Macroeconomic Theory

308 Pages·2012·1.726 MB·English

by  Dirk Krueger

Macroeconomic Theory
Dirk Krueger1
Department of Economics
University of Pennsylvania
January 26, 2012
1I am grateful to my teachers in Minnesota, V.V 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.
ii
Contents
1 Overview and Summary 1
2 A Simple Dynamic Economy 5
2.1 General Principles for Specifying a Model . . . . . . . . . . . . . 5
2.2 An Example Economy . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 De(cid:133)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(cid:146)s (1960) Method to Compute Equilibria . . . . . . 14
2.2.5 Sequential Markets Equilibrium . . . . . . . . . . . . . . . 18
2.3 Appendix: 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 Neoclassical Growth Model in Discrete Time 31
3.1 Setup of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Optimal 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(cid:133)ed Golden Rule . . . . . . . 52
3.2.6 A Remark About Balanced Growth . . . . . . . . . . . . 53
3.3 Competitive Equilibrium Growth . . . . . . . . . . . . . . . . . . 55
3.3.1 De(cid:133)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 Mapping the Model to Data: Calibration . . . . . . . . . . . . . 67
iii
iv CONTENTS
4 Mathematical Preliminaries 71
4.1 Complete Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . 72
4.2 Convergence of Sequences . . . . . . . . . . . . . . . . . . . . . . 73
4.3 The Contraction Mapping Theorem . . . . . . . . . . . . . . . . 77
4.4 The Theorem of the Maximum . . . . . . . . . . . . . . . . . . . 83
5 Dynamic Programming 85
5.1 The Principle of Optimality . . . . . . . . . . . . . . . . . . . . . 85
5.2 Dynamic Programming with Bounded Returns . . . . . . . . . . 92
6 Models with Risk 95
6.1 Basic Representation of Risk . . . . . . . . . . . . . . . . . . . . 95
6.2 De(cid:133)nitions 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 Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.4 Stochastic 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(cid:133)nition of Competitive Equilibrium . . . . . . . . . . . . . . . 114
7.4 The Neoclassical Growth Model in Arrow-Debreu Language . . . 115
7.5 A Pure Exchange Economy in Arrow-Debreu Language . . . . . 117
7.6 The First Welfare Theorem . . . . . . . . . . . . . . . . . . . . . 119
7.7 The Second Welfare Theorem . . . . . . . . . . . . . . . . . . . . 120
7.8 Type Identical Allocations . . . . . . . . . . . . . . . . . . . . . . 128
8 The Overlapping Generations Model 129
8.1 A Simple 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 Ricardian Equivalence Hypothesis . . . . . . . . . . . . . . . 160
8.2.1 In(cid:133)nite Lifetime Horizon and Borrowing Constraints . . . 161
8.2.2 Finite Horizon and Operative Bequest Motives . . . . . . 170
8.3 Overlapping Generations Models with Production. . . . . . . . . 175
8.3.1 Basic Setup of the Model . . . . . . . . . . . . . . . . . . 175
8.3.2 Competitive Equilibrium . . . . . . . . . . . . . . . . . . 176
CONTENTS v
8.3.3 Optimality of Allocations . . . . . . . . . . . . . . . . . . 183
8.3.4 The Long-Run E⁄ects of Government Debt . . . . . . . . 187
9 Continuous Time Growth Theory 193
9.1 Stylized Growth and Development Facts . . . . . . . . . . . . . . 193
9.1.1 Kaldor(cid:146)s Growth Facts . . . . . . . . . . . . . . . . . . . . 194
9.1.2 Development Facts from the Summers-Heston Data Set . 194
9.2 The Solow 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 Ramsey-Cass-Koopmans Model . . . . . . . . . . . . . . . . 215
9.3.1 MathematicalPreliminaries: Pontryagin(cid:146)sMaximumPrin-
ciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
9.3.2 Setup of the Model . . . . . . . . . . . . . . . . . . . . . . 215
9.3.3 Social Planners Problem . . . . . . . . . . . . . . . . . . . 217
9.3.4 Decentralization . . . . . . . . . . . . . . . . . . . . . . . 226
9.4 Endogenous 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 Bewley Models 261
10.1 Some Stylized Facts about the Income and Wealth Distribution
in the U.S.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
10.1.1 Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . 262
10.1.2 Main Stylized Facts . . . . . . . . . . . . . . . . . . . . . 263
10.2 The Classic Income Fluctuation Problem . . . . . . . . . . . . . 269
10.2.1 Deterministic Income . . . . . . . . . . . . . . . . . . . . 270
10.2.2 Stochastic Income and Borrowing Limits. . . . . . . . . . 278
10.3 Aggregation: Distributions as State Variables . . . . . . . . . . . 282
10.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
10.3.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . 289
11 Fiscal Policy 293
11.1 Positive Fiscal Policy . . . . . . . . . . . . . . . . . . . . . . . . . 293
11.2 Normative Fiscal Policy . . . . . . . . . . . . . . . . . . . . . . . 293
11.2.1 Optimal Policy with Commitment . . . . . . . . . . . . . 293
11.2.2 TheTimeConsistencyProblemandOptimalFiscalPolicy
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 (cid:133)rst part
ofthecoursewewilldiscussthetwoworkhorsesofmodernmacroeconomics,the
neoclassical growth model with in(cid:133)nitely lived consumers and the Overlapping
Generations (OLG) model. This (cid:133)rst part will focus on techniques rather than
issues; one (cid:133)rst has to learn a language before composing poems.
I will (cid:133)rst present a simple dynamic pure exchange economy with two in-
(cid:133)nitely lived consumers engaging in intertemporal trade. In this model the
connectionbetweencompetitive equilibriaandParetooptimalequilibriacanbe
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 (cid:133)rst by Negishi (1960) and discussed nicely by
Kehoe (1989).
This model with then enriched by production (and simpli(cid:133)ed by dropping
one of the two agents), to give rise to the neoclassical growth model. This
model will (cid:133)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 (cid:133)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(cid:133)nite-dimensional. We will draw on Stokey et al., chapter 15(cid:146)s discussion of
Debreu (1954).
Thenexttwotopicsarelogicalextensionsoftheprecedingmaterial. Wewill
(cid:133)rst discuss the OLG model, due to Samuelson (1958) and Diamond (1965).
The(cid:133)rstmainfocusinthismodulewillbethetheoreticalresultsthatdistinguish
theOLGmodelfromthestandardArrow-Debreumodelofgeneralequilibrium:
in the OLG model equilibria may not be Pareto optimal, (cid:133)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(cid:133)nitely lived agents in the standard
neoclassical growth model. Barro (1974) demonstrated under which conditions
(operativebequestmotives)anOLGeconomywillbeequivalenttoaneconomy
with in(cid:133)nitely lived consumers. One main contribution of Barro was to provide
a formal justi(cid:133)cation for the assumption of in(cid:133)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(cid:135)y discuss Diamond(cid:146)s (1965) analysis of government debt
in an OLG model.
In the next module we will discuss the neoclassical growth model in con-
tinuous time to develop continuous time optimization techniques. After having
learned the technique we will review the main developments in growth the-
ory and see how the various growth models fare when being contrasted with
the main empirical (cid:133)ndings from the Summers-Heston panel data set. We will
brie(cid:135)y discuss the Solow model and its empirical implications (using the arti-
cle 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. Wewillthenreviewthemaincontributionsofendogenousgrowththe-
ory, (cid:133)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
lotofinterestingquestionsinvolvingdistributionalaspectsofgovernmentpolicy.
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 realiza-
tionsexpost. Theseincomeshocksareassumedtobeuninsurable(wetherefore
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(cid:146)s analysis into a
full-blowndynamicgeneralequilibriummodel. Thestatevariableforthisecon-
omy 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 (de-
veloped in measure theory) for de(cid:133)ning and new computational techniques for
3
computing equilibria. The early papers therefore restricted attention to steady
state equilibria (in which the cross-sectional wealth distribution remained con-
stant). Very recently techniques have been developed to handle economies with
distributions as state variables that feature aggregate shocks, so that the cross-
sectional wealth distribution itself varies over time. Krusell and Smith (1998)
is the key reference. Applications of their techniques to interesting policy ques-
tions could be very rewarding in the future. If time permits I will discuss such
an application due to Heathcote (1999).
For the next two topics we will likely not have time; and thus the corre-
sponding lecture notes are work in progress. So far we have not considered
how government policies a⁄ect equilibrium allocations and prices. In the next
modules this question is taken up. First we discuss (cid:133)scal policy and we start
with positive questions: how does the governments(cid:146)decision to (cid:133)nance a given
streamofexpenditures(debtvs. taxes)a⁄ectmacroeconomicaggregates(Barro
(1974),Ohanian(1997))?;howdoesgovernmentspendinga⁄ectoutput(Baxter
andKing(1993))? Inthisdiscussiongovernmentpolicyistakenasexogenously
given. The next question is of normative nature: how should a benevolent gov-
ernment carry out (cid:133)scal policy? The answer to this question depends crucially
on the assumption of whether the government can commit to its policy. A gov-
ernmentthatcancommittoitsfuturepoliciessolvesaclassicalRamseyproblem
(not to be confused with the Ramsey model); the main results on optimal (cid:133)scal
policy are reviewed in Chari and Kehoe (1999). Kydland and Prescott (1977)
pointed out the dilemma a government faces if it cannot commit to its policy
-this is the famous time consistency problem. How a benevolent government
that cannot commit should carry out (cid:133)scal policy is still very much an open
question. Klein and Rios-Rull (1999) have made substantial progress in an-
swering this question. Note that we throughout our discussion assume that the
governmentactsinthebestinterestofitscitizens. Whathappensifpoliciesare
instead chosen by votes of sel(cid:133)sh individuals is discussed in the last part of the
course.
As discussed before we assumed so far that government policies were either
(cid:133)xedexogenouslyorsetbyabenevolentgovernment(thatcanorcan(cid:146)tcommit).
Nowwerelaxthisassumptionanddiscusspolitical-economicequilibriainwhich
peoplenotonlyactrationallywithrespecttotheireconomicdecisions, butalso
rationally with respect to their voting decisions that determine macroeconomic
policy. Obviouslywe(cid:133)rsthadtodiscussmodelswithheterogeneousagentssince
with homogeneous agents there is no political con(cid:135)ict and hence no interesting
di⁄erences between the Ramsey problem and a political-economic equilibrium.
This area of research is not very far developed and we will only present two
examples (Krusell et al. (1997), Alesina and Rodrik (1994)) that deal with the
question of capital taxation in a dynamic general equilibrium model in which
the capital tax rate is decided upon by repeated voting.
4 CHAPTER 1. OVERVIEW AND SUMMARY