On Core-Walras equivalence in Banach spaces when feasibility is defined by the Pettis integral
type: Item
instance of: scholarly article
title: On Core-Walras equivalence in Banach spaces when feasibility is defined by the Pettis integral
zbMATH Open document ID: 1096.91047
author: Konrad Podczeck
published in: Journal of Mathematical Economics
publication date: 2004-08-19T00:00:00Z
review text: The paper studies the equivalence between core allocations and Walrasian equilibrium allocations in a pure exchange economy with a continuum of agents represented by an atomless measure space. Consider the following framework: let \(E\) be an ordered Banach space and \(E^{\ast }\) its dual. An \textit{economy} \(\mathcal{E}\) is a pair \( [ ( T,\mathcal{T},\nu ) ,( X( t) ,\succ _{t},e( t) ) _{t\in T}] \) where \(( T,\mathcal{T} ,\nu ) \) is a complete positive finite measure space and the generic element \(t\in T\) represents an individual agent; \(X( t) \sqsubset E\) is the consumption set of agent \(t;\succ _{t}\subset X( t) \times X( t) \) is the (strict) preference relation of agent \(t\); and \(e( t) \in E\) is the initial endowment of agent \(t\). An \textit{allocation} is a (suitably defined) integrable function \( f:T\to E\) such that \(f( t) \in X( t) \) for almost all \(t\in T\) and \(\int_{T}f( t) d\nu ( t) =\int_{T}e( t) d\nu ( t) .\) An allocation \(f\) is a \textit{Walrasian} \textit{allocation} if there is a (price system) \(p\in E^{\ast }\backslash \{ 0\} \) such that \(( p,f) \) is a Walrasian equilibrium, i.e., for almost every \(t\in T\), (i) \(\langle p,f( t) \rangle \leq \langle p,e( t) \rangle \) and (ii) if \(x\in X( t) \) satisfies \(x\succ _{t}f( t) \) then \(\langle p,x\rangle >\langle p,e( t) \rangle .\) An allocation \(f\) is a \textit{core allocation} if there does not exist a coalition \(S\in \mathcal{T}\) with \(\nu ( S) >0\) and a (suitably defined) integrable function \(g:T\to E_{+}\) such that (i) \(\int_{S}g( t) d\nu ( t) =\int_{S}e( t) d\nu ( t) \) and (ii) \(g( t) \succ _{t}f( t) \) for almost all \(t\in S\). Denote by \(\mathcal{C}( \mathcal{E}) \) and \(\mathcal{W}( \mathcal{E}) \) the set of core allocations and Walrasian allocations, respectively. The Core-Walras equivalence theorem asserts that, under suitable assumptions, \(\mathcal{C}( \mathcal{E}) =\mathcal{W}( \mathcal{E}) .\) The paper deals with problems that arise in …
Mathematics Subject Classification ID: 91B54
zbMATH DE Number: 2092969
zbMATH Keywords: Core-Walras equivalence
zbMATH Keywords: Pettis integral
zbMATH Keywords: Bochner integral
zbMATH Keywords: measurability of preferences
zbMATH Keywords: non-separable space
zbMATH Keywords: Banach lattice
MaRDI profile type: Publication
full work available at URL: j.jmateco.2004.01.002
OpenAlex ID: W2016690586
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DOI: 10.1016/J.JMATECO.2004.01.002
description: scientific article; zbMATH DE number 2092969