We consider the computation of equilibria in two economic models that generalize the exchange model by including production. In the constant returns model, each producer has a convex, constant-returns-to-scale, technology. In particular, this means that if the technology can output a certain quantity of a good using as input certain quantities of other goods, then scaling all these quantities by a common, nonnegative, number also results in a technologically feasible plan. The technology also accomodates the no-free-lunch property, which says that it is not possible to produce something from nothing. At a given price, the producer picks a technologically feasible plan that maximizes her profit. Associated with each consumer is an initial endowment of goods and a utility function that describes her preferences between various bundles of goods. At a given price, the consumer sells her initial endowment, thus obtaining a certain income, and demands the bundle of goods maximizing her utility among all bundles that she can afford at the given price with her income. An equilibrium for such an economy is a set of prices, one for each good, such that utility maximization is well-defined for each consumer, profit maximization is well-defined for each producer, and the optimal choices made by the consumers and producers are such that for any good, its total demand, over all consumer choices as well as input choices of producers, is at most the total supply, over all the initial endowments and the output choices of producers.
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