Macroeconomic Theory
Dirk Krueger1
Department of Economics
Stanford University
September 25, 2002
I am grateful to my teachers inMinnesota, V.V Chari, Timothy Kehoe and Edward
Prescott, my colleagues at Stanford, Robert Hall, Beatrix Paal and Tom Sargent,
my co-authors Juan Carlos Conesa, Jesus Fernandez-Villaverde and Fabrizio Perri as
well as Victor Rios-Rull for helping me to learn modern macroeconomic theory. All
remaining errors are mine alone.
Contents
1 Overview and Summary 1
2 A Simple Dynamic Economy 5
2.1 General Principles for Specifying aModel . . . . . . . . . . . . . 5
2.2 An Example Economy . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Definition of Competitive Equilibrium . . . . . . . . . . . 7
2.2.2 Solving for the Equilibrium . . . . . . . . . . . . . . . . . 8
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 SequentialMarkets Equilibrium . . . . . . . . . . . . . . . 18
2.3 Appendix: Some Facts about Utility Functions . . . . . . . . . . 23
3 The Neoclassical Growth Model in Discrete Time 27
3.1 Setup of theModel . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Optimal Growth: Pareto Optimal Allocations . . . . . . . . . . . 28
3.2.1 Social Planner Problem in Sequential Formulation . . . . 29
3.2.2 Recursive Formulation of Social Planner Problem . . . . . 31
3.2.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.4 The Euler Equation Approach and Transversality Conditions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Competitive Equilibrium Growth . . . . . . . . . . . . . . . . . . 49
3.3.1 Definition of Competitive Equilibrium . . . . . . . . . . . 50
3.3.2 Characterization of the Competitive Equilibrium and the
Welfare Theorems . . . . . . . . . . . . . . . . . . . . . . 52
3.3.3 SequentialMarkets Equilibrium . . . . . . . . . . . . . . . 56
3.3.4 Recursive Competitive Equilibrium . . . . . . . . . . . . . 57
4 Mathematical Preliminaries 59
4.1 CompleteMetric Spaces . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Convergence of Sequences . . . . . . . . . . . . . . . . . . . . . . 61
4.3 The ContractionMapping Theorem . . . . . . . . . . . . . . . . 65
4.4 The Theoremof theMaximum . . . . . . . . . . . . . . . . . . . 71
5 Dynamic Programming 73
5.1 The Principle of Optimality . . . . . . . . . . . . . . . . . . . . . 73
5.2 Dynamic Programming with Bounded Returns . . . . . . . . . . 80
6 Models with Uncertainty 83
6.1 Basic Representation of Uncertainty . . . . . . . . . . . . . . . . 83
6.2 Definitions of Equilibrium . . . . . . . . . . . . . . . . . . . . . . 85
6.2.1 Arrow-DebreuMarket Structure . . . . . . . . . . . . . . 85
6.2.2 SequentialMarketsMarket Structure . . . . . . . . . . . . 87
6.2.3 Equivalence betweenMarket Structures . . . . . . . . . . 88
6.3 Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.4 Stochastic Neoclassical GrowthModel . . . . . . . . . . . . . . . 90
7 The Two Welfare Theorems 93
7.1 What is an Economy? . . . . . . . . . . . . . . . . . . . . . . . . 93
7.2 Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.3 Definition of Competitive Equilibrium . . . . . . . . . . . . . . . 99
7.4 The Neoclassical Growth Model in Arrow-Debreu Language . . . 99
7.5 A Pure Exchange Economy in Arrow-Debreu Language . . . . . 101
7.6 The FirstWelfare Theorem . . . . . . . . . . . . . . . . . . . . . 103
7.7 The SecondWelfare Theorem . . . . . . . . . . . . . . . . . . . . 104
7.8 Type Identical Allocations . . . . . . . . . . . . . . . . . . . . . . 113
8 The Overlapping Generations Model 115
8.1 A Simple Pure Exchange Overlapping GenerationsModel . . . . 116
8.1.1 Basic Setup of theModel . . . . . . . . . . . . . . . . . . 117
8.1.2 Analysis of the Model Using Offer Curves . . . . . . . . . 122
8.1.3 Inefficient Equilibria . . . . . . . . . . . . . . . . . . . . . 129
8.1.4 Positive Valuation of OutsideMoney . . . . . . . . . . . . 134
8.1.5 Productive Outside Assets . . . . . . . . . . . . . . . . . . 136
8.1.6 Endogenous Cycles . . . . . . . . . . . . . . . . . . . . . . 138
8.1.7 Social Security and Population Growth . . . . . . . . . . 140
8.2 The Ricardian Equivalence Hypothesis . . . . . . . . . . . . . . . 145
8.2.1 Infinite Lifetime Horizon and Borrowing Constraints . . . 146
8.2.2 Finite Horizon and Operative BequestMotives . . . . . . 155
8.3 Overlapping GenerationsModels with Production . . . . . . . . . 160
8.3.1 Basic Setup of theModel . . . . . . . . . . . . . . . . . . 161
8.3.2 Competitive Equilibrium . . . . . . . . . . . . . . . . . . 161
8.3.3 Optimality of Allocations . . . . . . . . . . . . . . . . . . 168
8.3.4 The Long-Run Effects of Government Debt . . . . . . . . 172
9 Continuous Time Growth Theory 177
9.1 Stylized Growth and Development Facts . . . . . . . . . . . . . . 177
9.1.1 Kaldor’s Growth Facts . . . . . . . . . . . . . . . . . . . . 178
9.1.2 Development Facts from the Summers-Heston Data Set . 178
9.2 The SolowModel and its Empirical Evaluation . . . . . . . . . . 183
9.2.1 TheModel and its Implications . . . . . . . . . . . . . . . 186
9.2.2 Empirical Evaluation of theModel . . . . . . . . . . . . . 189
9.3 The Ramsey-Cass-KoopmansModel . . . . . . . . . . . . . . . . 199
9.3.1 Mathematical Preliminaries: Pontryagin’sMaximum Principle
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
9.3.2 Setup of theModel . . . . . . . . . . . . . . . . . . . . . . 200
9.3.3 Social Planners Problem . . . . . . . . . . . . . . . . . . . 202
9.3.4 Decentralization . . . . . . . . . . . . . . . . . . . . . . . 210
9.4 Endogenous GrowthModels . . . . . . . . . . . . . . . . . . . . . 215
9.4.1 The Basic AK-Model . . . . . . . . . . . . . . . . . . . . 216
9.4.2 Models with Externalities . . . . . . . . . . . . . . . . . . 220
9.4.3 Models of Technological Progress Based on Monopolistic
Competition: Variant of Romer (1990) . . . . . . . . . . . 232
10 Bewley Models 245
10.1 Some Stylized Facts about the Income and Wealth Distribution
in the U.S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
10.1.1 Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . 246
10.1.2 Main Stylized Facts . . . . . . . . . . . . . . . . . . . . . 247
10.2 The Classic Income Fluctuation Problem . . . . . . . . . . . . . 253
10.2.1 Deterministic Income . . . . . . . . . . . . . . . . . . . . 254
10.2.2 Stochastic Income and Borrowing Limits . . . . . . . . . . 262
10.3 Aggregation: Distributions as State Variables . . . . . . . . . . . 266
10.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
10.3.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . 273
11 Fiscal Policy 279
11.1 Positive Fiscal Policy . . . . . . . . . . . . . . . . . . . . . . . . . 279
11.2 Normative Fiscal Policy . . . . . . . . . . . . . . . . . . . . . . . 279
11.2.1 Optimal Policy with Commitment . . . . . . . . . . . . . 279
11.2.2 The Time Consistency Problem and Optimal Fiscal Policy
without Commitment . . . . . . . . . . . . . . . . . . . . 279
12 Political Economy and Macroeconomics 281
13 References 283
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