高级宏观经济学Krueger最好的高宏学习笔记

Macroeconomic Theory

Dirk Krueger1

Department of Economics

University of Pennsylvania

January1,2015

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 and teach modern macroeconomic theory.These notes were tried out on numerous students at Stanford,UPenn,Frankfurt and Mannheim,whose many useful comments I appreciate.Alexander Bick,Kaiji Chen,Antonio Doblas-Madrid and Weilong Zhang provided many important corrections to these notes.

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Contents

1Overview and Summary1 2A Simple Dynamic Economy5

2.1General Principles for Specifying a Model (5)

2.2An Example Economy (6)

2.2.1De...nition of Competitive Equilibrium.. (8)

2.2.2Solving for the Equilibrium (9)

2.2.3Pareto Optimality and the First Welfare Theorem (11)

2.2.4Negishi’s(1960)Method to Compute Equilibria (14)

2.2.5Sequential Markets Equilibrium (19)

2.3Appendix:Some Facts about Utility Functions (26)

2.3.1Time Separability (26)

2.3.2Time Discounting (27)

2.3.3Standard Properties of the Period Utility Function (27)

2.3.4Constant Relative Risk Aversion(CRRA)Utility (27)

2.3.5Homotheticity and Balanced Growth (30)

3The Neoclassical Growth Model in Discrete Time33

3.1Setup of the Model (33)

3.2Optimal Growth:Pareto Optimal Allocations (34)

3.2.1Social Planner Problem in Sequential Formulation (35)

3.2.2Recursive Formulation of Social Planner Problem (37)

3.2.3An Example (39)

3.2.4The Euler Equation Approach and Transversality Condi-

tions (46)

3.2.5Steady States and the Modi...ed Golden Rule. (55)

3.2.6A Remark About Balanced Growth (56)

3.3Competitive Equilibrium Growth (58)

3.3.1De...nition of Competitive Equilibrium.. (59)

3.3.2Characterization of the Competitive Equilibrium and the

Welfare Theorems (61)

3.3.3Sequential Markets Equilibrium (67)

3.3.4Recursive Competitive Equilibrium (68)

3.4Mapping the Model to Data:Calibration (70)

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iv CONTENTS 4Mathematical Preliminaries75

4.1Complete Metric Spaces (76)

4.2Convergence of Sequences (77)

4.3The Contraction Mapping Theorem (81)

4.4The Theorem of the Maximum (87)

5Dynamic Programming89

5.1The Principle of Optimality (89)

5.2Dynamic Programming with Bounded Returns (96)

6Models with Risk99

6.1Basic Representation of Risk (99)

6.2De...nitions of Equilibrium. (101)

6.2.1Arrow-Debreu Market Structure (102)

6.2.2Pareto E￠ciency (104)

6.2.3Sequential Markets Market Structure (105)

6.2.4Equivalence between Market Structures (106)

6.2.5Asset Pricing (106)

6.3Markov Processes (108)

6.4Stochastic Neoclassical Growth Model (110)

6.4.1Social Planner Problem in Recursive Formulation (111)

6.4.2Recursive Competitive Equilibrium (112)

7The Two Welfare Theorems115

7.1What is an Economy? (115)

7.2Dual Spaces (118)

7.3De...nition of Competitive Equilibrium (120)

7.4The Neoclassical Growth Model in Arrow-Debreu Language (121)

7.5A Pure Exchange Economy in Arrow-Debreu Language (123)

7.6The First Welfare Theorem (125)

7.7The Second Welfare Theorem (126)

7.8Type Identical Allocations (134)

8The Overlapping Generations Model135

8.1A Simple Pure Exchange Overlapping Generations Model (136)

8.1.1Basic Setup of the Model (137)

8.1.2Analysis of the Model Using O¤er Curves (142)

8.1.3Ine￠cient Equilibria (149)

8.1.4Positive Valuation of Outside Money (154)

8.1.5Productive Outside Assets (156)

8.1.6Endogenous Cycles (158)

8.1.7Social Security and Population Growth (160)

8.2The Ricardian Equivalence Hypothesis (165)

8.2.1In...nite Lifetime Horizon and Borrowing Constraints (166)

8.2.2Finite Horizon and Operative Bequest Motives (175)

8.3Overlapping Generations Models with Production (180)

CONTENTS v

8.3.1Basic Setup of the Model (181)

8.3.2Competitive Equilibrium (181)

8.3.3Optimality of Allocations (188)

8.3.4The Long-Run E¤ects of Government Debt (192)

9Continuous Time Growth Theory197

9.1Stylized Growth and Development Facts (197)

9.1.1Kaldor’s Growth Facts (198)

9.1.2Development Facts from the Summers-Heston Data Set.198

9.2The Solow Model and its Empirical Evaluation (203)

9.2.1The Model and its Implications (206)

9.2.2Empirical Evaluation of the Model (208)

9.3The Ramsey-Cass-Koopmans Model (218)

9.3.1Mathematical Preliminaries:Pontryagin’s Maximum Prin-

ciple (219)

9.3.2Setup of the Model (219)

9.3.3Social Planners Problem (221)

9.3.4Decentralization (229)

9.4Endogenous Growth Models (234)

9.4.1The Basic AK-Model (235)

9.4.2Models with Externalities (239)

9.4.3Models of Technological Progress Based on Monopolistic

Competition:Variant of Romer(1990) (251)

10References263

vi CONTENTS

Chapter1

Overview and Summary Broadly,these notes will…rst provide a quick warm-up for dynamic general equilibrium models before we will discuss the two workhorses of modern macro-economics,the neoclassical growth model with in…nitely lived consumers and the Overlapping Generations(OLG)model.The focus of these notes is as much on developing a coherent language for formulating dynamic macroeconomic model and on the techniques to analyze them,as is it on the application of these models for basic questions in economic growth and business cycle research.

In chapter2I 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).Furthermore I will show the equivalence of equilibria in a market arrangement in which households trade dated consumption goods at the beginning of time,and equilibria under a market structure where trade takes place sequentially and in every period households exchange the current period consumption good and a simple…nancial asset.

This baseline model with then enriched,in chapter3,by production(and simpli…ed by dropping one of the two agents),to give rise to the neoclassi-cal growth model.This model will…rst be presented in discrete time to discuss discrete-time dynamic programming techniques;both theoretical as well as com-putational in nature.The main reference will be Stokey et al.,chapters2-4.On the substantive side,I will argue how this model can be mapped into the data and what it implies for economic growth in the short and in the long run.On the methodological side I will give the general mathematical treatment of dis-crete time dynamic programming in chapters??(where I discuss the required mathematical concepts)and chapter5(where the main general results in the theory of dynamic programming will be summarized).

In chapter6I will introduce models with risk.After setting up the appropri-ate notation I will…rst discuss a pure exchange version of the stochastic model

1

2CHAPTER1.OVERVIEW AND SUMMARY and show how this model(essentially a version of Lucas’(1978)asset pricing model)can be used to develop a simple theory of asset pricing.I will then turn to a stochastic version of the model with production in order to develop the Real Business Cycle(RBC)theory of business cycles.Cooley and Prescott(1995) are a good reference for this application.

Chapter7presents a general discussion of the two welfare theorems in economies with in…nite-dimensional commodity spaces,as is typical for macro-economic applications in which the economy extends forever and thus there are an(countably)in…nite number of dated consumption goods being traded.We will heavily draw on Stokey et al.,chapter15’s discussion of Debreu(1954)for this purpose.

The next two topics are logical extensions of the preceding material.In chapter8we 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 gen-eral equilibrium:in the OLG model equilibria may not be Pareto optimal,…at money may have 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 discus-sion 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…nal module of these notes,chapter9,we will discuss the neoclassi-cal growth model in continuous time to develop continuous time optimization techniques.After having learned the technique we will review the main devel-opments 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 impli-cations(using the article by Mankiw et al.(1992)and Romer,chapter2),then continue with the Ramsey model(Intriligator,chapter14and16,Blanchard and Fischer,chapter2).In this model growth comes about by introducing ex-ogenous technological progress.We will then review the main contributions of endogenous growth theory,…rst by discussing the early models based on exter-nalities(Romer(1986),Lucas(1988)),then models that explicitly try to model technological progress(Romer(1990).1

1Previous versions of these notes contained a chapter on models with heterogeneous house-holds.This material has been merged into my manuscript An Introduction to Macroeconomics

3 with Household Heterogeneity and signi…cantly expanded there.