Quantity constrained equilibria

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In a standard general equilibrium model it is assumed that there are no price restrictions and that prices adjust infinitely fast to their equilibrium values. Price rigidities may cause that a competitive equilibrium cannot be attained, and rationing on net demands or supplies is needed to clear the markets. BA©nassy, DrA¨ze and YounA¨s proved in the mid 1970s that there exist equilibria with rationing. Moreover, rationing is such that not both demand and supply of a good are rationed simultaneously, at least one commodity is not rationed at all, and there is rationing on the net supply or net demand of a good only if the price of that good is on its low or upper bound, respectively. In the 1980s disequilibrium models with only rationing on net supplies were introduced. In all these models prices are restricted by positive lower and upper bounds. In this paper the set of admissible prices is allowed to be an arbitrary convex set. For such an arbitrary set it cannot be guaranteed that there exists a constrained equilibrium satisfying that a price will be on its upper or lower bound in case of rationing. We introduce a more general equilibrium concept, called Quantity Constrained Equilibrium (QCE). At such an equilibrium the levels of supply and demand rationing are completely determined by the components of a direction in which the price system cannot be moved further without leaving the set of admissible prices. When the set is compact, we show the existence of a connected set of QCEs, containing two trivial no-trade equilibria. Moreover, the set contains for every commodity a generalized DrA¨ze equilibrium, begin a QCE at which this commodity is not being rationed, and also a generalized supply-constrained equilibrium without demand rationing. We apply this main result to several special cases, including the case of an unbounded set of admissible prices.