Contractual design in agency problems with non-monotonic cost and correlated information

We model an agency relationship in which the agent's cost is non-monotonic with respect to type and the type is correlated with a public ex-post signal. The principal can use lotteries to exploit the type-signal correlation within the limit of the agent's liability. We establish conditions for first-best implementation, highlighting two effects on contractual design. First, the structure of the optimal lottery varies across types and, for each type, it depends on whether the cost is U shaped or reverse U shaped with respect to type. Second, as compared to the case of monotonic cost, the design of incentive compatible lotteries is easier when the cost is U shaped, more difficult when the cost is reverse U shaped. The root of the second effect is that incentives are non-monotonic either below or above some interior types. The two effects involve that non-monotonicity is unfavorable to the principal when the cost is reverse U shaped. This conclusion is at odds with the wisdom, concerning settings without correlated information, that non-monotonicity, which triggers countervailing incentives, enhances contracting.