Convergence of a dynamic matching and bargaining market with two-sided incomplete information to perfect competition

Consider a decentralized, dynamic market with an infinite horizon in which both buyers and sellers have private information concerning their values for the indivisible traded good. Time is discrete, each period has length ä, and each unit of time a large number of new buyers and sellers enter the market to trade. Within a period each buyer is matched with a seller and each seller is matched with zero, one, or more buyers. Every seller runs a first price auction with a reservation price and, if trade occurs, both the seller and winning buyer exit the market with their realized utility. Traders who fail to trade either continue in the market to be rematched or become discouraged with probability äì (ì is the discouragement rate) and exit with zero utility. We characterize the steady-state, perfect Bayesian equilibria as ä becomes small and the marketin effect becomes large. We show that, as ä converges to zero, equilibrium prices at which trades occur converge to the Walrasian price and the realized allocations converge to the competitive allocation.