季刊 理論経済学
Online ISSN : 2185-4408
Print ISSN : 0557-109X
ISSN-L : 0557-109X
WALRAS' EXISTENCE THEOREM AND BROUWER'S FIXED-POINT THEOREM
HIROFUMI UZAWA
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1962 年 13 巻 1 号 p. 59-62

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THE PURPOSE of this note is to show the equivalence of two fundamental theorems-Walras' Existence Theorem on the one hand and Brouwer's FixedPoint Theorem on the other.
Walras' theorem1) is concerned with the existence of an equilibrium in the Walrasian system of general equilibrium and has been a problem of some importance in formal economic analysis since his work [12] appeared in 1874-7. It was, however, not until Wald's contributions, [10] and [11], that the existence problem was rigorously treated. Recent contributions, in particular those of Arrow and Debreu [2], McKenzie [6], NikaidÔ [7], and Gale [4], have shown that Walras' theorem is essentially a necessary consequence of Brouwer's FixedPoint Theorem. The latter theorem, 2)first proved by Brouwer [3] in 1911, also bears a fundamental importance in mathematics. It may be hence of some interest to see that Brouwer's theorem is in fact implied by Walras' theorem. It would indicate the reason that the general treatment of the existence pro blem in the Walrasian system had to wait for the development of the twentieth century mathematics.

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