Matching with Transfers

Pierre-Andre Chiappori is the E. Rowan and Barbara Steinschneider Professor of Economics at Columbia University and a Distinguished Fellow at the Becker Friedman Institute for Research in Economics at the University of Chicago. He is the author or coauthor of several books, including Economics of the Family.

• Preface xi Acknowledgments xv 1 Introduction: Matching Models in Economics 1 1.1 Motivation: Two Puzzles 1 1.1.1 Inequality 1 1.1.2 Demand for Higher Education 2 1.2 Matching Models: Main Features 4 1.2.1 Absence of Frictions 4 1.2.2 Are Transfers Relevant? 5 1.3 Matching and the Household 9 1.3.1 Household Behavior: Existing Models 9 1.3.2 Bargaining Models of the Household 12 1.4 Content 17 2 Matching with Transfers: Basic Notions 19 2.1 Bilateral, One-to-One Matching: Common Framework 19 2.2 The Three Types of Models 20 2.2.1 Defining the Problem 20 2.2.2 Defining the Solution 23 3 Matching under Transferable Utility: Theory 27 3.1 Definition and First Properties 27 3.1.1 Formal Definition 27 3.1.2 TU as an Ordinal Property 27 3.1.3 Specific Assumptions on Preferences 29 3.1.4 The Sources of Heterogeneity 33 3.1.5 A TU Model Is Unitary 34 3.1.6 An Example 35 3.1.7 Extensions 38 3.1.8 Testable Implications 39 3.1.9 A Bridge between NTU and TU: From Gale-Shapley to Kelso-Crawford-Knoer and Beyond 40 3.2 Optimal Transportation 43 3.2.1 Basic Duality Result 43 3.2.2 An Intuitive Illustration 46 3.2.3 Implications of the Basic Result 47 3.3 Supermodularity and Assortativeness 49 3.3.1 Supermodularity 49 3.3.2 Assortativeness 50 3.3.3 Two Simple Examples 53 3.3.4 Who Are the Singles? 55 3.3.5 The Twist Condition 58 3.4 Individual Utilities and Intrahousehold Allocation 60 3.4.1 Recovering Individual Utilities 60 3.4.2 Particular Case: Matching on Income 63 3.4.3 Exogenous versus Endogenous Sharing Rules: A Simple Example 69 3.5 Link with Hedonic Models 73 3.5.1 Hedonic Models 73 3.5.2 Hedonic Equilibrium and Stable Matching 75 3.5.3 Example 1: A Competitive IO Model 77 3.5.4 Example 2: Randomized Matching 81 4 Matching by Categories 87 4.1 Accounting for Unobservable Heterogeneity 87 4.1.1 The Separability Assumption 88 4.1.2 How Can Separability Be Justified? 89 4.1.3 Dual Structure under Separability 91 4.1.4 A Comparative Static Result 93 4.2 The Choo-Siow Model 96 4.2.1 The Basic Structure 96 4.2.2 The Matching Function 98 4.2.3 Heteroskedasticity: A Short Discussion 99 4.2.4 Covariates 103 4.2.5 Comparative Statics in the Choo-Siow Model 104 4.2.6 Testability and Identifiability of the Choo-Siow Model 108 4.2.7 Extension: Galichon and Salanie's Cupid Framework 112 5 Matching under Transferable Utility: Some Extensions 115 5.1 Preinvestment 115 5.1.1 The Issue 115 5.1.2 A Simple Example 116 5.1.3 What Was Wrong with the Previous Arguments? 117 5.1.4 The Main Result 119 5.1.5 Coordination Failures and Inefficient Equilibria 120 5.2 Risk Sharing 121 5.2.1 When Is TU Relevant? A Simple Example 121 5.2.2 When Is TU Relevant? A General Result 123 5.2.3 An Integrated Example 125 5.3 Multidimensional Matching 129 5.3.1 Index Models 130 5.3.2 The General Case: Equal Dimensions 133 5.3.3 The General Case: Many-to-One Matching 138 5.4 Roommate Matching 140 5.4.1 Existence of a Stable Matching: A Counterexample 140 5.4.2 The Cloned Bipartite Problem 141 5.5 Divorce and Remarriage 143 5.5.1 The Basic Model 143 5.5.2 Extensions 150 6 Matching under Transferable Utility: Applications 154 6.1 Roe v.Wade and Female Empowerment 154 6.1.1 The Model 156 6.1.2 Stable Matching on the Marriage Market 157 6.1.3 Changes in Birth Control Technology 167 6.1.4 Extensions 172 6.2 Gender and the Demand for Higher Education 177 6.2.1 The CIW Model 179 6.2.2 Equilibrium 184 6.2.3 Preferences for Singlehood 189 6.2.4 Comparative Statics 189 6.2.5 Empirical Implementation: What Do Data Say? 192 6.2.6 The Low Model 197 7 Matching under Imperfectly Transferable Utility 199 7.1 Basic Notions 199 7.1.1 Theoretical Framework 199 7.1.2 Recovering Individual Utilities 200 7.1.3 Positive Assortative Matching 202 7.1.4 Econometrics 204 7.2 Examples 205 7.2.1 Matching onWages 206 7.2.2 Endogenous ParetoWeights 210 8 Conclusion 219 References 227 Index 235

"With Matching with Transfers, Chiaporri provides a valuable contribution and summary of the matching and family economics literature."---Scott Alan Carson, Journal of Economic and Social Thought