Reachable Stability Lattices after Population Shocks

Abstract

This paper studies how one-sided population shocks, such as worker exits and firm entries in senior-level labor markets, transform stability lattices in many-to-many matching markets with contracts and substitutable preferences. Deferred acceptance induces a re-equilibration map from the pre-shock to the post-shock stability lattice. We show that this map preserves order and joins, maps the pre-shock worker-optimal stable allocation to the greatest reachable element, and maps the pre-shock firm-optimal stable allocation to the post-shock firm-optimal stable allocation. Consequently, the reachable post-shock set forms a finite lattice. The construction factors through the firm-quasi-stable lattice, where the deferred-acceptance outcome coincides with the limit of a monotone Tarski-type operator. Sequential shock invariance shows that the reachable lattice is determined by the initial market and the aggregate shock, not by the order of exits and entries.

27 pages