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 basestock, reserve-price-auction policy to a traditional basestock, list-price policy. We prove that in the limiting case of one buyer per period and in the limiting case of a large number of buyers per period and linear holding cost, list pricing is optimal. List pricing also becomes optimal as the holding cost tends to zero. Numerical comparisons confirm these theoretical results and show that auctions provide significant benefits 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.