Rationing Rules and Bertrand-edgeworth Equilibria in Large Markets

Allen and Hellwig (A-H hereafter) (1984a) show for a market where firms compete in prices with capacity constraints that as the number of firms increases, Nash equilibria (in pure or mixed strategies) converge in distribution to a (perfectly) competitive price, but monopoly prices persist for any number of firms. In other words, the supports of the equilibrium mixed strategies do not necessarily converge to a subset of the competitive price set. We have, thus, that in Bertrand-Edgeworth mixed strategy equilibria, ‘competition plays no role in the determination of the “highest” price in the market’ [A-H (1984a, p. 16)]. A-H assume that rationing at the lowest price is made through a queueing system. Consumers in front of the line obtain their entire demand while others obtain nothing. We will refer to this way of allocating unsatisfied demand the proportional rationing scheme since once the lowest priced firms have satisfied a proportion cr of the,consumers there remains a proportion 1 (Y of consumers to be served at higher prices. An alternative method is to have consumers buy first from the cheapest supplier and assume no income effects. In this case the high-priced portion of the demand curve is satisfied first. We will refer to this method as the surplus-maximizing rationing scheme. ’ We present below a model using the surplus-maximizing rationing scheme where the supports of the equilibrium mixed strategies do converge to the (unique) competitive price.