### Transcription of Macroeconomic Theory - SSCC

1 **Macroeconomic** **Theory** Dirk Krueger1. Department of Economics University of Pennsylvania January 26, 2012. 1. I am grateful to my teachers in Minnesota, Chari, Timothy Kehoe and Ed- ward 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 appreci- ate. Kaiji Chen and Antonio Doblas-Madrid provided many important corrections to these notes.

2 Ii Contents 1 Overview and Summary 1. 2 A Simple Dynamic Economy 5. General Principles for Specifying a Model .. 5. An Example Economy .. 6. De nition of Competitive Equilibrium .. 8. Solving for the Equilibrium .. 9. Pareto Optimality and the First Welfare Theorem .. 11. Negishi's (1960) Method to Compute Equilibria .. 14. Sequential Markets Equilibrium .. 18. Appendix: Some Facts about Utility Functions .. 24. Time Separability .. 24. Time Discounting .. 24. Standard Properties of the Period Utility Function .. 25. Constant Relative Risk Aversion (CRRA) Utility .. 25. Homotheticity and Balanced Growth .. 28. 3 The Neoclassical Growth Model in Discrete Time 31. Setup of the Model .. 31. Optimal Growth: Pareto Optimal Allocations .. 32. Social Planner Problem in Sequential Formulation.

3 33. Recursive Formulation of Social Planner Problem .. 35. An Example .. 37. The Euler Equation Approach and Transversality Condi- tions .. 44. Steady States and the Modi ed Golden Rule .. 52. A Remark About Balanced Growth .. 53. Competitive Equilibrium Growth .. 55. De nition of Competitive Equilibrium .. 56. Characterization of the Competitive Equilibrium and the Welfare Theorems .. 58. Sequential Markets Equilibrium .. 64. Recursive Competitive Equilibrium .. 65. Mapping the Model to Data: Calibration .. 67. iii iv CONTENTS. 4 Mathematical Preliminaries 71. Complete Metric Spaces .. 72. Convergence of Sequences .. 73. The Contraction Mapping Theorem .. 77. The Theorem of the Maximum .. 83. 5 Dynamic Programming 85. The Principle of Optimality.

4 85. Dynamic Programming with Bounded Returns .. 92. 6 Models with Risk 95. Basic Representation of Risk .. 95. De nitions of Equilibrium .. 97. Arrow-Debreu Market Structure .. 98. Pareto E ciency .. 100. Sequential Markets Market Structure .. 101. Equivalence between Market Structures .. 102. Asset Pricing .. 102. Markov Processes .. 104. Stochastic Neoclassical Growth Model .. 106. 7 The Two Welfare Theorems 109. What is an Economy? .. 109. Dual Spaces .. 112. De nition of Competitive Equilibrium .. 114. The Neoclassical Growth Model in Arrow-Debreu Language .. 115. A Pure Exchange Economy in Arrow-Debreu Language .. 117. The First Welfare Theorem .. 119. The Second Welfare Theorem .. 120. Type Identical Allocations .. 128. 8 The Overlapping Generations Model 129.

5 A Simple Pure Exchange Overlapping Generations Model .. 130. Basic Setup of the Model .. 131. Analysis of the Model Using O er Curves .. 136. Ine cient Equilibria .. 143. Positive Valuation of Outside Money .. 148. Productive Outside Assets .. 150. Endogenous Cycles .. 152. Social Security and Population Growth .. 154. The Ricardian Equivalence Hypothesis .. 160. In nite Lifetime Horizon and Borrowing Constraints .. 161. Finite Horizon and Operative Bequest Motives .. 170. Overlapping Generations Models with Production .. 175. Basic Setup of the Model .. 175. Competitive Equilibrium .. 176. CONTENTS v Optimality of Allocations .. 183. The Long-Run E ects of Government Debt .. 187. 9 Continuous Time Growth **Theory** 193. Stylized Growth and Development Facts.

6 193. Kaldor's Growth Facts .. 194. Development Facts from the Summers-Heston Data Set . 194. The Solow Model and its Empirical Evaluation .. 199. The Model and its Implications .. 202. Empirical Evaluation of the Model .. 204. The Ramsey-Cass-Koopmans Model .. 215. Mathematical Preliminaries: Pontryagin's Maximum Prin- ciple .. 215. Setup of the Model .. 215. Social Planners Problem .. 217. Decentralization .. 226. Endogenous Growth Models .. 231. The Basic AK-Model .. 231. Models with Externalities .. 235. Models of Technological Progress Based on Monopolistic Competition: Variant of Romer (1990) .. 248. 10 Bewley Models 261. Some Stylized Facts about the Income and Wealth Distribution in the .. 262. Data Sources .. 262. Main Stylized Facts.

7 263. The Classic Income Fluctuation Problem .. 269. Deterministic Income .. 270. Stochastic Income and Borrowing Limits .. 278. Aggregation: Distributions as State Variables .. 282. **Theory** .. 282. Numerical Results .. 289. 11 Fiscal Policy 293. Positive Fiscal Policy .. 293. Normative Fiscal Policy .. 293. Optimal Policy with Commitment .. 293. The Time Consistency Problem and Optimal Fiscal Policy without Commitment .. 293. 12 Political Economy and **macroeconomics** 295. 13 References 297. vi CONTENTS. Chapter 1. Overview and Summary After a quick warm-up for dynamic general equilibrium models in the rst part of the course we will discuss the two workhorses of modern **macroeconomics** , the neoclassical growth model with in nitely lived consumers and the Overlapping Generations (OLG) model.

8 This rst part will focus on techniques rather than issues; one rst has to learn a language before composing poems. I will rst present a simple dynamic pure exchange economy with two in- nitely lived consumers engaging in intertemporal trade. In this model the connection between competitive equilibria and Pareto optimal equilibria can be easily demonstrated. Furthermore it will be demonstrated how this connec- tion can exploited to compute equilibria by solving a particular social planners problem, an approach developed rst by Negishi (1960) and discussed nicely by Kehoe (1989). This model with then enriched by production (and simpli ed by dropping one of the two agents), to give rise to the neoclassical growth model. This model will rst be presented in discrete time to discuss discrete-time dynamic programming techniques; both theoretical as well as computational in nature.

9 The main reference will be Stokey et al., chapters 2-4. As a rst economic application the model will be enriched by technology shocks to develop the Real Business Cycle (RBC) **Theory** of business cycles. Cooley and Prescott (1995) are a good reference for this application. In order to formulate the stochastic neoclassical growth model notation for dealing with uncertainty will be developed. This discussion will motivate the two welfare theorems, which will then be presented for quite general economies in which the commodity space may be in nite-dimensional. We will draw on Stokey et al., chapter 15's discussion of Debreu (1954). The next two topics are logical extensions of the preceding material. We will rst discuss the OLG model, due to Samuelson (1958) and Diamond (1965).

10 The rst main focus in this module will be the theoretical results that distinguish the OLG model from the standard Arrow-Debreu model of general equilibrium: in the OLG model equilibria may not be Pareto optimal, at money may have 1. 2 CHAPTER 1. OVERVIEW AND SUMMARY. positive value, for a given economy there may be a continuum of equilibria (and the core of the economy may be empty). All this could not happen in the standard Arrow-Debreu model. References that explain these di erences in detail include Geanakoplos (1989) and Kehoe (1989). Our discussion of these issues will largely consist of examples. One reason to develop the OLG model was the uncomfortable assumption of in nitely lived agents in the standard neoclassical growth model. Barro (1974) demonstrated under which conditions (operative bequest motives) an OLG economy will be equivalent to an economy with in nitely lived consumers.