Tropical analysis: with an application to indivisible goods

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 BaldwinKlemperer2019's BaldwinKlemperer2019 characterization of demand in economies with indivisible goods.