PDF Macroeconomic Theory - SSCC

Macroeconomic Theory

Dirk Krueger1 Department of Economics University of Pennsylvania

January 26, 2012

1 I am grateful to my teachers in Minnesota, V.V Chari, Timothy Kehoe and Edward Prescott, my ex-colleagues at Stanford, Robert Hall, Beatrix Paal and Tom Sargent, my colleagues at UPenn Hal Cole, Jeremy Greenwood, Randy Wright and Iourii Manovski and my co-authors Juan Carlos Conesa, Jesus Fernandez-Villaverde, Felix Kubler and Fabrizio Perri as well as Victor Rios-Rull for helping me to learn modern macroeconomic theory. These notes were tried out on numerous students at Stanford, UPenn, Frankfurt and Mannheim, whose many useful comments I appreciate. Kaiji Chen and Antonio Doblas-Madrid provided many important corrections to these notes.

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Contents

1 Overview and Summary

1

2 A Simple Dynamic Economy

5

2.1 General Principles for Specifying a Model . . . . . . . . . . . . . 5

2.2 An Example Economy . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 De...nition of Competitive Equilibrium . . . . . . . . . . . 8

2.2.2 Solving for the Equilibrium . . . . . . . . . . . . . . . . . 9

2.2.3 Pareto Optimality and the First Welfare Theorem . . . . 11

2.2.4 Negishi's (1960) Method to Compute Equilibria . . . . . . 14

2.2.5 Sequential Markets Equilibrium . . . . . . . . . . . . . . . 18

2.3 Appendix: Some Facts about Utility Functions . . . . . . . . . . 24

2.3.1 Time Separability . . . . . . . . . . . . . . . . . . . . . . 24

2.3.2 Time Discounting . . . . . . . . . . . . . . . . . . . . . . 24

2.3.3 Standard Properties of the Period Utility Function . . . . 25

2.3.4 Constant Relative Risk Aversion (CRRA) Utility . . . . . 25

2.3.5 Homotheticity and Balanced Growth . . . . . . . . . . . . 28

3 The Neoclassical Growth Model in Discrete Time

31

3.1 Setup of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Optimal Growth: Pareto Optimal Allocations . . . . . . . . . . . 32

3.2.1 Social Planner Problem in Sequential Formulation . . . . 33

3.2.2 Recursive Formulation of Social Planner Problem . . . . . 35

3.2.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.4 The Euler Equation Approach and Transversality Condi-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.5 Steady States and the Modi...ed Golden Rule . . . . . . . 52

3.2.6 A Remark About Balanced Growth . . . . . . . . . . . . 53

3.3 Competitive Equilibrium Growth . . . . . . . . . . . . . . . . . . 55

3.3.1 De...nition of Competitive Equilibrium . . . . . . . . . . . 56

3.3.2 Characterization of the Competitive Equilibrium and the

Welfare Theorems . . . . . . . . . . . . . . . . . . . . . . 58

3.3.3 Sequential Markets Equilibrium . . . . . . . . . . . . . . . 64

3.3.4 Recursive Competitive Equilibrium . . . . . . . . . . . . . 65

3.4 Mapping the Model to Data: Calibration . . . . . . . . . . . . . 67

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CONTENTS

4 Mathematical Preliminaries

71

4.1 Complete Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . 72

4.2 Convergence of Sequences . . . . . . . . . . . . . . . . . . . . . . 73

4.3 The Contraction Mapping Theorem . . . . . . . . . . . . . . . . 77

4.4 The Theorem of the Maximum . . . . . . . . . . . . . . . . . . . 83

5 Dynamic Programming

85

5.1 The Principle of Optimality . . . . . . . . . . . . . . . . . . . . . 85

5.2 Dynamic Programming with Bounded Returns . . . . . . . . . . 92

6 Models with Risk

95

6.1 Basic Representation of Risk . . . . . . . . . . . . . . . . . . . . 95

6.2 De...nitions of Equilibrium . . . . . . . . . . . . . . . . . . . . . . 97

6.2.1 Arrow-Debreu Market Structure . . . . . . . . . . . . . . 98

6.2.2 Pareto E? ciency . . . . . . . . . . . . . . . . . . . . . . . 100

6.2.3 Sequential Markets Market Structure . . . . . . . . . . . . 101

6.2.4 Equivalence between Market Structures . . . . . . . . . . 102

6.2.5 Asset Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.3 Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.4 Stochastic Neoclassical Growth Model . . . . . . . . . . . . . . . 106

7 The Two Welfare Theorems

109

7.1 What is an Economy? . . . . . . . . . . . . . . . . . . . . . . . . 109

7.2 Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.3 De...nition of Competitive Equilibrium . . . . . . . . . . . . . . . 114

7.4 The Neoclassical Growth Model in Arrow-Debreu Language . . . 115

7.5 A Pure Exchange Economy in Arrow-Debreu Language . . . . . 117

7.6 The First Welfare Theorem . . . . . . . . . . . . . . . . . . . . . 119

7.7 The Second Welfare Theorem . . . . . . . . . . . . . . . . . . . . 120

7.8 Type Identical Allocations . . . . . . . . . . . . . . . . . . . . . . 128

8 The Overlapping Generations Model

129

8.1 A Simple Pure Exchange Overlapping Generations Model . . . . 130

8.1.1 Basic Setup of the Model . . . . . . . . . . . . . . . . . . 131

8.1.2 Analysis of the Model Using O?er Curves . . . . . . . . . 136

8.1.3 Ine? cient Equilibria . . . . . . . . . . . . . . . . . . . . . 143

8.1.4 Positive Valuation of Outside Money . . . . . . . . . . . . 148

8.1.5 Productive Outside Assets . . . . . . . . . . . . . . . . . . 150

8.1.6 Endogenous Cycles . . . . . . . . . . . . . . . . . . . . . . 152

8.1.7 Social Security and Population Growth . . . . . . . . . . 154

8.2 The Ricardian Equivalence Hypothesis . . . . . . . . . . . . . . . 160

8.2.1 In...nite Lifetime Horizon and Borrowing Constraints . . . 161

8.2.2 Finite Horizon and Operative Bequest Motives . . . . . . 170

8.3 Overlapping Generations Models with Production . . . . . . . . . 175

8.3.1 Basic Setup of the Model . . . . . . . . . . . . . . . . . . 175

8.3.2 Competitive Equilibrium . . . . . . . . . . . . . . . . . . 176

CONTENTS

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8.3.3 Optimality of Allocations . . . . . . . . . . . . . . . . . . 183 8.3.4 The Long-Run E?ects of Government Debt . . . . . . . . 187

9 Continuous Time Growth Theory

193

9.1 Stylized Growth and Development Facts . . . . . . . . . . . . . . 193

9.1.1 Kaldor's Growth Facts . . . . . . . . . . . . . . . . . . . . 194

9.1.2 Development Facts from the Summers-Heston Data Set . 194

9.2 The Solow Model and its Empirical Evaluation . . . . . . . . . . 199

9.2.1 The Model and its Implications . . . . . . . . . . . . . . . 202

9.2.2 Empirical Evaluation of the Model . . . . . . . . . . . . . 204

9.3 The Ramsey-Cass-Koopmans Model . . . . . . . . . . . . . . . . 215

9.3.1 Mathematical Preliminaries: Pontryagin's Maximum Prin-

ciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

9.3.2 Setup of the Model . . . . . . . . . . . . . . . . . . . . . . 215

9.3.3 Social Planners Problem . . . . . . . . . . . . . . . . . . . 217

9.3.4 Decentralization . . . . . . . . . . . . . . . . . . . . . . . 226

9.4 Endogenous Growth Models . . . . . . . . . . . . . . . . . . . . . 231

9.4.1 The Basic AK-Model . . . . . . . . . . . . . . . . . . . . 231

9.4.2 Models with Externalities . . . . . . . . . . . . . . . . . . 235

9.4.3 Models of Technological Progress Based on Monopolistic

Competition: Variant of Romer (1990) . . . . . . . . . . . 248

10 Bewley Models

261

10.1 Some Stylized Facts about the Income and Wealth Distribution

in the U.S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

10.1.1 Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . 262

10.1.2 Main Stylized Facts . . . . . . . . . . . . . . . . . . . . . 263

10.2 The Classic Income Fluctuation Problem . . . . . . . . . . . . . 269

10.2.1 Deterministic Income . . . . . . . . . . . . . . . . . . . . 270

10.2.2 Stochastic Income and Borrowing Limits . . . . . . . . . . 278

10.3 Aggregation: Distributions as State Variables . . . . . . . . . . . 282

10.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

10.3.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . 289

11 Fiscal Policy

293

11.1 Positive Fiscal Policy . . . . . . . . . . . . . . . . . . . . . . . . . 293

11.2 Normative Fiscal Policy . . . . . . . . . . . . . . . . . . . . . . . 293

11.2.1 Optimal Policy with Commitment . . . . . . . . . . . . . 293

11.2.2 The Time Consistency Problem and Optimal Fiscal Policy

without Commitment . . . . . . . . . . . . . . . . . . . . 293

12 Political Economy and Macroeconomics

295

13 References

297

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