Equilibrium and Matching under Price Controls∗ P. Jean-Jacques Herings†

The paper considers a one-to-one matching with contracts model in the presence of price controls. This set-up contains two important streams in the matching litera-ture, those with and those without monetary transfers, as special cases and allows for intermediate cases with some restrictions on the monetary transfers that are feasible. An adjustment process that ends with a stable outcome is presented, thereby prov-ing the existence of stable outcomes. The process contains the deferred acceptance algorithm of Gale and Shapley (1962) and the approximate auction mechanism of Demange, Gale, and Sotomayor (1986) as special cases. The paper presents a no-tion of competitive equilibrium, called Drèze equilibrium, for this class of models, an extension of the concept as developed by Drèze (1975) for economies with divisible commodities subject to price controls. It is shown that Drèze equilibrium allocations are equivalent to allocations induced by stable outcomes. One implication is the ex-istence of Drèze equilibria. Another implication is the equivalence of a competitive equilibrium concept and the concept of stable outcomes that is valid with and with-out monetary transfers as well as when monetary transfers are limited