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.

💡 Research Summary

The paper revisits the classic principal‑agent problem with moral hazard under limited‑liability constraints and asks a simple yet profound question: when can the first‑order approach (FOA), which replaces the global incentive‑compatibility (GIC) condition with the local first‑order condition (LIC), be relied upon? Existing literature (Rogerson 1985; Jewitt 1988; subsequent works) provides sufficient conditions that are often very restrictive—requiring strong concavity of the payment schedule, monotone likelihood‑ratio property (MLRP) together with additional curvature assumptions on the utility or cost functions. As a result, many applied papers either assume FOA without justification or restrict contracts to linear or binary‑effort forms.

The authors’ main contribution, Theorem 1, shows that a much weaker condition suffices: if the agent’s reservation utility is sufficiently high, the FOA is valid regardless of the shape of the contract, even when the agent’s problem exhibits multiple local maxima. The key intuition is that a higher reservation utility forces the principal to offer contracts that give the agent a relatively high payoff at the intended action. This raises the right‑most local maximum of the agent’s expected utility faster than any left‑hand side maximum, eventually making the intended action the unique global optimum. In technical terms, the “kink” of the relaxed optimal contract moves leftward as the reservation utility rises, which renders the agent’s payoff function more concave and eliminates profitable non‑local deviations.

The model assumes: (i) a risk‑neutral principal; (ii) a strictly concave, smooth utility u(·) with unbounded marginal utility at zero and zero marginal utility at infinity; (iii) a strictly convex, smooth cost c(a) with c′(0)=0; (iv) an output density f(y|a) that is smooth in a and satisfies MLRP—i.e., the score ∂ₐ log f(y|a) is strictly increasing in y and its range covers ℝ. The limited‑liability constraint is imposed by requiring the wage w(y)≥0, which is equivalently expressed through the inverse marginal utility function k = u⁻¹ and a truncation function g that sets wages to zero below a certain threshold.

Under these assumptions, the authors prove existence and uniqueness of the optimal contract by showing that the cost‑minimization problem is a continuous convex program over a compact feasible set. The optimal wage takes the closed‑form representation

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