Competitive Markdown Timing for Perishable Products

We study markdown timing decisions for perishable products in the presence of competition. We model as a duopoly two firms selling their fixed stocks of two substitutable items in a given selling horizon. Each firm starts with a common initial price, and has the option to decrease the price once during the selling season. The prices, corresponding demand rates and inventory levels are public information. The problem for each firm is to determine when to decrease its price (markdown) in order to maximize its revenue. We showthat the existence and characterization of a pure-strategy Nash equilibrium depend critically on the demand rate that a firm enjoys as a monopoly when its competitor runs out of stock relative to three thresholds. When the monopoly demand rate is smaller than all three thresholds, we show that neither firm has the incentive to force its rival to run out of stock and that if a firm marks its price down after the season starts, it ensures that its inventory runs out precisely at the end of the season. When the monopoly demand rate is larger than the first threshold, waiting until its rival runs out of inventory may be an equilibrium strategy for the larger firm. When the monopoly demand rate is larger than one of the last two thresholds, we show that there may be no pure-strategy equilibrium in the game. In all possible equilibria, the larger firm marks its price down earlier than the smaller firm. We also show that some of the usual monotonicity results may fail to hold in the presence of competition. In particular, a firm's revenue may be decreasing in its starting inventory. We also show that the difference between markdown times can be clearly represented as a function of inventory imbalance and level of demand interaction. Although our results hold for any arbitrary demand function, we also interpret them in the context of linear and attraction demand models