Selling Multiple Units to Strategic Customers

40 Pages Posted: 30 Mar 2020

See all articles by Chen Jin

Chen Jin

National University of Singapore

Qian Liu

Dept. of IEDA, HKUST, Hong Kong

Chenguang (Allen) Wu

Hong Kong University of Science and Technology

Date Written: March 4, 2020


Problem definition: We consider a monopolist firm selling products to strategic customers who may purchase more than one unit of the product. A key feature of multi-unit purchases is that customers do not value the additional purchases as highly as the first unit, implying the results of multi-unit purchases cannot simply replicate those of single-unit purchases already revealed in the literature. To uncover the implications of multi-unit demands on customers' strategic purchasing choices and the firm's pricing decisions, we consider a two-period model where each strategic customer demands up to two units of a product.

Academic/practical relevance: Although the existing literature has revealed the significant implications of strategic customer behavior on firms' pricing decisions, the majority of it assumes that strategic customers purchase at most one unit of a product, and little is known about situations involving multi-unit purchases, which is often the case for certain types of products such as wardrobe basics, for example, classical white shirts or jeans.

Methodology: We formulate the firm's profit optimization problem under two pricing schemes including price commitment and dynamic pricing, and provide closed-form solutions for the firm's optimal prices under both schemes.

Results: A thorough analysis of price commitment enables us to investigate the effects of two critical factors - the value of the second unit and the strategic level of customers - on the firm's prices and revenue, consumer surplus and social welfare. We find that the optimal prices in each period are non-monotonic in both factors. While a higher value for the second unit generally increases prices, the firm chooses to lower prices at certain points in order to induce multi-unit purchases. Unlike single-unit selling, the first-period price in the multi-unit setting can increase as customers become more strategic. We also find that the surplus of customers may not increase with either the second-unit value, or their strategic level. All these results are driven principally by the complex market segmentation caused by the convolution of multi-unit purchases and strategic customer behavior.

Managerial implications: Our study sheds light on how the multi-unit purchase generates different insights on the firm's profit, consumer surplus and social welfare. Our results provide the firm with guidance on the optimal pricing decisions.

Keywords: multi-unit purchase, strategic customer behavior, price commitment, dynamic pricing

Suggested Citation

Jin, Chen and Liu, Qian and Wu, Chenguang (Allen), Selling Multiple Units to Strategic Customers (March 4, 2020). Available at SSRN: or

Chen Jin

National University of Singapore ( email )

13 Computing Drive
Singapore, 117417

Qian Liu (Contact Author)

Dept. of IEDA, HKUST, Hong Kong ( email )

Chenguang (Allen) Wu

Hong Kong University of Science and Technology ( email )

Room 5559C, Academic Building
HongKong University of Science and Technology
Hong Kong
Hong Kong

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