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An application of fixed-point theory to equilibrium analysis. (English) Zbl 0559.90019
The main purpose of the present paper is to extend some basic results concerning equilibria of certain unbounded private ownership economies to the case of generalized preference structures, which need not be complete, convex or transitive. The corresponding classical theory is based on a theorem of Gale-Nikaido-Debreu, which is an equivalent version of Brouwer’s fixed-point theorem; see H. Nikaido [”Convex structures and economic theory” (1968; Zbl 0172.445)] for the classical background material. In the first part of the present paper two generalizations of the Gale-Nikaido-Debreu theorem are presented in order to handle the case of the most general type of preference structures in the sense of J. Greenberg [J. Math. Econ. 4, 163-165 (1977; Zbl 0368.90015)]. In the last part, these fixed-point type theorems are used to derive the existence of equilibria and of quasiequilibria in a rather general Arrow-Debreu model. The main theorem includes, in particular, a result of A. Mas-Colell [ibid. 1, 237-246 (1974; Zbl 0348.90033)]. Further information and some extensions of the results of the present note will be contained in a forthcoming monograph by H. König and the author [”Mathematische Wirtschaftstheorie mit einer Einführung in die konvexe Analysis”, Math. Syst. Econ. 100, Anton Hain bei Athenäum (1985)].

91B62 Economic growth models
54H10 Topological representations of algebraic systems
54H25 Fixed-point and coincidence theorems (topological aspects)