Customized Dynamic Pricing when Customers Develop a Habit or Satiation

We study a dynamic pricing problem in which a firm chooses prices over multiple periods when consumers are state dependent, i.e., they develop a habit or satiation from their past consumption. We first derive an inter-temporal demand function to capture how demand in one period depends on the price in that period and consumption in previous periods through habit or satiation. Subsequently, we formulate the optimal price setting problem for a firm over a multi-period horizon. We establish that this problem is jointly concave in prices and then characterize the temporal trends in the optimal prices. These trends in optimal prices are a net outcome of two opposite effects: (i) the progressive build up of habit or satiation from consumption, and (ii) the progressive deterioration of the habit or satiation developed in prior periods. Based on the relative strengths of these two effects, the optimal prices follow either a penetration policy (prices increase over time), or a skimming policy (prices decrease over time), or a skimming-penetration policy (U-shaped prices), or a penetration-skimming policy (Inverse-U-shaped prices). Subsequently, we provide several extensions including bounds on prices and optimal profit and non-stationary state dependence. Numerical studies show that ignoring habit and satiation effects in customers can be significantly costly for a firm.