On Core-Walras equivalence in Banach spaces when feasibility is defined by the Pettis integral
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 trying to extend the equivalence theorem to the case when the commodity space \(E\) is allowed to be an infinite dimensional space. In most existing extensions the commodity space is taken to be separable, an assumption which is open to the criticism that it entails the implicit restriction that the size of the commodity space, relative to the size of the space of agents, is ``not too large'' and, therefore, the preferences of the agents cannot be ``too dispersed''. Indeed recent work shows that in non-separable Hilbert spaces, with the notion of integrability that of the Bochner integral, the equivalence theorem fails. However, under this notion of integrability, blocking possibilities are very limited, when the commodity space is non-separable, implying a bias in favour of non-equivalence. This paper uses the weaker notion of Pettis integrability and then study the equivalence question in order to address this issue in a more appropriate setting. It turns out that results are sensitive not just to the notion of integrability itself but conjoint with measurability assumptions on the profile of preferences of agents. Measurability of the profile of preferences of agents can be defined in different ways, two of which are as follows. (M1) If \(x\) and \(y\) are any two consumption bundles then \( \{ t\in T:x\succ _{t}y\} \in \mathcal{T}\). (M2) If \(f\) and \(g\) are any two allocations then \(\{ t\in T:f( t) \succ _{t}g( t) \} \in \mathcal{T}\). The paper shows that: (1) The core-Walras non-equivalence results continue to hold when allocations are defined to be Pettis integrable but only (M1) is required on preferences. (2) One can find non-separable commodity spaces in which equivalence fails when allocations are defined to be Pettis integrable and the stronger (M2) is required on preferences. (3) However, if allocations are defined to be Pettis integrable together with requiring (M2) on the preferences, in the commodity space setting of Banach lattices with an order continuous norm and a weak unit, core-Walras equivalence holds, regardless of whether the commodity space is separable or not. This is interpreted to mean restrictions on the extent of heterogeneity of preferences allowed, as implied by the stronger (M2) as compared to (M1) in the setting of Pettis integrable allocations, can matter regarding the equivalence result.
scientific article; zbMATH DE number 2092969