AN EQUILJBRIUM EXISTENCE THEOREM FOR A GENERAL MODEL WITHOUT ORDERED PREFERENCES* D. GALE and A. MAS-COLELL

In a recent paper the second author has shown that some of the usual hypotheses on consumers’ preferences are not needed for the proof of existence of a Walrasian General Equilibrium [Mas-Cole11 (1974)]. Specifically, it is not necessary that preferences come from a preference ordering. The only order property required is irreflexivity (meaning that a given commodity bundle is not preferred to itself). The properties of non-satiation, continuity and convexity of preferred sets turn out to be sufficient to obtain the existence result. The main purpose of the present note is to give a second proof of this fact which seems simpler than that of Mas-Cole11 (1974), and no more lengthy or complicated than the known equilibrium existence proofs which use ordered preferences. In two additional respects the model studied here generalizes the usual equilibrium model. The standard Walras, Arrow-Debreu theory treats what might be called the laissez-faire model in which each agent’s income is whatever he gets from selling goods plus his share of the profits of any firm in which he may own stock. In the present model the income of a consumer may be any continuous function of the prices, so the laissez-faire income function is included, but so also would any rule for assigning income to consumers (e.g., according to his ability or his need or the color of his eyes). Another way of saying this is that the model includes the possibility of arbitrary lump sum transfers of income among consumers, as might be achieved, for example, by a program of income taxes and subsidies. This substantial economic generalization requires no change whatever in the mathematical argument. The second generalization concerns production. The only requirement on our production set, besides the usual closure, convexity, and free disposal, is that it intersect the positive orthant in a boundedset.This means that the usual assumption