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

Alfred Galichon, Yu-Wei Hsieh and Maxime Sylvestre

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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.

Date: 2023-04, Revised 2026-10
New Economics Papers: this item is included in nep-des
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