Topics in Modern Macroeconomicsonline.sfsu.edu/mbar/Xaimen Macro/LectureNotes.pdf · Topics in Modern Macroeconomics Michael Bar1 ... (ii) Macro-economics. ... The advantage of micro-foundations

Topics in Modern Macroeconomics

Michael Bar1

July 4, 2011

1San Francisco State University, department of economics.

ii

Contents

1 Introduction 11.1 The Scope of Macroeconomics . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Models in Economics and Science . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 What is a model? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Why Models? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Models are not realistic and are not supposed to be . . . . . . . . . . 2

1.3 Modern Macroeconomics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Business Cycles 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 The Classical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 The description of the model . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Important remarks about models in general . . . . . . . . . . . . . . 72.2.3 Working with the classical model . . . . . . . . . . . . . . . . . . . . 82.2.4 Real business cycle doctrine . . . . . . . . . . . . . . . . . . . . . . . 15

3 Unemployment 173.1 Labor Market De�nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 The Search Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.1 Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.2 Job O¤er Acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.3 Distribution of wage o¤ers . . . . . . . . . . . . . . . . . . . . . . . . 233.2.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.5 Experiments with the Search Model . . . . . . . . . . . . . . . . . . . 26

4 Saving and Investment 334.1 Saving and Investment Equation . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Saving and Investment in the U.S. . . . . . . . . . . . . . . . . . . . . . . . . 354.3 Intertemporal Choice Model (Saving Theory). . . . . . . . . . . . . . . . . . 37

4.3.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.3.2 Optimal Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3.3 Changes in income . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3.4 Changes in the real interest rate . . . . . . . . . . . . . . . . . . . . . 454.3.5 Changes in taxes and Ricardian equivalence . . . . . . . . . . . . . . 46

4.4 Two-Period Model of Investment . . . . . . . . . . . . . . . . . . . . . . . . 48

iii

iv CONTENTS

4.4.1 Optimal investment decision . . . . . . . . . . . . . . . . . . . . . . . 494.4.2 Changes in interest rate . . . . . . . . . . . . . . . . . . . . . . . . . 504.4.3 Changes in technology . . . . . . . . . . . . . . . . . . . . . . . . . . 504.4.4 Solving for optimal investment . . . . . . . . . . . . . . . . . . . . . . 51

4.5 Capital Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.5.1 Decline in government de�cit (SG ") . . . . . . . . . . . . . . . . . . 544.5.2 Increase future productivity at home (A2 ") . . . . . . . . . . . . . . 54

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.7 Appendix: Firm With Unlimited Life Span . . . . . . . . . . . . . . . . . . . 55

5 Economic Growth 575.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2 The Solow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2.1 Description of the model . . . . . . . . . . . . . . . . . . . . . . . . . 595.2.2 Working with the model . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.3 Endogenous Growth Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.3.1 Description of the model . . . . . . . . . . . . . . . . . . . . . . . . . 635.3.2 Working with the model . . . . . . . . . . . . . . . . . . . . . . . . . 645.3.3 Economic Policy and Growth . . . . . . . . . . . . . . . . . . . . . . 665.3.4 Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.3.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6 Money and Prices 696.1 What is Money? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.2 The Demand for Money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.2.1 Quantity Theory of Money . . . . . . . . . . . . . . . . . . . . . . . . 706.2.2 Money in the Utility Function . . . . . . . . . . . . . . . . . . . . . . 72

6.3 Money Supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.3.1 Example of Money Creation . . . . . . . . . . . . . . . . . . . . . . . 75

6.4 Illustration of the Money Multiplier . . . . . . . . . . . . . . . . . . . . . . . 77

7 Phillips Curve 817.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817.2 Pillips Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

7.2.1 The impact of the Phillips curve on monetary policy . . . . . . . . . 867.3 Expectations-Augmented Phillips Curve (Edmund Phelps) . . . . . . . . . . 88

7.3.1 The impact of the expectations-augmented Phillips curve on monetarypolicy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.4 Rational Expectation (Robert Lucas) . . . . . . . . . . . . . . . . . . . . . . 907.4.1 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.5 Credibility of Monetary Policy (Finn E. Kydland, Edward C. Prescott) . . . 927.6 Appendix: Estimating the Expectations-Augmented Phillips Curve . . . . . 93

CONTENTS v

8 International Macroeconomics 958.1 Balance of Payments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958.2 Exchange Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

8.2.1 Using the exchange rates . . . . . . . . . . . . . . . . . . . . . . . . . 978.3 Law of One Price and Purchasing Power Parity (PPP) . . . . . . . . . . . . 98

8.3.1 Predicting future trends in exchange rates . . . . . . . . . . . . . . . 998.3.2 Fixed vs. �oating exchange rate . . . . . . . . . . . . . . . . . . . . . 101

8.4 Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

9 Math Review 105

10 Micro Review 10710.1 Consumer�s Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

10.1.1 Budget Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10810.1.2 Indi¤erence Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11010.1.3 Optimal Choice: graphical illustration . . . . . . . . . . . . . . . . . 11210.1.4 Optimal Choice: mathematical treatment . . . . . . . . . . . . . . . . 11210.1.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11610.1.6 Invariance of utility functions . . . . . . . . . . . . . . . . . . . . . . 11810.1.7 Income and substitution e¤ects . . . . . . . . . . . . . . . . . . . . . 119

10.2 Producer�s Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12110.2.1 Firm�s pro�t maximization problem . . . . . . . . . . . . . . . . . . . 12210.2.2 Factor shares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

10.3 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12410.3.1 Transitivity assumption . . . . . . . . . . . . . . . . . . . . . . . . . 12410.3.2 The slope of indi¤erence curves . . . . . . . . . . . . . . . . . . . . . 12410.3.3 Demand with Cobb-Douglas Preferences and n goods . . . . . . . . . 124

11 Rates of Change 12711.1 Measuring Rates of change. . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

11.1.1 Discrete time variables . . . . . . . . . . . . . . . . . . . . . . . . . . 12711.1.2 Continuous time variables . . . . . . . . . . . . . . . . . . . . . . . . 128

11.2 Rate of change of a product and ratio . . . . . . . . . . . . . . . . . . . . . . 12911.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

11.3 Logarithmic scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

vi CONTENTS

Chapter 1

Introduction

Traditionally, economics is divided into two broad �elds: (i) Microeconomics, and (ii) Macro-economics. Microeconomics is the study of individual behavior of consumers and �rms. Amicroeconomist might study questions like what a¤ects the price of individual good, or whatcauses changes in output of particular �rm or industry? Macroeconomics is the study of ag-gregate behavior of consumers and �rms. A macroeconomist might study questions like whata¤ects the �aggregate price level" in the economy, or what causes changes in the �aggregateoutput level�?

1.1 The Scope of Macroeconomics

The two main areas of macroeconomics are: (i) business cycles and (ii) growth, as describedin the next diagram.

1

2 CHAPTER 1. INTRODUCTION

Each of these two areas has many sub�elds. For example, some business cycle economistsstudy the evolution of real GDP and unemployment rate, while others focus on money andprices. Growth economists study the long-term growth trend of an economy. Economics ofgrowth also has several sub�elds.

1.2 Models in Economics and Science

1.2.1 What is a model?

A model is a simpli�ed version of the real object that we study. Examples models include:(i) a map in geography, (ii) a rat in neuroscience, and (iii) supply and demand model ineconomics. Model is another word for theory. In any model, we distinguish between exoge-nous variables - determined outside of the model, and endogenous variables - determinedwithin the model. In any model, the endogenous variables are determined by the exogenousvariables. Models generate prediction about the endogenous variables for di¤erent valuesof exogenous variables.For example, in the model of supply and demand the endogenous variables are price (P)

and quantity traded (Q), and exogenous variables are those that determine the location ofsupply and demand curves (such as income, prices of related goods, tastes of consumers,prices of inputs, technology of �rms, etc.). The model generates a prediction about P andQ for any set of exogenous variables. The model�s prediction about P and Q is competitiveequilibrium.

1.2.2 Why Models?

Models can explain some features of the real world. Models don�t tell us what the worldlooks like. Instead, they tell us what we can expect to happen in the world if the worldwas like the model. For example, the supply and demand diagram doesn�t look anythinglike the markets in the real world. The diagram does not show the identities of the buyersand sellers, their feelings and emotions, their physical appearance. The supply and demanddiagram only captures two features of real markets: (1) buyers typically want to buy lesswhen the price goes up, and (2) sellers want to produce more when the price goes up. Itturns out that the supply and demand diagram is very useful for explaining why prices di¤eracross goods and why there are changes in prices over time. After testing the predictions ofthe model against the data we conclude that indeed the two features of buyers and sellersthat we included in the model were important.Models can be used to perform controlled experiments. In the real world many things

change at the same time; the technology changes, government policies change, etc. Inthe model we can perform controlled experiments of changing one thing at a time (ceterisparibus). This is impossible to do with actual economies.

1.2.3 Models are not realistic and are not supposed to be

When the object of study is very complicated, we need models that will highlight someimportant features of the object and leave out many other features. For example, when we

1.3. MODERN MACROECONOMICS 3

study the economy of an entire country with millions of people, thousands of markets and�rms, it is di¢ cult for us to understand the behavior of the economy by just looking at it.Moreover, if we don�t have any models to work with, we don�t even know what data shouldbe collected about the object of our study. For example, the model of supply and demandtells us that we don�t need to collect data of all the names of buyers and sellers in a marketin order to understand how it works.

1.3 Modern Macroeconomics

The common feature of modern macroeconomics, regardless of its �eld, ismicro-foundations.This means that in macroeconomic models we want to see the choices of individuals and �rms.The common elements of modern macroeconomics are: (i) consumers, who maximize utilitysubject to budget constraints, and (ii) �rms, who maximize pro�t. The old approach tomacroeconomics was to make assumptions about how individuals behave, while in modernmacroeconomics economists make assumptions about individuals�objectives, and then derivetheir behavior. The advantage of micro-foundations approach is that we get more economicintuition and see more clearly the choices that individuals make.For example, in the old (Keynessian) macroeconomic model, the consumers were rep-

resented by a demand for consumption function: C = C0 +MPC(Y � T ). According tothis model, the average consumer starts with some consumption level C0, and increasesconsumption byMPC for each dollar increase in the after-tax (disposable) income. In mod-ern macroeconomics, consumers choose optimal consumption, when facing time and budgetconstraints:

maxC;l;LS

U (C; l)| {z }utility

s:t:

[Budget constraint] : C � [wLS + �] (1� t)[Time constraint] : LS + l = h

The above says that consumers derive utility from consumption and leisure, and thereforethey wish to maximize their utility (objective function). Consumers however are facingconstraints and cannot choose arbitrary levels of consumption and leisure. Spending onconsumption cannot exceed the income from labor (wLS) and non labor (� - pro�t), netof taxes (t - is the tax rate). Time is also constrained, so that leisure l and worktimeLS must equal to the time endowment. Notice that this optimization problem must besolved in order to obtain the demand for consumption. This is the di¤erence from the oldmacroeconomics, which assumes a particular demand for consumption. The next sectionillustrates the modern approach to macroeconomics, with the the classical model, applied tothe study of real business cycles.

4 CHAPTER 1. INTRODUCTION

Chapter 2

Business Cycles

2.1 Introduction

Recall from the introduction that the output per capita in the U.S. is growing steady, butthere are �uctuations about the trend. These �uctuations are called business cycles. Figure2.1 shows the ln of real GNP per capita in the U.S. in the last century, together with a lineartrend. The linear trend �ts the data pretty well, which means that the original variable,GNP per capita, was growing at constant rate.

Figure 2.1: Ln of real GNP per capita in U.S.

ln(Real GNP per capita)

1

1.5

2

2.5

3

3.5

4

1900 1920 1940 1960 1980 2000 2020Year

The study of the long run growth trend belongs to the �eld of economic growth. In thesenotes we focus on the �uctuations of the output around the trend. Subtracting the growthtrend from the time series in �gure 2.1, results in a series of deviations from trend, displayedin �gure 2.2. The series of deviations from trend is called detrended real output, or thecyclical part of the real output.The questions that we want to ask in these notes are:

1. What causes business cycles?

5

6 CHAPTER 2. BUSINESS CYCLES

Figure 2.2: Cyclical part of GNP per capita in U.S.

Deviations from trend

0.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

1900 1920 1940 1960 1980 2000 2020

Year

2. Can the government smooth out the business cycles?

3. Should the government smooth out the business cycles?

In order to answer these questions, economists use models. We will see that di¤erentmodels give di¤erent answers to those questions.

2.2 The Classical Model

2.2.1 The description of the model

The model consists of a representative consumer, representative �rm, and a government.The consumer receives income from supplying his labor and from dividends from the �rm heowns. The consumer chooses his consumption and time allocation between labor and leisure.The �rm is owned by the consumer, it owns a �xed amount of capital, and it chooses theoptimal amount of labor to maximize pro�ts. The government consumption is exogenousto the model. The government balances its budget by collecting taxes at the amount ofexpenditures.The formal description of the model economy:

1. Consumer:

maxC;l

� lnC + (1� �) ln l

s:t:

C = [w(h� l) + �] (1� t)

where C is consumption, l is leisure, w is real wage, h is time endowment (say 100hours per week), � is the pro�ts or dividends from the �rm, t is the �at tax rate. Thus,

2.2. THE CLASSICAL MODEL 7

the time spent working (labor supply) is

LS = h� l

2. Firm:maxLD

� = AK�L1��D � wLD

where A is productivity parameter (called Total Factor Productivity, TFP), K is thecapital stock, and LD is labor employed by the �rm. The productivity parameter re-�ects the idea that with technological improvement (A ") more output can be producedwith the same inputs. The total output in the economy is thus Y = AK�L1��.

3. Government: collects taxes on all income at the rate of t, and spends them ongovernment consumption. The government budget is

G = t (wL+ �)

4. De�nition: Competitive equilibrium consists of (w;C;G; L; l; �; Y ) such that

(a) Given w, the values of (C; l) solve the consumer�s problem,

(b) Given w, the value of L solves the �rm�s problem,

(c) Markets are cleared:

i. LD = LS = L (labor market),ii. C +G = Y (�nal goods market).

2.2.2 Important remarks about models in general

This section is a philosophical discussion of our approach in general. It is essential to read itin order to understand the material of this entire course, and many other courses that youare taking. You should come back and read this again after you have practiced working withthe classical model.

1. The competitive equilibrium is the model�s prediction about the endogenous vari-ables. Endogenous variables are determined inside the model, i.e. the variables whichthe model is trying to explain. The exogenous variables are those that are determinedoutside of the model. For example, in the model of a market the endogenous variablesare price and quantity traded, and exogenous variables are those that determine thelocation of supply and demand curves (such as income, prices of related goods, etc.). Inthe classical model the exogenous variables are: (A; t;K), and the endogenous variablesare: (w;C;G; L; l; �; Y ).

2. Causality: what causes what? In any model, the exogenous variables are �causing�the endogenous variables. For example, we can change A (the technology level) andobserve the changes in real wage, employment, consumption, output, etc. All theendogenous variables are caused by the exogenous variables. If we don�t change any


of the exogenous variables, no change in the endogenous variables can occur. Thus,in this model we cannot say that �output causes employment�, since both output andemployment are endogenous variables, and cannot change unless we change some ofthe exogenous. Be very careful about making statements of causality in the real world.

3. Why models?

(a) Models can explain some features of the real world. Models don�t tell us whatthe world looks like. Instead, they tell us what we can expect to happen in theworld if the world was like the model. For example, the supply and demanddiagram doesn�t look anything like the markets in the real world. The diagramdoes not show the identities of the buyers and sellers, their feelings and emotions,their physical appearance. The supply and demand diagram only captures twofeatures of real markets: (1) buyers typically want to buy less when the pricegoes up, and (2) sellers want to produce more when the price goes up. It turnsout the supply and demand diagram is very useful in explaining why prices di¤eracross goods and why there are changes in prices. After testing the predictions ofthe model with the data we conclude that indeed the two features of buyers andsellers that we included in the model were important.

(b) Models can be used to perform controlled experiments. In the real world manythings change at the same time; the technology changes, government policieschange, etc. In the model we can perform controlled experiments of changing onething at a time. This is impossible to do with actual economies.

4. Models are not realistic and are not supposed to be. When the object of studyis very complicated, we need models that will highlight some important features of theobject and leave out many other features. For example, when we study the economy ofan entire country with millions of people, thousands of markets and �rms, it is di¢ cultfor us to understand the behavior of the economy by just looking at it. Moreover, if wedon�t have any models to work with, we don�t even know what data should be collectedabout the object of our study. For example, the model of supply and demand tells usthat we don�t need to collect data of all the names of buyers and sellers of the marketin order to understand how it works.

2.2.3 Working with the classical model

The de�nition of competitive equilibrium is instructive about how the model should besolved. The de�nition suggests the following steps: (1) solve the consumer�s problem toget the labor supply, (2) solve the �rm�s problem to get the labor demand, and (3) use themarket clearing conditions to �nd the real wage, the equilibrium employment, and the restof the endogenous variables.

Mathematical solution

Step 1: solving the consumer problem


The consumer�s problem can be written as

maxC;l

� lnC + (1� �) ln l

s:t:

C + w (1� t) l = (wh+ �) (1� t)

This is a standard consumer choice problem with two goods: C and l, the prices of the goodsare 1 and w (1� t) respectively, and the consumer�s income is (wh+ �) (1� t). We knowalready how to solve a consumer choice problem with Cobb-Douglas preferences. Thus, thedemand is

C = � (wh+ �) (1� t)

l = (1� �) (wh+ �) (1� t)w (1� t) = (1� �)

�h+

�

w

�and the labor supply is

LS = h� (1� �)�h+

�

w

�(2.1)

Observe that consumption is increasing in w and �, and decreases in t. The labor supplyis increasing in w, decreasing in � and does not depend on taxes. The intuition why thelabor supply is decreasing in � goes as follows. The dividend income is non labor income,so when it goes up the consumer does not need to work as much. Figure 2.3 shows thegraph of the labor supply curve. i.e. how much labor the consumer wants to supply at anygiven wage, holding everything else �xed. This means that changes in w are re�ected bymovements along the curve, while changes in � will shift the entire curve.

Figure 2.3: Labor supply curve

Labor Supply Curve

0

1

2

3

4

5

6

7

8

9

10

0 10 20 30 40 50 60 70Labor (N)

Rea

l wag

e (w

)

Ls

Step 2: Solving the �rm�s problem


The �rm�s problem ismaxLD

� = AK�L1��D � wLD

The �rst order condition

@�

@LD= (1� �)AK�L��D � w = 0

(1� �)AK�L��D = w (2.2)

which tells us that the �rm maximizes pro�t when it equates the marginal product of laborthe the real wage. Equation (2.2) thus gives us the labor demand of the �rm. We can solvefor LD explicitly from equation (2.2) to get

LD =

�(1� �)AK�

w

�1=�Observe that this curve is decreasing in w. Figure 2.4 shows the labor demand curve, i.e. howmuch labor the �rm wants to employ at any given wage, holding everything else constant.Thus, changes in w are re�ected by movements along the curve while changes in A will shiftthe entire curve.The pro�t is therefore given by

Figure 2.4: Labor demand curve

Labor Deamand curve

0

1

2

3

4

5

6

7

8

9

10

0 10 20 30 40 50 60 70Labor (N)

Rea

l wag

e (w

)

Ld

� = AK�L1��D � (1� �)AK�L��D � LD = �AK�L1��D (2.3)

Step 3: equilibrium in the labor marketLetting LS = LD = L and substituting equations (2.2) and (2.3) into equation (2.1) gives

L = h� (1� �)�h+

�AK�L1��

(1� �)AK�L��

�


Solving for equilibrium L:

L = h� (1� �)�h+

�

(1� �)L�

L = h� (1� �)h� (1� �) �(1� �) L

�h = L+(1� �) �(1� �) L

�h = L

�1 +

(1� �) �1� �

��h = L

�1� � + (1� �) �

1� �

��h = L

�1� ��1� �

�

Equilibrium employment: L� =� (1� �)h1� ��

Once we found the equilibrium employment L�, all the other endogenous variables can befound in terms of L�. Equilibrium leisure:

l� = h� L�

To solve for equilibrium wage, use equation (2.2):

w� = (1� �)AK�L��

Equilibrium output:

Y � = AK�L�1��

Equilibrium pro�t, using equation (2.3):

�� = �AK�L�1��

To �nd equilibrium consumption we use the budget constraint:

C� = [w�L� + ��] (1� t)C� =

�(1� �)AK�L�1�� + �AK�L1��D

�(1� t)

C� = (1� t)Y �

Equilibrium government expenditures:

G� = Y � � C� = tY �


Summary of equilibrium:

L� =� (1� �)h1� ��

l� = h� L�

w� = (1� �)AK�L��

Y � = AK�L�1��

�� = �AK�L�1��

C� = Y � (1� t)G� = tY �

As you can see, an increase in productivity A, causes an increase in equilibrium output,equilibrium real wage, equilibrium consumption, equilibrium government consumption, andequilibrium pro�t. Equilibrium employment does not depend on the level of technology, eventhough the real wage went up. The e¤ect of higher K is similar because A and K alwaysappear together in the equations.An increase in the tax rate a¤ects only the distribution of the total output between the

private sector and the government sector. If t = 30% for example, then the governmentconsumes 30% of the total output, while the private consumers get to consume the rest 70%.

Graphical analysis

The classical model can be analyzed graphically with only two diagrams, the labor marketand the production function, as shown in �gure 2.5.These graphs correspond to the following equations:

Production function : Y = AK�L1��

Labor supply curve : LS = h� (1� �)�h+

�

w

�, where � = �AK�L1��

Labor demand curve : LD =

�(1� �)AK�

w

�1=�It is important to repeat here that labor supply curve is increasing in w. On the other hand,if � increases, this leads to a shift of the entire supply curve to the left. The labor demandcurve is decreasing in w. If A or K increase, the entire labor demand curve will shift to theright.Now we use this graphical framework in order to perform 3 experiments with the model:

1. An increase in productivity (A ").Figure 2.6 shows the e¤ects of an increase in A in the classical model.

As A ", there is an increase in labor demand (shift of the labor demand curve tothe right) and a decrease in labor supply (shift of the supply curve to the left) andan increase in production function. The e¤ect of the increase in labor demand onemployment is an increase in employment, while the e¤ect of a decrease in labor supplyon employment is a decrease in employment. Thus, without solving the model with


Figure 2.5: Classical model: graphical illustration

w*

Prod

uctio

n fu

nctio

nLa

bor m

arke

t

L

L

Y

w

LS

LD

L*

L*

Y*

particular functional forms we cannot tell what is the e¤ect of A " on equilibriumemployment. In the pervious section however we solved the model with Cobb-Douglastechnology and preferences and found that the equilibrium employment does not changeas A ". In other words, the e¤ect on employment of a decline in labor demand andof an increase in labor supply cancel each other. Both e¤ects however increase theequilibrium real wage.

2. An increase in K.

The e¤ect of an increase in K is the same as the e¤ect of an increase in A. Notice thatA and K always appear together as AK�.

3. An increase in the tax rate (t ").

Neither the labor demand nor the labor supply depend on the tax rate, hence nothingwill change in the labor market. The production function does not depend on thetax rate as well, and therefore none of the curves in �gure 2.5 will shift. As we haveseen before, the only e¤ect that an increase in the tax rate has on the economy is theincrease in the government share of the total output.


Figure 2.6: An increase in productivity (A ")

L1

W2

W1

Prod

uctio

n fu

nctio

nLa

bor m

arke

t

L

L

Y

w

LS

LD

L1

Y2

Y1

Answering the questions

Now we are ready to answer the questions we posed in the beginning of these notes, withinthe framework of the classical model.

1. What causes business cycles?

The exogenous variables in this model are (A; t;K). As we have seen before, a positiveshock to productivity (A ") increases the equilibrium output while a negative shock toproductivity (A #) decreases it. We can think of shocks to productivity as agglomera-tion of many factors such as innovations, shocks to oil prices, weather, political events,etc., that change the amount produced with the same inputs1. So this model suggeststhat business cycles might be a result of productivity shocks. As we have seen before,changes in t do not a¤ect the equilibrium output. How about K? It is possible thata hurricane, or a terrorist attack would destroy part of the nation�s capital and causea decline in output. It is harder to think of how the stock of capital can experience asudden increase. In any case, when we look at the data on capital stock, it looks very

1For a more detailed discussion about productivity shocks see the next section.


smooth and does not exhibit �uctuations that can potentially be the cause of businesscycles.

2. Can the government smooth out the business cycles?

We have seen before that changes in the tax rate in this economy do not a¤ect theequilibrium output. Changes in the tax rate only a¤ect the fraction of total outputthat is consumed by the government. Recall that

C = (1� t)YG = tY

So the answer to the question is, NO, the government cannot smooth the businesscycles (in this model).

3. Should the government smooth out the business cycles?

It shouldn�t because it can�t (in this model).

2.2.4 Real business cycle doctrine

Real business cycle theory suggests that the main source of business cycles is shocks toproductivity. The real business cycle school is led by Edward Prescott and Finn Kydland.They were awarded a Nobel Prize in Economics in economics in 2004 "for their contributionsto dynamic macroeconomics: the time consistency of economic policy and the driving forcesbehind business cycles". Finn Kydland and Edward Prescott developed a methodology thatallows them to answer the following quantitative equation: "how much of the �uctuationsin output around a trend can be accounted for by random shocks to productivity?". Theiranswer was 2/3. Kydland and Prescott used a model that is a more complex version of theclassical model (their model is called "the Neoclassical Growth Model"). But the idea canbe illustrated with the classical model.Step 1: Choose functional forms for utility and production function.In the data, although the real wage went up in the last decades, the average worktime

did not change. Notice that in our model with Cobb-Douglas utility function, we get thesame result, i.e. in equilibrium the worktime is constant and does not depend on the realwage.In the data, the labor share of total output is roughly constant over time. The Cobb-

Douglas production function delivers this property. Recall that the capital share is � andthe labor share is 1� �, and these are constant.Step 2: Choose the parameter values for the utility and production functions.In the data the labor share is about 2=3 of the total output. Thus, set � = 1=3 so that

(1�� = 2=3). In the real world people have approximately 100 hours per week that they canallocate between labor and leisure activity (24 hours per day, minus 8 hours of sleep and 2hours of maintenance such as bathroom, eating, resting). In the data the average worktimeis 40 hours per week, so using our equilibrium equation for employment we can �nd � as


follows:

L� =� (1� �)h1� ��

40 =��1� 1

3

�100

1� �13

0:4 =�23

1� �13

0:4� � � 0:4 � 13= �

2

31:2� 0:4� = 2a

1:2 = 2:4a

� = 0:5

Thus, � = 0:5, � = 13.

Step 3: Estimate the shocks to productivityWe assume that aggregate output is produced with

Y = AK�L1�� (2.4)

We have data on real GDP (Y ), on capital (K) and labor employed (L). This means thatwe can �nd Afrom the above equation as a residual. Because of this procedure A is calledthe Solow Residual, since we �nd it as the residual that would equate the left hand side andthe right hand side of equation (2.4).Step 4: Model simulationHaving found the time series of A we can simulate the model and generate time series of

consumption and output. We have seen that an increase in A causes an increase in outputin the classical model and a decline in A will cause a decline in output. It turns out thatthe time series of Y generated by the model is very similar to the data in �gure 2.2. In fact,the variance of the output generated by the model is about 2/3 of the actual variance of thereal GDP/capita in the data. This means that random shocks to productivity can explainmost, but not all the variation in real GDP/capita over the business cycles.

Chapter 3

Unemployment

In the classical model the labor market is cleared by assumption. This means that all peoplelooking for a job were able to �nd a job. In practice, there are unemployed people. Eventhough in the U.S. unemployment rate is about 5% and does not represent a huge problem;there are countries in Europe whose unemployment rate is in the double digits.All policymakers agree on the importance of the objective to keep unemployment low.

High unemployment stands in the way of achieving full productive capacity and increasesinequality. In this chapter we focus on the labor market, and in particular, on the determi-nants of unemployment rate. We begin with key concepts related to the labor market, andlater present the search model of unemployment.

3.1 Labor Market De�nitions

� Civilian noninstitutional population - Included are persons 16 years of age andolder residing in the 50 States and the District of Columbia who are not inmates ofinstitutions (for example, penal and mental facilities, homes for the aged), and whoare not on active duty in the Armed Forces.

� Unemployed persons - Persons aged 16 years and older who had no employmentduring the reference week, were available for work, except for temporary illness, andhad made speci�c e¤orts to �nd employment sometime during the 4-week period end-ing with the reference week. Persons who were waiting to be recalled to a job fromwhich they had been laid o¤ need not have been looking for work to be classi�ed asunemployed.

� Employed persons - Persons 16 years and over in the civilian noninstitutional popu-lation who, during the reference week, (a) did any work at all (at least 1 hour) as paidemployees; worked in their own business, profession, or on their own farm, or worked15 hours or more as unpaid workers in an enterprise operated by a member of thefamily; and (b) all those who were not working but who had jobs or businesses fromwhich they were temporarily absent because of vacation, illness, bad weather, child-care problems, maternity or paternity leave, labor-management dispute, job training,or other family or personal reasons, whether or not they were paid for the time o¤ or

17

18 CHAPTER 3. UNEMPLOYMENT

were seeking other jobs. Each employed person is counted only once, even if he or sheholds more than one job. Excluded are persons whose only activity consisted of workaround their own house (painting, repairing, or own home housework) or volunteerwork for religious, charitable, and other organizations.

� Labor force - The labor force includes all persons classi�ed as employed or unem-ployed.

� Not in the labor force - Includes persons aged 16 years and older in the civiliannoninstitutional population who are neither employed nor unemployed.

The next diagram illustrates the breakdown of the population into di¤erent categories.

Population =

labor forcez }| {employed+unemployed+ not in labor force| {z } +

8<:age < 16in the militaryinstitutionalized

The next table shows the data for the U.S., January 2006 (in thousands).

Total Population 298048Civilian Noninstitutional Population 227553

Labor Force 150114Employed 143074Unemployed 7040

Not in the Labor Force 77439

The most important indicators of the labor market are: (1) Unemployment Rate, and(2) Labor Force Participation Rate.

Unemployment Rate =#Unemployed#Labor Force

Labor Force Participation Rate =#Labor Force

#Civilian Noninstitutional Population

Based on the above data,

Unemployment Rate =7040

150114= 4:7%

Labor Force Participation =150114

227553= 66%

Labor force participation in the U.S. is approximately 70% for men and 60% for women.In the last 50 years, participation rates for women more than doubled while for men, partici-pation rate slightly declined. Some of the prominent hypotheses for why women increasinglyentered the labor force include the closing of the gender wage gap, the declining price ofhome appliances, and the wide spread use of a contraceptive pill. All of these stories have

3.1. LABOR MARKET DEFINITIONS 19

the feature of increasing the return to women of educating themselves and working relativeto staying at home. Participation rate is a completely di¤erent thing from the unemploymentrate. It measures the degree of willingness of people to work for paid wage. Unemploymentrate measures the degree of di¢ culty of �nding a job. In these notes, we only focus on theunemployment rate.Some Determinants of the Unemployment Rate1. Aggregate economic activity. High levels of output are associated with lower unem-

ployment rates. In other words, unemployment is countercyclical, as the �gure 7.2 shows.

Figure 16.2 Deviations from Trend in theUnemployment Rate and Percentage Deviations fromTrend in Real GDP for 1948–2003

Figure 3.1: Unemployment rate is countercyclical.

2. Demographic structure of the population. For example, younger workers tend to switchjobs more often, they have less to loose by getting �red, etc. Hence, younger populations,all things equal, tend to have higher unemployment rates. For example, if during the 50�sthere was a baby boom in the U.S., then 20 years later when the baby boom cohort entersthe labor market, we expect the unemployment rate to increase.3. Sectorial Shifts. For example, a shift away from manufacturing has displaced many

workers. Finding a new job for these workers involves acquiring di¤erent skills. Hence,societies with a greater degree of restructuring tend to have higher unemployment rates.4. Government policies. These include unemployment insurance programs as well as wel-

fare, training programs and job matching services for the unemployed. The unemployment


insurance (UI) program in the U.S. is run by state governments. Typically, unemployedworkers in the U.S. draw bene�ts for 6 months and the replacement ratio (ratio of UI bene-�ts to the wage the unemployed worker used to receive) is 1/2. Existence of this governmentpolicy a¤ects the behavior of both, employed and unemployed.

The unemployment rate in the data exhibits both, �uctuations at business cycle frequen-cies (determinant #1) as well as longer run trends (determinants #2,3,4).

3.2 The Search Model

This model will provide us with some simple insight into how the unemployment rate is de-termined. This model will also allows us to study how the unemployment rate can be a¤ectedby government policy with respect to unemployment bene�ts, labor income or unemploymentincome taxation as well as changes in informational technology.

3.2.1 Preferences

There are many jobs with di¤erent real wages w. The only characteristics of a job thatpeople care about is the wage that it pays. People are either employed or unemployed. Thismeans that everybody is in the labor force. Let U represent the fraction of people that areunemployed. Then fraction 1� U of people are employed. A fraction s of all the employedwill be separated from their jobs at any given period. We call s the separation rate andassume that it is �xed and the same for all jobs. The separation rate is exogenously givenparameter and can be thought of as the probability that any employed worker will loose hisjob. A fraction p of all unemployed people get a job o¤er. Again, p is just a given parameter;people have no control over it. We can think of p as the probability that any unemployedworker will get a job o¤er.

Let Ve (w; s; tw) denote the utility of being employed at wage w, with separation rate sand taxes on labor income tw. We assume that Ve is increasing in w, but at a decreasingrate (which means that Ve is concave in w). Also assume that Ve is decreasing in separation

rate s and in taxes on labor income tw. Thus Ve

�w+; s�; tw�

�. For example, Ve could be of the

following form

Ve (w; s; tw) = (1� s)pw (1� tw)

If we plot Ve as a function of w (keeping s and tw �xed), the graph would look like thefollowing

3.2. THE SEARCH MODEL 21

),,( we tswV

w

Utility of Employed, as a Function of w

The notation �s and �tw means that the above graph was plotted for some �xed values of s andtw. Changes in these values will shift the entire curve. In particular, an increase in either sor tw will shift the entire curve down, as shown in the next picture

w

),,( we tswV

Shift in the Utility as s " or tw ".

In what follows, we will use the notation Ve (w) to denote the utility of employed person forgiven and �xed values of s and tw.Let Vu (b; p; tb) denote the utility of being unemployed, where b is the real unemploy-

ment bene�t, p is the probability of getting a job o¤er, and tb is the tax on income from

unemployment. We assume that Vu

�b+; p+; tb�

�, which means that Vu is increasing in the

unemployment bene�ts b, increasing in the chances of getting an o¤er p, and decreasing inthe taxes on unemployment bene�t. For example, the function Vu could be of the followingform

Vu (b; p; tb) = ppb (1� tb)

Plotting the graph of Vu against the real wage (for given values of b; p; tb) looks like ahorizontal curve since Vu does not depend on w, as shown in the next picture


),,( bu tpbV

w

In what follows, we will use the notation Vu as a shorthand for the utility of unemployed forgiven and �xed values of b; p and tb.

3.2.2 Job O¤er Acceptance

When an unemployed worker receives a job o¤er w he accepts it if Ve (w) � Vu: The minimumwage o¤er which an unemployed worker accepts is an o¤er w� such that Ve (w�) = Vu. Werefer to this w� as the reservation wage. For all job o¤ers w � w�; Ve (w) � Vu and thereforethe unemployed will accept those job o¤ers. For all job o¤ers w < w�; Ve (w) < Vu andtherefore the unemployed will reject those job o¤ers. The next graph illustrates the job o¤eracceptance decision.

*w

)(wVe

w

uV

The reservation wage w� is crucial for determining the unemployment rate. Suppose thatthere are 1000 job o¤ers, and 700 of them are � w�. Then we know that 70% of thosereceiving job o¤ers will accept them and become employed.Examples1. Suppose that Ve (w; s; t) = (1� s)

pw (1� tw), and Vu (b; p; tb) = 3p

pb (1� tb). Find

the reservation wage w�.


Solution: the reservation wage solves

(1� s)pw (1� tw) = 3p

pb (1� tb)

Thus

w� =

�3p

1� s

�2b (1� tb)(1� tw)

2. Explain how does w� depend on the exogenous parameters b; p; s; tw; tb and give someintuition for your results.Solution: The reservation wage, w�, is increasing in b; p; s and tw. This makes intuitive

sense. Higher unemployment bene�t b means that the unemployed are more comfortablewith being unemployed and it will take higher wage to induce them to accept the job o¤er.Higher p means that greater fraction of unemployed receive job o¤ers, so the chances of�nding a higher paid job (everything else equal) is higher and therefore w� is higher. Highers means greater separation rate, or greater risk of loosing the job. Thus, the unemployedwill demand higher wage to compensate for that risk. Finally, higher tax on labor tw lowersthe net of tax real wage and hence it takes higher before tax wage to induce the unemployedto accept a job.Also observe that w� is decreasing in the tax on unemployment bene�t tb, which is also

intuitive; higher tax on unemployment bene�t lowers the net of tax unemployment bene�tand hence lowers the utility from being unemployed. Thus, it will take lower wage to inducethe unemployed to accept the job.3. In the above example, suppose that all types of income are taxed at the same rate t,

how does the reservation rate w� depend on t?Solution: It doesn�t, t cancels out

w� =

�3p

1� s

�2b (1� t)(1� t) =

�3p

1� s

�2b

4. Suppose that in the above example we have b = 5, p = 0:6, s = 0:1, tw = tb = 0:3.Find the reservation wage w�.Solution:

w� =

�3p

1� s

�2b (1� tb)(1� tw)

=

�3 � 0:61� 0:1

�2 �5 � (1� 0:3)(1� 0:3)

�=

�1:8

0:9

�25 = 20

3.2.3 Distribution of wage o¤ers

We need one more piece of information in order to �nd the unemployment rate, namelythe distribution of wage o¤ers. We assume that the distribution of job o¤ers is given by a


function H (w) which gives the probability that an o¤er is at least w. For example, supposethat

H (w) = 1� 1

100w

The next �gure is the plot of this function.

Pr(offer >= w)

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100 120

w

H(w)

Important! H (w) does not give the probability of receiving an o¤er of at least w. What itdoes tell us is that if some unemployed person received an o¤er, thanH (w) is the probabilitythat that o¤er is at least w.Examples1. Suppose that the distribution of job o¤ers is as given above, and suppose that I am

unemployed who received and o¤er. What is the probability that this o¤er is above 20?Solution:

H (20) = 1� 1

10020 = 0:8

2. Suppose that I am an unemployed person and a fraction p = 0:6 receive job o¤ers.What is the probability that I get an o¤er of at least 20?Solution:

p �H (20) = 0:6 � 0:8 = 0:48

3. Explain the di¤erence between part 1 and 2.Solution: In part 1 it was already given that I received an o¤er, so H (w) tells us what

is the probability that an o¤er that was received is at least w. In part 2 it is not knownwhether I will receive an o¤er or not. In fact, there is only 60% chance that I will. Thus,the chances that I will get an o¤er of at least 20 are 0:6 times what they are in part 1.

3.2.4 Equilibrium

Now we have all the information we need in order to compute the law of motion of unem-ployment rate:

Ut+1 = Ut + s (1� Ut)� pH (w�)Ut


The unemployment rate in the next period Ut+1 is equal to the sum of 3 elements. The �rstis the current unemployment rate Ut.The term s (1� Ut) is the addition to the unemployment rate due to separation of cur-

rently employed from their jobs. Suppose that 90% of the labor force are currently employed(1�Ut = 0:9) and the separation rate is 0:1, which means that 10% of the currently employedwill loose their jobs. Thus the term s (1� Ut) = 0:1 � 0:9 = 0:09, i.e., the unemployment ratenext period will increase by 0:9% due to some of the currently employed loosing their jobs.The last term on the right hand side represents the decline in unemployment rate due to

some of the currently unemployed �nding jobs. Suppose that currently there is 10% unem-ployment rate. Suppose that p = 0:6, which means that 60% of the currently unemployedwill receive a job o¤er. What fraction of them will accept the o¤er? It is given by H (w�),which is the probability that an o¤er exceeds the reservation wage (remember that unem-ployed people accept an o¤er if it is at least as high as their reservation wage). Suppose thatin our example w� = 20, so H (w�) = 0:8. This means that 80% of the unemployed whoreceived and o¤er will accept it. Thus, pH (w�)Ut = 0:6 � 0:8 � 0:1 = 0:048%, which is thedecline in the unemployment rate due to unemployed �nding jobs. In this numerical exam-ple we see that the unemployment rate will increase from period t to period t + 1 becausethe addition to unemployment rate from employed who loose their jobs is greater than thedecline in unemployment rate from unemployed who �nd jobs.Long-run equilibriumRearranging the law of motion of unemployment rate gives

Ut+1 = Ut + s (1� Ut)� pH (w�)UtUt+1 = Ut + s� sUt � pH (w�)UtUt+1 = Ut [1� s� pH (w�)] + s

If 1� s� pH (w�) < 1 then the law of motion has a steady state, as shown in the next �gureLaw of motion of U_t

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.1 0.2 0.3 0.4

U_t

U_t

+1 U_t+145_deg

We can see that starting from any unemployment rate, the economy will converge to a steadystate U� such that Ut = Ut+1 = U� for all t. To �nd the steady state we solve

U = U + s (1� U)� pH (w�)Us (1� U) = pH (w�)U

U� =s

pH (w�) + s

Once we �nd the reservation wage w�; we can plot pH (w�)U as a function of U; it is justa linear function of U with slope pH (w�) : The intersection of this line with s (1� U) (also


a linear function of U with slope �s and intercept s) determines the long-run (or steadystate) equilibrium level of U�:

U*

s

U

UwpH *)(

)1( Us −

1

In our numerical example

U� =s

pH (w�) + s=

0:1

0:6 � 0:8 + 0:1 = 0:172413793

Intuitively, the term s (1� U) represents the "�ow in" to the unemployment as a result ofemployed people separating from their jobs, while the term pH (w�)U represents the "�owout" of the unemployment as a result of unemployed accepting job o¤ers.

3.2.5 Experiments with the Search Model

An increase in unemployment insurance bene�t (b ")

As b ", the value of being unemployed shifts up, hence, the reservation wage increases (theunemployed get more picky about job o¤ers). As a result, fewer of those who are o¤ered jobs(the same job o¤ers are made) accept their o¤er, i.e., H (w�) # : Hence, U� = s

pH(w�)+s goesup. Notice that in this experiment, s and p remained the same while H (w�) went down. Asthe denominator became smaller, the fraction became larger. Intuitively, the �ow out of theunemployed is reduced since fewer job o¤ers are accepted. The �gures below illustrate all ofthe steps we mentioned.


*2

*1 ww

)()(

*2

*1

wHwH

1

*2

*1 UU

*2

*1 ww

w)(wH

s

U

w

),,(−−+ we tswV

),,(−++ bu tpbV

UwpH *)(

)1( Us −

1

Utility


An increase in probability of getting a job o¤er (p ")

This increase can be a result of an improvement in information technology that facilitatesa job search, or government policy that is successful at increasing the chances of the unem-ployed of �nding a job, such as retraining programs.As p goes up, the value of being unemployed increases, driving the reservation wage up

(again the unemployed become more picky). As a result, a lower fraction of unemployedwith job o¤ers actually accept their job, that is, H (w�) #. Let�s consider the equilibriumunemployment rate U� = s

pH(w�)+s . What happens to this fraction? s remained unchanged.p went up but H (w�) went down. It is unclear whether the product pH (w�) increased ordecreased. Thus,

?

*2

*1 ww

)()(

*2

*1

wHwH

1

*2

*1 UU

*2

*1 ww

w)(wH

s

U

w

),,(−−+ we tswV

),,(−++ bu tpbV

UwpH *)(

)1( Us −

1

Utility


An increase in labor income taxes (tw ")

An increase in labor income taxes leads to a fall in the value of being employed for any givenwage. As Ve (w) shifts down, the reservation wage goes up and fewer of those unemployedwith job o¤ers actually accept their o¤er, that is, H (w�) goes down. So, the equilibriumunemployment rate U� = s

pH(w�)+s increases. Thus,

*2

*1 ww

)()(

*2

*1

wHwH

1

*2

*1 UU

*2

*1 ww

w)(wH

w

),,(−−+ we tswV

),,(−++ bu tpbV

Utility

s

U

UwpH *)(

)1( Us −

1


An increase in separation rate (s ")

This increase can be a result of moving from a command economy (operated by government)to a market economy where the jobs are less secured and there is a higher risk of beingseparated from the job (since people are employed based on their skill and not based ontheir connections to the ruling party).An increase in the separation rate decreases the utility from being employed, so the

value of being employed goes down and the reservation wage goes up w� ". As a result,the probability of getting o¤ers that are accepted goes down H (w�) #. Thus, the �ow outof unemployment is reduced (pH (w�) goes down). At the same time the �ow in to theunemployment goes up s (1� U) ". Thus,

2s

*2

*1 ww

)()(

*2

*1

wHwH

1

*2

*1 UU

*2

*1 ww

w)(wH

1s

U

w

),,(−−+ we tswV

),,(−++ bu tpbV

UwpH *)(

)1( Us −

1

Utility


Numerical ExamplesSuppose that Ve (w; s; t) = (1� s)

pw (1� tw), and Vu (b; p; tb) = 3p

pb (1� tb), b = 5,

p = 0:6, s = 0:1, tw = tb = 0:3, H (w) = 1� 1100w.

1. Find the steady state unemployment rate.Solution:Step 1: �nd the reservation wage w�

(1� s)pw (1� tw) = 3p

pb (1� tb)

Thus

w� =

�3p

1� s

�2b (1� tb)(1� tw)

=

�3 � 0:61� 0:1

�2 �5 � (1� 0:3)(1� 0:3)

�=

�1:8

0:9

�25 = 20

Step 2: Find the fraction of o¤ers that are at least w� (�nding H (w�))

H (20) = 1� 0:01 � 20 = 0:8

Step 3: Find the steady state unemployment rate (U�)

s (1� U) = pH (w�)U

U� =s

pH (w�) + s=

0:1

0:6 � 0:8 + 0:1 = 0:172413793

2. Suppose that p = 1, so that everybody receives an o¤er. Find the steady stateunemployment rate.Solution:Step 1: �nd the reservation wage w�

(1� s)pw (1� tw) = 3p

pb (1� tb)

Thus

w� =

�3p

1� s

�2b (1� tb)(1� tw)

=

�3 � 11� 0:1

�2 �5 � (1� 0:3)(1� 0:3)

�=

�3

0:9

�25 = 55:55555556

Step 2: Find the fraction of o¤ers that are at least w� (�nding H (w�))

H (20) = 1� 0:01 � 55:55555556 = 0:444444444


Step 3: Find the steady state unemployment rate (U�)

s (1� U) = pH (w�)U

U� =s

pH (w�) + s=

0:1

0:6 � 0:444444444 + 0:1 = 0:183673469

So that the unemployment rate increased.

Chapter 4

Saving and Investment

The process of economic growth depends, among other things, on the ability of �rms toexpand their productive capacity through investment in additional equipment. Firms can�nance their investment from retained earnings (also called undistributed pro�ts or businesssaving) or borrow funds from households who save. In this chapter we discuss the relationshipbetween saving and investment in the macroeconomy, and present a theory of saving andinvestment. We will examine what factors a¤ect investment decisions by the �rms and savingdecisions by the households, and how those decisions are a¤ected by government policies.Before we start the formal discussion of saving and investment, we need to introduce two

general concepts. A stock variable is a magnitude measured at a point in time (say at theend of the year), and a �ow variable is a variable measured over a given time interval (sayover the year). For example, the stock of capital in the U.S. on December 31st 2005, is astock variable. Investment that took place in 2005 is a �ow variable. As another example,the saving during 2005 is a �ow variable and the savings at the end of 2005 (the value of allthe balances of saving accounts) is a stock variable. The �ow variables determine the valueof the stock variable at the end

4.1 Saving and Investment Equation

In any economy there exists an accounting identity that relates saving and investment. Inthis section we derive this relationship, called the saving and investment equation. TheGDP is given by

GDP = C +G+ I +NX (4.1)

where C is personal consumption expenditure, G is government consumption expenditure,I is gross domestic investment, and NX = X � IM is net exports (exports minus imports).We de�ne disposable income as

Y D = GDP + TR� T (4.2)

where TR are transfer payments by the government (such as unemployment insurance ben-e�ts, social security, medicare,...). and T is taxes. We de�ne the private saving as

SP = Y D � C (4.3)

33

34 CHAPTER 4. SAVING AND INVESTMENT

That is, the private saving is the disposable income that is not consumed. Similarly, thegovernment saving is

SG = T � TR�G (4.4)

which is the government income that is not spent on government consumption or transferpayments. Government de�cit is then

Def = �SG = G+ TR� T (4.5)

Now add TR and subtruct T from equation (4.1)

GDP + TR� T = C +G+ TR� T + I +NX

Now using the de�nition of disposable income in equation (4.2)

Y D = C +Def + I +NX

Finally, using the de�nition of the government de�cit in equation (4.5) gives the saving andinvestment equation

S = SP + SG = I +NX (4.6)

The left hand side of (4.6) is the gross national saving S, which is the sum of private andgovernment saving. On the right hand side we have the gross domestic investment I and netexports NX, which is also known as Net Foreign Investment. In a closed economy wehave

SP + SG = I

that is, in a closed economy the total saving is equal to total investment. In an open economy,on the other hand, part of the domestic saving can fund investment abroad, if S > I. Itis also possible in an open economy that the domestic saving are insu¢ cient to fund allof the domestic investment, if S < I, and in this case part of the domestic investment isfunded by foreigners. If NX > 0, then the economy is exporting more goods and servicesthan what it imports, i.e. the country is experiencing a trade surplus. This means that thedomestic economy is accumulating foreign assets, since the rest of the world has to borrowfrom the domestic economy. If on the other hand, NX < 0, this means that the economy isimporting more goods and services than its exports to the rest of the world, i.e. the countryis experiencing trade de�cit (trade de�cit is de�ned as �NX). In this case the domesticeconomy has to borrow from the rest of the world and the rest of the world is accumulatingdomestic assets. Thus, in the open economy, total saving is equal to the domestic investmentplus net foreign investment, as equation (4.6) statesWhat can we learn from the saving and investment equation (4.6)?

1. The domestic investment can be �nanced by domestic saving and by foreign saving.Rewriting (4.6) gives

I = SP + SG �NXThe term �NX represents the investment of the rest of the world in the domesticeconomy. For example, if I = 20, SP + SG = 15 and NX = �5, then

I|{z}20

= SP + SG| {z }15

�NX|{z}�5

4.2. SAVING AND INVESTMENT IN THE U.S. 35

which means that part of the domestic investment is �nanced by domestic saving (15)and part is �nanced by foreign saving (5).

If on the other hand we have I = 20, SP + SG = 25 and NX = 5, then

I|{z}20

= SP + SG| {z }25

�NX|{z}5

which means that the domestic saving �nances not only the domestic investment, butalso �nances some of the investment in the rest of the world.

2. The government can �nance its de�cit in two ways: (1) borrowing from domesticresidents or (2) borrow from the rest of the world. Rewriting equation (4.6) gives

SP + SG = I +NX

Def = SP � I �NX

Thus, we can see that an increase in government de�cit has to be associated with eitherincrease in private saving, or decrease in gross domestic investment or an increase inthe trade de�cit (borrowing from abroad).

4.2 Saving and Investment in the U.S.

Lets take a look at the behavior of total investment in the U.S. and how it was �nanced.Figure (4.1) shows the total investment in the U.S. as a fraction of GDP.We see that the

Gross Domestic Investment as a fraction of GDP (I/GDP)

0

0.05

0.1

0.15

0.2

0.25

0.3

1929 1939 1949 1959 1969 1979 1989 1999

Time

Frac

tion

of o

f GD

P

Figure 4.1: Gross Domestic Investment as a fraction of GDP.

total investment since 1929 tends to �uctuate around 20% of GDP. In other words, theinvestment rate in the U.S. is about 20% of GDP, and this rate does not change much overtime. From equation (4.6) we know that domestic investment can be �nanced by domesticsaving (private and government), or carried out by foreigners. That is, I = SP + SG �NX.


Private Saving as a fraction of GDP

0

0.05

0.1

0.15

0.2

0.25

0.3

1929 1939 1949 1959 1969 1979 1989 1999

Time

Frac

tion

of o

f GD

P

Figure 4.2: Private Saving as a fraction of GDP.

Lets examine the funding sources of investment. Figure (4.2) shows the private saving as afraction of GDP.Notice that in the last data point (in 2004) the private investment is about 15% of GDP.

Thus, the private saving is about 75% of the domestic investment. The rest, according to theidentity (4.6) has to come from government saving or from foreign investment. Figure (4.3)shows the government saving as a fraction of GDP.Notice that recently the government is

Government Saving as a fraction of GDP

0.1

0.08

0.06

0.04

0.02

0

0.02

0.04

0.06

0.08

1929 1939 1949 1959 1969 1979 1989 1999

Time

Frac

tion

of o

f GD

P

Figure 4.3: Government saving as a fraction of GDP.

running a budget de�cit, and in 2004 the de�cit was about 2% of GDP. Thus, the governmentdoes not "help" to �nance the domestic investment.The rest of the funding for the domestic investment has to come from abroad. Figure

(4.4) shows the net exports as a fraction of GDP.We can see that in the last 30 years theU.S. is experiencing trade de�cit. This means that foreigners accumulate U.S. assets. In

4.3. INTERTEMPORAL CHOICE MODEL (SAVING THEORY). 37

Net Exports as a fraction of GDP (NX/GDP)

0.070.060.050.040.030.020.01

00.010.020.030.04

1929 1939 1949 1959 1969 1979 1989 1999

Time

Frac

tion

of o

f GD

P

Figure 4.4: Net Exports as a fraction of GDP.

particular, in 2004 the trade de�cit was about 6% of GDP, which corresponds to foreigninvestment in the U.S.

4.3 Intertemporal Choice Model (Saving Theory).

In the �rst section of these notes we showed the relationship between saving and investmentin the economy, called the saving and investment equation:

S = I +NX

Our next goal is to investigate the determinants of saving and investment. In this section webuild a model in which consumers make explicit decisions about consumption and saving.Before presenting the model, let�s take a moment to think about what factors might a¤ectthe saving decision of households. We point out three factors that might be importantdeterminates of saving.Why do people save? Saving is a process of giving up current consumption in order

to increase the future consumption. Therefore, we expect that our saving decisions woulddepend on our current and future income. Typically, individuals who work and expect adecrease in their income when they retire, tend to save some of their current income forretirement. In contrast, other individuals who expect an increase in their future income,tend to borrow (have negative saving). For example, many students take student loans whilethey are studying, and plan to repay the loan when they graduate and earn higher income.Therefore, current and future income, are among the most important factors that a¤ect thesaving decisions.Another important factor that a¤ects the saving decision is the interest rate. If an

individual gives up some of his current consumption and decides to save, he will be able toincrease his future consumption. But the question is by how much? The real interest rategives the answer to that question. If you walk into a bank and open a savings account, the


interest rate that the bank will o¤er you is a nominal interest rate. The nominal interestrate tells you how many extra dollars you will get in the next period when you save onedollar today. For example, if the annual nominal interest is 10%, this means that when youdeposit $1 today, you will receive your $1 back, plus $0:1 interest.What consumers care about though is not how much money they got, but how much

consumption they can buy with their money. Suppose that as before, the nominal interestrate is 10%. In addition, suppose that there is some consumption good, say burger, whichcurrently costs 1$. What you really care about is how many extra burgers you will get nextyear when you give up one burger this year. Suppose that the in�ation rate is 5%, so thatthe price of a burger next year is $1:05. Now, when you give up one burger today, and savethe $1 in the savings account, you will receive $1:1 in the next year, and with this moneyyou can buy 1:1=1:05 � 1:048 burgers. Thus, when you give up one burger today, you get0:048 extra burgers in the future. We therefore say that the real interest rate is 0:048 or4:8%. Formally, the real interest rate is the extra amount of consumption that one gets in thefuture when he gives up one unit of current consumption. In contrast, the nominal interestrate is the extra dollars that one gets in the future when he gives up one dollar today. Leti be the nominal interest rate, r be the real interest rate, and � be the in�ation rate. Thenthe relationship between the nominal interest rate and the real interest rate is given by:

1 + i

1 + �= 1 + r

If the nominal interest rate and the in�ation rates are small, then we can derive anapproximation formula to the above. Taking ln from both sides gives

ln (1 + i)� ln (1 + �) = ln (1 + r)If i; r and � are small, the above is approximately

r � i� �Thus, the real interest rate is approximately equal to the nominal interest rate minus thein�ation rate. In the burger example, the nominal interest rate 10% and the in�ation rate is5%, and then the real interest rate is approximately 5%. Recall that the exact real interestrate was 4:8%, which is close to 5%. To summarize this discussion, since real interest ratedetermines how much extra future consumption we expect to get when we save one unit ofcurrent consumption, then we suspect that the real interest rate would be one of the pivotalfactors that a¤ect our saving decisions.The third factor that determines saving is our preferences. An individual who values

current consumption a lot and does not value future consumption much (someone who �livesthe day�), will tend to save little or even borrow. On the other hand, someone who valuesfuture consumption or consumption of his children and grand children a lot will tend to savemore. The three factors that a¤ect the saving decision are therefore, preferences, currentand future income, and the real interest rate.

4.3.1 The Model

Consumers: There are N identical consumers that live for two periods (1 and 2) and deriveutility from consumption c1 and c2 in the two periods: U (c1; c2). Consumers receive income


y1 and y2 in the two periods and pay a lump sum tax t1 and t2 to the government. Theconsumers decide how much to consume in each period and how much to save in the �rstperiod. We denote the saving in the �rst period by s. Consumers can borrow and lend atreal interest rate r, which is assumed exogenously given. Thus the budget constraints in thetwo periods are:

[BC1] : c1 + s = y1 � t1[BC2] : c2 = y2 � t2 + (1 + r) s

The consumers�problem is therefore

maxc1;c2;s

U (c1; c2)

s:t:

[BC1] : c1 + s = y1 � t1[BC2] : c2 = y2 � t2 + (1 + r) s

Government: The government collects tax revenues T1 = N � t1 and T2 = N � t2 in thetwo periods and spends G1 and G2 in the two periods. The government can borrow and lendat real interest rate r with the constraint that the present value of spending = present valueof taxes

G1 +G21 + r

= T1 +T21 + r

This means that if the government runs a de�cit in the �rst period, it must borrow theamount of the de�cit and pay that amount with the second period�s surplus. And if thegovernment has a surplus, it can save the surplus at interest r and be able to a¤ord a de�citin the second period. To see this, rearrange the above condition

(G1 � T1) (1 + r) = T2 �G2

Suppose that the interest rate is r = 5% and in the �rst period the government runs ade�cit of 100, thus G1�T1 = 100. The above condition means that in the second period thegovernment must have a surplus of 105 to pay the debt, i.e. T2 �G2 = 105.Now that we have completed the description of the model we would like to analyze the

impact on consumers of the following changes:

1. Changes in income: y1 and y2

2. Changes in the real interest rate: r

3. Changes in government taxes: T1 and T2

To answer the above questions we need to solve the consumers�problem. It is convenientto derive the lifetime budget constraint of the consumer. Substitute s from the second periodbudget constraint into the �rst period�s budget constraint. It is easy to do when you divideboth sides of BC2 by 1 + r to get

BC2 :c21 + r

=y21 + r

� t21 + r

+ s


Now add the two budget constraints and get the lifetime budget constraint:

c1 +c21 + r| {z }

PV of lifetime consumption

= y1 � t1 +y2 � t21 + r| {z }

we =PV of lifetime wealth

Thus, the left hand side is the present value of lifetime consumption, and the right hand sideis the present value of lifetime net of taxes income, which we call the lifetime wealth (we).

The consumers�problem can then be rewritten as

maxc1;c2

U (c1; c2)

s:t:

c1 +c21 + r

= y1 � t1 +y2 � t21 + r

Interpretation: Recall the consumer choice model of choosing the optimal amounts of twogoods x and y, with budget constraint pxx+pyy = I (from the micro foundations appendix).In that model the slope of the budget constraint (in absolute value) is px=py, which is therelative price of good x in terms of good y. Notice that the two period model is very similarto the general model of consumer choice, where the two goods are current consumption andfuture consumption (c1 and c2). We can write the budget constraint concisely as

c1 +c21 + r

= we

which is similar to

pxx+ pyy = I

The price of c1 is 1, and the price of c2 is 11+r, thus the slope of the budget constraint is (1+r),

and it represents the relative price of current consumption in terms of future consumption.Indeed, consider the cost of increasing current consumption by 1 unit. If the consumer savedthat unit, then he would have enjoyed an increase of 1 + r units in the future consumption.Hence the cost of current consumption in terms of future consumption is 1 + r. Thus theconsumer can choose the optimal bundle of two goods (c1 and c2), given his preferences andgiven the prices of the two goods. The next �gure shows the graph of the lifetime budget


constraint.

)1( rwe +

E22 ty −

1cwe

11 ty −

Slope = )1( r+−

2c

With free borrowing and lending, it is feasible for this consumer to consume all his wealthin the �rst period and nothing in the second: (c1 = we; c2 = 0). Similarly, it is feasible forthis consumer not to consume anything in the �rst period and consume all his wealth in thesecond period: (c1 = 0; c2 = we (1 + r)). Also notice that it is feasible for the consumerto consume in each period the income (net of taxes) received in that period: (c1 = y1 � t1;c2 = y2 � t2). This bundle is denoted by E is the consumer�s endowment. If the consumerwas not allowed to borrow or lend, then he would be forced to consume his endowment,i.e., his net of taxes income in each period. Because the consumers are free to borrow andlend at real interest rate r, they can chose other points on the budget constraint. If theconsumer chooses a point above E on the lifetime budget constraint, then he is a lender(his current consumption is less then current income, so he is saving a positive amount).If the consumer chooses a point below E on the lifetime budget constraint, then he is aborrower (his current consumption is greater than his current income, so he has negativesaving).


4.3.2 Optimal Choice

The next �gure shows the optimal choice for a consumer who is a lender.

A

)1( r+−s*>0

*2c

)1( rwe +

E22 ty −

1c

2c

we11 ty −*

1c

At the optimal bundle (point A) we have the usual condition that the marginal rate ofsubstitution between and is equal the relative price. That is

U1 (c1; c2)

U2 (c1; c2)= 1 + r

The left hand side is the (absolute value of) the slope of the indi¤erence curves and the righthand side is the (absolute value of) the slope of the budget constraint. This should look veryfamiliar to you and similar to the optimality condition

Ux (x; y)

Uy (x; y)=pxpy


Notice that in the case of a lender, the saving is positive. The next �gure shows the optimalchoice for a consumer who is a borrower.

E

)1( r+−s*<0

22 ty −

)1( rwe +

A*2c

1c

2c

we*1c11 ty −

Notice that the saving is negative for a borrower.

4.3.3 Changes in income

In this section we want to analyze the impact of changes in y1 and y2 on the consumer�schoice (c�1; c

�2; s

�). The consumer�s lifetime budget constraint is

c1 +c21 + r

= y1 � t1 +y2 � t21 + r

An increase in current income (y1 ")

We see from the budget constraint that an increase in will shift the budget constraint tothe right. If we assume both goods (current consumption and future consumption) arenormal, the consumer will increase the consumption in both periods. In order to increasethe consumption in the second period the consumer must increase his saving. Thus, anincrease in the current income will increase the current consumption by less than the changein the current income. We call this result consumption smoothing.To summarize:

y1 "=) c�1 "; c�2 "; s� ";�c1 < �y1


An increase in future income

We see from the budget constraint that an increase in will shift the budget constraint to theright. Given that both goods (current consumption and future consumption) are normal, theconsumer will increase the consumption in both periods. In order to increase the consumptionin the �rst period the consumer must decrease his saving. Thus, an increase in the futureincome will increase the future consumption by less than the change in the future income.To summarize:

y2 "=) c�1 "; c�2 "; s� #;�c2 < �y2

An increase in current and future income .

The budget constraint will shift to the right and again consumption in both periods will goup. It is unclear however what will happen to the saving. The impact on saving depends onthe relative magnitudes of the changes in y1 and y2.To summarize:

y1 "; y2 "=) c�1 "; c�2 "; s�?

Temporary vs. Permanent changes in income

The main point of the above experiments was to show that if the increase in income happensonly in one period, then the consumer will increase his consumption in that period by lessthan the change in that period�s income. This is called consumption smoothing.Is there evidence in the data of consumption smoothing? The next �gure shows the

percentage deviation from trend of real consumption per capita and real GDP per capita inthe U.S. What we can see from the next �gure is that consumption is smoother than GDP.

Detrended GDP, Consumption

0.08

0.06

0.04

0.02

0

0.02

0.04

0.06

194

7I

195

3II

195

9III

196

5IV

197

2I

197

8II

198

4III

199

0IV

199

7I

200

3II

Time

% D

evia

tion

det_GDPdet_C


4.3.4 Changes in the real interest rate

Suppose that . What is the e¤ect of the increase in real interest rate on the budget constraint?First of all, the slope of the budget constraint will increase in absolute value (it will becomemore steep). Also notice that whatever the interest rate, since the incomes and taxes did notchange, the new budget constraint has to pass through the endowment point E. Regardlessof the interest rate it is always feasible to consume in each period that period�s income. Thenext graph shows the e¤ect of on the budget constraint.

A

B

11 ty −

2c

)1( rwe +

E22 ty −

1cwe

The dashed line is the new budget constraint, after r ".An increase in the real interest rate has two e¤ects on the consumer. On the one hand

the relative price of current consumption in terms of future consumption (1 + r) has goneup. As a result, the consumer would like to substitute (the now more expensive) currentconsumption with the (the now cheaper) future consumption. This is called the substitutione¤ect �the change in consumption that results from the change in the relative prices. As aresult of the substitution e¤ect the consumer will reduce current consumption and increasefuture consumption. Since current income did not change, the saving must increase. Thus,as a result of the substitution e¤ect we have: .But there is another e¤ect. Notice that if the consumer was lender before the change (for

example chose the point A), then after the change he can still a¤ord the original bundle, andeven bundles that contain more of both goods than A. In other words, his purchasing powerincreased. We call this increase in purchasing power a positive income e¤ect. For a borrower(one who consumed at point B for example) the opposite happened. After the change he


can no longer a¤ord the previous bundle. We call this decrease in the purchasing power anegative income e¤ect.Our de�nition of income e¤ect is not precise, but it is intuitive and su¢ cient for the

purpose at hand. So for us a positive income e¤ect occurs if at the new prices, even if youtake some of the consumer�s income he can still a¤ord the old bundle. We say that a negativeincome e¤ect occurs if at the new prices the old bundle is not a¤ordable.Assuming that both goods are normal implies that the positive income e¤ect will cause

for the lender and for the borrower.The next table summarizes the results:

Lender BorrowerSubstitution e¤ect c�1 #; c�2 " c�1 #; c�2 "Income e¤ect c�1 "; c�2 " c�1 #; c�2 #Total e¤ect c�1?; c

�2 " c�1 #; c�2?

4.3.5 Changes in taxes and Ricardian equivalence

We have already analyzed the impact on consumers of the changes in incomes and . Noticethat our analysis also covers the changes in and since the budget isSo an increase in is just like a decrease in and an increase in is just like a decrease in .

There is however an important result in public �nance �the Ricardian equivalence theorem.Theorem (Ricardian equivalence):If the present value of government spending remains unchanged, then changes in the

taxes do not a¤ect the households�optimal consumption choice ( ).Proof:The government budget constraint isThus,We see that any changes in and must be such that the present value of taxes that the

consumer has to pay remain constant. This means that the consumer�s budget constraintremains unchanged,,since the last term on the right hand side (the present value of taxes) is unchanged. This

implies that the optimal choice of consumption ( ) for each consumer will remain unchanged.Notice however that the saving decision of consumers will change, and the aggregate

saving can change as well. To see that recall that . If the government changes the taxes (and ), then since will not change, then the saving must change by the amount of the changein , but in the opposite direction. More formally,So if the government reduces the taxes in the �rst period for each consumer by 1 unit,

then the consumers will increase their saving exactly by 1 unit.Discussion:The Ricardian equivalence theorem is a useful starting point for thinking about the

e¤ects of government de�cit. Governments can �nance their de�cit by taxing people or byborrowing (i.e. issuing debt). However, the government must eventually pay the debt byraising taxes in the future. The choice is therefore between "tax now" and "tax later". Oursimple framework suggests that from the consumers� point of view there is no di¤erence


between �tax now�or �tax later�. What matters for the optimal choice of consumption isthe present value of the lifetime taxes, , but not the speci�c magnitudes of and .It is useful to list the assumptions of our two-period model, which are responsible for the

Ricardian equivalence theorem.1. We assume that all the households are taxed equally. In the real world it could

be that the tax burden is not shared equally, so that tax policies will have an e¤ect on thedistribution of wealth in the economy.2. In the model, the same people who receive a tax cut are the ones who have to pay

the government debt in the future. In the real world the government can postpone the taxincrease until long in the future, when consumers who received the tax cut are either retiredor dead. In this case, the government tax policy will involve intergenerational transfer.3. In the model the taxes were lump-sum. If we change this assumption and let

the taxes be a fraction of income and also tax the interest rate earnings in the secondperiod, then the timing of taxes will matter for the optimal choice of the household. To seethis, notice that if the government taxes the interest earnings in the second period, thenthe household�s net-of-taxes saving in the second period is . This means that the slope ofthe budget constraint is now , so if the government changes the timing of the taxes, thehousehold�s budget constraint will not remain una¤ected.4. Finally, in the model consumers can borrow or lend at the same interest rate as

much as they please. If we relax these assumptions, the timing of taxes would a¤ect theoptimal consumption choices. A simple example can illustrate why. Suppose that someconsumer is not allowed to borrow, so he is forced to consume his endowment. Changes intaxes will change the consumer�s endowment, and therefore the consumption in both periodswill change.Appendix: Solving the consumer choice problem with Cobb-Douglas preferencesSuppose that the utility function has the form , where . This is a version of Cobb-Douglas

preferences. The coe¢ cient is a weight on the utility from consumption in the second period.The greater is, the more patient is the consumer. The consumer�s problem that we need tosolve is thereforeThis is a standard problem, similar to the ones we have solved before.The Lagrange function isFirst order conditions:Thus the familiar condition of MRS = slope of the budget constraint is, orPlugging this into the budget constraint gives the demand forThe demand for is thereforeThus, the consumer spends a fraction of his lifetime income on �rst period�s consumption

and a fraction of his lifetime income on the second period�s consumption. It makes intuitivesense that the higher the relative weight on a particular good in the utility, the greater isthe demand for that good.Substituting the expression of we in the demand gives the demand for and :The saving is thenNotice that higher interest rate implies lower consumption in the �rst period, and higher

saving and consumption in the second period. This makes intuitive sense, because is therelative price of current consumption in terms of future consumption. If interest rate goes


up, the consumer will substitute future consumption for current consumption and thereforewill save more in the �rst period.Also, the higher the current net-of-tax income is, the greater is the saving, and the higher

is the second period net-of-tax income, the lower is the saving. This makes intuitive sense.If the current income is relatively high, we would like to save more for the future, and if thefuture income is relatively high, we would like to save less for the future.Finally notice that an increase in by 1 unit will increase by less than 1 unit (by to be

precise). The saving will increase by . Thus we see the consumption smoothing result here.

4.4 Two-Period Model of Investment

1. There is one �rm that can produce output in two periods according to

Y1 = A1K�1L

1��1

Y2 = A2K�2L

1��2

where A1, A2 are productivity parameters (TFP - Total Factor Productivity), K1,K2 are the levels of physical capital in the two periods, and L1, L2 are labor inputs(number of workers employed by the �rm).

2. The �rm owns the capital stock in each period, and consumers own the �rm. That is,the �rm belongs to the shareholders, who are entitled to the stream of pro�ts from the�rm.

3. The capital stock evolves according to

K2 = (1� �)K1 + I (4.7)

where � is depreciation rate of capital and I is investment in capital in the �rst period.

4. The capital in the �rst period, K1, is exogenously given in the model, while the secondperiod capital is a result of the �rm�s investment decision.

5. The �rm decides on the labor demand in each period (L1; L2) and on the magnitudeof the investment in the �rst period, I, and thus implicitly chooses K2.

6. The distributed pro�t (dividends) in each period is given by

�1 = Y1 � w1L1 � I�2 = Y2 + (1� �)K2 � w2L2

Notice that since the economy lasts for two periods, the �rm can sell the nondepreciatedcapital stock, so the revenue in the second period is Y2 + (1� �)K2.

4.4. TWO-PERIOD MODEL OF INVESTMENT 49

4.4.1 Optimal investment decision

Recall that the �rm is owned by the consumer and the lifetime value of the �rm to theconsumer is given by

V = �1 +�21 + r

where r is the real interest rate. Thus, the �rm�s problem is to choose L1; L2; I;K2 tomaximize the present value of the stream of dividends

maxL1;L2;I;K2

V = A1K�1L

1��1 � w1L1 � I +

A2K�2L

1��2 + (1� �)K2 � w2L2

1 + rs:t:

K2 = (1� �)K1 + I

Notice that choosing particular value of I essentially pins down K2, we can let the �rmdirectly choose K2. Substitute the constraint into the objective and obtain the following�rm�s problem:

maxL1;L2;K2

V = A1K�1L

1��1 � w1L1 �K2 + (1� �)K1 +

A2K�2L

1��2 + (1� �)K2 � w2L2

1 + r

The �rst order conditions with respect to L1 and L2 are

@V

@L1= 0, ) (1� �)A1K�

1L��1 = w1

@V

@L2= 0, ) (1� �)A2K�

2L��2 = w2

This means that in each period the �rm wants to hire labor up to the point where themarginal product of labor equals the wage. The �rst order condition with respect to K2 is

@V

@K2

= �1 + �A2K��12 L1��2 + 1� �1 + r

= 0

The interpretation is as follows. Increasing next period�s capital by 1 unit costs 1 unit ofcurrent dividends. The bene�t in the next period comes from two sources: (1) the revenuein the next period will increase by the marginal product of capital, �A2K��1

2 L1��2 , and (2)the nondepreciated capital can be sold in the next period and thus we have 1 � � units ofaditional revenue. The bene�t from investment is collected in the second period, so it isdiscounted by 1+ r to obtain its present value. Thus, the optimal level of K2 (and thereforeof I) is determined by the condition

�1 + �A2K��12 L1��2 + 1� �1 + r

= 0

�A2K��12 L1��2 � � = r (4.8)

which means that the real interest rate is equal to the marginal product of capital net ofdepreciation. This condition makes intuitive sense. In equilibrium the return in the �nancial


market must be equal to the return in the capital market. If we invests 1 unit in the physicalcapital, the net return will be the marginal product of capital net of depreciation. If oneinvests in the �nancial market, the net return is r. In equilibrium there should be no arbitrageopportunities, and thus the rates of return must be equalized. It should make intuitive sensethat an increase in r will decrease the investment in physical capital since the return inthe �nancial market becomes relatively higher.

4.4.2 Changes in interest rate

The next �gure illustrates the e¤ect of an increase in r on the optimal investment decision.

δ

δ

+

+

r

r '

22 ' KK 2K

KMP

θθθ −− 12

122 LKA

The downward slopping curve is the marginal product of capital MPK . The optimal K2 isobtained at the point where the marginal product of capital is equal to r + �. Higher realinterest rate leads to lower K2 and thus lower investment (since I = K2 � (1� �)K1). Theintuition is as follows. The real interest rate represents the opportunity cost of investmentin physical capital. Higher real interest rate means that the return in the �nancial marketis higher, which makes the investment in physical capital less attractive. As shown in the�gure, when interest rate goes up from r to r0, the optimal future capital falls from K2 toK 02.

4.4.3 Changes in technology

The next �gure illustrates the e¤ect of an increase in A on the optimal investment decision.

4.4. TWO-PERIOD MODEL OF INVESTMENT 51

'22 KK

δ+r

2K

KMP

θθθ −− 12

122 LKA

θθθ −− 12

122 ' LKA

Notice that the marginal product curve shifts upward, so that for any given level of K2 itsmarginal product increases. The optimal level of K2 (and also of investment) will thereforeincrease. The net return on investment in physical capital is the marginal product of capital(net of depreciation), hence it is intuitive that when the marginal product of capital goesup, the investment in physical capital should go up.

4.4.4 Solving for optimal investment

Solving equation (4.8) gives the optimal future capital K2 as a function of real interest rate.

�A2K��12 L1��2 = r + �

�A2L1��2

r + �= K1��

2

K2 =

��A2L

1��2

r + �

� 11��

Notice that from the above equation we clearly see that optimal K2 is decreasing in interestrate r and increasing in the productivity level A. Now substitute this into equation (4.7) to�nd the optimal investment

I =

��A2L

1��2

r + �

� 11��

� (1� �)K1 (4.9)

Observe that the optimal investment is decreasing in r and decreasing in current capital(K1). The intuition for the last observation is simple: if today�s capital stock is big, wedon�t need to invest as much in order to attain the optimal level of future capital.The next �gure show the investment demand curve as a function of real interest rate.


)(rI

r

I

The price of investing in physical capital is the real interest rate - the opportunity cost ofinvestment. Changes in the real interest rate is re�ected in a movement along the samedemand curve, but the curve itself will not shift. Changes in parameters other than r willshift the entire curve. For example, higher expected productivity (A2 ") will shift the entiredemand curve to the right. With higher future productivity the �rm would like to investmore at any given real interest rate. The next �gure shows the impact of (A2 ") on thedemand for investment.

)';( 2ArI);( 2ArI

r

I

In the above �gure A02 > A2 results in shift to the right of the demand for investment. Itis clearly seen from equation (4.9) that an increase in future productivity should increaseinvestment.

4.5 Capital Market

Now we are ready to put together the supply of saving with the demand for investment. Westart with closed economy �rst. The next �gure shows the capital market. For simplicity,we draw linear supply and demand curves.

4.5. CAPITAL MARKET 53

**, SI

*r

)(rS

)(rI

r

SI ,

The supply of saving curve is assumed upward slopping. Recall from our analysis of thesaving decisions of household, we concluded that a lender will not necessarily increase hissaving when the real interest rate goes up. Nevertheless, we assume that the substitutione¤ect is stronger than the income e¤ect, which ensures that the total saving of householdsis increasing in interest rate.In a closed economy, national saving and domestic investment are equal. The above graph

shows that the real interest rate and the amount of saving and investment is determined inequilibrium, at the intersection of supply and demand.In an open economy, national saving can di¤er from domestic investment. For example,

if the world interest rate is below r� then the national saving will not be enough to �nancethe domestic investment, and the di¤erence has to be borrowed. As we discussed before,borrowing from the rest of the world is the negative of trade de�cit. The next �gure illustratesan economy with S < I.

NX

** IS

*r

)(rS

)(rI

r

SI ,

In this economy (as in the U.S. currently) the national saving falls short of the domesticinvestment, and the economy borrows the amount of �NX (the trade de�cit).What a¤ects the trade de�cit according to our theory? It is obvious from the graph that

national saving and domestic investment together determine the size of the trade de�cit. Inthe next sections we will work with this model, to analyze the impact of di¤erent events onthe saving, investment and the trade de�cit in the economy.


4.5.1 Decline in government de�cit (SG ")The total saving curve will shift to the right, as shown in the next �gure.

NX

**** ISS

*r

)(rS

)(rI

r

SI ,

For simplicity we assume that this change will not a¤ect the world interest rate. We seethat an increase in government saving leads to an increase the national saving and lowersthe trade de�cit. As bigger fraction of the domestic investment is funded by national saving,there is less borrowing from the rest of the world.

4.5.2 Increase future productivity at home (A2 ")The demand for investment curve will shift to the right, as demonstrated in the next �gure.

NX

**** IIS

*r

)(rS

)(rI

r

SI ,

If initially the national saving were not su¢ cient to fund the domestic investment, then afteran increase in domestic investment the shortage is greater. The borrowing from the rest ofthe world is increasing, meaning higher trade de�cit. The domestic investment increases,where all of the increase is funded by foreigners.

4.6. SUMMARY 55

4.6 Summary

1. We showed that saving and investment in any economy are related. This relationshipis called the "Saving and Investment Equation".

2. We presented a theory of saving in a two period model of intertemporal choice. Themodel allows us to study the impact real interest rate, current and future income, andgovernment tax policies on consumption and saving behavior. The model delivers anintuitive result, that people tend to smooth consumption. Another important result isRichardian Equivalence Theorem, which says that under certain conditions changes ingovernment taxes do not a¤ect the consumption and saving decisions of households.

3. We presented a theory of Investment decision by �rms. The demand for investmentis decreasing in real interest rate and increasing in future productivity.

4. Putting together the saving and investment theories, allows us to analyze the capitalmarket. We demonstrated how real interest rate, the level of saving, investment andtrade de�cit are determined in the capital market.

4.7 Appendix: Firm With Unlimited Life Span

In these notes we analyzed the investment decision of �rms that live for two periods. Nowwe show that the same condition for optimal investment holds when the �rm lives unlimitednumber of periods. Suppose that the production function is more general than we usedabove, namely

Yt = F (Kt; Lt; t)

The present value of the stream of pro�ts is

V =1Xt=0

�t

(1 + r)t=

1Xt=0

F (Kt; Lt; t)� wtLt � It(1 + r)t

The �rm�s problem is

maxfKt+1;Ltg1t=0

=1Xt=0

F (Kt; Lt; t)� wtLt � It(1 + r)t

s:t:

Kt+1 = (1� �)Kt + It

Substituting the constraint into the objective, gives

maxfKt+1;Ltg1t=0

=

1Xt=0

F (Kt; Lt; t)� wtLt �Kt+1 + (1� �)Kt

(1 + r)t


F.O.C. with respect to Kt+1 is

�1(1 + r)t

+F1 (Kt+1; Lt+1; t) + 1� �

(1 + r)t+1= 0

F1 (Kt+1; Lt+1; t) + 1� � = 1 + r

F1 (Kt+1; Lt+1; t) = r + �

Thus, as before, the �rm will invest up to the point where the marginal product of capitalF1 (Kt+1; Lt+1; t) is equal to the sum of the real interest rate and depreciation. Intuitively,the cost of investment is forgone interest rate and depreciation, and that has to be balancedby the return to investment - marginal product of capital.

Chapter 5

Economic Growth

5.1 Introduction

The most important macroeconomic observation in the world is the huge di¤erences in output(and income) per capita across countries. More than 60% of the world population is at least7 times poorer than the average American. In the poorest countries the GDP per capita isat least 40 times smaller than that in the U.S. What accounts for such big di¤erences? Wewill not attempt to answer this question here, but rather focus on one important source ofcross country di¤erences - the di¤erence in capital per worker.The next �gure shows the scattergram of capital per worker and GDP/capita in a large

sample of countries. We see that there is a strong positive correlation between capital perworker and GDP/capita across countries. In other words, countries with high GDP/capitatend to have higher capital per worker.

57

58 CHAPTER 5. ECONOMIC GROWTH

But why some countries have much less capital per worker than others? To answerthis question recall that capital is created through investment in new capital. The nextgraph shows the average investment over 1960-2000 v.s. physical capital per worker in 2000.Observe that, not surprisingly, there is a strong positive correlation between the investmentrate and physical capital per worker.

These observations motivated the Solow growth model. Robert Solow received a NobelPrize in Economics in 1987 "for his contributions to the theory of economic growth". Hereis part of the press release which describes Solow�s contribution:

"The study of the factors which permit production growth and increased welfare has beena central feature in economic research for many years. Robert M. Solow�s prize recognizeshis exceptional contributions in this area.

It is eminently reasonable to imagine that increased per capita production in a countrymay be the result of more machines and more factories (a greater stock of real capital). Butthis increased production may also be due to improved machines and more e¢ cient produc-tion methods (which may be termed technical development). In addition, better education andtraining, and improved methods of organizing production may also give rise to increased pro-ductivity. The discovery of fresh natural resources, or improvements in a country�s positionon the world market, may also lead to higher standards of living. Solow has created a theoret-ical framework which can be used in discussing the factors which lie behind economic growthin both quantitative and theoretical terms. This framework can also be exploited to measureempirically the contributions made by various production factors in economic growth."

5.2. THE SOLOW MODEL 59

5.2 The Solow Model

5.2.1 Description of the model

� Output is produced according to Yt = AtK�t L

1��t , 0 < � < 1.

� Capital evolves according to Kt+1 = Kt (1� �) + It, where � is the depreciation rateand It is aggregate investment.

� People save a fraction s of their income. This fraction is exogenous1. Thus, the totalsaving and total investment in this (closed) economy is

St = It = sYt

� The population of workers grows at a constant rate of n, which is exogenous in thismodel. Thus, Lt+1 = (1 + n)Lt.

We neglect the di¤erences between population and population of workers for the sake ofsimplicity, and use the terms "output per worker" and "output per capita" Interchangeably.

5.2.2 Working with the model

Now we derive the predictions of the model. The output per worker is:

yt =YtLt=AtK

�t L

1��t

Lt= At

�Kt

Lt

��= Atk

�t

The law of motion of capital per worker is

Kt+1

Lt+1=

Kt (1� �)Lt+1

+ItLt+1

kt+1 =Kt (1� �)Lt (1 + n)

+sYt

Lt (1 + n)

kt+1 =kt (1� �)1 + n

+sAtk

�t

1 + n(5.1)

Equation (5.1) describes the law of motion of physical capital per worker. If At is �xed atsome level A, then the law of motion can be illustrated graphically, as in �gure (5.1).With �xed productivity it can be shown that the capital per worker converges to a steady

state level, such thatkt+1 = kt = k

ss 8tThe steady state level of capital per worker can be seen in the graph at the intersection ofthe law of motion equation with the 450 line. It can be shown that starting from any levelof capital per worker, it converges to the steady state level kss. Thus, the prediction of theSolow model is that with �xed A, the capital per worker will converge to kss.

1We call a variable endogenous if it is determined within the model and exhogenous if it is determinedoutside the model. For example, in the model of a market (supply and demand diagram), the price andquantity traded of the good are endogenous variables, while other variables that determine the location ofthe supply and demand curve, such as income and prices of other goods, are assumed exogenous.


Figure 5.1: Law of motion of physical capital per worker.

Law of motion of capital per worker

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2k_t

k_t+

1k_t+145_deg

Finding the steady state

Using the law of motion and the de�nition of the steady state

kt+1 =kt (1� �)1 + n

+sAk�t1 + n

k =k (1� �)1 + n

+sAk�

1 + n

k (1 + n) = k (1� �) + sAk�

k (n+ �) = sAk� (5.2)

The intuition behind the last equation is as follows. The left hand side shows the decline incapital per worker due to depreciation and growth in the number of workers. The right handside is the investment per worker, i.e. the increase in capital per worker. At the steady statethe decline in capital per worker due to depreciation and growth in the labor force must beo¤set by the increase in capital per worker due to investment.The steady state capital per worker is

kss =

�sA

n+ �

� 11��

(5.3)

The steady state output per worker is

yss = Ak�ss = A

�sA

n+ �

� �1��

= A1

1��

�s

n+ �

� �1��

(5.4)

The steady state consumption per worker is

css = (1� s)Ak�ss (5.5)

5.2. THE SOLOW MODEL 61

The predictions of the model

Observe that kss is increasing in the saving (=investment) rate and productivity, and de-creasing in the population growth rate and depreciation. This result is fairly intuitive. First,higher saving rate means that investment per worker is higher, and thus capital per workershould be higher. Higher productivity means that investment per worker is higher simplybecause more output is produced per worker. This also leads to higher capital per worker inthe steady state. Equation (5.2) can help with developing the intuition. Think of the righthand side of (5.2) as the "�ow in" the stock of capital per worker, i.e. the investment perworker. Higher s or higher A increase the "�ow in", and result in higher stock of capital perworker. Now from equation (5.3) we can see that steady state capital per worker is decreas-ing in depreciation and the growth rate of population. Higher depreciation means that the"�ow out" of the stock of capital is higher, so the stock of capital is lower. Similarly, highergrowth rate of the workers population also reduces the capital available per worker, and in asense works just like depreciation. Observe that n and � appear together in equations (5.3)and (5.4). Similarly, the steady state output per worker yss is increasing in the saving rateand productivity, and decreasing in the population growth rate and depreciation, just likekss. This is just because output per worker is increasing function of the capital per worker.The Solow model therefore predicts that countries with higher investment rates, should

on average, enjoy higher standard of living. This prediction is consistent with the data, ascan be seen from �gure (5.2).

Figure 5.2: Investment rate and GDP per capita.

The Solow model also predicts that countries with higher growth rate of population,should on average enjoy lower standard of living. This prediction is also consistent with the


data.According to the Solow model, higher saving rate leads to higher output per worker.

Does this mean that the policy recommendation implied by the model is to save as muchas possible? The answer to this question is absolutely no. Suppose that at the extreme theconsumers save all their income. If this happens, the consumers will not consume anything,and just starve and die. The next section discusses the optimal saving rate, i.e. the savingrate that maximizes the steady state consumption per capita.

Optimal saving rate

Notice that although higher saving rate leads to higher steady state level of capital perworker and output per worker, it does not necessary lead to higher consumption per worker.Observe from equation (5.5) that on the one hand higher s leads to higher income per worker,but on the other hand higher saving rate means that a smaller fraction of that income isconsumed. We can �nd the optimal saving rate, i.e. the saving rate that maximizes thesteady state consumption per worker. This saving rate is called the golden rule savingrate.

css = (1� s)Ak�ss = Ak�ss � (n+ �) kss

maxkss

css = Ak�ss � (n+ �) kss

First order condition:

�Ak��1GR = n+ �

kGR =

��A

n+ �

� 11��

Now comparing this with the steady state capital

kss =

�sA

n+ �

� 11��

implies thatsGR = �

5.3 Endogenous Growth Model

Recall that in the Solow growth model, when the TFP (= A) is �xed, capital per worker,output per worker and consumption per worker, all converge to a steady state level:

kss =

�sA

n+ �

� 11��

yss = Ak�sscss = (1� s) yss

5.3. ENDOGENOUS GROWTH MODEL 63

Higher TFP means higher levels of the endogenous variables (kss; yss; css), higher saving rate(s) results in higher kss and yss, but not necessarily higher css and higher n results in lower(kss; yss; css). In the Solow growth model, sustained growth in output per worker is possibleonly if A is growing all the time (perpetual growth).It is important to realize that the Solow growth model does not o¤er an explanation for

why growth occurs (i.e. why A grows). In the Solow model, country 1 will grow faster thancountry 2 because the A of country 1 grows faster than that in country 2. The question iswhat is A and what causes it to grow? This is the question that we address in this section.In addition to the TFP, we observed that di¤erences in saving rate in the Solow model

create di¤erences in the steady state output per worker. In the Solow model the saving rateis assumed to be �xed and exogenous. Is this a good assumption? People decide how muchof their current income they wish to consume and how much to save based on many factors.Among those factors one can list the interest rate on saving, expected future income, etc.Chapter 4 studies the consumption and saving decision of households in more depth.Now we modify the Solow model slightly such that the TFP and its growth depend on

other parameters in the model. In particular, the TFP re�ects the level of human capital inthe economy. Human capital is an abstract term which is typically associated with the skillsand education of workers in the economy. The resulting model will be called EndogenousGrowth Model. The human capital in this model depends on the fraction of time thatpeople spend on human capital accumulation (education), and the e¢ ciency of human capitalaccumulation. It is debatable whether the name �Endogenous� is justi�ed here, since thefactors that determine the TFP are themselves exogenous. Having said that, we will followthe convention of naming models in which the growth rate of endogenous variables dependson some other parameters of the model, as �Endogenous Growth Models�.The main results of the model are:

1. The greater the fraction of time spent on human capital accumulation, the greater isthe growth rate of endogenous variables.

2. The greater is the e¢ ciency of human capital accumulation, the greater is the growthrate of endogenous variables.

5.3.1 Description of the model

Consumers: Each consumer saves a fraction s of his income (and thus consumes a fraction1 � s of his income). Each consumer is endowed with 1 unit of time, with fraction u spenton work, and a fraction 1� u is spent on human capital accumulation.Technology: Output is produced using Capital and labor. Output at time t is given by:

Yt = K�t (uhtLt)

1��

where Kt is capital Lt is the population of workers, and ht is the level of human capital perworker. Thus, the labor input, �t � uhtLt, is not just the number of bodies, but dependson the time worked and the quality of workers (ht). The above production function can bewritten as

Yt = (uht)1��K�

t L1��t = AtK

�t L

1��t


This production technology is the same as in the Solow model, except now we have a storyabout the formation of the TFP, i.e., the TFP depends on the stock of human capital in theeconomy: At = (uht)

1��.Population: Population of workers grows at constant and exogenous rate n. Thus, the

population of workers evolves according to

Lt+1 = (1 + n)Lt

Physical Capital: Physical capital stock evolves according to

Kt+1 = (1� �)Kt + It

where � is depreciation rate and It is aggregate investment.Human Capital: The human capital per worker evolves according to

ht+1 = b (1� u)ht

where (1�u) is the fraction of time the people spend on human capital accumulation and theparameter b re�ects the e¢ ciency of human capital accumulation. We assume that b > 0,otherwise the human capital becomes negative. Notice that in order to enable growth inhuman capital, we must have b (1� u) > 1.

5.3.2 Working with the model

We start by deriving the growth rate of TFP.

At+1At

=(uht+1)

1��

(uht)1�� =

�ht+1ht

�1��= [b (1� u)]1��

It is important to realize that in this model the endogenous variables will not converge to asteady state since At is not �xed. It is possible to formally prove that in this model, all theendogenous per capita variables (kt; yt; ct) will be growing at a constant rate in the long run.We will postpone the formal proof for the appendix to this chapter, and here we only derivethe long run growth rate of endogenous variables. As a side comment, a situation in whichall the endogenous variables are growing at constant, but not necessarily the same, rate iscalled Balanced Growth.We �rst �nd the long run growth rate of capital per worker. The law of motion of capital

per worker is the same as in the Solow model

kt+1 =kt (1� �)1 + n

+sAtk

�t

1 + n

Dividing both sides by kt gives

kt+1kt

=(1� �)1 + n

+sAtk

��1t

1 + n


If capital per worker grows at constant rate, then the left hand side must be constant.This implies that the second term on the right hand side must be constant as well, and inparticular, Atk��1t must be constant. Thus,

At+1k��1t+1

Atk��1t

= 1

[b (1� u)]1��kt+1kt

��1= 1�

kt+1kt

�1��= [b (1� u)]1��

kt+1kt

= b (1� u)

We proved above that if capital per worker grows at constant rate, then this rate must be

kt+1kt

= b (1� u)

Next, we �nd the long run growth rate of output per worker.

yt+1yt

=At+1k

�t+1

Atk�t= [b (1� u)]1�� [b (1� u)]� = b (1� u)

Thus, output per worker gros at the same rate as physical capital per worker. Another wayto prove the above is to use tha fact that Atk��1t is constant, wich is the ratio of output tocapital

Atk��1t =

Atk�t

kt=ytkt

If the output/capital ratio is constant, this means that the output must grow at the samerate as capital.Finally, it should be clear that consumption per worker must grow at the same constant

rate as output per worker, since consumption is proportional to output:

ct = (1� s) yt

Thus, in the endogenous growth model, all the per worker variables grow (in the long run)at a constant rate of b (1� u).Summary:

At+1At

= [b (1� u)]1��

kt+1kt

=yt+1yt

=ct+1ct

= b (1� u)


5.3.3 Economic Policy and Growth

In this endogenous growth model, the higher the fraction of time that is allocated to humancapital accumulation (1� u), the higher will be the long run growth rate of the endogenousvariables (At; kt; yt; ct). Similarly, the higher is the e¢ ciency of human capital accumulation(b), the higher is the long run growth rate of the endogenous variables (At; kt; yt; ct). Twoquestions regarding government policy arise immediately: (1) can the government a¤ect band (1� u)? and (2) Should the government try to a¤ect b and (1� u)?Many believe that the answer to the �rst question is yes. The government can increase

b by implementing better incentives for performance in the school system, or by changingthe mix of private and public schools. The government can increase (1 � u) by subsidizingeducation.The answer to the second question is not clear. Notice that Yt = (uht)

1��K�t L

1��t . We

see right away that increasing (1 � u) means that u will drop and the current output willdrop. Thus, the short run e¤ect is a drop in (At; kt; yt; ct). True, in the long run thesevariables will grow faster, but this does not come without cost. What about increasing b?In this framework we do not model the cost of increasing b, but we can be sure that nothingcomes without cost in the real world, including reforming the educational system.

5.3.4 Evidence

The next �gure suggests that there is a positive correlation between educational attainmentand growth rate of real GDP. To me it seems that an inverse U curve can better �t the datathan a straight line with positive slope. This could mean that there is a level of educationalattainment that maximizes growth.


The next graph shows a clear positive trend between educational attainment and outputper worker. The two graphs thus support the prediction of the model that higher (1� u) isassociated with faster growth of output per worker.

5.3.5 Appendix

We now prove that in this model all the endogenous per capita variables, (kt; yt; ct), grow atconstant rate in the long run. The production function is

Yt = K�t �

1��t

where �t = uhtLt is the labor input. Notice that the growth rate of this quantity is:

�t+1�t

=uht+1Lt+1uhtLt

= b (1� u) (1 + n)

We de�ne the variables in e¢ ciency units, i.e mormalized by the total labor input, as follows:

k�t =Kt

�t; y�t =

Yt�t; c�t =

Ct�t

Next we show that all the � variables converge to a steady state. We derive the law ofmotion of k�t :

Kt+1 = (1� �)Kt + sK�t �

1��t

Kt+1

�t+1=

(1� �)Kt

b (1� u) (1 + n) �t+

sK�t �

1��t

b (1� u) (1 + n) �t

k�t+1 =(1� �)

b (1� u) (1 + n)k�t +

s

b (1� u) (1 + n) (k�t )�


The law of motion of k�t has the same shape as the one in the Solow model, and therefore inthe long run k�t converges to some constant level k

�ss. Similarly, is is easy to see that y

�t and

c�t converge to a steady state in the long run. Then, if k�t is constant, we have:

k�t+1k�t

=Kt+1=�t+1Kt=�t

=Kt+1=uht+1Lt+1Kt=uhtLt

=kt+1kt

� htht+1

= 1

) kt+1kt

=ht+1ht

= b (1� u)

Similarly, we show that the growth rate of yt and ct is also constant in the long run.

Chapter 6

Money and Prices

6.1 What is Money?

We de�ne money as the medium of exchange in the economy, i.e. a commodity or �nancialasset that is generally acceptable in exchange for goods and services. Currency which consistsof coins and bank notes1 are a medium of exchange. Checking accounts can also be used asa medium of exchange, since a consumer can write a check in exchange for goods. Travelerschecks are another example of money. There are other assets where it is not clear if theyshould be considered as money, for example saving accounts. A consumer can withdraw froma saving account and pay for goods, but the main purpose saving accounts is to store valuenot to serve as a medium of exchange.Are credit cards considered a form of money? The answer is no. If you buy goods from

a supermarket using a credit card, the credit card company will pay the shopkeeper todayand you will have an obligation to pay the credit card company when your credit card billcomes in. This obligation to the credit card company does not represent money. The moneypart of the transaction between you and the credit card company only comes into play whenyou pay your bill.There are several de�nitions money aggregates:

1. M1 - Measure of the U.S. money stock that consists of currency held by the public,travelers checks, demand deposits and other checkable deposits including NOW (ne-gotiable order of withdrawal) and ATS (automatic transfer service) account balancesand share draft account balances at credit unions.

2. M2 - Measure of the U.S. money stock that consists of M1, certain overnight repur-chase agreements and certain overnight Eurodollars, savings deposits (including moneymarket deposit accounts), time deposits in amounts of less than $100,000 and balancesin money market mutual funds (other than those restricted to institutional investors).

3. M3 - Measure of the U.S. money stock that consists of M2, time deposits of $100,000 ormore at all depository institutions, term repurchase agreements in amounts of $100,000

1Take a look at $1 bill, or any other U.S. paper money. It is called "Federal Reserve Note".

69

70 CHAPTER 6. MONEY AND PRICES

or more, certain term Eurodollars and balances in money market mutual funds re-stricted to institutional investors.

The most important monetary aggregate is M1 and is often referred to as the moneysupply in the economy.

6.2 The Demand for Money

There are many theories that try to explain how the quantity of money that households wantto hold is determined. Most of the modern theories are quite complicated and cannot bepresented in this class. We will describe two theories here: (1) Quantity Theory of Money2,and (2) Money in the Utility Function.

6.2.1 Quantity Theory of Money

According to this theory, households want to hold money in proportion to the dollar valueof goods produced in the economy. Let Yt be the real GDP at time t and let Pt be the pricelevel (GDP de�ator). Thus, the nominal GDP is PtYt. The demand for money according tothis theory is given by

MDt = kt � PtYt (6.1)

where MDt is the demand for money and kt is the propensity to hold money. Typically,

kt < 1 since each dollar can be used more than once every year, so if the households spendPtYt during year t, they need to keep only a fraction of their planned spending as money. In2005 kt = 0:11, which means that in 2005 households held money at the amount of 11% ofthe GDP. In other words, each dollar circulated 9 times during 2005 (1=0:11 = 9). We de�nethe velocity of money as the average number of times a piece of money circulates during theyear. The velocity is denoted by Vt and de�ned as

Vt =PtYtMDt

:

Thus, equation (6.1) can be written as

MDt Vt = PtYt (6.2)

Notice that Vt = 1=kt. In equilibrium, the money demand�MDt

�is equal to the money

supply�MSt

�, and denoted by Mt and the above can be written as

MtVt = PtYt (6.3)

Equation (6.3) is called the quantity equation. It is important to realize that the quantityequation always holds for every economy in the world. This is simply because we de�ne thevelocity to be such that the above equation holds.

2Sometimes called the Classical Theory.

6.2. THE DEMAND FOR MONEY 71

The quantity theory is silent about what determines the velocity Vt and real GDP Yt,but nevertheless this equation is useful for relating money, in�ation and real GDP. Divideequation (6.3) at time t+ 1 by the same equation at time t:

Mt+1Vt+1MtVt

=Pt+1Yt+1PtYt�

1 + M��1 + V

�=

�1 + P

��1 + Y

�(6.4)

where "hat" above a variable denotes its growth rate, for example M = (Mt+1 �Mt) =Mt.Taking logs of equation (6.4) gives

ln�1 + M

�+ ln

�1 + V

�= ln

�1 + P

�+ ln

�1 + Y

�For small growth rates the above is approximately

M + V = P + Y (6.5)

Suppose that velocity is constant, i.e. V = 0. Then we have

M = P + Y

RearrangingP = M � Y (6.6)

The growth rate of the price level is in�ation � = P . Equation (6.6) tells us that if velocityis constant, then the in�ation rate in the economy is approximately equal to the growth rateof money supply minus the growth rate of the real GDP. For example, suppose that during2005 the money supply increased by 4% and the growth rate of real GDP was 1.5%, thenthe in�ation rate must be 2.5%, if velocity did not change during 2005.The next �gure shows the velocity in the U.S. since 1959.

Velocity of M1 in the U.S.

012345

6789

10

1955 1965 1975 1985 1995 2005

Years

Velo

city

Notice that velocity has increased during the time period in question by a factor of 2.5.During the decade of 1995-2005 the velocity increased from 6 to 9, which is 50% increase.This means that each piece of money circulates more times than it used to in the past. Inother words, people economize on money holdings. In the next section we attempt to developa theory of velocity.


6.2.2 Money in the Utility Function

Consumers derive utility from consumption C and real balances M=P . We can think ofthe later as liquidity services, or the purchasing power of the nominal money holdings. Theopportunity cost of holding money is the nominal interest rate i. The consumer�s income isequal to the nominal GDP (PY ). The consumer�s problem is therefore

maxC;M

� lnC + (1� �) ln�M

P

�s:t:

PC + iM = PY

Now divide the budget constraint by the price level

maxC;M

� lnC + (1� �) ln (M)� (1� �) lnP

s:t:

C + iM

P= Y

Notice that the last term in the utility function can be dropped because utility is invariantwith respect to monotone transformations. Hence, the consumer�s problem is

maxC;M

� lnC + (1� �) ln (M)

s:t:

C + iM

P= Y

We know that when preferences are of the Cobb-Douglas form, the demand is

C = �Y

MD =(1� �)i

PY

The demand for money that we derived, is increasing GDP and decreasing in nominal interestrate - which represents the cost of holding money. Thus, according to this model the velocityis

V =i

1� �

This result makes intuitive sense: as interest rate goes up, the opportunity cost of holdingmoney goes up and the households economize on money holdings. As a result, each piece ofmoney is used more times during the year. The next �gure shows the graphs of velocity andfederal funds rate since 1959.

6.3. MONEY SUPPLY 73

Federal Funds rate (i) and Velocity (V)

02468

1012141618

1955 1965 1975 1985 1995 2005Year

Fede

ral F

unds

Rat

e

012345678910

Velo

city

i V Poly. (V) Poly. (i)

The dashed lines are polynomial trends (degree 6). As we see from the above �gure, untilearly 80�s the trends of velocity and interest rates move in the same direction, as the modelpredicts. After 1981 however, the trends move in the opposite directions. The only way thatour model can reconcile this observation is by increasing �. Recall that � is the weight ofconsumption in the utility function while 1� � is the weight on real balances in the utilityfunction. Decline in 1 � � means that liquidity services provided by the real balances arenot as valuable as before. Our model cannot o¤er any explanation why this might happenthough. One might conjecture that the sharp increase in velocity in the last decade (50%increase) has something to do with innovations in the banking and payment system. Theseinclude ATM machines, electronic transfers, etc. Anything that allows faster payments andsmaller average holdings of money will increase the velocity. For example, if householdsreceive income every two weeks instead of every month, the average money holding will belower and velocity higher.

6.3 Money Supply

In this section we explain how the Federal Reserve System3 (FED for short) and the com-mercial banks create money. We de�ne the Monetary Base (MB) as all the coins and papermoney that is created by the FED. The Monetary Base is held partly as reserves (R) of thecommercial banks and partly as currency in the hands of public (CU). Thus,

MB = R + CU

The money supply M consists of currency and checking deposits, which is the M1 moneyaggregate de�ned above. Thus

M = CU +D

The commercial banks are required to keep certain minimum percentage of deposits (D) asliquid reserves of cash and currency. This fraction is called the required reserve ratio and

3To learn about the history, the structure and the activities of the Federal Reserve System, visithttp://www.federalreserveeducation.org/fed101/index.htm.


denoted by rd, where

rd =R

D

Suppose that consumers want to hold certain amount of cash, as a fraction of their deposits.Let this fraction (currency/deposits ratio) be cd, that is

cd =CU

D

Under these assumptions, M , D, R, and CU are all proportional to the monetary base.

M

MB=D + CU

R + CU=1 + cd

rd+ cd

The last equality is obtained by dividing the numerator and the denominator byD. Similarly,

D

MB=

D

R + CU=

1

rd+ cdR

MB=

R

R + CU=

rd

rd+ cdCU

MB=

CU

R + CU=

cd

rd+ cd

Thus

�M =

�1 + cd

rd+ cd

��MB

�D =

�1

rd+ cd

��MB

�R =

�rd

rd+ cd

��MB

�CU =

�cd

rd+ cd

��MB

The magnitude mm =�1+cdrd+cd

�is called the money multiplier, and it gives us the change in

the money supply that results from $1 change in the monetary base.Now we are ready to illustrate how the FED, together with the commercial banks, a¤ects

the money supply. The FED has several policy instruments, the most popular of which isopen market operations. An open market operation is purchasing and selling governmentbonds. Every purchase of bonds by the FED increases the monetary base (injects money intothe economy). Because of the partial reserve requirements, the commercial banks can lendsome of the money received from the FED and thereby generating additional money. TheFED can also alter the reserve requirement ratio, but it rarely does so. We will demonstratethe two modes of operation in the next example.

6.3. MONEY SUPPLY 75

6.3.1 Example of Money Creation

Suppose the currency/deposit ratio that the public wants is 20% and the reserve requirementratio is 10%. The following is a consolidated balance sheet of the commercial banks.

Balance sheet of the commercial banksAssets LiabilitiesR = 10 D = 100BG = 25L = 65100 100

where R is the reserves, BG government bonds, L is loans, and D is deposits. Noticethat the balance sheet must always be balanced. Also observe that the banks conform tothe reserve requirement and indeed R=D = 0:1.

1. Find the monetary base in this economy.

MB = CU +R

CU = cd �D = 0:2 � 100 = 20R = 10

ThusMB = 20 + 10 = 30

2. Find the money supply in this economy

M = CU +D = 20 + 100 = 120

3. Find the money multiplier in this economy.

mm =

�1 + cd

rd+ cd

�=

�1 + 0:2

0:1 + 0:2

�= 4

4. Now suppose that the FED performs an open market operation and buys governmentbonds at the amount of 5. Find the new monetary base, the money supply and describethe new balance sheet of the commercial banks.

MB = 30 + 5 = 35

�M =

�1 + cd

rd+ cd

��MB =

�1 + 0:2

0:1 + 0:2

�� 5 = 20

�D =

�1

rd+ cd

��MB =

�1

0:1 + 0:2

�� 5 = 162

3

�R =

�rd

rd+ cd

��MB =

�0:1

0:1 + 0:2

�� 5 = 12

3

�CU =

�cd

rd+ cd

��MB =

�0:2

0:1 + 0:2

�� 5 = 31

3


Thus

M = 120 + 20 = 140

D = 100 + 162

3= 116

2

3

R = 10 + 12

3= 11

2

3

CU = 20 + 31

3= 23

1

3

Balance sheet of the commercial banksAssets LiabilitiesR = 112

3D = 1162

3

BG = 20L = 851162

31162

3

The loans are simply set to balance the balance sheet. Notice that when the FEDincreased the monetary base by 5, the money supply increased by 20. This illustrates thefact that the FED does not have direct control over the money supply, but rather thecommercial banks together with the FED create the money supply.

5. Suppose that instead of the open market operation, the FED sets the required reserveratio to 5%. Find the new monetary base, money multiplier, the money supply anddescribe the new balance sheet of the commercial banks.

The monetary base does not change, because the FED did not buy or sell any asset.

MB = 30

mm =1 + cd

rd+ cd=

1 + 0:2

0:05 + 0:2= 4:8

M =

�1 + cd

rd+ cd

�MB =

�1 + 0:2

0:05 + 0:2

�� 30 = 144

D =

�1

rd+ cd

�MB =

�1

0:05 + 0:2

�� 30 = 120

R =

�rd

rd+ cd

�MB =

�0:05

0:05 + 0:2

�� 30 = 6

CU =

�cd

rd+ cd

�MB =

�0:2

0:05 + 0:2

�� 30 = 24

Balance sheet of the commercial banksAssets LiabilitiesR = 6 D = 120BG = 25L = 89120 120

6.4. ILLUSTRATION OF THE MONEY MULTIPLIER 77

Just to check that we did not make an error, lets verify that the consumers hold cur-rency/deposits at the right ratio, and also that the banks hold reserves/deposits at the rightratio:

CU

D=

24

120= 0:2

R

D=

6

120= 0:05

As was mentioned before, the FED does not use the second type of policy (changingthe required reserve ratio) frequently. One of the goals of the FED is to maintain a stablebanking system and if the required reserve ratio changes, the banks have to adjust theirloans and deposits in a complicated way.

6.4 Illustration of the Money Multiplier

In this section we show in detail how an open market operation a¤ects the balance sheet ofthe commercial banks and the money supply. The following steps illustrate the working ofthe money multiplier when the FED buys bonds at the amount of x from the commercialbanks.

Balance Sheet of Banks0. CU = 20 Assets Liabilities3. +cd x

1+cd0. R = 10 0. D = 100

4. +cd (1�rd)x(1+cd)2

1. +x 3. + x1+cd

5. +cd (1�rd)2x

(1+cd)32. �x 4. + (1�rd)x

(1+cd)2

...... 3. +rd x

1+cd5. + (1�rd)2x

(1+cd)3

4. +rd (1�rd)x(1+cd)2

......

5. +rd (1�rd)2x

(1+cd)3

......

0. BG = 251. �x

0. L = 652. +x

3. + (1�rd)x1+cd

4. + (1�rd)2x(1+cd)2

5. + (1�rd)3x(1+cd)3

......


Each step is numbered. Step "0" is the initial balance sheet. It is important to makesure that the balance sheet is balanced after each step, i.e. assets are equal to liabilities. Instep 1 the FED buys bonds at the amount of x from the commercial banks. As a result,the bonds decreased by x and the reserves increased by x. Now after step 1, the commercialbanks have too much reserves and they can lend the extra reserves to households. As aresult of step 2 the reserves decreased by x while loans increased by x. In step 3, the loansat the amount of x are distributed between currency and deposits such that �CU=�D = cdand �CU + �D = x. Also, the commercial banks keep a fraction rd of the new depositsin reserves and lend the rest (a fraction 1 � rd of the new deposits). Observe that all thechanges in step 3 in the balance sheet keep it balanced. The same is repeated in step 4, thatis the extra loans are distributed between CU and D according to the currency/depositsratio, and a fraction rd of the new deposits is kept in reserves while the rest is used for extraloans. This process continues inde�nitely. In order to follow the above process more easily,color each step in a di¤erent color and make sure that the impact of each step on the balancesheet keeps it balanced.Suppose that the above steps continue forever. We can use the rule of summation of an

in�nite geometric series1Xt=0

qt =1

1� q (0 < q < 1)

to get the following results:

�D =x

1 + cd

1Xt=0

�1� rd1 + cd

�t=

x

1 + cd

1

1� 1�rd1+cd

!

=x

1 + cd

1

1+cd�1+rd1+cd

!=

�1

rd+ cd

�x

�CU = cd ��D =�

cd

rd+ cd

�x

�M = �CU +�D =

�1 + cd

rd+ cd

�x

�R = rd ��D =�

rd

rd+ cd

�x

Thus, we derived all the multipliers in section 3 as the limit when t!1 of the sequence ofloans and deposits generated by the open market operation.We can also compute all the magnitudes above after T steps. This is more realistic

because in the real world the money circulates only limited number of times per year. Inparticular, the money velocity in 2005 is 9. Thus, we would like to compute the summationsof the �rst 9 steps. This can be done using the rule summation of a �nite geometric series

TXt=0

qt =1� qT+11� q

6.4. ILLUSTRATION OF THE MONEY MULTIPLIER 79

This gives

�D =x

1 + cd

TXt=0

�1� rd1 + cd

�t=

x

1 + cd

1�

�1�rd1+cd

�T+11� 1�rd

1+cd

!

=x

1 + cd

1�

�1�rd1+cd

�T+11+cd�1+rd

1+cd

!=

1�

�1�rd1+cd

�T+1rd+ cd

!x

�CU = cd ��D = cd � 1�

�1�rd1+cd

�T+1rd+ cd

!x

�M = �CU +�D = (1 + cd) � 1�

�1�rd1+cd

�T+1rd+ cd

!x

�R = rd ��D = rd � 1�

�1�rd1+cd

�T+1rd+ cd

!x

For example, if the FED buys bonds at the amount of 5, then after 9 rounds of loans anddeposits we have

�D =

1�

�1�rd1+cd

�9rd+ cd

!� 5

�CU = cd � 1�

�1�rd1+cd

�9rd+ cd

!� 5

�M = (1 + cd) � 1�

�1�rd1+cd

�9rd+ cd

!� 5

�R = rd � 1�

�1�rd1+cd

�9rd+ cd

!� 5

This completes the illustration of money creation by the FED and the commercial banks.


Chapter 7

Phillips Curve

7.1 Introduction

In�ation and unemployment are both considered undesirable for an economy. Why is in�a-tion bad? High in�ation leads to decline in the purchasing power of nominal assets, suchas money and wages. It is also argued that in�ation brings with it a lot of uncertaintyabout future prices since not all the prices tend to rise at the same rate. Therefore �rmsare having hard time planning its future production and how the particular prices of itsinputs and production evolve relative to other prices. Fortunately for the U.S., we did notexperience very high in�ation rates, as can be seen in Figure 7.1.Some countries were not as

Inflation Rate in U.S.

420

2468

10

121416

Jan

47

Jan

51

Jan

55

Jan

59

Jan

63

Jan

67

Jan

71

Jan

75

Jan

79

Jan

83

Jan

87

Jan

91

Jan

95

Jan

99

Jan

03

Time

Infla

tion

Rat

e %

Figure 7.1: In�ation in the U.S.

81

82 CHAPTER 7. PHILLIPS CURVE

fortunate however. For example, Austria in 1921-22 had 10,000% annual in�ation rate, andArgentina during 1989-1999 had annual in�ation rate of 20,000%. Can you imagine livingin an economy where prices change by the minute.High unemployment is also considered undesirable for a number of reasons. The most

important reason is the waste of resources, since unemployed workers do not produce output.There is also emotional cost for those who loose their job or stay unemployed for longperiods of time. Figure 7.2 shows the unemployment rate in the U.S. since 1948.Compared

Unemployment Rate in U.S.

0

2

4

6

8

10

12

Jan

47

Jan

51

Jan

55

Jan

59

Jan

63

Jan

67

Jan

71

Jan

75

Jan

79

Jan

83

Jan

87

Jan

91

Jan

95

Jan

99

Jan

03

Time

Une

mpl

oym

ent R

ate

%

Figure 7.2: Unemployment rate in the U.S.

to other European countries during the same period, the unemployment rate in the U.S. isconsidered quite low. So both in�ation and unemployment are undesirable. The search modelof unemployment shed some light on the factors that determine the unemployment rate (e.g.separation rate, government policies about unemployment insurance bene�ts, etc.). We alsomentioned cyclical unemployment, which varies with the business cycle. So far however wedid not suggest that In�ation and Unemployment are related in any way.

7.2 Pillips Curve

The New Zealand-born economist A.W. Phillips, in his 1958 paper "The relationship betweenunemployment and the rate of change of money wages in the UK 1861-1957" published inEconomica, observed an inverse relationship between money wage changes and unemploy-ment in the British economy over the period examined. He concluded that government"demand" policies can move the economy along the curve, and thereby changing the level

7.2. PILLIPS CURVE 83

of unemployment in the economy. Today, what is known as "The Phillips Curve" is a graphthat shows the relationship between in�ation and unemployment over a particular period oftime.

The negative relationship between in�ation and unemployment was observed in othercountries during di¤erent periods. Figure 7.3 shows the Phillips Curve in the U.S. duringthe 60�s. We can see the during the 60�s there a signi�cant negative relationship betweenin�ation and unemployment was observed in the U.S.

Philips Curve (1960 1969) y = 1.0821x + 7.4364R2 = 0.6483

1

0

1

2

3

4

5

6

7

0 2 4 6 8

Unemployment Rate (in %)

Infla

tion

Rat

e (a

nnua

l %)

Figure 7.3: Phillips curve in the U.S., 1960�s

During other periods in the U.S., the negative relationship between in�ation and unem-ployment rate seem to weaken and sometimes even a positive relationship was observed. Thenext �gures show the Phillips curve in the U.S. for the 70�s, 80�s, 90-2005, and 1948-2005.



0

2

4

6

8

10

12

14

0 2 4 6 8 10


Infla

tion

Rat

e (a

nnua

l %)


0

2

4

6

8

10

12

14

0 2 4 6 8 10


Infla

tion

Rat

e (a

nnua

l %)



0

2

4

6

8

10

12

14

0 2 4 6 8 10 12


Infla

tion

Rat

e (a

nnua

l %)


0

1

2

3

4

5

6

7

0 2 4 6 8 10


Infla

tion

Rat

e (a

nnua

l %)



4202468

10121416

0 2 4 6 8 10 12


Infla

tion

Rat

e (a

nnua

l %)

Observe that the negative relationship between in�ation and unemployment that wasobserved during the 60�s breaks down during the subsequent decades. Moreover, for theentire period of 1948-2005 (last �gure), the observed relationship between in�ation andunemployment is negative.

7.2.1 The impact of the Phillips curve on monetary policy

Phillips� article in 1958 and subsequent empirical work on the Phillips curve convincedmany economists and policy makers that there exists a stable trade-o¤ between in�ationand unemployment. Some economists modi�ed the standard Keynesian model so that itwould predict a negative trade-o¤ between in�ation and unemployment. Some of you mayhave seen in your principle class the so called AD-AS (aggregate demand, aggregate supply)model, illustrated in the next �gure.


P2

P1

Q1 Q2

P

Q

AS

AD

In this model expansionary policies (�scal or monetary) lead to an outward of the AD curve,and equilibrium output goes up, but at a cost of higher price level. The obvious implication ofhigher output is lower unemployment. Equipped with this model and the empirical evidenceabout the negative relationship between in�ation and unemployment, some economists andpolicy makers concluded that it is possible to exploit the Phillips curve. In other words, theybelieved that there is a stable relationship between in�ation and unemployment rate, andthe policy makers have a choice of lowering unemployment at the cost of higher in�ation, orhaving higher unemployment rate but with lower in�ation. It was often argued that as longas in�ation is not too high, it does not pose much danger to the economy, so it is better tosu¤er some in�ation as long as this would lead to lower unemployment. Central banks ofsome countries (e.g. Israel) printed money with the hope that higher in�ation would lowerunemployment and thereby boost the economy.

In most cases, if not in all of them, the in�ationary policies that attempted to lowerunemployment resulted in higher in�ation but without lowering unemployment. In somecountries these policies led to hyper in�ation, and the collapse of the local currency, bankingsystem and ultimately led to high unemployment. Why did in�ationary policies fail somiserably? The key to answering this question is expectations, which is the topic of the nextsection.


7.3 Expectations-Augmented Phillips Curve (EdmundPhelps)

Edmund Phelps did most of his important work on In�ation and Unemployment during thelate 60�s, when the most compelling evidence that supported the downward slopping Phillipscurve was becoming available (see �gure 7.3). Contrary to the conventional wisdom thatthe downward slopping Phillips curve represents a stable and exploitable trade-o¤ betweenin�ation and unemployment, Phelps argued that this is not the case in the long-run. Phelpsintroduce an important notion of expected in�ation, which plays a key role in his argument.The expectation-augmented Phillips curve is given by:

�t = �et � � (ut � un) (7.1)

or�t � �et = �� (ut � un) (7.2)

where �t is the actual in�ation in period t, �et is the period t � 1 expectation of �t (orformally, �et = Et�1 (�t)), ut is the unemployment at time t and un is some natural rate ofunemployment or NAIRU (Non-Accelerating In�ation Rate of Unemployment). According tothe extended Keynesian model mentioned above, � > 0. Thus, if the monetary policy is suchthat the actual in�ation is equal to the expected in�ation, then the unemployment in thatperiod is by de�nition equal to the natural rate. That is if �t = �et then ut = un. To motivatethis formulation, think of workers and employers who set work contracts based on theirexpectations of future in�ation. The contracts are set in such a way that when they perfectlyanticipate the future in�ation, the labor markets are cleared (the only unemployment is thenatural one). If for some reason people made an error in predicting the future in�ation,then the unemployment di¤ers from the natural rate. Suppose that the realized in�ationis greater than the expected one, i.e. �t > �et . In this case we can see that the realizedunemployment rate will fall below the natural rate, i.e. ut < un.We need to make some assumption about the way people form their expectations about

future in�ation. The most simple assumption that we can make is backward-looking expec-tations. In words, this means that the expected in�ation at time t is the realized in�ationat time t � 1. Formally, backward-looking expectations mean that �et = �t�1. Phelps�ar-gument that the there is no long-run trade-o¤ between in�ation and unemployment can beeasily illustrated with Figure 7.4. Suppose that initially the Phillips curve is the curve la-beled �e = �1 and the actual in�ation is initially �1. This curve represents the Phillips curvefor expected in�ation of �1. Since the actual in�ation is equal to the expected in�ation,the actual unemployment rate is equal to the natural rate. Thus, at the initial point theeconomy has in�ation rate of �1 and unemployment rate of un. Suppose now that the centralbank decided to increase the in�ation to �2. In the short run people still expect the in�ationto be �1 (as it was in the last period), so the economy moves to the point labeled Short run,with higher in�ation and lower unemployment. In the following period however, people�sexpectations will adjust and they will expect in�ation to be �2. The Phillips curve will shiftup1 and the new Phillips curve is the one labeled �e = �2. At the long-run equilibrium, theunemployment rate returns to the natural rate un but the in�ation is higher �2.Thus, we

1The Pillips curve shifts up because �e is part of the intercept.

7.3. EXPECTATIONS-AUGMENTED PHILLIPS CURVE (EDMUND PHELPS) 89

tuut

2π

1π

nuu2

tπP

1ππ =e2ππ =e

Initial point

Longrun

Short run

Figure 7.4: Phelps�Expectations-Augmented Philips curve

see that even when we make the very simplistic assumption that expectations are backward-looking, the result of this model is that there can be at best a short-run trade-o¤ betweenin�ation and unemployment, but not in the long run. As we will see below, with morereasonable assumptions of expectation formation, Phelps�result becomes even more robust.

7.3.1 The impact of the expectations-augmented Phillips curve onmonetary policy

Phelps� insight and his emphasis of the expected in�ation revolutionized the way mone-tary policy is conducted. He switched the discussion from the permanent trade-o¤ betweenin�ation and unemployment to discussion about intertemporal trade-o¤ (between loweringunemployment now but su¤ering from high in�ation in the future). The theoretical under-pinnings for the policy of in�ation targeting, which many central banks have adopted sincethe early 1990s, are to a large extent derived from the framework developed in Phelps�1967paper. He demonstrated that expansionary monetary policy can lower in�ation only in theshort run, in the best case scenario of backward looking expectations. We will see in whatfollows that if we make more reasonable assumptions about expectation formation, thenin�ationary monetary policy becomes even less e¤ective and can fail boosting the economyeven in the short run.


7.4 Rational Expectation (Robert Lucas)

Rational Expectations theory assumes that people use all the information available to themto predict the future in�ation. According to the backward-looking expectations, people�s pre-diction of future in�ation is current in�ation. This is similar to predict tomorrow�s weatherto be exactly like today�s weather. When meteorologists predict tomorrows weather, theyuse a host of information other than yesterdays weather. Similarly, when people form ex-pectations about future in�ation they use the media, the FED announcements and all sortsof other information that is available.

What are the implications of the rational expectations assumption on monetary policy?Looking back at Figure 7.4, suppose that the public is able to predict that the FED isplanning to increase the in�ation. Then the public�s expectations will immediately become�e = �2 and the Phillips curve will shift upward at the same period when the in�ationincreases from �1 to �2. Thus, with rational expectations, in�ationary policy will not havean impact on unemployment even in the short run. The economy will jump from the initialpoint to the long run equilibrium with the same unemployment and higher in�ation.

Lucas argument strengthened Phelps�argument and implied that even in the short-run,the central banks would not be able to boost the economy with in�ationary monetary policies.

7.4.1 Numerical example

Suppose that � = 1, un = 5, and �et = �t�1 (backward-looking expectations).

1. Suppose that the FED creates in�ation of 1% until period 3 and then increases thein�ation permanently to 2%. Show the time path of in�ation, expected in�ation andunemployment from period 1 on.

Rewriting the expectations-augmented Phillips curve in equation (7.1) gives

�t = �et � � (ut � un)�t � �et�� = ut � un

ut = un +�t � �et�1

ut = un + �et � �t

The next table shows the time paths.

7.4. RATIONAL EXPECTATION (ROBERT LUCAS) 91

2. Now suppose that the FED creates in�ation of 1% until period 3 and then increasesthe in�ation permanently to 2%, but this time people have rational expectations andthey anticipate perfectly the increase in in�ation. Show the time path of in�ation,expected in�ation and unemployment from period 1 on.

3. Now suppose that the FED creates in�ation of 1% until period 3 and then increases thein�ation permanently to 2%, but this time people have almost rational expectations sothey are expecting that in period 3 the FED will increase the in�ation to 1:9%. Afterperiod 3 they learn that the in�ation is 2%. Show the time path of in�ation, expectedin�ation and unemployment from period 1 on.


Notice that when we make the assumption of backward-looking expectations, then in�a-tionary policy is e¤ective in reducing unemployment only in the short run. If expectationsare rational, then in�ationary policy is not e¤ective in reducing unemployment even in theshort run. The last example showed that if expectations are almost rational, then the mone-tary policy has some e¤ect in the short run, but this is getting smaller the better the publicsprediction of in�ation is.

7.5 Credibility of Monetary Policy (Finn E. Kydland,Edward C. Prescott)

Kydland and Prescott illustrated another problem with monetary policy - time inconsistency.To illustrate their argument, consider again the expectations-augmented Phillips curve. No-tice that if the public has expectations for low in�ation, then the central bank can reducethe unemployment rate by creating in�ation. In other words, there is an incentive for thecentral bank (or government) to promise a low in�ation rate in the next period, and whenthe public sets the wage contracts according to the low in�ationary expectations, there atemptation for the central bank to break the promise and create high in�ation. If economicpeacemakers lack the ability to commit in advance to a speci�c decision rule, they will oftennot implement the most desirable policy later on. Kydland and Prescott�s results o¤ereda common explanation for events that, until then, had been interpreted as separate policyfailures, e.g., that economies become trapped in high in�ation even though price stabilityis the stated objective of monetary policy. Their work established the foundations for anextensive research program on the credibility and political feasibility of economic policy.This research shifted the practical discussion of economic policy away from isolated policymeasures towards the institutions of policymaking, a shift that has largely in�uenced thereforms of central banks and the design of monetary policy in many countries over the lastdecade.Because of the time inconsistency problem, the recommended policy by Kydland and

Prescott is "Rules Rather Than Discretion"2. That is, they recommended that the central2Kydland, Finn E & Prescott, Edward C. 1977 "Rules Rather Than Discretion: The Inconsistency of

7.6. APPENDIX: ESTIMATINGTHEEXPECTATIONS-AUGMENTEDPHILLIPS CURVE93

banks should commit to a simple rule (for example in�ation targeting at certain level) andnot attempt to exercise discretionary policy. When the central bank commits to a certain ruleby law, this commitment is credible, while a promise without commitment is not. In otherwords, they recommended that the central banks should tie their hands and not conductmonetary policy which responds to economic events.

7.6 Appendix: Estimating the Expectations-AugmentedPhillips Curve

We want to test whether the assumption that � > 0 holds in the data. For simplicity weassume that expectations are backward-looking, �et = �t�1, i.e. we assume that people�sexpectation about in�ation at time t is the time t � 1 in�ation rate. Also, we can assumefor simplicity that un is the average unemployment rate over the sample period, say 5%.Let ��t � �t � �t�1 denote the di¤erence in in�ation rate between period t� 1 and t. Thestatistical relationship that we want to estimate is then

��t = �1 (ut � un) + "t (7.3)

Having obtained the estimate for the slope (�), we test whether � < 0. When we run thisregression, we need to impose the restriction that the constant is zero.Remark. The speci�cation in equation (7.3) is not the same as

��t = �0 + �1ut + "t (7.4)

Rearranging equation (7.3) gives

��t = ��1un + �1ut + "t

Since un = 5 is given, then the �rst speci�cation in eq. (7.3) is equivalent to the second,eq. (7.4), if we impose a restriction on the second speci�cation that the intercept is equal tothe slope times un. In other words, the speci�cation that is implied by the theory requiresestimating equation (7.3), which is the same as estimating the constrained relationship

��t = �0 + �1ut + "t

s:t:

�0 = ��1 � un

Optimal Plans", The Journal of Political Economy .


Chapter 8

International Macroeconomics

8.1 Balance of Payments

Almost all the economies are open economies and have interactions in trade and �nancewith other countries. These interactions are documented in the balance of payment ac-count which records the country�s trade with other countries in goods, services and assets.The balance of payments consists of three accounts: the current account, the �nancial ac-count, and the capital account. The next table shows the balance of payments for the U.S.in 2005. All the values are in millions of dollars.

The current account records the country�s exports and imports as well as income frominvestment and unilateral transfers. Any payments received by U.S. residents are positivenumbers and any payments made by the U.S. residents are negative numbers. For example,when the U.S. companies export goods or services, they receive payments, which are recorded

95

96 CHAPTER 8. INTERNATIONAL MACROECONOMICS

as positive entries in line 2 above. Similarly, when a U.S. resident receives dividends from acompany he owns in a foreign country, those payments are recorded as positive entries in line3 above. Conversely, when U.S. residents import goods and services from other countries,these payments are recorded as negative entries in line 5 and payments to foreigners who earnincome from the U.S. are negative entries in line 6. Finally, unilateral transfers includeforeign aid, grants, gifts, donations, etc.The �nancial account records purchases of assets that a country has made abroad and

foreign purchases of assets in the country. These assets include physical capital as well as�nancial assets, such as shares of stock and bonds. When investors in the U.S. buys foreignassets such as foreign government bonds, or when a U.S. �rm builds a factory in anothercountry, these payments are capital out�ow from the U.S. and are recorded as negativeentries in line 11. The capital in�ow into the U.S. occurs when a foreign investor buys abond issued by a U.S. company or government, or when a foreign �rm builds a factory inthe U.S. When �rms build or buy facilities in foreign countries they engage in foreign directinvestment, while purchases of �nancial assets are called foreign portfolio investment.The capital account is the part of the balance of payments that records relatively

minor transactions such as migrants transfers when they cross borders and also sales andpurchases of assets that are neither produced nor �nancial assets, such as copyrights, patents,trademarks or right to natural resource.The balance of payments is always balanced, up to statistical discrepancy. That is, the

sum of the balance on current account and the �nancial and capital accounts must be zero.Notice that in 2005 the U.S. spent $791.5 more on goods and services and transfers thanit received. This means that foreigners have accumulated $791.5 during 2005, which theyeither invested in the U.S. as purchases of assets or not spent at all. In the later case thenon-spent amount is added to foreign holding of dollars, which is a positive entry in the�nancial account in line 12.

8.2 Exchange Rates

The balance of payments that we have seen in the previous section showed the summary oftransactions between the U.S. and other countries in millions of dollars. However, when theU.S. residents are engaged in trade of goods, services and assets with other countries, thesetransactions involve other currencies. The price of one currency in term of another currencyis called the nominal exchange rate. For example, the exchange rate between the dollar($) and the British pound ($) can be expressed as how many pounds are required to buyone dollar:

e$

$= 0:5

$

$The most confusing part about the exchange rates is that they can be expressed also inanother way, for example how many dollars are needed in order to buy one pound:

e$

$= 2

$

$

The notation $=$ reads "pounds per dollar" and $=$ reads "dollars per pound". The nexttable shows a few selected exchange rates expressed in both ways. The middle column shows

8.2. EXCHANGE RATES 97

the price of the U.S. dollar in units of other currencies and the third column shows the priceof the foreign currencies in U.S. dollars.

Example.Q. Based on the above table, what is the exchange rate between the dollar and the euro?A. We can say that the price of one U.S. dollar is 0.76 euro or the price of one euro is

1.31 dollars. Using our notation, the two ways of describing the exchange rate between thedollar and the euro are

ee

$= 0:76

e

$or

e$

e= 1:31

$

e

In what follows, to avoid confusion, when we talk about exchange rate between the dollarand other currency, we will express it as the price of the dollar in units of that currency. Forexample, ee

$is the exchange rate between euro and the dollar expressed in euros per dollar,

and eU$is the exchange rate between the yen and the dollar, expressed in yens per dollar.

8.2.1 Using the exchange rates

The exchange rates are useful for converting prices in one currency into another. Supposethe price of a television in the U.S. is $200 and we want to convert this price into Japanese


yen. Then,

PU = eU$P$ = 116 � 200 = U23; 200

Since each dollar is worth 116 yens, then 200 dollars are worth 116 � 200 yens. Now supposethat the price of a car in Japan is 2; 088; 000 and we want to convert this price into dollars.Then,

P$ = e$

UPU =PU

eU$

=2; 088; 000

116= $18; 000

Lets look at another example - the Big Mac. The price of the big mac in the U.S. is $3:1and the price in Japan is U250. Which one is cheaper? In order to compare these prices weneed to convert them to common currency. So for example, lets convert the price in the U.S.into Japanese currency.

PU = eU$P$ = 116 � 3:1 = U359:6

So the Japanese big mac is much cheaper than the U.S. big mac. Is this surprising? Didwe expect the price of an identical good to be the same in all countries, when compared ata common currency? As we will show in the next section, the answer is "it depends"; itdepends on whether the good is traded or not.

8.3 Law of One Price and Purchasing Power Parity(PPP)

The law of one price is the notion that the price of traded goods has to be the same intwo countries when converted into a common currency. The reason why we expect the lawof one price to hold is because if the price in one place is cheaper than in the other thereis a possibility to perform an arbitrage: buying the good where it is cheaper and selling itwhere it is more expensive. In reality, the law of one price should hold when taking intoaccount the transportation costs and tari¤s. We do not expect the law of one price to holdfor non-traded goods, such as haircuts and restaurant meals, the big mac included. We doexpect the law of one price to hold for goods like crude oil, gold, and other traded goods.A related concept isPurchasing Power Parity is a method of comparing the purchasing

power of $1 in di¤erent countries. The PPP between the U.S. and another country holds if$1 has the same purchasing power in the other country. In other words, $1 when convertedto the other currency can buy the same goods in the foreign country as it can buy in the U.S.Suppose the price of a bundle of goods in the domestic economy (U.S.) is P and the priceof the same bundle in the foreign country (in foreign currency) is P �. Let the exchange ratebetween the dollar the foreign currency be e c

$, where c is the name of the foreign currency.

Then $1 in the U.S. buys 1=P goods and $1 when converted to the foreign currency can bye=P � goods. If the PPP holds for this bundle of goods then we should have

1

P= e

1

P �or

1 = eP

P �

8.3. LAW OF ONE PRICE AND PURCHASING POWER PARITY (PPP) 99

The right hand side of the last equation is called the real exchange rate (er), whichshows the relative price of a bundle of domestic goods in terms of foreign goods.

er = eP

P �(8.1)

If PPP holds for this speci�c bundle, then er = 1. Of course we don�t expect the PPP tohold for any bundle of goods, but we do expect it to hold for traded goods. For example, wehave seen that PPP does not hold for the big mac.We can check to what extent does the PPP hold for all the consumption goods as follows.

Let the consumer price index in the U.S. be P , the the consumer price index in the foreigncountry be P � and the exchange rate between the dollar the foreign currency be e. Thenequation (8.1) can be used to compute the real exchange rate. If er > 1 then the domesticbundle is more expensive than the foreign bundle, and if er < 1 then the domestic bundle ischeaper than the foreign bundle.If we though that there are forces that would drive the real exchange rate to 1 in the

long run, then we could predict the future trend in the exchange rates. For example, if rightnow we have

er = eP

P �> 1

and we believe that in the future the real exchange rate should be 1 (er ! 1) then we couldpredict that in the future the dollar should depreciate (e #) relative to the foreign currency.In the next section we will take a closer look at PPP.

8.3.1 Predicting future trends in exchange rates

A currency appreciates when its market value rises relative to another currency. A currencydepreciates when its market value falls relative to another currency. It would be nice if wecould predict the future trends of exchange rates. We realize that we should not expect thePPP to hold for all the consumption goods. But we do expect it to hold for traded goods.Let the price index in the U.S. be a weighted average of the price of traded goods P T andnon-traded goods PN as follows

P = �P T + (1� �)PN , where 0 � � � 1

Similarly, let the price index in the foreign country be

P � = �P �T + (1� �)P �N , where 0 � � � 1

Then the real exchange rate is

er = e�P T + (1� �)PN�P �T + (1� �)P �N

Rearranging the above

er = eP T

P �T

��+ (1� �)PN=P T� + (1� �)P �N=P �T

�(8.2)


The fractions PN=P T and P �N=P �T are ratios of prices of non-traded to traded goods in thehome country and in the foreign country. If those ratios are more or less stable over time,then we expect the term in the brackets to be some constant C. Also, because of the law ofone price we expect the PPP to hold for traded goods, that is

eP T

P �T= 1

Thus, the real exchange rate in equation (8.2) becomes constant

er = eP T

P �T| {z }1

��+ (1� �)PN=P T� + (1� �)P �N=P �T

�| {z }

C

= C

With these assumptions we can predict the future movement in exchange rates using thede�nition of the real exchange rate in equation (8.1)

er = eP

P �= C (8.3)

Expressing the equation (8.3) in terms of rates of change gives the following approximation1

e+ P � P � = C (8.4)

Since C is constant, we have C = 0. Then equation (8.4) becomes

e = P � � P = �� (8.5)

where � is domestic in�ation rate and �� is foreign in�ation rate. Equation (8.5) tells ussomething very intuitive. We expect the dollar to depreciate relative to the foreign currencyif the domestic in�ation is higher (� ") or if the foreign in�ation is lower (�� #). Higherdomestic in�ation means that the dollar is worth less while lower foreign in�ation meansthat the foreign currency is worth relatively more.To test equation (8.5) we can plot the expression e� (�� ) for some countries and see

how close it is to zero. Figure 8.1 plots e � (�� ) for Canada, U.K. and Japan. Noticethat although there are large �uctuations, the graphs tend to converge back to zero, whichis consistent with the result in equation (8.5).We can use the quantity theory of money to relate the movements in exchange rates

to money growth and the growth in real output. Recall that the quantity equation is

M =PY

V

where P is the price index, Y is the real GDP and V is the velocity of money. Expressed ingrowth rates, this is approximately

M = P + Y � Vor

P = M � Y + V1Recall that a hat above the variable represents the rate of change in a variable. Formally, x = xt+1�xt

xt.

8.3. LAW OF ONE PRICE AND PURCHASING POWER PARITY (PPP) 101

Figure 8.1: e� (�� ) for Canada, U.K. and Japan.

30

20

10

0

10

20

30

40

50

1949 1959 1969 1979 1989 1999

Years

perc

ent

Canada U.K. Japan

If we assume that in the short run the velocity is constant, then the above becomes P =M � Y . Using this expression for in�ation in equation (8.5) gives

e = M� � Y �| {z }��

��M � Y

�| {z }

�

or

e =�M� � M

�+�Y � Y �

�(8.6)

From the last equation we see that if the domestic money supply grows faster than the foreignmoney supply, then the value of the dollar will tend to depreciate. On the other hand, if thereal GDP in the domestic economy grows faster, the dollar will appreciate.

8.3.2 Fixed vs. �oating exchange rate

Sometimes central banks in some countries decide to intervene in the market for foreignexchange in order to keep the exchange rate at some �xed level. There are two advantagesto �xing the exchange rate. First, �xing the exchange rate reduces the uncertainty associatedwith exchange rate �uctuations. The second reason is that the �xed exchange rate helps tocontrol in�ation in the country. To understand how this is working recall equation (8.5)e = �� which describes the relationship between the exchange rate and foreign anddomestic in�ation rates. If the nominal exchange rate is �xed, then e = 0 and as a result thedomestic in�ation rate is equal to the foreign in�ation rate: � = ��. Sometimes �xing thenominal exchange rate for controlling in�ation is referred to as using a nominal anchor,because it help to anchor (stop) the in�ation rate.


Some people oppose to �xing the exchange rate because it neutralizes the discretionarymonetary policy. From equation (8.6) we see that if the nominal exchange rate is �xed(e = 0) then M = M� + Y � Y �, so the growth rate of domestic money is �xed by the thegrowth rate of the foreign currency and the growth of the real GDP in the home country andabroad. The concern is that shocks to the foreign money supply will be directly translatedto the shocks of the domestic money supply. In other words, the domestic economy hasno control over its money supply. This is why many economists advocate alternative waysto combat in�ation, other than �xing the exchange rates. The capter about the Phillipscurve and monetary policy commitment discusses in more length about the importance ofcommitment. Many economists propose solving the commitment problem of the central bankby adopting a monetary policy rule, i.e. specifying by law some in�ation target or in�ationinterval.

8.4 Review Questions

1. Suppose that the price of a television in the U.S. is $200 and the price of the sametelevision in India is INR8,000 (INR stands for Indian Rupee). If televisions are tradedgoods and assuming that the law of one price holds, what should be the exchange ratebetween the dollar and the Indian rupee?

2. Suppose that the price index in the domestic economy is P = $100 per basket of goodsand the price of the same basket of goods in the foreign country is P � = c170 (wherec is the name of the foreign currency). The nominal exchange rate is e = 2 c

$.

(a) Does the purchasing power parity hold between the two economies? Explain youranswer.

(b) If the answer to the previous question is "NO", then what should be the nominalexchange rate between the two countries for the PPP to hold?

3. Recall that equation (8.5) shows that under some assumptions the relationship betweenthe growth in the nominal exchange rate and domestic and foreign in�ation is givenby e = �� .

(a) Which assumptions were used to derive this result? Show your derivations.

(b) How would you test if e = �� holds in the data?

4. What are the consequesnces of �xing the nominal exchange rates on the e¤ectivenessof domestic monetary policy?

5. Discuss the arguments in favor and against �xing the exchange rates.

8.4. REVIEW QUESTIONS 103

Answers:

1. According to the law of one price, the price of a traded good should be the same in allcountries when converted to a single currency. Thus

$200 � eINR$

= INR8; 000

eINR

$=

8; 000

200= 40

INR

$

(a) To check if the PPP holds we need to compute the real exchange rate

er = eP

P �= 2 � 100

170=200

1706= 1

so the PPP does not hold since the real exchange rate is not equal to 1.

(b) In order for PPP to hold, the nominal exchange rate should be such that the realexchange rate is 1. Thus

er = 1 = e � 100170

) e = 1:7c

$

(a) The the assumptions are: (1) the PPP holds for traded goods, (2) the ratios ofthe price of non-traded to traded goods are approximately constant in the twocountries, and (3) the weights on traded and non-traded goods in each country(�; �) are �xed. Under these assumptions the real exchange rate is constant

er = eP T

P �T| {z }1

��+ (1� �)PN=P T� + (1� �)P �N=P �T

�| {z }

C

= C

and expressed in terms of approximate growth rates

er = eP

P �= C

e+ P � P � = 0

e = ��

(b) We can plot the time series of e� (�� ) and see if this time series is approxi-mately 0. See the graph in �gure (8.1).

2. See notes.

3. See notes.


Chapter 9

Math Review

105

106 CHAPTER 9. MATH REVIEW

Chapter 10

Micro Review

In these notes we brie�y discuss the two core topics of Microeconomics: (1) the theory ofconsumer choice and (2) the theory of producer choice. Modern Macroeconomic theory usesmodels with explicit decision making of households and �rms, and these notes will hopefullyprovide the necessary background for exploring such models.

10.1 Consumer�s Choice

The Model of Consumer Choice:

� There are two goods X, Y . A consumption bundle is a pair (x; y), where x is thequantity of good X and y is the quantity of good Y .

� Consumer�s preferences are represented by a utility function U such that the consumerlikes the bundle (x1; y1) at least as much as he likes the bundle (x2; y2) if and onlyif U (x1; y1) � U (x2; y2). If U (x1; y1) = U (x2; y2), then we say that the consumeris indi¤erent between the two bundles, and . If U (x1; y1) > U (x2; y2), we say thatthe consumer strictly prefers the bundle (x1; y1) over (x2; y2). We introduce someshorthand notation:

(x1; y1) % (x2; y2) is the same as U (x1; y1) � U (x2; y2)(x1; y1) � (x2; y2) is the same as U (x1; y1) = U (x2; y2)

(x1; y1) � (x2; y2) is the same as U (x1; y1) > U (x2; y2)

� Assumptions about preferences:

�Monotonicity or �more-is-better�: for all " > 0 we have

(x+ "; y) � (x; y)

(x; y + ") � (x; y)

107

108 CHAPTER 10. MICRO REVIEW

�Transitivity1:

if (x1; y1) % (x2; y2) and (x2; y2) % (x3; y3) then we must have (x1; y1) % (x3; y3)

� The prices of the goods are px; py and consumer�s income is I are taken as given.

10.1.1 Budget Constraint

The budget constraint is given bypxx+ pyy = I

This means that the spending on x and the spending on y must add up to the income2 I.Solving for y from the budget constraint gives

y =I

py� pxpyx

and it is illustrated in �gure 10.1

Figure 10.1: Budget constraint

x

y

ypI /

xpI /

Slope =y

x

pp

−

Points on or below the budget frontier are feasible, while points outside the budgetconstraint are not feasible. Notice that changes in income will not a¤ect the slope of thebudget constraint. When income goes up, the budget constraint will shift outward whileremaining parallel to the original one. Changes in prices will be re�ected in the slope ofthe budget constraint. An increase in the price of x for example will decrease the maximalquantity of x that the consumer can purchase, so the x-intercept will move to the left. Ifboth prices change by the same percent however, the slope of the budget constraint will notchange.

1In the appendix we will show that if a consumer has preferences that violate transitivity, then he canloose all his money very quickly.

2The more proper way of writing the budget constraint is pxx+ pyy � I, which means that the spendingon the two goods cannot exceed the income, but some income can be left over. We will always make theassumption of monotonicity though, that will guarantee that the consumer will always exhust his income.

10.1. CONSUMER�S CHOICE 109

Review question.

1. Draw a budget constraint that would result after a decline in income.

2. Draw a budget constraint that would result after a proportional increase in both prices,px and py.

3. Draw a budget constraint that would result after an increase in px.

4. Draw a budget constraint that would result after an increase in py.

5. Draw a budget constraint that would result after an increase of 50% in income andboth prices.

Answer. Figure 10.2 shows 4 budget constraints labeled A;B;C; and D. The origi-nal budget constraint, before any changes took place, is labeled A. The changes and thecorresponding budget constraints are listed below.

1. B

2. B

3. C

4. D

5. A

Figure 10.2: Changes in budget constraints

x

y

A

B

C

D


10.1.2 Indi¤erence Curves

We can represent preferences graphically with indi¤erence curves. An indi¤erence curve is acollection of all the bundles that are equivalent. Based on our assumptions about consumer�spreferences, what would the indi¤erence curves look like? Choose an arbitrary bundle, suchas bundle A in �gure 10.3. The �gure shows the regions of bundles that are strictly betterthan A and strictly worse than A. recall that the monotonicity assumption implies that allthe bundles that have more of at least one good are strictly better, and all the bundles thathave less of at least one good are strictly worse. Thus, the bundles that are equivalent to Acannot be in those regions. Thus, the indi¤erence curve that goes through the point A, i.e.the curve that contains all the bundles that are equivalent to A, must look like the curvelabeled IC1 or IC2 in �gure 10.3. Any indi¤erence curve must be decreasing, again by theassumption of monotonicity. In this course we will also assume that the indi¤erence curveshave a shape of IC1, i.e. that the indi¤erence curves are convex.

Figure 10.3: Indi¤erence curves

x

y

A

Bundlesstrictly betterthan A

Bundlesstrictly worsethan A 1IC

2IC

We could have chosen any bundle in the non-negative quadrant, and draw an indi¤erencecurve through it. Thus, preferences can be described graphically by an indi¤erence map,as shown in the �gure 10.4. You should be able to prove that higher indi¤erence curverepresents bundles that are strictly better than bundles on a lower indi¤erence curve. In�gure 10.4 for example, any bundle on indi¤erence curve labeled B is strictly better thatany bundle on A. Thus, the consumer would like to choose a bundle on his budget constraintthat attains the highest indi¤erence curve. Another property of indi¤erence curves that isvery important is that two distinct indi¤erence curves cannot intersect. You should be ableto prove this property as well.To summarize, there are three important properties of indi¤erence curves that you should

be able to prove:

1. Indi¤erence curves are downward slopping.

2. Higher indi¤erence curve contains bundles that are strictly better than those on lowerindi¤erence curves.


Figure 10.4: Indi¤erence map

x

y

AB

CD

3. Two distinct indi¤erence curves cannot intersect.

Convexity assumption

We assume that the indi¤erence curves are convex. This assumption means that the con-sumer prefers the average over extremes. Figure 10.5 illustrates this point. Consider bundlesB and C in �gure 10.5. These are extreme bundles; B contains very few X and a lot of Yand C contains a lot of X and very few Y . The picture shows that this consumer wouldprefer a bundle like D which is a convex combination of B and C. In other words, D is anaverage of B and C. This assumption makes sense. Suppose that you throw a party and youneed to buy drinks (X) and food (Y ). Bundles B corresponds to having too little drinks andlots of food, while bundle C corresponds to having too little food and lots of drinks.Observe

Figure 10.5: Convexity assumption

x

y

IC

B

C

D

that bundle D, which is an average of B and C is better that either of the extreme bundles.


10.1.3 Optimal Choice: graphical illustration

The consumer wants to choose the best bundle that is feasible. Figure 10.6 illustrates theoptimal choice for the consumer. Notice that the consumer would like to consume a bundleon indi¤erence curve D, but these bundles are not a¤ordable. The highest indi¤erence curvethat can be attained within the budget constraint is C.

Figure 10.6: Optimal Choice

x

y

A B

C D

Optimalconsumptionbundle

Under the assumption that indi¤erence curves are convex, the optimal bundle is at thepoint where the highest possible indi¤erence curve is tangent to the budget constraint. Thismeans that at the optimal point, the slope of the budget constraint is equal to the slope ofindi¤erence curves.

Condition for optimality of consumption bundle:

Slope of indi¤erence curve = Slope of the budget constraint

10.1.4 Optimal Choice: mathematical treatment

The consumer�s problem is

maxx;y

U (x; y)

s.t.

pxx+ pyy = I

This is a standard form of a constrained optimization problem. After the "max" follows theobjective function, which is what we attempt to maximize. In the consumer optimizationproblem the objective function is the utility. Under the max we write the choice variables,these are the variables that we choose. In the consumer optimization problem the choicevariables are the quantities of the goods purchased. The abbreviation "s.t." means subjectto or such that. The next component of the constrained optimization problem is the con-straint, which is the budget constraint in the consumer�s problem. Thus, in any constrained


optimization problem we maximize the objective function subject to some constraints. Inthe consumer�s problem we maximize the utility subject to the budget constraint3. We willshow two methods of solving the above problem.

Method 1: using the optimality condition

In this method we apply directly the optimality condition derived above (Slope of indi¤erencecurve = Slope of the budget constraint). The slope of the budget constraint is

�pxpy

What is the slope of the indi¤erence curves? Mathematically, an indi¤erence curve is de�nedas follows:

U (x; y) = �U

where �U is some constant utility level, say 17. Thus, an indi¤erence curve consists of all thebundles (x; y) that give the same utility of �U . U (x; y) is "implicit function", as opposed to"explicit", where y is described as a function of x. By the implicit function theorem, theslope of the indi¤erence curves is given by4

dy

dx= �Ux (x; y)

Uy (x; y)

where Ux (x; y) = @U (x; y) =@x and Uy (x; y) = @U (x; y) =@y are partial derivatives of U .These partial derivatives are called "marginal utility" in economics. Thus, Ux (x; y) is themarginal utility from x and Uy (x; y) is the marginal utility from y. The marginal utilityfrom x, Ux (x; y), measures the change in total utility as x changes a little. Recall thatmathematically, the partial derivative of U with respect to x is de�ned as

lim�x!0

U (x+�x; y)� U (x; y)�x

which measures the rate at which U changes when x changes by a small unit. Similarly,Uy (x; y) measures the change in U as we change y by a small unit.Thus, mathematically, the condition of optimality of consumption bundle is given by

�Ux (x; y)Uy (x; y)

= �pxpy

orUx (x; y)

Uy (x; y)=

pxpy

The term on the left hand side is called the Marginal Rate of Substitution between x and yand denoted byMRSx;y. It tells us how many units of y is the consumer willing to trade forone unit of x. Suppose that Ux (x; y) = 3 and Uy (x; y) = 4. This means that an extra unit of

3For more on constrained optimization see section 5.3 in the Math Review notes.4The proof is in the appendix


x increases the utility by 3 and extra unit of y increases the utility by 4. TheMRSx;y = 3=4which means that the consumer is willing to trade 1 unit of x for 3=4 units of y. This makessense because 3=4 units of y generate the same utility as 1 unit of x.To solve the consumer�s problem, we need to combine the optimality condition with the

budget constraint and solve for x and y:

1.Ux (x; y)

Uy (x; y)=

pxpy

2. pxx+ pyy = I

This is a system of two equations with two unknowns (x and y), and we can solve for theoptimal x and y as a function of the exogenous parameters px; py, and I. The solution to theconsumer optimization problem is called demand :

Demand for x : x (px; py; I)

Demand for y : y (px; py; I)

Method 2: Lagrange method

This method allows us to get rid of the constraint at the cost of introducing additionalvariable into the optimization problem. The consumer�s problem is

maxx;y

U (x; y)

s.t.

pxx+ pyy = I

The corresponding Lagrange function (Lagrangian) is

L = U (x; y)� � [pxx+ pyy � I]

The �rst term in the Lagrangian is the objective function, � is the Lagrange multiplier5,and it multiplies the di¤erence between the left hand side and the right hand side of thebudget constraint. Now we have a new function L to maximize. This function has 3 variables(x; y; �), but the advantage is that it does not have constraints. The �rst order conditionsare (di¤erentiating with respect to all 3 variables and equating to zero):

(1) Lx = Ux (x; y)� �px = 0(2) Ly = Uy (x; y)� �py = 0(3) L� = pxx+ pyy � I = 0

The last partial derivative is simply the budget constraint. Typically, people don�t botherto write it. The above system of equation has 3 equations and 3 unknowns (x; y; �). Letssolve it. Rewriting (1) and (2) gives

(1) Ux (x; y) = �px

(2) Uy (x; y) = �py

5See section 5.3 in the Math Review notes.


Dividing (1)/(2) givesUx (x; y)

Uy (x; y)=pxpy

Which is the same condition for optimality of consumption bundle as we derived abovegraphically and mathematically, and it says that at the optimal consumption bundle themarginal rate of substitution between x and y is equal to the absolute value of the slope ofthe budget constraint.Now we need to combine the optimality condition with the budget constraint and solve

for x and y:

1.Ux (x; y)

Uy (x; y)=

pxpy

2. pxx+ pyy = I

This is a system of two equations with two unknowns (x and y), and we can solve for theoptimal x and y as a function of the exogenous parameters px; py, and I. The solution to theconsumer optimization problem is called demand :

Demand for x : x (px; py; I)

Demand for y : y (px; py; I)

We can solve for � as well from the �rst order conditions:

� =Ux (x; y)

px=Uy (x; y)

py

The lagrange multiplier has economic meaning: at the optimum it is equal to the marginalutility from extra $1 spent on x or on y. It can be shown that the value of � also givesthe increase in the maximal utility as we relax the budget constraint by $1. Suppose thatinitially the income is $40,000 and we solve the consumer�s problem and �nd the value ofutility achieved. Now suppose that I want to know by how much will my utility change if myincome becomes $40,001. The answer is that the maximal utility will go up by approximately�. See the Math Review notes for additional illustration of this point.

The intuition behind the optimality condition

The condition for optimality of consumption bundle is

Ux (x; y)

Uy (x; y)=pxpy

This condition can be rewritten as

Ux (x; y)

px=Uy (x; y)

py

The left hand side is the marginal utility from extra $1 spent on x and the right hand sideis the marginal utility from extra $1 spent on y. To see this, observe that $1 buys 1=px


units of x or 1=py units of y. Thus the utility generated by extra $1 spent on x is equal tothe marginal utility from x times how many units of x that $1 can buy. Suppose that theconsumer purchased a bundle (x; y) for which the above condition does not hold, e.g.,

Ux (x; y)

px>Uy (x; y)

py

This means that by increasing the spending on x and decreasing the spending on y theconsumer is able to increase his utility and therefore the bundle is not optimal. Similarly if

Ux (x; y)

px<Uy (x; y)

py

the consumer can increase his utility by spending more on y and reducing the spending onx.Notice that in the discussion of the Lagrange method, the Lagrange multiplier is equal

at the optimum to the marginal utility from extra $1 spent on x or on y.

10.1.5 Examples

1. Suppose that consumer�s preferences are represented by U (x; y) = x�y�, �; � > 0.This is called Cobb-Douglas utility.

(a) Derive the consumer�s demand for x and y.The Lagrangian

L = x�y� � � [pxx+ pyy � I]The �rst order conditions

(1) Lx = �x��1y� � �px = 0(2) Ly = �x�y��1 � �py = 0

From (1) and (2) we obtain the condition for optimality of consumption bundle

(1) �x��1y� = �px

(2) �x�y��1 = �py

(1)/(2)�x��1y�

�x�y��1=pxpy

Thus, the condition for optimality is

�y

�x=pxpy

Here the LHS is MRSx;y and RHS is the absolute value of the slope of B.C.Solving for ygives

y =�

�

pxpyx (10.1)


Substituting into the budget constraint

pxx+ py

��

�

pxpyx

�= I

pxx+�

�pxx = I

pxx

�1 +

�

�

�= I

pxx

��+ �

�

�= I

This gives the demand for x

x =

��

�+ �

�I

px

Plug this into equation (10.1) to �nd the demand for y

y =�

�

pxpy

��

�+ �

I

px

�y =

��

�+ �

�I

py

Thus the demand for x and y is given by

x =

��

�+ �

�I

px, y =

��

�+ �

�I

py

(b) Find the optimal consumption bundle when � = 3, � = 7, px = 2, py = 4, andI = 1000.

x =

�3

3 + 7

�1000

2= 150

y =

�7

3 + 7

�1000

4= 175

Optimal consumption bundle is (150; 175).

(c) Based on the demand functions, what happens to the quantity demand for x andy if I goes up?The quantity demanded of both goods goes up.

(d) Based on the demand functions, what happens to the quantity demand for x andy if px goes up?The quantity demanded of x goes down, but nothing happens to the quantitydemanded of y.


(e) Based on the demand functions, what happens to the quantity demand for x andy if py goes up?

The quantity demanded of y goes down, but nothing happens to the quantitydemanded of x.

(f) In this example x and y

i. substitutes

ii. complements

iii. not related as substitutes or complementThey are not related (iii).

(g) What happens to the fraction of income spent on each good as the prices orincome change?

With Cobb-Douglas preferences, the fraction of income spent on each good is�xed and does not depend on prices or income. The demand is

x =

��

�+ �

�I

px, y =

��

�+ �

�I

py

The fraction of income spent on x is

pxx

I=

�

�+ �

and the fraction of income spent on y is

pyy

I=

�

�+ �

These fractions depend only on the parameters of the utility function, � and �,but not on prices or income. In this example, � = 3, � = 7, so the consumer willalways spend 30% of his income on x and the rest 70% on y, regardless what theprices and income are.

10.1.6 Invariance of utility functions

An important and very useful property of utility functions is invariance with respect tomonotone increasing transformations. What this means is that if U (�; �) is a utility functionthat represents some preferences, and f is some monotone increasing function, then

V = f (U)

represents the same preferences. In other words, consumer with utility function V will makethe same choices as consumer with utility function U .


Examples

1. Suppose that U (x; y) = xayb, a; b > 0. Let f (U) = U1

a+b , thus f is a monotoneincreasing function. Then

V (x; y) = f (U (x; y)) =�xayb

� 1a+b = x

aa+by

ba+b

represents the same preferences. Denoting � = aa+b

and thus (1� �) = ba+b

allows usto represent the Cobb-Douglas preferences with

V (x; y) = x�y1��

that is, we can normalize the exponents to sum up to 1. The demand for x and for yin the previous example becomes much simpler

x = �I

px, y = (1� �) I

py

2. Suppose that U (x; y) = xayb, a; b > 0. Let f (U) = ln(U). Then these same preferencescan be represented by

V (x; y) = � lnx+ (1� �) ln y

If you solve the consumer�s problem with U or V , the solution will be exactly the same,and the demand will be

x = �I

px; y = (1� �) I

py

3. Solve for the consumer�s demand with Cobb-Douglas preferences, when there are ar-bitrary number of goods. For the solution see the appendix.

10.1.7 Income and substitution e¤ects

What do we expect to happen to the quantity demanded of a good when the price of a goodincreases? We might think at �rst that the quantity demanded should necessarily go down,that is the demand curve is downward slopping. Economic theory however does not precludethe possibility of an upward slopping demand curve. There is no such thing as "Law ofDemand" which guarantees that the demand curve must be decreasing in price, but rathersome people make this as an assumption.Economists recognize that when the price of a good goes up, there are two e¤ects on the

quantity demanded:

1. Substitution e¤ect, the change in the quantity demanded due to the change inrelative prices, keeping the purchasing power constant.

2. Income e¤ect, the change in the quantity demanded due to changes in the purchasingpower.


Intuitively, when the price of some good goes up, this has a negative income e¤ect becausewith the same income we can no longer buy the same bundle as before the price increaseand we cannot achieve the level of utility as we had before the price change. Now, supposethat somebody compensated us for the loss of purchasing power by giving us enough incometo attain the same level of utility with the new prices as we had with the old prices. Then,this compensation would have neutralized the negative income e¤ect and the only changein consumption bundle would be due to the substitution e¤ect. Figure 10.7 illustrates theincome and substitution e¤ects.

Figure 10.7: Income and substitution e¤ects

x

y

A

B

C

1IC

2IC

The initial consumption bundle is A. Now suppose that px ", which is re�ected in theincrease in the slope of the budget constraint. The new budget constraint is below the oldone, and the new optimal consumption bundle is C. Thus, the total e¤ect of an increase inthe price of x is the move from bundle A to bundle C. Now we want to break this changeinto the income and substitution e¤ects. Suppose that we compensate the consumer for theprice increase and give him enough income so that with the new prices he can attain thesame utility level as before. This compensation would have resulted in the dashed budgetconstraint which is tangent to the initial indi¤erence curve (IC1) and has the same slope as

10.2. PRODUCER�S CHOICE 121

the new price ratio. Thus, the compensated budget constraint neutralized the income e¤ect,so that the only remaining e¤ect is the substitution e¤ect. With the compensated budgetconstraint the consumer would have chosen the bundle B. Thus the change A ! B is thesubstitution e¤ect. Therefore, the change B ! C is the income e¤ect. To summarize, wehave decomposed the total e¤ect of a price change A! C into two e¤ects: (1) substitutione¤ect A! B and (2) income e¤ect B ! C.Now we are ready to answer the question "what happens to x when px " ? The sub-

stitution e¤ect is always negative for a price increase, because when px ", the substitutione¤ect is re�ected in a movement up along the indi¤erence curve. We know that indi¤erencecurves must be downward slopping, so the substitution e¤ect always cause the consumer tosubstitute away from the good which became more expensive. In �gure 10.7, as a result of asubstitution e¤ect the consumer reduces x and increases y. What about the income e¤ect?From principles of microeconomics, we know that for a normal good the quantity demandedchanges in the same direction as income, while for inferior goods the quantity demandedchanges in the opposite direction from income. Thus, when px ", this is a negative incomee¤ect, so if x is normal the income e¤ect is to reduce x while if x is inferior, the income e¤ectis to increase x. The following table summarizes these results

px "x is normal x is inferior

Substitution e¤ect x # x #Income e¤ect x # x "Total e¤ect x # x?

Economic theory therefore has the following result: if the price of a normal good increases,then the quantity demanded of the good will go down for sure. If however the good is inferior,the quantity demanded will go down only if the substitution e¤ect is stronger than the incomee¤ect.

10.2 Producer�s Choice

We represented the consumer�s preferences with a utility function. In a similar fashion werepresent the production technology with a production function. We assume that there aretwo inputs, capital (K) and labor (L).

De�nition 1 A production function F (K;L) gives the maximal possible output that canbe produced when using K units of capital and L units of labor.

Example. A widely used production function in economics is the Cobb-Douglas produc-tion function

Y = AK�L1��, 0 < � < 1

where Y is the output, A is productivity parameter, K is the capital, L is labor, and �is called the capital share in output. We will discuss this parameter later. Suppose that


A = 10; K = 7; L = 20; � = 0:35. What is the maximal output that can be produced withthis technology and these inputs? Answer:

Y = 10 � 70:35 � 201�0:35 � 138:5

De�nition 2 A production F (K;L) exhibits constant returns to scale if

F (�K; �L) = �F (K;L) ; 8� > 0

This means if a function exhibits constant returns to scale, then when we double all theinputs, the output is also doubled. To see this let � = 2 in the above de�nition. Then

F (2K; 2L) = 2F (K;L)

Example. The Cobb-Douglas production function exhibits constant returns to scale.

A (�K)� (�L)1�� = ��1��AK�L1�� = �AK�L1��

De�nition 3 The marginal product of capital is FK (K;L) and the marginal product of laboris FL (K;L).

Thus, the marginal product of each input is the partial derivative of the productionfunction with respect to the input. In words, the marginal product of capital is the changein total output that results from a small change in capital input. The marginal product oflabor is the change in the total output that results from a small change in the labor input.The marginal product is completely analogous to the marginal utility in the consumer theory.

10.2.1 Firm�s pro�t maximization problem

We assume that the �rm is competitive in both the output market and the input markets.That is, the �rm takes the prices of output and inputs as given. Let P be the price of theoutput, and W and R be the wage and the rental rate of capital respectively. The �rm�smaximization problem is

maxY;K;L

P � Y �RK �WL

s:t:

Y = F (K;L)

Hence the �rm chooses the output level and how much inputs to employ, and it maximizespro�t subject to the technology constraint. The term P � Y is the revenue and thereforeP � Y � RK �WL is the pro�t (revenue - cost). In applications to Macro, we will oftenexpress all the magnitudes in real terms. Thus we will divide the pro�t by the price level Pand denote the real prices of inputs by

r =R

P; w =

W

P

10.2. PRODUCER�S CHOICE 123

The pro�t maximization problem then becomes

maxY;K;L

Y � rK � wL

s:t:

Y = F (K;L)

The easiest way to solve the pro�t maximization problem is to substitute the constraint intothe objective

maxK;L

F (K;L)� rK � wL

This is an unconstrained optimization problem, and the only choice of the �rm is the quan-tities of inputs K and L. The �rst order conditions for optimal input mix:

FK (K;L)� r = 0

FL (K;L)� w = 0

or

FK (K;L) = r

FL (K;L) = w

Thus, a competitive �rm pays each input its marginal product.Example. Write the pro�t maximization problem for a competitive �rm with Cobb-

Douglas technology and derive the �rst order conditions for optimal input mix.Answer:

Pro�t maximization problem: maxK;L

AK�L1�� rK � wL

First order conditions for optimal input mix:

�AK��1L1�� = r

(1� �)AK�L�� = w

Thus, in competition the rental rate of capital is equal to the marginal product of capitaland the wage is equal to the marginal product of labor.

10.2.2 Factor shares

Suppose that the aggregate output in the economy (real GDP) can be modeled as Cobb-Douglas production function:

Y = AK�L1��

And suppose that each input is paid its marginal product. Then a fraction � of the totaloutput is paid to capital and a fraction (1� �) of the total output is paid to labor. To seethis notice that the payment to capital is

rK = �AK��1L1�� K = �AK�L1�� = �Y

thusrK

Y= �


and the payment to labor is

wL = (1� �)AK�L�� L = (1� �)AK�L1�� = (1� �)Y

thuswL

Y= 1� �

10.3 Appendix

10.3.1 Transitivity assumption

Suppose that for some consumer we have A % B and B % C but C � A. Suppose he ownsbundle A. We o¤er him an exchange of C for A and since he strictly prefers C to A he willbe willing to pay 1 penny for the exchange. Then we o¤er him bundle B in exchange for Cand he will accept because B % C. Next we o¤er him to exchange B for A and again hewill accept since A % B. Now he has the bundle A, the one he had in the beginning, but wetook 1 penny from him. By repeating the scheme as many times as we want, he will loose allhis money. Thus, transitivity assumption is an assumption about consistency, and it seemsreasonable. For example, if you prefer BMW to In�nity and you prefer In�nity to Nissan,then the transitivity assumption implies that you will also prefer BMW to Nissan.

10.3.2 The slope of indi¤erence curves

The indi¤erence curve is given byU (x; y) = �U

Fully di¤erentiating both sides gives

Ux (x; y) dx+ Uy (x; y) dy = 0

The full di¤erential gives the total change in the function U as we change x by dx and yby dy. This change corresponds to moving slightly along the indi¤erence curve, so the totalchange in utility is 0. Rearranging the above gives the slope of the indi¤erence curves (andalso proves the part of the implicit function theorem that we used in the notes):

dy

dx= �Ux (x; y)

Uy (x; y)

10.3.3 Demand with Cobb-Douglas Preferences and n goods

In this appendix we solve the consumer�s problem with Cobb-Douglas preferences and ngoods. The consumer�s utility over the n goods is given by the utility function U (x1; x2; :::; xn) =x�11 � x�22 � ::: � x�nn , �i > 0 8i. It is convenient however to work with the logarithmic transfor-mation of this utility function:

~U (x1; x2; :::; xn) = �1 lnx1 + �2 lnx2 + :::+ �n lnxn =nXi=1

�i lnxi

10.3. APPENDIX 125

At this point it is important to recall that utility is invariant under monotone increasingtransformations. Thus, the consumer�s demand will be the same whether his preferences arerepresented by U or by ~U = ln (U).The consumer�s problem that we solve now is therefore

maxfxigni=1

nXi=1

�i lnxi

s:t:nXi=1

pixi = I

The Lagrangian is

L =nXi=1

�i lnxi � �"

nXi=1

pixi � I#

and the �rst order conditions are

Lxi =�ixi� �pi = 0, i = 1; :::; n

L� =nXi=1

pixi � I = 0

We now �nd the demand for an arbitrary good i. Take the �rst order conditions for twogoods, i and j, where the good j is can be any good other than i.

�ixi

= �pi

�jxj

= �pj

Dividing the �rst one by the second one gives

�i�j

xjxi=pipj

thusxj =

�j�i

pipjxi

Now we can substitute the above in the budget constraint

pixi +Xj 6=i

pjxj = I

pixi +Xj 6=i

pj

��j�i

pipjxi

�= I


and solving for xi

pixi +pixi�i

Xj 6=i

�j = I

pixi

1 +

1

�i

Xj 6=i

�j

!= I

pixi

��i +

Pj 6=i �j

�i

�= I

pixi

�Pni=1 �i�i

�= I

Thus, the demand for good i is given by

xi =

��iPni=1 �i

�I

pi

The interpretation is intuitive. The parameter �i is the weight on good i in the utilityfunction, and

Pni=1 �i is the sum of all the weights. Thus, the relative weight on good i is

(�i=Pn

i=1 �i), and the consumer spends this fraction of his income on good i.The next example illustrates the results further. Suppose that the consumers solves

maxx1;x2;x3;x4

0:2 ln x1 + 0:4 ln x2 + 0:6 ln x3 + 0:8 ln x4

s:t:

p1x1 + p2x2 + p3x3 + p4x4 = I

The weights on the four goods are 0:2; 0:4; 0:6; 0:8. The sum of the weights is 0:2 + 0:4 +0:6+0:8 = 2. Thus, the consumer will spend a fraction 0:2=2 of his income on the �rst good,a fraction 0:4=2 of his income on the second good, a fraction 0:6=2 of his income on the thirdgood and a fraction 0:8=2 of his income on the fourth good. The demand is then given by

x1 =0:1I

P1; x2 =

0:2I

P2; x3 =

0:3I

P3; x4 =

0:4I

P4

Chapter 11

Rates of Change

De�nition: Time series - the values of a variable recorded at di¤erent points in timeconstitutes a time series.Time series is collected by a number of di¤erent agencies in the economy. For example

the Bureau of Economic Analysis (BEA) collects data about National Income and Product.Federal Reserve System collects data on Monetary aggregates and Interest rates. The Bureauof Labor Statistics collects data on Employment and wages. Data are also measured indi¤erent time intervals, so we have annual data , which is recorded once a year, quarterlydata recorded four times a year. We also have data recorded every minute such as stockprices on the NYSE.

11.1 Measuring Rates of change.

We distinguish between 2 types of variables. Discrete time variable is a variable that we canmeasure only countable times per year. GDP is an example of such variable, it is measured4 times a year. Continuous time variable is a variable that can be measured at any instant.For example, the temperature in Minneapolis can be measured continuously. It is importantto distinguish between the nature of the variable and our ability to measure it. The GDP iscontinuous by nature since every instant something is being produced. However, we are notable to measure the GDP at the time it is produced. The BEA can estimate the output onlyafter it was sold to the buyers. Therefore, we are going to treat economic variables such asGDP as discrete variables.

11.1.1 Discrete time variables

Notations:Let yt = value a variable at time t.yt+1 = value of GDP at time t+ 1:The rate of change in y from period t to t+ 1 is given by

yt+1 � ytyt

(11.1)

127

128 CHAPTER 11. RATES OF CHANGE

Example: Suppose the price of a good was $76 at 2000 and $87 at 2001. What is therate of change in the price from 2000 to 2001?Solution:

pt+1 � ptpt

=87� 7676

= 0:14474 = 14:474%

Consider the special case when the rate of growth is a constant over time, say g. That isyt+1�ytyt

= g for all values of t. This implies that yt+1 = (1 + g)yt. Hence, we can express thevalue of y at time t in the following way

yt = (1 + g)ty0 (11.2)

where yo = Initial value y at time t = 0.Example: Suppose that you invest $1000 in a trust fund that promises 5% annual interest

rate. How much money will you have in the fund after 10 years?Solution:

yt = (1 + 0:05)101000 = $1628:9

Example: US GDP per capita grows at constant rate of 2% per year. After how manyyears will it double?Solution:

2y0 = (1 + 0:02)ty0

2 = (1:02)t

ln(2) = t ln(1:02)

t =ln(2)

ln(1:02)t 35

Example: Korea grows at 4%. How long would it take for Korea to catch up with theUS if the US GDP grows at a constant rate of 2% per annum and the Korean GDP is justhalf the size of US GDP?Solution:

(1 + 0:02)t2yo = (1 + 0:04)tyo

(1 + 0:02)t2 = (1 + 0:04)t

t ln(1:02) + ln(2) = t ln(1:04)

t =ln(2)

ln(1:04)� ln(1:02) t 35:7

11.1.2 Continuous time variables

Now we assume that the variable y is a di¤erentiable function of time, y(t): This impliesthat it is continuous function. The following formula gives the rate of change of a continuousvariable. This is the continuous time analog to formula 1.

d ln(y(t))

dt(11.3)

11.2. RATE OF CHANGE OF A PRODUCT AND RATIO 129

To show why this gives the rate of change, use the chain rule to get

d ln(y(t))

dt=

1

y(t)

dy(t)

dt

The term dy(t)dt(or in Newton�s notation _y) gives the change in y per �small�unit of time and

it is analogous to the numerator in equation 1, yt+1� yt: The denominator in both formulasis the same.Example: suppose that the population of �sh at time t is given by y(t) = 0:01t: Find the

rate of growth of the �sh population at time t = 7 and t = 8.Solution:

d ln(y(t))

dt=d ln(0:01t)

dt=d[ln(0:01) + ln(t)]

dt=1

t

Hence, after 7 periods the growth rate is 17and after 8 years it is 1

8(we have diminishing

growth rate).Example: Same as before, but now the population at time t is y(t) = e0:05ty0:Solution:

d ln(y(t))

dt=d[0:05t+ ln(y0)]

dt= 0:05 = 5%

Here we got constant growth rate. This example leads us to the continuous time analogto formula 2. This formula gives the value of y at time t under the constant growth rateassumption

y(t) = egty(0) (11.4)

where g is the constant growth rate.

11.2 Rate of change of a product and ratio

There are two important approximations for the growth rate of a product of two variablesand for the growth rate of ratio of two variables. Let a "hat" on top of the variable denoteits rate of change, i.e., bx = xt+1�xt

xt= xt+1

xt� 1. Then the two rules are:

1. The growth rate of a product is approximately the sum of the growth rates, i.e.

cxy � x+ y2. The growth rate of the ratio is approximately the di¤erence of the growth rates

d�xy

�� x� y

Proof. 1. From the de�nition of growth rate, we have

1 +cxy = xt+1yt+1xtyt


Taking ln of both sides

ln (1 +cxy) = ln�xt+1yt+1xtyt

�= ln

�xt+1xt

�+ ln

�yt+1yt

�= ln (1 + x) + ln (1 + y)

Recall that ln (1 + g) � g for small g. Thus, the above equation is approximately

cxy = x+ y2. From the de�nition of growth rate, we have

1 +d�xy

�=

�xt+1yt+1

�=

�xtyt

�=

�xt+1yt+1

��ytxt

�=

�xt+1xt

�=

�yt+1yt

�Taking ln of both sides

ln

1 +

d�xy

�!= ln

�xt+1xt

�� ln

�yt+1yt

�= ln (1 + x)� ln (1 + y)

Recall that ln (1 + g) � g for small g. Thus, the above equation is approximatelyd�xy

�= x� y

Remark: The above can be proved without resorting to the logarithms.Proof. 1. From the de�nition of growth rate, we have

1 +cxy = xt+1yt+1xtyt

= (1 + x) (1 + y) = 1 + x+ y + xy

For small growth rates, the product xy is negligible (e.g. 2% � 3% = 0:0006), so we have

cxy � x+ y2. Omitted.

11.2.1 Examples

1. Suppose that during the last year, the price of a product increased by 2% and thequantity sold increases by 1:5%. What is the approximate growth rate of the revenue?

Solution:[P �Q � P + Q = 2%+ 1:5% = 3%

Remark: If we wanted the exact growth rate of the revenue, then letting the initialvalues of price and quantity be P0 and Q0 we have

1 + [P �Q =

new pricez }| {(1 + 0:02)P �

new quantityz }| {(1 + 0:01)Q

P �Q = (1 + 0:02) (1 + 0:01) = 1:0302

which is close to the approximation.

11.3. LOGARITHMIC SCALE 131

2. Suppose that during the last year, the real GDP is grew at 2:5% and population grewat 1%. What is the approximate growth rate of GDP per capita?

Solution: d�YN

�� Y � N = 2:5%� 1% = 1:5%

Remark: if we wanted the exact growth rate, then letting the initial levels of GDP andpopulation be Y0 and N0, we have

Proof.

1 +d�YN

�=

new GDP per capitaz }| {�(1 + 0:025)Y0(1 + 0:01)N0

�=

�Y0N0

�=1 + 0:025

1 + 0:01= 1:014851485

which is close to the approximation.

11.3 Logarithmic scale

Suppose that a variable y grows at constant rate g and initial value of y0. Then the value ofy at time t is given by

yt = y0(1 + g)t

Now, if we take the natural logarithm of yt, we get that ln (yt) is a linear function of time:

ln(yt) = ln(y0) + t ln(1 + g)

This is a linear function of time, with slope of ln(1 + g) and intercept ln(y0). Now we canshow that the slope of this function is approximately equal to the growth rate, for small g,i.e. ln(1 + g) t gProof. We need to prove that

limg!0

ln (1 + g)

g= 1

Notice that when g ! 0, both the numerator and the denominator in the limit go to zero.In other words, we have a limit of the form of 0

0. Using L�Hopital�s rule we get

limg!0

ln (1 + g)

g= lim

g!0

1= (1 + g)

1= 1

Example. Suppose that you deposit $1000 in a savings account, with interest rateof 3%. The amount of money you have in the in the savings account at any time t isst = 1000 � (1 + 0:03)t, and is shown in the next graph.


Savings over time

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

t

S_t

Now the ln of savings is ln (st) = ln (1000) + t ln (1 + 0:03) and shown in the next graph.

ln(savings)

6.8

6.9

7

7.1

7.2

7.3

7.4

7.5

7.6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

t

ln(S

_t)

Notice that ln of savings is a linear function of time. Also observe that the slope of ln (st) isapproximately equal to

� 7:5� 6:920

=0:6

20= 3%

The properties of logarithmic scale are very useful every time we look at data that isgrowing over time. By looking at the original data we cannot tell whether it is growingat constant rate or not. But when we plot the ln of the variable, we can see right away ifthe variable is growing at constant rate or not. That is, if the ln of the variable looks likelinear function of time, we conclude that the original variable is growing at constant rate.Moreover, we can immediately compute the approximate growth rate of the original variablefrom the slope of the ln, as shown in the previous example.

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