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Quantity Discounts in Single Period Supply Contracts with Asymmetric Demand Information ∗ Apostolos Burnetas†, Stephen M. Gilbert ‡, Craig Smith § August 13, 2005 Abstract We investigate how supplier can use a quantity discount schedule to influence the stocking decisions of a downstream buyer that faces a single period of stochastic demand. In contrast to much of the work that has been done on single-period supply contracts, we assume that there are no interactions between the supplier and the buyer after demand information is revealed and that the buyer has better information about the distribution of demand than does the supplier. We characterize the structure of the optimal discount schedule for both all-unit and incremental discounts and show that the supplier can earn larger profits with an all-unit discount. (Supply Chain Management, Channel Coordination, Channels of Distribution, Asymmetric Information) ∗ The first author acknowledges support by the University of Athens Research Committee via program Kapodistrias † University of Athens, Department of Mathematics, Athens 15784, Greece, [email protected] ‡ The University of Texas at Austin, [email protected] § Management Department, CBA 4.202, Diageo, plc, 9 W. Broad St., 3rd Floor,Stamford, CT 06902 [email protected] Austin, TX 78712, 1 Introduction In many industries, such as fashion apparel, popular toys, etc., the combination of long lead times and short product life cycles forces retailers to make procurement decisions while there is still a great deal of uncertainty regarding demand. To make these newsvendor procurement decisions, the retailers attempt to maximize their own profits by balancing the potential opportunity costs associated with unsatisfied demand and excess stock. Unfortunately, there are a variety of reasons why the quantities the retailers choose fail to maximize either the profits of their suppliers or of the supply chain as a 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 EOQ environment are primarily tied to operational efficiencies. For a retailer, they naturally focus on their EOQ while CPG (Consumer Packaged Goods) suppliers routinely offer incentives to order in quantities based on factors such as their production batch size 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 mechanism design 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. In this environment, a fixed 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 approach, to which we refer as fixed package pricing, would be where the supplier offers the buyer a schedule of specific quantity-price pairs. 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 also note that in practice, we often see suppliers offering a discrete set of quantity discounts instead of fixed package pricing. In contrast to fixed package pricing which restricts the buyer to only as many quantity choices as there are options in the pricing schedule, a quantity discount always allows a buyer a full continuum of quantity choices, regardless of how many options are in the price schedule. Thus, a quantity discount may represent a compromise between the need to provide the buyer with the flexibility to choose from a continuum of order quantities and the simplicity of a discrete set of menu options. However, our intention 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. After introducing the mechanism design problem, we then examine two common forms of quantity discount: the all-unit and the incremental. In an all-unit discount, when the buyer’s quantity exceeds a given threshold, the corresponding price decrease applies to all of the units that he buys. In an incremental discount, when the buyer’s quantity exceeds a given threshold, the corresponding 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-agency issues in supply chains has tended to either ignore information asymmetry or to consider it with 3 respect to the buyer’s costs. For example, Corbett & deGroote (2000) and Corbett & Tang (1999) 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 non-linear pricing policy. Recently, 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 & Staelin (1984), Dolan (1987), Drezner & 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 & 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 & Lariviere (2001) use a similar representation of information asymmetry in a single period model of a capacity reservation contract. 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 is been based on situations in which there is on-going, deterministic demand. In the marketing literature, Jeuland & Shugan (1983) argue that quantity discounts can be used to overcome double marginalization, while Moorthy (1987) argues that they can be used for price discrimination. Ingene & Parry (1995) extended this work to multiple independent retailers. The operations literature on quantity discounts has tended to focus on Economic Order Quantity (EOQ) environments where it can be desirable to provide incentives for retailers to order in larger quantities to control the supplier’s setup costs. Heskett & Ballou (1967), Crowther (1967), and Monahan (1984) were 4 among the first to consider quantity discounts in EOQ environments. A good review of the operations literature on quantity discounts can be found in Viswanthan & 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 & Gallego (2002), who introduce the issue of inventory ownership into the determination of the quantity discount that coordinates the channel, and Viswanthan & Wang (2003) who allow for a quantity discount and a total volume discount to be used in combination. Chen, Federgruen, & Zheng (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 is based on a retailer’s annual sales volume, order quantity, and order frequency. Shin & Benton (2004) perform and extensive simulation analysis to test the effectiveness of quantity discounts under a variety of environmental 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 & Hwang (1988) and Weng (1995) consider incremental discounts, neither one examines the question of whether a supplier should prefer one to the other as we do. In fact, Weng (1995) shows that in an EOQ context with symmetric information, either an all-unit and incremental 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 wide-spread 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, Nahmias, & Agrawal (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 a quantity discount to best extract rents from a newsvending retailer who possesses better demand information 5 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 will be offered to a buyer who is procuring a product in order to satisfy a single period of uncertain demand. The buyer is an intermediary, either a retailer or value-adding manufacturer, who sells to end consumers. For each unit sold, the buyer receives an exogenously specified revenue of r. We assume that any additional costs incurred by the buyer, e.g. for shipping material handling, etc., are linear in the number of units sold, and these costs have been normalized to zero. We assume that the supplier faces a constant marginal cost of production, denoted c, and has less information about the distribution of end-demand for this product than does the buyer. We model this informational asymmetry by considering n buyer types. A buyer of type i = 1, ..., n faces uncertain demand that follows a continuous probability distribution with density fi (x) and cumulative distribution function Fi (x). Although the buyer knows his type, the supplier has only probabilistic information about the buyer’s type. Let pi be the probability that the buyer is of type i = 1, ..., n. Aside from having different demand distributions, the different buyer types are identical. We assume that the demand distributions have support only for x ≥ 0 and that distribution i stochastically dominates distribution j for j < i, i.e. Fi (x) ≤ Fj (x) for all x ≥ 0. This assumption of first-order stochastic dominance is similar to the one made by Porteus & Whang (1999) and Cachon & Lariviere (2001). To represent the fact that buyers are typically not captive to specific suppliers, we assume that the buyer has access to an alternative source of supply for the product at an exogenous per-unit price of u where c < u < r. Thus, the buyer will not purchase from the quantity discount schedule that is offered unless he can earn at least as large a profit as he could by purchasing from the alternative supplier at a constant wholesale price of u. After the supplier announces her pricing schedule, the buyer evaluates his purchasing options, including both the supplier’s pricing schedule and the alternative source, and responds by purchasing the quantity that maximizes his own expected profits. The supplier then produces the 6 quantity that has been ordered at a cost-per unit of c, and the buyer receives his entire order quantity prior to the start of the selling season. During the selling season, demand is realized and the buyer sells the product at the exogenous retail price r. To simplify the presentation, we assume that the product has zero salvage value at the conclusion of the selling season. We do not allow for returns, back-up agreements, or other mechanisms that might be used in conjunction with quantity discounts. This can be justified by the fact that it is often desirable to avoid additional informational and material flows following the initial transaction. 3.1 Preliminaries We first oberve that if a buyer of type i orders quantity q, then his expected revenue can be expressed as: Ri (q) = r Z q 0 xfi (x)dx + rq Z ∞ q fi (x)dx = r Z q 0 (1 − Fi (x))dx (3.1) Lemma 3.1 For buyer types i < j, the expected incremental revenue earned by a buyer of type i from having T 0 > T instead of T units satisfies the following: 0 Ri (T ) − Ri (T ) = r Z T0 T (1 − Fi (x))dx ≤ r Z T0 T (1 − Fj (x))dx = Rj (T 0 ) − Rj (T ) (3.2) Let us define πi (q, w) to be the expected gross profit earned by buyer i if he orders quantity q at a per-unit price of w, i.e. πi (q, w) = Ri (q) − qw. Thus, with no restrictions on the quantity ordered, the buyer faces a simple newsvendor problem. Let also qi (w) be the optimal solution to this unconstrained newsvendor problem, given a per-unit procurement cost of w. Thus: µ qi (w) = Fi−1 1 − w r ¶ (3.3) Let ²i represent the maximum profit that buyer i could earn if he procured the product from the external market at price u. It follows that ²i = πi (qi (u), u), and that ²i represents the threshold level of profits that a buyer of type i must be able to earn under a quantity discount scheme in order to be induced to participate. From the perspective of the total supply chain (the supplier and the buyer together), the marginal cost of production is c and the marginal revenue from an additional sale is r. It follows that the total expected supply chain profits that can be obtained from the market served by buyer type i are maximized when the order quantity is qi (c). If the buyer orders quantity qi (c), then the 7 total expected gross profit in the channel is: πi (qi (c), c) = Ri (qi (c)) − cqi (c) = r Z qi (c) 0 xfi (x)dx (3.4) where the final equality follows from the definition of qi () that is given in (3.3). Because u > c, we must have qi (c) > qi (u) and πi (qi (c), c) > ²i . Clearly, if the supplier could induce the buyer to order quantity qi (c) and extract all additional rents, this would maximize the supplier’s profit. Let Ti be the number of units that the supplier offers so that the buyer must purchase all Ti units or none of them. By using (3.4), it can be shown that the per-unit price corresponding to quantity Ti = qi (c) that allows the buyer to earn expected gross profit equal to his threshold profit ²i is: Ri (qi (c)) − ²i r w = = qi (c) ∗ R qi (c) 0 xf (x)dx − ²i +c qi (c) (3.5) If there were no asymmetry of information, i.e. the buyer is of type i = 1 with probability p1 = 1, then the supplier optimally offer the buyer an opportunity to order Ti = qi (c) for a total price of w∗ qi (c). This would coordinate the channel as well as insure that the buyer would weakly prefer our supplier over the alternative source. Thus, the supplier could maximize the total channel profits and extract all but the buyer’s threshold level of profits for herself. 3.2 Mechanism Design When the supplier is not certain about the buyer’s distribution of demand, then she may want to offer more than one purchasing option to the buyer. In fact, the problem can be framed as a classic mechanism design problem where, according to the revelation principle, the supplier can maximize her profits by establishing no more than n different quantity-price pairs to the buyer, i.e. one for each buyer type. Let Ti and wi be the quantity and per-unit price intended for buyer type i. Thus, the supplier’s problem can be expressed as: (MD) P max{wi ,Ti } { ni=1 pi (wi − c)Ti } s.t. πi (Ti , wi ) ≥ ²i πi (Ti , wi ) ≥ πi (Tj , wj ) wi ≥ 0 , Ti ≥ 0 i = 1, ..., n i = 1, ..., n; j 6= i i = 1, ..., n (IRi ) (ICi,j ) The first set of constraints, (IRi ), insure that each buyer type prefers the purchasing option that is intended for him over purchasing options designed for other types. The second set of constraints, ICi,j , insure that each buyer type earns at least as much under the supplier’s intended purchasing option as he would if he used his alternative procurement source. These are known as the Incentive 8 Compatibility (IC) and the Individual Rationality (IR) constraints respectively. (See Fudenberg & Tirole (1998) for more on these issues.) Because the solution to this mechanism design problem involves the design of a pricing schedule that makes a specific quantity-price pair attractive to each buyer type, let us refer to it as the fixed-package pricing problem. This mechanism design problem can be expressed for a continuum of buyer types, so long as their demand distributions are differentiated according to a single parameter and adhere to the assumption of first-order stochastic ordering. In the appendix we present the solution to a continuous version of this mechanism design problem in which each buyer type is assumed to have a uniform demand distribution, but higher indexed buyer types have larger mean demand. It is interesting to observe that, if we interpret quantity as a measure of product quality, and the buyer’s expected incremental revenue as the consumer’s marginal utility, then the structure of our problem is somewhat similar to that for the product-line design problem that has been studied by Mussa & Rosen (1978) and Moorthy (1984), among others. In that problem, the supplier must design a product-line in an environment in which each product offering is characterized by a unidimensional measure of quality. For each specific product offering, the supplier also specifies a price. The basic idea in these product-line design problems is that, by offering a product-line that is differentiated, it is possible for the supplier to exploit differences in consumers’ marginal utilities for quality. However, while it is quite natural for firm to offer consumers a discrete set of product variants, there may be less willingness among buyers to accept a discrete set of quantities, and this may be one reason why we see quantity discounts offered in practice. As previously mentioned, quantity discounts that are observed in practice generally come in two general forms: all-unit and incremental. Under either form of discount, multiple quantity thresholds can be offered. Because both of these types of discount are common in practice, it is of interest to see how well they work with respect to one another as well as with respect to the fixed package pricing problem. 3.3 An all-unit Discount Under Asymmetric Information In our original mechanism design problem, the supplier can offer a discrete set of price-quantity pairs from which the buyer must choose. Under a quantity discount, the supplier can offer a discrete set of prices (either all-unit or incremental), but she must allow the buyer to purchase from a continuum of quantities. Thus, a quantity discount allows the buyer greater flexibility in how he will respond. As we will show, this additional flexibility benefits the buyer at the expense of the supplier. 9 An all-unit quantity discount schedule can be represented by a series of threshold quantity breakpoints (T1 ≤ T2 ≤ ... ≤ Tn ), and per-unit wholesale prices (w1 ≥ w2 ≥ ... ≥ wn ) such that price wi is available to the buyer only if he orders a quantity at or above breakpoint Ti . According to the revelation principle, the supplier can maximize her profits from using an all-unit discount by establishing no more than n different break-points and prices so that the buyer self-selects the policy that has been designed for his type. Because πi (q, w) is concave in q, it follows that the optimal quantity for a type i buyer to order under menu option j is the maximum of Tj and qi (wj ). Let us denote this quantity by QAU i (Tj , wj ) = M ax{qi (wj ), Tj }. To satisfy the individual rationality constraint for a type i buyer, the supplier must set wi and Ti to insure that if a type i buyer purchases according to menu option i, his gross profit is at least ²i , the profit that he would earn by procuring the product from the external market at per-unit price u. In addition, the supplier must make sure that the incentive compatibility constraints are satisfied. Recall that, to satisfy the incentive compatibility constraints, the supplier must eliminate any incentive for the buyer to purchase QAU i (Tj , wj ) for any j 6= i. The supplier’s problem is to set w1 , ..., wn ; T1 , ..., Tn in order to maximize her own profits subject to the (IR) and (IC) constraints. Formally, the supplier’s problem can be written as follows: (S.1) P max{wi ,Ti } { ni=1 pi (wi − c)QAU i (Ti , wi )} s.t. πi (QAU (T , w ), w ) ≥ ²i i i i i AU πi (Qi (Ti , wi ), wi ) ≥ πi (QAU i (Tj , wj ), wj ) wi−1 ≥ wi ≥ 0 , Ti ≥ Ti−1 ≥ 0 i = 1, ..., n i = 1, ..., n; j 6= i i = 1, ..., n (IRi ) (ICi,j ) Note that, although the solution to the above model can have T1 > 0, i.e. a minimum purchase requirement, the buyer has the option of procuring any quantity he likes from the external source at a per-unit price of u. Because constraint (IR1 ) guarantees that the buyer weakly prefers quantity T1 at a per-unit price of w1 over buying any quantity at a price of u, the above model is equivalent to one in which our supplier also offers the buyer the opportunity to purchase an unrestricted quantity at a per-unit price of u. Theorem 3.1 There exists an optimal solution to problem (S.1) in which, for each j = 1, ..., n: QAU j (Tj , wj ) = Tj , i.e. the buyer’s order quantity is exactly equal to the quantity breakpoint specified for his type. AU ∗ ∗ ∗ Proof: From the definition of QAU j (Tj , wj ), we must have Qj (Tj , wj ) ≥ Tj . Suppose that for some ∗ ∗ ∗ ∗ AU ∗ ∗ j, the optimal Tj∗ and wj∗ are such that QAU j (Tj , wj ) > Tj . Note that for any Tj < Qj (Tj , wj ), 10 the profits of buyer type j are independent of Tj∗ . Thus, we can increase the quantity breakpoint associated with price wj∗ until the point where buyer type j’s optimal order quantity is equal to the threshold without affecting any of the constraints involving buyer type j’s preferences. Moreover, each of the ICij constraints for i 6= j continues to be satisfied since their right-hand sides are non-increasing in the breakpoint associated with price wj∗ . ♦ We can apply this result to problem (S.1) by substituting Ti for QAU i (Ti , wi ) in the objective function and in the left-hand-sides of the IR and IC constraints. Based on Theorem 3.1, this substitution affects neither the feasibility nor the objective value of any optimal solution to problem (3.1). Moreover, since the proposed substitution makes the IR and IC constraints more restrictive, the set of feasible solutions after making the substitutions will be a subset of the feasible solutions to problem (3.1). Therefore, we propose the following alternative formulation of the supplier’s problem: (S.2) P max{wi ,Ti } { ni=1 pi (wi − c)Ti } s.t. πi (Ti , wi ) ≥ ²i πi (Ti , wi ) ≥ πi (QAU i (Tj , wj ), wj ) wi−1 ≥ wi ≥ 0 , Ti ≥ Ti−1 ≥ 0 i = 1, ..., n i = 1, ..., n; j 6= i i = 1, ..., n (IRi ) (ICi,j ) It is of interest to compare the right-hand-sides of the incentive compatibility constraints in the above all-unit quantity discount problem to those in the original mechanism design problem. In problem (S.2), the right-hand side of (ICi,j ) is QAU i (T, w) which is the quantity greater than or equal to T that maximizes the expected profit of buyer i at a per unit price of w. In contrast, in problem (MD) the right-hand side of this constraint is just T . This is a non-trivial distinction between the two problems, and it alters the analysis considerably. We can now compare the quantity discount pricing problem to the fixed-package pricing problem. Theorem 3.2 For a given set of parameters (r, c, ², f1 (), ..., fn ()), the optimal value of the objective function to the mechanism design, i.e. fixed-package pricing, problem is at least as large as the optimal value of the objective function to (S.2). Proof: The fixed-package pricing problem differs from (S.2) only in the right-hand sides of the incentive compatibility constraints, where Tj is substituted for QAU i (Tj , wj ). From the definition AU of QAU i (T, w), we have that: πi (Qi (Tj , wj ), wj ) ≥ πi (Tj , wj ). Thus, the incentive compatibility constraints are less restrictive in the fixed-package pricing problem than they are in the quantity discount pricing problem as expressed in (S.2). ♦ It is interesting to note that, in the all-unit quantity discount problem, buyer i purchases 11 exactly the threshold quantity that is intended for his type by the supplier, just as in the fixedpackage problem. However, it is the buyer’s increased flexibility to do otherwise under the quantity discount that prevents the supplier from extracting as much of the total profits. From the results of Theorem 3.1, we can now focus our attention on solving the alternative representation of the quantity-discount-pricing problem in (S.2). Theorem 3.3 a) A necessary and sufficient condition for a set of prices and breakpoints, (w1 , ..., wn ; T1 , ..., Tn ), to be a feasible solution to (S.2) is for the following constraints to be satisfied: IR1 ; ICj,j−1 for j = 2, ..., n; and ICj,j+1 for j = 1, ..., n − 1. b) A sufficient condition for a set of prices and breakpoints, (w1 , ..., wn ; T1 , ..., Tn ), to be a feasible solution to (S.2) is for: Tj = QAU j (Tj , wj ) for j = 1, ..., n, and for IR1 and ICj,j−1 , for j = 2, ..., n to be satisfied at equality. c) If there is a feasible solution to (S.2), then there must be an optimal solution in which: i) IR1 and ICj,j−1 , for j = 2, ..., n, are binding, and ii) qj (wj ) ≤ Tj = QAU j (Tj , wj ) ≤ qj (c) for j = 1, ..., n, and Tj = qj (c) for j = n. The proof of this result is based on the approach outlined in Fudenberg & Tirole (1998). Because its details are quite tedious, we have not included it in this paper. However it is available upon request from the authors. The above result implies that to identify an optimal solution to (S.2), we can restrict our attention to solutions in which IR1 and ICj,j−1 , for j = 1, ..., n are binding. Moreover, as shown in Theorem 3.1, an optimal solution to (S.2) must also be an optimal solution to (S.1). Restricting our attention to this subset of the solution space has the following advantage: For any given set of quantity breakpoints (T1 ≤ T2 ≤ ... ≤ Tn ), the set of corresponding prices w1 , ..., wn can be determined as the solution to the set of equations consisting of IR1 and ICj,j−1 for j = 2, ..., n. Note also that the value of wj depends only on Tj−1 , Tj and wj−1 . Thus, if the quantities that buyers can request are limited to or can be approximated by a finite discrete set, then we can solve the problem using a dynamic programming approach. Let Wj (Tj , Tj−1 , wj−1 ) be the value of wj that solves: AU Rj (Tj ) − wj Tj = Rj (QAU j (Tj−1 , wj−1 )) − wj−1 Qj (Tj−1 , wj−1 ) (3.6) for j = 2, ..., n, and let W1 (T1 , 0, u) be the value of w1 that solves: R1 (T1 ) − w1 T1 = ²1 = R1 (q1 (u)) − uq1 (u) 12 (3.7) Define uj (T ) to be the conditionally optimal expected profits that the supplier can earn from buyer types 1, ..., j given that the quantity breakpoint for buyer type j is T . Define ωj (T ) to be the corresponding (conditionally) optimal per-unit price offered to buyer type j. These functions can be represented by the following recursive relationship: uj (T ) = M axτ ≤T {pj T (Wj (T, τ, ωj−1 (τ )) − c) + uj−1 (τ )} (3.8) where u0 (T ) = 0 for all T . Let τj−1 (T ) be the optimal quantity breakpoint offered to buyer type j − 1 given that the quantity designed for buyer type j is equal to T . Thus, τj−1 (T ) is the value of τ maximizing the right hand side in (3.8), and ωj (T ) = Wj (T, τj−1 (T ), ωj−1 (τj−1 (T ))) for j = 2, ..., n (3.9) ω1 (T ) = W1 (T, 0, u). The optimal solution to the n buyer-type problem is then: vn = M axT {un (T )}. (3.10) Note that if there is a finite number m of potential quantities that can be ordered, then vn can be solved in O(m2 n) time. In cases in which quantities must be ordered in discrete units, this assumption is reasonable. In other instances where the product can be produced in continuous quantities, we can use a discrete quantity space as an approximation to the continuous one. It is also worth noting that the above approach can also be used to solve the fixed-package pricing problem. The only thing that would be different is that we would substitute Tj−1 for QAU j (Tj−1 , wj−1 ) in equation (3.6) to represent the fact that in fixed-package pricing, the buyer must purchase quantity Tj−1 to obtain per-unit price wj−1 . 3.4 An Incremental Quantity Discount Under Asymmetric Information An incremental discount scheme can be represented by a series of thresholds (t1 , t2 , ..., tn ) and marginal prices (v0 , v1 , ..., vn ) such that the marginal price for units in the interval (ti , ti+1 ] is vi for i = 0, ..., n − 1, and the marginal price for units beyond tn is vn . Thus, the total cost of buying q ∈ (tk−1 , tk ] is Pi j=1 vj−1 (tj − tj−1 ) + vi (q − ti ), where t0 = 0. It is important to recognize that, in contrast to the all-unit problem, buyers will not tend to order at the thresholds since the marginal cost decreases at each threshold. Note that in the all-unit problem, the buyer’s cost function is not generally concave. For example, consider a single 13 breakpoint T , where the per-unit price is u for quantities less than T and is w1 for quantities q ≥ T . Thus, the buyer’s cost function, C(q), is equal to uq for q < T , and is equal to w1 q otherwise. If there exists q > T for which w1 q < u(T − 1), then for 0 < α < 1, we have αC(T − 1) + (1 − α)C(q) > C(α(T − 1) + (1 − α)q). With an incremental discount, the buyer’s cost function is concave so long as the vi ’s are decreasing. Moreover, in order to prevent buyers from pretending to be several small buyers, it is reasonable to assume that the vi ’s are decreasing. This guarantees a concave cost function for the buyer. The implication of this is that the buyer’s first-order condition will be satisfied for the purchasing option that he chooses. Thus, if buyer i purchases according to the conditions defined for buyer j, he will order the following quantity: QIi (tj , vj ) = M ax{tj , qi (vj )}. To simplify the notation, let us define the following function to represent the average per-unit price paid by a buyer purchasing quantity q ≥ ti under the purchasing option designed for buyer i: i 1 X ACi (q) = vj−1 (tj − tj−1 ) + vi (q − ti ) q j=1 (3.11) where t0 = 0 since there is no minimum quantity required to purchase from the exogenous source. The most obvious formulation of this problem is as follows: Incremental Discount Problem (I.1) P max{vi ,ti } { ni=1 pi (ACi (Qi (ti , vi )) − c)Qi (ti , vi )} s.t. πi (Qi (ti , vi ), ACi (Qi (ti , vi ))) ≥ πi (Qi (0, u), u) πi (Qi (ti , vi ), ACi (Qi (ti , vi ))) ≥ πi (Qi (tj , vj ), ACj (Qi (tj , vj ))) vi−1 ≥ vi ≥ 0 , ti ≥ ti−1 ≥ 0 i = 1, ..., n i = 1, ..., n; j 6= i i = 1, ..., n (IRi ) (ICi,j ) Note that because the supplier knows the demand distributions for each buyer type, he can anticipate the optimal response, QIi (tj , vj ), of any buyer type i to any menu option j, and design the quantity discount to target specific purchase quantities from each buyer type. Borrowing notation from the all-unit problem, the buyer can target purchase quantities of T1 , T2 , ..., Tn subject to the constraint that: Ti = qi (vi ). We will also borrow the notation wi to represent the average price per unit that would be paid by a buyer of type i. Using this notation, we can express the incremental discount problem in a form that lends itself to comparison with the all-unit problem: (I.2) P max{wi ,Ti ,vi ,ti } { ni=1 pi (wi Ti − c)Ti } s.t. πi (Ti , ACi (Ti )) ≥ πi (Qi (0, u), u) πi (Ti , ACi (Ti )) ≥ πi (Qi (tj , vj ), ACj (Qi (tj , vj ))) wi = ACi (Ti ) Ti = qi (vi ) Ti ≥ ti vi−1 ≥ vi ≥ 0 , ti ≥ ti−1 ≥ 0 14 i = 1, ..., n i = 1, ..., n; j 6= i i = 1, ..., n i = 1, ..., n i = 1, ..., n i = 1, ..., n (IRi ) (ICi,j ) Note that for a given set of marginal prices and breakpoints, (v1 , ..., vn ; t1 , ..., tn ), the wi and Ti variables in the above problem are uniquely determined. Theorem 3.4 a) A necessary and sufficient condition for a set of marginal prices and breakpoints, (v1 , ..., vn ; t1 , ..., tn ), to be a feasible solution to (I.2) is for either vj = vj−1 or Tj−1 ≤ tj for j = 2, ..., n, and the following constraints to be satisfied: IR1 ; ICj,j−1 for j = 2, ..., n; and ICj,j+1 for j = 1, ..., n − 1. b) A sufficient condition for a set of prices and breakpoints, (v1 , ..., vn ; t1 , ..., tn ), to be a feasible solution to (I.2) is for IR1 and ICj,j−1 , for j = 2, ..., n to be satisfied at equality. c) If there is a feasible solution to (I.2), then there must be an optimal solution in which: i) IR1 and ICj,j−1 , for j = 2, ..., n, are binding, and ii) Tj ≤ qj (c) for j = 1, ..., n, and Tj = qj (c) for j = n. The proof of this result is similar to that for Theorem 3.3, and is available upon request. Based on this result, we can adopt a dynamic programming approach similar to the one used for the all-unit discount to solve the incremental discount problem. Because we have converted the marginal costs into average costs per unit at various quantities, the buyer profit functions are identical to the ones in problem S.2: πi (q, ACj (q)) = Ri (q) − qACj (q) (3.12) Define πsAU to be the optimal solution to the all-unit quantity discount problem, and let πsI be the optimal solution to the incremental quantity discount problem. Theorem 3.5 For a given set of problem parameters, F1 , ...Fn , u, c, the supplier can earn at least as much under an all-unit discount than under an incremental discount. Specifically, πsI ≤ πsAU . Proof: Observe, that the two objective functions are identical. To prove the claim, it suffices to show that problem I.2 is more tightly constrained than problem S.2. First note in that any solution (w, T, v, t) that satisfies the last four constraints of I.2, (w, T ) must satisfy the constraints in S.2 requiring that wi ≤ wi−1 and Ti ≥ Ti−1 . Now consider the ICji constraints for i < j: From the assumption of stochastic dominance, we have that: Qj (ti , vi ) ≥ Qi (ti , vi ) = qi (vi ) (3.13) where the final equality follows from the requirement that Ti = qi (vi ), which implies that ti ≤ Ti . It follows from the definition of ACi (q) that ACi (Qj (ti , vi )) ≤ ACi (Ti ). From the constraint in I.2 15 requiring that wi = ACi (Ti ). It follows that: πj (Qj (ti , vi ), ACi (Qj (ti , vi ))) ≥ πj (Qj (ti , vi ), ACi (Ti )) ≥ πj (Ti , ACi (Ti )) (3.14) and because the right-hand-side of the above inequality is identical to the right hand side of the ICji constraint in problem S.2, the corresponding constraint in problem I.2 is more restrictive. Let us now consider the ICij constraints for i < j. From stochastic dominance, we have: Qi (tj , vj ) ≤ Qj (tj , vj ) = Tj (3.15) where the final equality follows from the definition of Qj (tj , vj ) and the requirements Tj = qj (vj ) and tj ≤ Tj . By the definition of Qi (tj , vj ) as buyer i’s optimal response to (tj , vj ), it follows that: πi (Qi (tj , vj ), ACj (Qi (tj , vj ))) ≥ πi (Tj , ACj (Tj , vj )) (3.16) and this demonstrates that the right hand side of the upward IC constraints in problem I.2 are no smaller than the right hand sides of the corresponding constraints in problem S.2. Since for given values of (w, T ), the left hand sides of these constraints are identical, the upward IC constraints are more difficult in I.2 than in S.2. ♦ Intuitively, the above result can be explained as follows. In the all-unit discount, the downward ICji constraint requires the buyer to prefer buying quantity Tj at average per-unit cost wj over buying any quantity q ≥ Ti at average per-unit price wi . In the incremental discount problem, this constraint is more severe because the buyer’s average per-unit price under menu option i falls below wi when q > Ti . ( Recall that, by construction, ACi (q) = wi when q = Ti , and is decreasing in q.) Similar intuition exists for the upward ICij constraints. In the all-unit discount, buyer i purchases exactly Tj under menu option j, since by stochastic dominance, qi (wj ) ≤ qj (wj ) and qj (wj ) ≤ Tj . In the incremental discount, tj ≤ Tj , and this allows the buyer more flexibility to purchase less than the targeted amount under menu option j. 3.5 Exclusion of Buyer Types In many situations where a firm offers a menu of purchasing options to customers who differ in their marginal valuation for a product, there is an issue as to whether the firm should attempt to serve all segments of the market. For example, Moorthy (1984), Mussa & Rosen (1978), Porteus & Whang (1999) all provide analysis that demonstrates that, by serving the low end of the market, the firm cannot extract as much rent from customers with the highest valuation. All three of these papers provide conditions in which, by ignoring the low end of the market, the firm can increase 16 the amount that it earns from the high end consumers by more than it gives up by not serving the low end consumers. In a paper that is more closely related to ours, Ha (2001) shows that in a newsvendor environment in which there is asymmetric information about the buyer’s costs, it is optimal for the supplier to not service buyers whose costs exceed a threshold. The main reason for this is that it is assumed that the buyer’s participation constraint is independent of his type, i.e. a high cost buyer requires just as much profit as does a low cost buyer. However, in our model this is not the case, and the supplier has no reason to exclude the low end segment of the market. To see why this is the case, recall that we assume that the buyer has an alternative source of supply at market price u (u). Our individual rationality (participation) constraints require that each buyer type weakly prefer purchasing from the supplier over purchasing from the market. However, by offering the low end buyer type(s) a per-unit price of u (u), the supplier can trivially satisfy this requirement without adversely affecting its ability to extract rents from the higher indexed buyer types. 4 Numerical Experiment We consider several numerical examples in order to assess the benefit of quantity discounts to the supplier as well as to explore the sensitivity of the optimal discount policies with respect to the model parameters. To estimate the benefit of quantity discounts to the supplier we first must analyze her optimal pricing policy without quantity discounts, i.e., when a single common price is offered to all buyer types. We refer to this case as the single-price problem. This problem was analyzed in Lariviere & Porteus (2001). Because we apply this to examples in which demand is normally distributed, let us briefly review the solution to the single-price problem for this special case of normally distributed demand: Assume that the demand distribution of buyer i is normal N (µi , σi2 ). Let w be the single per-unit price that the buyer offers to all buyer types. Thus, buyer i will solve a newsvendor problem and order: Qi (w) = Fi−1 (1 − w/r) = µi + σi Φ−1 (1 − w ), i = 1, . . . , n. r where P hi() denotes the unit-normal cumulative distribution function. The supplier’s expected profit as a function of w is: ΠPS (w) = n X pi (w − c)Qi (w) = (w − c)(µ̄ + σ̄Φ−1 (1 − i=1 17 w )), r where µ̄ = Pn i=1 pi µi and σ̄ = Pn i=1 pi σi . By making the transformation z = Φ−1 (1 − w r ), i.e., w = rΦ̄(z), the supplier’s profit can be written as: ΠPS (z) = (rΦ̄(z) − c)(µ̄ + σ̄z). Because a normal distribution has an increasing generalized failure rate (IGFR), ΠPS (z) is unimodal and is maximized at the point where the first-order conditions are satisfied. (See Lariviere & Porteus (2001) for details.) Thus, it is easy to determine the maximizing value z ∗ numerically. The optimal price w∗ and profit ΠPS ∗ can then be obtained using the above transformation. So far, we have not considered the possibility that buyers may have an alternative source of the product at an exogenous price of w0 per-unit. When such an opportunity exists, the optimal price for the supplier to charge becomes: w̃ = min(w∗ , w0 ) and the supplier’s optimal profit is: Π̃PS ∗ = ΠpS (w̃). In our first numerical experiment we consider a case where there are five different buyer types, all perceived by the supplier as equally likely. Each buyer type has normally distributed demand with common standard deviation σ = 10, and the following means: µ1 = 50, µ2 = 60, µ3 = 70, µ4 = 80, µ5 = 90. The selling price is r = 50 and the supplier’s cost c = 10. Table 1 shows the optimal price and profit under the single-price policy for w0 varying from 25 to 50, the optimal profits under all-unit and incremental discounts, as well as the profit improvement for each quantity discount policy with respect to the single-price standard. Table 2 presents the optimal pricing policy for all-unit discounts. Tables 3 and 4 present the optimal pricing policy for the incremental discount case, in terms of the order quantities/average prices and breakpoints/marginal prices, respectively. Note that when the alternative price (w0 or x0 ) equals the retail price of 50, the buyer’s profit from using the alternative supply is equal to zero. In this case the individual rationality constraints require only that the buyer earn non-negative profits. As this price for the alternative source of supply decreases, the intensified competition from the alternative source of supply makes it more difficult for our supplier to extract profits from the buyer, and this is confirmed by the supplier’s profits in these two tables. Note that as w0 (x0 ) increases, the quantities decrease and the average per-unit prices increase for both types of discount. It can also be observed in Table 1 that, when external pressure on price intensifies, 18 i.e. w0 (x0 ) decreases, the benefit from using either an all-unit or an incremental quantity discount instead of a single-price policy becomes very small. However, when external pressure on price is non-existent, i.e. w0 (x0 ) = 50, then the two types of quantity discount are most advantageous, respectively allowing for 14.68% and 12.27% increases in profit over a single price policy. From Tables 2 and 3, it can be observed that for the lowest indexed buyer type (i = 1), the incremental discount results in smaller quantities and higher average per-unit prices than does the all-unit discount. Moreover, the magnitude of the gap between the quantity (average per-unit price) from the all-unit discount versus that for the incremental discount is increasing in the external price w0 = x0 . These two tables also confirm that, regardless of which type of discount is used, the quantity sold to the highest indexed buyer type is the one that maximizes channel profits for this buyer type’s distribution of demand, i.e. q5 (c) = 98.5. In the second numerical experiment we explore the sensitivity of the profits with respect to variability of demand. There are various ways we could change demand variability. For example we could assume that all types of buyers have normally distributed demands with different means but a common variance σ 2 , and then investigate how the policies are affected by different values of σ. Alternatively, we assume that all types of buyers have a common coefficient of variation, i.e. σj /µj = C for all j = 1, ..., n, and then investigate how the policies are affected by different values of C. We use an approach that lies between these two alternatives. In our approach, we assume that the difference in demand distributions among buyer types is due to the different market sizes they serve. Specifically, buyer j serves a market of Nj individual customers. Each customer in every market has demand that is independent and identically distributed with mean µ0 and variance σ02 . Therefore, when the market sizes Nj are large, the demand that is faced by a buyer of type j is approximately normally distributed with mean µj = Nj µ0 , variance σj2 = Nj σ02 , and coefficient of variation Cj = p Nj (σ0 /µ0 ). In this approach, the differentiation of variance across buyer types is justified on the basis of our assumptions about the behavior of individual consumers. For the numerical experiment, we consider a situation with n = 4 buyers, r = 50, c = 10, w0 = v0 = 30. The market sizes are N1 = 9, N2 = 25, N3 = 36, N4 = 49 and the mean demand for each customer is equal to µ0 = 2. Tables 5 to 8 show the optimal quantity discount policies and supplier profits as well as the comparison to the single price benchmark case for various values of σ0 . From these results it can be observed that, regardless of the pricing policy employed, the supplier’s profit decreases with σ. This is consistent with the fact that, in any newsvendor setting, an increase in demand uncertainty will increase the total expected cost of over and under-stocking. 19 However, it can be observed that increasing the standard deviation of demand does not have the same effect upon the all-unit discount that it does on the incremental discount. In Table 6, it can be seen that, under the all-unit discount, increasing σ causes the quantities to increase and the average per-unit prices to decrease. However, in Table 7, it can be seen that, under the incremental discount, the quantities actually decrease for the lower indexed buyer types (i = 1, 2, 3). Although these numerical computations confirm our analytical result that the supplier’s expected profit is higher under an all-unit discount than under an incremental discount, it is interesting to examine the magnitude of the difference which is shown in the far right column of Table 7. It can be observed that the incremental discount generally results in a loss of no more than about 1.5% compared to the all-unit discount. In our last numerical experiment, we investigate the sensitivity of the profits of the supplier and of the total channel to the parameters of normally distributed demand when there are two equally likely buyer types. Specifically, we investigate how the parameters of the normally distributed demands for the two buyer types affect the percentage gap between the expected profits earned by the supplier (total channel) under each form of quantity discount and the expected profits that would be earned under symmetric information. Recall that under symmetric information, the supplier would know the buyer’s type and could use either form of discount to induce the (channel) profit maximizing quantity, leaving the buyer with only enough profit to satisfy his individual rationality constraint. Since we assume that the two buyer types are equally likely, the expected SI profits is the average of the amounts that the supplier could extract from each type of buyer with perfect information about his types. Obviously, the SI profits represent an upper bound on the supplier’s and the channel expected profits, respectively. To perform this last experiment, we consider two different cases for the means of the two buyer types: one in which the buyer types have means µ1 = 53 and µ2 = 58, and another in which they have means µ1 = 40 and µ2 = 70. We consider a range of standard deviations, σ, and assume that this is the same for both buyer types. We assume that the retail price r = 40, the production cost c = 10, and the price from the alternative source of supply w0 = x0 = 25. Figure 1 shows the percentage gap between the supplier’s expected profits under symmetric information versus those under either the incremental (IN vs. SI) or the all-unit (AU vs. SI) discount. Similarly, Figure 2 shows the percentage gap between the channel expected profits under symmetric information versus those under either the incremental (IN vs. SI) or the all-unit (AU vs. SI) discount. The lines are labelled to reflect the percent profit gap that they represent ( IN vs SI 20 or AU vs SI ), and the assumption about the means for the two buyer types (53 − 58 or 40 − 70). Figure 1 confirms our result that the supplier can earn more under an all all-unit discount than under an incremental discount, since the profit gap with (with respect to symmetric information) is higher for the all-unit than the incremental discount. Moreover, the gap between the supplier’s symmetric information profits and her profits under either type of discount increases with the magnitude of the difference between the means for the two buyer types, and the size of the standard deviation. It is also interesting to observe how the standard deviation affects the relative performance of the all-unit and incremental discounts when the buyer types have mean demands of µ1 = 53 and µ2 = 58, versus when their mean demands are µ1 = 40 and µ2 = 70. When the buyer types have means that are close together, i.e. 53 and 58, the relative gap between the incremental and the all-unit discount decreases with demand uncertainty. However, when the buyer types’ demands are more differentiated, i.e. means of 40 and 70, the relative gap between the two types of discount increases with demand uncertainty. In fact, for this case, when the standard deviation of demand is 25, the gap between the incremental discount and the symmetric information profit is 67% larger than that between the all-unit discount and the symmetric information profit. For the channel profits, we first observe from Figure 2 that both quantity discount policies yield lower channel profits than the symmetric information case, which is the result of the supplier trading off channel efficiency against the information rents paid to the buyer. However, it is worth noting that the gaps in Figure 2 are smaller than those in Figure 1, indicating that the lack of symmetric information hurts the supplier more than the supply chain as a whole. Finally, we can observe that when the buyer types differ significantly, i.e. demand means (40,70), the all-units discount results in greater channel efficiency (lower profit gaps) than does the incremental. On the other hand, when the buyer types are more similar, i.e. demand means (53, 58), the channel efficiency is very high (profit gaps are less that 1% ), and the specific type of discount makes very little difference. 5 Conclusions and Extensions In this paper, we have considered how incremental and all-unit discounts can be used to increase a supplier’s profits when selling to a newsvending downstream buyer who has more information about the distribution of demand than does the supplier. Understanding how quantity discounts can be used in such environments is important because they are commonly used in practice and because they are a natural mechanism for a supplier to use to overcome informational asymmetry regarding 21 12 10 Profit Gap (%) 8 6 IN vs SI 53-58 4 AU vs SI 53-58 IN vs SI 40-70 AU vs SI 40-70 2 8 10 12 14 16 18 20 22 24 26 Standard Deviation Figure 1: Supplier Profit Gap as a Function of Standard Deviation and Difference in Means 7 6 Profit Gap (%) 5 4 3 2 IN vs SI 53-58 AU vs SI 53-58 1 IN vs SI 40-70 0 AU vs SI 40-70 8 10 12 14 16 18 20 22 24 26 Standard Deviation Figure 2: Channel Profit Gap as a Function of Standard Deviation and Difference in Means 22 the buyer’s distribution of demand. One important feature associated with a quantity discount is the fact that, after the initial transaction, there is no need for additional flows of information or logistics between the buyer and seller. Certainly this could be advantageous in many environments. Our main results include characterizing the structure of the all-unit and incremental discounts and demonstrating that the all-unit discount always dominates the incremental discount from the perspective of the supplier. This compliments previous results in EOQ settings with symmetric information in which the supplier has been shown to be indifferent between incremental and all-unit discounts. In our model, the quantity discount is used to counter the informational asymmetry between the supplier and the buyer. This is different than the role that is typically played by returns policies, quantity flexibility, price protection, and revenue sharing in the existing literature. Most of the results for these other mechanisms have been obtained in symmetric information environments in which the primary purpose of the mechanism is to overcome double marginalization. It is worth noting that when the retail price r is exogenous and demand information is symmetric, a quantity discount is equivalent to a two-part tariff and allows a supplier to simultaneously induce the order quantity that maximizes total profits and extract all but the participation profits from buyer. The fact that this is inconsistent with the reality that newsvendor-like buyers do earn excess profits may be one of the reasons why so little attention has been paid to quantity discounts in the literature. However, when there is asymmetric demand information, the supplier can no longer extract all of the profits from the buyer with a quantity discount, even if the buyer cannot control the retail price. Our purpose is to recognize the importance of quantity discounts in environments in which there is asymmetric demand information, and show that one form of quantity discount, i.e. the allunit discount, dominates the incremental discount. 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Discount pricing decisions in distribution channels with price sensitive demand. European Journal of Operational Research 149, 571–587. Weng, Z. K. (1995). Channel coordination and quantity discounts. Management Science 41 (9), 1509–1522. 26 Appendix Continuum of Buyer Types with Uniform Demand Distributions Let us consider a situation in which all consumers have demand that is uniformly distributed, but they differ in terms of one of the parameters of the Uniform distribution. Specifically, let us index consumer types by θ, and assume that a consumer of type θ has demand that is distributed U [θ, θ + a]. Thus, fθ (x) = x a , and Fθ (x) = x−θ a for x ∈ [θ, θ + a]. The density of the parameter θis g(θ), and assume that g(θ) has positive support on the interval [θ, θ̄]. The expected revenue function for a consumer of type θ is the following function of Q: à R(Q, θ) = r θ + Z Q θ+a−x a θ ! ¡ r 2Q(θ + a) − θ2 − Q2 dx = 2a ¢ (A.1) and the consumer’s profit is equal to the total expected revenue less the total amount that he pays to receive Q units, which we denote by w(Q). Thus, if the supplier offers a menu of contracts, (Q(θ̂), w(θ̂)), then a retailer of type θ earns profits of: πR (θ) = o M ax n R(Q(θ̂), θ) − w(Q(θ̂)) θ̂ (A.2) Differentiating with respect to θ, we have: dπR (θ) dR(Q, θ) r(Q − θ) = = dθ dθ a Integrating, we can obtain an alternative expression for the profit of a type θ retailer: πR (θ) = r a Z θ θ (Q − µ)dµ − πR (θ) = r a Z θ θ (Q − µ)dµ (A.3) By definition, w(Q(θ)) = R(Q(θ), θ) − πR (θ). The supplier’s profit can now be expressed as follows: πM = = Z θ θ Z θ θ (w(Q(θ) − cQ(θ))g(θ)dθ (R(Q(θ), θ) − πR (θ) − cQ(θ))g(θ)dθ where the last equality was obtained by substituting (A.1). Substituting (A.3) for πR (θ) into the above, we have: πM = Z θà θ r R(Q(θ), θ) − a Z θ θ ! (Q − µ)dµ − cQ(θ) g(θ)dθ We can apply integration by parts to the middle term in this integrand to obtain: πM = = Z θ· θ ¸ r (R(Q(θ), θ) − cQ(θ)) g(θ) − (1 − G(θ)) (Q − θ) dθ a Z θ ·µ r θ ¶ ¸ r (2Q(θ + a) − Q2 − θ2 − cQ(θ) g(θ) − (1 − G(θ)) (Q − θ) dθ 2a a 27 The manufacturer wants to choose Q(θ) to maximize the integrand in the above expression for every value of θ. Differentiating the integrand and setting the rusult equal to zero implies: Q(θ) = θ + a − ca 1 − G(θ) − r g(θ) Note that this has the desirable properties that: 1) Q(θ) is increasing in θ(So long as g() is IFR), and 2) when θ = θ, we have Q(θ) = θ + a − ca r so that 1 − Fθ (Q(θ) = rc . 28 29 w̃ 25.00 30.00 35.00 40.00 45.00 46.56 Π˜PS 1050.00 1349.33 1618.90 1847.51 2001.46 2016.36 All-Unit 1062.44 1374.32 1658.77 1918.05 2145.60 2312.46 Improvement (%) 1.18 1.85 2.46 3.82 7.20 14.68 Incremental 1055.58 1365.43 1646.50 1903.72 2101.49 2263.74 Improvement (%) 0.53 1.19 1.70 3.04 5.00 12.27 T1 53.75 49.25 51.25 50.25 49.25 49.25 T2 63.75 57.50 61.50 60.50 59.50 59.25 T3 73.75 67.50 71.50 70.50 69.75 69.25 T4 84.00 77.75 81.75 80.75 80.00 79.50 T5 98.50 98.50 98.50 98.50 98.50 98.50 w1 24.74 29.94 34.23 38.74 43.02 46.32 w2 24.54 29.94 33.61 37.81 41.69 44.49 w3 24.37 29.94 33.13 37.11 40.67 43.19 w4 24.22 29.94 32.72 36.52 39.86 42.16 w5 23.59 28.76 31.21 34.48 37.29 39.19 Profit 1062.44 1374.32 1658.77 1918.05 2145.60 2312.46 50.13 47.50 45.00 42.00 42.00 42.00 25.00 30.00 35.00 40.00 45.00 50.00 60.25 57.75 55.25 52.25 52.25 52.25 T2 70.38 68.00 65.50 62.50 62.50 62.50 T3 80.50 78.25 75.75 77.75 77.75 77.75 T4 98.50 98.50 98.50 98.50 98.50 98.50 T5 25.00 30.00 35.00 40.00 44.71 48.57 w1 24.96 29.99 34.92 39.88 43.67 46.77 w2 24.89 29.91 34.79 39.74 42.91 45.50 w3 24.81 29.78 34.64 39.23 41.78 43.86 w4 24.06 28.49 32.74 36.05 38.06 39.70 w5 1055.58 1365.43 1646.50 1903.72 2101.49 2263.74 Profit Profit Loss(%) .65 .65 .74 .75 2.06 2.11 Table 3: Incremental Discount (Quantities and Average Prices-per-unit) for Normally Distributed Demands and Varying External Price; r = 50; c = 10; σ1 = σ2 = σ3 = σ4 = σ5 = 10 ; µ1 = 50, µ2 = 60, µ3 = 70, µ4 = 80, µ5 = 90 T1 x0 Table 2: All-Unit Discount for Normally Distributed Demands and Varying External Price; r = 50; c = 10; σ1 = σ2 = σ3 = σ4 = σ5 = 10 ; µ1 = 50, µ2 = 60, µ3 = 70, µ4 = 80, µ5 = 90 w0 25.00 30.00 35.00 40.00 45.00 50.00 Table 1: Profits Under All-Unit and Incremental Quantity Discounts vs. Single Price for Normally Distributed Demands and Varying External Price; r = 50; c = 10; σ1 = σ2 = σ3 = σ4 = σ5 = 10 ; µ1 = 50, µ2 = 60, µ3 = 70, µ4 = 80, µ5 = 90 w0 25.00 30.00 35.00 40.00 45.00 50.00 30 t1 50.06 47.48 44.88 41.79 39.79 36.33 t2 60.19 57.63 55.13 52.13 52.13 52.13 t3 70.31 67.88 65.38 62.38 62.38 62.38 t4 80.44 78.13 75.63 75.24 75.24 75.24 t5 94.27 93.09 91.85 92.84 92.84 92.84 v1 24.75 29.94 34.57 39.41 39.41 39.41 v2 24.50 29.45 34.13 39.04 39.04 39.04 v3 24.25 28.96 33.68 38.67 38.67 38.67 v4 24.00 28.47 33.23 29.45 29.45 29.45 v5 9.88 9.88 9.88 9.88 9.88 9.88 Profit 1055.58 1365.43 1646.50 1903.72 2101.49 2263.74 w̃ 30.00 30.00 30.00 30.00 Π˜PS 1176.70 1163.40 1150.10 1136.80 All-Unit 1188.13 1186.76 1183.09 1182.90 Improvement (%) .97 2.01 2.87 4.06 Incremental 1178.44 1170.50 1169.98 1167.68 Improvement (%) .15 .61 1.73 2.72 T1 18.00 18.00 18.25 18.25 T2 50.50 51.00 51.50 52.00 T3 72.50 73.25 74.00 74.50 T4 101.00 104.00 107.00 110.00 w1 29.95 29.90 29.77 29.72 w2 29.85 29.70 29.49 29.35 w3 29.78 29.54 29.25 29.06 w4 29.38 28.78 28.17 27.69 Profit 1188.13 1186.76 1183.09 1182.90 Table 6: All-Unit Discount for Normally Distributed Demands and Varying Standard Deviation; r = 50; c = 10; w0 = v0 = 30; N1 = 9, N2 = 25, N3 = 36, N4 = 49; µ0 = 2 σ0 .50 1.00 1.50 2.00 Table 5: Profits Under All-Unit and Incremental Quantity Discounts vs. Single Price for Normally Distributed Demands and Varying Standard Deviation; r = 50; c = 10; w0 = v0 = 30; N1 = 9, N2 = 25, N3 = 36, N4 = 49; µ0 = 2 σ0 .50 1.00 1.50 2.00 Table 4: Incremental Discount (Breakpoints and Marginal Prices) for Normally Distributed Demands and Varying External Price; r = 50; c = 10; σ1 = σ2 = σ3 = σ4 = σ5 = 10 ; µ1 = 50, µ2 = 60, µ3 = 70, µ4 = 80, µ5 = 90 x0 25.00 30.00 35.00 40.00 45.00 50.00 31 17.63 17.25 16.88 16.50 .50 1.00 1.50 2.00 49.41 48.88 48.25 47.63 T2 71.31 70.75 70.00 69.25 T3 100.97 104.00 106.88 109.88 T4 30.00 30.00 30.00 30.00 w1 29.96 29.96 29.96 29.96 w2 29.88 29.80 29.85 29.87 w3 29.39 28.85 28.62 28.33 w4 1178.44 1170.50 1169.98 1167.68 Profit Profit Loss(%) .82 1.37 1.11 1.29 t1 17.62 17.24 16.87 16.49 t2 49.39 48.81 48.19 47.56 t3 71.30 70.70 69.95 69.20 t4 98.98 100.06 100.97 101.93 v1 29.94 29.94 29.94 29.94 v2 29.69 29.45 29.61 29.69 v3 29.53 29.13 29.40 29.53 v4 9.91 9.78 9.95 9.91 Profit 1178.44 1170.50 1169.98 1167.68 Table 8: Incremental Discount (Breakpoints and Marginal Prices) for Normally Distributed Demands and Varying Standard Deviation; r = 50; c = 10; w0 = v0 = 30; N1 = 9, N2 = 25, N3 = 36, N4 = 49; µ0 = 2 σ0 .50 1.00 1.50 2.00 Table 7: Incremental Discount (Quantities and Average Prices-per-unit) for Normally Distributed Demands and Varying Standard Deviation; r = 50; c = 10; w0 = v0 = 30; N1 = 9, N2 = 25, N3 = 36, N4 = 49; µ0 = 2 T1 σ0