Expected Profit of Fixed Price Policy Decays Exponentially in the Lead Time

35 Pages Posted: 25 Oct 2023

See all articles by Stefanus Jasin

Stefanus Jasin

University of Michigan, Stephen M. Ross School of Business

Alys Jiaxin Liang

University of Michigan, Stephen M. Ross School of Business

Yanzhe (Murray) Lei

Queen's University - Smith School of Business

Lai Wei

Boston College - Carroll School of Management

Date Written: September 20, 2023

Abstract

We revisit the joint inventory and pricing problem with backlogging and positive lead time, and ask, “What is the value of dynamic pricing?” While it is intuitive that dynamic pricing can yield a better expected profit than the best Fixed Price (FP) policy because the former is more flexible than the latter, the magnitude of this improvement is not well understood. In this paper, we shed partial light on this question by focusing on the impact of delivery lead time in a setting with a linear purchase rate and a Normal-like demand. We analytically derive an upper bound for the expected profit under the best FP policy (i.e., the policy that applies the optimal sequence of fixed prices, which may not be stationary over time, in combination with the best adaptive replenishment policy) and also a lower bound for the expected profit under the optimal joint inventory and pricing policy. Under a mild assumption, we show that the expected profit under the best FP policy decays exponentially in the length of lead time and, for all sufficiently large lead times, grows at most logarithmically in the length of the selling horizon, whereas the expected profit under the optimal joint inventory and pricing policy grows linearly in the length of the selling horizon regardless of the length of lead time. These results hold even in the setting where demand variability is smaller than its mean, which is in contrast to the well-known result in the Revenue Management (RM) literature regarding the near-optimality of the best FP policy in such a setting. More precisely, whereas in the aforementioned classic RM setting, dynamic pricing has been shown to only capture the second-order magnitude of the optimal revenue, in the presence of lead time and holding cost, dynamic pricing is sometimes needed even to capture the first-order magnitude of the optimal profit. This highlights the necessity of dynamic pricing in such a setting.

Keywords: Dynamic Pricing, Inventory, Asymptotic Analysis

JEL Classification: C44

Suggested Citation

Jasin, Stefanus and Liang, Alys and Lei, Yanzhe (Murray) and Wei, Lai, Expected Profit of Fixed Price Policy Decays Exponentially in the Lead Time (September 20, 2023). Available at SSRN: https://ssrn.com/abstract=4577059 or http://dx.doi.org/10.2139/ssrn.4577059

Stefanus Jasin

University of Michigan, Stephen M. Ross School of Business ( email )

701 Tappan Street
Ann Arbor, MI MI 48109
United States

Alys Liang (Contact Author)

University of Michigan, Stephen M. Ross School of Business ( email )

Ann Arbor, MI
United States
7348829063 (Phone)
48104 (Fax)

Yanzhe (Murray) Lei

Queen's University - Smith School of Business ( email )

Smith School of Business - Queen's University
143 Union Street
Kingston, Ontario K7L 3N6
Canada

Lai Wei

Boston College - Carroll School of Management ( email )

140 Commonwealth Avenue
Chestnut Hill, MA 02467
United States

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