Optimal contracts with a risk-taking agent

Consider an agent who can costlessly add mean-preserving noise to his output. To deter such risk-taking, the principal optimally offers a contract that makes the agent's utility concave in output. If the agent is risk-neutral and protected by limited liability, this concavity constraint binds and so linear contracts maximize profit. If the agent is risk averse, the concavity constraint might bind for some outputs but not others. We characterize the unique profit-maximizing contract and show how deterring risk-taking affects the insurance-incentive trade-off. Our logic extends to costly risk-taking and to dynamic settings where the agent can shift output over time.