Consider a firm that sells products over repeated seasons, each of which includes a full-price period and a markdown period. The firm may deliberately understock products in the markdown period to induce high-value customers to purchase early at full price. Customers cannot perfectly anticipate availability. Instead, they use observed past capacities to form capacity expectations according to a heuristic smoothing rule. Based on their expectations of capacity, customers decide to buy either in the full-price period or in the markdown period. We embed this customer learning process in a dynamic program of the firm's capacity choices over time. One main result demonstrates the existence of a monotone optimal path of customers' expectations, which converges to either a rationing equilibrium or a low-price-only equilibrium. Further, there exists a critical value of capacity expectation such that the market converges to a rationing equilibrium if customers' initial expectations are less than that critical value; otherwise, a low-price-only equilibrium is the limiting outcome. These results show how firms can be stuck with unprofitable selling strategies from incumbent customer expectations. We also examine numerically how this critical value is affected by the firm's discount factor and customers' learning speed and risk aversion. Last, we show that the equilibrium under adaptive learning converges to that under rational expectations as the firm's discount factor approaches one.
van Ryzin, Garrett, and Qian Liu. "Strategic Capacity Rationing When Customers Learn." Manufacturing & Service Operations Management 13, no. 1 (Winter 2011): 89-107.
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