Optimal dynamic pricing of perishable products with stochastic demand and a finite set of prices

Consider a continuous-time inventory problem in which a retailer sets the price on a fixed number of a perishable asset that must be sold prior to the time at which it perishes. The retailer can dynamically adjust the price between any of a finite number of allowable prices. Demand for the product is Poisson with an intensity that is inversely related to the price. The optimal policy is piecewise-constant. The maximum expected revenue is nondecreasing and concave in both the remaining inventory and the time-to-go. For a given inventory level the optimal price declines as the time at which the products perish approaches. At any given time the optimal price is nonincreasing in the number of items remaining unsold. These results are extended to (i) the case in which the prices and corresponding demand intensities depend on the time-to-go; and (ii) the case in which the retailer can restock to meet demand at a unit cost after the initial inventory has been sold.