more carefully at what determines investment demand. When we modeled
consumption demand we posited a very simple, highly tractable model of consumption: consumption demand depends only on current disposable income.
When modelling investment we follow the same strategy: we posit that investment demand only depends on the real interest rate r and we write
I = e ¡ dr
where e and d are positive constants. Our reason for why investment demand
depends negatively on the interest rate is the following. Most businesses don’t
have the funds available to …nance a new factory, an expensive new machine
and so forth. Therefore they have to take out a loan from a bank to …nance
this new investment. The higher the real interest rate, the more expensive it
is for …rms to borrow and the less investment projects are actually undertaken.
Therefore investment demand depends negatively on the real interest rate.9
In Figure 36 we draw the aggregate investment function. It is a straight line
that is downward sloping since aggregate investment demand depends negatively
on the real interest rate.
We now have all the ingredients together to analyze the determination of
income and interest rates jointly. Still we assume that the price level P is …xed.
Also the components G and X of total spending are assumed to be exogenously
given …xed numbers. Aggregate consumption is given by
C = a + b(1 ¡ ¿)Yh
Aggregate investment is given by
I = e ¡ dr
9 We use the real interest rate since, although banks pay the nominal interest rate speci…ed
in the loan contract, in the period of repayment one dollar is worth less than in the period
where the contract was agreed upon, due to in‡ation. Hence the real return on the loan for the
bank (and the real cost for the …rm) is given by the nominal interest rate minus the in‡ation
rate, i.e. the real interest rate.
4.2. THE IS-LM FRAMEWORK
107
Aggregate
Investment
(in billion $)
800
Slope -d
400
3
6
9
Real Interest
Rate (in %)
Figure 4.9:
Aggregate imports are given by
M = mY
Spending balance requires
Yh = Y
Therefore total spending is given by
Y = a + b(1 ¡ ¿ )Yh + e ¡ dr + G + X ¡ mYh
Using the identity that income equals spending we get
Y = a + b(1 ¡ ¿)Y + e ¡ dr + G + X ¡ mY
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
108
or
Y =
r
a+e+G+X
¡d
1 ¡ b(1 ¡ ¿ ) + m
1 ¡ b(1 ¡ ¿ ) + m
(4.6)
We can also solve for the interest rate. This yields
r=
1 ¡ b(1 ¡ ¿ ) + m
a+e+G+X
¡
Y
d
d
(4.7)
Remember that the only two variables in this equation are income Y and the
real interest rate r; all the other stu¤ are …xed numbers. Equation (4:7) or (4:6)
is called the IS-curve (for income=spending): it is a relation between income Y
and the real interest rate r and consists of all points (Y; r) so that income equals
spending and consumption is described by the aggregate consumption function,
investment by the aggregate investment function and imports by the aggregate
import function.
Figure 37 draws the IS-curve. It is downward sloping since a higher interest
rate decreases investment demand (by d) and therefore spending (income) by an
d
amount given by d times the investment multiplier, i.e. by 1¡b(1¡¿
)+m : Since
)+m
:
we draw Y on the x-axis, the slope is the inverse of this, 1¡b(1¡¿
d
We can derive the IS-curve directly from the Keynesian Cross diagram. This
is done in Figure 38
The top graph is our typical Keynesian cross from before. We start with a
given real interest rate r: For this interest rate investment demand is given by
I = e ¡ dr and the resulting spending and income is given by Y: So in the lower
graph we found one point on the IS curve: the smily face corresponding to the
point (Y; r): Now we want to construct a second point on the IS curve, so we
vary the real interest rate. In particular we reduce the interest rate to r0 : This
increases investment demand from I = e ¡ dr to I 0 = e ¡ dr0 : Since investment
increases with lower interest rates, I 0 > I: In the Keynesian Cross diagram
the spending curve shifts upwards and income (spending) increases to Y 0 (by
¢I
d
¢Y = 1¡b(1¡¿
)+m = 1¡b(1¡¿ )+m ¢r: In the bottom graph we mark a second
point (Y 0 ; r0 ) on the IS-curve. Continuing to do this we can trace out the entire
IS-curve by varying the interest rate and determining the income (spending)
level corresponding to this interest rate from the Keynesian Cross diagram.
We can now investigate what happens to the IS curve if the government
increases government spending by ¢G from G to G0 : We already did the analysis
in the Keynesian Cross diagram, so now our life is easy. Figure 39 shows what
happens. Again we draw two graphs. Suppose that in the bottom graph we
…gured out the IS-curve for a given level of government spending G: This line
is labeled as old IS-curve. Now G increases from G to G0 : What happens to the
IS-curve. Let is look at a single point on the new curve. Fix the interest rate at
r: For this interest rate and the old level of government spending G; the point
on the old IS-curve is the smily face corresponding to (Y; r): But where is the
point corresponding to the same interest rate r and the new level of government
spending G0 : Fixing r and increasing G by ¢G to G0 shifts the spending curve
4.2. THE IS-LM FRAMEWORK
109
Real Interest
Rate (in %)
10%
Slope (1-b(1-τ)+m)/d
5%
Income Y
5000
6000
7000
(GDP)
Figure 4.10:
in the Keynesian Cross diagram up by ¢G: The new income level is given by
Y 0 : We remember from above that
¢Y
Y0
¢G
1 ¡ b(1 ¡ ¿ ) + m
¢G
= Y +
1 ¡ b(1 ¡ ¿ ) + m
= Y0¡Y =
i.e. the new income level associated with the old r; but new G0 is exactly ¢G
times the government spending multiplier higher than the old income level. We
found one point on the new IS-curve: it is the smily face corresponding to
(Y 0 ; r): Again doing this for all possible interest rates yields the new IS-curve.
¢G
The new IS-curve looks like the old, but is shifted by 1¡b(1¡¿
)+m to the right,
exactly because income increases by ¢G times the multiplier for every interest
rate.
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
110
Y =Y
h
Spending Y
Y=a+(1-τ)bY +e-r’d+G+X-mY
h
h
Y=a+(1-τ)bY +e-rd+G+X-mY
h
h
Income Y
Y
Y’
h
Real Interest
Rate r
r
r’
Income Y
Y
Y’
h
=Spending Y
Figure 4.11:
Obviously a decline in government spending shifts the IS-curve to the left;
similar shifts are caused by changes in exports X or changes in the parameters
a and e:
4.2.3
The Demand for Money and the LM-Curve
Our macroeconomic model so far consists of the following equations. Some of
them are behavioral equations, i.e. describe the behavior of consumers or …rms.
These are the aggregate consumption, investment and import functions.
C
I
M
= a + b(1 ¡ ¿)Yh
= e ¡ dr
= mYh
4.2. THE IS-LM FRAMEWORK
111
Y =Y
h
Spending Y
Y=a+(1-τ)bY +e-rd+G’+X-mY
h
h
Y=a+(1-τ)bY +e-rd+G+X-mY
h
h
ΔG
Y
ΔY
Income Y
Y’
h
Real Interest
Rate r
ΔY
r
New IS Curve
Old IS Curve
Income Y
Y
Y’
h
=Spending Y
Figure 4.12:
We also have equations that are true by de…nition or by accounting rules. These
are the de…nition for total spending and the identity that income always equals
spending.
Y
Y
= C + I + G + (X ¡ M )
= Yh
We still assume that G; X are just given numbers and that the price level
is …xed at some predetermined level P: Hence we have …ve equations and
six endogenous variables to be determined, namely Yh ; Y; C; I; M; r (note that
a; b; d; e; m; ¿ are parameters, i.e. numbers that we will treat as …xed for all future purposes). That means that we cannot yet solve for the equilibrium values
of our variables altogether (by equilibrium values I mean values of endogenous
variables that satisfy all equation that describe our economy, given some values
for the parameters and exogenous variables G; X; P ). So far the best we can do
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
112
is to substitute C; I; M; Yh into the spending equation and derive the collection
of all income (spending) levels and interest rates (Y; r) that satisfy all equations. This was the IS-curve. The formula for the IS-curve is given (see the last
section) as
1 ¡ b(1 ¡ ¿ ) + m
a+e+G+X
¡
Y
d
d
Now we will add one additional equation that will enable us to solve exactly
for one single combination (Y ¤ ; r¤ ) of equilibrium income and interest rates.
Now we bring in money into our analysis. Note that so far we measured
all our variables in real terms, i.e. in physical units: Y is real GDP and so
forth. But in order to spend, people need money, at least in general.10 By
money in this course I will mean …at currency, i.e. pieces of paper issued by
the government that have no intrinsic value.11 These pieces of paper are money
because the government decrees that they are money. Although …at money is
the primary form of money in modern societies, historically most societies have
used as money a commodity with intrinsic value. This type of money is called
commodity money. A famous example are WW II prison camps where cigarettes
became the common form of money between inmates. Cigarettes were used as
medium of exchange to trade soap for food, but they were also consumed. The
most prevalent form of commodity money historically was gold. In the early
20-th century a lot of countries used pieces of paper as money, but these pieces
of paper were backed by gold: everybody could go to the bank and exchange
these pieces of paper for gold at a rate that was …xed and guaranteed by the
government. Such a monetary system is referred to as the gold standard. The
US left the gold standard when the Breton Woods system collapsed in 1973.
We will now add a behavioral equation to our economy that intends to describe the market for money. Let us …rst think about the demand for money.
People need money to purchase goods, i.e. to make transactions. We will develop three hypotheses about money demand. To understand these hypotheses
it is crucial to understand that households can hold their wealth in di¤erent
forms: in money or in assets that bear interest. So the question here is not
how much money households want (everybody prefers more to less), but how
households divide their wealth into money holdings and other assets (stocks,
bonds) that, in contrast to money, yield interest rate. Such a decision is called
a portfolio decision. Back to our three hypotheses.
r=
1. People want to hold more money when the price level is higher and less
money when the price level is lower. Since people do not care about money
10 Sometimes goods are exchanged for goods. For example in college I traded tutoring
sessions against cases of beer (instead of for money). Such trades are called barter. Barter
trade requires “double coincidence of wants”, i.e. my collegue wanted tutoring lessions and I
wanted beer. If there is no double coincidence of wants for trade to happen we need a medium
of exchange - money.
11 When measuring money economists include as money all assets that are readily available to make transactions, which includes not only currency, but also checking accounts that
households hold with private banks. For now it is conceptually easier to think of money just
as currency in circulation.
4.2. THE IS-LM FRAMEWORK
113
per se, but only as a medium of exchange for real consumption goods, if
all goods double in prices, households need a doubled amount of money to
purchase the same consumption goods. If we let M d denote the demand
for money and P the (…xed) price level, this hypothesis just states that
M d and P are proportional to each other.
2. Suppose people want to spend more in consumption goods, so that total
spending in real terms Y (or real GDP) increases, then people need more
money to carry out the additional trades. Therefore we assume that money
demand M d increases in Y; the desired real spending in the economy.
3. What is the opportunity cost for holding money, instead of interest bearing
assets? Money does not pay any interest rates, whereas interest bearing
assets pay the nominal interest rate. Therefore we assume that money
demand is decreasing in the nominal interest rate. Since, for the moment,
we assume that the price level is …xed and therefore the in‡ation rate
is zero in the short run, this translates into the assumption that money
demand is decreasing in the real interest rate r: For now we follow Hall
and Taylor and disregard the di¤erence between the nominal and the real
interest rate for the moment and denote by r just the interest rate.
We therefore model the demand for money as
M d = P ¤ L(Y; r)
or
Md
= L(Y; r)
P
The function L is called the real money demand function and gives the demand
for real money balances, i.e. for money adjusted by the price level. We assume
that the function L is linear, i.e.
Md
= L(Y; r) = kY ¡ hr
P
where k and h are positive constants. The constant k measures by how much
real money demand goes up if real spending goes up by one dollar, the constant
h measures by how much real money demand goes down if the interest rate goes
up by 1%: This completes our description of the demand for money.
What about the supply of money. The supply of money, M s ; is determined
by the Federal Reserve System, by the government agency that is responsible for
conduction monetary policy. We have to postpone a discussion of how exactly
the FED goes about conducting monetary policy. For now we assume that the
supply of money is …xed and exogenously given (as is the price level).
We assume that the money market is always in equilibrium, so that
Ms = Md
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
114
or
Ms
P
Ms
P
=
Md
P
= kY ¡ hr
(4.8)
Equation (4:8) is called the LM-curve (since it relates the demand for real money
balances L to the supply of money, M s : It is important to distinguish which
variables are endogenous and which are exogenous in this equation. We assume
that money supply M s is …xed by the FED and therefore exogenously given.
Also the price level P is …xed by assumption. The only endogenous variables in
the LM-curve are Y and r: Rewriting (4:8) yields
r=
1 Ms
k
Y ¡
h
h P
This closes our economic model: the IS-curve and the LM- curve can be used
jointly to determine the equilibrium values of (Y; r). Once we have these we
can deduce all other endogenous variables C; I; M; Yh from the other equations.
Therefore, given a price level P and a money supply M s (and given G; X), we
can …gure out total spending, income, consumption, investment, imports and
interest rates that prevail in the economy in the short run. We will do this
graphically in a bit. But …rst let’s analyze the LM-curve in more detail.
In Figure 40 we draw the LM-curve. The LM curve shows the interest rate
as a function of spending (GDP). The slope of this curve is given by hk > 0:
What is the intuition for this? Remember that since
the money supply and the
s
price level are …xed, the real supply of money MP is …xed. Now suppose that
spending Y goes up, so real money balances demanded increase. But the supply
is …xed. The only way to bring demand and supply to equilibrium again is a
rise in the interest rate, making the amount of real balances demanded decline,
o¤setting the increase due to higher Y:
Changes in the money supply M s and the price level P shift the LM-curve.
So let us consider what happens to the LM-curve if M s increases (but P stays
constant). This is important for the analysis of the e¤ects of monetary policy.
Suppose the money supply increases from M s to M s0 : How does GDP have to
change to leave the interest rate unchanged? Since real money supply increases,
real money demand must increase. If the interest rate is unchanged, to increase
real money demand, Y must increase. The LM-curve shifts to the right.12 This
is shown in Figure 41.
12 By
how much does the LM curve shift to the right? Suppose money supply increases by
¢M s . To leave the interest rate unchanged it has to be the case that the change in GDP, ¢Y
has to satisfy
0
=
=
=
¢r
¶ µ
¶
µ
k
1 M s0
k 0
1 Ms
¡
Y ¡
Y ¡
h
h P
h
h P
k
¢M s
¢Y ¡
h
hP
4.2. THE IS-LM FRAMEWORK
115
Interest
Rate (in %)
LM-curve
10%
Slope k/h
5%
Income Y
5000
6000
7000
(GDP)
Figure 4.13:
An increase in the price level has thes opposite e¤ect. Keeping the money
supply …xed, an increase in P decreases MP ; the real supply of money. Therefore
the demand for real money balances has to decrease. For a …xed interest rate
now GDP Y has to decrease to bring the money market back into equilibrium.
The LM-curve shifts to the left for an increase in the price level. This is crucial
for the derivation of the aggregate demand curve below. Figure 42 shows this
e¤ect.13
Therefore
¢Y =
s
¢M s
kP
i.e. the LM-curve shifts to the right by ¢M
kP
13 By how much does the LM curve shift to the left? Suppose the price level increases by
¢P . To leave the interest rate unchanged it has to be the case that the change in GDP, ¢Y
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
116
Interest
Rate (in %)
Old LM-curve
10%
New LM-curve
Slope k/h
5%
M
s
Income Y
5000
6000
7000
(GDP)
Figure 4.14:
has to satisfy
0
=
=
=
¢r
¶
µ
¶ µ
k 0
k
1 Ms
1 Ms
Y ¡
Y
¡
¡
h
h P0
h
h P
µ
¶
k
M s ¢P
¢Y ¡
h
h
P 0P
Therefore
¢Y = ¡
i.e. the LM-curve shifts to the left by
Ms
k
³
Ms
k
¢P
P 0P
µ
´
:
¢P
P 0P
¶
4.2. THE IS-LM FRAMEWORK
117
Interest
Rate (in %)
New LMcurve
10%
Old LM-curve
Slope k/h
5%
P
Income Y
5000
6000
7000
(GDP)
Figure 4.15:
4.2.4
Combination of IS-Curve and LM-Curve: ShortRun Equilibrium
We can combine the IS-curve and the LM-curve to determine short-run GDP
(income, spending) and interest rates. Remember that the IS-curve is given by
r=
1 ¡ b(1 ¡ ¿ ) + m
a+e+G+X
¡
Y
d
d
(4.9)
whereas the LM-curve is given by
r=
k
1 Ms
Y ¡
h
h P
(4.10)
These are two equations in the two unknowns (Y; r): Given that the IS-curve is
downward sloping and the LM-curve is upward sloping these to curves intersect
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
118
once and only once, as shown in Figure 43. This intersection determines the
short run level of GDP, Y ¤ and the short run interest rate r¤ :
Interest
Slope
Rate (in %)
(1-b(1-τ)+m)/d
LM-curve
10%
Slope k/h
r*
5%
IS-curve
Income Y
5000
6000
Y* 7000
(GDP)
Figure 4.16:
Let us solve for (Y ¤ ; r¤ ) algebraically. Combining the IS-curve and the LMcurve yields
1 Ms
a+e+G+X
1 ¡ b(1 ¡ ¿) + m
k
Y ¡
=
¡
Y
h
h P
d
d
Solving this mess for Y yields
µ
¶
k 1 ¡ b(1 ¡ ¿ ) + m
+
Y
h
d
Y¤
=
=
1 Ms
a+e+G+X
+
d
h P
¢
¡ a+e+G+X
1 Ms
+h P
d
³
´
1¡b(1¡¿ )+m
k
h +
d
4.2. THE IS-LM FRAMEWORK
119
Note that GDP (or total spending, total income) in the short run increases with
the level of government spending G and exports X as well as with money supply
M s and decreases with the price level P: Remember that in the short run real
GDP equals aggregate demand. So the fact that Y ¤ decreases with increases in
P justi…es that the aggregate demand curve is downward sloping as drawn in
Figure 29. It now follows that
Yh¤
r¤
I¤
C¤
M¤
= Y¤
k ¤ 1 Ms
Y ¡
=
h
h P
= e ¡ dr¤
= a + b(1 ¡ ¿)Y ¤
= mY ¤
These formulas give the short run equilibrium values of the endogenous variables
Y; Yh ; r; I; C; M as functions of the exogenous variables G; X; M s ; P and the parameters a; b; d; e; m; ¿: We have now formulated and solved our complete model
of the macroeconomy in the short run. Now we can address policy questions in
the next section.
4.2.5
Monetary and Fiscal Policy in the IS-LM Framework
Monetary Policy
Let us start with monetary policy. In our simple model monetary policy amounts
to the FED picking the money supply M s : Suppose we want to analyze statements of the form (which could be found in recent issues of the economist)
The US is going to a recession. A possible remedy: increase real
GDP by softening monetary policy
Let us try to analyze this statement with the tools we have. First we assume
that the US economy is well-described by the macroeconomic model we developed in the last section. Second, we focus on the short run e¤ects of monetary
policy (remember that in the long run money did not a¤ect real output, according to the classical dichotomy). Third, we translate “softening monetary policy”
to mean an increase in the money supply M s : We use our IS-LM diagram to
see what is going on. As usual, we …rst ask ourselves which curves, if any, shift.
The IS-curve (4:9) does not shift, but sthe LM-curve (4:10) shifts, as we argued
in the last section to the right (by ¢M
kP ). From Figure 44 we see that short-run
equilibrium real GDP Y ¤ increases to Y ¤0 and the interest rate r¤ falls to r¤0 .
Given values for the exogenous variables and parameters we can also compute
by how much real GDP and the interest rate change in response to an increase
in the money supply. This is straightforward and I will leave this for a problem
set.
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
120
Slope
Interest
(1-b(1-τ)+m)/d
Rate (in %)
Old LM-curve
New LM-curve
10%
Slope k/h
r*
r*’
5%
IS-curve
s
M
Income Y
Y* Y’*
(GDP)
Figure 4.17:
What is the economic intuition for this (this intuition is somewhat loose,
and in order to make it tighter we would need a fully dynamic model, which to
develop is beyond the scope of this course): An increase in M s increases the
supply of money. For households to be willing to hold this additional money
the interest rate must fall. A lower interest rate spurs higher investment and
the multiplier sets in, leading to higher real GDP Y ¤ :
From the …gure we also see what determines the magnitude of the response
of real GDP to an increase in money supply. Suppose the IS-curve is really
steep, almost vertical. Then a given increase in the money supply has very little
e¤ects on real GDP Y ¤ and a strong e¤ect on the interest rate. Why this. A
steep IS-curve means a low d; i.e. investment demand is not very responsive
to declines in the interest rate. So an increase in the money supply leading
to a drop in the interest rate does not increase investment by a whole lot and
therefore real GDP does not increase by much. On the other hand, if investment
4.2. THE IS-LM FRAMEWORK
121
demand is very sensitive to the interest rate (a high d), the IS curve is very ‡at
and an increase in money supply and the resulting drop in the interest rate have
a large e¤ect on investment and hence real GDP.
The e¤ect on real GDP induced by an increase in the money supply is also
the bigger the steeper the LM curve is. The LM-curve is steep when h is
low, i.e. when real money demand responds only weakly to the interest rate.
In this situation a large drop in r is required for money demand to absorb the
additional money supply. But large drops in interest rates induce large increases
in investment demand and hence real GDP.
Therefore the e¤ectiveness of monetary policy to increase real GDP (by increasing money supply) depend on how sensitive investment is to the interest
rate and how sensitive money demand is to the interest rate. The e¤ect on real
GDP of an increase in money supply are weak (but positive) if investment demand is insensitive to changes in the interest rate and/or money demand is very
sensitive to the interest rate. The e¤ect is strong (and positive) if investment
demand is very sensitive to the interest rate and real money demand is relatively
insensitive to changes in the interest rate. You should convince yourself of that
by drawing several IS-LM diagrams with di¤erent slopes of the IS-curve and the
LM-curve (or by looking at Hall/Taylor, pp. 194-95).
We can do the reverse experiment of a decline in money supply. I will leave
this as an exercise for a problem set, but it is worth mentioning that the two
recessions in 1980-82, the so-called Volcker recessions, are attributed to the
tight monetary policy that the FED carried out under then new chairman Paul
Volcker.
Fiscal Policy
Let us again study the Reagan SDI policy experiment. This program was probably not primarily designed to move the economy out of the Volcker recessions,
but rather motivated by strategic national defense reasons, but let us analyze
its e¤ect on the US economy anyway. Again we assume that the US is described
well by our model and that we only analyze the short-run e¤ects of the policy.
We also ignore the question how SDI was …nanced. Fiscal policy in our model
basically amounts to the government choosing how much to spend, i.e. how to
pick G: So initiating the SDI program amounts to an increase in G in our model.
Let us use IS-LM analysis to see what happens. Again, what curves shift? It
is obvious that the LM-curve (4:10) does not shift, but that the IS-curve (4:9)
shifts to the right (we actually saw this in the section where we developed the
¢G
¤
IS-curve) by 1¡b(1¡¿
)+m . We see from Figure 45 that real GDP Y increases
to Y ¤0 and the interest rate r¤ increases to r¤0 :
Again, what is the economic intuition? An increase in government spending
starts the multiplier process and increases total spending. We discussed that
when we talked about the multiplier. But now our model is richer, it includes
money and has investment depending on interest rates. So when consumption
spending increases, money demand increases. But money supply is …xed, so the
interest rate has to increase to bring the money market back into equilibrium.
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
122
Interest
Rate (in %)
Old IS-curve
LM-curve
New IS-curve
10%
Slope
Slope k/h
(1-b(1-τ)+m)/d
r*’
r*
G
Income Y
Y* Y’*
(GDP)
Figure 4.18:
But higher interest rates mean a reduction in investment demand. So part of
the stimulus of real GDP due to an increase in G and the multiplier process is
o¤set by a fall in private investment demand, induced by rising interest rates.
This process is called crowding-out: higher government spending leads to
higher interest rates and therefore crowds out private investment. Nevertheless
real GDP on net increases with an increase in government spending, but by less
than what is predicted by the naive multiplier analysis.
Again, the magnitude of the increase in real GDP induced by an increase
in government spending depends on how steep the IS-curve and the LM-curve
are. This has good economic intuition again. The e¤ects of an increase in
government spending are strong if the IS-curve is steep and/or the LM-curve is
‡at and are week if the IS-curve is ‡at and/or the LM-curve is steep.
The IS-curve is steep if d is small. Small d means that investment does not
react strongly to an increase in the interest rate. If this is the case, then the
4.3. THE AGGREGATE DEMAND CURVE
123
crowding out-e¤ect is small. Even though higher government spending leads
to higher interest rates, this does not reduce private investment by much. The
LM-curve is ‡at if h is big, i.e. money demand responds strongly to the interest
rate. Then only a small increase in the interest rate is needed to bring the
money market back into equilibrium (money demand had increased because of
higher consumption spending induced by higher G and the multiplier process).
But if interest rates rise only modestly, not much investment is crowded out and
the e¤ects of an increase in G are large. Reverse arguments hold if d is large
and h is small.
The previous discussion also explains why the model we developed so far
was so popular until the 70’s. It gave monetary and …scal policy an active
role in managing the business cycle. If the economy is in a recession, then
the model prescribes soft monetary policy and/or expansionary …scal policy
(high government spending). The economist and the politician is like a social
engineer that can …ne-tune the economy with the appropriate policy, and the
only problem left is to …gure out when and by how much exactly to change
monetary and …scal policy. The Keynesian model of business cycles was so
popular that even Nixon confessed that “we are all Keynesians now”. But it
was also in the mid-70’s that these simple recipes started to fail, which not
only led to a change in economic policies in the 80’s and 90’s, but also to a
dramatic change in economics as a science, away from Keynesianism and back
to neoclassical ideas (back to the future, so to speak). We will pick up this
theme in more detail in a bit.
4.3
The Aggregate Demand Curve
Given our IS-LM apparatus it is now simple to derive the aggregate demand
curve from Figure 29. For a …xed price level P we know how to derive aggregate
demand Y ¤ (which equals real GDP in the short run), using the IS-LM diagram.
Now suppose we want to …nd aggregate demand for a di¤erent price level, say
P 0 > P: If the price level increases, what happens in the IS-LM-diagram? As
we saw in the last section, the LM-curve shifts to the left. The IS-curve remains
unchanged (the price level does not enter the IS-curve). Therefore the aggregate
demand (GDP) associated with the higher price level P 0 ; Y ¤0 is lower (and the
interest rate is higher) than before. Doing this exercise for a lot of di¤erent
price levels one can trace out the entire aggregate demand curve. Figure 46
exempli…es this.
Again, what is the economics? A higher price level decreases real money
supply. Therefore real money demand has to fall which requires an increase
in the interest rate. A higher interest rate reduces investment and real GDP,
partly because of the direct e¤ect, partly because the multiplier kicks in.
There is one big question remaining: what is the process that lead us from
a short-run situation, where aggregate demand (and hence realized GDP) is
di¤erent from potential output (or aggregate supply) to the long run equilibrium
in which potential output equals to aggregate demand. The answer obviously
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
124
Interest
Rate r
New LM-curve
Old LM-curve
for P
for P’>P
P
IS-curve
Y*’
Y*
Price Level P
Real GDP Y
(Aggregate Demand)
P’
P
Aggregate
Demand Curve
Y*’
Y*
Real GDP Y
(Aggregate Demand)
.
Figure 4.19:
must have something to do with adjustment of prices. We will discuss this after
we are …nished with a little digression. When aggregate demand falls below
potential output, some labor is left unutilized and there is unemployment. We
…rst want to discuss some basic facts about unemployment before we turn to
the price adjustment mechanism.
4.4
Unemployment
The Keynesian business cycle theory can explain unemployment. In the short
run prices are sticky, realized GDP equals aggregate demand, which may very
well be below potential GDP. Factor inputs, in the short run, are left unutilized:
machines are left idle and some workers who desire to work for the market wage
can’t …nd a job. In this section we will look at the data about the labor market.
Even though the news usually reports only one number from the labor market,
4.4. UNEMPLOYMENT
125
namely the unemployment rate, there is much more going on. Even in good
times a large number of workers are …red or voluntarily leave their job and a
large number of new jobs are created and workers are hired. We will look at
some numbers from the US labor market and then we will build a simple, purely
descriptive model of the ‡ows into and out of unemployment. A fantastic source
of information about the ‡ow of workers into and out of jobs is the book “Job
Creation and Destruction” by Steven Davis, John Haltiwanger and Scott Schuh.
We will report their main …ndings.
4.4.1
Concepts and Facts
Let us start with some basic de…nitions
De…nition 3 The labor force is the number of people, 16 or older, that are
either employed or unemployed but actively looking for a job. We denote the
labor force at time t by Nt
De…nition 4 Let W Pt denote the total number of people in the economy that
are of working age (16 - 65) at date t : The labor force participation rate ft is
de…ned as the fraction of the population in working age that is in the labor force,
i.e. ft = WNPt t :
Note that for the U.S., in 1994 the labor force consisted of about 131 million
people whereas about 197 million people were of working age. That gives a labor
force participation rate of about 66.5%. This number has not changed much
over the last 7 years. It has become a bit higher since the prospectus of entering
a very good labor market in the second half of the 90’s has persuaded some
people to make themselves available for a job.
De…nition 5 The number of unemployed people are all people that don’t have
a job. We denote this number by Ut : Similarly we denote the total number of
people with a job by Lt : Obviously Nt = Lt + Ut : We de…ne the unemployment
rate ut by
ut =
Ut
Nt
De…nition 6 The job losing rate bt is the fraction of the people with a job which
is laid o¤ during a particular time, period, say one month (it is crucial for this
de…nition to state the time horizon). The job …nding rate et is the fraction of
unemployed people in a month that …nd a new job.
Note that we use one month as our time horizon. This is due to the fact
that new employment data become available each month. The agency responsible for measuring and reporting labor market data is the BLS, the Bureau
of Labor Statistics. Between 1967 and 1993 the average job losing rate was
2.7% per month, whereas the average job …nding rate was 43%. The average
unemployment rate during this time period was about 6.2%.
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
126
In Figure 47 we plot the unemployment rate for the US from 1967 to 1999.14
We see that in recessions the unemployment rate increases, whereas in expansion
it decreases. A variable that shows such a behavior is called “countercyclical”:
it is high when real GDP is low (relative to trend) and it is low when real GDP
is high. Also note that unemployment in 2000 was on its lowest level since 1970.
Unemployment Rate for the US 1967-2001
12
1980-82
back-to-back recessions
11
1974-75
recession
Unemployment Rate
10
9
1990-91
recession
8
7
1970-71
recession
6
5
4
3
2
1970
1975
1980
1985
Year
1990
1995
Figure 4.20:
But where does it come from that in recessions the unemployment rate is
higher than in booms. At …rst sight this seems obvious: less is produced, hence
less workers are needed in recessions. But the net decline in job masks what
happens to gross ‡ows out of and into unemployment. High unemployment in
recessions can be due to the fact that more people are …red in recessions or that
less people are hired in recessions. So let us look more closely.
14 The unemployment rate is measured by the Bureau of Labor Statistics (BLS). Go to their
homepage at http://stats.bls.gov/top20.html if you want to have a look at the original data.
2000
4.4. UNEMPLOYMENT
127
Let us de…ne four more concepts that will help getting a handle at these
questions.
De…nition 7 We have the de…nition of the following concepts:
1. The gross job creation Crt between period t¡1 and t equals the employment
gain summed over all plants that expand or start up between period t ¡ 1
and t:
2. The gross job destruction Drt between period t ¡ 1 and t equals the employment loss summed over all plants that contract or shut down between
period t ¡ 1 and t:
3. The net job creation N ct between period t ¡ 1 and t equals Crt ¡ Drt :
4. The gross job reallocation Rat between period t ¡ 1 and t equals Crt + Drt :
Note the following things. Job creation and destruction measures are derived
from plant level information, i.e. by asking …rms. Unemployment data are
derived from household data, i.e. by asking individual households. Obviously
these data are related, but one set of data cannot be reconstructed from the
other. And both data sets are immensely important in discussing what goes on
in the labor market, so we will report facts from both data sets. Let us start
with the plant level data examined in detail by Davis et al. They use data from
all manufacturing plants in the US with 5 or more employees from 1963 to 1987.
In the years they have data available, there were between 300,000 and 400,000
plants. Studying these data four major …ndings emerge:
² Gross job creation Crt and job destruction Drt are remarkably large. In
a typical year 1 out of every ten jobs in manufacturing is destroyed and a
comparable number of jobs is created at di¤erent plants. This implies a
large number for gross job reallocation Rat , but a modest number for net
job creation N ct .
² Most of the job creation and destruction over a twelve-month interval
re‡ects highly persistent plant-level employment changes. This persistence
implies that most jobs that vanish at a particular plant in a given twelvemonth period fail to reopen at the same location within the next two
years.
² Job creation and destruction are concentrated at plants that experience
large percentage employment changes. Two-thirds of job creation and
destruction takes place at plants that expand or contract by 25% or more
within a twelve-month period. About one quarter of job destruction takes
place at plants that shut down.
² Job destruction exhibits greater cyclical variation than job creation. In
particular, recessions are characterized by a sharp increase in job destruction accompanied by a mild slowdown in job creation.
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
128
The last point answers our earlier question: in recessions the unemployment
rate goes up because unusually many people get …red, not because unusually
few people are newly hired.
4.4.2
Some Theory and the Natural Rate of Unemployment
Let us formulate a little descriptive model of the unemployment rate. Suppose
last month the number of unemployed people was Ut¡1 and the number of
employed people was Lt¡1 = Nt¡1 ¡ Ut¡1 : Suppose for simplicity that the labor
force grows at the population growth rate n; so that Nt = (1 + n)Nt¡1 : Let us
compute the unemployment rate at date t: How many people are unemployed
at date t? A fraction e (the job …nding rate) of the previously unemployed …nd
a job, so this leaves (1 ¡ e)Ut¡1 previously unemployed still unemployed. In
addition a fraction b (the job losing rate) of the people with work Lt lose their
job and augment the pool of unemployed. Hence
Ut
= (1 ¡ e)Ut¡1 + bLt¡1
= (1 ¡ e)Ut¡1 + b(Nt¡1 ¡ Ut¡1 )
Dividing both sides by Nt = (1 + n)Nt¡1 yields
ut
=
=
=
Ut
(1 ¡ e)Ut¡1
b(Nt¡1 ¡ Ut¡1 )
=
+
Nt
(1 + n)Nt¡1
(1 + n)Nt¡1
1¡e
b(1 ¡ ut¡1 )
ut¡1 +
1+n
1+n
1¡e¡b
b
ut¡1 +
1+n
1+n
This is a …rst order di¤erence equation that gives the unemployment rate this
month as a function of the unemployment rate of last month.
Remember that we loosely de…ned the natural rate of unemployment as
the unemployment rate in normal times. In the light of our simple theory we
now de…ne it more concisely as that unemployment rate that would prevail , if
the population growth rate n; the job …nding rate e and the job losing rate b
are at their normal, long run average level and would not change over time. We
can then de…ne the natural rate of unemployment as the steady state u¤ of our
di¤erence equation; as that unemployment rate that, in the long run, would be
attained in the economy, absent any shocks to n; e; b:
Let us solve for u¤ : Set ut¡1 = ut = u¤ to get
u¤
=
n+e+b ¤
u
1+n
=
u¤
=
1¡e¡b ¤
b
u +
1+n
1+n
b
1+n
b
n+e+b
4.4. UNEMPLOYMENT
129
Using the long run average numbers from before, i.e. b = 2:7%; e = 43% and
n = 0:09% (note that the time period is one month here). Hence, according to
the data, the natural rate of unemployment is 5:9%; which is almost identical
to the average unemployment rate during the last 30 years (which justi…es the
de…nition of the natural rate of unemployment as unemployment rate in normal
times).
In Figure 48 we show the dynamics of the unemployment rate. Suppose the
economy starts at an unemployment level u0 lower than the natural rate. Then,
barring any changes in b; e; n over time the unemployment rate approaches the
natural rate of unemployment, where it remains forever, if there are no changes
in job …nding or losing rates.
u
t
u=u
t t-1
slope (1-e-b)/(1+n)
b/(1+n)
u
0
u
1
u u*
2
u
t-1
Figure 4.21:
From our dynamic equation it is also clear what factors determine the natural unemployment rate: the natural unemployment rate increases with the job
loosing rate b and declines with the job …nding rate e: But what are the fac-
130
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
tors that determine these numbers. Answers to these questions could provide
us with some explanation of why, for example, Europe had consistently higher
unemployment in the last 15 years than the U.S. Since this does not appear to
be a temporary phenomenon, one may conjecture that the natural rate of unemployment is higher in Europe than in the U.S. So what are the determinants
of job …nding and job losing rates?
1. Unemployment Insurance: Workers that get laid o¤ receive unemployment
insurance. Length and generosity of unemployment insurance vary greatly
across countries. Whereas in the US the replacement rate (the fraction of
the last net wage that the unemployment insurance covers) is about 34%,
and this only for the …rst six months, in countries like Germany, France
and Italy the replacement rate is about 67%, with duration well beyond
the …rst year of unemployment. Given these di¤erent incentives to …nd a
new job it seems clear that job …nding rates are higher in the US than in
Europe. To the extent that voluntary quits do happen, job losing rates
may also be higher in Europe than in the US.
2. Minimum Wages: High minimum wages would mainly a¤ect job …nding
rates. If the minimum wage is so high that it makes certain jobs unprofitable, less jobs are o¤ered and job …nding rates decline. I would think
that in the US the minimum wage has no bite (at least now) since even industries which tend to be low-wage industries these days pay wages above
the minimum wage (for example fast food chains -you may imagine which
companies I mean). In other countries this may be more of a factor, but
I think the importance of the minimum wage is hugely overstated.
3. Union Wage Premiums: The classical insider-outsider theory posits that
unions maximize the well-being of their members, meaning high wages
and good working conditions in highly unionized sectors. To the extent
that …rms in these sectors have to pay higher wages, less jobs are profitable, reducing the possibility of …nding a good job for the outsiders, the
unemployed. Furthermore the prospect of …nding a good job may lead
unemployed workers to forgo other, not so good job o¤ers. Both e¤ects
reduce job …nding rates. Unionization is much more prevalent in Europe,
so this may explain part of the European unemployment dilemma, or “Eurosclerosis”.
4. E¢ciency Wages: The e¢ciency wage theory starts with the presumption
that worker-employer matches work best when the worker knows what he
has to lose. Therefore employers may want to pay more than the market
wage to make workers perform well, since, if they wouldn’t they know they
could get …red and lose their privilege to work for a high wage, with others
standing in line for the job. But higher wages mean less pro…table jobs.
Hence, although each existing job is well paid, there are relatively few of
those jobs, so although job losing rates are low (no voluntary quits), job
…nding rates are extremely low as everybody that sits on a good job does
everything to keep it.
4.4. UNEMPLOYMENT
131
So far we have discussed the main determinants of the natural rate of unemployment -roughly the unemployment rate in the long run. Now let’s turn to
the behavior of the unemployment rate over the business cycle.
4.4.3
Unemployment and the Business Cycle
So why is the unemployment rate high in a recession and low in a boom. The
plant level data from Davis et al. indicated that during recession it is not the
case that fewer than normal new workers are hired by establishments. What
is the case is that much more workers get …red during recessions than booms.
So gross job creation is relatively stable over the business cycle, whereas gross
job destruction moves strongly countercyclical: it is high in recessions and low
in booms. In severe recessions such as the 74-75 recession or the 80-82 back to
back recessions up to 25% of all manufacturing jobs are destroyed within one
year, whereas in booms the number is below 5%. For our model this implies
that in recessions b increases, whereas in booms it decreases.
The time a worker spends being unemployed also varies over the business
cycle, with unemployment spells being longer on average in recession years than
in years before a recession. Note that we said earlier that job creation rates do
not vary much over the business cycle. These two facts are not contradictory,
since in recessions there are much more people being laid o¤ and looking for a
new job, so even though …rms hire at a roughly normal pace it takes longer for
the average person to …nd a new job.
In Table 10 we show how the length of unemployment spells vary across the
business cycle. We show data from 2 years, 1989 and 1992. The year 1989 was
the last good year before the 90-92 recession (that cost George Bush his job),
the year 1992 is the last bad year of the recession.
Table 10
Unemployment Spell
< 5 weeks
5 - 14 weeks
15 - 26 weeks
> 26 weeks
1989
49%
30%
11%
10%
1992
35%
29%
15%
21%
We see that the average unemployment spells increase during a recession.
In the recession year 1992 one …fth of all unemployed worker was unemployed
for longer than half a year, whereas in the decent year 1989 only one out of 10
unemployed workers faced that situation. If we compare this to other countries,
for example in Germany, France or the Netherlands about two thirds of all
unemployed workers in 1989 were unemployed for longer than six months!!
Why are more people …red in recessions than in booms? Our Keynesian
business cycle model gives the answer: in recessions aggregate demand is below
potential GDP because prices are sticky, …rms need less workers to satisfy the
demand of their customers and therefore lay o¤ part of their workforce. In fact,
132
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
the relation between the unemployment rate and the GDP gap (the gap between
potential GDP and realized GDP (or aggregate demand)) is so strong that it
has its own name. Okun’s law, named after economist Arthur Okun, asserts
that for every percentage point that the unemployment rate is above its natural
rate, real GDP is about 2.5-3% below potential GDP. Note that Okun’s law is
a classical misnomer: it is not a law in that it always has to hold, it is more
like an empirical regularity that happens to roughly hold for the US in the last
50 years or so (and does not perform too badly for other countries as well).
Formally stated, Okun’s law says that
Y ¡ Yp
= ¡3(u ¡ u¤ )
Y
where Yp is potential GDP, Y is actual GDP, u¤ is the natural rate of unemployment and u is the actual unemployment rate. Note that it is not straightforward
to measure this relation in the data, since data on the natural rate of unemployment and on potential GDP are required. Nevertheless we plotted Okun’s law
from US data in Figure 49. On the x-axis we have the unemployment rate in
deviation from 6%, on the y-axis we have the percentage deviation of realized
GDP from long term trend (i.e. we identi…ed long term trend GDP with potential GDP, which is somewhat problematic, but can’t be easily avoided). We
see that indeed unemployment and output gap are negatively correlated, with
a coe¢cient of roughly 2.5-3. The data are from 1967 to 1999. We also so that,
although Okun’s law holds on average, it is far from a law in the strict sense:
in single years reality may be quite far from Okun’s law.
4.5
The Price Adjustment Process
Our model of the macroeconomy so far consists of two parts: the neoclassical
growth model that determines potential GDP and the Keynesian business cycle
model (the IS-LM model) that determines aggregate demand and hence real
GDP in the short run, under the assumption that the price level is …xed and
may not be at a level for which aggregate demand equals potential output. By
assuming price stickiness we could also explain unemployment.
The missing ingredient of our model is the process by which the price level,
assumed to be …xed in the short run, in the medium run adjusts so that eventually the economy returns to a situation in which aggregate demand equals
potential GDP, i.e. to the long run equilibrium of the economy.
It is obviously somewhat unrealistic to assume that …rms will not change
their prices if demand is below the output that they can produce. What we
have really assumed so far is that producers do not change their prices immediately in reaction, but rather meet all the demand by consumers at the
pre-speci…ed …xed price level. But in situations in which aggregate demand is
below potential output, by cutting prices …rms may increase demand for their
products and therefore improve their utilization of capacities and increase profits/reduce losses. Similarly, in situations in which aggregate demand is above
4.5. THE PRICE ADJUSTMENT PROCESS
133
Real GDP in Deviation from Potential GDP
Okuns Law for the US between 1967-99
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-4
-3
-2
-1
0
1
2
3
4
Unemployment Rate in Deviation from Natural Rate
Figure 4.22:
output, instead of increasing capacity to higher levels …rms may just increase the
price. The Keynesian model rules immediate price adjustment out, but rather
assumes that the price that …rms charge in the next period will react to the
gap between potential output and aggregate demand. Two caveats are in order:
…rst, it is really crucial to specify the length of a period. In order for the price
stickiness assumption of the Keynesian model to have any bite, the period has
to be long enough, say at least a quarter, or better a year. Second, we lead our
discussion from the perspective of a single …rm, but talk about macroeconomic
aggregates like aggregate demand and potential output. So the …rm that we are
implicitly invoking in our discussion is the “average …rm”. On average, …rms
are assumed to behave as described, which, when averaging over …rms, give rise
to the aggregate behavior. As with the aggregate consumption, the issue of
aggregation is a di¢cult one and we can’t explicitly deal with it in this course.
So for the price adjustment process we assume that the (percentage) change
5
6
134
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
in the price level from last period to today (the in‡ation rate) is a function of
the (percentage) gap between yesterday’s realized output (aggregate demand)
and potential output. The relationship can be written mathematically as
¼t = f
Yt¡1 ¡ Yp;t
Yp;t
(4.11)
where f is a positive constant. Remember that Yp;t was potential output. Now
Y
¡Y
from Okun’s law we can substitute for t¡1Yp;t p;t the term ¡3(ut¡1 ¡u¤ ); so that
(4:11) becomes
¼t = ¡g(ut¡1 ¡ u¤ )
(4.12)
where g = 3f is a constant. Equation (4:12) is called the Phillips curve, named
after British economist A.W. Phillips.15 It states that the in‡ation rate depends
negatively on the unemployment rate: higher unemployment brings about lower
in‡ation and vice versa. Up until the early seventies, the Phillips curve was probably the single most important empirical relationship between two macroeconomic variables and a great deal of research was done in writing down economic
models whose outcome was a relation like the Phillips curve. It also seems to
provide an intriguing problem for policy makers: if there is a trade-o¤ between
in‡ation and employment, then the policy maker has a choice: does she accept
a little more in‡ation in order to bring down the unemployment rate? And,
believing in the Keynesian business cycle model we know how to increase aggregate demand for a given price level: expansionary monetary or …scal policy will
do it. Figure 50 plots the Phillips curve (i.e. unemployment rates against in‡ation rates) for the year 1967-1973. One can clearly see the negative relationship
between the unemployment and the in‡ation rate - a relation that was also quite
stable in the 50’s and early 60’s. In Figure 51 we plot the Phillips curve for the
entire sample from 1967 to 1999. There is no systematic relationship between
in‡ation and unemployment rate whatsoever. For some, yet to be explained reason the Phillips curve broke down completely and has not reappeared (at least
not in its original form) since. Even worse, the 70’s were a period of so-called
“Stag‡ation”, high unemployment with high in‡ation. The two oil price shocks
provide a partial explanation for this misery, but expansionary monetary and
…scal policy to combat the high unemployment rates have done their share of
bringing in‡ation up.
One remark: both (4:11) and (4:12) are called the Phillips curve, which, given
Okun’s law, is justi…ed since unemployment and the gap between potential and
actual GDP have such a stable relationship.
In the late 60’s, before the simple Phillips curve actually broke down in the
data, Milton Friedman from Chicago and Edmund Phelps from Columbia criticized the Phillips curve on theoretical grounds, arguing that it ignores in‡ation
15 Phillips himself studied the relationship between percentage changes in wages and the
unemployment rate, rather than percentage change in prices. In his study for the UK from
1861-1957 he found that the Phillips curve …t the data extremely well.
4.5. THE PRICE ADJUSTMENT PROCESS
135
Phillips Curve for the US between 1967-73
0.08
0.07
Inflation Rate
0.06
0.05
0.04
0.03
0.02
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
Unemployment Rate in Deviation from Natural Rate
Figure 4.23:
expectations. The so called expectations-augmented Phillips curve reads (we
augment (4:11) rather than (4:12))
¼t = ¼et + f
Yt¡1 ¡ Yp
Yp
where ¼et is the in‡ation rate that households and …rms expect for period t in
period t ¡ 1: One of the justi…cations for including in‡ation expectations as
determinant for actual in‡ation goes like this: if …rms and unions expect the
in‡ation rate to be 5% rather than 2%; then in their bargaining over wages they
will agree on a 3% higher wage increase to compensate for higher in‡ation (which
so far is just expected, not realized in‡ation). But if wages rise by 3% more
(due to higher in‡ation expectations), then …rms, in order to get reimbursed
for the increasing costs, have to increase their prices for next period by 3%; so
that in fact realized in‡ation rises by 3%: In this sense do in‡ation expectations
1
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
136
Phillips Curve for the US between 1967-99
0.18
0.16
0.14
Inflation Rate
0.12
0.1
0.08
0.06
0.04
0.02
0
-0.02
-4
-3
-2
-1
0
1
2
3
4
Unemployment Rate in Deviation from Natural Rate
Figure 4.24:
help determine actual in‡ation. The work by Friedman and Phelps basically
marks the …rst instance in macroeconomics where expectations explicitly enter
a macroeconomic model.
But now we face a dilemma: we have to model how people form in‡ation
expectations. Early contributors to the literature, including Phelps and Friedman, made their lives somewhat easy and assumed “adaptive expectations”: the
expectation for the in‡ation rate for time t at time t ¡ 1 is assumed to equal the
actual in‡ation rate at date t ¡ 1 (or a weighted average of past in‡ation rates
in a more sophisticated model), so the Phillips curve becomes
¼t = ¼t¡1 + f
Yt¡1 ¡ Yp
Yp
If we plot this relationship for 1967 to 1999, as in Figure 52 we see that our
negative conclusion from Figure 52 disappears: we now can see somewhat of an
5
6
4.5. THE PRICE ADJUSTMENT PROCESS
137
(expectation-augmented) Phillips curve.
Inflation Rate in Deviation from Inflation Expectation
Phillips Curve for the US between 1967-99
0.08
0.06
0.04
0.02
0
-0.02
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Real GDP in Percentage Deviation from Potential GDP
Figure 4.25:
Two remarks: to model the expected in‡ation rate as being determined by
past experience is clearly unsatisfactory: it is the future that should count for
your expectations, not the past. A lot of work as been done to overcome this
shortcoming, because it assumes that households are somewhat dum in making
their in‡ation forecasts.
A second, even more important point is that the expectations-augmented
Phillips curve is not an easy-to-exploit policy menu anymore, as monetary and
…scal policy may a¤ect in‡ation expectations and hence the realized in‡ation
rate. That is what a lot of economists believe happened in the 70’s. By the
70’s households by and large had roughly …gured out how the government does
Keynesian business cycle policy. Given a recession people expected that the
government will try to exploit the simple Phillips curve and curb unemployment, taking into account a bit higher in‡ation. But this now entered in‡ation
138
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
expectations of private households and …rms: they expected higher in‡ation and
this resulted in higher in‡ation -in fact much higher than the simple Phillips
curve would have predicted, because of the expectation e¤ect. And this is the
crux with the simple Phillips curve: once the public understands it and the
governments’ intention to exploit it, it can’t be exploited any longer successfully. Realizing that people are not stupid after all was a painful experience for
policy makers and led to a complete paradigm shift in macroeconomics, away
from Keynesian macroeconomics to “Rational Expectations Macroeconomics”.
Real Business Cycle theory is the business cycle part of Rational Expectations
Macroeconomics. Before discussing this, however, let us proceed and see how
the Keynesian model, augmented by the Phillips curve, works.
4.5.1
Aggregate Demand, Potential GDP and the Price
Adjustment Process
For simplicity we keep the discussion to the simple Phillips curve
¼t = f
Yt¡1 ¡ Yp
Yp
Suppose that, as in Figure 53, we start at a situation with price level P0 and a
Y ¡Y
corresponding percentage output gap 0Yp p : Then the Phillips curve indicates
0
< 0 and therefore P1 < P0 ; i.e. the
that, since Y0 ¡ Yp < 0 we have ¼ 1 = P1P¡P
0
price level falls (a process that economists call de‡ation). But for price level
P1 we still have a gap between aggregate demand and potential GDP (although
smaller) since Y1 ¡ Yp < 0: So ¼2 < 0 and prices fall further until, absent
any other shocks, over time the economy approaches the point where aggregate
demand equals potential GDP.
The same analysis can be applied for the study of the e¤ects of monetary
and …scal policy on output and the price level and the adjustment process over
time.
4.5.2
Monetary Policy
We have already done half of the work in the IS-LM analysis. Suppose we
want to analyze the e¤ect of a monetary expansion; i.e. suppose that the FED
increases the money supply M s : What happens in our economy? Let’s proceed
in steps
1. Fix the price level P: An increase in M s shifts the LM-curve to the right.
The IS-curve does not shift. Hence aggregate demand Y increases, for the
given price level. We did this analysis before, nothing new here.
2. The previous argument is true for every given price level P: Hence, in
response to loosening monetary policy (increasing M s ) the aggregate demand curve shifts to the right (since aggregate demand is higher now for
any given price level).
4.5. THE PRICE ADJUSTMENT PROCESS
Price Level (P)
139
Potential Output
Gap between
Potential
Output and
Aggregate
Demand
P
0
P
1
P*
Aggregate
Demand
Output (Y)
Y Y
0 1
Y
p
Figure 4.26:
3. The rest of the analysis is new and uses the price adjustment process. In
Figure 54 we show what happens. The aggregate demand curve shifts to
the right, due to the monetary expansion. We assume that before the policy change the economy was at its long run equilibrium where aggregate
demand equals potential output and the associated long run equilibrium
price level is P0 : Since the price level is …xed, immediately after the expansion aggregate output jumps up to Y0 : Why this happens is answered
by the IS-LM model: for a …xed price level the increase in money supply
increases real money supply, the interest rate in the money market has to
fall, this induces higher investment, the consumption multiplier sets in and
aggregate expands. So far nothing new. Now the price adjustment via the
Phillips curve comes into play. After the monetary expansion aggregate
demand is above potential output, …rms will start increasing prices, say to
P1 aggregate demand declines to Y1 : This decline is due to the fact that
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
140
with an increasing price level the real supply of money declines, the interest starts increasing and investment falls below the initial level after the
money injection. The …rst round e¤ect is partly reversed. This process
of price adjustment continues until aggregate demand eventually equals
potential output again (this of course assumes that there are no further
policy changes which would again shift the aggregate demand curve). So
in the long run monetary policy is ine¤ective; in the long run the monetary
expansion just leads to an increase in the price level without any e¤ect on
real GDP, just as the classical dichotomy predicts. Along the adjustment
process however, the expansion in monetary policy does a¤ect real GDP.
Therefore sometimes potential output is also called the natural (rate of )
output since in the long run it is the level of output that the economy
will return to.
Price Level (P)
Potential Output
New Aggregate
P
1
P
0
Demand Curve
Old Aggregate
Demand Curve
Output (Y)
Y =Y
0 p
Y
2
Figure 4.27:
Y
1
4.6. STABILIZATION POLICY
4.5.3
141
Fiscal Policy
The analysis of a change in …scal policy is almost identical to that of monetary policy. Suppose there is a …scal expansion so that government spending
increases. Let us repeat our three steps of reasoning
1. Fix the price level P: An increase in G shifts the IS-curve to the right.
The LM-curve does not shift. Hence aggregate demand Y increases, for
the given price level. We did this analysis before, nothing new here.
2. The previous argument is true for every given price level P: Hence, in
response to expanding …scal policy (increasing G) the aggregate demand
curve shifts to the right (since aggregate demand is higher now for any
given price level).
3. Again the rest of the analysis is almost identical to the process induced
by a monetary expansion. Again refer to Figure 54. The aggregate demand curve shifts to the right, due to the …scal expansion. We assume
that before the policy change the economy was at its long run equilibrium
where aggregate demand equals potential output and the associated long
run equilibrium price level is P0 : Since the price level is …xed, immediately
after the expansion aggregate output jumps up to Y1 : Why this happens
is answered by the IS-LM model: for a …xed price level the increase in
government spending induces the multiplier process and hence increases
aggregate demand. In the process the interest rises and the crowding-out
of private investment reduces the …rst round e¤ect somewhat. So far nothing new. Now the price adjustment via the Phillips curve comes into play.
After the …scal expansion aggregate demand is above potential output,
…rms will start increasing prices, say to P1 aggregate demand declines to
Y1 : This decline is due to the fact that with an increasing price level the
real supply of money declines, the interest starts increasing and private
investment falls even further. The …rst round e¤ect is partly reversed.
This process of price adjustment continues until aggregate demand eventually equals potential output again (this of course assumes that there are
no further policy changes which would again shift the aggregate demand
curve). So in the long run also …scal policy is ine¤ective; in the long run
private investment is crowded out one for one by government spending.
Along the adjustment process however, the expansion in …scal policy does
a¤ect real GDP as before did monetary policy
Given that we (hopefully) have understood how monetary and …scal policy
work in the complete model, including the adjustment process, we can now
analyze both types of policies more systematically.
4.6
Stabilization Policy
What brings the economy away from the long run equilibrium in which aggregate
demand equals potential GDP? We saw in the last section that monetary and
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CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
…scal policy can do so, but why should they. After all, monetary and …scal
policy should be used to smooth out the business cycle, not to create them.
In this section we will …rst identify shocks hitting the economy that may
lead to deviations of aggregate demand form potential GDP and then discuss
how monetary and …scal policy can counteract these shocks to smooth out or
even prevent business cycles.
Hall and Taylor identify two sources of shocks to the economy
1. Aggregate Demand Shocks: these are shocks that shift the entire aggregate
demand curve and therefore move the economy temporarily out of its
short-run equilibrium. Examples include all sources of shocks that either
shift the IS-curve or the LM-curve: a decline in exports due to a recession
in other countries, say in Asia, a decline in autonomous consumption
spending due to a sudden drop in the stock market, a sudden decline
in real money demand (due to the arrival of credit cards, for example),
etc.
2. Price Shocks: these shocks do not shift the aggregate demand curve, but
induce a jump along the aggregate demand curve. The most famous examples of price shocks are the oil price shocks in the 70’s and early 80’s.
There are two steps to analyzing these shocks. First, we have to …nd out how
they a¤ect the position of the economy in the aggregate demand - potential GDP
graph, holding monetary and …scal policy …xed, and then we have to …gure out
what monetary or …scal policy can do to counteract them. In all our analyzes
we assume that we start at the long-run equilibrium in which aggregate demand
equals potential GDP and that the shocks are permanent.
4.6.1
Aggregate Demand Shocks and Their Stabilization
Every Shock that shifts the IS-curve to the right or the LM-curve to the right
shifts the aggregate demand curve to the right. Examples include increases in
exports, autonomous consumption or investment spending and so forth. Every
shock that shifts the IS-curve or the LM-curve to the left shifts the aggregate
demand curve to the left. Examples were given before. Since the major concern
about stabilization policy is avoiding severe recessions we focus on examples
which, without government intervention, would lead to recessions.
So suppose that there is a …nancial crisis in Asia in 1997 and as a result Japan
and other countries fall into a severe recession. This, in turn, leads to a decline
of U.S. exports to Asia in 1997. Suppose the US government and the Federal
Reserve Bank do not react. What happens? A decrease in exports shifts the
IS-curve to the left, therefore for each price level aggregate demand falls, hence
the aggregate demand curve shifts to the left. In the initial period of the decline,
1997, US real GDP drops from potential output Yp to Y0 (see Figure 55). The
US falls into a recession. Over time prices decline and the economy returns out
of the recession back to potential GDP. Can …scal (or monetary policy) be used
to avoid the recession? The answer is yes, and there is an easy recipe. Suppose
4.6. STABILIZATION POLICY
143
the government increases government spending G by exactly the amount by
which exports fall, immediately once Japan’s problem becomes public. Then,
as indicated in Figure 56, the shift of the aggregate demand curve to the left
is immediately o¤set by a shift back to the right, due to increased government
spending. The economy remains at potential GDP and full employment (the
unemployment rate equals its natural rate). Neither a recession nor an increase
in the price level (higher in‡ation has occurred).
Price Level (P)
Potential Output
P
0
Old Aggregate
P
1
Demand Curve
New Aggregate
Demand Curve
Output (Y)
Y Y
0 1
Y
p
Figure 4.28:
A similar story can be told when, for example, money demand increases.
This would shift the LM-curve to the left, hence the aggregate demand curve to
the left and push the economy into a recession if not the FED would increase
the money supply by exactly the right amount to counteract this initial shift
and avoid the recession. The fact that consumers, by developing a stronger
preference for holding cash, could cause a recession was a major concern for
Keynes.
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
144
Price Level (P)
Potential Output
Decline in Exports
Increase in Government
Spending
Old Aggregate
Demand Curve
New Aggregate
Demand Curve
Output (Y)
Figure 4.29:
So how can any sensible mind dispute the usefulness of stabilization policy?
Almost all economists would agree that it would be very desirable to eliminate
business cycles with monetary and …scal policy if we could. So what is the
criticism of Non-Keynesians of stabilization policy. It is two-fold:
1. First, all policies occur with lags. It takes time for politicians and central
bankers to realize when an adverse demand shock has hit the economy
(presumably politicians longer that central bankers). Then it takes time
to decide on an appropriate policy, because congress or the Federal Open
Market Committee (FOMC) has to assemble, deliberate and take a decision. Finally it takes time to implement the decision. Congress agreeing
on SDI does not mean that the orders for the …rst satellites go out the next
morning, there is a rather lengthy bureaucratic process involved. Given
these time lags the stabilization policy may hit the economy when it is already recovering from the recession and may create the opposite problem,
4.6. STABILIZATION POLICY
145
an overheated economy.
2. Not only timing is di¢cult, but also to …nd the right magnitude of the
policy is not a trivial task to …nd out. Friedrich August Hayek, an important neoclassical economist criticized the belief that politicians and central
bankers can overcome these practical problems and carry out e¤ective stabilization policy as hubris.
These points do not dispute the principle usefulness of stabilization policy,
but question its implementability. In contrast real business theorists question
the usefulness of stabilization policy, in particular monetary policy, altogether.
Both fractions of opponents suggest instead that the best the government and
the central bank can do is keep monetary policy transparent and stable so as
not to cause additional shocks over and above the ones already present in the
economy; and otherwise trust the magic of free markets to bring the economy
back to its long-term equilibrium.
4.6.2
Price Shocks and Their Stabilization
Now suppose an adverse shock hits the US economy that increases the price level
suddenly. The two oil price shocks in 1973-74 and 1979-80 are classic examples
of such events. Without any policy intervention Figure 57 shows what happens.
A sudden increase in the price level, brought about by the increase in oil prices,
lets the price level jump up from P ¤ to P0 . Output declines and the economy
goes into a recession. Over time the price level starts declining and output
comes back to potential output, but not without a recession in the meantime.
For the two speci…c episodes the numbers are the following: in 1973-74 the price
of gasoline increased by 35%, the CPI increased by 4.8% and real GDP from
1974 to 1975 shrank by 0.8%. For 1978-79 the gasoline price increased by 35%,
the CPI by 3.7% and real GDP shrank 0.5% from 1979 to 1980.
Can monetary or …scal do something in this case. Let us focus on monetary
policy. Suppose monetary policy does not react at all. Such a monetary policy
is called nonaccomodative. The situation is as in Figure 57: a severe recession,
but real GDP and the price level …nally come back to their initial levels P ¤ and
Yp:
Now suppose the FED reacts to the price shock and increases the money
demand. Such a policy is called accomodative. An increase in the money supply shifts the LM-curve to the right and hence the aggregate demand curve to
the left. Again, due to the price shock, the price level jumps up to P0 ; but
output declines only to Y0 ; a smaller decline compared with the nonaccomodative policy. This is shown in Figure 58. Hence the accomodative policy softens
the recession. But this comes at a price. Over time in the nonaccomodative
policy case the economy goes back to the original price level, whereas with the
accomodative price level it goes to a price level P1 > P ¤ , with higher in‡ation
rates (lower disin‡ation rates) along the way. Hence for a price shock not even
the Keynesians have an easy answer what to do: one may use monetary policy
to soften the recession, but this comes at the cost of higher in‡ation.
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
146
Price Level (P)
Potential Output
Gap between
Potential
Output and
Aggregate
Demand
P
0
P*
Aggregate
Demand
Output (Y)
Y
0
Y
p
Figure 4.30:
This concludes our discussion of the Keynesian business cycle model. To
recapitulate:
1. In the short run prices are sticky and aggregate demand determines GDP.
Aggregate demand may fall short of potential GDP in which case there is
unemployment.
2. In the medium run prices adjust, according to the Phillips curve. Prices
go up if aggregate demand is higher than potential output and go down if
aggregate demand is lower than potential output.
3. The adjustment process described by the Phillips curve in the long run
leads prices back to a level at which aggregate demand equals potential
GDP.
4. Active monetary and …scal policy are able to prevent or soften recessions
4.6. STABILIZATION POLICY
147
Price Level (P)
Potential Output
P
0
New Aggregate
Demand
P*’
P*
Old Aggregate
Demand
Output (Y)
Y
0
Y
p
Figure 4.31:
that may arise because of adverse aggregate demand or price shocks. Severe information and implementation problems have to be solved to use
these policies e¤ectively, though.
Overall the Keynesian model was unambiguously successful until the 70’s
when the Phillips curve broke down. Still today, a signi…cant fraction of practitioners and academic researchers trust the Keynesian model as their model of
business cycles, which, I guess rightly so, is re‡ected in macroeconomics textbooks, in which this model is still the workhorse to explain business cycles.
Before leaving business cycles completely, let’s have a brief look at a competing
paradigm for business cycle research.
148
4.7
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
Real Business Cycle Theory
Real business cycle theory builds on the basic insight that whatever is good for
explaining economic growth should be good for explaining business cycles. In
stark contrast to Keynesian business cycle theory it is assumed that prices are
fully ‡exible even in the short run and that aggregate demand never falls short
of potential GDP. How then can business cycles arise. The answer: technology
shocks. In particular, let us assume that total output Y in the economy can be
produced by the aggregate production function
Yt = zt Kt® L1¡®
t
where zt is a technology shock and equals 0:95 with probability 0:5 and equals
1:05 with probability 0:5: But this is not the end of the story. In real business
cycle theory we have households that live for, say 60 periods. These households
like to eat consumption goods ct and like to have leisure. They have 16 hours
of time in a day, 365 days a year and can decide how much of this time to work.
Let by N denote the total hours in a year that a household can work and by
lt the number of hours the household actually decides to work. Their utility
function is then
u(c0 ; N ¡ l0 ) + ¯u(c1 ; N ¡ l1 ) + : : : + ¯ T u(cT ; N ¡ lT )
So what happens if zt is low? The return from working (the real wage) is low
and households optimally decide to work less in the current period and more
later. So the e¤ect of the technology shock on output is ampli…ed by the labor
supply decision of the households. There is no involuntary unemployment in
this model: all households can work at the equilibrium wage, but this wage may
be so low that some people don’t …nd it worthwhile to work or work full hours.
By making the technology shocks really persistent (if today is bad, then
the likelihood of tomorrow being bad is very high) Kydland and Prescott (1982)
showed that around 70% of all business cycle ‡uctuation can be accounted for by
technology shocks and the ensuing e¤ects on labor supply. It is also important
to note that in this model there is no role for monetary policy, since the source
of the ‡uctuations, technology shocks, can’t be cured with monetary policy,
and conditional on having the shocks in the economy everybody, …rms and
households are behaving optimally and there are no market failures. Monetary
policy would just make matters worse.
My assessment of the model: is has a big methodological plus: it is soundly
based on the microeconomic principles of consumer and …rm maximization and
market clearing. No ad-hoc assumptions as in the Keynesian model are needed.
The big problem is: what are these technology shocks exactly and how can
they be identi…ed in the data? Given that these shocks are at the heart of the
model one would expect the RBC’ers to have a satisfactory answer for this, but
a convincing explanation is missing so far. For this course we have to leave it
here. But some of the material covered next, namely a more detailed look at
consumption and investment behavior shares the same principles with the RBC
4.7. REAL BUSINESS CYCLE THEORY
149
model: an explicit model of the decision problem that single households and
…rms face.
150
CHAPTER 4. BUSINESS CYCLE FLUCTUATIONS
Chapter 5
Microeconomic Foundations
of Macroeconomics
In this section we discuss the foundations for some of the behavioral equations
we wrote down when developing the IS-LM model. In particular we will subject
to more detailed analysis the aggregate consumption function and the aggregate
investment function. We will discuss how good or bad they perform empirically
and present other, more involved theories of consumption demand and investment demand.
5.1
Consumption Demand
Consumption is the sole end and purpose of all production [Adam
Smith]
But consumption is not only the …nal purpose of all economic activity, but
also constitutes about two thirds of GDP. Therefore economists have spent
a great deal of time trying to understand the determinants of consumption
demand. In this section we want to accomplish three things: we …rst want to
look more carefully at the data on consumption, we then want to investigate
empirically whether the simple Keynesian consumption function is in fact a
good approximation to reality. Finally we will look at an alternative model of
consumption demand, the life-cycle permanent income model, which is more
soundly based on microeconomic principle, to see what other determinants of
consumption beyond current disposable income there are.
5.1.1
Data on Consumption
In this section we will describe the basic facts about aggregate consumption. In
Figure 59 we plot real GDP, personal disposable income and total consumption
expenditures for 1959 to 1999. The data are in billion 1996 chained dollars,
151
152CHAPTER 5. MICROECONOMIC FOUNDATIONS OF MACROECONOMICS
i.e. real quantities and of quarterly frequency, i.e. we have one new observation
every quarter. What are the main observations? All three time series trend
upward, due to growth in population and due to economic growth as described
in the third section of this lecture. With respect to the cyclical properties of the
data, we see that there are substantial ‡uctuations of all variables around their
long term growth trend. More importantly, real GDP tends to ‡uctuate more
than both disposable personal income and consumption expenditures, which exhibit a smoother time series. We also see that consumption makes up the bulk
of GDP, a fraction that varies but is about 60-70% of GDP. Finally we see that
consumption seems to track personal disposable income rather closely. This observation, after all, was the motivation for the Keynesian aggregate consumption
function, which speci…ed consumption solely as a function of disposable income.
In the next section we will see how well this consumption function does in the
data.
GDP, Disposable Income, Consumption
8000
Real GDP, Personal Disposable Income, Consumption Expenditures, 1959-99
7000
6000
5000
Personal Disposable Income
4000
Real GDP
3000
Consumption
2000
1000
1955
1960
1965
1970
1975
1980
Year
Figure 5.1:
1985
1990
1995
2000
5.1. CONSUMPTION DEMAND
153
We can break consumption down into its components, expenditures on a)
nondurable consumption items, durable consumption items and services. In
1997, 13.6% of all consumption expenditures were accounted for by purchases of
consumer durables, 30.2% were due to purchases of nondurable goods and 56.2%
accrued to services. Over time, the share of consumption expenditures going
to services has increased substantially, as has the share of consumer durables,
whereas the share of nondurables has declined over time.
When plotting these components over time (see Hall/Taylor’s Figure 10.2)
we see that, although consumer durables are the smallest item among total
consumption expenditures, it is by far the most volatile part: purchases of
consumer durables are particularly low during recessions and particularly high
during booms, whereas purchases of nondurables and services are relatively stable over the business cycle. This fact is quite intuitive since consumer durables
(cars, furniture) have investment goods character; they require a large outlay,
are usually …nanced by credit and provide services for a prolonged period of
time. This investment good character of consumer durables has led economists
to think that we in fact mismeasure consumption by looking at consumption
expenditures. When you buy a new car in 1999, the whole price for the car
is counted in consumption expenditures for 1999. But the car delivers services
for many years (unless you bought a real lemon). Therefore from a theoretical
point of view the price of the car should be split up into, say, ten pieces (for ten
years of usage), and only that part of the price that corresponds to the services
that the car provides in the …rst year should be counted as consumption expenditures. This method is obviously somewhat hard to implement in practice.
But once implemented it seems almost certain that expenditures on consumer
durables, measured this way, would be way less volatile than it is now with the
conventional measurement technique. Hence consumption expenditures would
be even smoother over the business cycle that they already are.
5.1.2
The Keynesian Aggregate Consumption Function and
the Data
Now we will look in more detail at the Keynesian aggregate consumption function. Remember that in its simplest form it was given as
C
= a + b(1 ¡ ¿ )Yh
= a + bYd
where Yd is disposable income. Let us see how the function does in practice. In
Figure 60 we plot total consumption expenditures as measured in the data and
consumption expenditures predicted by the aggregate consumption function for
1959 to 1999. All data is in billion 1996 US dollars.
We pick the parameters (a; b) in such a way as to minimize the di¤erence
between actual and predicted data, i.e. we give the aggregate consumption
function its best shot. Estimating the parameters that gives the best …t1 yields
1 Technically,
we estimate the parameters by ordinary least squares, i.e. in order to minimize
154CHAPTER 5. MICROECONOMIC FOUNDATIONS OF MACROECONOMICS
6000
Consumption Expenditures, Actual and Predicted, 1959-99
Consumption, Actaul and Predicted
5500
5000
4500
4000
3500
3000
2500
2000
Consumption Expenditures in the Data
1500
Consumption Expenditures Predicted by Aggregate Consumption Function
1000
1955
1960
1965
1970
1975
1980
1985
1990
1995
Year
Figure 5.2:
(a; b) = (¡106; 0:923): The fact that the estimated a is negative is a bit bothersome, but let’s ignore this for a second. The marginal prospensity to consume
is estimated at 0:92; i.e. on average the US households spend 92 cents out of
every additional dollar disposable income.
Figure 60 may indicate that the Keynesian consumption function does rather
well. But the magnitudes on the y-axis a re substantial. Let us in Figure 61
plot the deviation of actual consumption expenditures from the ones predicted
by the consumption function.
We see that the deviations are quite sizeable, amounting to under-or overestimation of actual consumption by up to 200 billion US dollar, or about 6%
of total consumption expenditures in given periods. Given that consumption
expenditures make up about two thirds of total real GDP, this under- or overprediction may easily lead to an under -or overprediction of real GDP by 3-4%.
the sum of squared deviations of actual data from predicted ones.
2000
5.1. CONSUMPTION DEMAND
155
Deviation of Consumption Expenditures from Predicted, 1959-99
250
200
Consumption Deviations
150
100
50
0
-50
-100
-150
-200
1955
1960
1965
1970
1975
1980
Year
1985
1990
1995
Figure 5.3:
Given that economic policy is carried out based on economic forecasts, these are
huge numbers because the prediction may easily show a healthy economy when
in fact the true data afterwards indicate that the economy was well under way
into a recession. For example, in the 70’s and early 80’s realized consumption
expenditures were quite smaller than predicted ones. According to the prediction …scal or monetary expansionary policy was not called for, but ex-post it
turned out to be the case that the economy was or was going into a recession and
an expansionary …scal or monetary policy, uncalled for based on the prediction,
may have been able to prevent or at least soften the recession. The apparent
malfunction of the Keynesian aggregate consumption function, plus its weak
foundation on microeconomic principles of consumer optimization led to the
development of an alternative model of consumption, the life cycle/permanent
income model of consumption.
2000
156CHAPTER 5. MICROECONOMIC FOUNDATIONS OF MACROECONOMICS
5.1.3
The Life Cycle/Permanent Income Model of Consumption
In this subsection we will present and then apply a simple version of AndoModigliani’s life-cycle model and Friedman’s permanent income model. The
simple model we present is due to Irving Fisher (1867-1947), and the life-cycle
as well as the permanent income model are relatively straightforward generalizations of Fisher’s model. At the end of this section we will apply Fisher’s model
to the analysis of how a social security system a¤ects consumption demand of
households.
Consider a single individual, for concreteness call this guy Freddy Krueger.
Freddy lives for two periods (you may think of the length of one period as 30
years, so the model is not all that unrealistic). He cares about consumption in
the …rst period of his life, c1 and consumption in the second period of his life,
c2 : His utility function takes the simple form
u(c1 ; c2 ) = log(c1 ) + ¯ log(c2 )
(5.1)
where the parameter ¯ is between zero and one and measures Freddy’s degree
of impatience. A high ¯ indicates that consumption in the second period of his
life is really important to Freddy, so he is patient. On the other hand, a low ¯
makes Freddy really impatient. In the extreme case of ¯ = 0 Freddy only cares
about his consumption in the current period, but not at all about consumption
when he is old.
Freddy has income y1 > 0 in the …rst period of his life and y2 ¸ 0 in the
second period of his life (we want to allow y2 = 0 in order to model that Freddy
is retired in the second period of his life and therefore, absent any social security
system, has no income in the second period). Income is measured in units of
the consumption good, not in terms of money. As with the Keynesian aggregate
production function we abstract from money in this analysis. Freddy also starts
his life with some initial wealth A ¸ 0; due to bequests that he received from his
parents. Again A is measured in terms of the consumption good, not in terms
of money. Freddy can save some of his income in the …rst period or some of his
initial wealth, or he can borrow against his future income y2 : We assume that
the interest rate on both savings and on loans is equal to r; and we denote by s
the saving (borrowing if s < 0) that Freddy does. Hence his budget constraint
in the …rst period of his life is
c1 + s = y1 + A
(5.2)
Freddy can use his total income in period 1, y1 + A either for eating today c1 or
for saving for tomorrow, s: In the second period of his life he faces the budget
constraint
c2 = y2 + (1 + r)s
(5.3)
i.e. he can eat whatever his income is and whatever he saved from the …rst
period. The problem that Freddy faces is quite simple: given his income and
5.1. CONSUMPTION DEMAND
157
wealth he has to decide how much to eat in period 1 and how much to save
for the second period of his life. The is a very standard decision problem as
you have studied left and right in microeconomics, with the only di¤erence that
the goods that Freddy chooses are not apples and bananas, but consumption
today and consumption tomorrow. In micro our people usually only have one
budget constraint, so let us combine (5:2) and (5:3) to derive this one budget
constraint, a so-called intertemporal budget constraint, because it combines
income and consumption in both periods. Solving (5:3) for s yields
s=
c2 ¡ y2
1+r
and substituting this into (5:2) yields
c1 +
c2 ¡ y2
= y1 + A
1+r
or
c1 +
c2
y2
= y1 +
+A
1+r
1+r
(5.4)
Let us interpret this budget constraint. We have normalized the price of the
consumption good in the …rst period to 1 (remember from micro that we could
multiply all prices by a constant and the problem of Freddy would not change.
1
; which is also the relative
The price of the consumption good in period 2 is 1+r
price of consumption in period 2; relative to consumption in period 1: Hence the
gross interest rate 1 + r is really a price: it is the relative price of consumption
goods today to consumption goods tomorrow (note that this is a de…nition). So
the intertemporal budget constraint says that total expenditures on consumpc2
; measured in prices of the period 1 consumption good, have
tion goods c1 + 1+r
y2
; measured in units of the period 1 consumption
to equal total income y1 + 1+r
y2
good, plus the initial wealth of Freddy. The sum of all labor income y1 + 1+r
y2
is sometimes referred to as human capital. Let us by I = y1 + 1+r + A denote
Freddy’s total income, consisting of human capital and initial wealth.
Now we can analyze Freddy’s consumption decision. He wants to maximize
his utility (5:1); but is constrained by the intertemporal budget constraint (5:4):
To let us solve
max flog(c1 ) + ¯ log(c2 )g
c1 ;c2
s:t:
c2
c1 +
1+r
= I
One option is to use the Lagrangian method, which you should have seen in
Microeconomics, and you should try it out for yourself. The second option is to
substitute into the objective function for c1 to get
¾
½ µ
¶
c2
max log I ¡
+ ¯ log(c2 )
c2
1+r
158CHAPTER 5. MICROECONOMIC FOUNDATIONS OF MACROECONOMICS
This is an unconstrained maximization problem. Let us take …rst order conditions with respect to c2
1
¡ 1+r
¯
=0
c2 +
I ¡ 1+r
c2
or
c2
1+r
c2
(1 + ¯)c2
c2
Since c1 = I ¡
c2
1+r
µ
¶
c2
= ¯ I¡
1+r
= ¯ ((1 + r)I ¡ c2 )
= ¯(1 + r)I
¯
(1 + r)I
=
1+¯
¯
=
((1 + r)(y1 + A) + y2 )
1+¯
(5.5)
(5.6)
we …nd
c1
c2
1+r
¯
= I¡
I
1+¯
I
=
1+¯
¶
µ
1
y2
+A
=
y1 +
1+¯
1+r
= I¡
(5.7)
Since saving s equals y1 + A ¡ c1 we …nd
s=
¯
y2
(y1 + A) ¡
1+¯
(1 + ¯)(1 + r)
which may be positive or negative, depending on how high …rst period income
and initial wealth is compared to second period income. So Freddy’s optimal
1
of total lifetime
consumption choice today is quite simple: eat a fraction 1+¯
income I today and save the rest for the second period of your life.
So on what variables does current consumption depend on? According to our
model it is income today, income next period, initial wealth A and the interest
rate r: All those variables, apart from income today, did the simple Keynesian
aggregate consumption function ignore. But even the simplest model that has
consumers deciding optimally on their consumption predicts that future income,
the intertemporal price of consumption (the interest rate) and initial wealth
holdings should enter the consumption function. More complex models based
on consumer optimization add even more variables.
5.1. CONSUMPTION DEMAND
159
For now let us stick with our simple model. As in microeconomics we can
analyze the decision problem of Freddy graphically, using budget lines and indi¤erence curves. First we plot the budget line (5:4): This is the combination
of all (c1 ; c2 ) Freddy can a¤ord. We draw c1 on the x-axis and c2 on the y-axis.
Looking at the left hand side of (5:4) we realize that the budget line is in fact
a straight line. Now let us …nd two points on the line. Suppose c2 ; i.e. Freddy
y2
does not eat in the second period. Then he can a¤ord c1 = y1 + A + 1+r
is the
y2
a a
…rst period, so one point on the budget line is (c1 ; c2 ) = (y1 + A + 1+r ; 0): Now
suppose c1 : Then Freddy can a¤ord to eat c2 = (1+r)(y1 +A)+y2 in the second
period, so a second point on the budget line is (cb1 ; cb2 ) = (0; (1+ r)(y1 +A)+ y2 ):
Connecting these two points with a straight line yields the entire budget line.
We can also compute the slope of the budget line as
cb2 ¡ ca2
cb1 ¡ ca1
(1 + r)(y1 + A) + y2
´
³
y2
¡ y1 + A + 1+r
slope =
=
= ¡(1 + r)
Hence the budget line is downward sloping with slope (1 + r): Now let’s try
to remember so microeconomics. The budget line just tells us what Freddy
can a¤ord. The utility function (5:1) tells us how Freddy values consumption
today and consumption tomorrow. Remember that an indi¤erence curve is a
collection of bundles (c1 ; c2 ) that yield the same utility, i.e. between which
Freddy is indi¤erent. Let us …x a particular level of utility, say u (which is just
a number). Then an indi¤erence curve consists of all (c1 ; c2 ) such that
u = log(c1 ) + ¯ log(c2 )
Solving for c2 yields
log(c2 ) =
c2
u ¡ log(c1 )
¯
= e
u¡log(c1 )
¯
u
¡1
= e ¯ c1¯
Hence as c1 becomes bigger and bigger, c2 approaches 0: As c1 approaches 0; c2
becomes bigger and bigger. See Figure 62 for a typical shape of an indi¤erence
curve. The slope of the indi¤erence curve is given as
dc2
dc1
=
=
=
¡1
¡1 u¯ ¡1
e c1¯
¯
¡1 u¯ ¡1
e c1¯
¯c1
¡c2
¯c1
160CHAPTER 5. MICROECONOMIC FOUNDATIONS OF MACROECONOMICS
c
2
Slope c / βc
2
1
(1+r)(y +A)+y
1
2
Slope
1+r
c*
2
Indifference curve
log(c )+βlog(c )=constant
1
2
y
2
Saving s
c*
1
Budget line
y +A
1
y +A+y /(1+r)
1
2
c
1
Figure 5.4:
Incidently this slope equals the (negative of the) marginal rate of substitution
(as always)
MRS =
c2
uc1 (c1 ; c2 )
=
uc2 (c1 ; c2 )
¯c1
where uci indicates the partial derivative of u with respect to ci : From the …gure
we note that Freddy should pick his consumption such that the indi¤erence curve
is tangent to the budget line. This means that at the optimal consumption choice
the slope of the indi¤erence curve and the budget line are equal or
uc1 (c1 ; c2 )
= 1+r
uc2 (c1 ; c2 )
uc1 (c1 ; c2 ) = (1 + r)uc2 (c1 ; c2 )
(5.8)
This equation has a nice interpretation. At the optimal consumption choice the
5.1. CONSUMPTION DEMAND
161
cost, in terms of utility, os saving one more unit should be equal to the bene…t
of saving one more unit (if not, Freddy should either save more or less). But
the cost of saving one more unit, and hence one unit lower consumption in the
…rst period, in terms of utility equals uc1 (c1 ; c2 ): Saving one more unit yields
(1 + r) more units of consumption tomorrow. In terms of utility, this is worth
(1+ r)uc2 (c1 ; c2 ): Equality of cost and bene…t implies (5:8): Using the particular
from of the utility function yields as the condition for an optimal consumption
choice
c2
=1+r
¯c1
This, together with the intertemporal budget constraint (5:4) can be solved
for the optimal consumption choices, which obviously gives the same result as
before.
Income and Interest Changes
Now we can investigate how changes in today’s income y1 ; next period’s income
y2 and initial wealth A change the optimal consumption choice. From (5:7) and
(5:5) we see that both c1 and c2 increase with increases in either y1 ; y2 or A:
In contrast to the Keynesian consumption function, an increase in tomorrow’s
income will increase today’s consumption as well as tomorrow’s consumption.
The marginal prospensity to consume out of today’s income or initial wealth is
dc1
1
dc1
=
>0
=
dA
dy1
1+¯
and the marginal prospensity to consume today out of tomorrows income equals
1
dc1
>0
=
dy2
(1 + ¯)(1 + r)
We see this e¤ect graphically in Figure 63. The e¤ect of increases in both c1
and c2 in reaction to increases in y1 ; y2 or A is called an income e¤ect.
More complicated are changes in the interest rate, since this will entail income e¤ects and substitution e¤ects. A substitution e¤ect comes about since the
gross interest rate 1 + r is the relative price of consumption in period 1; relative
to consumption in period 2: So as the interest changes, not only does income
y2
changes), but also the relative price of consumption goods
change (because 1+r
in the two periods.
Let us analyze an increase in the interest rate from r to r0 and let us start
graphically. What happens to the curves in Figure 62 as the interest rate increases? The indi¤erence curves do not change, as they do not involve the
interest rate. But the budget line changes. Since we assume that the interest
rate increases, the budget line gets steeper. And it is straightforward to …nd
a point on the budget line that is a¤ordable with old and new interest rate.
Suppose Freddy eats all his …rst period income and wealth in the …rst period,
162CHAPTER 5. MICROECONOMIC FOUNDATIONS OF MACROECONOMICS
c
2
Slope c / βc
2
1
(1+r)(y +A)+y
1
2
Slopes
1+r
’
c*’
2
c*
2
y
2
Budget lines
c*
1
c*’
1
y +A
1
1
c
Figure 5.5:
c1 = y1 + A and all his income in the second period c2 = y2 ; in other words,
he doesn’t save or borrow. This consumption pro…le is a¤ordable no matter
what the interest rate (as the interest rate does not a¤ect Freddy as he neither
borrows nor saves). This consumption pro…le is sometimes called the autarkic
consumption pro…le, as Freddy needs no markets to implement it: he just eats
whatever he has in each period. Hence the budget line tilts around the autarky
point and gets steeper, as shown in Figure 64.
Consumption in period 2 increases and consumption in period 1 decreases.
Saving increases. This is also apparent from equations (5:7) and (5:5). What
is the reason? There are two e¤ects from an increase in the interest rate. First
there is an income e¤ect: if Freddy is a saver (as we assume in the picture)
then a higher interest rate, for given savings, increases his income in the second
period. The in‡uence of this e¤ect on both c1 and c2 is positive and is called the
income e¤ect. Also, an increase in the interest rate makes consumption today
5.1. CONSUMPTION DEMAND
163
c*’
2
Slope c / βc
2
1
Slope
1+r
c*
2
y
2
Budget lines
c*’ c*
1
1
y +A
1
c
1
Figure 5.6:
more expensive compared to consumption tomorrow, so individuals substitute
substitute consumption today with consumption tomorrow. This is the substitution e¤ect: it is negative for c1 and positive for c2 : Hence c2 unambiguously
increases; for c1 it depends on the size of the income and the substitution e¤ect.
For the particular utility function we chose and the assumptions on income we
made c1 decreases and saving increases. Note that if the consumer is a borrower
then the income e¤ect is negative rather than positive: a higher interest rate
increases the interest payments on his loan. The substitution e¤ect works as
before. Hence for a borrower we can conclude that consumption in the …rst
period declines in a response to an increase in the interest rate (both income
and substitution e¤ects are negative). Consumption in the second period may
increase or decline, depending on whether the positive substitution e¤ect is
stronger or weaker than the negative income e¤ect (again for our assumptions
the substitution e¤ect is stronger).
164CHAPTER 5. MICROECONOMIC FOUNDATIONS OF MACROECONOMICS
So far we assumed that Freddy could borrow freely at interest rate r: But
we all (at least some of us) know that sometimes we would like to take out a
loan from a bank but are denied it. Let us analyze how the presence of so-called
borrowing constraints a¤ect the consumption choice. So let us assume that
Freddy cannot borrow, so he is constrained by s ¸ 0: Obviously if Freddy is a
saver anyway, nothing changes for him since the constraint on borrowing is not
binding. The situation is di¤erent if, without the borrowing constraint, Freddy
would be a borrower. Now with the borrowing constraint, the best he can do is
set c1 = y1 + A; c2 = y2 : He would like to have even bigger c1 ; but since he is
borrowing constrained he can’t bring any of his second period income forward by
taking out a loan. Note that in this situation Freddy’s …rst period consumption
does not depend on second period income or the interest rate. In particular, if
y2 goes up, c1 remains unchanged if Freddy is borrowing-constrained. This can
be seen from Figure 65.
c
2
Slope c / βc
2
1
Slope
1+r
c’ = y’
2
2
Budget line
c =y
2 2
Indifference
Curves
c = y +A
1 1
Figure 5.7:
c
1
5.1. CONSUMPTION DEMAND
165
The budget line with the presence of borrowing constraints has a kink at
(y1 + A; y2 ): For c1 < y1 + A we have the usual budget constraint, as here
s > 0 and the borrowing constraint is not binding. But with the borrowing
constraint Freddy cannot a¤ord any consumption c1 > y1 + A; so the budget
constraint has a vertical segment at y1 + A; because regardless of what c2 ;
the most Freddy can a¤ord in period 1 is y1 + A: If Freddy was a borrower
without the borrowing constraint, then his optimal consumption is at the kink.
And with an increase of second period income y2 ; Freddy just increases second
period consumption, with …rst period consumption unchanged. Also not that
(as long as the borrowing constraint remains binding, Freddy will consume every
cent of an income increase in the …rst period immediately in the …rst period,
i.e. his marginal prospensity to consume out of current income is 1 if Freddy is
borrowing-constrained.
Hence with borrowing constraints the consumption function of Freddy looks
much more Keynesian: consumption only depends on current income and is
independent of the interest rate and future income. Since in the overall economy
there are individuals that face borrowing constraints and others who do not, we
can expect the aggregate consumption function to depend heavily on current
income, but also on future income and the interest rate. We saw in the …rst
section how the Keynesian aggregate consumption function fared with respect
to the data. A huge amount of empirical work has been done to test more
elaborate versions of the simple Fisher model. We come back to this later.
Borrowing constraints may be one explanation of why the Japanese saving
rate is higher than the US saving rate. Individuals that are borrowing constrained consume less (and save more) than they otherwise would, without the
borrowing constraint. The biggest expense, particularly for young families is
usually the purchase of the …rst home. In the US, a down payment of about
10% on a house is quite common, the rest is borrowed. In Japan down payments of 40% or higher are common, hence households are much more borrowing constrained in Japan than in the US, at least with respect of this particular
transaction. Hence Japanese have to save more in advance to …nance home purchases, which explains part of their higher saving rate. Note that, although a
lot of economists argue that a high saving rate is good for growth, the particular
feature of the Japanese economy that brings the higher saving rate about (high
down payments) is usually not regarded as desirable.
Social Security in the Life-cycle model
Now we use the model to analyze a policy issue that has drawn large attention in
the public debate. The personal saving rate -the fraction of disposable income
that private households save- has declined from about 7-10% in the 60’s and
70’s to 2.1% in 1997. Since saving provides the funds for investment a lower
saving rate, so a lot of people argue, harms growth be reducing investment.2
Some economists argue that the expansion of the social security system has
2 This argument obviously ignores increased government saving in the US and the increased
in‡ow of foreign funds into the US.
166CHAPTER 5. MICROECONOMIC FOUNDATIONS OF MACROECONOMICS
led to a decline in personal saving. We want to analyze this claim using our
simple model. We look at a pay-as-you go social security system, in which the
currently working generation pays payroll taxes, whose proceeds are used to pay
the pensions of the currently retired generation. The key is that current taxes are
paid out immediately, and not invested. We make the following simpli…cations
to our model. We interpret the second period of a person’s life as his retirement,
so in the absence of social security he has no income apart from his savings, i.e.
y2 = 0: For simplicity we also assume A = 0: Without social security3 we have
from before
c1
=
c2
=
s =
y1
1+¯
¯(1 + r)y1
1+¯
¯y1
1+¯
Now suppose we introduce a pay as-you-go social security system. As a consequence in the …rst period of his life Freddy has to pay payroll taxes. Let us
assume that the tax rate on labor income is ¿ ; so Freddy’s after tax wage is
(1 ¡ ¿ )y1 : Note that currently the payroll tax for social security is 15:3%; paid
half by the employer and half by the employee. This includes contributions
to medicare and disability insurance. In the second period of his life he now
receives social security payments SS: Let us assume that the population grows
at rate n; so when Freddy is old there are (1 + n) as many young guys around
compared when he was young. Also assume that pre-tax-income grows at rate
g; so the income of the young people, when Freddy is old, equals (1 + g)y1 the
income that Freddy had when he was young. Finally assume that the social security system balances its budget, so that total social security payments equal
total payroll taxes. This implies that
SS = (1 + g)(1 + n)¿ y1
Freddy bene…ts from the fact that population and wages grow over time since
when he is old there are more people around to pay his pension from higher
wages of theirs. Now Freddy has the budget constraints
c1 + s = (1 ¡ ¿ )y1
c2 = (1 + r)s + SS
Again we can write this as a single intertemporal budget constraint
c1 +
c2
SS
= (1 ¡ ¿)y1 +
=I
1+r
1+r
(5.9)
3 Conceptually a fully funded system is as if everybody saves for him- or herself. We abstract
from uncertainty about the length of life and hence from insurance aspects of a socail security
system.
5.1. CONSUMPTION DEMAND
167
Maximizing (5:1) subject to (5:9) yields, by the same logic as before
c1
=
c2
=
I
1+¯
¯
(1 + r)I
1+¯
Now we use the fact that SS = (1 + g)(1 + n)¿ y1 since the budget of the social
security system has to be balanced. Therefore
I
SS
1+r
(1 + g)(1 + n)¿ y1
= (1 ¡ ¿ )y1 +
1+r
µ
¶
(1 + g)(1 + n)
= y1 ¡ 1 ¡
¿y1
1+r
= y~1
= (1 ¡ ¿ )y1 +
where we de…ned y~1 to be the mess on the right hand side. Hence
c1
=
c2
=
y~1
1+¯
¯
(1 + r)~
y1
1+¯
Comparing this with the result from before we see that consumption in both
periods is higher with social security than without if and only if y~1 > y1 ; i.e. if
> 1: Hence people are better o¤ with social security if
and only if (1+g)(1+n)
1+r
(1 + g)(1 + n) > 1 + r
This condition makes perfect sense. If people save by themselves for their retirement, the return on their savings equals 1 + r: If they save via a social
security system ( are forced to do so), their return to this forced saving consists
of (1 + n)(1 + g) (more people with higher wages pay for the old guys). This
result makes clear why a pay-as-you-go social security system may make sense
in some countries (those with high population and wage growth), but not in
others, and that it may have made sense in the US in the 60’s and 70’s, but not
in the 90’s. Just some numbers: the current population growth rate is about
n = 1%; growth of wages and salaries is about g = 2%; and the average return
on the stock market for the last 100 years is about r = 7% (and obviously much
higher recently). This is the basis for many economists to call for a reform of the
social security system, most prominently Martin Feldstein, chief of the National
Bureau of Economic Research, the most important economic think tank in the
US. There is an intense debate over how one could privatize the social security
system, i.e. create individual retirement funds so that basically each individual
would save for her own retirement, with return 1+r > (1+g)(1+n): The biggest
168CHAPTER 5. MICROECONOMIC FOUNDATIONS OF MACROECONOMICS
problem is one missing generation: at the introduction of the system in the 30’s
there was one old generation that received social security but never paid taxes
for it. Now we face the dilemma: if we abolish the pay-as-you go system, either
the currently young pay double, for the currently old and for themselves, or we
just default on the promises for the old. Both alternatives seem to be di¢cult
to implement politically and problematically from an ethical point of view. The
government could pay out the old by increasing government debt, but this has
to be …nanced by higher taxes in the future, i.e. by currently young and future
generations. Hence this is problematic, too. The issue is very much open, and
since I did research on this issue in my thesis I am happy to talk to whoever is
interested in more details.
But back to the original question: what does pay-as-you go social security
do to saving? Without social security saving was given as
s=
¯y1
1+¯
with social security it is given by
s=
¯(1 ¡ ¿ )y1 ¡
1+¯
SS
1+r
and obviously private saving falls. Note that the social security system as part
of the government does not save, it pays all the tax receipts out immediately
as pensions. So saving unambiguously goes down with social security. To the
extent that this harms investment, capital accumulation and growth the payas-you-go social security system may have substantial negative long-run e¤ects,
over and above the e¤ects due to its lower return as compared to private saving
for retirement.4
This analysis shows that, although or model is very simple, it is quite powerful in addressing an array of interesting policy questions. Now we turn to a
description of more involved models of consumption choice that build on this
simple model.
Extensions of the Basic Model
In the mid-50’s Franco Modigliani, jointly with Albert Ando and Richard Brumberg developed the life-cycle hypothesis of consumption. The basic insight of
the simple model above builds the corner stone of the life-cycle hypothesis: individuals want a rather smooth consumption pro…le over their life, but their
labor income varies substantially over their lifetime, starting out low, increasing
up until about the 50’th year of a person’s life and then slightly declining until
65, with no labor income after 65. The life-cycle hypothesis then states that by
saving (and borrowing) individuals achieve it that they turn a very nonsmooth
labor income pro…le into a very smooth consumption pro…le. Therefore the life
4 In this simple model there is really no bene…cial role for a pay-as-you-go system. This
changes as one introduces mortality risk or income distribution considerations into the model.
5.1. CONSUMPTION DEMAND
169
cycle hypothesis predicts that current consumption (as well as future consumption) depends on total lifetime income and given initial wealth, as in the simple
model. The life-cycle model stresses the importance of saving: in particular saving should follow a very pronounced life-cycle pattern with little saving (or even
borrowing) in the early periods of an economic life (which usually is assumed
to begin around 16-20), signi…cant saving in the high earning years from 35-50
and dissaving in retirement years as the accumulated wealth is used to provide
consumption in old age.
The life-cycle version of the model seems to fare quite well when confronted
with data from the Consumer Expenditure Survey or other data sources that
record individual households incomes and consumption expenditures. One empirical fact that puzzles life-cyclers is the observation that older, retired household do not dissave to the extent predicted by the theory. There are several
explanation for this puzzle. One is that, contrary to the assumptions of the
theory, individuals are altruistic and want to leave bequests to their children.
A di¤erent explanation is that it is highly uncertain how long one lives and
whether one stays healthy. If older households are extremely risk-averse and
fear the possibility of living very long and hence not having saved enough -or
if they fear the risk of getting sick and the resulting huge medical bills, then it
may be rational to keep almost all savings intact to be prepared for this very
unlikely, but very deeply feared event.
Milton Friedman’s permanent income hypothesis is also an immediate extension of the basic model discussed above. Instead of stressing the life-cycle
aspect of consumption and saving Friedman focussed on the fact that future
labor income is uncertain to a certain degree. He posited that the income of
an individual household, y consists of a permanent part, yp and a transitory
part yt ; i.e. y = yp + yt : One may think of the permanent part as expected
average future income and of the temporary part as the random ‡uctuations
around this average income. Examples may help: your usual salary makes up
the largest fraction of your permanent income. A win in the lottery is the
typical component of transitory income, or a particularly good summer for an
ice-cream vendor, something that increases (or decreases) your income, but is
not a permanent event. Friedman observed correctly that individuals would
react quite di¤erently to an increase in permanent and an increase in transitory
income. Suppose you start a new, permanent job that doubles your salary up
into the inde…nite future. By how much would you increase your consumption
expenditures? Now suppose you win $1,000 in the lottery, and the chances of
that happening again are very small. By how much would you increase your
consumption expenditures? Friedman claimed that an increase in the permanent component of income would bring about an (almost) equal response in
consumption, whereas individuals would smooth out transitory income shocks
over time: you take the 1,000 bucks and spend $50 to see Stanford beat Berkeley
and the rest you put in your saving account for future usage. It then follows that
individual consumption is almost entirely determined by permanent income, i.e.
170CHAPTER 5. MICROECONOMIC FOUNDATIONS OF MACROECONOMICS
by the average income you think you will make for the rest of your life. Formally
c = ®yp
where ® is a parameter close to 1: Again we have the insight that consumption
today depends on future income (expectations) rather than on current income,
which may be unusually high (because of a positive transitory shock) or unusually low (because of a negative transitory shock).
A large empirical literature has investigated the life-cycle and permanent
income theories of consumption demand.5 Although the book is not closed
yet, it appears that the data seem quite favorable to these theories. Because
of this and because of the fact that these theories have sound foundations in
microeconomics they are the leading theories in current research on consumption
and saving behavior.
5.2
Investment Demand
Before turning to the theoretical analysis of investment demand let us have
a look at the data. Although investment is a much smaller fraction of total
GDP than consumption, the analysis of investment demand is is crucial for the
analysis of business cycles as investment demand is much more volatile than
consumption demand and GDP. Remember that we could divide total gross
investment into three categories:
1. Residential Fixed investment: this is the spending of private households
on the construction of new houses and apartments
2. Nonresidential Fixed Investment; this is the spending of …rms on new
plants and equipment
3. Inventory investment: this is the change of the value of inventories held
by businesses. Inventory investment can be positive (inventories increase)
or negative (inventories decline).
5.2.1
Facts about Investment
In Figure 66 we plot real GDP and real gross investment over tiem for the US.
Note that the scale on the two sides of the graph is di¤erent. The scale on the
left side is the relevant scale for real GDP, whereas the scale on the right side is
relevant scale for gross investment. This techinque of plotting the time series is
chosen to enable better comparison between the ‡uctuations of GDP and gross
investment. Comparing the two plots we observe the following features
1. Gross Investment is about 15% of real GDP on average. This fraction, the
so called investment-output ratio ‡uctuates over the business cycle, going
down in recessions and up in booms, but is fairly constant in the long run.
5 An excellent book that discusses the theories as well as their empirical tests is Angus
Deaton’s (1992) “Understanding Consumption”.
5.2. INVESTMENT DEMAND
Real GDP, Gross Investment, 1959-99
10000
GDP and Gross Investment
171
2000
5000
1000
Real GDP
Gross Investment
0
1955
1960
1965
1970
1975
1980
Year
1985
1990
Figure 5.8:
2. Gross Investment ‡uctuates much more severely than real GDP, with
more pronounced declines in recessions and more pronounced increases
in booms. In this sense gross investment is that part of GDP that is
mostly responsible for the business cycle.
Now we break down gross investment into its components, residential …xed
investment, nonresidential …xed investment and changes in business inventories.
In Figure 67 we plot gross investment and its …rst two components over time,
leaving for Figure 68 the plot changes in inventories. From Figure 67 we see
that over time nonresidental …xed investment (plant and equipment purchases
of …rms) have become relatively more important compared to residential …xed
investment. Whereas in 1959 made up about 50% of total gross investment, in
1999 nonresidential …xed investment made up around 74% of total gross investment and residential …xed investment around 23% (the rest going to changes
in inventories). It also appears from this …gure that both residential as well as
1995
0
2000
172CHAPTER 5. MICROECONOMIC FOUNDATIONS OF MACROECONOMICS
Real Gross Investment and Components, 1959-99
1800
1600
Investment and Components
1400
1200
1000
800
Gross Investment
600
Nonresidential Fixed Investment
400
200
0
1955
Residential Fixed Investment
1960
1965
1970
1975
1980
Year
1985
1990
1995
Figure 5.9:
nonresidential …xed investment ‡uctuate less over time as total gross investment.
The last fact obviously implies that the remaining part of investment, namely
changes in business inventories, has to ‡ucutate a lot over the business cycle.
This conjecture is veri…ed in Figure 68, where we plot total investment and
inventory investment. Again notice that we have used di¤erent cales for both
variabl;es to enable easier comparison.
In paricular the scale on the left side is for total investment, whereas the scale
on the right side is for inventory investment. We see that inventory investment
‡ucuates much more than total investment, or, for that matter, much stronger
than any ohter component of real GDP. Hence, although inventory investment
makes up only about 1% of GDP, it is a strong contibutor to business cycles
and inventory investment of …rms is heavily studied by both theoretical as well
as empirical economists trying to explain the business cyle. Also note that
during recessions inventory investment typically becomes (or at least gets close
2000
5.2. INVESTMENT DEMAND
Gross Investment and Change in Inventories, 1959-99
2000
Gross Investment, Change in Inventories
173
100
Change in Inventories
1000
0
Gross Investment
0
1955
1960
1965
1970
1975
1980
Year
1985
1990
1995
Figure 5.10:
to) negative: during recessions …rms tend not to produce for inventory.
After this little tour overviewing the basic facts with respect to investment
data let us now look at soem theory trying to explain the investment behavior
of …rms.
5.2.2
The Theory of Investment
Nonresidential Fixed Investment Demand
To start our study of investment demand of a single we proceed in two steps. We
…rst assume that our …rm rents all capital that it uses in the production process
from other …rms that are in the business of equipment and plant renting for
industrial purposes. Although in reality most equipment and plants used are in
fact owned by the …rms who use it, this assumption will turn out not to matter.
Let the rental price for one unit of the capital good be denoted by rk and the
-100
2000
174CHAPTER 5. MICROECONOMIC FOUNDATIONS OF MACROECONOMICS
wage for one worker be denoted by w: We normalize the price of the output
good to 1, so rk and w are the real rental price of capital and the real wage,
respectively. Let us assume that the …rm can produce output Y according to
the following production function
Y = K ® L1¡®
where K is the amount of capital rented and L is the number of workers hired.
To determine the …rm’s demand for rented capital we have to solve the …rm’s
pro…t maximization problem
max K ® L1¡® ¡ rk K ¡ wL
K;L
The …rst part of the above equation is the revenue the …rm takes in (remember
that we normalized the price of the …nal output good to 1), and the second and
third part are the total costs for renting capital and hiring workers, respectively.
Taking the …rst order condition with respect to capital yields
rk = ®K ®¡1 L1¡® =
®Y
®K a L1¡®
=
K
K
solving this for K gives the optimal demand for capital to be rented, K; as
K=
®Y
rk
(5.10)
Hence, if the …rm decides to produce output Y and faces a rental price of capital
rk ; then the optimal amount of capital to rent out is given by K = ®Y
rk : This
is the standard pro…t maximization condition from micro: the …rm should hire
inputs, in particular capital, to the point where the additional cost for one
unit rented, rk , equals the additional bene…t, the marginal product of capital,
®K ®¡1 L1¡® : This gives a demand curve for capital that is increasing in the
desired amount of output produced, Y; and decreasing in the rental rate of
capital rk :
In Figure 69 we plot the demand for rented capital as a function of the rental
rate of capital. As good (actually bad!) tradition in microeconomics we plot
the price (the rental rate rk ) on the y-axis and the quantity of rental capital
demanded on the x-axis. In this graph we hold the desired level of output
constant. As indicated in (5:10) the quantity demanded of capital decreases
with the rental rate rk : The optimal quantity demanded at the price rk is given
by K ¤ ; because at this level of the capital stock the marginal cost from renting
an additional unit, rk is equal to the marginal product ®Y
K ; for a given level of
output Y:
Figure 70 shows what happens to the demand curve for rented capital if the
planned level of output increases. Again as (5:10) shows an increase in Y; for
…xed rk increases K: But this is true for every rk ; indicating that the entire
demand curve shifts to the right. If the rental price of capital doesn’t change,
5.2. INVESTMENT DEMAND
175
Rental price
of capital
r
Marginal Cost of
Capital
k
Marginal Product of
Capital
K*
Demand for Rented Capital (K)
Figure 5.11:
the new optimal choice of capital is now K 0¤ > K ¤ ; i.e. the …rm reacts to higher
desired output by demanding more rented capital (and more workers).
So far we have ignored the fact how the rental price of capital is determined.
So let us consider the hypothetical problem of a …rm engaged in the business of
renting out capital, i.e. equipment and plants. For concreteness let us consider
the choice of such a …rm buying a particular piece of equipment. Let pk denote
the relative price of equipment (relative to the price of the …nal output good),
r denote the real interest rate and ± the depreciation rate. What are the costs
and what are the revenues from purchasing this machine and renting it out in
the current period? The revenues in the current period equal rk : The costs are
composed of two parts. The …rm has to …nance the purchase by borrowing the
money to purchase the machine. The interest on the loan is a cost, equal to
rpk : Furthermore a part ± of the machine wears out in the production process.
The loss of value due to this wearing out amounts to ±pk in the current period.
176CHAPTER 5. MICROECONOMIC FOUNDATIONS OF MACROECONOMICS
Rental price
of capital
r
Marginal Cost of
Capital
k
Y
K*
K’*
Marginal Product of
Capital
Demand for Rented Capital (K)
Figure 5.12:
Since there is free entry into the business of renting out capital pro…ts are bit
down to zero and therefore it must be the case that
rk = (r + ±)pk
So the rental price of capital equals the interest rate plus the rate of depreciation, times the relative price of investment goods to consumption goods, pk :
This relative price in turn depends on the technology that speci…es how much
of the …nal output good is needed to produce one unit of the investment good.
In a lot of macro models we assume that output can be used both for consumption and investment on a 1-1 basis (remember the Solow model) in which case
pk = 1: These types of models are called one-sector models as there is only one
production sector that produces both consumption and investment goods. We
will focus on these types of models as they are most tractable. We then have
a direct relation between the rental rate of capital and the real interest rate of
5.2. INVESTMENT DEMAND
177
the form rk = r + ± and the demand for rented capital depends negatively on
the real interest rate.
The remaining step is to relate the desired rented capital stock and investment demand. Consider a hypothetical …rm that does two things: it purchases
capital goods and rents it to itself and it produces output. Suppose this …rm
enters the period with capital stock K¡1 : The demand of the …rm for rented
capital services is given by
K=
®Y
r+±
and the investment demand of the consolidated …rm (taking the output producing and the capital renting division together) equals
I
= K ¡ (1 ¡ ±)K¡1
®Y
¡ (1 ¡ ±)K¡1
=
r+±
This is the investment demand for a single …rm. It depends positively on the
output that this …rm produces and negatively on the real interest rate. Summing over all …rms in the economy we get the total demand for nonresidential
…xed investment. As for the individual …rm the aggregate nonresidential …xed
investment demand depends positively on the level of output in the economy
and negatively on the real interest rate. Hence our more careful study of investment demand has revealed that our simple investment function from above was
correct in that the real interest rate entered negatively, but it disregarded the
in‡uence of current output Y on aggregate investment demand.
Residential Fixed Investment Demand
For residential …xed investment we can carry out a similar analysis. We start
by assuming that the demand for housing H decreases as the rent rh increases.
The relation between rh and H is identical to the relation between rk and K;
just relabel the axes in Figure 69. Also by the same reasoning the price for a
new apartment building ph ; the depreciation rate for buildings ± h and the real
interest rate r are related by
rh = (r + ± h )ph
and the investment demand for residential …xed investment is, as above, a negative function of the real interest rate. To the extent that the demand for housing
depends positively on income (equal to spending), the demand for residential
…xed investment also depends positively on Y; hence has the same qualitative
features as nonresidential …xed investment demand.
Inventory Investment Demand
A small fraction of total investment demand (usually not more than 1% of GDP)
comes from changes in inventories. Although the change in inventories may be
178CHAPTER 5. MICROECONOMIC FOUNDATIONS OF MACROECONOMICS
small (but very volatile over the business cycle), the total inventories held in
the entire economy are quite substantial. Hall and Taylor report that in 1995
inventories amounted to about 17% of GDP. Note that holding inventories is
not costless for …rms. Suppose the production of the goods in inventories has
been …nanced by credit, then if the price all for the goods held in the inventory
is pk ; the current period cost of holding the inventory is (r + ± i ) pk ; i.e. equals
to the cost of capital bound in the inventory as well as the depreciation of the
goods being held in inventories. Note that ± i may be small (in the case of highly
durable goods), very large (in the case of, say, vegetables) or even negative (for
goods that appreciate, like wine).
What are the bene…ts of inventories. One …rst observation is that inventories
may be required in the production process. Whiskey is an example. Whiskey
has to be stored for a while before it reaches its best quality. So putting Whiskey
into inventory for some time is a requirement of the production process. Oil is
a second example. Unavoidably large fractions of all oil produced and sold is
in transit in pipelines, in involuntary inventory, so to speak. More traditional
examples includes inventories in the manufacturing industry, where certain intermediate goods are stored in inventories before being used in the production
process. Just-in-time production techniques have sharply reduced inventories of
this kind in the last 15 years or so.
Secondly inventories serve a bu¤er function against unexpected ‡uctuations
of demand. Final goods are put into inventory so that they are available upon
demand. The bene…t from having an inventory is to be able to serve demand
immediately and hence to avoid losing the customer to a competing supplier.
Of course these bene…ts have to be balanced against the cost of holding the
inventory, as discussed above.
Empirically changes in inventory investment is a strongly procyclical variable, it tends to increase with overall production and tends to decline with
overall production. A higher level of production requires more goods “in the
pipeline” and in intermediate inventories. Booms are also times where …rms
expect high demand that they want to bu¤er with high inventory of …nal goods.
Occasionally …rms are caught by surprise in that their sales expectations are
not met and inventories are accumulated involuntarily. This explains the few
occasions where we observed a strong positive change in inventory investment
and a recession (as in 1974). For more details see Hall/Taylor’s Figure 11.8 on
p. 321.
To summarize, our analysis of investment demand has recon…rmed our previous assumption that aggregate investment demand depends negatively on the
real interest rate. It has added the insight that investment demand should depend positively on the level of output, a fact that was ignored in the traditional
aggregate investment function and the IS-LM analysis based on it.
Chapter 6
Trade, Exchange Rates &
International Financial
Markets
Foreign trade is a central policy issue. The high and increasing US trade de…cit
is of immediate concern to policy makers and there is a lot of controversy how to
reduce it. Since the trade de…cit is closely related to the exchange rate (the value
of the dollar compared to other currencies), some economists believe that, in
order to control the trade de…cit, the exchange rate has to be controlled. Hence
the discussion of the trade de…cit leads us directly to the discussion about …xed
vs. ‡exible exchange rates and the international …nancial system.
6.1
Terms of Trade, the Nominal and the Real
Exchange Rate
In order to organize our thoughts we need a few de…nitions
De…nition 8 The trade balance is total value of exports minus the total value
of imports of the US with all its trading partners. In symbols
TB = X ¡ M
Hence the trade balance is an important component of total spending in the
economy. For the US the trade balance has been negative for the last 20 years,
as can be seen from Figure 71. For 1997 the trade balance for the US is around
-100 billion dollar, i.e. the US had a trade de…cit of about 1.2% of GDP. For
1999 the trade de…cit is projected to reach about 3.5% of GDP
De…nition 9 The current account balance equals the trade balance plus net
179
180CHAPTER 6. TRADE, EXCHANGE RATES & INTERNATIONAL FINANCIAL MARKETS
Trade Balance for the US 1967-2001 (in Constant Prices)
0
-50
Trade Balance
-100
-150
-200
-250
-300
-350
-400
1970
1975
1980
1985
Year
1990
1995
Figure 6.1:
unilateral transfers
CAB = T B + N U T
The main ingredient of net unilateral transfers are interest payments to
people living in the US holding government bonds of foreign countries, net
of interest payments on US government debt to foreigners. For the US net
unilateral transfers are slightly negative, about 0.3% of GDP in 1997. For some
highly indebted countries, in particular in South America and South East Asia,
net unilateral transfers can be signi…cantly negative, amounting to about 5% to
10% of GDP. Remember again what a negative current account balance means.
For this we have to understand another de…nition
De…nition 10 The Capital Account Balance is the change in the net wealth
position of the US during a year.
2000
6.1. TERMS OF TRADE, THE NOMINAL AND THE REAL EXCHANGE RATE181
It follows from basic rules of accounting that the current account balance
always equals the capital account balance. A negative current account balance
means a negative capital account balance, and this means that the net wealth
position of the US, the amount that the US (the government and its citizens)
is owed, net of what it owes, decreases. The persistent current account de…cits
of the US have led to the fact that in the early 80’s the US, traditionally a
net creditor (having a positive net wealth position with the rest of the world)
turned into a net debtor (having a negative net wealth position with the rest of
the world). This tendency seems to continue without sign of reversal.
An important determinant of the trade balance is the relative price of US
goods to foreign goods. If US goods are expensive relative to Japanese goods,
a lot of Japanese goods will be imported by the US and few US goods will be
exported to Japan. Therefore, in order to understand the trade balance we have
to understand exchange rates
De…nition 11 The nominal exchange rate e is the relative price of two
currencies.
For example, if the exchange rate between the dollar and the euro is 0:98;
then one has to pay 0:98 euros to purchase one dollar, or reversely, one has to
pay 1:02 dollar to buy one euro. These days most exchanged rates are ‡exible:
they are determined on international capital markets beyond the direct control
of national governments (obviously monetary and …scal policy will in‡uence the
exchange rate of the domestic currency, but under a ‡exible exchange rate regime
the government does not directly …x the exchange rate). The opposite is a regime
of …xed (sometimes called pegged) exchange rates: via international agreements
exchange rates between certain countries are …xed. Before the collapse of the
Breton Woods system in 1973, for example, the exchange rates of the western
industrialized countries were pegged. These days, for example, the Argentinian
peso is pegged to the dollar: the exchange rate between the dollar and the peso
is 1 and the Argentinian government committed to defend this exchange rate.1
De…nition 12 The real exchange rate " is the relative price of goods in two
countries.
As it turns out it is the real exchange rate that is the key for net exports,
i.e. the trade balance. To see this, consider the following example. Think of a
good that is produced in many countries, say cars. Suppose a Ford Escort costs
$12,000 and a similar car, a Honda Civic costs 1,890,000 yen. How expensive is
a Ford relative to a Honda, i.e. how many Fords do we have to exchange for one
Honda Civic. This is exactly what the real exchange rate tells us (if all that is
traded between the US and Japan were Ford Escorts and Honda Civics). Now
we have to bring in the nominal exchange rate, since the price of the Japanese
car is measured in yen, the price of US cars in dollars. Suppose the exchange
1 By buying Argentinian pesos at exchange rate 1:1 if necessary. Obviously this requires
potentially substantial dollar reserves on the side of the Argentinian government. It is not
clear whether this peg would survive a major speculative attack against the peso.
182CHAPTER 6. TRADE, EXCHANGE RATES & INTERNATIONAL FINANCIAL MARKETS
rate between the yen and the dollar is 105; i.e. one needs 105 yen to buy one
dollar. Then the price of a Honda Civic, in dollar terms is
1; 890; 000 yen
= $18; 000
105 yen per $
Hence the real exchange rate (for the two cars) is
"cars
=
=
$12; 000 per US car
$18; 000 per Jap. car
2
Japanese car per US car
3
To summarize
"cars
=
=
=
$12; 000 per US car
1; 890; 000 yen per Jap. car/105 yen per $
(105 yen per $) ¤ ($12; 000 per US car)
1; 890; 000 yen per Jap. car
2
Japanese car per US car
3
In other words, in order to buy 2 Honda Civic one has to exchange in return 3
Ford Escort. We can generalize this example to obtain a formal relation between
the nominal and the real exchange rate. Obviously not only cars are sold in the
US and Japan. Let P denote the price level (i.e. the price of a representative
basket of goods) in the US, measured in $: Similarly, denote by P ¤ the price
level in the foreign country, in terms of the foreign currency. For concreteness
take Japan and the yen and denote by " the real exchange rate between the US
and Japan and by e the nominal exchange rate. Then
"=e¤
P
P¤
These are all the de…nitions we need in this section.2 .
There are two obvious questions to be answered:
1. How do real exchange rates a¤ect the trade balance?
2. What are the determinants of the real exchange rate?
2 Sometimes the real exchange rate is referred to as “Terms of Trade” (abbreviated t.o.t.).
This is usually done when P is interpreted as the price of export goods and P ¤ as the price
of import goods (rather than the price for a basket of goods that also includes goods that
are nontraded, like services). The terms of trade indicate at what exchange rate the US can
exchange their goods against foreign goods.
6.2. EFFECTS OF THE REAL EXCHANGE RATE ON THE TRADE BALANCE183
6.2
E¤ects of the Real Exchange Rate on the
Trade Balance
The …rst question appears to have an obvious answer. If the real exchange rate
increases US products become expensive relative to foreign products. This leads
to an increase in imports and a decline in exports. Hence the trade balance
should be a decreasing function of the real exchange rate, (X ¡ M ) = (X ¡
M)("). Note that this argument often provides the rationale for countries to
devalue their currency. Suppose price levels P and P ¤ are …xed in the short
run, as mostly assumed during this course. Then a decline in the nominal
exchange rate ( a devaluation of the currency) leads to corresponding decline
in the real exchange rate and an increase in net exports. In particular for
small, export-oriented countries this used to be a popular method to avoid or
get out of a recession. With exchange rates mostly ‡exible and determined on
world capital markets governments cannot directly devalue their currencies, so
this type of policy has become signi…cantly more di¢cult to implement under
‡exible exchange regimes.
If we look at the data net exports and the real exchange rate are in fact
negatively related (see Hall/Taylor, Figure 12.4). One reason why the e¤ect
described above may not be so direct as asserted is the following. Fix the price
levels P; P ¤ and think about a decline in the nominal exchange rate. For the
Japanese customers it becomes cheaper to acquire dollars to purchase American
goods. But prices of US goods sold in Japan are usually quoted in yen, and
unless Ford, say, doesn’t cut the yen price for its cars, nothing will happen
to their sales. For a given yen price, a decline in the exchange rate increases
Ford’s revenue in dollar terms, allowing them to sell their cars cheaper or make
a higher pro…t on their existing sales. In a world with perfectly competitive
markets the former should happen, but in particular with our assumption of
sticky prices it is not clear why, at least in the short run, US …rms would not
just take the windfall pro…ts from a better exchange rate.
Leaving aside these concerns we accept that net exports depend negatively
on the real exchange rate. For concreteness we use as our equation for net
exports
X ¡M
= g ¡ mY ¡ n"
eP
= g ¡ mY ¡ n ¤
P
= g ¡ mY ¡ n"
(6.1)
where g; m; n are positive constants. We now turn to the question of what
determines the real exchange rate.
184CHAPTER 6. TRADE, EXCHANGE RATES & INTERNATIONAL FINANCIAL MARKETS
6.3
6.3.1
Determinants of the Real Exchange Rate
Purchasing Power Parity
One of the basic principles in economics is the law of one price: absent transportation costs the same good cannot sell at di¤erent prices in di¤erent locations.
If a bushel of wheat sold for less in New York than in Chicago, then arbitrageurs
would take advantage of this riskless opportunity to make money and buy wheat
in New York and sell it in Chicago. Prices in New York will go up and/or prices
in Chicago down, removing this arbitrage opportunity.
The law of one price, applied to the international marketplace is called purchasing power parity: absent transportation costs a BMW should cost the same
in New York and Munich, once we converted the dollar price in New York into
Deutsche Mark, using the nominal exchange rate. Otherwise there would be
again arbitrage opportunities, buying the car in one place and selling it in another place a making a riskless pro…t. This principle provides us with a theory
for the real exchange rate and the nominal exchange rate.
Suppose all goods are traded and there are no transportation cost. Then the
real exchange rate should equal one: the same good sold in di¤erent countries
should have the same price. Second, changes in price levels in the US and
abroad, i.e. changes in PP¤ should be fully re‡ected in the nominal exchange
rate. This can formally be expressed as follows. The real exchange rate is given
by
"=e
P
P¤
If purchasing power parity were to hold the real exchange rate should not change
and (taking logs and di¤erentiating with respect to time)
g(e) = ¼¤ ¡ ¼
i.e. the percentage change of the nominal exchange rate should be equal to the
di¤erence between in‡ation rates in the two countries. Suppose, for example,
that the in‡ation rate between 1999 and 2000 in Germany is 2% and in the
US it is 5%: Then, according to purchasing power parity, the exchange rate
between the dollar and the Mark should decline by 3% between 1999 and 2000,
i.e. more dollars are required to buy one mark and less mark are required to
buy one dollar. Again, the intuition is simple: suppose a Ford costs $10; 000
in 1999 in New York and 20; 000 mark in Berlin and suppose (as purchasing
power parity would predict) that the exchange rate is 2 (2 mark per dollar).
The same car sells at the same price in both locations. Now there is in‡ation:
the 5% in‡ation rate in New York implies that in 2000 the car costs $10; 500
and the 2% in‡ation rate in Berlin implies that the car costs 20; 400 mark. But
the absence of arbitrage requires that both cars sell for the same price, hence
mark
= 1:943; a drop of
in 2000 the nominal exchange rate has to be 20;400
$10;500
1:943¡2
¼
0:03
=
3%:
Here
we
have
a
simple
theory
of
the
nominal
and the real
2
exchange rate.
6.3. DETERMINANTS OF THE REAL EXCHANGE RATE
185
Let us look at the data. Consider the example of the Big Mac. This high
point of American cuisine is sold in just about every country in the world by
now. Making all the assumptions needed for purchasing power parity (no transportation costs, most importantly) the price of the Big Mac should be the same
all over the world, once local currencies are converted into US currency. Let
us apply the theory and predict nominal exchange rates, based on the Bic Mac
price. Again applying the formula for the real exchange rate yields
PBM;U S
PBM;Abroad
PBM;Abroad
PBM;US
1 = e
e =
In order to predict the nominal exchange rate between the US and an arbitrary
country \Abroad" we just need to know the price of a Big Mac in the US,
PBM;US and the price of a Big Mac abroad, PBM;Abroad : The economist did this
in 1993 and got the results summarized in Table 11. The price of a Big Mac in
the US was about $2:28
This table demonstrate that the purchasing power parity theory is not completely out of line, but that there are substantial deviations. Obviously we
looked at only one example, Big Macs, and this particular commodity does not
make up a major fraction of GDP of the countries we considered. But looking
at plots for real exchange rates (see Hall/Taylor, Figure 12.3) we see that real
exchange rates ‡uctuate quite heavily, in contrast to what the purchasing power
parity theory predicts. So what are the problems that prevent the law of one
price from applying.
² Transportation costs: it may be quite costly to ship, say, cars from Europe
to the US and vice versa I would guess around $500 to $1,000 per car)
² Nontraded goods: a lot of goods that enter GDP (and hence the price levels
P; P ¤ ) are not traded across borders. Services are the most important
example. Hence the law of one price holds only for traded goods, and
the purchasing power parity theory of exchange rate is more successful if
P; P ¤ are taken to be price indices for exports and imports
² Trade restrictions as tari¤s and quotas: these things act like transportation
costs, they drive a wedge between the price of a good domestically and
the same good sold in other countries.
Although the purchasing power parity theory has limited success with respect to the data it is an important benchmark. And its most basic prediction
that real exchange rates should be somewhat stable in the long run is born out
in the data.
186CHAPTER 6. TRADE, EXCHANGE RATES & INTERNATIONAL FINANCIAL MARKETS
Table 11
Country
US
Argentina
Australia
Belgium
Brazil
Britain
Canada
China
Denmark
France
Germany
Hong Kong
Hungary
Ireland
Italy
Japan
Malaysia
Mexico
Netherlands
Russia
South Korea
Spain
Sweden
Switzerland
Thailand
6.3.2
Currency
Dollar
Peso
Dollar
Franc
Cruzeiro
Pound
Dollar
Yuan
Crown
Franc
Mark
Dollar
Forint
Pound
Lira
Yen
Ringgit
Peso
Goulder
Ruble
Won
Peseta
Crown
Franc
Baht
Price of BM
2.28
3.60
2.45
109.00
77,000.00
1.79
2.76
8.50
25.75
18.50
4.60
9.00
157.00
1.48
4,500.00
391.00
3.35
7.09
5.45
780.00
2,300.00
325.00
25.50
5.70
48.00
e (predicted)
1.00
1.58
1.07
47.81
33,772.00
0.79
1.21
3.73
11.29
8.11
2.02
3.95
68.86
0.65
1,974.00
171.00
1.47
3.11
2.39
342.00
1,009.00
143.00
11.18
2.50
21.05
e (actual)
1.00
1.00
1.39
32.45
27,521.00
0.64
1.26
5.68
6.06
5.34
1.58
7.73
88.18
0.65
1,523.00
113.00
2.58
3.10
1.77
686.00
796.00
114.00
7.43
1.45
25.16
Real Exchange Rates and Interest Rates
We will pursue a di¤erent explanation of the nominal, and hence the real exchange rate that is based on international …nancial markets. Think of a big
player in international …nancial markets, a George Soros or the manager of a
big mutual fund. Given that money can travel borders almost without any cost
in the western world, these investors face the choice of where, i.e. in what country to invest. Suppose these investors hold a certain portfolio and now the real
interest rate in the US, compared to other countries where the investors hold
positions, goes up. At the prevailing nominal exchange rate it becomes more
attractive to invest in the US, and this would cause huge (and fast) in‡ows
of …nancial capital into the US and out of other markets (because it is almost
costless to transfer money from one market to the other).
Flows of funds between countries are substantial, but not as large as one
would expect, following increases in the interest rate, say, in the US after the
6.4. THE INTERNATIONAL FINANCIAL SYSTEM
187
FED raised interest rates. What prevents foreign and domestic investors to move
their portfolio into US interest bearing securities. The answer: an appreciation
of the dollar, i.e. an increase in the nominal exchange rate. Investors have to
acquire dollars to purchase US securities, the demand for US dollars increases
and hence the price increases. But if the dollar gets more expensive, then, even
if US securities now earn a higher interest rate, investors may not be tempted
to buy more of them. Hence the reaction of the nominal interest rate keeps
international capital ‡ows in check.
Hence we theorize that the nominal exchange rate is determined by the real
interest rate, both domestic and foreign. A higher domestic real interest rate
leads to a higher nominal exchange rate. Taking price levels as sticky in the short
run yields a positive relation between the real exchange rate and the domestic
real interest rate.3 Formalizing this we posit
"=
eP
= q + vr
P¤
(6.2)
where q; v are positive constants.
Combining (6:2) and (6:1) we see that net exports are a negative function of the interest rate. The intuition: a higher real domestic real interest
rate increases the nominal and hence the real interest rate, therefore makes
US products more expensive relative to products from the rest of the world,
hence reduces net exports. Now that net exports depend negatively on the interest rate, this modi…es our IS-curve and hence our policy analysis using the
IS-LM framework. For example, the latest increase in interest rates by the FED
(accomplished by a reduction in money supply) should, in theory, lead to an
increase in the nominal exchange rate. This has already happened. It should
translate into an increase in the real exchange rate (which has happened, too,
from what we know yet) and a reduction of net exports, i.e. a widening of the
already big US trade de…cit. This shows that sometimes economic policy is
quite problematic, and to accomplish one goal (preventing the economy from
overheating) compromises another goal (bringing down the large trade de…cit).
6.4
The International Financial System
[To be completed]
3 See, for example, Figure 12.3 in Hall/Taylor, for the fact that the real excahnge rate
tracks the nominal exchange rate very closely.
188CHAPTER 6. TRADE, EXCHANGE RATES & INTERNATIONAL FINANCIAL MARKETS
Chapter 7
Fiscal and Monetary Policy
in Practice
Economic policy can be broadly divided into monetary and …scal policy. Fiscal
policy is carried out by the government at di¤erent levels: by the President and
Congress at the federal level, by the governor and the state congresses at the
state level and by majors on the local level. The …scal policy instruments include
government purchases and transfers as well as taxes. In contrast to …scal policy
monetary policy is conducted by appointed bureaucrats, not elected politicians.
The instruments of monetary policy include the money supply as well as certain
interest rates. In the following chapter will discuss how monetary and …scal
policy are conducted in practice. As usual we will look both at some theory and
at data.
7.1
Fiscal Policy
7.1.1
Data on Fiscal Policy
The Structure of Government Budgets
We start our discussion with the federal budget. The federal budget surplus is
de…ned as
Budget Surplus = Total Federal Tax Receipts
¡Total Federal Outlays
Federal outlays, in turn are de…ned as
Total Federal Outlays = Federal Purchases of Goods and Services
+Transfers
+Interest Payments on Fed. Debt
+Other (small) Items
189
190
CHAPTER 7. FISCAL AND MONETARY POLICY IN PRACTICE
The entity “government spending” that we considered so far equals to federal,
state and local purchases of goods and services, but does not include transfers,
such as social security bene…ts, unemployment insurance and welfare payments.
The US federal budget had a de…cit every year since 1969 until 1998, and in
fact it seemed so unlikely that this would change in the near future that Hall
and Taylor, on p. 362 conjectured that the federal budget would be in de…cit at
least until the turn of the century. How can the federal government spend more
than it takes in? Simply by borrowing, i.e. issuing government bonds that are
bought by private banks and households, both in the US and abroad. The total
federal government debt that is outstanding is the accumulation of past budget
de…cits. The federal debt and the de…cit are related by
Fed. debt at end of year = Fed. debt at end of year
+Fed. budget de…cit
Hence when the budget is in de…cit, the outstanding federal debt increases, when
it is in surplus (as in 1999), the government pays back part of its outstanding
debt. Now let us look at the federal government budget for the latest year we
have …nal data for, 1997. See Table 12
7.1. FISCAL POLICY
191
Table 12
1997 Federal Budget (in billion $)
Receipts
1719.9
Pers. Income Taxes
769.1
Corporate Income Taxes
210.0
Indirect Business Taxes
93.8
Social Security Contrib.
518.5
Outlays
1741.0
Fed. Gov. Purchases
460.4
National Defense
306.3
Other Purchases
154.1
Transfer Payments
791.9
Grants to Local Gov.
225.0
Interest Paym. on Debt
231.2
Subsidies less Pro…ts
32.5
Surplus
-21.1
We see that the bulk of the federal government’s receipts comes from income taxes and social security contributions paid by private households, and,
to a lesser extent from corporate income taxes (taxes on pro…ts of private companies). The role of indirect business taxes (i.e. sales taxes) is relatively minor
for the federal budget as most of sales taxes go to the steady are the city in
which it is levied. On the outlay side the two biggest posts are national defense, which constitutes about two thirds of all federal government purchases
(G) and transfer payments, mainly social security bene…ts (about 550 billion if
one includes Medicare) and unemployment (about 220 billion). About 13% of
federal outlays go as transfer to states and cities to help …nance projects like
highways, bridges and the like. About 2% go as subsidies to public enterprises,
net of pro…ts (if any) of public enterprises. A sizeable fraction (13%) of the
federal budget is devoted to interest payments on outstanding federal government debt. The outstanding government debt at the end of 1997 was $5369; 7
billion, or about 67% of GDP. In other words, if the federal government could
expropriate all income of all households for the whole year of 1997, it would
need to thirds of this in order to repay all debt at once. The ratio between total
government debt (which, roughly, equals federal government debt) and GDP is
called the (government) debt-GDP ratio, and is the most commonly reported
statistics (apart from the budget de…cit as a fraction of GDP) with respect to
the indebtedness of the federal government. It makes sense to report the debtGDP ratio instead of the absolute level of the debt because the ratio relates
the amount of outstanding debt to the governments ability to generate revenue,
namely GDP.
Let’s have a brief look at the budget on the state and local level. The de…nitions apply as before. The main di¤erence between the federal and state and
local governments is the type of revenues and outlays that the di¤erent levels
of government have, and the fact that states usually have a balanced budget
192
CHAPTER 7. FISCAL AND MONETARY POLICY IN PRACTICE
amendment: they are by law prohibited from running a de…cit, and correspondingly have no debt outstanding. The only state in the US that currently does not
have a balanced budget amendment is Vermont. But let’s look at the numbers
in Table 13
Table 13
1997 State and Local Budgets (in billion $)
Receipts
1094.3
Personal Taxes
219.9
Corporate Income Taxes
36.0
Indirect Business Taxes
533.4
Social Insurance Contrib.
79.9
Federal Grants
225.0
Outlays
960.1
State and Local Purchases
758.8
Transfer Payments
304.1
Interest Paid Less Dividends
-92.2
Subsidies less Pro…ts
-10.6
Surplus
134.1
The main observations from the receipts side are that the main source of
state and local government revenues stems from indirect sales taxes. Personal
Taxes are mostly the income taxes paid to the state and property taxes paid
by homeowners. Also about 25% of all revenues of state and local governments
come from federal grants that help …nance large infrastructure projects. On
the outlay side the biggest item are purchases, which are basically comprised of
outlays for paying government employees, notably public school teachers, police
o¢cers and local bureaucrats and outlays for infrastructure. Transfer payments
on the state and local level basically consists of welfare bene…ts. As mentioned
above almost all states and cities have balanced budget requirements prohibiting
running government de…cits. Consequently these governments have positive
assets rather than debt in general, hence their interest payments are outweighed
by their interest receipts and a negative entity appears on the spending side.
Also state and city-owned enterprises seem to make more pro…ts than losses, so
the net subsides to these enterprises are negative.
Fiscal Variables and the Business Cycle
In our discussion of …scal policy in the IS-LM framework we asserted that in
recessions …scal policy may be called upon to increase government spending to
lead the economy out of the recession. In this section we will investigate to
what extent actual …scal policy is correlated with the business cycle. Since in
this section we will only look at data, all the statements we can make are bout
correlations, not about causality. In Figure 72 we plot the unemployment rate
as prime indicator of business cycle and purchases of the government (federal,
7.1. FISCAL POLICY
193
state and local) as a fraction of GDP over time. One feature that appears in
the data is that government spending, as a fraction of GDP, has declined over
time from about 30% of GDP in the late 50’s to below 20% in the late 90’s
(see the right scale). One also can detect that in recessions (in times where
the unemployment rises, see the left scale) government spending as a fraction
of GDP increases. This is consistent with the view that government spending
is being used to a certain degree -successfully or not- to smoothen out business
cycles. A similar, even more accentuated picture appears if one plots government transfers (such as unemployment compensation and welfare against the
unemployment rate). The fact that government transfers are countercyclical
follows almost by construction: in recessions by de…nition a lot of people are
unemployed and hence more unemployment compensation (and once this runs
out, welfare) is paid out. These welfare programs are sometimes called automatic stabilizers, as these programs provide more transfers in situations where
incomes of households tend to be low on average, hence softening the decline in
consumption expenditures and therefore the recession.
In Figure 73 we plot the unemployment rate and government tax receipts as a
fraction of GDP against time. We see that tax receipts are strongly procyclical,
they increase in booms and decline during recessions. In this sense taxes act
as automatic stabilizers, too, since, due to the progressivity of the tax code, in
good times households on average are taxed at a higher rate than in bad times.
In this sense the tax system stabilizes after-tax incomes and hence spending. A
second reason for declines of taxes in recessions is discretionary tax policy: if
we believe the IS-LM analysis then cutting taxes provides a stimulus for private
consumption and may lead the economy out of a recession. For example, the tax
cuts in the early 60’s under President Kennedy were designed for this purpose.
So rather than being automatic stabilizers, taxes may be used deliberately to
control the business cycle.
Now let us look at the government de…cit over the business cycle. Figure 74
plots the federal budget de…cit as a fraction of GDP and the unemployment rate
over time. The …rst observation is (see the right scale) that the federal budget
had small surpluses in the late 50’s, then went into (heavy) de…cit for the next 35
years or so and only very recently showed surpluses again. One clearly sees the
large de…cits during the oil price shock recession and the large de…cit during the
early Reagan years, due to large increases of defense spending. Overall one sees
that the budget de…cit is clearly countercyclical: the de…cit is large in recessions
(as tax revenues decline and government spending tends to increase) and is small
in booms. In fact the extremely long and powerful expansion during the 90’s
resulted, in combination if federal government spending cuts, in the current
budget surplus.
How does one determine whether the federal government is loose or tight
on …scal policy. Just looking at the budget de…cit may obscure matters, since
the current government may either have generated a large de…cit because of
loose …scal policy or because the economy is in a recession where taxes are
typically low and transfer payments high, so that the large de…cit was beyond
the control of the government. Hence economists have developed the notion of
CHAPTER 7. FISCAL AND MONETARY POLICY IN PRACTICE
194
Government Purchases and Unemployment Rate, 1959-99
15
10
0.4
0.3
Unemployment
Gov. Spending as % of GDP
5
0.2
Unemployment Rate
0
1955
1960
1965
1970
1975
1980
Year
1985
1990
1995
Figure 7.1:
the structural government de…cit: it is the government de…cit that would arise
if the economy’s current GDP equals its potential (or long run trend) GDP. The
structural part of the de…cit is not due to the business cycle, it is the de…cit
that on average arises given the current structure of taxes and expenditures.
The cyclical government de…cit is the di¤erence between the actual and the
structural de…cit: it is that part of the de…cit that is due to the business cycle.
How loose or restrictive monetary policy is can then be determined by looking
at the structural (rather than the actual) de…cit. Unfortunately the structural
de…cit is not easily available in the data and we have to leave its discussion for
later.
Finally lets have a look at the government debt, the accumulated de…cits
of the federal government in Figure 75. What is striking is the explosion of
the government debt outstanding in the last 70 years. The picture is obviously
somewhat misleading, since it does not take care of in‡ation (in‡ation numbers
0.1
2000
7.1. FISCAL POLICY
195
Taxes and Unemployment Rate, 1959-99
15
0.35
0.3
Unemployment
10
Government Tax Receipts as % of GDP
5
0.25
Unemployment Rate
0
1955
1960
1965
1970
1975
1980
Year
1985
1990
1995
Figure 7.2:
before the turn of the century are somewhat hard to come by). But clearly
visible is the sharp increase during World War II. Somewhat more informative
is a plot of the debt-GDP ratio in Figure 76.
The main facts are that during the 60’s the US continued to repay part of
its WWII debt as debt grows slower than GDP, then, starting in the 70’s and
more pronounced in the 80’s large budget de…cit led to a rapid increase in the
debt-GDP ratio, a trend that only recently has been stopped and reversed
7.1.2
A Few Theoretical Remarks
The standard IS-LM analysis indicates that current tax cuts should have expansionary e¤ects: current disposable income increases, hence consumption spending increases, hence income and output increases. This is the Keynesian rationale for active …scal policy. There is a powerful theoretical counter argument
0.2
2000
CHAPTER 7. FISCAL AND MONETARY POLICY IN PRACTICE
196
Federal Deficit and Unemployment Rate, 1959-99
15
0.05
Federal Budget Deficit as % of GDP
0
Unemployment
10
5
-0.05
Unemployment Rate
0
1955
1960
1965
1970
1975
1980
Year
1985
1990
1995
Figure 7.3:
against this reasoning, known as the Ricardian Equivalence Hypothesis. Originally formulated by the classical 19’th century economist David Ricardo and
rediscovered by Robert Barro from Harvard in 1974 the hypothesis states that
for a given stream and timing of government spending the timing of taxes does
not a¤ect real activity in the economy, i.e. consumption, saving, output or the
real interest rate. You already reconstructed the argument in HW5: a current
tax cut has to be …nanced by a higher budget de…cit today and hence higher
taxes in the future. But, at least according to the life cycle/permanent income
theory of consumption, what really matters for the intertemporal consumption
choice is total discounted lifetime income, not when it comes. The private
households, according to the Ricardian Equivalence Theorem, see through the
government budget planning, anticipate the future tax hikes, adjust their savings accordingly to exactly o¤set the change in tax policy. In other words it is
irrelevant whether the government …nances it expenditures with current taxes or
-0.1
2000
7.1. FISCAL POLICY
US Nominal Government Debt, 1791-1998
12
6
197
x 10
US Government Debt
5
4
3
2
1
0
1750
1800
1850
Year
1900
1950
Figure 7.4:
a higher government de…cit (future taxes). Therefore the Ricardian Equivalence
Theorem is also often called the Debt Neutrality Theorem.
Note that the Ricardian Equivalence Theorem is in fact a theorem: given its
assumptions the debt neutrality result follows. The main assumptions are:
1. Consumers behave as rational life-cyclers: if they were myopic Keynesians,
obviously Ricardian equivalence breaks down
2. No borrowing constraints: you have seen in HW5 that temporary tax cuts
may have real e¤ects on consumption for consumers that are right on their
borrowing constraint
3. Consumers are in…nitely lived, i.e. they never die. Otherwise, if the future
tax hikes needed to …nance current tax cuts come after the agent has died
he does not take these tax hikes into account. Is it crazy to assume that
2000
CHAPTER 7. FISCAL AND MONETARY POLICY IN PRACTICE
198
Debt-GDP-ratio for US, 1960-1998
70
65
Debt-GDP-ratio in %
60
55
50
45
40
35
30
1960
1965
1970
1975
1980
Year
1985
1990
1995
Figure 7.5:
people life forever. Here comes Barro’s contribution: if people care about
their children as well as about themselves, then this is equivalent to them
living forever. In some sense altruistic agents live on in their children.
4. No uncertainty with respect to future income or perfect insurance markets
against future income uncertainty
5. Lump-sum taxation is possible
The last two points are a bit too involved to explain at this point, but talk
to me if you are curious about this.
The real question is whether the Ricardian Equivalence theorem is a good
description of reality. Almost certainly an actual economy like the US economy will not satisfy all the assumptions exactly. The question really is whether
the theorem (once we think about it as a hypothesis about the real world, we
2000
7.2. MONETARY POLICY
199
shouldn’t really call it a theorem anymore) is a good approximation to the
real world. Economists are split right through the middle. Keynesians don’t
like Ricardian equivalence since it defeats tax cut as useful stabilization policy,
neoclassical economists tend to like it for exactly the same reason. Not so surprisingly empirical analyses of the issue yield results all over the map, depending
on the exact method (sometimes economists are quite creative in generating results they like, sometimes it is not so clear what the right method is). Since I
do active research in this area feel free to come by for a chat if this issue is of
interest to you.
7.2
Monetary Policy
[To be completed]