Optimal Assortment and Pricing Decisions for Vertically Differentiated Products

Abstract

We consider a joint assortment, inventory, and pricing problem of a retailer selling multiple vertically differentiated products of a single category. Consumers are heterogeneous in their willingness to pay for product quality. But the retailer only knows the distribution of consumer preferences. A consumer observes qualities and prices of products in the assortment and chooses a product that provides her the maximum utility. Further, we consider consumer substitution behavior to be assortment based or, static. Depending upon whether or not the demand is stochastic, we consider two scenarios: risky (demand is stochastic) and riskless (demand is deterministic). We formulate a single-period price-dependent newsvendor problem to determine the optimal stocking quantities and prices along with assortment decisions. Under the risky case, we prove that given an assortment the optimal profit function is a strictly pseudo-concave function of the product prices. We also develop the Increasing Convex Envelope Algorithm, a polynomial-time algorithm (O(nlogn), where n is the number of products) to determine the optimal riskless assortment from a given set of available products. We find that as compared to the riskless scenario, under the risky scenario, the optimal prices increase and the optimal assortment size reduces, in order to reduce the effect of inventory costs arising from random demand. Further, we show that the optimal assortment in the risky case is a subset of the optimal assortment in the riskless case. This leads to a reduction in computational time in many practical cases. Through a numerical study, we show that the optimal assortment and the profit thereof are stable even when demand parameters are erroneously estimated. However, both optimal assortment and profit are highly sensitive to errors in the specification of the quality of the products, suggesting that retailers need to have better knowledge about how consumers perceive and evaluate product quality and product attributes.