General Equilibrium Model with a Convex Cone as the Set of Commodity Bundles

S. Schalk

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Abstract

In this paper, we present a model for an exchange economy which is an extension of the classical model as introduced by Arrow and Debreu.In the classical model, there is a nite number of commodi- ties and a nite number of consumers.The commodities are treated separately, and so a commodity bundle is an element of the positive orthant of the Euclidean space IR l , wherelis the number of com- modities.A closer look at Arrow and Debreu's model shows that this Euclidean structure is used only indirectly.Instead of using the Euclidean structure, we allow for just the exis- tence of commodity bundles, and do not take into consideration indi- vidual commodities.More speci cally, we model the set of all possible commodity bundles in the exchange economy under consideration, by a pointed convex cone in a nite-dimensional vector space.This vec- tor space is used only to de ne the suitable topological concepts in the cone, and therefore is not part of the model.Since we do not consider separate commodities, we do not intro- duce prices of individual commodities.Instead, we consider price systems, which attach a positive value to every commodity bundle. These price systems are modelled by the linear functionals on the vec- tor space that are positive on the cone of commodity bundles.The set of price systems is a cone with similar properties as the commodity cone.More precisely, the price cone is the polar cone of the commodity cone.The commodity cone introduces a partial ordering on the commod- ity bundles and the price systems are compatible with this ordering.If we take the positive orthant of the Euclidean space IR l as the pointed convex cone then the partial ordering coincides with the Euclidean order relation on IR l taken in the classical approach.In this setting, given a nite number of consumers each with an ini- tial endowment and a preference relation on the commodity cone, we prove existence of a Walrasian equilibrium under assumptions which are essentially the same as the ones in Arrow and Debreu's model.We introduce the new concept of equilibrium function on the price system cone; zeroes of an equilibrium function correspond with equilibrium price systems.So proving existence of a Walrasian equilibrium comes down to constructing an equilibrium function with zeroes.
Original languageEnglish
Place of PublicationTilburg
PublisherMicroeconomics
Number of pages41
Volume740
Publication statusPublished - 1996

Publication series

NameFEW Research Memorandum
Volume740

Fingerprint

Convex cone
General equilibrium model
Commodities
Walrasian equilibrium
Exchange economy
Preference relation
Endowments
Equilibrium price

Keywords

  • general equilibrium
  • commodities

Cite this

Schalk, S. (1996). General Equilibrium Model with a Convex Cone as the Set of Commodity Bundles. (FEW Research Memorandum; Vol. 740). Tilburg: Microeconomics.
Schalk, S. / General Equilibrium Model with a Convex Cone as the Set of Commodity Bundles. Tilburg : Microeconomics, 1996. 41 p. (FEW Research Memorandum).
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Schalk, S 1996, General Equilibrium Model with a Convex Cone as the Set of Commodity Bundles. FEW Research Memorandum, vol. 740, vol. 740, Microeconomics, Tilburg.

General Equilibrium Model with a Convex Cone as the Set of Commodity Bundles. / Schalk, S.

Tilburg : Microeconomics, 1996. 41 p. (FEW Research Memorandum; Vol. 740).

Research output: Book/ReportReportProfessional

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AB - In this paper, we present a model for an exchange economy which is an extension of the classical model as introduced by Arrow and Debreu.In the classical model, there is a nite number of commodi- ties and a nite number of consumers.The commodities are treated separately, and so a commodity bundle is an element of the positive orthant of the Euclidean space IR l , wherelis the number of com- modities.A closer look at Arrow and Debreu's model shows that this Euclidean structure is used only indirectly.Instead of using the Euclidean structure, we allow for just the exis- tence of commodity bundles, and do not take into consideration indi- vidual commodities.More speci cally, we model the set of all possible commodity bundles in the exchange economy under consideration, by a pointed convex cone in a nite-dimensional vector space.This vec- tor space is used only to de ne the suitable topological concepts in the cone, and therefore is not part of the model.Since we do not consider separate commodities, we do not intro- duce prices of individual commodities.Instead, we consider price systems, which attach a positive value to every commodity bundle. These price systems are modelled by the linear functionals on the vec- tor space that are positive on the cone of commodity bundles.The set of price systems is a cone with similar properties as the commodity cone.More precisely, the price cone is the polar cone of the commodity cone.The commodity cone introduces a partial ordering on the commod- ity bundles and the price systems are compatible with this ordering.If we take the positive orthant of the Euclidean space IR l as the pointed convex cone then the partial ordering coincides with the Euclidean order relation on IR l taken in the classical approach.In this setting, given a nite number of consumers each with an ini- tial endowment and a preference relation on the commodity cone, we prove existence of a Walrasian equilibrium under assumptions which are essentially the same as the ones in Arrow and Debreu's model.We introduce the new concept of equilibrium function on the price system cone; zeroes of an equilibrium function correspond with equilibrium price systems.So proving existence of a Walrasian equilibrium comes down to constructing an equilibrium function with zeroes.

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Schalk S. General Equilibrium Model with a Convex Cone as the Set of Commodity Bundles. Tilburg: Microeconomics, 1996. 41 p. (FEW Research Memorandum).