Matching Function Equilibria : Existence, Uniqueness and Estimation

In this paper, we argue that models coming from a variety of fields share a common structure that we call matching function equilibria. This structure revolves around an aggregate matching function and a system of nonlinear equations. This encompasses search and matching models, matching models with transferable, non-transferable and imperfectly transferable utility models, BLP models, trade gravity models, and matching with peer effects. We provide a proof of existence and uniqueness of an equilibrium as well as an efficient algorithm to compute it. We show how one can estimate parameters in matching functions of these models by maximum likelihood and obtain closed-form formulas to compute their confidence interval for models with constant return to scale. We also propose an approach to construct counterfactuals without estimating their matching functions for a subclass of these models. We illustrate our estimation approach by analyzing the impact of the elimination of the Social Security Student Benefit Program in 1982 on the marriage market in the United States