Equivalence of Canonical Matching Models

This paper offers expected revenue and pricing equivalence results for canonical models of pricing and matching. The equivalence of these models is centered on the assumption that there are large numbers of buyers and sellers and the assignment of buyers within a submarket of sellers is random. Therefore, the distribution of buyers to sellers is approximated by the Poisson distribution. The list of canonical matching models includes the models developed by Burdett and Judd (1983), Shimer (2005), and McAfee (1993). In the Burdett and Judd (1983) model, buyers post prices and the equilibrium features price dispersion because identical buyers play mixed strategies. In the Shimer (2005) model, sellers post a vector of prices corresponding to different buyer types. In equilibrium, all identical buyers pay the same price. In the McAfee (1993) model, equilibrium pricing is determined by simple second price auctions. McAfee’s model also features price dispersion, because the number of bidders at each auction is stocastic.