Approximating welfare in large efficient markets

We consider the efficient outcome of a canonical economic market model involving buyers and sellers with independent and identically distributed random valuations and costs, respectively. When the number of buyers and sellers is large, we show that the joint distribution of the equilibrium quantity traded and welfare is asymptotically normal. Moreover, we bound the approximation rate. The proof proceeds by constructing, on a common probability space, a representation consisting of two independent empirical quantile processes, which in large markets can be approximated by independent Brownian bridges. The distribution of interest can then be approximated by that of a functional of a Gaussian process. This methodology applies to a variety of mechanism design problems.