Optimal Auctioning and Ordering in an Infinite Horizon Inventory-Pricing System

We consider a joint inventory-pricing problem in which buyers act strategically and bid for units of a firm’s product over an infinite horizon. The number of bidders in each period, as well as the individual bidders’ valuations, are random but stationary over time. There is a holding cost for inventory and a unit cost for ordering more stock from an outside supplier. Backordering is not allowed. The firm must decide how to conduct its auctions and how to replenish its stock over time to maximize its profits. We show that the optimal auction and replenishment policy for this problem is quite simple, consisting of running a standard first-price or second-price auction with a fixed reserve price in each period and following an order-up-to (basestock) policy for replenishing inventory at the end of each period. Moreover, the optimal basestock level can be easily computed. We then compare this optimal auction mechanism and replenishment policy to a traditional fixed price, basestock policy. We prove that in the limiting case of one buyer per period and the limiting case of a large number of buyers per period, list pricing is optimal. List pricing is also optimal as the holding cost tends to zero. Numerical comparisons confirm these theoretical results and show that auctions only provide a significant benefit when: 1) the number of buyers is moderate, 2) holding costs are high or 3) there is high variability in the number of buyers per period. Indeed, one can argue that most retail and industrial trade corresponds to cases where our results suggest auctions are of minimal benefit. However, under the right conditions, an optimal auction and replenishment policy can provide significant improvements in profits over list pricing.