Constrained optimal transport with an application to large markets with indivisible goods

TL;DR

This paper develops a duality theory for constrained optimal transport problems with applications to large markets with indivisible goods, correcting previous proofs and providing a new way to compute equilibrium prices.

Contribution

It introduces a novel duality framework for constrained optimal transport with continuum agents and applies it to establish equilibrium existence and computation in large markets.

Findings

Established a Monge--Kantorovich duality variant for constrained transport.

Corrected a flaw in the equilibrium existence proof for large markets.

Provided a method to compute equilibrium prices via potential minimization.

Abstract

We establish a variant of Monge--Kantorovich duality for a constrained optimal transport problem with a continuum of agents, a finite set of alternatives, and general linear constraints. As an application, we revisit the large-market model of indivisible goods in Azevedo et al. (2013), identify a flaw in the original equilibrium-existence proof stemming from an incorrect compactness claim, and recover equilibrium existence via our duality approach. We also characterize equilibrium prices as minimizers of a potential function, which yields a method for computing equilibrium prices.