Up Then Down: Bid‐Price Trends in Revenue Management

In the classic revenue management (RM) problem of selling a fixed quantity of perishable inventories to price‐sensitive non‐strategic consumers over a finite horizon, the optimal pricing decision at any time depends on two important factors: consumer valuation and bid price. The former is determined exogenously by the demand side, while the latter is determined jointly by the inventory level on the supply side and the consumer valuations in the time remaining within the selling horizon. Because of the importance of bid prices in theory and practice of RM, this study aims to enhance the understanding of the intertemporal behavior of bid prices in dynamic RM environments. We provide a probabilistic characterization of the optimal policies from the perspective of bid‐price processes. We show that an optimal bid‐price process has an upward trend over time before the inventory level falls to one and then has a downward trend. This intertemporal up‐then‐down pattern of bid‐price processes is related to two fundamental static properties of the optimal bid prices: (i) At any given time, a lower inventory level yields a higher optimal bid price, which is referred to as the resource scarcity effect; (ii) Given any inventory level, the optimal bid price decreases with time; that is referred to as the resource perishability effect. The demonstrated upward trend implies that the optimal bid‐price process is mainly driven by the resource scarcity effect, while the downward trend implies that the bid‐price process is mainly driven by the resource perishability effect. We also demonstrate how optimal bid price and consumer valuation, as two competing forces, interact over time to drive the optimal‐price process. The results are also extended to the network RM problems.     

不确定环境下的期权价格上下界研究

传统的期权定价理论总是建立在标的资产价格分布的严格假设下,而没有考虑分布的不确定性。本文对标的资产价格分布的严格假设进行放松,分别在仅知到期日标的资产价格的前二阶矩及前三阶矩,而不知道其具体分布的条件下,对期权进行定价。由于信息不充分及分布不确定,推导出的期权价格为一个区间。我们针对有限信息条件下求解期权价格上下界的问题,建立数学规划模型,并将其转化为对偶规划问题进行求解。对此上下界和Black-Scholes价格进行对比分析后发现,Black-Scholes价格介于此上下界之间,相对于采用前二阶矩推导的上下界,采用前三阶矩信息推导的上下界更窄。在使用香港恒生指数权证数据进行的时序分析及横截面分析中发现,市场价格确实介于上下界之间,上下界区间随波动率及剩余存续期的减小而缩小。采用本文的定价方法,不需要对资产价格分布进行严格假设,故可提高定价模型的稳健性,有助于投资者结合期权价格上下界及自己的主观判断进行投资决策。     

Distributing a Product Line in a Decentralized Supply Chain

Although there is a rich literature on single product distribution in decentralized supply chains, the incentive problems that arise in distributing a product line have largely not been investigated. In practice, most manufacturers distribute a line of products with different features and qualities and not just a single product. Consider a manufacturer who distributes a product line through competing downstream retailers. In this setting, we investigate how and why the retailers' price and inventory decisions deviate from the centrally optimal decisions. Due to substitution between different product variants, as well as between different retailers, the incentive problems associated with distributing a product line are more complicated than that of distributing a single product. We characterize retailers' incentive distortions under a residual‐claimancy contract, and construct contracts that achieve channel coordination. We show that retail price floors or inventory buybacks, appropriately tailored to each product variant, are among the contracts that can achieve coordination. Using numerical simulations, we demonstrate how the optimal contract terms (such as wholesale prices and buyback prices) for each variant are influenced by the parameters of an underlying consumer choice model.     

A Choice‐Based Dynamic Programming Approach for Setting Opaque Prices

Opaque pricing is a form of pricing where certain characteristics of the product or service are hidden from the consumer until after purchase. In essence, opaque selling transforms a differentiated good into a commodity. Opaque pricing has become popular in service pricing as it allows firms to sell their differentiated product at higher prices to regular brand loyal customers while simultaneously selling to non‐brand loyal customers at discounted prices. We use a nested logit model in combination with logistic regression and dynamic programming to illustrate how a service firm can optimally set prices on an opaque sales channel. The choice model allows the characterization of consumer trade‐offs when purchasing opaque products while the dynamic programming approach allows the characterization of the optimal pricing policy as a function of inventory and time remaining. We compare optimal prices and expected revenues when dynamic pricing is restricted to daily price changes. We provide an illustrative example using data from an opaque selling mechanism ( Hotwire.com ) and a Washington DC‐based hotel.     

非流动性市场中的跨期最优消费和投资策略

本文在Merton跨期最优消费和资产组合的理论框架内引入非流动性资产,构造了一个三资产的连续时间经济模型,探讨流动性约束对投资者最优消费和投资决策的影响。本文用动态规划方法给出了跨期优化问题的解析解。数值分析表明对投资者而言,非流动性资产的真实价值低于其市值,必须引入影子价格来刻划这种流动性效应;流动性约束降低了投资者的福利,并且显著地影响到投资者的消费和投资策略。     

Impact of Price Postponement on Capacity and Flexibility Investment Decisions

Investments in dedicated and flexible capacity have traditionally been based on demand forecasts obtained under the assumption of a predetermined product price. However, the impact on revenue of poor capacity and flexibility decisions can be mitigated by appropriately changing prices. While investment decisions need to be made years before demand is realized, pricing decisions can easily be postponed until product launch, when more accurate demand information is available. We study the effect of this price decision delay on the optimal investments on dedicated and flexible capacity. Computational experiments show that considering price postponement at the planning stage leads to a large reduction in capacity investments, especially in the more expensive flexible capacity, and a significant increase in profits. Its impact depends on demand correlation, elasticity and diversion, ratio of fixed to variable capacity costs, and uncertainty remaining at the times the pricing and production decisions are made.     

Optimal Policies for a Two‐Product Inventory System under a Flexible Substitution Scheme

We study a two‐product inventory model that allows substitution. Both products can be used to supply demand over a selling season of N periods, with a one‐time replenishment opportunity at the beginning of the season. A substitution could be offered even when the demanded product is available. The substitution rule is flexible in the sense that the seller can choose whether or not to offer substitution and at what price or discount level, and the customer may or may not accept the offer, with the acceptance probability being a decreasing function of the substitution price. The decisions are the replenishment quantities at the beginning of the season, and the dynamic substitution‐pricing policy in each period of the season. Using a stochastic dynamic programming approach, we present a complete solution to the problem. Furthermore, we show that the objective function is concave and submodular in the inventory levels—structural properties that facilitate the solution procedure and help identify threshold policies for the optimal substitution/pricing decisions. Finally, with a state transformation, we also show that the objective function is ‐concave, which allows us to derive similar structural properties of the optimal policy for multiple‐season problems.     

Committed Versus Contingent Pricing Under Competition

Should capacitated firms set prices responsively to uncertain market conditions in a competitive environment? We study a duopoly selling differentiated substitutable products with fixed capacities under demand uncertainty, where firms can either commit to a fixed price ex ante, or elect to price contingently ex post, e.g., to charge high prices in booming markets, and low prices in slack markets. Interestingly, we analytically show that even for completely symmetric model primitives, asymmetric equilibria of strategic pricing decisions may arise, in which one firm commits statically and the other firm prices contingently; in this case, there also exists a unique mixed strategy equilibrium. Such equilibrium behavior tends to emerge, when capacity is ampler, and products are less differentiated or demand uncertainty is lower. With asymmetric fixed capacities, if demand uncertainty is low, a unique asymmetric equilibrium emerges, in which the firm with more capacity chooses committed pricing and the firm with less capacity chooses contingent pricing. We identify two countervailing profit effects of contingent pricing under competition: gains from responsively charging high price under high demand, and losses from intensified price competition under low demand. It is the latter detrimental effect that may prevent both firms from choosing a contingent pricing strategy in equilibrium. We show that the insights remain valid when capacity decisions are endogenized. We caution that responsive price changes under aggressive competition of less differentiated products can result in profit‐killing discounting.     

基于异质价格信念的金融资产泡沫形成机制

假设风险资产未来价格变化由其过去价格变化外推形成,基于现金流股息冲击提出一个新的资产价格泡沫模型研究金融资产泡沫形成机制、提炼其内在特征、获得金融市场稳定发展的相关启示。假设异质价格信念投资者具有常绝对风险规避效用偏好,基于期望效用模型获得基本面投资者和外推型投资者的最优风险资产需求函数;进一步,假设外推型投资者在资产交易过程中部分关注风险资产基本价值,修正其最优风险资产需求函数,在市场出清状态下获得资产价格泡沫模型;基于模型参数设定模拟金融资产泡沫形成机制,并结合金融市场证据分析其内在特征;最后,从投资者异质性和无风险资产收益率层面设定参数值对比讨论影响金融资产泡沫强弱程度的因素。结果表明：在正的现金流股息冲击下,外推交易行为导致金融资产泡沫,且具有滞后性;金融资产泡沫具有典型的三阶段特征,量价齐升是金融资产泡沫产生的显著标志;无风险资产供给是金融资产泡沫的抑制器;投资者结构及其价格信念差异与金融资产泡沫的强弱程度密切相关。研究不但为金融资产泡沫形成机制提供了一个分析框架,而且为金融市场稳定发展提出相关决策参考。     

The Impact of Capacity Costs on Product Differentiation in Delivery Time,Delivery Reliability,and Price

We develop an analytical framework for studying the role capacity costs play in shaping the optimal differentiation strategy in terms of prices, delivery times, and delivery reliabilities of a profit‐maximizing firm selling two variants (express and regular) of a product in a capacitated environment. We first investigate three special cases. The first is an existing model of price and delivery time differentiation with exogenous reliabilities, which we only review. The second focuses on time‐based (i.e., length and reliability) differentiation with exogenous prices. The third deals with deciding on all features for an express variant when a regular product already exists in the marketplace. We subsequently address the integrative framework of time‐and‐price‐based differentiation for both products in a numerical study. Our results shed light on the role that customer preferences towards delivery times, reliabilities and prices, and the capacity costs (absolute and relative) have on the firm's optimal product positioning policy.     

Coordinating pricing and production decisions for multiple products

We consider a dynamic problem of joint pricing and production decisions for a profit-maximizing firm that produces multiple products. We model the problem as a mixed integer nonlinear program, incorporating capacity constraints, setup costs, and dynamic demand. We assume demand functions to be convex, continuous, differentiable, and strictly decreasing in price. We present a solution approach which is more general than previous approaches that require the assumption of a specific demand function. Using real-world data from a manufacturer, we study problem instances for different demand scenarios and capacities and solve for optimal prices and production plans. We present analytical results that provide managerial insights on how the optimal prices change for different production plans and capacities. We extend some of the earlier works that consider single product problems to the case of multiple products and time variant production capacities. We also benchmark performance of proposed algorithm with a commercial solver and show that it outperforms the solver both in terms of solution quality and computational times.     

Scalable Dynamic Bid Prices for Network Revenue Management in Continuous Time

This study develops an approximate optimal control problem to produce time‐dependent bid prices for the airline network revenue management problem. The main contributions of our study are the analysis of time‐dependent bid prices in continuous time and the use of splines to modify the problem into an approximate second‐order cone program (ASOCP). The spline representation of bid prices permits the number of variables to depend solely on the number of resources and not on the size of the booking horizon. The advantage of this framework is the ASOCP's scalability, which we demonstrate by solving for bid prices on an industrial‐sized network. The numerical experiments highlight the ASOCP's ability to solve industrial sized problems in seconds.     

Optimal algorithms for online time series search and one-way trading with interrelated prices

The basic models of online time series search and one-way trading are introduced by El-Yaniv et al. in Algorithmica 30(1), 101–139 (2001) where it is assumed that the prices are bounded within interval [m,M] (0<m<M). In this paper, we consider another case where every two consecutive prices are interrelated, that is, the variation range of each price depends on its preceding price. We present optimal deterministic online algorithms for the two problems, respectively. According to one conclusion in Algorithmica 30(1), 101–139 (2001), we further point out that for the case we considered, an optimal deterministic algorithm for the one-way trading problem can be regarded as an optimal randomized one for the time series search problem, and randomization is useless for the one-way trading problem.     

Managing Clearance Sales in the Presence of Strategic Customers

We study the effect of strategic customer behavior on pricing and rationing decisions of a firm selling a single product over two periods. The seller may limit the availability of the product (that is, ration) in the second (clearance) period. Some customers are strategic and respond to the firm's decisions by timing their purchases. When capacity is nonconstraining and the seller has pricing flexibility, we show that rationing in the clearance period cannot improve revenue. However, when prices are fixed in advance, rationing can improve revenue. In the latter case, we conduct a detailed analysis for linear and exponential demand curves and derive explicit expressions for optimal rationing levels. We find that the policy of doing the better of not restricting availability at the clearance price or not offering the product at the clearance price is typically near optimal. Our analysis also suggests that rationing—although sometimes offering considerable benefit over allowing unrestricted availability in the clearance period—may allow the seller to obtain only a small fraction of the optimal revenue when the prices are chosen optimally without rationing. We extend the analysis to cases where the capacity is constraining and obtain similar results.     

消费者导向类型与电商全渠道决策研究

从现实出发，基于不同消费者导向类型研究了电商的全渠道决策问题。首先建立电商单渠道销售的基础模型，并得出最优产品组合、价格和交付时间。在此基础上，给出了消费者为产品导向型和渠道导向型情况下电商采取全渠道决策的条件，得出双渠道产品组合、价格及交付时间的最优决策；并进行了数值验证。研究发现，同一渠道提供的产品其最优价格相同，最优交付时间只和消费者耐心程度及交付成本函数有关。若消费者为产品导向型，仅当双渠道运营成本之差较小且消费者耐心程度较低时，电商进入线下渠道才有利可图，且在线下渠道提供最受欢迎的产品，线上渠道提供剩余产品。若消费者为渠道导向型，电商进入线下渠道必然有利可图，且在线下渠道提供最受欢迎的产品，线上渠道提供所有产品。     

EXcess Idle Time

We introduce a novel economic indicator, named excess idle time (EXIT), measuring the extent of sluggishness in financial prices. Under a null and an alternative hypothesis grounded in no‐arbitrage (the null) and market microstructure (the alternative) theories of price determination, we derive a limit theory for EXIT leading to formal tests for staleness in the price adjustments. Empirical implementation of the theory indicates that financial prices are often more sluggish than implied by the (ubiquitous, in frictionless continuous‐time asset pricing) semimartingale assumption. EXIT is interpretable as an illiquidity proxy and is easily implementable, for each trading day, using transaction prices only. By using EXIT, we show how to estimate structurally market microstructure models with asymmetric information.     

Insider Trading With a Random Deadline

We consider a model of strategic trading with asymmetric information of an asset whose value follows a Brownian motion. An insider continuously observes a signal that tracks the evolution of the asset's fundamental value. The value of the asset is publicly revealed at a random time. The equilibrium has two regimes separated by an endogenously determined time T. In [0, T), the insider gradually transfers her information to the market. By time T, all her information has been transferred and the price agrees with the market value of the asset. In the interval [T, ∞), the insider trades large volumes and reveals her information immediately, so market prices track the market value perfectly. Despite this market efficiency, the insider is able to collect strictly positive rents after T.     

考虑维修服务质量的产品质保联合决策

在售后服务竞争日益激烈的大环境下,维修服务质量对于企业的意义越来越重要。经典的质保决策问题通常关心质保期和产品价格的联合决策问题,本文在此基础上进一步考虑了质保维修的服务质量问题。质保期、产品价格以及维修服务质量在产品生命周期内存在动态的关联关系,目前同时考虑此三类因素的研究尚不多见。本文以具有重复购买行为的大众消费品为研究对象,构建了包括成本模型、需求模型、利润模型在内的最优控制模型,并应用最大化原理对模型进行求解分析,在此基础上进一步对最优策略在静态市场和动态市场的应用分别做了具体分析和数值试验。     

Two Backorder Compensation Mechanisms in Inventory Systems with Impatient Customers

We study a compensation mechanism design problem with customer‐choice behavior in a continuous review setting where the production and demand processes are stochastic. When a stockout occurs, the firm controls backorders on the basis of certain compensation policies. Customers make decisions to maximize their utility, which is decreasing in the price, the waiting time, and the customer's impatience factor. We assume that the impatience factor is private information held by the customer only. Two compensation mechanisms are designed to control backorders, namely uniform compensation and priority auction with an admission price. Under uniform compensation, the firm offers the same discount to all customers, whereas under auction compensation, priority is granted according to the customers' bid prices. We obtain the optimal stockout price and base stock level under each mechanism, and analyze the properties of the respective optimal policies. Assuming linear waiting costs with uniformly distributed impatience factor, we find that the auction mechanism (1) maintains a lower base stock level and results in greater profit and (2) benefits customers with relatively lower or higher impatience factors, but customers with a medium impatience factor may be rendered worse off. We further show that both compensation mechanisms are suitable for products with a high unit profit, a high lost sales penalty cost, and a high holding cost.     

Pricing Decisions During Inter‐Generational Product Transition

How should companies price products during an inter‐generational transition? High uncertainty in a new product introduction often leads to extreme cases of demand and supply mismatches. Pricing is an effective tool to either prevent or alleviate these problems. We study the optimal pricing decisions in the context of a product transition in which a new‐generation product replaces an old one. We formulate the dynamic pricing problem and derive the optimal prices for both the old and new products. Our analysis sheds light on the pattern of the optimal prices for the two products during the transition and on how product replacement, along with several other dynamics including substitution, external competition, scarcity, and inventory, affect the optimal prices. We also determine the optimal initial inventory for each product and discuss a heuristic method.