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...procurement decisions, retailers attempt maximize their own profits by balancing the potential opportunity costs associated with unsatisfied demand and excess stock. In general, the quantities that are chosen by the retailers fail to maximize the profits of either their suppliers or the supply chain as whole.
In recent years, a large amount of attention has been devoted to studying mechanisms for addressing agency issues in newsvendor environments. However, most of this attention has been directed toward overcoming the effects of double marginalization with various forms of returns policies, quantity flexibility, price protection and revenue sharing. A quantity discount schedule is another mechanism that is often used in practice. One of the reasons that small independent retailers have struggled to match the prices of the big box retailers is that they lack the volume necessary to obtain quantity discounts from manufacturers. Order sizes for retailers (and quantity-based pricing schemes for suppliers) in an Economic Order Quantity (EOQ) environment are primarily tied to operational efficiencies. For a retailer, they naturally focus on their EOQ while Consumer Packaged Goods suppliers routinely offer incentives to order in quantities consistent with their production batch sizes and efficient transportation sizes (full ocean container, full truckload, etc.) In a newsvendor environment, the pricing is based on the quantity for an entire selling season which relegates the importance of operational issues to a simple matter of how product is released and delivered against that contracted quantity. Although there is a wide literature on the role of quantity discounts in EOQ (long product life cycle) environments, little attention has been paid to understanding how such discount schedules can be used in newsvendor environments.
In this paper, we address the pricing problem faced by a supplier in her interactions with a buyer who faces a single period of uncertain demand. We assume that there is asymmetric information between the supplier and the buyer with respect to demand. This asymmetry can be represented in terms of the supplier's uncertainty over which probability distribution, from a discrete or continuous set, characterizes the buyer's demand type. In this environment, a single-price contract is unable to achieve the maximum profit for the supply chain, nor extract the maximum profit for the supplier. We explore several approaches that a supplier might take that would dominate a single-price contract. One of these involves the supplier offering the buyer a schedule of specific quantity-price pairs. Such an approach, which we refer to as fixed package pricing, can be derived as the solution to a mechanism design problem. From the revelation principle, we know that the supplier would need to offer no more than one quantity-price pair for each buyer type in order to maximize her profits. Although we confirm that this fixed package pricing approach is indeed the optimal solution to the supplier's mechanism design problem, we discuss the reasons why it may be difficult to operationalize in practice. The other two approaches that we explore are all-unit and incremental quantity discounts, both of which are easily operationalized and are commonly observed in practice. Our primary objective is not to explain why quantity discounts are used in practice, but rather to evaluate how well they allow a supplier to deal with asymmetric demand information and whether one form of quantity discount is more effective than another.
The remainder of our paper is organized as follows. In Section 2 we review the literature on quantity discounts, returns policies, and back-up agreements. In Section 3, we develop a model of asymmetric demand information in a supply chain involving a supplier and a newsvending retailer. We first approach the supplier's problem as one of classic mechanism design, the solution to which would take the form of fixed package pricing. We then examine two common forms of quantity discounts: the all-unit and the incremental. In an all-unit discount, when the buyer's quantity exceeds a given threshold, the price decrease applies to all of the units that he buys. In an incremental discount, the price decrease applies to only the additional, i.e., incremental, units that he buys beyond the threshold. Our analysis shows that the supplier will always prefer an optimal all-unit discount over an incremental discount, but that either form of quantity discount places more restrictions on the supplier's pricing schedule than would a fixed package pricing approach. After presenting numerical results in Section 4, we discuss our results and draw conclusions. Throughout the paper, we arbitrarily adopt the convention of referring to the supplier with female pronouns and to the buyer(s) with male pronouns.
2. Related literature
A major objective of our research is to understand how asymmetry with respect to demand information can be addressed in a decentralized supply chain. Typically, research on principal-agent issues in supply chains has tended to either ignore information asymmetry or to consider it with respect to the buyer's costs. For example, Corbett and Tang (1999) and Corbett and deGroote (2000) develop an EOQ-based model in which the buyer and the supplier have asymmetric information regarding the buyer's holding costs, and develop an optimal nonlinear pricing policy. Subsequently, Ha (2001) considers asymmetric cost information between a supplier and retailer in a newsvendor context. Several authors have addressed the issue of designing a quantity discount scheme for heterogeneous buyer types or asymmetric information in EOQ environments, including Lal and Staelin (1984), Dolan (1987), Drezner and Wesolowsky (1989), and Martin (1993).
Less work has been done on asymmetric demand information, in spite of the fact that it is arguably a more common practical problem than is asymmetric cost information. Atkinson (1979) was among the first to recognize and model the issue of asymmetric demand information in a stochastic demand environment. His approach is consistent with the traditional principal-agent framework in which the objective is to allow an owner to provide proper incentives to a manager who has better demand information. Specifically, he studies a contract in which a newsvending manager receives a share of the additional profits that are generated as a result of using his order quantity instead of the one that would have been chosen by the owner.
Porteus and Whang (1999) develop a model of informational asymmetry regarding demand in the context of long-term contracting and stochastic demand. Specifically, they assume that there are two stochastically ordered demand distributions for the buyer, and that the supplier knows only the relative probability with which the buyer's demand is from either distribution. In their model, the supplier offers a discrete set of prices, each one corresponding to an amount of installed capacity. Cachon and Lariviere (2001) use a similar representation of information asymmetry in a single-period model of a capacity reservation contract. More recently, Arya and Mittendorf (2004) show that the use of a returns policy can be beneficial to a manufacturer that sells to a retailer who has private information about demand. The model used by Arya and Mittendorf assumes that the only source of uncertainty is the valuation that a finite number of consumers have for the product. If the retailer has private information about this valuation, then the manufacturer can do a better job of discriminating among different private information values if she includes a returns allowance with each price/quantity pair offered to the retailer.
Much of our research addresses the issue of how quantity discounts can be used to address asymmetric demand information. Although there is a large literature on quantity discounts, nearly all of it has been based on situations in which there is on-going, deterministic demand. In the marketing literature, Jeuland and Shugan (1983) argue that quantity discounts can be used to overcome double marginalization, whereas Moorthy (1987) argues that they can be used for price discrimination. Ingene and Parry (1995) extended this work to multiple independent retailers. The operations literature on quantity discounts has tended to focus on EOQ environments where it can be desirable to provide incentives for retailers to order in larger quantities to control the supplier's setup costs. Crowther (1967), Heskett and Ballou (1967) and Monahan (1984) were among the first to consider quantity discounts in EOQ environments. A good review of the operations literature on quantity discounts can be found in Viswan than and Wang (2003), who classify the literature according to whether there is one or multiple buyers and whether there is price-sensitive demand. Recent contributions to the quantity discount literature include Boyaci and Gallego (2002), who introduce the issue of inventory ownership into the determination of the quantity discount that coordinates the channel, and Viswanthan and Wang (2003) who allow for a quantity discount and a total volume discount to be used in combination. Chen et al. (2001) show that although a traditional quantity discount cannot be used to achieve the first best solution in an environment in which a single supplier serves heterogenous retailers, coordination can be achieved with a discount which is based on a retailer's annual sales volume, order quantity, and order frequency. Shin and Benton (2004) perform an extensive simulation analysis to test the effectiveness of quantity discounts under a variety of conditions.
Since our research involves a comparison between all-unit and incremental discounts, it is worth pointing out that, with two notable exceptions, nearly all of the work on quantity discounts has focused on all-unit discounts. Although both Kim and Hwang (1988) and Weng (1995) consider incremental discounts, neither one examines the question of whether a supplier should prefer them over all-units discounts. In an EOQ context with symmetric information, Weng (1995) shows that either form of discount can be used to achieve channel coordination, and that the supplier would be indifferent between the two. Our work complements this result by demonstrating that, in a newsvendor context with asymmetric information, a supplier will prefer the all-unit discount.
Our work is also related to the potential incentive problem that exists between a supplier and a newsvending retailer that results in understocking. This issue was first identified by Pasternack (1985), and has received widespread attention. Several excellent review papers have been published. Lariviere (1999) provides a general overview of research that addresses the relationship between a supplier and a newsvending retailer. Tsay et al. (1999) focus on papers that seek to offer guidance to a supplier regarding the negotiation of the terms of trade with a newsvending retailer. Another review paper, Cachon (1998), addresses the issues related to the interactions that may occur among independent newsvendors that are supplied by a single supplier. A more recent and very thorough review of the literature in this area is provided by Cachon (2004).
The contribution of our paper is that we examine how a supplier can employ quantity discounts to best extract rents from a newsvending retailer who possesses better demand information than does the supplier. In contrast to returns policies, back-up agreements, or quantity flexibility, quantity discounts require no additional flow of information or logistics between the supplier and the retailer following the initial transaction. Since this can be a significant operational advantage for some products, we feel that this is a topic that merits attention.
3. The model
We consider the situation faced by a supplier who wants to design a pricing schedule that...
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