Monotone comparative statics under dual submodularity, with an application to aggregate deferred acceptance

Abstract

We develop monotone comparative statics for constrained quantity choices generated by convex costs satisfying dual submodularity, meaning that their Legendre-Fenchel conjugates are submodular. We introduce the Q-order, an exchange order for possibly set-valued quantity choices, and prove a dual counterpart of Topkis's theorem. A closed proper convex function is exchangeable in the Q-order if and only if its conjugate is submodular. Exchangeability implies Q-monotonicity of rejected quantities under price and capacity changes; when the conjugate has a sublattice domain, the comparative-statics property is also a characterization. We apply the result to two-sided matching with divisible masses and welfare-generated, possibly set-valued choices. An aggregate deferred-acceptance algorithm is well defined; the availability sequence and the least waiting-time multiplier sequences converge; and every proposal limit selected by the theorem yields an aggregate equilibrium matching. The construction covers classical stable nontransferable-utility matching and aggregate random-utility specifications.

arXiv commentSubstantially revised version. 38 pages