Note on solving one-to-one matching models with linear transferable utility

TL;DR

This paper develops a fixed-point approach to solve one-to-one matching models with transferable utility, proving convergence under certain conditions, which ensures reliable computation of equilibrium transfers.

Contribution

It introduces a contraction mapping framework for equilibrium transfers in matching models with linear transferable utility, guaranteeing convergence of fixed-point iterations.

Findings

Derived a system of fixed-point equations for equilibrium transfers.

Proved that the system is a contraction under bounded substitution.

Fixed-point iterations converge to a unique equilibrium transfer distribution.

Abstract

We derive a system of fixed-point equations for the equilibrium transfers in a class of one-to-one matching models with linear transferable utility. We then show that, when the degree of substitution between alternatives is bounded from above, the derived system of equations constitutes a contraction mapping. As a result, fixed-point iterations are guaranteed to converge to the unique distribution of equilibrium transfers.