Existence of a Competitive Equilibrium with Substitutes, with Applications to Matching and Discrete Choice Models

TL;DR

This paper establishes new conditions for the existence and uniqueness of competitive equilibrium in models with substitutes, applicable to matching and discrete choice models, and provides an algorithm for computation.

Contribution

It introduces three properties ensuring equilibrium existence and uniqueness, extending prior results and applying to important classes of matching and choice models.

Findings

Proves existence of equilibrium under weak, pivotal, and responsive substitutes.

Reformulates matching and discrete choice models as competitive systems.

Provides an algorithm to compute the unique equilibrium.

Abstract

We propose new results for the existence and uniqueness of a general nonparametric and nonseparable competitive equilibrium with substitutes. These results ensure the invertibility of a general competitive system. The existing literature has focused on the uniqueness of a competitive equilibrium assuming that existence holds. We introduce three properties that our supply system must satisfy: weak substitutes, pivotal substitutes, and responsiveness. These properties are sufficient to ensure the existence of an equilibrium, thus providing the existence counterpart to Berry, Gandhi, and Haile (2013)'s uniqueness results. For two important classes of models, bipartite matching models with full assignment and discrete choice models, we show that both models can be reformulated as a competitive system such that our existence and uniqueness results can be readily applied. We also provide an algorithm to compute the unique competitive equilibrium. Furthermore, we argue that our results are particularly useful for studying imperfectly transferable utility matching models with full assignment and non-additive random utility models.