Product Design Under Multinomial Logit Choices: Optimization of Quality and Prices in an Evolving Product Line

Abstract

Problem definition: We study a product-line design problem in which customer choice among multiple products is given by a multinomial logit (MNL) model. A firm determines product quality and prices in an evolving product line to maximize profit. In particular, given the prices and quality of products that already exist in a product line, the firm optimizes prices and/or quality of the new products. Academic/practical relevance: We extend the literature on discrete choice models to include the interaction between product quality and product price and consider two variations of the problem, each mirroring a relevant decision setting found in practice: Variation i is a price-optimization problem in which the firm determines prices of the new products given the quality. Variation ii is a joint price and quality optimization of the new products. Methodology: We apply convex optimization techniques and analyze properties of optimal solutions. Results: We establish concavity of the profit function under price optimization and present tractable solution approaches for the joint quality–price optimization. For each problem variation, we characterize the optimal solution and develop efficient algorithms. We show that the interaction of price and quality is central not only to reconciling the divergence of the existing literature’s equal-markup price prediction from differentiated markups observed in practice, but also for explaining differentiated quality measures across products; this empirically observed strategy can now be quantified and optimized with the model developed in this paper. In addition, we show that the presence of existing products tends to drive the firm to offer new products with both higher quality and prices because of the price–quality interaction. Managerial implications: Findings of this paper offer not only managerial guidelines, but also tools for decision support because of the wide empirical applicability of the MNL model. An important managerial implication is that the lack of realism in the linear utility of the MNL model can be addressed by including price–quality interaction, which is central to understanding the quality and price decision in product-line design. The interaction rationalizes the matching of high markup with high quality and justifies differentiated offering of new products in the presence of existing products.