Dynamic Pricing via Dynamic Programming<Superscript>1</Superscript>

This article specifies an efficient numerical scheme for computing optimal dynamic prices in a setting where the demand in a given period depends on the price in that period, cumulative sales up to the current period, and remaining market potential. The problem is studied in a deterministic and monopolistic context with a general form of the demand function. While traditional approaches produce closed-form equations that are difficult to solve due to the boundary conditions, we specify a computationally tractable numerical procedure by converting the problem to an initial-value problem based on a dynamic programming formulation. We find also that the optimal price dynamics preserves certain properties over the planning horizon: the unit revenue is linearly proportional to the demand elasticity of price; the unit revenue is constant over time when the demand elasticity is constant; and the sales rate is constant over time when the demand elasticity is linear in the price. ¹We acknowledge professor robert e. kalaba for initiating this work and suggesting solution methods.     

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.     

Signaling Quality: Dynamic Price-Advertising Model

This paper extends the existing quality-signaling literature by investigating the roles of price and advertising levels as quality indicators in a dynamic framework. Considering perceived quality as a form of goodwill, we modify the well-known Nerlove-Arrow dynamic model to include price effects. In our model, price is used both as a monetary constraint and as a signal of quality, while advertising spending is used only as a signaling device, and thus purely as a dissipative expense. Utilizing optimal control, we determine optimal decision rules for a firm regarding both price and advertising over time as functions of perceived quality. The results indicate that, when prices act as monetary constraints and are reduced to increase demand, the firm should use the signaling role of advertising by increasing spending to accelerate perceived quality increases. In cases when the value of the perceived quality goes up together with the increase in the perceived quality by more than the demand, in percentage terms, the firm should increase the price (use its signaling role). At steady-state, we find that the level of optimal profit margin relative to price decreases with the elasticity of demand with respect to the brand price. However, higher elasticity of demand with respect to the firm’s perceived quality and/or a higher impact of price (advertising) lead/leads to a higher optimal profit margin (advertising spending) relative to price (revenue).     

Optimal strategies for general price-quality decision models of new products with learning production costs

The question of product quality permeates every level of business and is becoming crucial for the survival of modern manufacturing firms in automotive and high-tech industries. In this paper, we deal with the optimal price and quality policies for the introduction of a new product. On the supply side, the firm wants to determine the unit price and quality level over time given that unit cost declines along a learning curve, and increases if quality is made greater. On the demand side, dynamic demand is related to price and quality, as well as to cumulative sales (which represent diffusion and saturation effects). We will model this problem in a general framework that includes several previous results as special cases.By applying the maximum principle, we will derive the optimal price and quality policies and discover the interactions between these two major strategic marketing instruments, and the diffusion process. Several fundamental theoretical results will be established for the model. Under certain specified conditions higher prices do imply higher quality, and under other conditions the optimal price declines over time while the product quality improves. To illustrate these results, the theoretical results are applied to two specific-cases: the first one is a simple nonseparable demand growth function in price and quality, the other is a separable demand function of price and quality.     

Coordinating advertising and pricing in a manufacturer–retailer channel

Cooperative advertising is a practice that a manufacturer pays retailers a portion of the local advertising cost in order to induce sales. Cooperative advertising plays a significant role in marketing programs of channel members. Nevertheless, most studies to date on cooperative advertising have assumed that the market demand is only influenced by advertising expenditures but not by retail price. This paper addresses channel coordination by seeking optimal cooperative advertising strategies and equilibrium pricing in a two-member distribution channel. We establish and compare two models: a non-cooperative, leader–follower game and a cooperative game. We develop propositions and insights from the comparison of these models. The cooperative model achieves better coordination by generating higher channel-wide profits than the non-cooperative model with these features: (a) the retailer price is lower to consumers; and (b) the advertising efforts are higher for all channel members. We identify the feasible solutions to a bargaining problem where the channel members can determine how to divide the extra profits.     

Optimal normative policies for marketing of products with limited availability

We develop optimal normative policies for pricing and advertising of products with limited availability by including the traditional product diffusion parameters (Bass, 1969)–innovation and imitation, and the scarcity effects generated due to limited product availability (Swami and Khairnar, 2003). Using optimal control methodology, our pricing policy results suggest that a profit-maximizing firm gradually increases the price as the sales approach the product availability. The optimal normative advertising policy recommends gradually decreasing the expenditure on the awareness advertising and increasing the expenditure on the availability advertising as the product diffusion progresses. These results are illustrated with suitable numerical examples.     

Contract type and decision right of sales promotion in supply chain management with a capital constrained retailer

From the practices of Chinese consumer electronics market, we find there are two key issues in supply chain management: The first issue is the contract type of either wholesale price contracts or consignment contracts with revenue sharing, and the second issue is the decision right of sales promotion (such as advertising, on-site shopping assistance, rebates, and post-sales service) owned by either manufacturers or retailers. We model a supply chain with one manufacturer and one retailer who has limited capital and faces deterministic demand depending on retail price and sales promotion. The two issues interact with each other. We show that only the combination (called as chain business mode) of a consignment contract with the manufacturer’s right of sales promotion or a wholesale price contract with the retailer’s right of sales promotion is better for both members. Moreover, the latter chain business mode is realized only when the retailer has more power in the chain and has enough capital, otherwise the former one is realized. But which one is preferred by customers? We find that the former is preferred by customers who mainly enjoy low price, while the latter is preferred by those who enjoy high sales promotion level.     

考虑退货的电商平台销售策略

本文研究当存在无缺陷退货时,电商平台如何选择合适的销售模式。根据销售价格和退货价格对市场需求以及退货的影响,电商平台可以策略性地选择转售模式和代理模式。我们构建了一个斯塔克伯格博弈模型,电商平台作为领导者选择销售模式,制造商在既定销售模式下进行决策。研究发现,当需求对价格的敏感程度较低、需求对退货价的敏感程度较高及退货量对退货价的敏感程度较低时,平台应提供代理模式,反之应提供转售模式,并探究平台的最优销售策略对制造商利润的影响,从而对平台和制造商的销售策略选择起到一定的指导意义。     

Coordination of cooperative advertising in a two-level supply chain when manufacturer offers discount

We studied the coordination of cooperative advertisement in a manufacturer–retailer supply chain when the manufacturer offers price deductions to customers. With a price sensitive market, the expected demand with cooperative advertising and price deduction is demonstrated. When the manufacturer is a leader, we obtained the optimal national brand name investment, local advertisement and associated manufacturer’s allowance with any given price deduction. When the manufacturer offers more price deduction to customers, the retailer will increase local advertisement if the manufacturer provides the same portion of the local advertising allowance. We obtained the necessary and sufficient condition for the price deduction to ensure an increase of manufacturer’s profit, and a search procedure for determining such an optimal price deduction is provided as well. When the manufacturer and retailer are partners, we obtained the optimal national brand name investment and local advertisement. For any given price deduction, the total profit for the supply chain with cooperative scheme is always higher than that with the non-cooperative scheme. When price elasticity of demand is larger than one, the resulting closed form optimal price deduction with partnership is also obtained. To increase profits for both parties in a supply chain, we recommend that coordination in local and national cooperative advertising with a partnership relationship between manufacturer and retailer is the best solution. The bargaining results show how to share the profit gain between the manufacturer and the retailer, and determine the associated pricing and advertising policies for both parties.     

风险中性零售商的合作广告策略与订货模型

考虑了由一个制造商与一个零售商构成的单期二阶段供应链是否进行合作广告的博弈问题.面对市场需求的不确定性,零售商从制造商处订购报童类型产品销售给消费者,零售商具有风险中性的行为特征.通过不合作广告与合作广告两种情形,制造商与零售商进Stackelberg主从博弈,得到了均衡解,比较后发现,合作广告下的最优解及利润总是优于不合作广告下的最优解和利润,告诉了上下游企业采用合作广告的广告策略.最后,通过数值算例,给出了需求敏感系数对最优决策的影响,同时也论证了有关结论.     

Joint pricing and production decisions for new products with learning curve effects under upstream and downstream trade credits

Today's leading companies generate a substantial portion of their sales from new products. For example, Apple Inc. generates almost 60% of its sales revenue from products launched in the past four years. Hence, the importance of new products cannot be overestimated. Pricing strategy is particularly important during the introductory stage of a new product, when the learning curve effect is most pronounced. Although the learning curve effect reduces production cost significantly during the introduction period of a new product, most researchers have assumed that the production cost remains constant throughout a product's lifecycle. In our opinion, failing to consider this learning curve phenomenon may lead to biased solutions. Furthermore, suppliers often offer manufacturers a short-term interest-free loan (i.e., trade credit) to stimulate sales, while manufacturers adjust price to influence consumer purchasing decisions. However, most researchers have not considered the combined effects of pricing strategy on demand, learning curve effect on production cost, and trade credit influence on lot size. Therefore, we propose an inventory model to determine optimal lot-sizing and pricing strategies with both upstream and downstream trade credit, manufacturer's production cost which follows a learning curve effect, and production quantity influenced by both selling price and trade credit. Then we derive the conditions of the optimal solution. Numerical examples and sensitivity analysis are performed to examine results and provide managerial insights. These results show that the learning curve effect significantly lowers selling price, while tremendously increasing both profit and demand.     

Pricing and two-tier advertising with one manufacturer and one retailer

This paper considers the pricing decisions and two-tier advertising levels between one manufacturer and one retailer where customer demand depends on the retail price and advertisement by a manufacturer and a retailer. We solve a Stackelberg game with the manufacturer as the leader and the retailer as the follower. With price sensitive customer demand and a linear wholesale contract, we obtain the optimal decisions by the manufacturer and the optimal responses by the retailer. Our results show that cost sharing of local advertising does not work well, it is better for the manufacturer to advertise nationally and offer the retailer a lower wholesale price.     

A fractional programming approach for retail category price optimization

We present a new mixed-integer programming (MIP) approach to study certain retail category pricing problems that arise in practice. The motivation for this research arises from the need to design innovative analytic retail optimization techniques at Oracle Corporation to not only predict the empirical effect of price changes on the overall sales and revenue of a category, but also to prescribe optimal dynamic pricing recommendations across a category or demand group. A multinomial logit nonlinear optimization model is developed, which is recast as a discrete, nonlinear fractional program (DNFP). The DNFP model employs a bi-level, predictive modeling framework to manage the empirical effects of price elasticity and competition on sales and revenue, and to maximize the gross-margin of the demand group, while satisfying certain practical side-constraints. This model is then transformed by using the Reformulation–Linearization Technique in tandem with a sequential bound-tightening scheme to recover an MIP formulation having a relatively tight underlying linear programming relaxation, which can be effectively solved by any commercial optimization software package. We present sample computational results using randomly generated instances of DNFP having different constraint settings and price range restrictions that are representative of common business requirements, and analyze the empirical effects of certain key modeling parameters. Our results indicate that the proposed retail price optimization methodology can be effectively deployed within practical retail category management applications for solving DNFP instances that typically occur in practice.     

A dynamic model for advertising and pricing competition between national and store brands

We study the relationship between the pricing and advertising decisions in a channel where a national brand is competing with a private label. We consider a differential game that incorporates the carryover effects of brand advertising over time for both the manufacturer and the retailer and we account for the complementary and competitive roles of advertising. Analysis of the obtained equilibrium Markov strategies shows that the relationship between advertising and pricing decisions in the channel depends mainly on the nature of the advertising effects. In particular, the manufacturer reacts to higher competitive retailer’s advertising levels by offering price concessions and limiting his advertising expenditures. The retailer’s optimal reaction to competitive advertising effects in the channel depends on two factors: (1) the price competition level between the store and the national brands and (2) the strength of the competitive advertising effects. For example, in case of intense price competition between the two brands combined with a strong manufacturer’s competitive advertising effect, the retailer should lower both the store and the national brands’ prices as a reaction to higher manufacturer’s advertising levels. For the retailer, the main advantage from boosting his competitive advertising investments seems to be driven by increased revenues from the private label. The retailer should however limit his investments in advertising if the latter generates considerable competitive effects on the national brand’s sales.     

Temporary price increase during replenishment lead time

Pricing and inventory management make up together revenue management, which is a significant effort to boost revenues out of available resources. Firms use various forms of dynamic pricing, including personalized pricing, markdowns, promotions, coupons, discounts, and clearance sales, to respond to market fluctuations and demand uncertainty. In this paper, we study a temporary price increase policy, a form of dynamic pricing, for a non-perishable product, a practice used by several giant retailers such as Amazon, Walmart, and Apple. We develop a continuous review inventory model that allows for joint replenishment and pricing decisions, where the lead time is not zero. A replenishment decision controls supply, while a pricing decision controls demand. A manager exercises a temporary price increase to slow demand and avoid a stock-out situation while waiting for a shipment, which may not necessarily increase revenues, but decrease stock-out costs. The problem is to solve for the optimal replenishment and the pricing policy parameters that maximize the long-run expected profit. That is, when and how much to order and when to raise the price. In this paper, the inventory level and time trigger a price increase. We solve many numerical examples and perform extensive sensitivity analyses. Our results show that compared to a model that focuses on fixed pricing, our model brings an additional increase in profit of about 13%.     

Optimal trade execution under endogenous pressure to liquidate: Theory and numerical solutions

We study optimal liquidation of a trading position (so-called block order or meta-order) in a market with a linear temporary price impact (Kyle, 1985). We endogenize the pressure to liquidate by introducing a downward drift in the unaffected asset price while simultaneously ruling out short sales. In this setting the liquidation time horizon becomes a stopping time determined endogenously, as part of the optimal strategy. We find that the optimal liquidation strategy is consistent with the square-root law which states that the average price impact per share is proportional to the square root of the size of the meta-order (Bershova & Rakhlin,2013; Farmer et?al., 2013; Donier et?al., 2015; Tóth (2016).Mathematically, the Hamilton–Jacobi–Bellman equation of our optimization leads to a severely singular and numerically unstable ordinary differential equation initial value problem. We provide careful analysis of related singular mixed boundary value problems and devise a numerically stable computation strategy by re-introducing time dimension into an otherwise time-homogeneous task.     

Optimal dynamic marketing-mix policies for frequently purchased products and services versus consumer durable goods: A generalized analytic approach

This paper deals with the qualitative characterization of optimal pricing and advertising policies together with the optimal ratio of the advertising elasticity of demand to its price elasticity over time. The problem is studied for frequently purchased products and services (FPS) as well as consumer durable goods (CDG) in both monopolistic and duopolistic markets. Demand dynamics, cost learning and discounting of future profits are taken into consideration. In addition, both the open-loop and feedback methodologies are pursued to characterize and compare the derived optimal policies.The paper uses an analytical approach to characterize the optimal dynamic policies in a general setting as is mathematically tractable, followed by the analysis of more specific models to gain additional managerial insights while maintaining a certain degree of generality. Optimal FPS marketing-mix policies are shown to be different from their CDG counterparts for both monopolistic and duopolistic markets. While the ratio of advertising elasticity to price elasticity appears to have been governed by similar set of rules for FPS and CDG, the direction of change of such ratio over time looks different from each other. Managerial implications and directions for future research are also discussed.     

Optimal Control of a Logarithmic Advertising Model

A logarithmic advertising model is posed and solved for an optimal dynamic advertising policy for both finite and infinite horizon cases by using optimal control theory. In the case of the infinite horizon, the optimal long-run stationary equilibrium rate of advertising is obtained. It is shown that the optimal advertising policy is independent of the initial level of sales for the logarithmic model.     

Competitive Advertising Under Uncertainty: A Stochastic Differential Game Approach

We analyze optimal advertising spending in a duopolistic market where each firm's market share depends on its own and its competitor's advertising decisions, and is also subject to stochastic disturbances. We develop a differential game model of advertising in which the dynamic behavior is based on the Sethi stochastic advertising model and the Lanchester model of combat. Particularly important to note is the morphing of the sales decay term in the Sethi model into decay caused by competitive advertising and noncompetitive churn that acts to equalize market shares in the absence of advertising. We derive closed-loop Nash equilibria for symmetric as well as asymmetric competitors. For all cases, explicit solutions and comparative statics are presented.     

Stochastic optimal budget decision for advertising considering uncertain sales responses

Firms are faced with uncertain sales responses even though they advertise appropriately. To help marketing managers make optimal budget decisions in this situation, we develop a stochastic model, depicting the problem of advertising budget decision as a special Markov decision process where a new objective, maximizing expected market utility, is proposed. In the model we introduce a two-dimension state variable including accumulative sales, which vary randomly with advertising budget, and the predicted probability that an advertising campaign obtains a full sales response. We make an analysis of the model on the premise of growing infinite market potential, deriving the property of optimal policies and that of optimal value function. These results are successfully used to make advertising budget decisions for a private university in Xi’an, China.