In the classical principal-agent problem, a principal hires an agent to perform a task. The principal cares about the task's output but has no control over it. The agent can perform the task at different effort intensities and that choice affects the task's output. To provide an incentive to the agent to work hard and since his effort intensity cannot be observed, the principal ties the agent's compensation to the task's output. If both the principal and the agent are risk-neutral and no further constraints are imposed, it is well-known that the outcome of the game maximizes the social welfare.

In this talk, we quantify the potential social-welfare loss due to the existence of limited liability, which takes the form of a minimum wage constraint. To do so we rely on the concept of worst-case welfare loss (also known as the Price of Anarchy), which quantifies the efficiency

of a system when its players act selfishly versus choosing a socially-optimal solution. In our setting the worst-case welfare loss is defined as the largest possible ratio between the social welfare when the agent chooses the effort that is optimal for the system and that of the sub-game perfect equilibrium of the game. Our main result establishes that under the monotone likelihood-ratio property and limited liability constraints, the worst-case welfare loss in the principal-agent model is exactly equal to the number of efforts available. This suggests that the inefficiency introduced by the principal-agent relationship depends on the discretion the agent has in choosing how to perform the delegated task.

This is joint work with F. Balmaceda, S.R. Balseiro, and J.R. Correa.