Envy-Free Pricing in Versioning Problems

86 Pages Posted: 22 Dec 2022

See all articles by Jianqing "Fisher" Wu

Jianqing "Fisher" Wu

California State University, Long Beach

Mohit Tawarmalani

Purdue University - Krannert School of Management

Karthik Natarajan Kannan

Purdue University

Date Written: December 11, 2022

Abstract

In this paper, we treat versioning problems where the reservation prices satisfy the Spence-Mirlees condition, although we do not require that consumers derive a higher value from a higher product type. The prior versioning literature makes additional assumptions in order to decompose the problems by consumer types. We relax such assumptions and find different optimal structures. Moreover, we consider development costs, span of consumers costs, versioning costs, and limits on the number of offered products. We show that, with these costs, the problem can be solved as a shortest path problem resolving a previous complexity conjecture. Using this insight, we prove various structural results about how the optimal policy changes with cost changes. For example, we show that as development cost decreases, all consumers and seller are better off, the seller's profit is a convex decreasing function of versioning cost, and the size of the market served weakly reduces with an increase in span of consumers cost. Surprisingly, we find that seller profit can reduce when reservation prices increase. When the consumer and product types are continuous, we develop a discretization approach that identifies the optimal policy to any desired accuracy. We show that, for the functional forms of reservation prices studied before, the problem simplifies in that it decomposes by consumers. For this case, we derive new analytical solutions. Our results show that when span of consumers costs are present, the optimal product allocation profile may be discontinuous, a solution structure that has not been identified in previous studies.

Keywords: Versioning, envy-free pricing, product line design, shortest-path problem

Suggested Citation

Wu, Jianqing "Fisher" and Tawarmalani, Mohit and Kannan, Karthik Natarajan, Envy-Free Pricing in Versioning Problems (December 11, 2022). Available at SSRN: https://ssrn.com/abstract=4299532 or http://dx.doi.org/10.2139/ssrn.4299532

Jianqing "Fisher" Wu (Contact Author)

California State University, Long Beach ( email )

1250 Bellflower Blvd., MS 5101
Long Beach, CA 90840
United States

Mohit Tawarmalani

Purdue University - Krannert School of Management ( email )

1310 Krannert Building
West Lafayette, IN 47907-1310
United States

Karthik Natarajan Kannan

Purdue University ( email )

Krannert School of Management
West Lafayette, IN 47907
United States

Do you have a job opening that you would like to promote on SSRN?

Paper statistics

Downloads
78
Abstract Views
406
Rank
635,330
PlumX Metrics