Tropical Analysis: With an Application to Indivisible Goods

TL;DR

This paper develops a theoretical framework using tropical analysis to characterize demand and equilibrium conditions in economies with indivisible goods, extending classical duality results to non-concave settings.

Contribution

It establishes the Subgradient, Potential, and Duality Theorems for monotone correspondences, extending demand analysis to non-concave utility scenarios with indivisible goods.

Findings

A new test for the existence of Walrasian equilibrium in quasi-linear economies.

Extension of Fenchel's Duality to non-concave aggregate utility.

Reinterpretation of demand characterization in economies with indivisible goods.

Abstract

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.