Equilibria in economies with countably many commodities
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; Zbl 0122.377)], they conclude that \(C_ r\) can be identified with a nonempty compact subset \(C^*_ r\) of \((\Lambda(P)_ +)^ m\). Finally, the main proposition of the article states that if \(x\in\bigcap^ \infty_{r=1} C^*_ r\), then there exists \(p\in P\) with \(p_ k>0\) for \(k=1,2,\dots\) such that \((x,p)\) is a competitive equilibrium of \(E\).
scientific article; zbMATH DE number 130359