Salesforce Compensation and Two-Sided Ambiguity: Robust Moral Hazard with Moment Information

We analyze a salesforce principal-agent model where both the firm and sales agent have limited information on the effort-dependent demand distribution, creating two-sided ambiguity. Under the max-min decision criteria, the firm offers a contract to the agent who exerts unobservable effort to influence the demand distribution. We formulate the problem as a semi-infinite program and use the agent's shadow prices to construct the least expensive contract. Next, we use the least expensive contract to create a non-linear optimization model, which provides the firm's optimal robust contract. Due to the problem's complexity, we focus our attention on the class of distribution-free contracts. We show that using a distribution-free contract is a necessary condition for achieving the first-best outcome. Our analysis reveals that the index of dispersion determines whether the optimal distribution-free contract is linear or quadratic. Finally, we extend our model to incorporate quota-bonus contracts and inventory considerations. Overall, our results demonstrate that variance information plays a critical role in designing contracts under distributional ambiguity and provides justification for the application of quadratic contracts in practice.