Optimal Dynamic Pricing for Perishable Assets with Nonhomogeneous Demand

Management Insights

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

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

We study dynamic pricing and inventory control of substitute products for a retailer who faces a long supply lead time and a short selling season. Within a multinomial logit model of consumer choice over substitutes, we develop a stochastic dynamic programming formulation and derive the optimal dynamic pricing policy. We prove that dynamic pricing converges to static pricing as inventory levels of all variates approach the number of remaining selling periods (assuming at most one customer arrival within each period). Our extensive numerical study of the effects of time and inventory depletion on the optimal pricing reveals two fundamental underlying driving forces of the complex price behavior: the level of inventory scarcity and the quality difference among products. We also compare the performance of three restricted pricing strategies: static, unified dynamic, and mixed dynamic pricing. We find that full-scale dynamic pricing is of great value in the presence of inventory scarcity, and initial inventory decisions are quite robust in the pricing scheme employed in the selling season. Based on the above insights, we propose a computationally efficient approach to the initial inventory decision, which delivers close-to-optimal inventory levels for all testing cases.

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.

Dynamic selling of quality-graded products under demand uncertainties

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 .

Motivated by the recent rapid growths of freemium services, we study freemium service providers’ dynamic operating policies to maximize their lifetime profits. We adopt a hybrid of the Bass diffusion model and the replicator equation from evolutionary game theory to capture the user base growth, and model the provider’s dynamic decisions as an optimal control problem. We analyze two variants of the freemium business model inspired by two prominent examples, Dropbox and Spotify, which, respectively, adopt the monetization strategies of limiting features and advertising (and its removal). We establish the optimal dynamic operating policies for the Dropbox-inspired ad-free freemium model and the Spotify-inspired ad-supported freemium model and identify their distinctive operational characteristics: the optimal policies as well as the sustainability of the Dropbox model critically depend on the network effect among users, whereas the optimal policies of the Spotify model show that advertising could be a more effective monetization strategy than limiting features. The latter observation is confirmed in a general model allowing both monetization strategies where the optimal policies consistently adopt advertising without limiting features. We then show that the freemium model significantly outperforms the ad-supported free model in weak advertising markets and that offering multiple tiers of premium subscription is unnecessary and confirm our core insights’ robustness under a generalized network effect, customized advertisements, and liquidity constraints. Our study establishes the versatility and value of the freemium model to suit distinct needs of a service in different stages from growth to monetization, and provides guidelines for service providers adopting this business model.

We consider a general class of network revenue management problems, where mean demand at each point in time is determined by a vector of prices, and the objective is to dynamically adjust these prices so as to maximize expected revenues over a finite sales horizon. A salient feature of our problem is that the decision maker can only observe realized demand over time but does not know the underlying demand function that maps prices into instantaneous demand rate. We introduce a family of “blind” pricing policies that are designed to balance trade-offs between exploration (demand learning) and exploitation (pricing to optimize revenues). We derive bounds on the revenue loss incurred by said policies in comparison to the optimal dynamic pricing policy that knows the demand function a priori, and we prove that asymptotically, as the volume of sales increases, this gap shrinks to zero.

We study a joint implementation of price‐ and availability‐based product substitution to better match demand and constrained supply across vertically differentiated products. Our study is motivated by firms that utilize dynamic pricing as well as customer upgrades, as ex ante and ex post mechanisms, respectively, to mitigate inventory mismatches. To gain insight into how offering product upgrades impacts optimal price selection, we formulate a multiple period, nested two‐stage model where the firm first sets prices and replenishment levels for each product while the demand is still uncertain, and after observing the demand, decides how many (if any) of the customers to upgrade to a higher quality product. We characterize the structure of the optimal upgrade, pricing and replenishment policies and find that firms having greater flexibility to offer product upgrades can restrain their reliance on dynamic pricing, enabling them to better protect the price differentiation between the products. We also show how the quality differential between the products or changes in the replenishment cost structures influence the optimal policy. Using insights gained from the optimal policy structure, we construct a heuristic policy and find that it performs well across various parameter values. Finally, we consider an extension in which the firm dynamically sets upgrade fees in each period. Our results overall help further our understanding of the intricate relationship among a firm's decisions on pricing, replenishment, and product upgrades in an effort to better match demand and constrained supply.

Dynamic Pricing of Relocating Resources in Large Networks

Is there a role for governments in emerging countries to accelerate economic development by intervening in product and factor markets? To address this question, we study optimal dynamic Ramsey policies in a standard growth model with financial frictions. The optimal policy intervention involves pro-business policies like suppressed wages in early stages of the transition, resulting in higher entrepreneurial profits and faster wealth accumulation. This, in turn, relaxes borrowing constraints in the future, leading to higher labor productivity and wages. In the long run, optimal policy reverses sign and becomes pro-worker. In a multi-sector extension, optimal policy subsidizes sectors with a latent comparative advantage and, under certain circumstances, involves a depreciated real exchange rate. Our results provide an efficiency rationale, but also identify caveats, for many of the development policies actively pursued by dynamic emerging economies.

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

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

In this paper, we deal with dynamic pricing strategies for connection-oriented services in wireless systems. Dynamic pricing policies allow the network operator to charge a cost per time unit depending on the network usage. In this way, the users behavior can be regulated and the network management is significantly improved. We model the user demand and the call duration as functions of the service price. By using standard Markovian techniques to represent the system evolution, we devise an optimal linear pricing scheme, which can be easily computed and controlled. When compared with a flat-rate policy, where a constant price for the network services is fixed, the proposed solution is able to provide a better quality of service to the users as well as a greater revenue to the network operator. For example, when eight radio channels are available and the traffic load is equal to 0.8, we obtain a 25% improvement in the network revenue with respect to the flat-rate policy, while the blocking probability is halved.