Dynamic Pricing of Limited Inventories When Customers Negotiate

Although take-it-or-leave-it pricing is the main mode of operation for many retailers, a number of retailers discreetly allow price negotiation when some haggle-prone customers ask for a bargain. At these retailers, the posted price, which itself is subject to dynamic adjustments in response to the pace of sales during the selling season, serves two important roles: (i) it is the take-it-or-leave-it price to many customers who do not bargain, and (ii) it is the price from which haggle-prone customers negotiate down. In order to effectively measure the benefit of dynamic pricing and negotiation in such a retail environment, one must take into account the interactions among inventory, dynamic pricing, and negotiation. The outcome of the negotiation (and the final price a customer pays) depends on the inventory level, the remaining selling season, the retailer's bargaining power, and the posted price. We model the retailer's dynamic pricing problem as a dynamic program, where the revenues from both negotiation and posted pricing are embedded in each period. We characterize the optimal posted price and the resulting negotiation outcome as a function of inventory and time. We also show that negotiation is an effective tool to achieve price discrimination, particularly when the inventory level is high and/or the remaining selling season is short even when implementing negotiation is costly.