  In this paper, we argue that models coming from a variety of fields share a
common structure that we call matching function equilibria with partial
assignment. 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, 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 to estimate parametric versions of these models by
maximum likelihood. We also propose an approach to construct counterfactuals
without estimating the matching functions for a subclass of 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.
