Broad Validity of the First-Order Approach in Moral Hazard

TL;DR

This paper shows that the first-order approach (FOA) to the standard moral hazard problem with limited liability is broadly valid when the agent’s reservation utility is sufficiently high, even in settings with multiple local maxima. By solving a relaxed problem via Lagrangian methods and exploiting a canonical contract form, it proves existence and uniqueness of the optimal contract and derives a simple FOA-based characterization: the optimal wage is $w(y)=k\circ g\bigl(\mu+\lambda S(y|a_0)\bigr)$, where $S(y|a)=\partial_a \log f(y|a)$. Under log utility and distributions from the exponential family with linear sufficient statistics, optimal contracts are piecewise linear option contracts; the paper also provides a practical algorithm to compute contracts in both FOA-valid and FOA-invalid regimes and discusses when FOA may fail at low reservation utility. The results connect to and extend prior literature by emphasizing high reservation utility as the key condition for FOA validity and by delivering tractable closed-form forms and robust numerical methods for contract design in a broad class of distributions (Gaussian, Poisson, Gamma, etc.). The work has practical implications for contract design under moral hazard by showing that simple, tractable incentive schemes remain optimal in a wide range of realistic settings.

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 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.

Figures (8)

• Figure 1: Optimal contracts with Gaussian distribution and log utility.
• Figure 2: Optimal contracts with Poisson distribution and log utility.
• Figure 3: Optimal contracts with Gaussian distribution and CARA utility.
• Figure 4: Pareto frontier in the Gaussian-log utility example
• Figure 5: Optimal contracts with Gaussian distribution and log utility in cost minimization problem with intended action $a_0 = \$100,000$.

Theorems & Definitions (29)

• Theorem 4.1
• Corollary 4.1
• Definition 5.1
• Proposition 5.1
• Proposition 5.2
• proof : Proof of Theorem \ref{['thm:main']}
• Remark 1: Piecewise Linear Option Contracts
• Remark 2
• proof
• Lemma A.1