Summary. The existence of Nash and Walras equilibrium is proved via Brouwer's Fixed Point Theorem, without recourse to Kakutani's Fixed Point Theorem for correspondences. The domain of the Walras fixed point map is confined to the price simplex, even when there is production and weakly quasi-convex preferences. The key idea is to replace optimization with “satisficing improvement,” i.e., to replace the Maximum Principle with the “Satisficing Principle.”
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Received: July 9, 2001; revised version: February 25, 2002
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ID="*" I wish to thank Ken Arrow, Don Brown, and Andreu Mas-Colell for helpful comments. I first thought about using Brouwer's theorem without Kakutani's extension when I heard Herb Scarf's lectures on mathematical economics as an undergraduate in 1974, and then again when I read Tim Kehoe's 1980 Ph.D dissertation under Herb Scarf, but I did not resolve my confusion until I had to discuss Kehoe's presentation at the celebration for Herb Scarf's 65th birthday in September, 1995.
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ID="*"Correspondence to: C. D. Aliprantis
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Geanakoplos, J. Nash and Walras equilibrium via Brouwer. Econ Theory 21, 585–603 (2003). https://doi.org/10.1007/s001990000076
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DOI: https://doi.org/10.1007/s001990000076