Optimum pricing policy for dynamic markets

It is common for a firm to make use of multiple suppliers of different delivery lead times, reliabilities, and costs. In this study, we are concerned with the joint pricing and inventory control problem for such a firm that has a quick‐response supplier and a regular supplier that both suffer random disruptions, and faces price‐sensitive random demands. We aim at characterizing the optimal ordering and pricing policies in each period over a planning horizon, and analyzing the impacts of supply source diversification. We show that, when both suppliers are unreliable, the optimal inventory policy in each period is a reorder point policy and the optimal price is decreasing in the starting inventory level in that period. In addition, we show that having supply source diversification or higher supplier reliability increases the firm's optimal profit and lowers the optimal selling price. We also demonstrate that, with the selling price as a decision, a supplier may receive even more orders from the firm after an additional supplier is introduced. For the special case where the quick‐response supplier is perfectly reliable, we further show that the optimal inventory policy is of a base‐stock type and the optimal pricing policy is a list‐price policy with markdowns.

Optimal peak load pricing with time-additive consumer preferences

In this paper we extend the results of the dynamic market optimum pricing policy of monopoly markets, given in Part 1, to the competitive case. We show that the most important contributing factors for differences between static and dynamic, competitive market optimum pricing policy as in the monopoly case are: consumer response time, planning period and the discount rate. Additionally the price of the competitor enters to the decision rule.

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

<p style='text-indent:20px;'>In the real world, the demand cannot be depicted exactly because of customer behavior cannot be forecasted without error. In this paper, we study the effect of the error of the estimated price-demand parameters by analyzing the sensitivity of the optimal joint pricing and ordering policy on the price-demand parameters based on a periodic-review, multi-period and lost sale inventory model for perishable products with constant quantity decay rate and price-sensitive demand. Firstly, we formulate the joint pricing and inventory control problem and find the optimal ordering quantity and the optimal price for deterministic price-demand function. The optimal solutions show that the retailer tends to set a lower price in early periods of each ordering cycle in order to reduce the inventory holding costs. Furthermore, the sensitivity of the optimal joint pricing and inventory control system with respect to the price-demand parameters is examined analytically and evaluated numerically. The sensitivity analysis reveals that compared to the optimal ordering quantity, the optimal prices are less sensitive in the demand-price parameters. Finally, according to the findings of the sensitivity analysis, a heuristic method of regulating the estimated demand-price parameters is employed to improve the average profit. <b>185</b> words.</p>

Optimal pricing and inventory strategies for fashion products under time-dependent interest rate and demand

Designing Shipping Policies with Top-up Options to Qualify for Free Delivery

In this paper we develop an economic order quantity model to investigate the optimal replenishment policies for instantaneous deteriorating items under inflation and trade credit. Demand rate is a linear function of selling price and decreases negative exponentially with time over a finite planning horizon. Shortages are allowed and partially backlogged. Under these conditions, we model the retailer’s inventory system as a profit maximization problem to determine the optimal selling price, optimal order quantity and optimal replenishment time. An easy-to-use algorithm is developed to determine the optimal replenishment policies for the retailer. We also provide optimal present value of profit when shortages are completely backlogged as a special case. Numerical examples are presented to illustrate the algorithm provided to obtain optimal profit. And we also obtain managerial implications from numerical examples to substantiate our model. The results show that there is an improvement in total profit from complete backlogging rather than the items being partially backlogged.

This article considers the dynamic pricing of two substitutable products over a predetermined, finite selling season. The initial inventory levels of the products are fixed exogenously and there are no replenishment opportunities during the season. It is assumed that each arriving customer chooses from available products based on the multinomial logit choice model, which captures the effect of prices on consumer choice. Every time a product runs out of stock, the set of choices shrinks, capturing the effect of stockouts on consumer choice. It is shown that, under the optimal pricing policy, the marginal value of an item is increasing in the remaining time and decreasing in its own stock level and the other product's stock level. Despite such non-surprising behavior on the part of marginal values, the optimal price itself is not simply monotonic in the remaining time or the other product's stock level. For example, a product's optimal price may increase if the remaining time decreases or if the total inventory grows. It is shown that such optimal behavior can be understood through alternative gauges such as the optimal price difference between the two products and the optimal purchase probabilities.

We consider a dynamic pricing model for selling a given stock of a perishable product over a finite time horizon. Customers, whose reservation price distribution changes over time, arrive according to a nonhomogeneous Poisson process. We show that at any given time, the optimal price decreases with inventory. We also identify a sufficient condition under which the optimal price decreases over time for a given inventory level. This sufficient condition requires that the willingness of a customer to pay a premium for the product does not increase over time. In addition to shedding managerial insight, these structural properties enable efficient computation of the optimal policy. Numerical studies are conducted to show the revenue impact of dynamic price policies. Price changes are set to compensate for statistical fluctuations of demand and to respond to shifts of the reservation price. For the former, our examples show that using optimal dynamic optimal policies achieves 2.4–7.3% revenue improvement over the optimal single price policy. For the latter, the revenue increase can be as high as 100%. These results explain why yield management has become so essential to fashion retailing and travel service industries.

Problem definition: We consider the dynamic pricing problem of multiple products under (asymmetric) reference effects over an infinite horizon. Unlike existing literature, which is mostly focused on the single-product setting, our multiproduct setting takes into account the cross-product effects among substitutes and incorporates the memory-based reference prices into the multinomial logit (MNL) demand model. Even with the single-product logit demand, the structure of the optimal pricing policy is intractable. Therefore, we focus on the long-run patterns of the optimal pricing policy and also discuss the performance of the myopic pricing policy. Methodology/results: We first provide a comprehensive characterization of the myopic pricing policy, including its solution, long-run convergence behavior, and optimality gap. For the optimal pricing policy, we show an intricate connection between its long-run dynamics and types of reference effects. We demonstrate that the presence of any gain-seeking product renders a long-run constant pricing policy suboptimal. Conversely, the constant policy (or optimal steady state) can exist in both loss-neutral and loss-averse scenarios, where we provide a sufficient condition for such existence and give the analytical expression for the optimal steady state. We further show that when pricing perfect substitutes, the true optimal policy under the multiproduct framework is more likely to yield a long-run cyclic pattern than the policy derived from the single-product framework, a phenomenon that aligns well with the periodic discounts in real-world markets. Managerial implications: This discrepancy in the long-run behaviors between multi- and single-product-based policies highlights the importance of employing the multiproduct framework and addressing the cross-product effects, as sticking to the single-product framework while managing multiple substitutes can misrepresent long-run dynamics and result in suboptimality. In the multiproduct domain, our model suggests that retailers are more likely to benefit from appropriate price variations than maintaining a constant pricing policy. Funding: H. Jiang acknowledges support from the Natural Sciences and Engineering Research Council of Canada [NSERC Discovery Grant RGPIN 2024 05796]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2024.0801 .

Management Insights

The objective of this paper is to find the key factors that affect a firm's optimal transfer pricing policy. It examines two minimalist vertical models — one consisting of a vertically integrated firm monopolizing an intermediate input for its own and rival's downstream division, and the other comprising two vertically integrated firms competing in a final goods market. Four modes of competition are considered — Cournot, Bertrand, Stackelberg quantity, and price. The paper shows that, in addition to the usual tax considerations, the optimal transfer pricing policy depends on competition mode, demand and strategic characteristics, vertical structure, and production technology. For example, under the same demand structure and competition mode, the two models can yield diametrically opposite outcomes; within a given vertical model, different competition modes may yield different optimal strategies; and within a given competition mode, the four pairings of ordinary substitutes/complements and strategic substitutes/complements can also produce quite different results. The general structure analyzed in this paper can be applied to other transfer pricing models involving uncertainty, cost sharing, asymmetric information, etc. that have been mainly studied in the literature under specific competition modes and demand and strategic characteristics.

The Journal of FinanceVolume 41, Issue 5 p. 1141-1148 Note A Note on Optimal Credit and Pricing Policy under Uncertainty: A Contingent-Claims Approach CHUN H. LAM, CHUN H. LAMSearch for more papers by this authorANDREW H. CHEN, ANDREW H. CHEN Both authors from Edwin L. Cox School of Business, Southern Methodist University. We are indebted to the comments and suggestions of Chris Barry, Sam Dennis, and two anonymous referees on earlier versions of the paper. The authors also acknowledge the financial support from the Center for the Study of Financial Institutions and Markets at Southern Methodist University.Search for more papers by this author CHUN H. LAM, CHUN H. LAMSearch for more papers by this authorANDREW H. CHEN, ANDREW H. CHEN Both authors from Edwin L. Cox School of Business, Southern Methodist University. We are indebted to the comments and suggestions of Chris Barry, Sam Dennis, and two anonymous referees on earlier versions of the paper. The authors also acknowledge the financial support from the Center for the Study of Financial Institutions and Markets at Southern Methodist University.Search for more papers by this author First published: December 1986 https://doi.org/10.1111/j.1540-6261.1986.tb02536.xCitations: 3 Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume41, Issue5December 1986Pages 1141-1148 RelatedInformation

Prior work has investigated time- and inventory-level-dependent pricing of limited inventories with finite selling horizons. We consider a third dimension—in addition to time and inventory level—that the firms can use in setting their prices: the information that the firm has at the individual customer level. An arriving customer provides a signal to the firm, which is an imperfect indicator of the customer's willingness to pay, and the firm makes a personalized price offer depending on the signal, inventory level, and time. We consider two different models: full personalization and partial personalization. In the full personalization model, the firm charges any price it wishes given the customer signal, while in the partial personalization model, the firm can charge one of two prices. We find that a mere correlation between the signals and customers' willingness to pay is not sufficient to ensure intuitive relationships between the signal and the optimal prices. We determine a stronger condition, which leads to several structural properties, including the monotonicity of the optimal price with respect to the signal in the full personalization model. For the partial personalization model, we show that the optimal pricing policy is of threshold-type and that the threshold is monotonic in the inventory level and time.