WELFARE ECONOMICS AND EXISTENCE OF AN EQUILIBRIUM FOR A COMPETITIVE ECONOMY
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I. The proof of the existence of an equilibrium for a competitive economy is given by Arrow and Debreu [I] and many others such as Gale [4], Kuhn [6], McKenzie [8], [ g ] , and Nikaido [IO]. In this note, we shall give another proof of the existence of an equilibrium, putting emphasis on the welfare aspect of the competitive equilibrium (1). As is well known, an equilibrium point of an economic system under perfect competition is an efficient state in Pareto's sense in which we cannot make anyone better off without making someone worse off. In other words, i t can be said that a competitive equilibrium is a maximum point of some properly defined social welfare function subject to the resource and technological constraints. In the following, we shall show that a competitive equilibrium is a maximum point of a social welfare function which is a linear combination of utility functions of consumers, with the weights in the Combination in inverse proportion to the marginal utilities of income. Then, the existence of an equilibrium is equivalent to the existence of a maximum point of this special welfare function. Therefore, we can prove the former by showing the latter. 2. Let us construct our economic model, the existence of whose equilibrium we shall prove, as follows. Let there be m goods, n consumers, and I firms. Let x , be a consumption vector (whose element is x t t 2 o), xi be an initial holding vector (whose element is ;, 3> o), and U c (x,) be the utility (function) of the i th consumer. Let y k be a production vector of the kth firm whose element y k r > o (< 0) is the output (input) of the i th good, and Y k be the possible set of y k , i. e., the set of y k which satisfies the restriction on production F k (ye) >_ 0. Let P (whose element P , 2 0) be the price vector. For a non-free good, P, > 0 . Let h i k be the proportion of profit o i the k t h firm distributed to the i t h consumer. We define an equilibrium point under perfect competition: Definition I. The following are the conditions of an equilibrizcm point (x i j y k , P):
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