Technical Note—A Stationary Infinite-Horizon Supply Contract Under Asymmetric Inventory Information

We consider a decentralized supply chain in which a supplier sells goods to a retailer facing general random demand over an infinite horizon. The retailer satisfies the demand to the extent of the inventory on hand. The retailer has private information about the retailer’s stock in each period, and the supplier offers the retailer a supply contract menu to account for the information asymmetry. We obtain a necessary condition for optimizing a long-term stationary truth-telling contract under general demand and belief distributions. We apply it to a batch-order contract, which replenishes a prespecified inventory quantity for a fixed payment in each period only when the retailer’s beginning inventory becomes zero. Methodologically, we formulate the supplier’s contract design as a calculus of variations problem and apply the concept of Gâteaux derivative to obtain these results. This methodology can potentially be applied to other dynamic contracting problems.

Funding: A. Bensoussan acknowledges support from the National Science Foundation [Grant NSF-DMS 220 47 95].

Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2020.0495.