Substitutability, equilibrium transport, and matching models

TL;DR

This chapter analyzes how substitutability influences equilibrium transport and matching models, emphasizing mathematical properties, convergence of algorithms, and applications in economic theory and computation.

Contribution

It provides a detailed mathematical framework for substitutability in matching and transport models, including convergence analysis and computational methods.

Findings

Jacobi's algorithm converges under certain substitutability conditions

Theoretical insights into matching models with transferable and non-transferable utility

Practical applications of computational methods like Sinkhorn and Gale--Shapley

Abstract

This chapter explores the role of substitutability in economic models, particularly in the context of optimal transport and matching models. In equilibrium models with substitutability, market-clearing prices can often be recovered using coordinate update methods such as Jacobi's algorithm. We provide a detailed mathematical analysis of models with substitutability through the lens of Z- and M-functions, in particular regarding their role in ensuring the convergence of Jacobi's algorithm. The chapter proceeds by studying matching models using substitutability, first focusing on models with (imperfectly) transferable utility, and then on models with non-transferable utility. In both cases, the text reviews theoretical implications as well as computational approaches (Sinkhorn, Gale--Shapley), and highlights a practical economic application.