A Stationary Infinite-Horizon Supply Contract under Asymmetric Inventory Information

We consider a decentralized supply chain where a supplier sells goods to a retailer facing a 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 his inventory in each period and the supplier offers him a supply contract menu to account for the information asymmetry. We develop a novel framework of a long-term stationary truth-telling contract that requires the supplier to form a belief only about the retailer's first-period inventory and still induces the retailer to order based on his true inventory level in each period. We obtain a necessary condition for the optimality of such a contract. We specialize the necessary condition for a batch-order contract, which replenishes a pre-specified inventory quantity for a fixed payment in each period only when the retailer's beginning inventory becomes zero in that period. The supplier's contract design problem is to maximize the payoff functional of the replenishment-quantity function of the retailer's current inventory. Meanwhile, the retailer's incentive compatibility constraint is expressed as a functional Bellman equation. Mathematically, the problem belongs to the calculus of variations and requires us to use a powerful methodology that replaces the usual concept of the gradient with that of the classical Gâteaux derivative. This methodology can potentially be applied to other dynamic contracting problems.REVISED Dec. 28, 2021