Competitive equilibria in markets with a continuum of traders and incomplete preferences. (English) Zbl 0184.45201

Schmeidler, David

Econometrica 37, No. 4, 578-585 (1969).

From the text: Existence of competitive equilibrium in a market model introduced by R. J. Aumann is proved. This paper generalized Aumann’s result and simplifies his proof.
R. J. Aumann [Econometrica 32, 39–50 (1964; Zbl 0137.39003)] defined a “market with a continuum of traders”. He proved existence of a competitive equilibrium in such a market [ibid. 34, 1–17 (1966; Zbl 0142.17201)] without using any convexity assumption on the preference relation. In this paper we shall show that we may dispense with an additional assumption, namely, that of the completeness of the preference relations, in proving the existence of a competitive equilibrium. Aumann used the Brouwer fixed point theorem in his proof in the same way as L. W. McKenzie [ibid. 27, 54–71 (1959; Zbl 0095.34302)]. In this paper we use Aumann’s definition of the market (Section 2 is taken almost literally from Section 2 of Aumann’s second paper cited above), but the proof is based on the Kakutani fixed point theorem, as used by K. J. Arrow and G. Debreu [ibid. 22, 265–290 (1954; Zbl 0055.38007)]. We also make considerable use of Aumann’s paper [J. Math. Anal. Appl. 12, 1–12 (1965; Zbl 0163.06301)], which provides the basic tools for dealing with integrals of set-valued functions in economic applications.

MSC:

91B50	General equilibrium theory
91B26	Auctions, bargaining, bidding and selling, and other market models
91B08	Individual preferences
28B20	Set-valued set functions and measures; integration of set-valued functions; measurable selections

Citations:

Zbl 0137.39003; Zbl 0142.17201; Zbl 0095.34302; Zbl 0055.38007; Zbl 0163.06301