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...

Consider an agent who can costlessly add mean-preserving noise to his output. To deter such risk-taking, the principal optimally offers a contract that makes the agent's utility concave in output. If the agent is risk-neutral and protected by limited liability, this concavity constraint binds...

We provide a new class of counter-examples to existence in a simple moral hazard problem in which the first-order approach is valid. In contrast to the Mirrlees example, unbounded likelihood ratios on the signal technology are not central. Rather, our examples center around the behavior of the...

We study the moral hazard problem without the first-order approach or other common structure. We present sufficient conditions under which the shadow value of simultaneously tightening the minimum payment and individual rationality constraints has a simple and intuitive expression. We then show...