An Implementation Relaxation Approach to Principal-Agent Problems

The classic first-order approach (FOA) relaxes the principal-agent problem by replacing the incentive compatibility (IC) constraint with its first-order condition. We show that FOA is not a valid relaxation when the support of the outcome distribution shifts with the agent's effort, as in well-studied additive-noise models. In such cases, the optimal effort may occur at a kink point that the first-order condition cannot capture, causing FOA to miss optimal contracts, including the widely adopted bonus schemes. Motivated by this limitation, we introduce the Implementation Relaxation Approach (IRA), which accommodates nondifferentiable optima and is straightforward to apply across settings. Rather than directly relaxing IC, IRA relaxes the set of implementable agent efforts and utilities, reducing the problem to identifying the effort-utility pair from which the optimal contract can be constructed. This inverse perspective is particularly convenient for analyzing simple contracts. Using IRA, we derive an optimality condition for quota-bonus contracts that is more general than FOA-based conditions, including those established in the literature under fixed-support assumptions. This also fills a gap where the optimality of quota-bonus contracts in shifting-support settings has been examined only under endogenous assumptions, and it highlights the broader applicability of IRA as a methodological tool.