Revisiting the First-Order Approach to Principal-Agent Problems

Abstract

Principal-agent problems are typically solved by the first-order approach (FOA) that replaces the incentive-compatibility constraint in the original problem with its first-order condition, leading to a relaxed problem. The most celebrated condition that validates FOA is the Mirrlees-Rogerson Condition, developed for a risk-averse agent. We show that it fails to generalize to a widely studied setting of risk-neutral players and limited liability. Instead of justifying FOA, we propose a less stringent notion that only requires FOA to be valid for quota-bonus contracts (FOAVQB). This proposal is rationalized by our finding that quota-bonus contracts are optimal for the relaxed problem. We identify exogenous sufficient conditions that justify FOAVQB, thus ensuring the optimality of quota-bonus contracts for the original problem. These sufficient conditions are economically interpretable, reasonably easy to verify, and flexible enough to accommodate common instances where the agent’s expected utility is non-unimodal in effort under optimal contracts (besides those with unimodal utility functions), contrasting the literature that typically justifies FOA by requiring the agent’s utility function to be not only unimodal but also concave.

Keywords: Principal-agent problems, Moral hazard, Limited liability, Quota-bonus contracts, First-order approach