Broad Validity of the First-Order Approach in Moral Hazard

Abstract: The first-order approach (FOA) is the main tool for the moral hazard principal-agent problem. Although many existing results rely on the FOA, its validity has been established only under relatively restrictive assumptions. We demonstrate in examples that the FOA frequently fails when the agent's reservation utility is low (such as in principal-optimal contracts). However, the FOA broadly holds when the agent's reservation utility is at least moderately high (such as in competitive settings where agents receive high rents). Our main theorem formalizes this point. The theorem 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. We use the theorem to derive tractable optimal contracts across several settings. Under log utility, option contracts are optimal for numerous common output distributions (including Gaussian, exponential, binomial, Gamma, and Laplace).