On the existence of equilibria in economies with an infinite dimensional commodity space.

*(English)*Zbl 0535.90020This 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.

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.

##### MSC:

91B50 | General equilibrium theory |

##### Keywords:

infinite dimensional commodity space; fixed point theorems; correspondences; ordered Banach spaces; locally convex topological vector spaces; existence of equilibria; pure exchange economy; production economy
Full Text:
DOI

##### References:

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