Notes on Dynamic Methods in Macroeconomics

Nicola Pavoni.
This version: October 2008.
Very Preliminary and Largely Incomplete.

Contents

1 Introduction to Recursive Methods 7
1.1 The Optimal Growth Model . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.1 Sequential Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.2 Recursive Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Finite Horizon Problems as Guided Guesses . . . . . . . . . . . . . . . . . 14
2 Useful Mathematics 19
2.1 Metric and Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 The Inmum and the Supremum of a Set, the Completeness of IR . . . . . 21
2.3 Sequences: Convergence, liminf, limsup and the Cauchy Criterion . . . . . 23
2.4 Closed and Compact Sets: Maxima and Minima . . . . . . . . . . . . . . . 28
2.5 Complete Metric Spaces and The Contraction Mapping Theorem . . . . . 29
2.6 Continuity and the Maximum Theorem . . . . . . . . . . . . . . . . . . . . 34
3 Deterministic Dynamic Programming 43
3.1 The Bellman Principle of Optimality . . . . . . . . . . . . . . . . . . . . . 43
3.2 The BPO under Bounded Returns: Continuity, Concavity and Dierentiability
of the Value Function . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 The Euler's Variational Approach and the Bellman Principle of Optimality 55
3.4 Optimal Control and the Maximization Principle of Pontryagin . . . . . . 62
4 Stochastic Dynamic Programming 69
4.1 The Axiomatic Approach to Probability: Basic Concepts of Measure Theory 69
4.2 Markov Chains and Markov Processes . . . . . . . . . . . . . . . . . . . . . 72
4.3 Bellman Principle in the Stochastic Framework . . . . . . . . . . . . . . . 79
4.4 The Stochastic Model of Optimal Growth . . . . . . . . . . . . . . . . . . 83
5 A Quick Introduction to Numerical Methods 87
5.1 Value Function Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1.0.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.1.0.2 Smooth Approximation . . . . . . . . . . . . . . . . . . . 92
5.2 Solving Directly the Functional Equation: Projection Methods . . . . . . . 93
6 Risk Aversion, Insurance and Asset Pricing in Macroeconomics 97
6.1 Risk and Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2 Arrow-Debreu Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2.1 Properties of the ecient allocation: Insurance . . . . . . . . . . . . 105
6.3 Asset Pricing Under Complete Markets . . . . . . . . . . . . . . . . . . . . 107
6.3.1 Radner Sequential Trade Equilibrium . . . . . . . . . . . . . . . . . 107
6.3.2 Introducing Proper Assets . . . . . . . . . . . . . . . . . . . . . . . 108
6.3.2.1 Equilibrium price of time . . . . . . . . . . . . . . . . . . 111
6.4 The Lucas Tree Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.4.1 The Consumption CAPM and the Security Market Line . . . . . . 120
6.4.2 Pricing assets from fundamentals . . . . . . . . . . . . . . . . . . . 121
6.5 The Equity Premium Puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7 Introduction to Consumption Theory 127
7.1 Quadratic Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.1.1 Liquidity Constraints: Excess Sensitivity . . . . . . . . . . . . . . . 128
7.1.2 Quadratic Utility: Certainty Equivalence . . . . . . . . . . . . . . . 129
7.2 Permanent Income Hypothesis (PIH) . . . . . . . . . . . . . . . . . . . . . 130
7.2.1 Excess Smoothness versus Excess Sensitivity . . . . . . . . . . . . . 130
8 Theory of Fixed Investment and Employment Dynamics 133
8.1 The Value of the Firm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.2 Problem of the Representative Firm . . . . . . . . . . . . . . . . . . . . . . 135
8.3 The Neoclassical Theory of Investment . . . . . . . . . . . . . . . . . . . . 136
8.4 Convex Adjustment Costs: The q-Theory of Investment . . . . . . . . . . . 138
8.4.1 Marginal versus Average Tobin's q . . . . . . . . . . . . . . . . . . . 142
8.5 Linear Adjustment Costs and Employment Dynamics under Uncertainty . 143
8.5.1 The Option Value Effect . . . . . . . . . . . . . . . . . . . . . . . . 145
9 Introduction to Search Theory 149
9.1 Search and the labor market . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9.2 The Representative Worker . . . . . . . . . . . . . . . . . . . . . . . . . . 150
9.3 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
9.4 Search and Asset Accumulation . . . . . . . . . . . . . . . . . . . . . . . . 153
10 General Equilibrium, Growth, and Real Business Cycles 155
10.1 Business Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
10.1.1 A bit of History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
10.2 The Neoclassical Growth Model Again . . . . . . . . . . . . . . . . . . . . 159
10.3 Competitive Equilibrium and the Welfare Theorems . . . . . . . . . . . . . 160
10.4 Calibration and Estimation: The RBC Methodology . . . . . . . . . . . . . 162
10.5 The Basic RBC Model and its Performance . . . . . . . . . . . . . . . . . . 164
10.5.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
10.5.2 Evaluating Quantitatively the model . . . . . . . . . . . . . . . . . 169
10.6 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
10.7 Recursive General Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 174
10.8 General Equilibrium with Heterogeneous Agents . . . . . . . . . . . . . . . 174
10.8.1 Finitely Many Agents and Ex-Ante Heterogeneity: The Negishi's
Weights Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
10.8.2 Decentralization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
11 Endogenously Incomplete Markets 177
11.1 A Model with Endogenous Liquidity Constraints (from Kehoe and Levine,
Econometrica 2001) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
11.1.0.1 Model (Physical Environment) . . . . . . . . . . . . . . . 177
11.1.1 Complete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
11.1.2 Liquidity Constrained Economy . . . . . . . . . . . . . . . . . . . . 179
11.1.3 Debt Constrained Economy . . . . . . . . . . . . . . . . . . . . . . 181
11.1.4 Equilibrium and Eciency . . . . . . . . . . . . . . . . . . . . . . . 181
11.2 Stochastic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
12 Recursive Contracts 185
12.1 Default Contracts with one sided commitment . . . . . . . . . . . . . . . . 185
12.1.1 Set up and contracts . . . . . . . . . . . . . . . . . . . . . . . . . . 185
12.1.2 The Recursive formulation . . . . . . . . . . . . . . . . . . . . . . . 186
12.1.3 Characterization of the contract in the innite horizon case . . . . . 189