Open AccessThe no‐upward‐crossing condition, comparative statics, and the moral‐hazard problem
Author(s) -
Chade Hector,
Swinkels Jeroen
Publication year - 2020
Publication title -
theoretical economics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 4.404
H-Index - 32
eISSN - 1555-7561
pISSN - 1933-6837
DOI - 10.3982/te2937
Subject(s) - comparative statics , parameterized complexity , monotone polygon , smoothness , moral hazard , mathematical economics , order (exchange) , exponential function , computer science , derivative (finance) , mathematics , mathematical optimization , economics , mathematical analysis , microeconomics , algorithm , geometry , finance , incentive , financial economics
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.