Equilibria in matching markets with soft and hard liquidity constraints

We consider a one-to-one matching with contracts model in the presence of liquidity constraints on the buyer's side. Liquidity constraints can be either soft or hard. Competitive equilibria do exist in economies with soft liquidity constraints, but not necessarily in the presence of hard liquidity constraints. The limit of a convergent sequence of competitive equilibria in economies with increasingly stringent soft liquidity constraints may fail to be a competitive equilibrium in the limit economy with hard liquidity constraints. We establish equivalence and existence results for two alternative notions of competitive equilibrium, quantity-constrained competitive equilibrium and expectational equilibrium, together with stable outcomes and core outcomes, in economies with both types of liquidity constraints. We argue that these notions of equilibrium and stability do not suffer from discontinuity problems by showing appropriate limit results.