Stable allocations in matching markets with contracts form a complete lattice under substitutable preferences, but little is known about how this structure transforms when the population changes. We study re-equilibration following structured population perturbations. In particular, when workers exit or firms enter, a stable allocation of the original market, when restricted to the perturbed market, typically fails to remain stable but becomes firm-quasi-stable. The set of firm-quasi-stable allocations forms a nonempty complete lattice under the Blair order. On this lattice we define a monotone operator whose fixed points coincide with the stable allocations of the perturbed market. Firm-proposing deferred acceptance corresponds to asynchronous iteration of this operator and converges, from any initial condition, to the minimal stable allocation that dominates the initial state. Our main contribution is to characterize equilibrium transformations across markets. Viewing stability sets as join-semilattices, we show that the re-equilibration map induced by deferred acceptance is order-preserving and, under the law of aggregate demand, becomes a join-semilattice homomorphism between the stability lattices of the original and perturbed markets, admitting an explicit lattice characterization. These results establish that, under structured population changes, re-equilibration can be understood as a structure-preserving transformation between equilibrium lattices.
Categories: econ.TH