Bertrand Equilibrium with Capacity Constraints and Restricted Mobility

Abstract

This paper considers price competition among firms when there are capacity constraints and buyers have limited ability to visit firms. A natural method of allocating buyers among firms arises in the equilibrium of the buyers' search game. Sufficient conditions are given under which the buyers' equilibrium varies continuously with the prices charged by firms. Capacity constraints are used to guarantee that this ensures existence of (mixed strategy) equilibria for the pricing game played by sellers. We show that natural pure strategy equilibria arise when the game is made large in appropriate ways. SINCE BERTRAND'S CRITICISM OF COURNOT, it has been known that models where sellers compete with prices may have undesirable properties when the commodities traded are homogenous. In the simplest models with constant marginal costs, price may be driven down to the competitive level, even when there are only two firms. More problematic, if sellers are subject to capacity constraints, Nash (in prices) equilibrium may not exist at all. These problems all arise because of a discontinuity in firms' profit functions that arises when the prices of two or more firms are equal. There have been many attempts to resolve this problem. The simplest is to constrain sellers' prices to be equal, and to force them to compete in quantities. This is the basis of most of the literature on monopolistic competi