  In a recent and ongoing work, Baldwin and Klemperer explored a connection
between tropical geometry and economics. They gave a sufficient condition for
the existence of competitive equilibrium in product-mix auctions of indivisible
goods. This result, which we call the Unimodularity Theorem, can also be traced
back to the work of Danilov, Koshevoy, and Murota in discrete convex analysis.
We give a new proof of the Unimodularity Theorem via the classical
unimodularity theorem in integer programming. We give a unified treatment of
these results via tropical geometry and formulate a new sufficient condition
for competitive equilibrium when there are only two types of product.
Generalizations of our theorem in higher dimensions are equivalent to various
forms of the Oda conjecture in algebraic geometry.
