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

Hector Chade*, Jeroen Swinkels

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


We define and explore 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.

Original languageEnglish (US)
Pages (from-to)445-476
Number of pages32
JournalTheoretical Economics
Issue number2
StatePublished - May 1 2020


  • D81
  • D86
  • Log-supermodularity
  • first-order approach
  • moral hazard
  • quasi-concavity

ASJC Scopus subject areas

  • Economics, Econometrics and Finance(all)


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