Quantity Premiums and Discounts in Dynamic Pricing

Abstract: We consider a dynamic pricing problem for a monopolistic company selling a perishable product when customer demand is uncertain and occurs in batches which must be fulfilled as a whole. The seller can price-discriminate between the batches of different size by setting different unit prices. The problem is modeled as a stochastic optimal control problem to find an inventory-contingent dynamic pricing policy to maximize the expected total revenues. We find the optimal pricing policy and prove several monotonicity results. First, we establish stochastic order conditions on the unit willingness-to-pay distributions that determine when quantity discounts or premiums take place as a batch purchase is broken up into a rapid sequence of smaller purchases. Second, we give a sufficient condition for prices to be monotonically decreasing in inventory when the maximum batch size is two. Third, we establish sufficient stochastic order and value function convexity conditions for the perceived quantity discounts and premiums which result from comparing unit prices for different batch sizes under a particular inventory level. Numerical experiments reveal further details of interactions between stochastic order and convexity conditions, and demonstrate that a company can generate significant additional revenues though explicit dynamic non-linear pricing.