Approximating welfare in large efficient markets

Abstract

We consider the efficient outcome of a canonical economic market model which involves \(N\) buyers with unit demand and i.i.d. random valuations, and \(M\) sellers with unit supply whose costs are also i.i.d. random variables independent of the valuations. We approximate the joint distribution of the quantity \(K_\alpha\) of units traded and the gains \(W_\alpha\) from trade when there is a large number of market participants, i.e. both components of \(\alpha:=(N,M)\) tend to infinity. The problem is reduced to studying a process expressed in terms of two independent empirical quantile processes which, in large markets, can be approximated by appropriately weighted independent Brownian bridges. That allows us to approximate the distribution of \((K_\alpha, W_\alpha)\) by that of a functional of a Gaussian process. Moreover, we give upper bounds for the approximation rate.