The no-upward-crossing condition, comparative statics, and the moral-hazard problem - Hector Chade; Jeroen M. Swinkels (Theoretical Economics, 2020)
We define and explore the No-Upward-Crossing NUC, a condition satisfied by every parameterized family of distributions commonly used in economic applications. Under smoothness assumptions, NUC is equivalent to log-supermodularity of the negative of the derivative of the distribution with respect to the parameter. It is characterized by a natural monotone comparative static, and is central in establishing quasi-concavity in a family of decision problems. As an application, we revisit the first-order approach to the moral hazard problem. NUC simplifies the relevant conditions for the validity of the first-order approach and gives them an economic interpretation. We provide extensive analysis of sufficient conditions for the first-order approach for exponential families.
Keywords: D81; D86; 330; Log-supermodularity; quasi-concavity; moral hazard; first-order approach