as . Here is the number of commodities that exists in the economy. It is a finite number. The price vector is a vector of length , with each coordinate being the price of a commodity. The prices may be zero or positive. households The households are indexed as . Each household begins with an endowment of commodities . Each household begins with a tuple of ownerships of the producers . The ownerships satisfy . The budget that the household receives is the sum of its income from selling endowments at the market price, plus profits from its ownership of producers:( stands for money) Each household has a Consumption Possibility Set . Each household has a preference relation over . With assumptions on (given in the next section), each preference relation is representable by a utility function by the Debreu theorems. Thus instead of maximizing preference, we can equivalently state that the household is maximizing its utility. A consumption plan is a vector in , written as . is the set of consumption plans at least as preferable as . The budget set is the set of consumption plans that it can afford:. For each price vector , the household has a demand vector for commodities, as . This function is defined as the solution to a constraint maximization problem. It depends on both the economy and the initial distribution.It may not be well-defined for all . However, we will use enough assumptions such that that it is well-defined at equilibrium price vectors. producers The producers are indexed as . Each producer has a Production Possibility Set . Note that the supply vector may have both positive and negative coordinates. For example, indicates a production plan that uses up 1 unit of commodity 1 to produce 1 unit of commodity 2. A production plan is a vector in , written as . For each price vector , the producer has a supply vector for commodities, as . This function will be defined as the solution to a constraint maximization problem. It depends on both the economy and the initial distribution.It may not be well-defined for all . However, we will use enough assumptions such that that it is well-defined at equilibrium price vectors. The profit is aggregates aggregate consumption possibility set . aggregate production possibility set . aggregate endowment aggregate demand aggregate supply excess demand the whole economy An economy