The normative appeal of the market mechanism derives from the pareto optimality of competitive equilibrium allocations. Its positive appeal derives from the generic determinacy of competitive allocations : the requirement of market clearing generically suffices in order to determine the allocation of resources up to a finite set; also, the variation in the allocation of resources following a small variation in exogenous parameters, tastes, endowments or technologies. The existence of competitive equilibria guarantees the consistency both of the model and of the market as an allocation mechanism. Within the Arrow-Debreu model, these textbook properties of competitive equilibria hold under assumptions that are well understood, even standard. A most demanding requirement of the Arrow-Debreu model is the requirement that the asset market be cbmplete : that for each date-event a portfolio of securities exist that pays off one unit of revenue at that date-event and zero otherwise ; also that all securities be traded at an initial period with all individuals present. When the asset market is incomplete, prices acquire a role in addition to that of conveying the aggregate scarcity of commodities: in conjunction with the asset structure, prices determine the attainable reallocations of revenue across date-events. This additional role of the price mechanism underlies the divergence between economies with a complete and an incomplete asset structure. Competitive equilibrium allocations are fully pareto optimal when the asset market is complete: 1 no variation in the distribution of assets or commodities can improve on a competitive allocation. The argument for optimality is analytically simple and requires a minimum of assumptions: it derives simply from the fact that all potential reallocations can be valued unambiguously at the equilibrium prices. What is demanding is the appropriate interpretation of the notion of a commodity or a market ; equivalently, the requirement that the asset market be complete. The appropriate criterion of optimality when the asset market is incomplete has been a vexing question since the construction of examples with an incomplete asset market and