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 Fenchel dual whose subdifferential is another conservative monotone correspondence. We use these results to reinterpret and extend Baldwin and Klemperer's (2019) characterization of demand in economies with indivisible goods. We introduce a simple test for existence of Walrasian equilibrium in quasi-linear economies. Fenchel's Duality Theorem implies this test is met when the aggregate utility is concave, which is not necessarily the case with indivisible goods even if all consumers have concave utilities.