A New Class of Non–Existence Examples for the Moral Hazard Problem∗

We provide a class of counter-examples to existence in a simple moral hazard problem in which the first-order approach is valid. These examples involve utility functions such as ln(w) in which utility diverges to negative infinity at a finite wealth level. In contrast to the Mirrlees example, unbounded likelihood ratios are not needed, and no sequence of contract can approach the full information first best. The examples expose a hidden (and not always correct) assumption in the existence proof of Jewitt, Kadan, and Swinkels (2008). We show that their result (and proof method) is valid under very mild conditions outside of this class of utility functions. In particular, if utility only goes to negative infinity when wealth does as well, then any utility function where the agent continues to dislike risk as wealth diverges to negative infinity will suffice.