Existence and non-existence in the moral hazard problem

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 utility function as utility diverges to negative infinity. For any utility function, such as ln(w), in which utility diverges to negative infinity at a finite wealth level, existence will fail for some specifications of the agent's cost of effort. When utility diverges to negative infinity only as wealth does as well, existence holds for all specifications of the agent's cost of effort if and only if the agent continues to dislike risk as wealth diverges to negative infinity. When there is a finite lower bound on utility, existence is assured. For those cases where existence fails, we characterize the limit of near optimal contracts.