A Supply and Demand Framework for Two-Sided Matching Markets

Abstract

We propose a new model of two-sided matching markets, which allows for complex heterogeneous preferences, but is more tractable than the standard model, yielding rich comparative statics and new results on large matching markets. We simplify the standard Gale and Shapley (1962) model in two ways. First, following Aumann (1964) we consider a setting where a finite number of agents on one side (colleges or firms) are matched to a continuum mass of agents on the other side (students or workers). Second, we show that, in both the discrete and continuum model, stable matchings have a very simple structure, with colleges accepting students ranked above a threshold, and students demanding their favorite college that will accept them. Moreover, stable matchings may be found by solving for thresholds that balance supply and demand for colleges. We give general conditions under which the continuum model admits a unique stable matching, in contrast to the standard discrete model. This stable matching varies continuously with the parameters of the model, and comparative statics may be derived as in competitive equilibrium theory, through the market clearing equations. Moreover, given a sequence of large discrete economies converging to a limit economy, the set of stable matchings of the discrete economies converges to the stable matching of the limit economy. We bound the rate of convergence of the set of stable matchings of large discrete economies to the continuum approximation, and show that comparative statics regarding the unique stable matching of the continuum model extend to strong set ordering of the sets of stable matchings of approximating discrete economies. We model the transferrable utility case, as in Becker (1973). We characterize the limit of school choice mechanisms used in practice, generalizing previous results of Che and Kojima (2010). Finally, we illustrate the model's applicability by quantifying how competition induced by school choice gives schools incentives to invest in quality. Specifically, we show that schools have muted, and possibly even negative incentives to invest in quality dimensions that benefit lower ranked students.

Stable Matching in Large Economies

“…We also identify a condition under which a side‐optimal stable matching can be found via a generalized Gale–Shapley algorithm. Finally, we also find a condition, richness , that guarantees the uniqueness of the stable matching, thus generalizing the uniqueness result of Azevedo and Leshno () beyond the special case of responsive preferences. When firms have responsive preferences but face general group‐specific quotas (e.g., affirmative actions), our richness condition is implied by a full support assumption on firms' preferences, leading to a unique stable matching under that assumption.…”

“…While the set of stable matchings can be large in finite economies, there is a sense in which the set shrinks as the market grows large. Indeed, Azevedo and Leshno () established that a stable matching is generically unique in a continuum economy when firms have so‐called responsive preferences, a special case of substitutable preferences. To what extent such a result applies to more general preferences is an interesting issue that can be explored in a large market setting.…”

“…Our study was inspired by recent research on matching with a continuum of agents due to Abdulkadiroğlu, Che, and Yasuda () and Azevedo and Leshno (). As in the present study, these authors assumed that there are a finite number of firms and a continuum of workers.…”

“…As in the present study, these authors assumed that there are a finite number of firms and a continuum of workers. In particular, Azevedo and Leshno () showed the existence and uniqueness of a stable matching in that setting. However, as opposed to the present study, these authors assumed that firms have responsive preferences—which is a special case of substitutability.…”

Matching of like rank and the size of the core in the marriage problem

“…Lee (2011) introduces cardinal utilities which are chosen at random within a fixed interval, and measures the size of the core in these cardinal utility units. 1 Azevedo and Leshno (2012) consider a many-to-one setting with a constant number of schools on one side, and an increasing number of students, modeling the limit as having a continuum of students. Bodoh-Creed (2013) also studies a continuum of agents, using a type-space to describe their characteristics.…”