Abstract
To prove the existence of an equilibrium price vector for the equilibrium problem (EP) on Sn, also called the zero point problem (ZPP) on Sn, with respect to a continuous function from Sn to Rn+1 \(\sum\nolimits_{i\, = \,1}^{n\, + \,1}\)pi zi (p) = 0 satisfying for all p while zi (p) ≥ 0,if p =0, one can use Brouwer’s fixed point theorem that each continuous function f from a nonempty, convex, and compact set into itself has at least one fixed point.This because the fixed point problem and the zero point problem are equivalent as was shown in Uzawa [1962]. Any continuous excess demand function z:Sn → Rn + 1 can be transformed into a continuous function f:Sn → Sn such that a fixed point of f is a zero of z, and conversely.
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© 1988 Springer-Verlag Berlin Heidelberg
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Doup, T.M. (1988). Definitions and Existence Theorems. In: Simplicial Algorithms on the Simplotope. Lecture Notes in Economics and Mathematical Systems, vol 318. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46651-9_2
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DOI: https://doi.org/10.1007/978-3-642-46651-9_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-50233-3
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