Robust Contract Designs: Linear Contracts and Moral Hazard

We consider incentive compensation where the firm has ambiguity on the effort-contingent output distribution: The parameters of the output probability distribution are in an ellipsoidal uncertainty set. The firm evaluates any contract by its worst-case performance over all possible parameters in the uncertainty set. Similarly, the incentive compatible condition for the agent must hold for all possible parameters in the uncertainty set. The firm is financially risk neutral and the agent has limited liability. We find that when the agent is financially risk neutral, the optimal robust contract is a linear contract—paying the agent a base payment and a fixed share of the output. Moreover, the linear contract is the only type of contract that is robust to the parameter uncertainty. When there is model uncertainty over a general effort-contingent output distribution, we show that a generalized linear contract is uniquely optimal. When the agent is risk averse and has a piecewise linear utility, the only optimal contract is a piecewise linear contract that consists of progressive fixed payments and linear rewards with progressive commission rates. We also provide the analysis for the trade-off between robustness and worst-case performance and show that our results are robust to a variety of settings, including cases with general ℓ^p -norm uncertainty sets, multiple effort levels, and so on. Our paper provides a new explanation for the popularity of linear contracts and piecewise linear contracts in practice and introduces a flexible modeling approach for robust contract designs with model uncertainty.