Style Goods Pricing with Competition

We study timing decisions for price change of style goods in the existence of competition. We model as a duopoly two firms which are selling their fixed stocks of an item in a given period of time. Our model differs from existing single-firm inventory models in that we explicitly consider the demand interactions through pricing decisions by two competing firms. Each firm starts with the same initial price, and has the option to decrease the price once during the selling season. The demand for the product at one firm depends on the prices offered by both firms. The price levels and corresponding demand rates are known in advance. The problem for each firm is to decide when to decrease its price (mark-down) in order to maximize its revenue. We find the unique equilibrium switching times for both firms. We show that the firm with higher inventory will lower its price first but each firm will sell off its inventory at the end of the horizon. We also analyze the effect of demand interaction on price switching decisions. As the demand for one firm gets more dependent on the price offered by the other firm, the larger firm will tend to lower its price later, while the smaller firm will tend to lower its price earlier. Furthermore, with intense demand interaction, the revenue of the larger firm will increase, while the revenue of the smaller firm will decrease. Although our results hold for general symmetric demand functions, we interpret our results in the context of a linear demand model. We then extend our model to the mark-up problem in which either company has one opportunity to increase its price to a preset level. We apply the mark-up model to a discount fare allocation problem for airlines.