Bounds on the Welfare Loss from Moral Hazard with Limited Liability

Abstract

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 and the social welfare arising from the principal's optimal pay-for-performance contract, i.e., the welfare loss ratio. In the presence of moral hazard, we provide simple parametric bounds on the welfare loss ratio of a given instance of the problem, and then study the worst-case welfare loss ratio among all problem instances with a fixed number of effort and outcome levels. Key parameters for these bounds are the number of possible effort levels and the likelihood ratio evaluated at the highest outcome. As extensions, we also look at linear contracts and at cases with multiple identical tasks. Our work constitutes an initial attempt to quantify the losses arising from moral hazard problems when the agent is subject to limited liability, and shows that these losses are non-negligible in the worst case (e.g., the welfare loss can approach 100% when the number of possible effort levels is large).

Keywords: Contract Theory, Principal-Agent Problem, Price of Anarchy, First-Order Method