Matching in Closed-Form: Equilibrium, Identification, and Comparative Statics

TL;DR

This paper derives closed-form formulas for a multidimensional two-sided matching model with transferable utility, enabling efficient computation and analysis of equilibrium, identification, and comparative statics in complex matching markets.

Contribution

It provides explicit closed-form solutions for the matching distribution and surplus function in models with quadratic surplus and continuous logit heterogeneity, simplifying analysis and computation.

Findings

Optimal matching distribution is jointly normal under specified conditions.

Quadratic surplus function can be identified from the matching distribution.

Formulas facilitate analysis of large matching markets and their evolution.

Abstract

This paper provides closed-form formulas for a multidimensional two-sided matching problem with transferable utility and heterogeneity in tastes. When the matching surplus is quadratic, the marginal distributions of the characteristics are normal, and when the heterogeneity in tastes is of the continuous logit type, as in Choo and Siow (J Polit Econ 114:172-201, 2006), we show that the optimal matching distribution is also jointly normal and can be computed in closed form from the model primitives. Conversely, the quadratic surplus function can be identified from the optimal matching distribution, also in closed-form. The closed-form formulas make it computationally easy to solve problems with even a very large number of matches and allow for quantitative predictions about the evolution of the solution as the technology and the characteristics of the matching populations change.