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An Introduction to Macroeconomics with Household Heterogeneity

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An Introduction to Macroeconomics with Household Heterogeneity An Introduction to Macroeconomics with Household Heterogeneity Dirk Krueger1 Department of Economics University of Pennsylvania March 6, 2018 1I wish to thank Orazio Attanasio, Richard Blundell, Juan Carlos Conesa, Jesus Fernandez-Villaverde, Robert Hall, Tim Kehoe, Narayana Kocherlakota, Felix Kubler, Fabrizio Perri, Luigi Pistaferri, Ed Prescott, Victor Rios-Rull and Thomas Sargent for teaching me much of the material presented in this manuscript. c by Dirk Krueger. All comments are welcomed, please contact the author at [email protected]. ii Contents I Introduction 1 1 Overview over the Monograph 3 2 Why Macro with Heterogeneous Households (or Firms)? 7 3 Some Stylized Facts and Some Puzzles 9 3.1 Household Level Data Sources . 9 3.1.1 CEX . 9 3.1.2 SCF . 10 3.1.3 PSID . 10 3.1.4 CPS . 11 3.1.5 Data Sources for Other Countries . 11 3.2 Main Findings . 12 3.2.1 Organization of Facts . 12 3.2.2 Aggregate Time Series: Means over Time . 12 3.2.3 Means over the Life Cycle . 16 3.2.4 Dispersion over the Life Cycle . 18 3.2.5 Dispersion (Inequality) at a Point in Time . 19 3.2.6 Changes in Dispersion over Time . 19 3.2.7 Other Interesting Facts (or Puzzles) . 19 4 The Standard Complete Markets Model 23 4.1 Theoretical Results . 25 4.1.1 Arrow-Debreu Equilibrium . 25 4.1.2 Sequential Markets Equilibrium . 34 4.2 Empirical Implications for Asset Pricing . 39 4.2.1 An Example . 48 4.3 Tests of Complete Consumption Insurance . 50 iii iv CONTENTS 4.3.1 Derivation of Mace's (1991) Empirical Specifications . 51 4.3.2 Results of the Tests . 54 4.3.3 The Problem of Measurement Error . 57 4.3.4 Other Empirical Issues and Solutions . 59 4.3.5 Self-Selection into Occupations with Differential Ag- gregate Risk . 59 4.4 Appendix A: Discussion of the No Ponzi Condition . 67 4.5 Appendix B: Proofs . 69 4.6 Appendix C: Properties of CRRA Utility . 71 4.6.1 Constant Relative Risk Aversion . 72 4.6.2 Constant Intertemporal Elasticity of Substitution . 73 4.6.3 Homotheticity and Balanced Growth . 75 II The Standard Incomplete Markets Model (SIM) 77 5 The SIM in Partial Equilibrium 81 5.1 A Simple 2 Period Toy Model . 82 5.1.1 Household Decision Problem . 82 5.1.2 Comparative Statics . 83 5.2 The General Model with Certainty Equivalence . 85 5.2.1 Nonstochastic Income . 86 5.2.2 Stochastic Income and Quadratic Preferences . 90 5.3 Prudence . 101 5.3.1 A Simple Model and a General Result . 103 5.3.2 A Parametric Example for the General Model . 114 5.4 Liquidity Constraints . 118 5.4.1 The Euler Equation with Liquidity Constraints . 119 5.4.2 Precautionary Saving Due to Liquidity Constraints . 121 5.4.3 Empirical Tests of Liquidity Constraints . 122 5.5 Prudence and Liquidity Constraints: Theory . 123 5.5.1 Finite Lifetime . 124 5.5.2 Infinite Horizon . 125 5.6 Prudence and Liquidity Constraints: Computation . 139 5.6.1 IID Income shocks . 141 5.6.2 Serially Correlated, Mean-Reverting Income Shocks . 141 5.6.3 Permanent Shocks . 143 5.6.4 From Policy Functions to Simulated Time Series . 148 CONTENTS v 5.6.5 Implementation of Specific Algorithms . 149 5.7 Prudence and Liquidity Constraints: Empirical Implications . 150 5.7.1 Time Series Implications: How Does Consumption Re- spond to Income Shocks . 150 5.7.2 Cross-Sectional Implications: How Does Consumption Inequality Evolve over the Life Cycle . 154 5.8 Appendix A: The Intertemporal Budget Constraint . 154 5.9 Appendix B: Derivation of Consumption Response to Income Shocks . 156 6 A Digression: Stochastic Earnings or Wage Processes 159 6.1 Estimation . 162 6.2 Results . 163 6.3 Interpretation . 165 6.3.1 RIP vs. HIP . 165 6.3.2 Size of ρ ..........................167 7 The SIM in General Equilibrium 169 7.1 A Model Without Aggregate Uncertainty . 170 7.1.1 The Environment . 170 7.1.2 Theoretical Results: Existence and Uniqueness . 178 7.1.3 Computation of the General Equilibrium . 183 7.1.4 Qualitative Results . 184 7.1.5 Numerical Results . 185 7.2 An Incomplete Markets Model with Unsecured Debt and Equi- librium Default . 189 7.2.1 Model Overview . 189 7.2.2 Institutional Details of Bankruptcy . 189 7.2.3 Household Problem in Recursive Formulation . 190 7.2.4 Production Firms . 192 7.2.5 Financial Intermediaries . 192 7.2.6 Stationary Recursive Competitive Equilibrium . 192 7.2.7 Characterization of Household Default Decision . 194 7.2.8 Characterization of Equilibrium Loan Interest Rate Func- tion . 194 7.2.9 Bringing the Model to the Data: Calibration and Es- timation . 196 7.2.10 Quantitative Predictions of the Model . 196 vi CONTENTS 7.3 Unexpected Aggregate Shocks and Transition Dynamics . 196 7.3.1 Definition of Equilibrium . 197 7.3.2 Computation of the Transition Path . 199 7.3.3 Welfare Consequences of the Policy Reform . 201 7.4 Aggregate Uncertainty and Distributions as State Variables . 203 7.4.1 The Model . 204 7.4.2 Computation of the Recursive Equilibrium . 209 7.4.3 Calibration . 211 7.4.4 Numerical Results . 214 7.4.5 Why Quasi-Aggregation? . 217 7.4.6 Rich People are Not Rich Enough . 218 III Complete Market Models with Frictions 221 8 Limited Enforceability of Contracts 225 8.1 The Model . 226 8.2 Constrained Efficient Allocations . 228 8.3 Recursive Formulation of the Problem . 229 8.4 Decentralization . 230 8.5 An Application: Income and Consumption Inequality . 232 8.6 A Continuum Economy . 237 9 Private Information 245 9.1 Partial Equilibrium . 245 9.1.1 Properties of the Recursive Problem . 249 9.2 Endogenous Interest Rates in General Equilibrium . 253 9.3 Applications . 253 9.3.1 New Dynamic Public Finance . 253 IV Conclusions 255 Part I Introduction 1 Chapter 1 Overview over the Monograph This monograph is meant to be an introduction into the research field of macroeconomics with household heterogeneity. It is concerned with con- structing, computing and empirically evaluating dynamic stochastic macroe- conomic models in which individual households differ according their abili- ties, wages, incomes, preferences or market opportunities in a way such that the household sector cannot be aggregated into a representative agent. In these models, therefore, aggregate allocations and prices will depend on the cross-sectional household distribution of household characteristics. We will argue below that the macroeconomic implications of these models differ, often (but not always) in a quantitatively significant way, from their representative agent counterpart. In addition, these models are in principle suitable to ask and answer positive and normative distributional questions about which the representative household paradigm is silent by construction. In the next chapter we will briefly summarize selected empirical obser- vations on wages, earnings, income, consumption and wealth from cross- sectional, household level data sets that have motivated this literature, and briefly describe the data sources from which these observations are drawn. We will also document some empirical puzzles that we aim at explaining with the models to be developed below. Note that to call an empirical finding a puzzle requires to take a stand on the standard economic theory relative to which the finding is puzzling; on of the goals of the course is to develop extensions or, if needed, radical departures, of standard theory to explain the empirical puzzles. The remainder of the monograph is then devoted to the construction, analysis and applications of theoretical models aimed at explaining these facts that also can be used for applications (e.g. to the 3 4 CHAPTER 1. OVERVIEW OVER THE MONOGRAPH positive or normative analysis of fiscal policy). In order to provide a taxonomy of the models discussed in this mono- graph, consider a world in which an infinitely-lived households is faced with income risk and aims at maximizing expected lifetime utility. Let s 2 S de- note a state of nature, where S is a finite set. For ease of exposition1 let the shock s be iid over time, with probabilities π(s): Household income is given by y(s): Households desire smooth consumption in the presence of stochastic income fluctuations, and the key distinguishing feature of the models dis- cussed in this monograph is the set of financial assets that households have access to, and the extent to which they can go short in these assets. On one extreme, households have no access to any formal or informal ways to smooth income fluctuations, and thus their consumption fully inherits the stochastic properties of the income process. I will call these households hand-to-mouth consumers, and the associated (non-) market structure financial autarchy. On the other extreme households may be able to trade a full set of state- contingent short-term bonds, so that their budget constraint reads as X c + q(s0)a0(s0) ≤ y(s) + a(s) (1.1) s02S where c is current consumption, y(s) is current income, a0(s0) is the number of bonds bought today that pay of one unit of consumption in state s0 tomor- row, and q(s0) is its price. If households face no other trading constraints2 I will call the resulting model the standard complete markets model. The next chapter is devoted to the review of its theoretical predictions and em- pirical tests. With this market structure (and under suitable assumptions on household preferences) household consumption is fully insured against household-specific (idiosyncratic) income shocks and the household sector can be represented by a representative consumer.
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