Expected Profit of Fixed Price Policy Decays Exponentially in the Lead Time
35 Pages Posted: 25 Oct 2023
Date Written: September 20, 2023
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: Suggested Citation