关键词: 垂直差异化产品, 捆绑设计, 定价, 非线性整数规划

Abstract: Bundling refers to the marketing strategy of offering two or more products or services as a specially priced package. It is widely adopted by manufacturers and retailers, as it reduces production and transaction costs, captures more consumer surplus, achieves economies of scale, and raises entry barriers for competing firms.
This paper aims to investigate how a monopolistic firm determines the composition and pricing of multiple bundles. Component suppliers provide the retailer with m categories of components, each containing n products of distinct quality levels. The retailer selects appropriate products from the upstream component portfolio to bundle and sells to heterogeneous consumers, with the goal of maximizing profits. In this paper, we assume that each bundle must include exactly one component from each category, that is, the m categories of components are assembled in a 1∶1∶…∶1 ratio, resulting in a total of n^m technically feasible bundled products available for the retailer to choose from. Considering that consumers exhibit heterogeneous perceived quality toward bundled products, this paper explores the retailer’s optimal bundle design and pricing strategies under two scenarios: super-additive perceived quality and sub-additive perceived quality.
The customer’s demand across different bundles is developed based on the utility maximization theory, and a mixed integer non-linear program is proposed to solve this problem. Firstly, a two-step solution approach is developed to obtain the optimal decisions of the retailer: in the first stage, the optimal prices can be obtained based on the assumption that the composition and assortment of bundles are known; in the second stage, the optimal composition and assortment of bundles can be recognised using the optimal price generated in the first step. Secondly, we propose an efficient algorithm to solve the problem based on the optimal properties. Finally, the validity of the algorithm is tested and the effect of perceived quality and consumer quality valuation on the retailer’s decisions is examined by numerical analysis.
The results reveal the following key findings: (1)The optimal assortment of bundles consists of the Pareto frontier of all feasible bundles, and the components constituting the optimal bundles also lie on their respective Pareto frontiers. Additionally, the optimal set of bundles depends only on the cost/perceived quality ratios of the bundles and is independent of the distribution of consumers’ valuation for quality. (2)The optimal prices of bundles, the retailer’s market share and profits are all monotonically decreasing functions of b. Specifically, when most consumers in the market have a low marginal valuation for quality, the retailer will reduce the selling price to stimulate consumption. However, this price reduction does not lead to an increase in market share. Under the dual impact of declining marginal revenue and reduced demand, the retailer’s profits decrease. (3)The optimal prices of bundles and the retailer’s profits both increase in α, which implies that when consumers have a higher perceived quality, the retailer can appropriately raise product prices to capture more consumer surplus.
The above conclusions provide the following practical implications for retailers in bundle design and pricing: (1)In making pricing decisions, retailers should take into account not only the cost and quality of the bundles but also the distribution of consumers’ quality valuation. (2)In the design of bundles, replacing dominated components with dominant ones can either improve the qualities of the bundles or reduce their costs, thereby increasing profits. (3)For retailers, the key to securing stable future returns lies in enhancing consumers’ perceived quality of their products rather than continuously expanding the scale of their product lines.
Notably, to the best of our knowledge, this is the first study to provide practitioners with optimization approaches for the design and pricing of vertically differentiated bundles. However, this research has certain limitations that are worth noting. For instance, consumers may engage in dynamic substitution if their first-choice bundle is out of stock. Therefore, a promising direction for future research is to solve the problem of optimal composition and prices of multiple bundles under stockout-based substitution.

Key words: vertically differentiated products, bundle design, pricing, mixed non-linear integer program