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Consistent solutions in atomless economies. (English) Zbl 0816.90021

A group of agents collectively owns a bundle of commodities. A solution is a rule for allocating the commodities among the members of the group, that also can be applied to the allocation of a given bundle to the members of any subgroup. Solutions should satisfy the requirement of (i) efficiency, (ii) equity (no agent should prefer equal division of the endowment) and (iii) consistency (if a solution allocates certain commodity bundles to the members some subgroup, the solution applied to the subgroup should allocate the same bundles to the members of the subgroup, given that the total is available that was originally allocated to them). The economy studied consists of a measure space \((\Omega, {\mathcal B},\mu)\) of consumers; each consumer has a preference and there is a total endowment vector \(e\). A feasible allocation \(f\) satisfies \(\int_ \Omega f(\omega)= e\). The preferences are assumed to be continuous, locally nonsatiated at non satiation points, locally representable by a differentiable utility and measurable (but need not be convex, transitive, monotone or nonsatiated).
The theorem of the paper asserts that a solution, satisfying the above requirements, is the allocation of some Walrasian equilibrium with equal budgets (possibly with slack: satiated consumers do not exhaust there budget restriction). Equilibrium prices may be negative.

MSC:

91B50 General equilibrium theory
91B14 Social choice
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