Dynamic Pricing for Vertically Differentiated Products Ren´ e Caldentey ∗ Ying Liu † Abstract This paper studies the seller’s optimal pricing policies for a family of substitute perishable products. The seller aims to maximize her expected cumulative revenues over a finite selling horizon. At each demand epoch, the arriving customer observes the set of substitute products with positive inventory together with their prices. Based on this information as well as the customer’s own budget constraint, he either buys one of the available products, or leaves the system without making any purchase. We propose a choice model where a fixed ranking of the products is decided by the quality-price combination. We show the monotonicity property of the optimal prices with respect to quality, inventory and time-to-go. We derive the distribution-free pricing methodology and obtain the robust bounds on the price increment through the first-order Taylor approximation. Our work also sheds light on the assortment design in terms of choosing the breadth of the product quality range as well as the number of products in the assortment. Keywords: Dynamic pricing, demand substitution, consumer choice model, budget constraint, approximations. 1 Introduction 1.1 Motivation In this paper, we study the firm’s dynamic pricing problems of differentiated but substitutable products. The firm aims to maximize her expected cumulative revenue over a finite selling season. The products could be differentiated in one or more attributes, but we aggregate these attributes into the “quality” index for the purpose of our analysis. The products are substitutable in the sense that customer can pick any product within the in-stock offering if the product has an appealing quality-price combination. Dynamic pricing is a valuable tool for products with short selling season and limited capacity. However, it is largely complicated by the noticeable substitution behavior by customers based on quality-price trade-offs. For example, in retailing sector, a clothing company many operate multiple brands that differ in design, material quality and fashion. How to make the promotional decisions for one brand during the seasonal or holiday sales depends on the brand’s own inventory level, how closed it is to the end of the selling season, as well as the substitution * Booth School of Business, University of Chicago, Chicago, IL 60637, Rene.Caldentey@chicagobooth.edu † Stern School of Business, New York University, New York, NY 10012, yliu2@stern.nyu.edu