Broad Validity of the First-Order Approach in Moral Hazard

Eduardo Azevedo; Ilan Wolff

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arXiv · 2506.18873

Broad Validity of the First-Order Approach in Moral Hazard

Eduardo Azevedo·Ilan Wolff

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Abstract

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 a linear sufficient statistic (including Gaussian, exponential, binomial, geometric, and gamma distributions). We provide an algorithm that finds the optimal contracts at negligible computational cost, whether or not the FOA is valid.

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Eduardo Azevedo, Ilan Wolff. 2026-09-14. Broad Validity of the First-Order Approach in Moral Hazard. https://arxiv.org/abs/2506.18873

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