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.

Modeling and simulating the multi-generation product sales, production and inventory system within the context of quality upgrades

“…In contrast to the benchmark model, enterprises upgrade the quality of multiple product generations to maximize their profits. Product pricing will improve because the upgrading of product quality leads to an increase in product development costs, and enterprises must increase their pricing when next-generation products are launched, which is also consistent with the conclusion of the literature [28], which provides high-quality products at high prices. In terms of inventory operations, enterprises use low inventory to reduce inventory costs.…”

“…In addition, quality plays a pivotal role in the entire product diffusion process, as it serves as a metric for gauging product advancement, influencing the cost and pricing of products [27], and subsequently impacting consumers' purchasing decisions. Li [28] studied the interactive relationship between the price and quality of new products using a multinomial logit model and found that the interaction relationship between quality and price encouraged enterprises to provide products with high quality and price through joint optimization of the two. Bala [29] formulated a game-theoretic model to determine the upgrade range and pricing of a product and found that the upgrade cost is a critical factor in determining consumers' purchase decisions.…”

Modeling Multi-Generation Product Diffusion in the Context of Dual-Brand Competition and Sustainable Improvement

“…Therefore, in the spread of multi-generation products, the quality factor must be considered. Some previous studies jointly optimized price and quality [30][31][32] and analyzed enterprises' quality and price competition strategies for different consumers by constructing consumer functions. Some scholars have introduced quality level as a key factor to the optimal decision-making problem of product renewal design and determined the optimal quality upgrade level by constructing a non-competitive model, in which new and old products coexist in competition and old products are gradually withdrawn.…”

Product‐line pricing with dual objective of profit and consumer surplus

“…Finally, we remark that our work sits on the shoulders of numerous researchers who studied pricing under various discrete choice models, including the MNL and nested logit models (Davis et al., 2014; Gallego & Wang, 2014; Huh & Li, 2015; Kouvelis et al., 2015; Kök & Xu, 2011; Li & Huh, 2011; Li et al., 2015; Rayfield et al., 2015), and the mixed MNL model (Hanson & Martin, 1996; Li et al., 2018). There are a number of multi‐product pricing studies considering various features: the MNL model with search costs (Wang & Sahin, 2018), restricted consideration set (Wang, 2022), impatient customers (Gao et al., 2021), network effect (Du et al., 2016; Wang & Wang, 2017), price‐quality interaction (Li et al., 2020), multi‐product market diffusion (Li, 2020), market expansion (Wang, 2021), and sequential options for cross‐category products (Ke & Wang, 2021). Pricing has also been studied under the paired combinatorial logit model (Li & Webster, 2017), generalized extreme value model (Zhang et al., 2018), the Markov Chain choice model (Blanchet et al., 2016; Dong et al., 2019), exponomial choice model (Alptekinoğlu & Semple, 2016, 2021), and nonparametric choice model (Jagabathula & Rusmevichientong, 2017; Paul et al., 2018).…”