is a tuple . That is, it is a tuple specifying the commodities, the consumer preferences, consumption possibility sets, and the producers' production possibility sets. An economy with initial distribution is an economy, along with an initial distribution tuple for the economy. A state of the economy is a tuple of price, consumption plans, and production plans for each household and producer: . A state is feasible iff each , each , and . The feasible production possibilities set, given endowment , is . Given an economy with distribution, the state corresponding to a price vector is . Given an economy with distribution, a price vector is an equilibrium price vector for the economy with initial distribution, iffThat is, if a commodity is not free, then supply exactly equals demand, and if a commodity is free, then supply is equal or greater than demand (we allow free commodity to be oversupplied). A state is an equilibrium state iff it is the state corresponding to an equilibrium price vector. Assumptions Imposing an artificial restriction The functions are not necessarily well-defined for all price vectors . For example, if producer 1 is capable of transforming units of commodity 1 into units of commodity 2, and we have , then the producer can create plans with infinite profit, thus , and is undefined. Consequently, we define "restricted market" to be the same market, except there is a universal upper bound , such that every producer is required to use a production plan , and each household is required to use a consumption plan . Denote the corresponding quantities on the restricted market with a tilde. So for example, is the excess demand function on the restricted market. is chosen to be "large enough" for the economy, so that the restriction is not in effect under equilibrium conditions (see next section). In detail, is chosen to be large enough such that: For any consumption plan such that , the plan is so "extravagant" that even if all the producers coordinate, they would still fall short of meeting the demand. For any list of production plans for the economy , if , then for each . In other words, for any attainable production plan under the given endowment , each producer's individual production plan must lie strictly within the restriction. Each requirement is satisfiable. Define the set of attainable aggregate production plans to be ,