Static Pricing Policy Research Articles

Technical Note—Pricing in On-Demand and One-Way Vehicle-Sharing Networks

In this paper, we introduce a new method for evaluating the performance of static pricing in one-way vehicle-sharing systems. Our approach, based on a well-known recursive relationship, leads to a series of increasingly tight bounds on the performance of the static pricing policy. These bounds are valid for systems with multiple locations, nonzero travel times, and an arbitrary number of vehicles. They also apply to systems where the static pricing policy does not lead to a fully connected network. Our method results in a family of asymptotically optimal static pricing policies that improve upon previous results in the literature. The approach applies to the case of a single location and yields a bound that is at least as tight as the best known bound.

Real-Time Spatial–Intertemporal Pricing and Relocation in a Ride-Hailing Network: Near-Optimal Policies and the Value of Dynamic Pricing

In “Real-Time Spatial–Intertemporal Pricing and Relocation in a Ride-Hailing Network: Near-Optimal Policies and The Value of Dynamic Pricing,” Chen, Lei, and Jasin consider a dynamic pricing problem faced by a ride-hailing service provider who manages a fixed number of servers and serves price-sensitive customers within a network. Servers serve arriving customers by relocating from the requested origins to destinations within a certain travel time. The authors first propose a static pricing policy based on the optimal solution to a deterministic relaxation of the original stochastic problem. They show that the proposed static policy matches the best possible asymptotic performance of any static policy. The authors further propose a dynamic pricing policy that adaptively changes the prices in a way that reduces the impact of past demand randomness on the balance of future distributions of servers and customers across the network. They show that the dynamic pricing policy achieves significantly better asymptotical performance. The proposed policies and their performance guarantees are further extended to a case where the firm jointly decides the relocation of vacant servers

An Approximate Analysis of Dynamic Pricing, Outsourcing, and Scheduling Policies for a Multiclass Make-to-Stock Queue in the Heavy Traffic Regime

A new study by Ata and Barjesteh proposes an effective joint dynamic pricing, outsourcing, and scheduling policy for a multiclass make-to-stock manufacturing system. The authors approximate this control problem with a Brownian control problem. They solve the Brownian problem explicitly by exploiting the solution to a Riccati equation and propose a policy for the manufacturing system based on its solution. The proposed policy is a two-sided barrier policy. Outsourcing and idling processes are used to maintain the workload above the lower and below the upper barriers, respectively. Dynamic prices are used to control the workload process between the two barriers. The authors show using a simulation study that the optimality gap of the proposed policy is small, and their proposed policy outperforms a long list of static pricing policies. Moreover, the gap between their proposed policy and the static pricing policies increases with the server utilization and the outsourcing cost.

Wait Time–Based Pricing for Queues with Customer-Chosen Service Times

This paper studies a pricing problem for a single-server queue where customers arrive according to a Poisson process. For each arriving customer, the service provider announces a price rate and system wait time. In response, the customer decides whether to join the queue, and, if so, the duration of the service time. The objective is to maximize either the long-run average revenue or social welfare. We formulate this problem as a continuous-time control model whose optimality conditions involve solving a set of delay differential equations. We develop an innovative method to obtain the optimal control policy, whose structure reveals interesting insights. The optimal dynamic price rate policy is not monotone in the wait time. That is, in addition to the congestion effect (the optimal price rate increases in the wait time), we find a compensation effect, meaning that the service provider should lower the price rate when the wait time is longer than a threshold. Compared with the prevalent static pricing policy, our optimal dynamic pricing policy improves the objective value through admission control, which, in turn, increases the utilization of the server. In a numerical study, we find that our revenue-maximizing pricing policy outperforms the best static pricing policy, especially when the arrival rate is low, and customers are impatient. Interestingly, the revenue-maximizing policy also improves social welfare over the static pricing policy in most of the tested cases. We extend our model to consider nonlinear pricing and heterogeneous customers. Nonlinear pricing may improve the revenue significantly, although linear pricing is easier to implement. For the hetergeneous customer case, we obtain similar policy insights as our base model. This paper was accepted by Baris Ata, stochastic models and simulation. Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2022.4474 .

Fair and profitable: How pricing and lead-time quotation policies can help

In this paper, we propose four policies to serve price and lead-time sensitive customers with a single type of product produced in an M/GI/1 type make-to-stock queueing system. The policies are developed to observe certain principles of fairness: if a customer is quoted a longer lead-time, she/he must be charged a lower price and a significant proportion of the deliveries has to be made during the quoted lead times. Although handling non-exponential service times in this setting presents difficulties, our analysis of the proposed policies is exact. Two of the policies operate with static prices, while one of the two dynamic pricing policies also quotes dynamic lead times. By construction of the policies, we show that the dynamic pricing policies are more profitable. Numerical examples bring additional light by showing that in small markets with oversensitive customers, dynamic policies can be profitable while static pricing policies can fail. In a larger market, a simple dynamic policy charging two prices depending on the stock availability can be a reasonable compromise. Dynamic policies tend to suffer less against high production time variability as well.

Technical Note—Static Pricing: Universal Guarantees for Reusable Resources

A Simple and Near-Optimal Static Pricing Policy for Reusable Resources

Technical Note—On Revenue Management with Strategic Customers Choosing When and What to Buy

In practice, when thinking about their purchasing decisions, customers usually strategize along two dimensions: (1) when to buy and (2) what to buy. That is, they might delay a purchase in anticipation of future price reductions, and/or they might purchase cheaper substitutes. Despite this, the literature has thus far dealt exclusively with either one of the two extremes whereby one of the two strategic dimension is missing. For example, a large body of work has studied forward-looking customers strategizing on when to buy but has done so merely within models in which customers have no alternatives to choose from. Conversely, another large body of work on assortment optimization and choice modeling has studied customers who choose what to buy from multiple product offerings but acting myopically. This paper takes a first step toward analyzing dynamic pricing when customers are fully strategic and choose both when and what to buy. Using a novel decomposition approach for the underlying multidimensional mechanism design problem, the paper presents theoretical and numerical performance analyses of static pricing policies.

Optimal pricing policies for tandem queues: Asymptotic optimality

We study the optimal pricing problem for a tandem queueing system with an arbitrary number of stations, finite buffers, and blocking. The problem is formulated using a Markov decision process model with the objective to maximize the long-run expected time-average revenue or gain of the service provider. Our interest lies in comparing the performances of static and dynamic pricing policies in maximizing the gain. We show that the optimal static pricing policies perform as well as the optimal dynamic pricing policies when the buffer size at station 1 becomes large and the arrival rate is either small or large. More importantly, we propose two specific static pricing policies for systems with small and large arrival rates, respectively, and show that each proposed policy produces a gain converging to the optimal gain with an approximately exponential rate as the buffer size before station 1 becomes large. We learn from numerical results that the proposed static policies perform as well as optimal dynamic policies even for a moderate-sized buffer at station 1. We also learn that there exist cases where optimal static pricing policies are, however, neither optimal nor near-optimal.

Dynamic Pricing of Wireless Internet Based on Usage and Stochastically Changing Capacity

Problem definition: Inspired by new developments in dynamic spectrum access, we study the dynamic pricing of wireless Internet access when demand and capacity (bandwidth) are stochastic. Academic/practical relevance: The demand for wireless Internet access has increased enormously. However, the spectrum available to wireless service providers is limited. The industry has, thus, altered conventional license-based spectrum access policies through unlicensed spectrum operations. The additional spectrum obtained through these operations has stochastic capacity. Thus, the pricing of this service by the service provider has novel challenges. The problem considered in this paper is, therefore, of high practical relevance and new to the academic literature. Methodology: We study this pricing problem using a Markov decision process model in which customers are posted dynamic prices based on their bandwidth requirement and the available capacity. Results: We characterize the structure of the optimal pricing policy as a function of the system state and of the input parameters. Because it is impossible to solve this problem for practically large state spaces, we propose a heuristic dynamic pricing policy that performs very well, particularly when the ratio of capacity to demand rate is low. Managerial implications: We demonstrate the value of using a dynamic heuristic pricing policy compared with the myopic and optimal static policies. The previous literature has studied similar systems with fixed capacity and has characterized conditions under which myopic policies perform well. In contrast, our setting has dynamic (stochastic) capacity, and we find that identifying good state-dependent heuristic pricing policies is of greater importance. Our heuristic policy is computationally more tractable and easier to implement than the optimal dynamic and static pricing policies. It also provides a significant performance improvement relative to the myopic and optimal static policies when capacity is scarce, a condition that holds for the practical setting that motivated this research.

Pricing policy selection for a platform providing vertically differentiated services with self-scheduling capacity

In this article, we study three pricing policies for a monopoly platform, such as Uber or Gett, who offers vertically differentiated services to customers via multiple types of self-scheduling providers. Ideally, the platform can employ a “dynamic pricing” policy, which pays providers wages and charges customers prices for the transactions of different services that both adjust based on prevailing demand conditions, to maximize its profit. However, since it is challenging for the platform to implement and for providers to understand this policy, the other two pricing policies are commonly adopted in practice, that is, “surge pricing” policy (adopted by Uber) which pays providers a fixed commission of its dynamic prices, and “static pricing” policy (applied by Gett) which pays providers a fixed commission of its fixed prices. By observing these phenomena, we propose to study and discuss the platform’s profit performance of these three pricing strategies. We show that the surge pricing policy does not always perform well, which can explain why some on-demand platforms would implement the static pricing policy in practice. Also, although the dynamic pricing policy will significantly improve the platform’s profit, we find that the profitability of the static (surge) pricing policy would approach that of the dynamic pricing policy if the platform can balance the number of different types of providers and/or reduce the commission rate.

On Revenue Management with Strategic Customers Choosing When and What to Buy

We study a network revenue management model in which a decision maker offers multiple products, which consume capacitated resources, for sale to customers over a finite horizon. The decision maker employs an anonymous posted-price policy, and arriving customers strategize on (a) when, and (b) which product to purchase so as to maximize their utility. Customers’ choices are driven by heterogeneous product valuations that decay over time at heterogeneous rates. Both initial valuations and decay rates are private information. We derive for all nonanticipating dynamic pricing policies an upper bound to expected revenues. We use our bound to conduct theoretical and numerical performance analyses of static pricing policies. In our theoretical analysis, we derive a constant factor guarantee for the performance of static pricing, for the classical fluid-type regime where inventory and demand grow large. Our numerical analysis shows static pricing to be able capture at least 75%-90% of maximum possible expected revenue under a wide range of realistic problem parameters, and suggests that static pricing is likely to perform better under higher product proliferation and lower load factor in the presence of strategic customers.

Real-Time Dynamic Pricing with Minimal and Flexible Price Adjustment

We study a standard dynamic pricing problem where the seller (a monopolist) possesses a finite amount of inventories and attempts to sell the products during a finite selling season. Despite the potential benefits of dynamic pricing, many sellers still adopt a static pricing policy because of (1) the complexity of frequent reoptimizations, (2) the negative perception of excessive price adjustments, and (3) the lack of flexibility caused by existing business constraints. In this paper, we develop a family of pricing heuristics that can be used to address all these challenges. Our heuristic is computationally easy to implement; it requires only a single optimization at the beginning of the selling season and automatically adjusts the prices over time. Moreover, to guarantee a strong revenue performance, the heuristic only needs to adjust the prices of a small number of products and do so infrequently. This property helps the seller focus his effort on the prices of the most important products instead of all products. In addition, in the case where not all products are equally admissible to price adjustment (due to existing business constraints such as contractual agreement, strategic product positioning, etc.), our heuristic can immediately substitute the price adjustment of the original products with the price adjustment of similar products and maintain an equivalent revenue performance. This property provides the seller with extra flexibility in managing his prices. This paper was accepted by Noah Gans, stochastic models and simulation.

A joint dynamic pricing and advertising model of perishable products

Advertising and dynamic pricing play key roles in maximizing profit of a firm. In this paper a joint dynamic pricing and advertising problem for perishable products is investigated, where the time-varying demand rate is decreasing in sales price and increasing in goodwill. A dynamic optimization model is proposed to maximize total profit by setting a joint pricing and advertising policy under the constraint of a limited advertising capacity. By solving the dynamic optimization problem on the basis of Pontryagin’s maximum principle, the analytical solutions of the optimal joint dynamic pricing and advertising policy are obtained. Additionally, to highlight the advantage of the joint dynamic strategy, the case of the optimal advertising with static pricing policy is considered. Numerical examples are presented to illustrate the validness of the theoretical results, and some managerial implications for the pricing and advertising of the perishable products are provided.

Optimal dynamic pricing problem considering patient and impatient customers’ purchasing behaviour

We divide customers into two types according to their purchasing behaviour: one is impatient and the other is patient. An impatient customer will be immediately lost if his reservation price of the product is below the current price, while a patient customer will wait until the price drops below his reservation price and buy one item. The seller’s objective is to maximise his expected discounted overall revenue by dynamically pricing. We prove that the optimal pricing policy has a control limit structure, examine the impact of the proportion parameter on the seller’s expected discounted overall revenue and show that the limiting behaviour of the optimal pricing policy when there are little patient customers is the same as the optimal static pricing policy. We also show that the seller will make more profit if he partially knows the number of waiting customers by providing subscription service to customers. A set of numerical results illustrates the value of dynamic pricing and information with respect to problem parameters such as the arriving probability and the subscription probability.

Pricing in Ride-Share Platforms: A Queueing-Theoretic Approach

We study optimal pricing strategies for ride-sharing platforms, such as Lyft, Sidecar, and Uber. Analysis of pricing in such settings is complex: On one hand these platforms are two-sided -- this requires economic models that capture the incentives of both drivers and passengers. On the other hand, these platforms support high temporal-resolution for data collection and pricing -- this requires stochastic models that capture the dynamics of drivers and passengers in the system.In this paper we build a queueing-theoretic economic model to study optimal platform pricing. In particular, we focus our attention on the value of dynamic pricing: where prices can react to instantaneous imbalances between available supply and incoming demand. We find two main results: We first show that performance (throughput and revenue) under any dynamic pricing strategy cannot exceed that under the optimal static pricing policy (i.e., one which is agnostic of stochastic fluctuations in the system load). This result belies the prevalence of dynamic pricing in practice. Our second result explains the apparent paradox: we show that dynamic pricing is much more robust to fluctuations in system parameters compared to static pricing. Thus dynamic pricing does not necessarily yield higher performance than static pricing -- however, it lets platforms realize the benefits of optimal static pricing, even with imperfect knowledge of system parameters.

Optimal production and pricing policies in a combined make-to-order/make-to-stock system

In this paper, we consider a production system which is capable to produce two types of products. The first type of products is make-to-order, while the second type is make-to-stock whose demand is satisfied by the on-hand inventory. The demand arrival rates of both types of products are price-sensitive. The excess demand that cannot be satisfied immediately is either backlogged or lost. The system costs include the holding costs of product inventories and shortage costs of unsatisfied demand. The objective is to maximise the total discounted profit over an infinite planning horizon by coordinating the production process and pricing decisions. By analysing the properties of objective functions, we characterise the optimal control policy by two switch curves and the optimal price is also given for different ordering and inventory levels. We also explore the monotonicity of both switch curves which will reduce the computation effort. Numerical experiments are conducted to demonstrate the use of the switch curves in managing the production system and illustrate that compared with the static pricing policy, the optimal integrated price and inventory control policy can result in a significant profit improvement in the make-to-order/make-to-stock system that is much higher than in a single-product system.

OPTIMAL DYNAMIC PRICING FOR USED PRODUCTS IN REMANUFACTURING OVER AN INFINITE HORIZON

The return of used products (cores) is the beginning of remanufacturing. Although an appropriate pricing policy can effectively manage the returns, a static pricing policy cannot match the returns and demands because of the high uncertainties in both sides, which in turn results in high inventory cost or lost-sale cost. In this paper, we apply a dynamic pricing policy commonly used in retail setting to the core acquisition management in remanufacturing and study the pricing problem for used products with the objective of minimizing average cost over an infinite horizon. We formulate the pricing problem as a continuous-time Markov decision process and characterize the structural properties of the optimal policy. We also conduct a numerical study to investigate the benefit of dynamic pricing.

Dynamic pricing vs. acquiring information on consumers’ heterogeneous sensitivity to product freshness

We propose a model of an inventory system in which a perishable product is periodically replenished, and the retailer is unaware of consumer heterogeneity in consumers’ sensitivity to freshness of a perishable product with a fixed shelf life (though it exists). Using an analytical approach, we optimally solve the problem and evaluate the extent to which unawareness is likely to detract from a retailer’s profit and the extent to which it is likely to affect the price that consumers pay. In addition, we evaluate the conditions in which a dynamic pricing policy is beneficial either to the retailer or to the consumer, as compared with a static pricing policy. It is proven that the retailer should assign products a lower price at the early stages of their shelf life and then raise the price as the products approach expiration. A numerical illustration combined with sensitivity analysis demonstrates the applicability of the modelling approach. Key parameters such as volatility of consumer sensitivity to freshness and the retailer’s estimation regarding consumers’ sensitivity are investigated. A conclusion is that, in monetary terms, it is preferable from the retailer’s perspective to invest in dynamic pricing technology rather than in technologies for gathering information on consumers’ purchases.

A continuous-time dynamic pricing model knowing the competitor’s pricing strategy

In this paper we consider a dynamic pricing model for a firm knowing that a competitor adopts a static pricing strategy. We establish a continuous time model to analyze the effect of dynamic pricing on the improvement in expected revenue in the duopoly. We assume that customers arrive to purchase tickets in accordance with a geometric Brownian motion. We derive an explicit closed-form expression for an optimal pricing policy to maximize the expected revenue. It is shown that when the competitor adopts a static pricing policy, dynamic pricing is not always effective in terms of maximizing expected revenue compared to a fixed pricing strategy. Moreover, we show that the size of the reduction in the expected revenue depends on the competitor’s pricing strategy. Numerical results are presented to illustrate the dynamic pricing policy.

Dynamic pricing model with logarithmic demand

The demand of an item depends on time and price both. For example, electric fan in winter season has low demand than summer. The competing company with lower price gets benefit of better sale during this season. The offer of discount in price of product is an effective tool to increase the demand in the declining market. In this paper, we develop a dynamic pricing model for the product having logarithmic decline price sensitive demand. A useful solution procedure is presented to determine the optimal number of price settings and respective selling price. It proved there exist an optimal number of price settings to provide optimal profit for varying business setup. Results are supported by a numerical example and a computational procedure is appended to illustrate the performance of the model. It is shown that the dynamic pricing policy has better performance over the static pricing policy.