Optimal Transport Methods in Economics

Alfred Galichon

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English

Princeton University Press
23 October 2018

Economic theory & philosophy; Microeconomics; Econometrics; Linear programming

Optimal Transport Methods in Economics is the first textbook on the subject written especially for students and researchers in economics. Optimal transport theory is used widely to solve problems in mathematics and some areas of the sciences, but it can also be used to understand a range of problems in applied economics, such as the matching between job seekers and jobs, the determinants of real estate prices, and the formation of matrimonial unions. This is the first text to develop clear applications of optimal transport to economic modeling, statistics, and econometrics. It covers the basic results of the theory as well as their relations to linear programming, network flow problems, convex analysis, and computational geometry. Emphasizing computational methods, it also includes programming examples that provide details on implementation. Applications include discrete choice models, models of differential demand, and quantile-based statistical estimation methods, as well as asset pricing models.

Authoritative and accessible, Optimal Transport Methods in Economics also features numerous exercises throughout that help you develop your mathematical agility, deepen your computational skills, and strengthen your economic intuition.

The first introduction to the subject written especially for economistsIncludes programming examplesFeatures numerous exercises throughoutIdeal for students and researchers alike

By: Alfred Galichon
Imprint: Princeton University Press
Country of Publication: United States
Dimensions: Height: 235mm, Width: 152mm,
ISBN: 9780691183466
ISBN 10: 0691183465
Pages: 184
Publication Date: 23 October 2018
Audience: College/higher education , A / AS level , Further / Higher Education
Format: Paperback
Publisher's Status: Active

Preface xi 1 Introduction 1 1.1 A Number of Economic Applications 1 1.2 A Mix of Techniques 3 1.3 Brief History 4 1.4 Literature 5 1.5 About These Notes 6 1.6 Organization of This Book 7 1.7 Notation and Conventions 9 2 Monge-Kantorovich Theory 11 2.1 Couplings 11 2.2 Optimal Couplings 13 2.3 Monge-Kantorovich Duality 14 2.4 Equilibrium 18 2.5 A Preview of Applications 19 2.6 Exercises 22 2.7 References and Notes 23 3 The Discrete Optimal Assignment Problem 24 3.1 Duality 25 3.2 Stability 26 3.3 Pure Assignments 27 3.4 Computation via Linear Programming 29 3.5 Exercises 32 3.6 References and Notes 33 4 One-Dimensional Case 34 4.1 Copulas and Comonotonicity 35 4.2 Supermodular Surplus 36 4.3 The Wage Equation 40 4.4 Numerical Computation 42 4.5 Exercises 43 4.6 References and Notes 44 5 Power Diagrams 45 5.1 Hotelling's Location Model 45 5.2 Capacity Constraints 48 5.3 Computation via Convex Optimization 53 5.4 Exercises 54 5.5 References and Notes 56 6 Quadratic Surplus 57 6.1 Convex Analysis from the Point of View of Optimal Transport 57 6.2 Main Results 60 6.3 Vector Quantiles 63 6.4 Polar Factorization 65 6.5 Computation by Discretization 68 6.6 Exercises 69 6.7 References and Notes 70 7 More General Surplus 72 7.1 Generalized Convexity 72 7.2 The Main Results 76 7.3 Computation by Entropic Regularization 78 7.4 Exercises 79 7.5 References and Notes 80 8 Transportation on Networks 81 8.1 Setup 82 8.2 Optimal Flow Problem 87 8.3 Integrality 90 8.4 Computation via Linear Programming 91 8.5 Exercises 93 8.6 References and Notes 94 9 Some Applications 95 9.1 Random Sets and Partial Identification 95 9.2 Identification of Discrete Choice Models 98 9.3 Hedonic Equilibrium 101 9.4 Identification via Vector Quantile Methods 104 9.5 Vector Quantile Regression 106 9.6 Implementable Mechanisms 110 9.7 No-Arbitrage Pricing of Financial Derivatives 115 9.8 References and Notes 117 10 Conclusion 118 10.1 Mathematics 118 10.2 Computation 119 10.3 Duality 120 10.4 Toward a Theory of Equilibrium Transport 122 10.5 References and Notes 123 A Solutions to the Exercises 125 A.1 Solutions for Chapter 2 125 A.2 Solutions for Chapter 3 128 A.3 Solutions for Chapter 4 130 A.4 Solutions for Chapter 5 133 A.5 Solutions for Chapter 6 137 A.6 Solutions for Chapter 7 139 A.7 Solutions for Chapter 8 141 B Linear Programming 144 B.1 Minimax Theorem 144 B.2 Duality 144 B.3 Link with Zero-Sum Games 146 B.4 References and Notes 148 C Quantiles and Copulas 149 C.1 Quantiles 149 C.2 Copulas 151 C.3 References and Notes 153 D Basics of Convex Analysis 154 D.1 Convex Sets 154 D.2 Convex Functions 155 D.3 References and Notes 159 E McFadden's Generalized Extreme Value Theory 160 E.1 References and Notes 160 References 161 Index 169

Alfred Galichon holds joint appointments in the Department of Economics and the Courant Institute of Mathematical Sciences at New York University. He is on the editorial boards of the Review of Economic Studies and Economic Theory and is a research fellow at the Center for Economic and Policy Research (CEPR) and the Institute for the Study of Labor (IZA).

Reviews for Optimal Transport Methods in Economics

As the author develops it, the theory is almost always very clear. The approach that Galichon has developed, based on linear programming, is receiving increased attention from economists and mathematicians in spite of its obvious complexity. . . . [A] clear and succinct analysis and discussion. --Choice Very timely. Until now, there was no focused yet coherent introduction to the subject available to economic theorists. --Rakesh Vohra, University of Pennsylvania Economic applications of optimal transport problems have recently flourished. This book, written by one of the best specialists of the field, provides a detailed presentation of the mathematical tools involved, from the abstract foundations of the theory to the practical implementation of optimizing algorithms, with applications ranging from the analysis of the labor market to the resolution of difficult problems in econometrics. For any student or researcher interested in the topic, this is a must-read. --Pierre-Andr Chiappori, Columbia University Since its inception, the mathematical theory of optimal transportation has served as a secret weapon in the hands of a few scholars for tackling outstanding challenges in economics, ranging from matching and price theory to econometrics and finance. With the publication of Galichon's engaging and accessible text, the secret is revealed to a new generation of students and researchers; I expect much progress to result. --Robert J. McCann, University of Toronto I am aware of no other book on the subject written especially for economists. --Philipp Kircher, University of Edinburgh This book is a valuable contribution to a rapidly expanding literature, written by a leader in the field. --James J. Heckman, University of Chicago, winner of the Nobel Prize in economics