Optimal Pricing Policy Research Articles

Advance selling to strategic consumers

Advance selling of goods and services is a form of separating purchase from consumption. It is often employed when consumers are uncertain about their consumption utilities until a short time period before consumption. A book to be released, a concert to attend, or a cruise to take are some examples. Invariably, in consumers’ mind inventory availability (of copies, seats, or rooms) is a concern. In this paper we study a retailer’s inventory and pricing decisions in an advance selling scenario that involves consumers who are strategic. Some consumers not only consider advance and spot prices, but also the uncertainty in future availability of the product (during the spot period) and in their consumption utility from it. We characterize the optimal inventory management and pricing policies, and discuss several interesting aspects of the solution. For example, it can be optimal for the retailer to limit advance sales even if there is more demand for it, and it can be optimal for the retailer to limit its inventory even though there is more capacity to keep it, but not both.

Are Markets with Loss-Averse Consumers More Sensitive to Losses?

Behavioral pricing and revenue management aims to incorporate realistic consumer behavior into firms' pricing and inventory models. The key input to these models is market demand, which is often assumed to inherit the characteristics of consumer behavior --- as when, for example, one assumes that a market consisting of loss-averse consumers is more responsive to losses than to gains. Yet market demand and consumer behavior need not be related and so, we argue, that approach to modeling market demand is misguided. This paper proposes an approach that accounts for the heterogeneity in consumer valuation: we aggregate the demand of individual loss-averse consumers to obtain market demand and show that market demand may be more responsive to gains, to losses or be equally responsive. Our results have profound implications not only for how best to characterize the market demand of behaviorally biased consumers but also for determining the firm's optimal pricing policy.

Ordering and pricing model of retailers’ preventive transshipment dominated by manufacturer with conditional return

Retailers often fail to implement the transshipment policy due to lack of the trust or information asymmetry among them in the process of transshipment, especially in the case that return becomes a norm. Therefore, we develop a framework about the preventive transshipment which is dominated by the manufacturer between two independent retailers to cope with the mismatch between demand and inventory. In this research, a two-period ordering and pricing model about the preventive transshipment with conditional return is formulated. To simplify the implementation of preventive transshipment, a dominant preventive transshipment policy is recommended at the beginning of the second period. So, one retailer may adjust the inventory without considering the impact of the other retailer’s inventory quantity and the transshipment strategy. To analyze the mutual effect between two retailers, the existence and uniqueness of the ordering Nash Equilibrium and the optimal pricing policy is identified and analyzed. Furthermore, we obtain the relationship between the ordering quantity and transshipment price. Finally, a numerical analysis is carried out to examine the sensitivity of the transshipment price, wholesale price, ordering quantity, profit of the retailer and manufacture to the return rate.

The Revenue Impact of Dynamic Pricing Policies in Major League Baseball Ticket Sales

We study the operational implementation of multiproduct dynamic pricing by a Major League Baseball franchise for single game tickets. We develop and apply a comprehensive customer choice model to help design dynamic pricing policies for the franchise. Our model encompasses all relevant aspects of customer demand generation process, including ticket quantity and stadium seat section choice. Furthermore, our demand model incorporates external factors that drive customer valuation of sports tickets such as the home team's on-field performance and observed overall attendance level at the time of purchase. Our counterfactual results show potential revenue improvement of up to 15% through the effective use of dynamic pricing. We also find that a properly calibrated fixed pricing policy based on a detailed customer demand model can achieve similar levels of revenue improvement as the optimal dynamic pricing policy.

Dynamic Pricing of Preemptive Service for Secondary Demand

Motivated by recent regulatory evolutions that pave the way to secondary spectrum markets, we investigate profit maximization in a loss network that accommodates calls of two classes of users: 1) primary users (PUs) and 2) secondary users (SUs). PUs have preemptive priority over SUs, i.e., when a PU arrives to the system and finds all channels busy, it preempts an SU. We assume that SU demand is sensitive to price whereas PU demand is inelastic. We study the optimal pricing policy of SUs to maximize the average profit by introducing a finite horizon discounted dynamic programming formulation. Our main contribution is to show that the optimal pricing policy depends only on the total number of users, i.e., the total occupancy. We also demonstrate that optimal prices increase with the total occupancy and show that the optimal pricing policy structure of the original system is not preserved for systems with price-sensitive PUs. Finally, we extend the results to non-preemptive loss systems and establish a connection with results obtained for such models in the literature.

Joint optimal dynamic pricing and replenishment policies for items with simultaneous quality and physical quantity deterioration

This paper discusses optimal dynamic pricing and replenishment policies for items with simultaneous deterioration of quality and physical quantity. Qualitative deterioration is assumed to be instantaneous, while physical deterioration follows non-instantaneous pattern. In order to tackle dynamic essence of the problem, selling price is defined as a time-dependent function of the initial price and discount rate. The product is sold at the initial price value in the time period with no physical quantity deterioration; subsequently it is exponentially discounted to boost customer's demand. In addition to price, the demand rate is dependent on the quality of inventory and changes in price over time. This consideration has enhanced dynamic characteristic of the proposed model. The model seeks to maximize total profit of the system by determining the optimal replenishment cycle, initial price and discount rate. In order to characterize the optimal solution several theoretical results are derived which demonstrate existence and uniqueness of the optimal solution. Then an iterative solution algorithm is developed based on these theoretical results. Finally, in order to analyze the behavior of model and illustrate the solution procedure numerical results accompanied by sensitivity analyses of key parameters of the model are provided.

Optimal markdown policy of perishable food under the consumer price fairness perception

Markdown policies are widely used by retailers to sell perishable food. Consumers purchase food at different prices during different sales periods. Some consumers may compare their experience with others who purchase the same items. Price unfairness or inequity is perceived when different prices are quoted without reasonable explanations. This study develops an optimal markdown model for perishable food pricing to optimise the food retailer revenue and enable a maximum aggregated consumer utility considering individuals’ price fairness perception. The model serves as the first step in evaluating trade-offs between food retailer revenue and consumer utility. In addition, it enables consumer utility to be depicted through perceived price fairness by including the effects of food perishability and scarcity. Another innovative feature is the equalisation of the consumer average aggregated utility during different sales periods as a condition of intertemporal price fairness perception. The proposed model is compared with two benchmark models to justify the effectiveness and advantages in the example. Finally, a sensitive analysis based on the food deterioration rate, consumer food desire rate and consumer average reservation price is conducted to justify the manner in which these factors influence the optimal pricing policy, and managerial insights are suggested for food retailers.

An EOQ model with price-dependent supply and demand

This paper considers a generalization of the Economic Order Quantity (EOQ) model for an end-item that requires an input component for production. The quantity of input components available for production is a nondecreasing function of the price offered by the producer to its suppliers, while the end-item demand is also price-dependent. We analyze the behavior of the optimal replenishment and pricing policy, which depends on the economics of production and procurement costs and the prices associated with input components and end-items. This analysis sheds light on how to deal with price-dependent supply in production planning, as well as on the value of supplier heterogeneity and information on the relationship between component price and supply.

Technical Note—Intertemporal Price Discrimination via Reference Price Effects

We consider the dynamic pricing problem a monopolistic seller faces when customers arrive in heterogeneous time periods and their purchase decisions are affected by reference prices formed from their past observations of prices. We illustrate that a new form of price discrimination opportunity exists in such situations, where the seller’s optimal pricing strategy is a cyclic one, even when the customers are loss-neutral and their demand functions are identical. This result differs from those of prior studies where the optimal price paths are shown to be asymptotically constant and thus is unique due to the interaction between the heterogeneous arrivals and the reference price effects. We also provide the cycle length of the optimal pricing policy when the demand function is linear and when customers are loss-neutral. In this era when customer information is easier accessible, our results suggest the sellers consider this new dimension of price discrimination in conjunction with the old ones, to take advantage of the full power of customer data.

Option pricing and coordination in the fresh produce supply chain with portfolio contracts

This paper studies a fresh produce supply chain that consists of a supplier and a retailer in a newsvendor framework. The supplier is the Stackelberg leader and the retailer is the follower. The retailer can obtain products from the supplier by wholesale price and call option portfolio contracts. The fresh produce incurs a circulation loss in quantity during its transportation. The retailer’s optimal ordering policy and the supplier’s optimal pricing policy are derived in the presence of portfolio contracts and circulation loss. It is demonstrated that, as the prices of option increase toward their optimal, the supplier’s expected profit increases whereas the retailer’s expected profit decreases, and the retailer is more sensitive to the price change. It is also found that the fresh produce supply chain can be coordinated by the portfolio contracts, and Pareto improvement for both chain members can also be achieved as compared with the non-coordinated contracts. However, when the supply chain is coordinated, the supplier cannot realize its optimal pricing strategy. Finally, it is shown that the supplier’s optimal option pricing policy is independent to the demand risk and wholesale price, and the circulation loss of fresh produce increases the management risks of the fresh produce supply chain.

Strategic Price, Warranty and Profit Maximization Model of a Software Product Using Dynamic Optimization

Technological advancements have affected each and every aspect of our day to day life. Over the last few decades our dependence on software products has increased immensely. At the same time, software technologies have been advancing at an electrifying pace to achieve new milestones. With this rapid development, competition between the developers is gaining new heights. In such a competitive business arena, customer satisfaction plays a vital role to decide on the survival of product in the market. So, it becomes essential for the firms to work on strategies of product portfolios to attract more customers in order to elevate profit levels. In this regard, firms are using different strategies to obtain a significant market share. An appropriate price becomes one of the most effective factors to influence customers demand. On the other hand, warranty length also acts as an attribute of quality and forms another factor that may sway the customer towards or away from the product. In a warranty contract, the seller or manufacturer promises for a free repair or renewal policy for failed products. In this paper, we have investigated the marketing and production problem for a software firm considering price and warranty as simultaneous dynamic decision variables under the condition of a dynamic demand. We have considered a demand function dependent upon price, warranty and cumulative sales. We have used optimal control theoretic approach for the proposed profit maximization model to derive optimal price and warranty policies. The results specify several optimal policies with which decision makers would gain insight into the consequence of their decisions. Further, genetic algorithm is used to find the optimal results. Numerical example is provided to illustrate the applicability of model.

An EOQ inventory system of ameliorating items for price dependent demand rate under retailer partial trade credit policy

This paper studies the optimal ordering decision in the economic ordering quantity framework under two levels of trade credit policy where demand rate is a selling price dependent with ameliorating items whose value or utility or quantity increase over the time. In this paper, it is assumed that the retailer maintains a powerful decision-making right and can obtain the full trade credit offered by the supplier yet retailer just offers the partial trade credit to his/her customers. Furthermore, we consider that the items which were already sold but not yet paid for by customers would also incur interest charges or capital cost, which is not considered in the existing studies concerning retailer partial trade credit. For the objective function sufficient conditions for the existence and uniqueness of the optimal solution are provided. An efficient algorithm is designed to determine the optimal pricing and inventory policies for the retailer. Finally we obtain a lot of managerial insights from numerical examples. The results would provide valuable references for retailer in controlling the inventory of ameliorating items. The results also show that it is very necessary and realistic to consider the capital cost incurred by the items which were already sold but not yet paid for by customers under retailer partial trade credit policy.

Inventory model with variable demand rate under stochastic inflation for deteriorating and ameliorating items with permissible delay in payment

In this paper, an inventory policy for the production units is presented with stochastic inflation, variable demand rate, deteriorating and ameliorating items in warehouse and permissible delay in payment. Shortages, backlogging and lost sale, are allowable. Also, inflation is constantly changing due to economic, political, social and cultural issues in the real world and we assume it uniform distribution. The demand rate is depended on selling price and permissible delay in payment and increases as delay in payment increases, however, decreases as selling price increases. The objective function considers optimal pricing policy, the number of production cycles and time fraction without shortage. So, the purpose is to derive the optimal replenishment policy that minimises the net present value of total cost per unit time. The study shows that the optimal solution not only exists but also is unique. Finally, some numerical examples and sensitivity analysis on parameters are made to validate the results of the proposed inventory system.

Spatial Pricing in Ride-Sharing Networks

We explore spatial price discrimination in the context of a ride-sharing platform that serves a network of locations. Riders at different locations are heterogeneous in terms of their destination preferences, as captured by the demand pattern of the underlying network. Drivers decide whether, when, and where to provide service so as to maximize their expected earnings given the platform’s pricing policy. Our findings highlight the impact of the demand pattern of the underlying network on the platform’s optimal profits and aggregate consumer surplus. In particular, we establish that both profits and consumer surplus are maximized when the demand pattern is “balanced” across the network’s locations. In addition, we show that profits and consumer surplus are monotonic with the balancedness of the demand pattern (as formalized by the pattern’s structural properties). Furthermore, we explore the widely adopted compensation scheme that allocates a constant fraction of the fare to drivers and identify a class of networks for which it can implement the optimal equilibrium outcome. However, we also showcase that generally this scheme leads to significantly lower profits for the platform than the optimal pricing policy especially in the presence of heterogeneity among the demand patterns in different locations. Together, these results illustrate the value of accounting for the demand pattern across a network’s locations when designing the platform’s pricing policy, and complement the existing focus on the benefits of dynamic (surge) pricing to deal with demand fluctuations over time.

Optimal Ordering and Pricing Policies for Seasonal Products: Impacts of Demand Uncertainty and Capital Constraint

With a stochastic price-dependent market demand, this paper investigates how demand uncertainty and capital constraint affect retailer’s integrated ordering and pricing policies towards seasonal products. The retailer with capital constraint is normalized to be with zero capital endowment while it can be financed by an external bank. The problems are studied under a low and high demand uncertainty scenario, respectively. Results show that when demand uncertainty level is relatively low, the retailer faced with demand uncertainty always sets a lower price than the riskless one, while its order quantity may be smaller or larger than the riskless retailer’s which depends on the level of market size. When adding a capital constraint, the retailer will strictly prefer a higher price but smaller quantity policy. However, in a high demand uncertainty scenario, the impacts are more intricate. The retailer faced with demand uncertainty will always order a larger quantity than the riskless one if demand uncertainty level is high enough (above a critical value), while the capital-constrained retailer is likely to set a lower price than the well-funded one when demand uncertainty level falls within a specific interval. Therefore, it can be further concluded that the impact of capital constraint on the retailer’s pricing decision can be influenced by different demand uncertainty levels.

Studies on inventory model for deteriorating items with Weibull replenishment and generalised Pareto decay having time dependent demand

Inventory models play an important role in determining the optimal ordering and pricing policies. Much work has been reported in literature regarding inventory models with finite or infinite replenishment. But in many practical situations the replenishment is governed by random factors like procurement, transportation, environmental condition, availability of raw material, etc. Hence, it is needed to develop inventory models with random replenishment. In this paper an EPQ model for deteriorating items is developed and analysed with the assumption that the replenishment is random and follows a Weibull distribution. It is further assumed that the life time of a commodity is random and follows a generalised Pareto distribution and demand is a function of time. Using the differential equations the instantaneous state of inventory is derived. With suitable cost considerations the total cost function is obtained. By minimising the total cost function the optimal ordering policies are derived. Through numerical illustrations the sensitivity analysis is carried. The sensitivity analysis of the model reveals that the random replenishment has significance influence on the ordering and pricing policies of the model. This model also includes some of the earlier models as particular cases for specific values of the parameters.

The online seller's optimal price and inventory policies under different payment schemes

We develop an analytical model to determine the seller's optimal price and inventory decisions under different e-commerce payment schemes. There are two types of payment schemes for consumers to choose from: pay-to-order, with payment occurring when an order is taken online; and pay-on-delivery, in which payment occurs after delivery of the product. Considering different consumer behaviours after consumers evaluate the products, we characterize the rational expectations equilibrium for the online seller. We obtain the optimal price and order quantity to maximise the online seller's profit under different payment schemes. We provide the conditions when it is feasible for online sellers to offer every payment scheme. Numerical examples are provided to illustrate the comparison of online seller's performance between the two payment schemes. [Received 19 March 2015; Revised 12 July 2015; Revised 6 October 2015; Accepted 5 November 2015]

Dynamic joint pricing and production policy for perishable products

AbstractA joint dynamic pricing and production problem for perishable products without shortages is considered. The demand rate is price‐dependent and time‐varying. This paper constructs an optimal control model to maximize the total profit under a general nonlinear production cost function. The feature of the optimal joint dynamic pricing and production policy is analyzed by solving the corresponding optimal control problem on the basis of improved Pontryagin's maximum principle. Then, an effective algorithm is designed to obtain the optimal joint policy. The case of the joint static optimal policy is also investigated and compared with the dynamic one. Finally, numerical examples are presented to illustrate the effectiveness of the proposed methods, and some managerial implications are provided for the management of perishable items.

Optimal Dynamic Pricing with Patient Customers

We consider an infinite-horizon single-product pricing problem in which a fraction of customers is patient and the remaining fraction is impatient. A patient customer will wait up to some fixed number of time periods for the price of the product to fall below his or her valuation at which point the customer will make a purchase. If the price does not fall below a patient customer’s valuation at any time during those periods, then that customer will leave without buying. In contrast, impatient customers will not wait, and either buy immediately or leave without buying. We prove that there is an optimal dynamic pricing policy comprised of repeating cycles of decreasing prices. We obtain bounds on the length of these cycles, and we exploit these results to produce an efficient dynamic programming approach for computing such an optimal policy. We also consider problems in which customers have variable levels of patience. For such problems, cycles of decreasing prices may no longer be optimal, but numerical experiments nevertheless suggest that such a decreasing cyclic policy (suitably chosen) often performs quite well.

Inter-temporal price discrimination and satiety-driven repeat purchases

We discuss customers’ inter-temporal repeat-purchasing behavior when their valuations for a product are subject to transitory satiety. The satiety can be increased by purchases and gradually decays in between. Repeat-purchasing customers optimize their inter-temporal purchasing schedules, including the purchasing time and the purchasing quantity, to maximize their time-average payoffs, and the monopoly seller chooses the optimal cyclic pricing policy to maximize its time-average profit. We derive the conditions under which firms earn the same profit under the cyclic pricing policy as under the fixed-price pricing policy. When the inter-temporal price discrimination is more profitable, we show that the optimal pricing policy depends on the relation between customer valuation and customer satiety. Our results imply that overlooking the effect of transitory satiety tends to cause firms to underprice. We also consider extensions in which there exists a fixed purchasing cost or in which purchasing and consumption decisions are separate.