Theoretical Economics 20 (2025), 815–829

We establish the Subgradient Theorem for monotone correspondences -- a monotone correspondence is equal to the subdifferential of a potential if and only if it is conservative, i.e. its integral along a closed path vanishes irrespective of the selection from the correspondence along the path. We prove two attendant results: the Potential Theorem, whereby a conservative monotone correspondence can be integrated up to a potential, and the Duality Theorem, whereby the potential has a dual whose subdifferential is a conservative monotone correspondence that is the inverse of the original correspondence. We use these results to reinterpret and extend Baldwin and Klemperer’s (2019) characterization of demand in economies with indivisible goods.

Keywords: Conservative correspondences, subgradient theorem, potential theorem, Fenchel duality, Fenchel's duality theorem, tropical geometry, convex analysis, normally labeled polyhedral subdivisions, subdifferentials, indivisible goods

JEL classification: C65