Theoretical Economics, Volume 15, Number 2 (May 2020)

Theoretical Economics 15 (2020), 445–476

The no-upward-crossing condition, comparative statics, and the moral-hazard problem

Hector Chade, Jeroen M. Swinkels


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: Log-supermodularity, quasi-concavity, moral hazard, first-order approach

JEL classification: D81, D86

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