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On the existence of equilibria in economies with an infinite dimensional commodity space. (English) Zbl 0535.90020
This paper states an infinite dimensional extension of the Gale-Nikaido- Debreu lemma (or market equilibrium theorem) to locally convex topological vector spaces, in order to give, under standard assumptions, a proof of the existence of equilibria in an economy with infinitely many commodities which exactly parallels the proof of Debreu for the finite dimensional case.
The commodity space is a Banach space, conjugate to another Banach space, and ordered by a positive cone with a non-empty interior. The proof is given for a pure exchange economy but could be extended without any difficulty to a production economy. The idea of the argument is to use the duality between the commodity space and its predual and to truncate consumption sets to weak-star compact sets, in order to define excess demand correspondences when the consumers’ preferences satisfy a weak- star continuity assumption; Debreu’s argument is applied to the truncated economy and then a limit argument allows the truncation to go to infinity. \(L_{\infty}(M,{\mathcal M},\mu)\) is of course a suitable commodity space.
The paper points out the assumptions which prevent from applying the approach to some Banach spaces as ca(K) and \(L_ p(M,{\mathcal M},\mu)\), \(1\leq p<\infty\) which have been found useful in practice to study some classical situations for economic theory: intertemporal equilibrium, uncertainty, differentiation of commodities. Subsequent results have proved that an extra assumption on preferences is then necessary for the existence of an equilibrium.

91B50 General equilibrium theory
Full Text: DOI
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