Equilibria in economies with countably many commodities
type: Item
instance of: scholarly article
title: Equilibria in economies with countably many commodities
zbMATH Open document ID: 0759.90008
DOI: 10.1016/0165-1765(88)90135-8
author: Margarita Estévez
author: Carlos Hervés
author: M. Besada Morais
published in: Economics Letters
publication date: 1993-03-10T00:00:00Z
review text: Studies of the existence of equilibria for economies with infinite- dimensional commodity spaces have different theoretical motivations --- e.g. models with uncertainty, commodity differentiation, infinite-horizon processes --- and have become more and more frequent in economic analysis. Among the pioneer studies in the field there is a well-known article by \textit{B. Peleg} and \textit{M. E. Yaari} [Rev. Econ. Stud. 40, 391-401 (1973; Zbl 0266.90017)] on an economy with a finite number of traders and where the commodity space is \(s\), the space of all real sequences, endowed with the product topology. The article we are reviewing here focuses also on an exchange economy \(E\) with \(m\) agents and countably many commodities. The commodity space is the nonnegative cone \(\Lambda(P)_ +\) of the linear space \(\Lambda(P)=\{x\in s: \sum^ \infty_{k=1}| p| | x_ k|<\infty\) for all \(p\in P\}\) where \(P=\{p\in s: \sum^ \infty_{k=1}| p_ k| | w_{i,k}|<\infty\), \(i=1,2,\dots,m\}\) and \(w=(w_{i,k})^ \infty_{k=1}\) is the initial endowment of the \(i\)th agent. If one defines the seminorm \(u_ p(x)=\sum^ \infty_{k=1}| p_ k| | x_ k|\), then the family \(\{u_ p: p\in P\}\) defines a locally convex topology on \(\Lambda(P)\). \(\Lambda(P)\) with such a topology constitutes a Köthe or perfect space. The authors of the article consider their result as an improvement over Yaari and Peleg's result [op. cit.] inasmuch as their chosen topology (1) admits more continuous preferences than the product topology, and (2) permits the price systems to belong to the duality of \(\Lambda(P)\). In their argument the authors appeal to an infinite-dimensional version of \textit{H. Scarf's} theorem [Econometrica 35, 50-69 (1967; Zbl 0183.240)] on the nonemptiness of the core \(C_ r\) of the \(r\)-replica \(E_ r\) of the original economy \(E\), and then, adopting a line of reasoning that goes back to a paper by \textit{G. Debreu} and \textit{H. Scarf} [Internat. Econom. Rev. 4, 235-246 (1963; Z…
Mathematics Subject Classification ID: 91B50
zbMATH DE Number: 130359
zbMATH Keywords: countably many commodities
zbMATH Keywords: infinite-dimensional commodity spaces
zbMATH Keywords: competitive equilibrium
MaRDI profile type: Publication
full work available at URL: 0165-1765(88)90135-8
OpenAlex ID: W1984895556
cites work: Q5588318
cites work: A Limit Theorem on the Core of an Economy
cites work: Q5569793
cites work: Markets with Countably Many Commodities
cites work: The Core of an N Person Game
description: scientific article; zbMATH DE number 130359