Broad Validity of the First-Order Approach in Moral Hazard

We consider the standard moral hazard problem with limited liability. The first-order approach (FOA) is the main tool for its solution, but existing sufficient conditions for its validity are restrictive. Our main result shows that the FOA is broadly valid, as long as the agent's reservation utility is sufficiently high. In basic examples, the FOA is valid for almost any positive reservation wage. We establish existence and uniqueness of the optimal contract. We derive closed-form solutions with various functional forms. We show that optimal contracts are either linear or piecewise linear option contracts with log utility and output distributions in an exponential family with linear sufficient statistic (including Gaussian, exponential, binomial, geometric, and Gamma). We provide an algorithm for finding the optimal contracts both in the case where the FOA is valid and in the case where it is not at trivial computational cost.