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.

Research Design Meets Market Design: Using Centralized Assignment for Impact Evaluation

“…The two lemmas in the appendix di er from Azevedo and Leshno (2014)'s results in that they use Assumption 1 and are proved using the extended continuous mapping theorem. The characterization of the DA propensity score in Theorem 1 does not appear to have an analog in the Azevedo andLeshno (2014) framework. al.…”

“…We consider markets with a nite number of schools, but with either nite (n) or in nitely many students. As in Abdulkadiro §lu et al (2015) and Azevedo and Leshno (2014), the latter setting is referred to as a continuum economy. In a continuum economy, I = [0, 1] and school capacities are de ned as the fraction of the continuum that can be seated at each school.…”

“…Examples include Abdulkadiro §lu et al (2015); Azevedo and Leshno (2014); Budish (2011);Che and Kojima (2010); Kesten and Ünver (2015). Student i's preferences over schools constitute a partial ordering of schools, i , where a i b means that i prefers school a to school b.…”

“…How should the resulting IV estimates be interpreted? Our IV procedure identi es causal e ects for applicants treated when DA produces a charter o er but not otherwise; in the local average treatment e ects (LATE) framework of Imbens and Angrist (1994) and Angrist et 5 See also Azevedo and Leshno (2014), who provide convergence results for the cuto s and conditional-on-type probabilities of assignment generated by a sequence of stable matchings, showing that the empirical assignment rates for types in a nite market converge to the continuum probability of assignment. The two lemmas in the appendix di er from Azevedo and Leshno (2014)'s results in that they use Assumption 1 and are proved using the extended continuous mapping theorem.…”

Improving Community Cohesion in School Choice via Correlated-Lottery Implementation

“…000-000, c 0000 INFORMS This model is similar to the continuum two-sided matching model in Azevedo and Leshno (2012). The main difference is that preference structure is one-sided in our model and two sided in their model: only students have preferences over schools in our model, while schools also have strict preferences over students in their model.…”