A Fractiles Perspective to the Joint Price/Quantity Newsvendor Model

Pricing and quantity decisions are critical to many firms across different industries. We study the joint price/quantity newsvendor model where only a single quantity and price decision is made, such as a fashion or holiday product that cannot be replenished and where the price is advertised nationally and cannot be changed. Demand is uncertain and sensitive to price. We develop a method for easily finding the optimal price and quantity that applies to more general cases than the usual one in which uncertainty is either additive, multiplicative, or a combination of the two. We represent a quantity by its fractile of the probability distribution of demand for a given price. We use a standard approach to approximating a given distribution with a finite number of representative fractiles and assume that these fractile functions are piecewise linear functions of the price. We identify effects that are not usually seen in a joint price/quantity newsvendor model. For example, although the optimal quantity is a decreasing function of the unit cost, the optimal price can be nonmonotone in the unit cost and we shed insight into why. We illustrate that using a simplified structure of demand uncertainty can result in substantially lower profits.