Ordinal and cardinal solution concepts for two-sided matching

We characterize solutions for two-sided matching, both in the transferable- and in the nontransferable-utility frameworks, using a cardinal formulation. Our approach makes the comparison of the matching models with and without transfers particularly transparent. We introduce the concept of a no-trade stable matching to study the role of transfers in matching. A no-trade stable matching is one in which the availability of transfers does not affect the outcome.