Hedonic price equilibria , stable matching , and optimal transport : equivalence , topology , and uniqueness 1

Hedonic pricing with quasilinear preferences is shown to be equivalent to stable matching with transferable utilities and a participation constraint, and to an optimal transportation (Monge-Kantorovich) linear programming problem. Optimal assignments in the latter correspond to stable matchings, and to hedonic equilibria. These assignments are shown to exist in great generality; their marginal indirect payo¤s with respect to agent type are shown to be unique whenever direct payo¤s vary smoothly with type. Under a generalized Spence-Mirrlees condition the assignments are shown to be unique and to be pure, meaning the matching is one-to-one outside a negligible set. For smooth problems set on compact, connected type spaces such as the circle, there is a topological obstruction to purity, but we give a weaker condition still guaranteeing uniqueness of the stable match. An appendix resolves an old problem (# 111) of Birkho¤ in probability and statistics [5], by giving a necessary and su¢ cient condition on the support of a joint probability to guarantee extremality among all joint measures with the same marginals.