Optimal dynamic pricing problem considering patient and impatient customers' purchasing behaviour

We divide customers into two types according to their purchasing behaviour: one is impatient and the other is patient. An impatient customer will be immediately lost if his reservation price of the product is below the current price, while a patient customer will wait until the price drops below his reservation price and buy one item. The seller's objective is to maximise his expected discounted overall revenue by dynamically pricing. We prove that the optimal pricing policy has a control limit structure, examine the impact of the proportion parameter on the seller's expected discounted overall revenue and show that the limiting behaviour of the optimal pricing policy when there are little patient customers is the same as the optimal static pricing policy. We also show that the seller will make more profit if he partially knows the number of waiting customers by providing subscription service to customers. A set of numerical results illustrates the value of dynamic pricing and information with respect to problem parameters such as the arriving probability and the subscription probability.