Relational contracts and the first-order approach

This paper justifies the first-order approach (FOA) to relational contract models. Optimal relational contracts pay a bonus if an agent passes an evaluation, where the cutoff point is independent of signal distribution or the agent's cost function. Based on this independence, I find a weak FOA-justifying condition, which requires convexity of the underlying distribution-cost structure only at the cutoff point. Prominent examples (e.g., the normal or generalized error distribution with various cost functions) are consistent with this condition, but are inconsistent with existing conditions such as the Mirrlees-Rogerson condition.