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

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.