My dissertation consists of two papers covering distinct topics within Microeconomic Theory. The first chapter is drawn from Matching Theory. One of the oldest but least understood matching problems is Gale and Shapley's (1962) "roommates problem": is there a stable way to assign 2N students into N roommate pairs? Unlike the classic marriage problem or college admissions problem, there need not exist a stable solution to the roommates problem. However, the traditional notion of stability ignores the key physical constraint that roommates require a room, and it is therefore too restrictive. Recognition of the scarcity of rooms motivates replacing stability with Pareto optimality as the relevant solution concept. This paper proves that a Pareto optimal assignment always exists in the roommates problem, and it provides an efficient algorithm for finding a Pareto improvement starting from any status quo. In this way, the paper reframes a classic matching problem, which previously had no general solution, to become both solvable and economically more meaningful.

The second chapter focuses on the role networks play in market and social organization. In network theory, externalities play a critical role in determining which networks are optimal. Adding links can create positive externalities, as they potentially make distant vertices closer. On the other hand, links can result in negative externalities if they increase congestion or add competition. This paper will completely characterize the set of optimal and equilibrium networks for a natural class of negative externalities models where an agent's payoff is a function of the degree of her neighbors. These results are in sharp contrast to the optimal and equilibrium networks for the standard class of positive externalities models where payoff is a function of the distance two agents are apart. This highlights the role externalities play in optimal and equilibrium network structure.