Models of matching with contracts can elegantly capture the discreteness and heterogeneity of interactions that arise in many real-world markets.However, the indivisibility of contracts makes the existence of equilibrium in matching models a subtle issue.This dissertation develops new results on the existence of equilibrium in matching models, as well as other settings with indivisible goods—with applications to the design of matching markets and auctions.Chapter 1, which is joint work with Tamás Fleiner, Zsuzsanna Jankó, and Alexander Teytelboym, shows how frictions affect equilibria in a model of matching in trading networks.It proves that, when contracts are fully substitutable for firms, competitive equilibria exist and coincide with outcomes that satisfy a cooperative solution concept called trail stability.However, competitive equilibria are generally neither stable nor Pareto-efficient.Chapter 2, which is joint work with Elizabeth Baldwin, Omer Edhan, Paul Klemperer, and Alexander Teytelboym, addresses the issue of when competitive equilibria exist in markets for indivisible goods with income effects.It shows that the existence of equilibrium fundamentally depends on agents' substitution effects, not their income effects.One consequence is that net substitutability—which is a strictly weaker condition than gross substitutability—is sufficient for the existence of equilibria.Chapter 3, which is joint work with Karolina Vocke, addresses the issue of the choice of solution concept in matching markets by considering large markets.In the context of complex, finite markets, it is generally unclear what solution concept to use, as no general microfoundations have been given for any of the solution concepts proposed in the matching literature, and most of the proposed solution concepts suffer from existence issues.In contrast, Chapter 3 shows that, in the context of large markets, a solution concept called tree stability has a microfoundation and tree stable outcomes are guaranteed to exist for arbitrary preferences and network structures.Chapter 4 simplifies and clarifies the theoretical analysis of the cadet-branch matching problem.Sönmez (2013) and Sönmez and Switzer (2013) used matching theory with unilaterally substitutable priorities to propose mechanisms to match cadets to military branches.Chapter 4 shows that alternatively, the Sönmez and Sönmez—Switzer mechanisms can be constructed as descending salary adjustment processes in Kelso—Crawford (1982) economies in which cadets are substitutable.
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