Existence of Competitive Equilibrium with a system of Complete Prices

In classical equilibrium analysis it is assumed that consumers may purchase a given amount of good in any fraction he (she) wishes from any of a xed number of rms. Sometimes, as with ski-lifts, at any one time it is only possible for consumers to purchase from one rm. An equilibrium price system is a complete price system. In this paper, a complete price system is a system of prices that varies between rms even though the product each rm sells may be identical. In addition as any consumer switches the rm that he (she) purchases from the price faced by all consumers may change and it is sometimes necessary to impose lump sum transfers between consumers and between rms and between consumers and rms. Congestion is allowed. Even if there are no non-convexities associated with the choice of rm by consumers then lump sum transfers may be needed to ensure that rst best allocations are supported. The presence of non-convexities and congestion ensures that proof of existence of equilibrium is di cult. A weak proof of existence is given.