🤖 AI Summary
This study addresses the optimal transport problem between a continuum of agents and a finite set of discrete options under linear constraints, aiming to rectify a flaw in the existence proof of equilibria in large markets with indivisible goods. By developing a Monge–Kantorovich duality theory tailored to this setting and leveraging tools from functional analysis, convex analysis, and potential function optimization, the authors correct a prior erroneous claim regarding compactness. The main contributions are twofold: first, they rigorously reestablish the existence of market equilibria; second, they characterize equilibrium prices as minimizers of a dual potential function, thereby providing a computationally tractable method for equilibrium computation.
📝 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.