Re-equilibration in Matching Markets with Contracts: Firm-Quasi-Stable Allocations
Yi-You Yang
TL;DR
The paper addresses how matching markets with contracts re-equilibrate after shocks such as firm entry or worker exit. It develops a lattice-theoretic framework in which firm-quasi-stable allocations form a complete Blair lattice and stable allocations are the fixed points of a monotone $T$-operator on this domain. The authors then link static structure to dynamics by showing that decentralized vacancy chains via a firm-proposing deferred acceptance algorithm asymptotically reproduce the centralized $T$-operator, yielding order-independence and convergence to the worker-pessimal stable outcome dominating the initial state. Under the law of aggregate demand, the re-equilibration delivers a new-entrant advantage, with entering firms achieving their firm-optimal stable outcomes and incumbent workers gaining welfare while incumbent firms may lose relative to incumbents in the pre-entry market.
Abstract
We analyze the re-equilibration dynamics of matching markets with contracts following disruptions such as firm entry or worker exit. In a many-to-many framework with substitutable preferences, we identify firm-quasi-stable allocations as the canonical state space for out-of-equilibrium adjustment. We prove that this set forms a complete lattice and characterize stable allocations as the fixed points of a monotone Tarski operator defined on it, thereby establishing the lattice structure of stable allocations. We link this static characterization to dynamics by showing that decentralized vacancy chains approximate the centralized Tarski operator asynchronously. Specifically, we prove that the trajectory of a generalized firm-proposing deferred acceptance algorithm is bounded by the operator's iterations. This structural containment ensures both processes converge to the same limit. Consequently, the adjustment process exhibits order independence, with the final outcome invariant to the sequence of proposals. Finally, we derive welfare implications for market participants, and, under the law of aggregate demand, generalize the new-entrant advantage theorem to matching markets with contracts, establishing that entering firms achieve their optimal stable outcomes.