Macroeconomic Theory

Dirk Krueger¹ Department of Economics University of Pennsylvania

September 12, 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	Ove	rview	and Summary	1
2	A Si	mple	Dynamic Economy	5
	2.1	Gener	al Principles for Specifying a Model . . . . . . . . . . . . .	5
	2.2	An Ex	ample 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	Appen	dix: 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	Neoc	lassical Growth Model in Discrete Time	31
	3.1	Setup	of the Model . . . . . . . . . . . . . . . . . . . . . . . . . .	31
	3.2	Optim	al 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	Comp	etitive 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	Mappi	ng the Model to Data: Calibration . . . . . . . . . . . . .	67

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iv			CONTENTS
4	Mat	hema	tical Preliminaries 71
	4.1	Comp	lete Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . 72
	4.2	Conve	rgence of Sequences . . . . . . . . . . . . . . . . . . . . . . 73
	4.3	The C	ontraction Mapping Theorem . . . . . . . . . . . . . . . . 77
	4.4	The T	heorem of the Maximum . . . . . . . . . . . . . . . . . . . 83
5	Dyn	amic	Programming 85
	5.1	The P	rinciple of Optimality . . . . . . . . . . . . . . . . . . . . . 85
	5.2	Dyna	mic Programming with Bounded Returns . . . . . . . . . . 92
6	Mo	dels w	ith Risk 95
	6.1	Basic	Representation of Risk . . . . . . . . . . . . . . . . . . . . 95
	6.2	De…ni	tions 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	Mark	ov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
	6.4	Stoch	astic 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…ni	tion of Competitive Equilibrium . . . . . . . . . . . . . . . 114
	7.4	The N	eoclassical Growth Model in Arrow-Debreu Language . . . 115
	7.5	A Pu	re Exchange Economy in Arrow-Debreu Language . . . . . 117
	7.6	The F	irst Welfare Theorem . . . . . . . . . . . . . . . . . . . . . 119
	7.7	The S	econd Welfare Theorem . . . . . . . . . . . . . . . . . . . . 120
	7.8	Type	Identical Allocations . . . . . . . . . . . . . . . . . . . . . . 128
8	The	Over	lapping Generations Model 129
	8.1	A Sim	ple 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 R	icardian 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	Overl	apping Generations Models with Production . . . . . . . . . 175
		8.3.1	Basic Setup of the Model . . . . . . . . . . . . . . . . . . 175
		8.3.2	Competitive Equilibrium . . . . . . . . . . . . . . . . . . 176

CO	NTE	NTS 8.3.3	v Optimality of Allocations . . . . . . . . . . . . . . . . . . 183
		8.3.4	The Long-Run E¤ects of Government Debt . . . . . . . . 187
9	Con	tinuou	s Time Growth Theory 193
	9.1	Stylize	d 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 S	olow 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 R	amsey-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	Endog	enous 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	Bew	ley M	odels 261
	10.1	Some in the	Stylized Facts about the Income and Wealth Distribution U.S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
		10.1.1	Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . 262
		10.1.2	Main Stylized Facts . . . . . . . . . . . . . . . . . . . . . 263
	10.2	The C	lassic Income Fluctuation Problem . . . . . . . . . . . . . 269
		10.2.1	Deterministic Income . . . . . . . . . . . . . . . . . . . . 270
		10.2.2	Stochastic Income and Borrowing Limits . . . . . . . . . . 278
	10.3	Aggreg	ation: Distributions as State Variables . . . . . . . . . . . 282
		10.3.1	Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
		10.3.2	Numerical Results . . . . . . . . . . . . . . . . . . . . . . 289
11	Fisc	al Poli	cy 293
	11.1	Positiv	e Fiscal Policy . . . . . . . . . . . . . . . . . . . . . . . . . 293
	11.2	Norma	tive 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	Poli	tical E	conomy and Macroeconomics 295
13	Refe	rence	s 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. 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 connection 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. 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). 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

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CHAPTER 1. OVERVIEW AND SUMMARY

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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. One main contribution of Barro was to provide a formal justi…cation for the assumption of in…nite lives. As we will see this methodological contribution has profound consequences for the macroeconomic e¤ects of government debt, reviving the Ricardian Equivalence proposition. As a prelude we will brie‡y discuss Diamond’s (1965) analysis of government debt in an OLG model.

In the next module we will discuss the neoclassical growth model in continuous time to develop continuous time optimization techniques. After having learned the technique we will review the main developments in growth theory and see how the various growth models fare when being contrasted with the main empirical …ndings from the Summers-Heston panel data set. We will brie‡y discuss the Solow model and its empirical implications (using the article by Mankiw et al. (1992) and Romer, chapter 2), then continue with the Ramsey model (Intriligator, chapter 14 and 16, Blanchard and Fischer, chapter 2). In this model growth comes about by introducing exogenous technological progress. We will then review the main contributions of endogenous growth theory, …rst by discussing the early models based on externalities (Romer (1986), Lucas (1988)), then models that explicitly try to model technological progress (Romer (1990).

All the models discussed up to this point usually assumed that individuals are identical within each generation (or that markets are complete), so that without loss of generality we could assume a single representative consumer (within each generation). This obviously makes life easy, but abstracts from a lot of interesting questions involving distributional aspects of government policy. In the next section we will discuss a model that is capable of addressing these issues. There is a continuum of individuals. Individuals are ex-ante identical (have the same stochastic income process), but receive di¤erent income realizations ex post. These income shocks are assumed to be uninsurable (we therefore depart from the Arrow-Debreu world), but people are allowed to self-insure by borrowing and lending at a risk-free rate, subject to a borrowing limit. Deaton (1991) discusses the optimal consumption-saving decision of a single individual in this environment and Aiyagari (1994) incorporates Deaton’s analysis into a full-blown dynamic general equilibrium model. The state variable for this economy turns out to be a cross-sectional distribution of wealth across individuals. This feature makes the model interesting as distributional aspects of all kinds of government policies can be analyzed, but it also makes the state space very big. A cross-sectional distribution as state variable requires new concepts (developed in measure theory) for de…ning and new computational techniques for