An Implementation Relaxation Approach to Principal-Agent Problems

Abstract: The first-order approach (FOA) is the standard tool for solving principal-agent problems, replacing the incentive compatibility (IC) constraint with its first-order condition to obtain a relaxed problem. 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 widely adopted bonus schemes. Motivated by this limitation, we introduce the Implementation Relaxation Approach (IRA), which relaxes the set of agent actions and payoffs that feasible contracts can induce, rather than directly relaxing IC. IRA accommodates non-differentiable optima and is straightforward to apply across settings, particularly for deriving optimality conditions for simple contracts. Using IRA, we derive an optimality condition for quota-bonus contracts that is more general, encompassing a broader range of scenarios 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.