The no‐upward‐crossing condition, comparative statics, and the moral‐hazard problem
定义了无上交叉条件，该条件被经济应用中常用的参数化分布族满足，并证明其等价于对数超模性，可用于简化道德风险问题中一阶方法的有效性条件。
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.