Existence of a Pareto-optimal beneficial allocation

Establish whether there exists an exchange vector Y in the allowed set of exchanges Y such that Y is both Pareto optimal with respect to the agents’ indirect-utility-induced preferences and beneficial, i.e., U^i(X^i;Y^i) ≥ U^i(X^i;0) for every agent i=1,…,N with strict inequality for at least one agent, and there is no other \tilde{Y} in Y with U^i(X^i;\tilde{Y}^i) ≥ U^i(X^i;Y^i) for all i and strict inequality for some j. Conduct this analysis within the paper’s segmented-market framework under the Standing Assumptions and Assumption 1, where U^i(X^i;\cdot) denotes the indirect utility of agent i.

Background

The paper defines beneficial exchanges as allocations Y in the allowed set of exchanges that weakly improve each agent’s indirect utility U^i(X^i;Y^i) relative to no exchange and strictly improve at least one agent’s utility. Pareto optimality is defined via the indirect-utility-induced preferences: an allocation is Pareto optimal if no other feasible allocation makes all agents weakly better off and at least one strictly better off.

While weighted-sum optimizers of the form sup_{Y\in\mathcal{Y}} \sum_i \lambda_i U^i(X^i;Y^i) (with \lambda_i>0) are Pareto optimal, the authors note that it is challenging to guarantee the simultaneous attainment of Pareto optimality and the beneficial property. The difficulty stems from the fact that the relevant upper-level sets are closed in the product of weak topologies but need not be compact, impeding standard existence arguments.