Broad Validity of the First-Order Approach in Moral Hazard

2025-06-23

AI Summary

This paper investigates the validity boundary of the first-order approach (FOA) in moral-hazard principal–agent models. While conventional FOA requires stringent assumptions—such as vanishingly low agent reservation utility—it often fails under realistic settings (e.g., competitive markets) where reservation utility is non-negligible. We establish, theoretically, that under standard limited-liability constraints, FOA holds universally whenever the agent’s reservation utility exceeds a well-defined threshold, ensuring existence and uniqueness of the optimal contract. This result relaxes prior restrictive requirements on output distribution families, agent utility specifications, and incentive strength. Furthermore, for logarithmic utility, we derive closed-form optimal option-type contracts under Gaussian, exponential, and gamma output distributions—substantially enhancing the operationality and empirical applicability of FOA.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Reasoning under Uncertainty: Decision/Utility Theory

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Responsible Web: Machine-in-the-loop, human agency and autonomy

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).

Problem

• Validates first-order approach in moral hazard problems
• Examines FOA failure with low agent reservation utility
• Proves FOA validity under high agent reservation utility

Innovation

• Validates FOA for high reservation utility
• Proves existence and uniqueness of contracts
• Derives optimal contracts for log utility