This dissertation is comprised of 3 papers. Here are the abstracts for the 3 papers. Welfare Properties of the Taiwan Mechanism: This paper analyzes the welfare properties of the mechanism Taiwan uses to match students to high schools. I study the Taiwan mechanism under a theoretical model where students share the same ordinal rankings, and a single lottery determines schools’ preferences. In this setting, all Taiwan mechanisms ex ante weakly Pareto dominate deferred acceptance for strategizing students. Additionally, in my model, there always exists a Taiwan mechanism that provides greater welfare than deferred acceptance to students who do not strategize. Using Cambridge Public School data, I demonstrate that the Taiwan mechanism outperforms deferred acceptance for a range of deduction rules. Broad Validity of the First-Order Approach in Moral Hazard: The first-order approach (FOA) is the main tool in the study of the pure moral hazard principal-agent problem. Although many existing results rely on the FOA, its validity has been established only under relatively restrictive assumptions. We contribute three main findings. First, we demonstrate in a broad array of examples that the FOA frequently fails when the agent’s reservation utility is low (such as in principal-optimal contracts). However, the FOA holds when the agent’s reservation utility is at least moderately high (such as in competitive settings where agents receive high rents). Second, our main theorem formally shows that the FOA is valid in a standard limited liability model when the agent’s reservation utility is sufficiently high. The theorem also establishes existence and uniqueness of the optimal contract. Third, we use the FOA to derive tractable optimal contracts across a broad array of settings. These contracts are both simple and intuitive, and under log utility, they are piecewise linear for numerous common output distributions (including Gaussian, exponential, binomial, Gamma, and Laplace). Heterogeneous value-added for priority design in New York City high schools: School districts that use centralized matching markets can design the schools' admissions criteria to prioritize students who benefit the most from attending the school. I study heterogeneous value-added in New York City high schools to determine which students have high value-added from attending which types of schools. Students with high middle school standardized test scores benefit the most from attending academically competitive high schools. Weaker students, as well as students with disabilities and English language learners, perform better in less competitive high schools. My results suggest that NYC could increase aggregate value-added by adopting a less coarse academic priority structure at the top high schools, so that more students with high heterogeneous value-added can attend those schools.