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Optimal contract under double moral hazard and limited liability

This paper investigates optimal contracts between risk-neutral parties when both exert efforts and the agent faces limited liability. We identify a sufficient and necessary condition for any contract to implement the second-best outcome, i.e., the best possible outcome in double moral hazard even when the agent faces unlimited liability. It is shown that a simple share-or-nothing with bonus contract (SonBo for short) is optimal and implements the second-best outcome when the condition holds. SonBo contracts have one degree of freedom, which is very useful in dealing with heterogeneous circumstances while still maintaining consistency in contracting. SonBo admits as special cases the option-like and step bonus contracts, which are widely used in dealing with limited liability. Nevertheless, we demonstrate that a step bonus contract is more powerful because an option-like contract can be problematic in some situations. The paper also discusses the performance of SonBo when the principal also faces liability constraint and investigates the optimal contract when the second-best outcome is not achievable.