I found an interesting section in a nice intermediate textbook on General Equilibrium Theory by Ross M. Starr:
In Section 18.4 titled "Uzawa-Equivalence Theorem" shows the equivalence of two existence theorems:
- The existence of equilibrium in an economy characterized by a continuous excess demand function fulfilling Walras' Law
- The Brouwer Fixed-Point Theorem
Interestingly, the two apparently distinct results are mathematically equivalent, which is originally shown by Hirofumi Uzawa (1962) in his short note: "Walras' Existence Theorem and Brouwer's Fixed Point Theorem." Economic Studies Quarterly, 8: 59-62.
The importance of the theorem is stressed by Prof. Starr as follows:
What are we to make of the Uzawa Equivalence Theorem? It says that use of the Brouwer Fixed-Point Theorem is not merely one way to prove the existence of equilibrium. In a fundamental sense, it is the only way. Any alternative proof of existence will include, inter alia, an implicit proof of the Brouwer Theorem. Hence, this mathematical method is essential; one cannot pursue this branch of economics without the Brouwer Theorem. If Walras (1874) provided an incomplete proof of existence of equilibrium, it was in part because the necessary mathematics was not yet available.The paper is included in this volume of Uzawa's collected papers. (I thank Prof. Kawamura for the information.)