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Substitutability, equilibrium transport, and matching models

Alfred Galichon, Antoine Jacquet

TL;DR

The chapter investigates how substitutability shapes equilibrium computation in economic models, highlighting that coordinate-update methods like Jacobi's algorithm and Sinkhorn's algorithm converge under Z- and M-function structures. It develops transferable-utility matching as a fixed-point problem using a distance-to-frontier transform to eliminate transfers, enabling Jacobi-based solutions with convergence guarantees. For non-transferable utility, it ties stable matching theory to monotone operator frameworks via Adachi's formulation and DALM, establishing fixed-point characterizations and convergence results for equilibrium matchings. The text offers both theoretical convergence results and practical algorithms (Sinkhorn, Gale–Shapley, DALM, Adachi) and illustrates them with applications to expenses, housingMarket rent control, and dynamic ride-hailing pricing. Collectively, it provides a unified, mathematically rigorous toolkit for analyzing substitutability-driven matching under both transferable and non-transferable utilities.

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.

Substitutability, equilibrium transport, and matching models

TL;DR

The chapter investigates how substitutability shapes equilibrium computation in economic models, highlighting that coordinate-update methods like Jacobi's algorithm and Sinkhorn's algorithm converge under Z- and M-function structures. It develops transferable-utility matching as a fixed-point problem using a distance-to-frontier transform to eliminate transfers, enabling Jacobi-based solutions with convergence guarantees. For non-transferable utility, it ties stable matching theory to monotone operator frameworks via Adachi's formulation and DALM, establishing fixed-point characterizations and convergence results for equilibrium matchings. The text offers both theoretical convergence results and practical algorithms (Sinkhorn, Gale–Shapley, DALM, Adachi) and illustrates them with applications to expenses, housingMarket rent control, and dynamic ride-hailing pricing. Collectively, it provides a unified, mathematically rigorous toolkit for analyzing substitutability-driven matching under both transferable and non-transferable utilities.

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.
Paper Structure (20 sections, 17 theorems, 88 equations)

This paper contains 20 sections, 17 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. M-functions and Jacobi's algorithm
  3. Jacobi's algorithm
  4. Nondecreasing Jacobi sequence
  5. M- and M0-functions
  6. Convergence of the Jacobi sequence
  7. Application: a toy hedonic model
  8. References and notes
  9. Models of matching with transfers
  10. Microfoundation of the matching model
  11. Distance-to-frontier function
  12. Matching equilibrium
  13. Full assignment case
  14. Models of matching without transfers
  15. Introduction
  16. ...and 5 more sections

Key Result

Proposition 2.1

Assume $Q$ is diagonal isotone. If $p^t$ is a subsolution, then $p^t \leq p^{t+1}$.

Theorems & Definitions (41)

  • Definition 2.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Example 2.1
  • Definition 2.2
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • ...and 31 more