Decentralized Re-equilibration and Comparative Statics in Matching Markets with Contracts

Abstract

This paper studies decentralized re-equilibration following population shocks, such as worker exits or firm entries, in many-to-many matching markets with contracts under substitutable preferences. We show that restricting any pre-shock stable allocation to the surviving agents yields an element of a complete lattice of firm-quasi-stable allocations. On this lattice, the deferred acceptance algorithm operates as an asynchronous iteration of a monotone operator to restore stability. The induced re-equilibration map defines a join-semilattice homomorphism between the stability lattices of the original and perturbed markets, preserves the firm-optimal stable allocation across markets, and establishes an opposition of interests whereby incumbent workers are weakly better off and incumbent firms weakly worse off. Under the law of aggregate demand, the outcome simplifies to the join of the restricted pre-shock allocation and the firm-optimal stable allocation of the perturbed market. Consequently, each entering firm obtains its firm-optimal assignment, independent of the pre-shock equilibrium selection.