Real Macroeconomic Theory

Contents

1 Introduction 7
2 Motivation: Solow’s growth model 9
2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Business Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.3 Other topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Where next? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Dynamic optimization 15
3.1 Sequential methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.1 A finite horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.2 Infinite horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Dynamic programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 The functional Euler equation . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 Steady states and dynamics under optimal growth 41
4.1 Properties of the capital accumulation function . . . . . . . . . . . . . . 43
4.2 Global convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Dynamics: the speed of convergence . . . . . . . . . . . . . . . . . . . . . 47
4.3.1 Linearization for a general dynamic system . . . . . . . . . . . . . 49
4.3.2 Solving for the speed of convergence . . . . . . . . . . . . . . . . 50
4.3.3 Alternative solution to the speed of convergence . . . . . . . . . . 54
5 Competitive Equilibrium in Dynamic Models 57
5.1 Sequential competitive equilibrium . . . . . . . . . . . . . . . . . . . . . 58
5.1.1 An endowment economy with date-0 trade . . . . . . . . . . . . . 59
5.1.2 The same endowment economy with sequential trade . . . . . . . 61
5.1.3 The neoclassical growth model with date-0 trade . . . . . . . . . . 62
5.1.4 The neoclassical growth model with sequential trade . . . . . . . 64
5.2 Recursive competitive equilibrium . . . . . . . . . . . . . . . . . . . . . . 66
5.2.1 The neoclassical growth model . . . . . . . . . . . . . . . . . . . . 66
5.2.2 The endowment economy with one agent . . . . . . . . . . . . . . 69
5.2.3 An endowment economy with two agents . . . . . . . . . . . . . . 70
5.2.4 Neoclassical production again, with capital accumulation by firms 71
6 Uncertainty 75
6.1 Examples of common stochastic processes in macroeconomics . . . . . . . 75
6.1.1 Markov chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.1.2 Linear stochastic difference equations . . . . . . . . . . . . . . . . 77
6.2 Maximization under uncertainty . . . . . . . . . . . . . . . . . . . . . . . 78
6.2.1 Stochastic neoclassical growth model . . . . . . . . . . . . . . . . 82
6.3 Competitive equilibrium under uncertainty . . . . . . . . . . . . . . . . . 91
6.3.1 The neoclassical growth model with complete markets . . . . . . . 92
6.3.2 General equilibrium under uncertainty: the case of two agent types
in a two-period setting . . . . . . . . . . . . . . . . . . . . . . . . 94
6.3.3 General equilibrium under uncertainty: multiple-period model with
two agent types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.3.4 Recursive formulation . . . . . . . . . . . . . . . . . . . . . . . . 101
6.4 Appendix: basic concepts in stochastic processes . . . . . . . . . . . . . . 102
7 Aggregation 107
7.1 Inelastic labor supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.2 Valued leisure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.2.1 Wealth effects on labor supply . . . . . . . . . . . . . . . . . . . . 110
7.2.2 Wealth effects on labor supply . . . . . . . . . . . . . . . . . . . . 110
8 The overlapping-generations model 111
8.1 Definitions and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.2 An endowment economy . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.2.1 Sequential markets . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.2.2 Arrow-Debreu date-0 markets . . . . . . . . . . . . . . . . . . . . 115
8.2.3 Application: endowment economy with one agent per generation 117
8.3 Economies with intertemporal assets . . . . . . . . . . . . . . . . . . . . 123
8.3.1 Economies with fiat money . . . . . . . . . . . . . . . . . . . . . . 123
8.3.2 Economies with real assets . . . . . . . . . . . . . . . . . . . . . . 127
8.3.3 A tree economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.3.4 Storage economy . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.3.5 Neoclassical growth model . . . . . . . . . . . . . . . . . . . . . . 130
8.4 Dynamic efficiency in models with multiple agents . . . . . . . . . . . . . 132
8.5 The Second Welfare Theorem in dynastic settings . . . . . . . . . . . . . 135
8.5.1 The second welfare theorem in a 1-agent economy . . . . . . . . . 135
8.5.2 The second welfare theorem in a 2-agent economy . . . . . . . . . 138
8.6 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.7 Hybrids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.7.1 The benchmark perpetual-youth model . . . . . . . . . . . . . . . 142
8.7.2 Introducing a life cycle . . . . . . . . . . . . . . . . . . . . . . . . 144
9 Growth 145
9.1 Some motivating long-run facts in macroeconomic data . . . . . . . . . . 145
9.1.1 Kaldor’s stylized facts . . . . . . . . . . . . . . . . . . . . . . . . 145
9.1.2 Other facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9.2 Growth theory I: exogenous growth . . . . . . . . . . . . . . . . . . . . . 147
9.2.1 Exogenous long-run growth . . . . . . . . . . . . . . . . . . . . . 147
9.2.2 Choosing to grow . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
9.2.3 Transforming the model . . . . . . . . . . . . . . . . . . . . . . . 153
9.2.4 Adjustment costs and multisector growth models . . . . . . . . . 155
9.3 Growth theory II: endogenous growth . . . . . . . . . . . . . . . . . . . . 155
9.3.1 The AK model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
9.3.2 Romer’s externality model . . . . . . . . . . . . . . . . . . . . . . 159
9.3.3 Human capital accumulation . . . . . . . . . . . . . . . . . . . . . 160
9.3.4 Endogenous technological change . . . . . . . . . . . . . . . . . . 161
9.3.5 Directed technological change . . . . . . . . . . . . . . . . . . . . 165
9.3.6 Models without scale effects . . . . . . . . . . . . . . . . . . . . . 165
9.4 What explains long-run growth and the world income distribution? . . . 165
9.4.1 Long-run U.S. growth . . . . . . . . . . . . . . . . . . . . . . . . 165
9.4.2 Assessing different models . . . . . . . . . . . . . . . . . . . . . . 165
9.4.3 Productivity accounting . . . . . . . . . . . . . . . . . . . . . . . 168
9.4.4 A stylized model of development . . . . . . . . . . . . . . . . . . . 168