Bounds on the Welfare Loss of Moral Hazard with Limited Liability

This article studies a principal-agent problem with discrete outcome and effort level spaces. The principal and the agent are risk neutral and the latter is subject to limited liability. We consider the ratio between the first-best social welfare to the social welfare arising from the principal’s optimal pay-for-performance contract, i.e., the welfare loss. In the presence of moral hazard, we provide simple parametric bounds on the welfare loss of a given instance, and then study the worst-case welfare loss among all instances with a fixed number of effort and outcome levels. Key parameters to these bounds are the number of possible effort levels, the likelihood ratio evaluated at the highest outcome, and the ratio between costs of the highest and the lowest effort levels. As extensions, we also look at linear contracts and at cases with multiple identical tasks. Our work constitutes an initial effort to analyze losses arising from moral hazard problems when the agent is subject to limited liability, and shows that these losses can be costly in the worst case.