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.

中文翻译：

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非向上交叉条件、比较静力学和道德风险问题

我们定义并探索了无向上交叉 (NUC)，这是经济应用中常用的每个参数化分布族都满足的条件。在平滑假设下，NUC 等效于分布的导数相对于参数的负数的对数超模。它的特点是自然单调的比较静态，并且是在一系列决策问题中建立准凹性的核心。作为一个应用，我们重新审视道德风险问题的一阶方法。NUC 简化了一阶方法有效性的相关条件，并给出了经济解释。我们为指数族的一阶方法提供了充分条件的广泛分析。