Efficient replenishment in the distribution channel

Journal of Retailing 83 (3, 2007) 253–278

Efficient replenishment in the distribution channel Yan Dong a,∗ , Venkatesh Shankar b,1 , Martin Dresner c a

c

Marketing and Logistics Management, Carlson School of Management, University of Minnesota, Minneapolis, MN 55455, United States b Mays Business School, Texas A&M University, College Station, TX 77843, United States Logistics, Business, and Public Policy, Robert H. Smith School of Business, University of Maryland, College Park, MD 20742, United States

Abstract Efficient replenishment (ER), a business process that involves the reduction of order cost to facilitate deliveries of goods from the manufacturer to the retailer, is becoming increasingly important in distribution channel management. While a well-executed ER program is expected to lower total channel costs and increase channel profit, very little is known about how this incremental channel profit is distributed between the manufacturer and the retailer and how it varies across the two common channel relationship structures, retailer price leadership and manufacturer price leadership. In this paper, we develop the conditions under which the manufacturer and the retailer gain more or less from the adoption of ER based on a game theoretic channel model of bilateral monopoly under the two channel relationship structures. We develop analytic results on the impact of ER on purchase quantity, price and the distribution of profits in three cases, namely, (1) when only the retailer adopts ER, (2) when both the manufacturer and the retailer adopt ER, and (3) when the manufacturer and the retailer are vertically integrated in the distribution channel, which adopts ER. The results, which can be generalized for all demand functions, show that the manufacturer benefits from the retailer’s adoption of ER only when the manufacturer’s holding cost relative to the retailer’s is sufficiently large, relative to its order cost relative to the retailer’s. By adopting ER, the retailer gains more than what the manufacturer gains even if the manufacturer is the price leader. Both the parties are likely to gain more if they both adopt ER than if only the retailer adopts ER. The incremental channel profit due to the retailer’s ER adoption is highest in a vertically integrated distribution channel and is greater in a retailer-led channel relationship than in a manufacturer-led relationship. © 2007 New York University. Published by Elsevier Inc. All rights reserved. Keywords: Retailer–manufacturer relationship; Channel management; Efficient consumer response (ECR); Efficient replenishment; Game theory

Introduction Efficient consumer response (ECR) has become an increasingly important business practice for manufacturers and retailers to attain competitive advantage in the distribution channel (Levy and Grewal 2000). Efficient replenishment (ER) is a program within ECR that has gained widespread attention among manufacturers and retailers. ER can be defined as a business process that involves the reduction of order cost to facilitate of goods from the manufacturer to the retailer in the channel (McKinney and Clark 1995). ER can be undertaken by either the retailer or the manufacturer or both, but its impact extends to the whole channel. The order cost reduction in ER comes from benefits such as negotiation simplification, paper work reduction, order lead time reduction, faster manufacturer/retailer evaluation and selection, automated manufacturer/retailer rating systems, and speedier quality assurance. This view of ER is consistent with that in the consulting industry (e.g., IBM Report 1999; Sykes Enterprises 1999).

∗ 1

Corresponding author. Tel.: +1 612 625 2903; fax: +1 612 624 8804. E-mail addresses: [email protected] (Y. Dong), [email protected] (V. Shankar). Tel.: +1 979 845 3246; fax: +1 979 862 2811.

0022-4359/$ – see front matter © 2007 New York University. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.jretai.2007.03.003

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Anecdotal evidence suggests that the adoption of ER is associated with industry cost savings and firm performance. It is estimated that the annual savings due to ER are about $11.9 billion in the food industry (Supermarket Business 1998) and about $11 billion in the healthcare industry (Transportation and Distribution 1997). With regard to firm performance, a firm like CVS pharmacy has managed to reduce its backroom inventory to 10–15 percent of total inventory through ER (Chain Store Age 2001). Furthermore, in a survey of business executives, Myers et al. (2000) found that the use of automatic replenishment service programs was positively related to firm performance. Although anecdotal evidence indicates that channel profit may rise as a result of ER, the incremental channel profit may be unequally distributed between the manufacturer and the retailer depending on the relationship structure, that is, whether the channel price leader is the retailer or the manufacturer, and on other factors. It is important for managers to know which party gains more and what conditions are associated with such gains. By better understanding the allocation of incremental channel profit between the manufacturer and the retailer due to ER and channel relationship structure, managers can make more informed pricing, inventory holding, and ordering decisions. From a theoretical standpoint, researchers need to better understand when and why the manufacturer’s or the retailer’s profit rises due to ER adoption by one or more channel members. The unequal distribution across the manufacturer and the retailer of the additional profit due to ER adoption may depend on several factors such the channel relationship structure (who is the price leader? the manufacturer or the retailer?), the entity adopting ER (the retailer or both), the cost function (e.g., linear, nonlinear), the demand function (e.g., linear, nonlinear), and the relationship among the holding and order costs of the manufacturer and the retailer. There is an important need for a theoretical model that would address these issues. Despite the need for a theoretical model of ER adoption, there has been little research on ER among marketing or operations management academics. Marketing academics have predominantly focused on channel coordination pricing (e.g., Choi 1991; Ingene and Parry 1995, 2000; Jeuland and Shugan 1983; McGuire and Staelin 1983), or discount policy (e.g., Lal and Staelin 1984), but not on ER. Scholars in operations research and supply chain management have focused on either Just-in-Time (JIT) programs or inventory policy in the channel (e.g., Weng 1995), or Quick Response (QR) initiative (e.g., Iyer and Bergen 1997), but have not examined the impact of ER. While JIT is aimed at minimizing inventory and QR is focused on reducing order lead time, ER is about reducing order cost. In addition to pricing decisions, ordering decisions are important in the marketing channel (Hall et al. 2005). The purpose of this paper is to extend prior research by developing a game theoretic model of the impact of ER on channel outcomes for both the manufacturer and the retailer. We consider a bilateral channel monopoly with two alternative channel relationship structures: (a) a dominant retailer acting as a Stackelberg price leader, and (b) a dominant manufacturer acting as a Stackelberg price leader. We derive analytic results on the impact of ER adoption in three cases: (1) when only the retailer adopts ER, (2) when both the manufacturer and retailer adopt ER, and (3) when the manufacturer and the retailer are vertically integrated and jointly implement ER. Our research is distinct from Weng (1995) and Choi (1991) and offers important contributions over their work. Although our modeling framework is consistent with Weng (1995), unlike Weng who focuses on quantity discounts and cooperative behavior, we focus on ER and examine Stackelberg leadership by both the manufacturer and the retailer. Similar to Choi (1991), we develop two models representing different retailer–manufacturer price relationship structures, retailer-led and manufacturer-led Stackelberg models.1 However, while Choi examines the effects of production cost on the equilibrium price and profit levels in a two product situation, we include inventory costs in our model and examine the effects of a reduction in the order cost (or ER) on the equilibrium quantity, price, and profit levels of the manufacturer and the retailer for a single product scenario. To our knowledge, our work is the first to model the impact of ER on channel outcomes in different channel relationship structures, offering important insights into the role of ER in the distribution channel. Our analysis offers important and counter-intuitive insights that can be generalized for all demand functions. The results show that the manufacturer will benefit from the retailer’s ER adoption only when the manufacturer’s holding cost relative to the retailer’s is sufficiently higher than its order cost relative to the retailer’s. By adopting ER, the retailer gains more than what the manufacturer gains even if the manufacturer is the price leader. Both the retailer and manufacturer will gain more if they both adopt ER than if only the retailer adopts ER. The incremental channel profit due to the retailer’s ER adoption is highest in a vertically integrated distribution channel and is greater in a retailer-led channel relationship than in a manufacturer-led relationship.

1 By Stackelberg leadership, we refer to the leader (manufacturer or retailer) first making a decision on price, followed by the follower (retailer or manufacturer) making a decision on quantity, consistent with Wang and Seidmann (1995). We do not focus on Nash equilibrium between the manufacturer and the retailer because it can be shown that the results for such a game structure fall in between the two interesting extremes of results from the two Stackelberg leadership structures.

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Relevant literature and models Although research in channel management and retailing is vast (e.g., Brown et al. 2005; Ingene and Parry 2000; Inman et al. 2004; Levy et al. 2005), two streams of channel management literature are relevant to our research. One stream focuses on pricing, inventory, and ordering policies in a retailer–manufacturer framework (e.g., Lal and Staelin 1984; Weng 1995), while the other stream primarily examines channel coordination (e.g., Choi 1991; Ingene and Parry 1995, 2000; Jeuland and Shugan 1983; McGuire and Staelin 1983; Moorthy 1987). The stream of research on pricing and inventory policies generally addresses situations where a large manufacturer sells to a number of retailers and where the manufacturer’s product does not represent a major component of the retailer’s final product. The channel coordination literature focuses on optimal pricing decisions in a manufacturer-retailer channel with margin as the key decision variable. While we know much about the impact of inventory policies on channel costs and about the optimal prices and discounts that coordinate a channel, we know little about the impact of cost reduction initiatives such as ER on purchase quantity, price and the distribution of profits in a channel. We, therefore, extend the two research streams by examining the impact of ER on purchase quantity, price, and profits in the channel. Retailer-led Stackelberg model We first consider a bilateral channel monopoly, comprising a single manufacturer selling a single product to a single retailer. The retailer–manufacturer relationship is described by a Stackelberg model in which the retailer first determines the purchase price and then the manufacturer decides the quantity it wants to supply, given this price and its own cost structure. Although price setting by the retailer has received considerable attention the empirical literature (e.g., Ailawadi et al. 2005; Bolton and Shankar 2003; Shankar and Bolton 2004), it has not been adequately investigated by analytic models. Our model captures the reality of the present situation in many industries (e.g., electronics and grocery), where some retailers are regarded as more powerful than any single manufacturer (Wang and Seidmann 1995). For example, Wal-Mart generally sets the price for manufacturers, who decide the quantity to supply at the given price (Schiller 1992).2 Similarly, in the electronics industry, retailers such as Circuit City and Best Buy typically determine the prices for their manufacturers, who act as pricetakers. Indeed, Fisher et al. (1994) note that consolidation in retailing has given the surviving retailers more power over the manufacturers in some industries. The growing power of giant retailers such as Wal-Mart and Target has been discussed by Bradley and Ghemawat (1994) and Bradley et al. (2003). There are several reasons for growth in the retailer’s pricing power over the manufacturer. First, although many manufacturers are relying more on the sales through these dominant retailers, these retailers are depending less on those manufacturers, leading to their leadership. For example, in 1993, Wal-Mart accounted for 10 percent of P&G’s total revenues, but in 2003, this figure became 17 percent. However, despite being Wal-Mart’s largest supplier, P&G accounted for less than 3 percent of Wal-Mart’s total revenues in 2003. Second, Wal-Mart minimizes or eliminates price negotiation with suppliers (Bradley and Ghemawat 1994). For example, since the late 1990s, Wal-Mart has set the price of pickles from Vlasic, the manufacturer (Fishman 2003). “If the supplier company doesn’t sell its goods at the price Wal-Mart sets, Wal-Mart denies them shelf space at its stores” (Freeman 2003). Third, an interview with a merchandizing manager at a top retailer by the authors indicates that this firm uses a similar approach to determining the purchase price. Fourth, an empirical analysis by Kadiyali et al. (2000) shows that there is significant retailer pricing power for grocery product categories. The retailer uses a purchasing model in which there is a tradeoff between inventory holding and order costs. Following Lal and Staelin (1984) and Weng (1995), we use the Economic Order Quantity (EOQ) as a basis for our purchasing model given that: (1) the EOQ principle is embedded in the cost and profit functions of both the retailer and manufacturers (Weng 1995) and (2) it is widely used in distribution channel management decisions. It should be noted, however, that the results from our analysis are not dependent on the EOQ model, but hold as long as there is a similar tradeoff between inventory holding and order costs in the purchasing process. Assumptions. • • • • •

The retailer acts as the Stackelberg leader and the manufacturer acts as the follower. The retailer tradesoff inventory holding and order set up costs by following an EOQ model. The manufacturer’s (production) order size is an integer multiple (n) of the retailer’s (purchase) order size.3 Other costs of the retailer are normalized to zero, allowing us to focus on ER’s impact. Unit holding cost is fixed and does not change with short-run fluctuations in price.

2 Farris and Ailawadi (1992) suggest that some retailers may not be more powerful than manufacturers, but they note that retailers like Wal-Mart, Home Depot and Toys-R-Us wield more power than manufacturers. We also recognize that some manufacturers may jointly negotiate prices with some retailers. 3 Any cost economies of scale for the manufacturer from production runs will be reflected by the order size.

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• Costs other than inventory-related cost are captured through a conventional cost function cx2 (e.g., Eliashberg and Steinberg 1991) A basic EOQ order size (Q) from the retailer is given by:   2xSR 1/2 Q= , hR

(1)

where x: annual quantity purchased by the retailer from the manufacturer, SR : retailer’s unit order cost, hR : retailer’s unit holding cost, SM : manufacturer’s unit order cost, that is, the cost of executing a order from the retailer, hM : manufacturer’s unit holding cost of the finished product, and w: purchase/wholesale price paid by the retailer to the manufacturer. In the EOQ model, for given values of S, h and x, the holding (order) cost increases (decreases) with order size. Because total costs are the sum of the holding and order costs, it is convex in order size, so the order size at which the total costs are minimized is a square root function of the ratio of the unit order and holding costs. We assume that the non-inventory costs follow cx2 for several reasons. First, it is consistent with the scenario involving many products which exhibit decreasing profit returns to scale (quantity). Second, non-inventory costs are often subject to production and distribution capacity constraints, and can be conveniently modeled as a quadratic function. Third, much research in operations has found this functional form to be very reasonable (e.g., Eliashberg and Steinberg 1991). We define the inverse demand function for the retailer as p(x) = αx−β , consistent with Jeuland and Shugan (1988) and Moorthy (1987), where α and β are parameters of the inverse demand function. The demand curve is a simplified variant of Choi’s (1991) model, which has two products. We explicitly consider the retailer’s inventory cost and the cost of supplies. The manufacturer’s costs consist of inventory cost (order cost and holding cost) and other (non-inventory) cost, which typically includes purchase cost (if the manufacturer purchases the products or raw materials), and/or production cost (if the manufacturer produces the products). The manufacturer’s inventory holding and order costs are hR Q/2 and SR x/Q, respectively, similar to Weng (1995). The retailer, being the Stackelberg leader, sets the purchase price with perfect knowledge of the quantity the manufacturer is willing to sell at the price set by the retailer. The manufacturer then maximizes its profits by adjusting the quantity it sells to the retailer under the given price. This situation is akin to a deliver-to-order process by the manufacturer that is common in several markets where retailers are typically more powerful. All information needed to make decisions is certain and costless for both the parties. We model ER by examining the comparative statistics of each channel outcome (quantity, price and profit) with respect to reduction in order cost. 4 To solve for the equilibrium purchase quantity and price, we start with the manufacturer’s (or the follower’s) maximization of its objective function, as is the case in a Stackelberg model. The manufacturer determines the quantity to sell by taking the price (w) set by the retailer as given. It maximizes its profits (πM ), as shown in Eq. (2).5     SR hR 1/2 SM hM x1/2 ; πM = wx − C(x) − + (2) 2 SR hR From the first order conditions for the manufacturer’s profit function, the following relationship between the purchase quantity and purchase price can be established.     1 SR hR 1/2 SM hM w = 2cx + x−1/2 . (3) + 2 2 SR hR The retailer, the Stackelberg leader, uses its knowledge of the price-quantity relationship from Eq. (3) in its objective function of total profits (πR ) in Eq. (4) below. πR = αx1−β − wx − (2hR SR x)1/2 The first order condition for the retailer can be expressed as      SM 1 −1/2 SR hR 1/2 hM −β 4− =0 −w − 2cx + α(1 − β)x − x + 4 2 SR hR

(4)

(5)

The retailer’s optimal price and the associated purchase quantity can be solved from the first order condition in Eq. (5), after substituting for w from Eq. (3) into Eq. (4). Closed-form solutions can be obtained for purchase quantity and price, namely, 4 This nonlinear demand model can be extended to multiple manufacturers if the costs of the manufacturers are identical. A linear demand model, however, is not subject to this restriction. 5 The derivations of this equation and subsequent expressions in the paper can be found in Appendix A available with the authors. The costs are consistent with constant returns to scale as in Weng (1995).

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x* (SM , SR , hR , hM ) and w∗ (SR , SM , hR , hM ) given certain values of β. Our objective, however, is to examine the impact of ER on distribution channel outcomes, so comparative statics can be analyzed for the key characteristics through the implicit functions. Manufacturer-led Stackelberg model The manufacturer led Stackelberg model differs from the retailer led model in that the manufacturer dictates the purchase price with the retailer (e.g., Ailawadi et al. 2005; Mela and Kopalle 2002). This structure is consistent with the channel coordination literature (e.g., Ingene and Parry 1995, 2000; Messinger and Narasimhan 1995). The model still uses the EOQ inventory principles and the retailers determine the order size. Assumptions. The assumptions are slightly different from those in the retailer led model and are as follows: • • • •

The manufacturer acts as the Stackelberg leader and the retailer acts as the follower. The manufacturer’s order size is an integer multiple (n) of the retailer’s order size. Other costs of the retailer are normalized to zero, allowing us to focus on ER’s impact. Costs other than inventory-related cost are captured in a conventional cost function cx2 .

Because the retailer is the follower in this structure, we first solve the retailer’s optimization problem. By maximizing the retailer’s profit function with respect to quantity, Eq. (4), while holding w constant, we can obtain: 1 w = α(1 − β)x−β − (2SR hR )1/2 x−1/2 . 2

(3 )

Similar to the retailer-led Stackelberg model, we substitute for purchase quantity x in the manufacturer profit function, Eq. (2), with the optimal quantity x as a function of w obtained from the first order condition of the retailer, Eq. (3 ). The optimal quantity and purchase price can be solved from the above equations for specific values of β and the second order condition can be shown to be satisfied for a certain range of values of β. The retailer–manufacturer relationships in the two models are thus captured by two systems—the traditional price-quantity mechanism and the inventory tradeoffs, consistent with Lal and Staelin (1984). In the retailer-led Stackelberg model, the retailer determines the purchase price, but the manufacturer decides the purchase quantity, while in the manufacturer-led model, the retailer decides the purchase quantity, but the manufacturer sets the purchase price.

The impact of ER when the retailer is the leader We now examine the impact of ER on the distribution channel outcomes. We first present the impact of ER in the retailer-led model. To study the impact of ER on the equilibrium of the distribution channel, we examine comparative statics of the optimal purchase quantity, purchase price, and profits of the retailer and the manufacturer with respect to a reduction in the retailer’s order cost. We examine three cases, (1) when only the retailer adopts ER, (2) when both the manufacturer and the retailer reduce their order costs through the adoption of ER, and (3) when the distribution channel is integrated and the retailer and the manufacturer jointly implement ER. A summary of the statements of the key lemmas and results appear in Table 1. Only the retailer adopts ER Purchase quantity Lemma 1A ((Purchase quantity)). The purchase quantity from the manufacturer will increase due to the adoption of ER by the retailer if the ratios of the holding and order costs satisfy the following inequality: hM SM − > −4. hR SR

(6)

The intuition behind Lemma 1A is as follows. When the retailer reduces its order cost, the order cost of the manufacturer relative to the retailer (the ratio of manufacturer’s and retailer’s order costs) increases. As long as this ratio remains within an upper bound (the holding cost ratio plus four), the increase in the manufacturer’s order cost relative to the retailer’s order cost, can be compensated by an increase in the purchase price so that the manufacturer will still be willing to supply a higher quantity to the retailer. When the condition does not hold, even if the retailer offers a higher price, the increase in the inventory cost and

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Table 1 Summary of key lemmas and results Lemma/result

Statement

Lemma 1B

A retailer adopting ER will purchase more quantity from a manufacturer if the manufacturer’s holding cost relative to the retailer’s is sufficiently large, relative to the manufacturer’s order cost relative to the retailer’s

Result 1B

A retailer adopting ER will pay a higher price to the manufacturer when the manufacturer’s holding cost relative to the retailer’s is adequately greater than the manufacturer’s order cost relative to the retailer’s

Result 2B

The profit of a retailer adopting ER will increase if purchase quantity increases The profits of both the manufacturer and the retailer will increase due to retailer adoption of ER when the manufacturer’s holding cost relative to the retailer’s is adequately greater than the manufacturer’s order cost relative to the retailer’s The profit of a retailer adopting ER will decrease while the manufacturer’s profit may increase or decrease if the manufacturer’s holding cost relative to the retailer’s is moderately smaller than the manufacturer’s order cost relative to the retailer’s The profits of both the manufacturer and the retailer will decrease due to retailer adoption of ER when the manufacturer’s holding cost relative to the retailer’s is much smaller than the manufacturer’s order cost relative to the retailer’s

Result 3B

The distribution channel profits will increase due to retailer adoption of ER if the manufacturer’s holding cost relative to the retailer’s is adequately greater than the manufacturer’s order cost relative to the retailer’s The distribution channel profits will decrease due to retailer adoption of ER if the manufacturer’s holding cost relative to the retailer’s is much smaller than the manufacturer’s order cost relative to the retailer’s

Result 4

The distribution channel profits for the integrated retailer and the manufacturer will increase unconditionally due to ER

Note. We have presented only Lemma 1B and Results 1B, 2B, and 3B because Lemma 1A and Results 1A, 2A, and 3A are special cases of Lemma 1B and Results 1B, 2B, and 3B, respectively. Lemma 1A and Results 1A , 2A , and 3A are similar and are omitted to save space.

the upward sloping cost function for the manufacturer will result in a lower quantity. This lemma is consistent with Lal and Staelin (1984) who find that a higher purchase quantity is associated with lower holding and order costs. The lemma has interesting implications for the manufacturer and the retailer. If the manufacturer is interested in increasing its market share with and through the retailer, then it needs to ensure that its relative order cost is sufficiently lower than its relative holding cost. The retailer also has to satisfy the condition in the lemma if it desires to purchase greater quantity of the product, which might be appropriate if the product is a traffic-builder or market leader. Price Result 1A ((Price)). The change in the purchase price that the retailer will pay the manufacturer due to the retailer’s adoption of ER depends on the ratios of the holding and order costs. Specifically, (i) if 0 > (hM /hR − SM /SR ) > −4, then the purchase price will increase; (ii) if (hM /hR − SM /SR ) > 0, or if (hM /hR − SM /SR ) < −4, then the purchase price could increase or decrease, depending on not only the difference in the ratios of the costs and the values of the costs, but also the change in purchase quantity. The change in price due to the retailer’s adoption of ER comes from two sources: (1) the direct inventory cost change since ER contributes to a purchase price increase under the condition of Result 1A (i) and (2) the increase in other costs due to a higher purchase quantity condition in Lemma 1A. When the purchase price increases to the level at which the cost increases from both sources are sufficiently compensated, purchase quantity increases as well. This is the case indicated by Lemma 1A and Result 1A (i). From Lemma 1A, when (hM /hR − SM /SR ) > 0, that is, when the manufacturer’s order cost relative to the retailer’s is less than the manufacturer’s holding cost relative to the retailer’s, the purchase quantity increases; but from Result 1A, this increase does not necessarily result in a purchase price increase. In fact, when the manufacturer’s order cost is significantly less than the retailer’s order cost, for example, while their holding costs are similar, a reduction in the retailer’s order cost would decrease the retailer’s order size and make it closer to the quantity that would minimize the manufacturer’s inventory cost. In this case, the manufacturer’s inventory cost will be reduced and the manufacturer may be able to provide a higher overall quantity even with a lower purchase price. In, Result 1A (i), when 0 > (hM /hR − SM /SR ) > −4 is satisfied, the cost increase on the manufacturer’s side prevails; thus with greater purchase quantity, the purchase price has to increase. On the other hand, in Result 1A (ii) when (hM /hR − SM /SR ) > 0, while purchase quantity will increase (from Lemma 1A), purchase price could decrease because a decrease in inventory cost may offset an increase in other costs. Both the manufacturer and the retailer can better plan their strategies by knowing these conditions under which the purchase price increases or decreases. By better understanding these conditions, they can make more informed tradeoffs between high prices and high quantities.

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Profit ER’s impact on the distribution channel may be judged as “favorable” if its implementation results in increases in the profits of both the manufacturer and the retailer. Result 2A shows the conditions under which ER increases or decreases retailer and manufacturer profits. Result 2A ((Manufacturer and retailer profits)). The retailer’s and the manufacturer’s profits change due to the retailer’s adoption of ER according to the following conditions: (i) If (hM /hR − SM /SR ) > 0, the profits of both the retailer and the manufacturer will increase. (ii) If 0 > (hM /hR − SM /SR ) > −4, the retailer’s profit will increase, while the manufacturer’s profit may increase or decrease. (iii) If (hM /hR − SM /SR ) < −4, the profits of both the retailer and the manufacturer will decrease. Result 2A shows that the conditions for the manufacturer to benefit from the adoption of ER are stronger than those for the retailer. Thus, the retailer is more likely to benefit from ER than the manufacturer. Specifically, ER will bring benefits to both distribution channel members if the manufacturer prefers a smaller order size. In this case, a reduction in the retailer’s order cost will make the order size determined by the retailer closer to the one the manufacturer would prefer. The retailer still benefits, while the manufacturer may have lower profits if the manufacturer prefers a larger order size while the retailer initiates a smaller order size into the system. However, if the order size is not small enough to cause the manufacturer’s inventory cost to increase substantially (or, the difference in the ratios of order and holding costs is no greater than four), while the retailer will still be able to gain, the manufacturer’s profit will decrease. Interestingly, if the cost ratios are very different and the reduction in the retailer’s order cost further enlarges the gap, the large increase in costs to the manufacturer will result in a decrease in the profits of both the members as in Result 2A. From Result 2A, the retailer is more likely to gain than the manufacturer. The condition for the retailer to gain is the same as that for purchase quantity increase. In other words, the retailer gains when the quantity purchased in the channel under ER increases (Lemma 1A). The manufacturer, however, may not be able to gain under the same condition. The reason is that the retailer has the advantage of being the price leader. Recall that the order size depends on the retailer’s order and inventory holding costs, as well as purchase quantity which, in this case is determined by the retailer-decided wholesale price. Therefore, the retailer has the dominant position in controlling both the price and physical product flow (order size). The position allows the retailer to more likely benefit from ER than would a manufacturer. Comparing Results 1A and 2A, we can see that the profit the manufacturer could realize from ER operations does not always move in the same direction as the purchase price. Under conditions such as 0 > (hM /hR − SM /SR ) > −4, although the purchase price increases, the manufacturer’s profits may decline due to other cost increases. The channel profits are defined as the total profits of the retailer and the manufacturer. Result 3A ((Distribution channel profits)). The distribution channel profits change due to the retailer’s adoption of ER as follows: (i) if (hM /hR − SM /SR ) > −2, the distribution channel profits will increase; (ii) if −2 > (hM /hR − SM /SR ) > −4, the change in the distribution channel profits is undetermined; (iii) if (hM /hR − SM /SR ) < −4, the distribution channel profits will decrease. When the cost reduction at the retailer does not increase the inventory (order plus holding) costs for the entire channel (given total purchase quantity), the entire channel gains. It is more interesting to see that the entire channel may still gain due to the impact of ER on purchase quantity which increases within this range. This potential channel gain, however, is at the expense of the manufacturer—the retailer gains through increased quantity and reduced cost more than offsets the manufacturer’s increased cost. Both the retailer and the manufacturer adopt ER Let us consider a more likely case where both the manufacturer and retailer are able to reduce their order costs. Let the rate of change of the manufacturer’s order cost with respect to change in the retailer’s order cost be represented by a constant m. When ∂SM /∂SR = m > 0, the manufacturer reduces its own order cost when the retailer reduces its order cost. When ∂SM /∂SR = m = 0, the manufacturer does not change its order cost when the retailer reduces its order cost. Finally, when ∂SM /∂SR = m < 0, the manufacturer actually increases its order cost when the retailer is reducing its order cost. This is the case when the manufacturer is counteractive to the retailer in the latter’s efforts to implement ER. This case can also be viewed as functional shifting and may not be very uncommon. It can be shown that the corresponding conditions in Lemma 1A and Results 1A, 2A and 3A (in the case without manufacturer order cost reduction) become as follows.

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Lemma 1B ((Purchase quantity)). The purchase quantity from the manufacturer will increase due to the adoption of ER by the retailer if the ratios of the holding and order costs satisfy the inequality: (hM /hR − SM /SR ) > −4 – 2m Result 1B( (Price)). The change in the purchase price that the retailer will pay the manufacturer due to the retailer’s adoption of ER depends on the ratios of the holding and order costs. Specifically, (i) if −2m > (hM /hR − SM /SR ) > −4 – 2m, then the purchase price will increase. (ii) if (hM /hR − SM /SR ) > − 2m, or (hM /hR − SM /SR ) <−4 – 2m, then the purchase price could increase or decrease, depending on not only the difference in the ratios of the costs and the values of the costs, but also the change in purchase quantity. Result 2B ((Manufacturer and retailer profits)). The retailer’s and the manufacturer’s profits change due to the retailer’s adoption of ER according to the following conditions: (i) if (hM /hR − SM /SR ) > −2m, the profits of both the retailer and the manufacturer will increase. (ii) if −2m > (hM /hR − SM /SR ) > −4 − 2m, the retailer’s profit will increase, while the manufacturer’s profit may increase or decrease. (iii) if (hM /hR − SM /SR ) < −4 − 2m, the profits of both the retailer and the manufacturer will decrease. Result 3B ((Distribution channel profit)). The distribution channel profits change due to the retailer’s adoption of ER as follows: (i) if (hM /hR − SM /SR ) > −2 − 2m, the distribution channel profits will increase; (ii) if (−2 − 2m) > (hM /hR − SM /SR ) > −4 − 2m, the change in distribution channel profits is undermined; (iii) if (hM /hR − SM /SR ) < −4 − 2m, the distribution channel profits will decrease. These lemma and results are variations on corresponding previous lemma and results. It can be seen that the conditions in the case when the manufacturer cannot reduce its order cost (m = 0) are special cases of the above conditions. From the above results, when the manufacturer reduces order cost along with the retailer, that is, m > 0, it can be shown that the range of values for which either the retailer or the manufacturer would benefit from ER is greater than that in the case where the manufacturer does not adopt ER. Thus, this interesting aspect shows that both the retailer and the manufacturer will more likely benefit from the retailer’s adoption of ER when the manufacturer also reduces order cost. On the other hand, however, when m < 0, that is, when the manufacturer increases order cost or is counteractive to the retailer or when there is functional shifting, the retailer will be less likely to benefit from ER, as a negative m reduces the range in which its profits can increase. However, more importantly, m < 0 is detrimental for the manufacturer as it increases the range in which the manufacturer’s profit will decrease. Therefore, it is not rational for a manufacturer to oppose the retailer’s adoption of ER. These results are important for both the manufacturer and the retailer. The conditions under which the manufacturer gets a greater share of the gains from ER when it also adopts ER are useful for it to decide on ER adoption. They also help the retailer understand its incremental gains when the manufacturer also adopts ER. ER in a channel in which the retailer and the manufacturer are vertically integrated In many markets, retailers and manufacturers have made extensive efforts to integrate their operations. Implemented successfully, such integration can have huge payoffs to all companies involved. Furthermore, such coordination is also deemed as the essence of various industry initiatives such as ER (Transportation and Distribution 1997). Some manufacturers of computing and electronics products have integrated operations with retailers. Apple computer, which has a rapidly growing number of stores, is a good example (Lohr 2006). Johnston and Murphy, a large maker of shoes and accessories, now owns a large number of its own outlets (http://www.johnstonmurphy.com/). An integrated distribution channel represents the strongest level of coordination between retailers and manufacturers, while not possessing some of the shortcomings of vertical integration (e.g., required capital outlays to acquire a manufacturer or a customer). In addition, an integrated distribution channel presents the possibility of global supply chain optimization, allowing retailers and manufacturers to collectively allocate their inventories and efficiently establish production and distribution plans. In this research, distribution channel integration refers to the integration of inventory decisions, purchasing, and other related functions, of the parties involved with the common goal of jointly maximizing profits. Specifically, the two parties maximize their joint profits with an integrated inventory system in which purchase quantity and inventory levels are determined jointly (Weng 1995). Purchase prices become the internal transaction/transfer prices and are no longer included in either party’s decision making process. Purchase quantities are not determined by either party, but by the cost characteristics of the combined system and by the retailer’s market demand. However, this integration does not consolidate or centralize the physical channel inventory systems. In other words, the products, the ordering and distribution processes and related costs remain decentralized in the channel members’ respective locations.

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The total profits of the distribution channel under integration can be written as follows: πT = πR + πM = αx1−β − cx2 − (2hT ST x)1/2

(7)

where ST = SR + SM and hT = hR + hM are total order cost and total holding cost respectively. Since the inventory system is also integrated, the order size the retailer purchases from the manufacturer is given by:   2ST x 1/2 QT = (8) hT ER in an integrated channel is represented by the joint effort the two parties make in reducing the total order cost (ST ). The order cost reduction can take place on either the retailer’s or the manufacturer’s side (SR or SM ). Purchase quantity Lemma 2 ((Purchase quantity)). The quantity that the retailer purchases from the manufacturer in the integrated channel will increase with the implementation of an integrated ER. Unlike Lemma 1, which shows that purchase quantity increases when certain conditions are satisfied, this lemma provides a much stronger implication for ER’s impact on purchase quantity. With ER, the retailer will purchase more from its integrated manufacturer regardless of the cost structure of both the parties, with the market share of the integrated manufacturer increasing unconditionally. This lemma is consistent with Lal and Staelin (1984) in that a reduction in order and holding costs are associated with an increase in purchase quantity. In the integrated channel model, the decisions are made for the entire channel rather than for each party. The order cost reduction at the retailer under ER reduces the channel cost and therefore leads to a larger purchase quantity. Unlike the results in the retailer led model, where an order size decision is made based on the retailer’s cost structure (order and holding costs) and this decision may have negative impact on the manufacturer’s costs, this result under the integrated framework is due to the impact of channel alignment in terms of inventory structures. Under integration, there is no conflict between the retailer and the manufacturer’s interests. The total channel cost is the basis for the order size decision. Therefore a cost reduction under ER benefits the entire channel. Profit Result 4( (Distribution channel profits)). The profits for the integrated retailer and manufacturer distribution channel will increase unconditionally with the adoption of ER. When the distribution channel is integrated, ER is implemented based on the mutual interest of both the members. Order costs are reduced where needed—cost minimization is carried out on the integrated distribution channel. If this strategy can be successfully implemented, the distribution channel as a whole will become more profitable, consistent with Weng (1995). Thus, Lemma 2 and Result 4 are useful to the manufacturer and the retailer for benchmarking the channel’s performance when it is not integrated. Comparing the impacts of ER on the traditional and the integrated retailer–manufacturer distribution channels, we conclude that order cost reductions by only one party (e.g., the retailer) may limit both the retailer and manufacturer profits while the second party (e.g., manufacturer) is likely to experience increasing costs. The dominant power of the retailer (assumed in this model) cannot solve this problem because the manufacturer will not continue to supply the retailer if its costs grow faster than its revenues. However, integrating the retailer and manufacturer within the distribution channel can unconditionally increase profits. A comparison of the direct impact of ER on purchase quantity under the two types of channel structures against the integrated channel provides some insights into channel coordination under ER for two reasons. First, an integrated channel represents a coordinated channel for benchmarking purposes. Second, from the standpoint of the ordering decision, the difference between an integrated channel and a non-integrated channel is the purchase quantity. Thus, ER could move, ceteris paribus, the purchase quantity in a non-integrated channel closer to that in an integrated channel, improving channel coordination. In fact, comparing Eqs. (1) and (8),   Q 1 + (hm / hR ) 1/2 = QT 1 + (SM /SR )

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and Q ≤ QT when 1+

hm SM ≤1+ . hR SR

Under ER, it can be shown that ∂(Q/QT ) > 0. ∂SR Therefore, when 1+

hm SM ≤1+ , hR SR

Q ≤ QT ,

ER moves the purchase quantity in a non-integrated channel to a level closer to that in the integrated channel, improving channel coordination. Correspondingly, the channel profits in a non-integrated channel could be brought closer to the profits in an integrated channel.

The impact of ER when the manufacturer is the leader In this section, we discuss the impact of ER when the manufacturer is the leader. We do not discuss the case when both the retailer and the manufacturer are integrated because it is the same as in the Section “ER in a channel in which the retailer and the manufacturer are vertically integrated”. Only the retailer adopts ER Purchase quantity and price Lemma 1A ((Purchase quantity and price)). When the manufacturer is the leader, the quantity purchased by the retailer will increase, but the purchase price may decrease, due to the retailer’s adoption of ER if the ratios of the holding and order costs satisfy the following inequality: hM SM − > −1 SR hR

(6 )

Comparing Lemmas 1A and 1A , one can find that the quantity purchased from the manufacturer by the retailer is less likely to increase with ER under manufacturer leadership than under retailer leadership in the distribution channel. Since the manufacturer has an increasing marginal cost, it would exercise its price leadership with respect to the retailer to offset a possible increase in the manufacturer’s inventory cost and other costs. The fact that the manufacturer has more power to control its quantity commitment by dictating the purchase price, compared with when the retailer is the leader, leads to a weaker condition under which ER may result in an increase in purchase quantity. By the same token, the purchase price may decrease under the same condition. Profit Result 2A ((Manufacturer and retailer profits)). When the manufacturer is the leader, both the retailer’s and the manufacturer’s profits change due to the retailer’s adoption of ER according to the following conditions. (i) if (hM /hR − SM /SR ) > −1, the profits of both the retailer and the manufacturer will increase. (ii) if (hM /hR − SM /SR ) < −1, the manufacturer’s profit will decrease while the retailer’s profit may increase or decrease. Under the manufacturer-led structure, the manufacturer has greater leverage against the retailer than in the retailer-led structure, which enables it to set a higher price and move more of the retailer side cost savings to the manufacturer. This cost shift will help compensate for the higher inventory and other costs for the manufacturer. Such interaction between the pricing mechanism and product flows balances the power in a distribution channel and limits the retailer’s ability to gain from ER. However, as was the case under the retailer-led model, the retailer is still more likely to profit from ER than the manufacturer.

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The intuition behind this result is similar to that in Result 2A, in that the pricing mechanism and the order decision drive the impact of ER on the two parties. In this case, however, since the manufacturer decides the wholesale price, he is able to affect the price and part of the order size decision (the retailer still places orders based on its own order and holding costs, but the purchase quantity which is part of the order size is determined by the wholesale price decided by the manufacturer). This balances the control in the channel when compared with the retailer-led model. In the manufacturer-led model, while the manufacturer is more likely to gain, the retailer is less likely to benefit from ER, compared with the retailer-led model. However, even in this manufacturer-led model, the retailer is still in a slightly better position to gain from ER than is the manufacturer. The reason is that although the manufacturer is the price leader, the wholesale pricing mechanism offers limited power in the relationship. The manufacturer control on wholesale price and to a certain extent purchase quantity is inadequate to overcome the retailer control on order size, which results in a more favorable condition for the retailer to gain from ER. Result 3A ( (Distribution channel profits)). When the manufacturer is the leader, the distribution channel profits due to the retailer’s adoption of ER changes as follows: (i) if (hM /hR − SM /SR ) > −1, the distribution channel profit will increase; (ii) if −1 > (hM /hR − SM /SR ) > −2, the change in the distribution channel profit is undetermined; (iii) if (hM /hR − SM /SR ) < −2, the distribution channel profit will decrease. Result 3A indicates that the impact of ER on the distribution channel profits is subject to the physical characteristics of the partners involved. The ratios of both parties’ inventory holding and order costs, to a large extent, determine the success of an ER implementation. ER can be less profitable under manufacturer leadership than under retailer leadership because the manufacturer may leverage some of the balance in the distribution channel’s physical product flows and pricing systems in the manufacturer-led model relative to the retailer-led model. Result 3A is similar to that for the retailer-led distribution channel structure although the points at which profits start to decrease are different between the two channel structures. When the retailer is the leader, the manufacturer’s profit starts to decrease when the retailer’s order cost is reduced below SM (hR /hM ) and the retailer’s profit starts to decrease if the retailer’s order cost becomes less than SM (hR /hM + 4). When the manufacturer is the leader, its profit begins to decrease when the retailer’s order cost is reduced below SM (hR /hM + 1). Lemma 1A and Results 1A –3A are valuable for guiding the manufacturer and the retailer when the manufacturer is the price leader. The conditions are similar to the case where the retailer is the price leader, only the threshold values and a few other aspects are different in the two cases. The reason for these differences is the asymmetric ordering structure in the channel relationship—the retailer is the entity placing the orders and deciding the purchase quantity, regardless of who the price leader is. Both the retailer and the manufacturer adopt ER In the case when both retailer and manufacturer reduce order costs to embrace ER, as discussed in Section “Both the retailer and the manufacturer adopt ER” above, the results are quite consistent with those shown in that section. In fact, Lemma 1B ((Purchase quantity)). When the manufacturer is the leader, the quantity purchased by the retailer will increase, but the purchase price may decrease, due to the retailer’s adoption of ER if the ratios of the holding and order costs satisfy the inequality (hM /hR − SM /SR ) > −1 − 2m. Result 2B ((Manufacturer and retailer profits)). When the manufacturer is the leader, both the retailer’s and the manufacturer’s profits change due to the retailer’s adoption of ER according to the following conditions. (i) if (hM /hR − SM /SR ) > −1 − 2m, the profits of both the retailer and the manufacturer will increase. (ii) if (hM /hR − SM /SR ) < −1 − 2m, the manufacturer’s profit will decrease while the retailer’s profit may increase or decrease. Result 3B ((Distribution channel profit)). When the manufacturer is the leader, the distribution channel profits due to the retailer’s adoption of ER changes as follows: (i) if (hM /hR − SM /SR ) > −1 − 2m, the distribution channel profit will increase; (ii) (ii) if −1 − 2m > (hM /hR − SM /SR ) > −2 − 2m, the change in the distribution channel profit is undetermined; (iii) if (hM /hR − SM /SR ) < −2 − 2m, the distribution channel profit will decrease.

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Table 2 Comparison of results between retailer-led and manufacturer-led structures Variable

Retailer-led model

Manufacturer-led model

Intuition for the difference

Only the retailer adopts ER

Purchase quantity

Will increase if (hM /hR − SM /SR ) > −4

Will increase if (hM /hR − SM /SR ) > −1

Retailer profit

Will increase if (hM /hR − SM /SR ) > −4

Will increase if (hM /hR − SM /SR ) > −1

Will decrease if (hM /hR − SM /SR ) < −4 Will increase if (hM /hR − SM /SR ) > 0

Reducing the retailer’s order cost may increase the manufacturer’s inventory cost. When the retailer is the price leader, since the manufacturer’s cost increase is not completely reflected in its marginal cost (order cost is part of the marginal cost while holding cost is not), the retailer can order more and force the manufacturer to absorb the increase in its holding cost. As a result, the threshold level for the purchase quantity to increase is lower in the retailer-led model than in the manufacturer-led model The retailer has the upper hand while being the price leader and deciding the order size based on its own cost structure (so that the quantity reflects its optimal tradeoff between order and holding costs). This position allows the retailer to benefit more from order cost reduction, resulting in a lower threshold level for increase in retailer’s profit in the retailer-led model than in the manufacturer-led model

Will increase if (hM /hR − SM /SR ) > −1

Will decrease if (hM /hR − SM /SR ) < −4 Will increase if (hM /hR − SM /SR ) > −2

Will decrease if (hM /hR − SM /SR ) < −1 Will increase if (hM /hR − SM /SR ) > −1

Will decrease if (hM /hR − SM /SR ) < −4

Will decrease if (hM /hR − SM /SR ) < −2

Purchase quantity

Will increase if (hM /hR − SM /SR ) > −4 − 2m

Will increase if (hM /hR − SM /SR ) > −1 − 2m

Retailer profit

Will increase if (hM /hR − SM /SR ) > −2m Will decrease if (hM /hR − SM /SR ) < −4 − 2m Will increase if (hM /hR − SM /SR ) > −2m Will decrease if (hM /hR − SM /SR ) < −4 − 2m Will increase if(hM /hR − SM /SR )> −2 − 2m Will decrease if (hM /hR − SM /SR ) < −4 − 2m

Will increase if (hM /hR − SM /SR ) > −1 − 2m

Manufacturer profit

Channel profits

Both the manufacturer adopt ER

Manufacturer profit

Channel profits

m = ∂SM /∂SR .

Will increase if (hM /hR − SM /SR ) > −1 − 2m Will decrease if (hM /hR − SM /SR ) < −1 − 2m Will increase if (hM /hR − SM /SR ) > −1 − 2m Will decrease if (hM /hR − SM /SR ) > −2 − 2m

The manufacturer’s profit is more likely to increase with retailer adoption of ER when it is the price leader than when the retailer is the price leader because the manufacturer’s decision on wholesale price balances the retailer’s advantageous position in replenishment. Therefore, the threshold level is lower in the manufacturer-led model than in the retailer-led model When this condition is satisfied, the channel is better off when the retailer leads the price game because the retailer also decides the order size. Even when this condition is not satisfied and the total channel inventory cost increases, the channel is more likely to benefit from the retailer’s order cost reduction in the retailer-led structure than in the manufacturer-led structure due to the impact on reducing channel inefficiency in its pricing game (double marginalization). The manufacturer-led model balances the retailer’s advantageous position in determining order size by allowing the manufacturer to determine price. This will reduce the likelihood of the channel benefiting from ER even when the total inventory cost is lower The channel is more likely to lose (i.e., higher threshold for decrease in profits) under ER when the manufacturer is the price leader. Since the retailer still decides the order size, both parties have some leverage to protect their own interests. While the manufacturer should fare better under its own price leadership, the channel may lose the opportunity to gain under certain circumstances when the manufacturer needs to lose for the channel as a whole to gain. The channel is more likely to lose under manufacturer-led structure than retailer-led structure because the manufacturer uses its pricing control to offset the cost reduction under ER and protect its own interests The intuitions for this set of results are very similar to those for the other configuration. The difference in the mathematical conditions is the term, -2m, which reflects the extent to which the manufacturer responds by reducing its order cost to the retailer’s reduction of order cost. The impact of ER is similar across the two structures

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Configuration

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In summary, the manufacturer-led model yields important results. First, the results show that even under this arrangement, the retailer is likely to profit more from its adoption of ER than is the manufacturer, especially if the manufacturer does not modify its distribution or manufacturing cost structures to accommodate ER. This result reflects the fact that retailers usually are in a position to determine the product flows (order lot size) between manufacturer(s) and themselves, and such an advantageous position allow them to gain from programs such as ER. A manufacturer’s leadership in pricing allows it to partially offset its potential cost increases due to ER, and enables it to gain more or lose less than when the retailer is the price leader. Second, it is important to both manufacturer and retailer profits to have the manufacturer change distribution and manufacturing processes to cut order costs, whether or not the manufacturer has price leadership in the distribution channel. Third, the distribution channel profits will less likely increase with ER when the manufacturer is the price leader. In other words, ER will more likely improve the profits of the whole distribution channel when the retailer decides both the order size and the purchase price. A comparison of the results from the two leadership structures, retailer-led model and manufacturer-model appears in Table 2. For each channel configuration, the conditions for directional changes in quantity, retailer profit, manufacturer profit and channel profit due to ER are shown in the columns. The intuitions for the differences in the results between the two channel structures are also presented. The retailer is more likely to gain from ER than the manufacturer, particularly, in the retailer-led structure. The manufacturer is in a better position to gain from ER if it can also reduce its order cost. The demand function in our model is one that entails vertical strategic complementarity (VSC), that is, a situation in which the follower’s margin is higher if the Stackelberg leader’s margin is higher. A demand curve that entails vertical strategic substitutability (VSS), a situation in which the leader’s and the follower’s margins are negatively correlated, better reflects reality (Lee and Staelin 1997). To check if our results hold when we consider a demand curve that evinces VSS, we tried two alternative demand functions. First, we analyzed the model based on a linear demand function. The results under this linear demand function are the same as those under the constant elasticity demand function. Second, we analyzed the model based on a general downward sloping demand function (under both VSC and VSS) that satisfies the second order condition (SOC). The results for the downward sloping demand function are consistent with those for the multiplicative demand that we use in our model. Therefore, we are comfortable with the robustness of our results.

Contributions, implications, limitations, and extensions In this research, we have developed a game theoretic channel model to study the impact of ER on channel outcomes. By analyzing both retailer-led and manufacturer-led Stackelberg models, we have developed a series of analytic results. These results identify the conditions under which the retailer and the manufacturer may profit from ER and show how the incremental channel profits are divided between the manufacturer and the retailer. Our most significant insights include the following: • ER benefits the retailer under a wider range of conditions and assumptions than it benefits the manufacturer. By comparing the results from the retailer-led and manufacturer-led models, one can infer that the retailer is in an advantageous position to gain from ER even if the manufacturer is the price leader. This result explains in part why the retailer gains more from ER than does the manufacturer. ER benefits both retailers and manufacturers only if manufacturers prefer smaller order sizes, or if they are adequately compensated by retailers to cover the manufacturer’s cost increases due to ER. • When the manufacturer also reduces its own order cost along with the retailer, the range of outcomes under which both the retailer and manufacturer benefit from ER increases; that is, the ER partners have a greater chance of benefiting from an ER program if the manufacturer is also able to reduce its order cost. This result does not change with channel relationships (retailer-led or manufacturer-led). • If the manufacturer and retailer act together to jointly implement an integrated ER program, profits for both the retailer and manufacturer increase unconditionally. • The performance of the whole distribution channel, measured by the joint profits by the manufacturer and the retailer, is more likely to be higher with a retailer-led channel relationship than it is with a manufacturer-led relationship. This result arises mainly from the retailer having the power to determine both the order size and the purchase price in a retailer-led channel relationship. Our results are consistent with the observed outcomes of order cost reduction programs. For example, Quaker Oats undertook an order and inventory cost reduction program with its retailers in the 1990s (UPS Report 2005). While the retailers enjoyed much of the gains in channel profits, Quaker Oats was able to experience increases in its profit mainly when it reduced its own order costs. In a study of ECR programs in Europe, Corsten and Kumar (2005) found that retailers gained a lot and manufacturers gained little and that retailers gained more if manufacturers were more committed to cost reduction. They cite Sainsbury, a UK supermarket chain as an example of a retailer proactively working with the manufacturer so that the latter could also adopt the ER program and benefit from it.

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The results significantly extend prior research on inventory policies and channel coordination. The channel coordination literature (e.g., Ingene and Parry 1995, 2000; Jeuland and Shugan 1983; McGuire and Staelin 1983) identifies the optimal price discounts that serve as channel control mechanism and the general conditions for quantity discounts to coordinate the channel by taking into account cost reduction (Weng 1995). Unlike Jeuland and Shugan (1983) and McGuire and Staelin (1983) who focus on demand changes due to price changes, Lal and Staelin (1984) identify a unified price discount policy motivated by cost justifications. Ingene and Parry (1995) identify a number of non-coordinating two-part tariffs might dominate a coordinating quantity discount schedule from the standpoint of manufacturer profit. Ingene and Parry (2000) show further that for a manufacturer, channel coordination may be inferior to non-coordinating sophisticated Stackelberg pricing. We have identified the effects of order cost reduction, the focus of ER, on purchase quantity, price, and profit under different channel leadership scenarios and offered important managerial implications. Our models and results also shed some light on the interaction of a channel’s pricing mechanism and its product flow characteristics. Since the nature of many channel initiatives, such as ER, is to align the product flows between the channel partners, it is important to better understand how different aspects of channel relationships function with each other. In our results, for example, price leadership of a manufacturer does not automatically guarantee it to benefit more from a channel initiative such as ER given that the retailer generally determines the order size and the way in which orders should be shipped. The results are generally consistent with Lal and Staelin (1984) who found that a higher purchase quantity is associated with lower manufacturer and retailer ordering and holding costs. They are also in line with Ingene and Parry (1995, 2000) in that the manufacturer does not always gain in a coordinated channel. Importantly, the results show how the additional channel profits due to ER are distributed between the manufacturer and the retailer. The results are applicable to any situation that involves reduction in order costs. These results have important implications for both retailers and manufacturers. First, our results indicate that since order cost reduction by manufacturers can benefit retailers as well as themselves, it may be in the best interest of a retailer, even if it is dominant, to help its manufacturers implement ER programs. This help may be especially important since manufacturers often encounter difficulties in adopting ER. A retailer could help its manufacturers in ER implementation by adopting technologies such as Electronic Data Interchange (EDI) and Extranet that can be integrated into a manufacturer’s information system, thus enabling a manufacturer to reduce its own order cost. Second, since an integrated ER approach is almost certain to be a more profitable alternative for the channel, integrated and collaborative approaches such as inter-organizational joint teams and collaborative planning, forecasting and replenishment (CPFR) initiative could be pursued to realize ER benefits (see Dong et al. 2006a, 2006b for more information on such emerging practices). Third, channel structures have significant influences on ER performance across channel partners. Although retailers generally have an advantage over manufacturers in benefiting from ER because of their ability to decide the lot sizes of their orders, their ability to dictate purchase prices determines how likely they can benefit from ER. From the perspective of the whole channel, it appears that a retailer-led distribution channel will more likely produce greater benefits from ER than would a manufacturer-led channel. Our research has certain limitations that can be addressed by future research. First, we have analyzed ER in a single manufacturer-single retailer setting. Although extending this to multiple retailers and manufacturers might increase the modeling complexity, it would be a useful extension. Second, we have focused our study on the physical aspect of ER, by applying an inventory tradeoff in modeling the product flow in the channel. The aspect of information flow is left for further research (e.g., Dong et al. 2006b). The issue of demand uncertainty, particularly the bullwhip effect—the phenomenon by which distortion in orders destabilizes demand upstream in the channel—has received much attention (e.g., Lee et al. 1997). Although a phenomenon is more relevant to innovative products than to functional products that we have studied (Fisher 1997), it would be interesting to study the role of ER in mitigating the bullwhip effect. Third, we have examined a linear price contract that is commonly used in marketing literature and in practice. For certain nonlinear contracts, however, the Stackelberg leader may be able to extract much of the gains arising from ER, so it would be useful to extend our analysis to nonlinear price contracts such as two-part tariff and quantity discount schedule. Fourth, we have not examined the impact of CR, continuous replenishment, a process by which the order cost is taken to zero because of the complexity associated with CR in the real world. Finally, in our model, the manufacturer’s holding cost reflected the tradeoff between economies of scale and ER adoption for the manufacturer. The tradeoff could be more deeply examined by explicitly considering the manufacturer’s production runs and costs.

Appendix A. For a nonlinear demand function A.1. When the retailer is the Stackelberg price leader The order quantity follows the Economic Order Quantity principle, from which:   2xSR 1/2 Q= . hR

(A.1)

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Following Weng (1995, p. 1512) for inventory holding and ordering costs, the profit function of manufacturer is given by: πM = wx − cx2 −

hM Q S M x − . 2 Q

Substituting for Q from the EOQ result in Eq. (A.1) into this profit function, we get:     SR hR 1/2 SM hM πM = wx − cx2 − x1/2 . + 2 SR hR

(A.2)

(A.3)

The manufacturer maximizes their profits taking the purchase price as given and the retailer then maximizes its profit by taking into consideration the manufacturer’ reactions with respect to purchase price. From the manufacturer’s first-order condition, the following relationship can be obtained.     1 SR hR 1/2 SM hM w − 2cx − x1/2 = 0. (A.4) + 2 2 SR hR The second order condition requires     1 SR hR 1/2 SM hM −2c + x−(3/2) < 0, + 4 2 SR hR

(A.5)

at the local optimal point indicated by Eq. (A.4). This can be derived from the manufacturer self-selection condition πM ≥ 0. Note that when x → 0, The LHS in the SOC → +∞; and when x → +∞, the LHS in the SOC → −2c. This indicates a convex–concave manufacturer profit function that has either (0, 0) or the local optimal point from Eq. (A.4) as its global solution. The retailer’s profit is given by: πR = αx1−β − wx −

h R Q SR x − . 2 Q

(A.6)

Again, substituting for Q from Eq. (A.1) into the retailer’s profit function, we get: πR = αx1−β − wx − (2hR SR x)1/2 .

(A.7)

The retailer then determines the purchase price it has to offer based on the manufacturers’ possible responses.From Eq. (A.4),   −1   ∂x 1 −(3/2) SR hR 1/2 SM hM = 2c − x + >0 (A.8) ∂w 4 2 SR hR Maximizing the retailer’s profits (A.7) with respect to wholesale price w subject to (A.4), one can obtain the equilibrium (w∗ , x∗ ) by solving Eqs. (A.4) and (A.9) simultaneously. Specifically, the first order condition when maximizing (A.7) is   1/2  ∂x ∂πR −β −(1/2) SR hR = 0. = −x + α(1 − β)x − w − x ∂w ∂w 2 Substituting (A.8) into this equation, and since ∂x/∂w > 0, this condition becomes      1/2 1 −(1/2) SR hR 1/2 SM hM −β −(1/2) SR hR −2cx + x + α(1 − β)x − w − x + = 0. 4 2 SR hR 2 Combining the terms, we get      SR hR 1/2 SM 1 hM −w − 2cx + α(1 − β)x−β − x−(1/2) 4− =0 + 4 2 SR hR

(A.9)

The local second order condition is satisfied when     1 w ∂x −1 −β −3cx − α(1 − β) β − x − − 1 < 0, x 2 2 ∂w which is satisfied when 1/2 < β < 1. β < 1 has been assumed in many situations when a constant elasticity of demand is used (see Jeuland and Shugan 1983; Moorthy 1987). When w → +∞, the second order condition becomes positive, indicating a convex–concave profit function with maximum profits at either zero or the local stationary point.

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Proof of Lemma 1A ((Purchase quantity)). Replacing x, p, and w with the optimal values as functions of SR , and combining the retailer’s and the manufacturer’s first order conditions,      SR hR 1/2 1 SM hM 4cx − α(1 − β)x−β + x−(1/2) 1+ = 0. (A.10) + 2 4 SR hR Taking the partial derivative with respect to SR and rearranging terms, we get:      ∂x∗ hR 1/2 1 hM 1 SM 1+ = − x−(1/2) − ∂SR 2 2SR 4 hR SR    −1    1 SM 1 −(3/2) SR hR 1/2 hM −β−1 × 4c + αβ(1 − β)x 1+ − x + 2 2 4 SR hR where  4c + αβ(1 − β)x

−β−1

  −1    1 SM 1 −(3/2) SR hR 1/2 hM 1+ − x + >0 2 2 4 SR hR

from the retailer’s local second order condition. Therefore, ∂x* /∂SR < 0, when   hM SM > −4 − hR SR Proof of Result 1A ((Purchase price)). Given the result of Lemma 1A, the manufacturer’s first order condition can be represented in terms of purchase price w as a function of x. Thus,    ∗       ∂w∗ ∂x hR 1/2 hM 1 −(3/2) SR hR 1/2 SM hM 1 SM . = 2c − x + + x−(1/2) − 4 2 SR hR ∂SR 4 2SR hR SR ∂SR Since

    1 SR hR 1/2 hM SM 2c − x−(3/2) >0 − 2 hR SR 4

from the manufacturer second order condition, when   hM ∂x∗ SM > −4, or − < 0, hR SR ∂SR and



hM SM − hR SR

 < 0,

∂w∗ < 0, ∂SR

that is, both purchase quantity and purchase price increase under ER. On the other hand, when     hM hM SM SM < −4, or >0 − − hR SR hR SR the direction is undetermined. Proof of Result 2A ((Manufacturer and retailer profits)). Given the optimal profits of the retailer and the manufacturer, one can derive implicitly the impact of SR by using the above results. Using the first order conditions of the retailer and the manufacturer, we get:    ∗       ∗ ∂πM ∂x 1 −(1/2) SR hR 1/2 SM hM 1 1/2 hR 1/2 hM SM . = 2cx − x + − x − ∂SR 4 2 SR hR ∂SR 4 2SR hR SR     ∂πR∗ 1 1/2 hR 1/2 hM SM =− x − +4 . ∂SR 4 2SR hR SR

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Again, note     hM 1 −(3/2) SR hR 1/2 SM >0 + 2c − x 4 2 SR hR from the manufacturer second order condition, (∂πR∗ /∂SR ) < 0, when   ∂x∗ SM hM > −4 or − <0 hR SR ∂SR ∗ /∂S < 0 because when (h /h − S /S ) < 0, the second (from Lemma 1A). However, this condition is not sufficient for ∂πM R M R M R 1/2 ∗ 1/2 term in ∂πM /∂SR , −(1/4)x (hR /2SR ) (hM / hR − SM /SR ) is positive and can be greater than the first term, which is negative, ∗ /∂S can be positive. so ∂πM R ∗ , and the impact of ER on Channel profits Proof of Result 3A ((Channel profits)). The optimal channel profits, π∗ = πR∗ + πM can be obtained by combining the separate impact of ER on each party’s profits. From the previous result,    ∗       ∗ ∂π∗ ∂x hR 1/2 hM ∂πR∗ ∂πM 1 −(1/2) SR hR 1/2 SM hM 1 SM = + = 2cx − x + − x1/2 − +2 ∂SR ∂SR ∂SR 4 2 SR hR ∂SR 2 2SR hR SR

When (hM /hR − SM /SR ) > −2, the condition in Lemma 1A is also satisfied so that ∂x* /∂SR < 0. Since     ∂π∗ hM 1 −(1/2) SR hR 1/2 SM > 0, + <0 2cx − x 4 2 SR hR ∂SR when (hM /hR − SM /SR ) > −2. On the other hand, when   hM ∂x∗ SM ∂π∗ < −4, − > 0 and > 0. hR ∂SR SR ∂SR It is clear that in these two ranges, the direction of ER impact on Channel profits is undecided. A.2. When the manufacturer is the Stackelberg price leader The retailer maximizes its profit function given by: πR = αx1−β − wx − (2hR SR x)1/2

(A.11)

For a given w, the retailer optimizes its profits by selecting a purchasing quantity x. The retailer’s first order condition can be expressed by the following equation. 1 α(1 − β)x−β − w − (2SR hR )1/2 x−(1/2) = 0, 2

(A.12)

The second order condition requires the following inequality to hold. 1 αβ(β − 1)x−β−1 + (2SR hR )1/2 x−(3/2) < 0 4

(A.13)

Similar to the retailer-led Stackelberg case, the retailer profit function is a convex and concave function and a necessary condition for the maximum profits to exist is β < 1. Verifying that this local second order condition holds, we get   1 1 1/2 −(3/2) −2 1−β 1/2 1/2 −β−1 αβ(β − 1)x , αβ(β − 1)x + (2SR hR ) x =x + (2SR hR ) x 4 4 From the manufacturer’s self-selection condition, 1 αβx1−β − (2SR hR )1/2 x1/2 ≥ 0, 2

or

1 − αβx1−β + (2SR hR )1/2 x1/2 ≤ 0. 2

Since 1 1 −αβ(1 − β)x1−β + (2SR hR )1/2 x1/2 ≤ −αβx1−β + (2SR hR )1/2 x1/2 4 2

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when β < 1/2, the local second order condition is satisfied. This condition can also found in Choi (1991) and Weng (1995), where a constant elasticity demand function is used and a similar inventory and pricing structure is applied. The manufacturer’s profit function is:     SR hR 1/2 SM hM πM = wx − cx2 − x1/2 . + (A.14) 2 SR hR From the retailer’s first order condition, −1  ∂x 1 −β−1 1/2 −(3/2) = αβ(β − 1)x + (2SR hR ) x < 0. ∂w 4

(A.15)

Again, one can solve for (w∗ , x∗ ) by maximizing the manufacturer’s profit function with respect to w, subject to the retailer’s first order condition. From the manufacturer’s profit function (A.14), the first order condition is       ∂x 1 SR hR 1/2 SM ∂πM hM −(1/2) =x+ w − 2cx − = 0. x + ∂w ∂w 2 2 SR hR Applying (A.15), and rearranging, we get       SR hR 1/2 1 SR hR 1/2 SM 1 hM −αβ(1 − β)x−β + x−(1/2) = 0. + w − 2cx − x−(1/2) + 2 2 2 2 SR hR Combining the terms, and we get      SR hR 1/2 SM 1 hM w − 2cx − αβ(1 − β)x−β − x−1/2 − 1 = 0. + 2 2 SR hR

(A.16)

Again, the second order condition is given by:       SM ∂x 1 −(3/2) SR hR 1/2 hM 2 −β−1 −1 > 0. 1 − 2c − αβ (1 − β)x − x + 4 2 SR hR ∂w Or

   1/2  S S ∂x h 1 h R M M R +1 > 0, + −αβ(1 − β)2 x−β−1 − 2c + x−(3/2) ∂w 4 2 SR hR



and 2 −β−1

−αβ(1 − β) x

     SM 1 −(3/2) SR hR 1/2 hM + 1 < 0, − 2c + x + 4 2 SR hR

since ∂x/∂w < 0.From the manufacturer’s profit function,      SR hR 1/2 SM 1 hM c + αβ(1 − β)x−β−1 − x1(3/2) + 1 ≥ 0, + 2 2 SR hR it can be shown that the above second order condition is satisfied with the condition β < 1/2 holds. Proof of Lemma 1A ((Purchase quantity and price)). Combining the first order conditions of the retailer and the manufacturer and inserting the optimal value of purchase quantity, we get:      SR hR 1/2 SM 1 hM α(1 − β)2 x−β − 2cx − x−(1/2) + 1 = 0. + 2 2 SR hR Taking the derivative with respect to SB , we get         ∗  1 −(3/2) SR hR 1/2 SM hM ∂x 1 −(1/2) hR 1/2 hM SM 2 −β−1 − 2c + x + +1 + x − + 1 = 0. −αβ(1 − β) x 4 2 SR hR ∂SR 4 2SR hR SR

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Applying the manufacturer second order condition, we get      1 −(3/2) SR hR 1/2 SM hM 2 −β−1 −αβ(1 − β )x − 2c + x + +1 < 0. 4 2 SR hR Therefore, ∂x∗ <0 ∂SR when



hM SM − hR SR

 > −1.

 ∗    ∂w∗ ∂x 1 1 2hR 1/2 −(1/2) = αβ(β − 1)x−β−1 + (2SR hR )1/2 x−(3/2) − x , ∂SR 4 ∂SR 4 SR where 1 αβ(β − 1)x−β−1 + (2SR hR )1/2 x−(3/2) < 0. 4 When ∂x∗ ≥ 0, ∂SR

∂w∗ < 0. ∂SR

∂x∗ < 0, ∂SR

∂w∗ ∂SR

When

is undetermined. Proof of Result 2A ((Manufacturer and retailer profits)). From the previous result regarding the impact of ER on purchase quantity, we substitute for the optimal value of purchase quantity in the manufacturer and retailer profit functions, and apply their respective first and second order conditions to get the following.     ∗ ∂πM 1 hR 1/2 1/2 hM SM =− x − +1 ∂SR 2 2SR hR SR  ∗    ∂x ∂πR∗ 1 1 hR 1/2 1/2 = −x αβ(β − 1)x−β−1 + (2SR hR )1/2 x−(3/2) − x . ∂SR 4 ∂SR 2 2SR Since 1 αβ(β − 1)x−β−1 + (2SR hR )1/2 x−(3/2) < 0, 4 ∗ /∂S < 0 or (h /h − S /S ) > −1, ∂π ∗ /∂S < 0. the first term in ∂πR∗ /∂SR is positive. Therefore, when ∂πM R M R M R R R

Proof of Result 3A ((Channel profits)). Again, adding the profits of channel members, the optimal channel profit is π∗ = ∗ . From the previous result, πR∗ + πM  ∗      ∗ ∂π∗ ∂πR∗ ∂πM 1 1 hR 1/2 1/2 hM SM −β−1 1/2 −(3/2) ∂x = + = −x αβ(β − 1)x + (2SR hR ) x − x − +2 . ∂SR ∂SR ∂SR 4 ∂SR 2 2SR hR SR When



hM SM − hR SR

 > −1,

∗ ∂πM < 0, ∂SR

and

∂π∗ ∂π∗ ∂π∗ = R + M < 0. ∂SR ∂SR ∂SR

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A.3. When the channel is integrated (coordinated) The joint ordering decision is made in such a manner that the following equation holds.   2xST 1/2 , QT = hT where ST = SR + SM , and hT = hR + hM . Substituting the joint order quantity into the channel profit function, we get π = αx1−β − cx2 − (2hT ST x)1/2 The first order condition can be expressed as 1 α(1 − β)x−β − 2cx − (2hT ST )1/2 x−(1/2) = 0. 2 The second order condition is given by: 1 −αβ(1 − β)x−β−1 − 2c + (2hT ST )1/2 x−(3/2) < 0, 4 which is satisfied when 1/2 < β< 1. Proof of Lemma 2( (Purchase quantity)). Substituting the optimal purchase quantity into the first order condition and taking the derivative with respect to ST , we get  ∗    1 1 hT 1/2 −(1/2) −β 1/2 −(3/2) ∂x − x =0 −αβ(1 − β)x − 2c + (2hT ST ) x 4 ∂ST 4 2ST Applying the second order condition, ∂x* /∂ST < 0. Proof of Result 5 ((Channel profits)). Taking the derivative of optimal channel profits with respect to ST and applying the first order condition for channel profits, we get:   hT 1/2 1/2 ∂π∗ =− x < 0. ∂ST 2ST

Appendix B. For a linear demand function B.1. When the retailer is the Stackelberg price leader The order quantity follows the Economic Order Quantity principle, from which:   2xSR 1/2 Q= hR

(B.1)

Following Weng (1995, p. 1512) for inventory holding and ordering costs, the profit function of manufacturer is given by: πM = wx − cx2 −

hM Q S M x − . 2 Q

Substituting for Q from the EOQ result in Eq. (B.1) into this profit function, we get:     SR hR 1/2 SM hM 2 πM = wx − cx − x1/2 . + 2 SR hR

(B.2)

(B.3)

The manufacturer maximizes their profits taking the purchase price as given and the retailer then maximizes its profit by taking into consideration the manufacturer’ reactions with respect to purchase price. From the manufacturer’s first-order condition, the following relationship can be obtained.     1 −(1/2) SR hR 1/2 SM hM w − 2cx − x = 0. (B.4) + 2 2 SR hR

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The second order condition requires     1 SR hR 1/2 SM hM −2c + x−(3/2) < 0, + 4 2 SR hR

273

(B.5)

at the local optimal point indicated by Eq. (B.4). This can be derived from the manufacturer self-selection condition πM ≥ 0. Note that when x → 0, The LHS in the SOC → +∞; and when x → +∞, the LHS in the SOC → −2c. This indicates a convex–concave manufacturer profit function that has either (0, 0) or the local optimal point from Eq. (B.4) as its global solution. The local optimal is the global solution when the optimal profit is greater than zero. Given a reverse demand function p = a − bx, the retailer’s profit is given by: πR = (a − bx)x − wx −

h R Q SR x − . 2 Q

(B.6)

Again, substituting for Q from Eq. (B.1) into the retailer’s profit function, we get: πR = (a − bx)x − wx − (2hR SR x)1/2 . The retailer then determines the purchase price it has to offer based on the manufacturers’ possible responses. From Eq. (B.4),   −1   ∂x hM 1 −(3/2) SR hR 1/2 SM + > 0. = 2c − x 4 2 SR hR ∂w

(B.7)

(B.8)

Maximizing the retailer’s profits (B.7) with respect to wholesale price w subject to (B.4), one can obtain the equilibrium (w∗ , x∗ ) by solving Eqs. (B.4) and (B.9) simultaneously. Specifically, the first order condition when maximizing (B.7) is   1/2  ∂x ∂πR −(1/2) SR hR = 0. = −x + a − 2bx − w − x ∂w ∂w 2 Substituting (B.8) into this equation, and since ∂x/∂w > 0, this condition becomes      1/2 1 −(1/2) SR hR 1/2 SM hM −(1/2) SR hR −2cx + x + a − 2bx − w − x + = 0. 4 2 SR hR 2 Combining the terms, we get      1 −(1/2) SR hR 1/2 SM hM −w − 2cx + a − 2bx − x 4− =0 + 4 2 SR hR

(B.9)

The local second order condition is satisfied when       SM ∂x hM 1 −(3/2) SR hR 1/2 4− + − 1 < 0, −2(b + c) + x 8 2 SR hR ∂w which becomes

     1 −(3/2) SR hR 1/2 SM hM −2b − 4c + x +4 <0 + 8 2 SR hR

from (B.8). When x → 0,

     1 SR hR 1/2 SM hM −2b − 4c + x−(3/2) + 4 → +∞; + 8 2 SR hR

and when x → +∞,

     1 −(3/2) SR hR 1/2 SM hM −2b − 4c + x + 4 → −2b − 4c. + 8 2 SR hR

This indicates a convex–concave manufacturer profit function that has either (0, 0) or the local optimal point from Eqs. (B.4) and (B.9) as its global solution. The local optimal is the global solution when the optimal profit is greater than zero.

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Proof of Lemma 1A( (Purchase quantity)). Replacing x, p, and w with the optimal values as functions of SR , and combining the retailer’s and the manufacturer’s first order conditions,      SR hR 1/2 1 SM hM 1+ = 0. (B.10) 4cx − a + 2bx + x−(1/2) + 2 4 SR hR Taking the partial derivative with respect to SR and rearranging terms, we get:        −1     1 hM 1 −(3/2) SR hR 1/2 1 SM ∂x∗ 1 −(1/2) hR 1/2 SM hM 1+ 4c + 2b − x 1+ =− x − + ∂SR 2 2SR 4 hR SR 2 2 4 SR hR where 

    −1 1 −(3/2) SR hR 1/2 1 SM hM 4c + 2b − x 1+ ( + ) >0 2 2 4 SR hR

from the retailer’s local second order condition. Therefore, ∂x* /∂SR < 0, when ((hM /hR − SM /SR ) > −4). Proof of Result 1A( (Purchase price)). Given the result of Lemma 1A, the manufacturer’s first order condition can be represented in terms of purchase price w as a function of x. Thus,    ∗       ∂w∗ ∂x hR 1/2 hM 1 −(3/2) SR hR 1/2 SM hM 1 SM . = 2c − x + + x−(1/2) − ∂SR 4 2 SR hR ∂SR 4 2SR hR SR Since

    1 −(3/2) SR hR 1/2 hM SM 2c − x >0 − 2 hR SR 4

from the manufacturer second order condition, when     hM ∂x∗ SM hM SM > −4, or < 0, − < 0, and − hR SR ∂SR hR SR

∂w∗ < 0, ∂SR

that is, both purchase quantity and purchase price increase under ER. On the other hand, when     hM hM SM SM < −4, or > 0, − − hR SR hR SR the direction is undetermined. Proof of Result 2A ((Manufacturer and retailer profits)). Given the optimal profits of the retailer and the manufacturer, one can derive implicitly the impact of SR by using the above results. Using the first order conditions of the retailer and the manufacturer, we get:    ∗       ∗ ∂πM ∂x 1 −(1/2) SR hR 1/2 SM hM 1 1/2 hR 1/1 hM SM . = 2cx − x + − x − 4 2 SR hR ∂SR 4 2SR hR SR ∂SR     ∂πR∗ hR 1/2 hM 1 SM = − x1/2 − +4 . ∂SR 4 2SR hR SR Again, note     1 SR hR 1/2 SM hM 2c − x−(3/2) >0 + 4 2 SR hR from the manufacturer second order condition, ∂πR∗ /∂SR < 0, when ((hM /hR − SM /SR ) > −4), or ∂x* /∂SR < 0 (from Lemma 1A). ∗ /∂S < 0 because when ((h /h − S /S ) < 0), the second term in ∂π ∗ /∂S , However, this condition is not sufficient for ∂πM R M R M R R M  1/2   hR hM SM 1 − − x1/2 2SR hR SR 4 ∗ /∂S can be positive. is positive and can be greater than the first term, which is negative, so ∂πM R

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∗ , and the impact of ER on Channel profits Proof of Result 3A ((Channel profits)). The optimal channel profits, π∗ = πR∗ + πM can be obtained by combining the separate impact of ER on each party’s profits. From the previous result,    ∗       ∗ ∂x hR 1/2 hM ∂πR∗ ∂πM 1 −(1/2) SR hR 1/2 SM hM 1 SM ∂π∗ = + = 2cx − x + − x1/2 − +2 . ∂SR ∂SR ∂SR 4 2 SR hR ∂SR 2 2SR hR SR

When ((hM /hR − SM /SR ) > −2), the condition in Lemma 1A is also satisfied so that ∂x* /∂SR < 0. Since     1 −(1/2) SR hR 1/2 SM hM 2cx − x > 0, + 4 2 SR hR ∂π* /∂SR < 0 when ((hM /hR − SM /SR ) >−2). On the other hand, when ((hM /hR − SM /SR ) < −4), ∂x* /∂SR > 0 and ∂π* /∂SR > 0. It is clear that in these two ranges, the direction of ER impact on Channel profits is undecided. B.2. When the manufacturer is the Stackelberg price leader The retailer maximizes its profit function given by: πR = (a − bx)x − wx − (2hR SR x)1/2

(B.11)

For a given w, the retailer optimizes its profits by selecting a purchasing quantity x. The retailer’s first order condition can be expressed by the following equation. 1 a − 2bx − w − (2SR hR )1/2 x−(1/2) = 0, 2

(B.12)

The second order condition requires the following inequality to hold. 1 −2b + (2SR hR )1/2 x−(3/2) < 0. 4

(B.13)

Again, this indicates a convex–concave profit function with either (0, 0) or the local optimal solution from (B.12) as its global solution. The local solution is the global solution when its optimal profit is greater than zero. The manufacturer’s profit function is:     SR hR 1/2 SM hM πM = wx − cx2 − x1/2 . + (B.14) 2 SR hR From the retailer’s first order condition, −1  1 ∂x 1/2 −(3/2) = −2b + (2SR hR ) x < 0. ∂w 4

(B.15)

Again, one can solve for (w∗ , x∗ ) by maximizing the manufacturer’s profit function with respect to w, subject to the retailer’s first order condition. From the manufacturer’s profit function (B.14), the first order condition is       ∂x 1 SR hR 1/2 SM hM ∂πM −(1/2) = 0. x =x+ w − 2cx − + ∂w ∂w 2 2 SR hR Applying (B.15), and rearranging, we get       1 SR hR 1/2 SR hR 1/2 SM 1 hM −2bx + x−(1/2) = 0. + w − 2cx − x−(1/2) + 4 2 2 2 SR hR Combining the terms, and we get      SM 1 −(1/2) SR hR 1/2 hM − 1 = 0. w − 2cx − 2bx − x + 2 2 SR hR Again, the second order condition is given by:       SM ∂x h 1 −(3/2) SR hR 1/2 −1 > 0. + 1 − 2c + 2b − x 4 2 SR hR ∂w

(B.16)

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Or

     SM ∂x 1 −(3/2) SR hR 1/2 hM +1 −4b − 2c + x + > 0, 4 2 SR hR ∂w



and

     SM hM 1 −(3/2) SR hR 1/2 + 1 < 0, + −4b − 2c + x 4 2 SR hR

since ∂x/∂w < 0.From the manufacturer’s profit function,      SR hR 1/2 SM 1 hM + 1 ≥ 0, c + 2b − x−(3/2) + 2 2 SR hR the above second order condition is satisfied. Proof of Lemma 1A ((Purchase quantity)). Combining the first order conditions of the retailer and the manufacturer and inserting the optimal value of purchase quantity, we get:      SR hR 1/2 SM hM 1 + 1 = 0. + a − 4bx − 2cx − x−(1/2) 2 2 SR hR Taking the derivative with respect to SB , we get       ∗    1 −(3/2) SR hR 1/2 SM hM ∂x 1 −(1/2) hR 1/2 hM SM −4b − 2c + x + +1 + x − +1 =0 4 2 SR hR ∂SR 4 2SR hR SR Applying the manufacturer second order condition, we get      1 −(3/2) SR hR 1/2 SM hM −4b − 2c + x + +1 < 0. 4 2 SR hR Therefore, ∂x* /∂SR < 0 when ((hM /hR − SM /SR ) > −1). Proof of Result 2A ((Manufacturer and retailer profits)). From the previous result regarding the impact of ER on purchase quantity, we substitute for the optimal value of purchase quantity in the manufacturer and retailer profit functions, and apply their respective first and second order conditions to get the following.     ∗ 1 hR 1/2 1/2 hM SM ∂πM =− x − +1 ∂SR 2 2SR hR SR  ∗    ∂x 1 1 hR 1/2 1/2 ∂πR∗ = −x −2b + (2SR hR )1/2 x−(3/2) − x . ∂SR 4 ∂SR 2 2SR Since 1 −2b + (2SR hLR)1/2 x−(3/2) < 0, 4 ∗ /∂S < 0 or ((h /h − S /S ) > −1), ∂π ∗ /∂S < 0. the first term in ∂πR∗ /∂SR is positive. Therefore, when ∂πM R M R M R R R Proof of Result 3A ((Channel profits)). ∗ . From the previous result, πR∗ + πM  ∗ ∂π∗ ∂πR∗ ∂πM = + = −x −2b + ∂SR ∂SR ∂SR

Again, adding the profits of channel members, the optimal channel profit is π∗ =  ∗     1 1 hR 1/2 1/2 hM SM 1/2 −(3/2) ∂x (2SR hR ) x − x − +2 4 ∂SR 2 2SR hR SR

∗ /∂S < 0, and When ((hM /hR − SM /SR ) > −1), ∂πM R

∂π∗ ∂π∗ ∂π∗ = R + M <0 ∂SR ∂SR ∂SR

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B.3. When the channel is integrated (coordinated) The joint ordering decision is made in such a manner that the following equation holds.   2xST 1/2 QT = , hT where ST = SR + SM , and hT = hR + hM . Substituting the joint order quantity into the channel profit function, we get π = (a − bx)x − cx2 − (2hT ST x)1/2 The first order condition can be expressed as 1 a − 2bx − 2cx − (2hT ST )1/2 x−(1/2) = 0. 2 The second order condition is given by: 1 −2b − 2c + (2hT ST )1/2 x−(3/2) < 0. 4 Proof of Lemma 2 ((Purchase quantity)). Substituting the optimal purchase quantity into the first order condition and taking the derivative with respect to ST , we get   1 ∂x∗ 1 hT 1/2 −(1/2) (−2b − 2c + (2hT ST )1/2 x−(3/2) ) − x = 0. 4 ∂ST 4 2ST Applying the second order condition, ∂x* /∂ST < 0. Proof of Result 5 ((Channel profits)). Taking the derivative of optimal channel profits with respect to ST and applying the first order condition for channel profits, we get:   ∂π∗ hT 1/2 1/2 =− x < 0. ∂ST 2ST

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