Strategic Role of Retailer Bundling in a Distribution Channel

Journal of Retailing 91 (1, 2015) 50–67

Strategic Role of Retailer Bundling in a Distribution Channel Qingning Cao a,∗,1 , Xianjun Geng b , Jun Zhang c a

b

Owen Graduate School of Management, Vanderbilt University, Nashville, TN 37203, United States Jindal School of Management, The University of Texas at Dallas, Richardson, TX 75080, United States c School of Business, Renmin University of China, Beijing, China

Abstract We study retailer bundling in a distribution channel when the manufacturer for one bundled product can strategically set the wholesale price. We show that the retailer can use a bundling option as a strategic leverage to extract concessions from the manufacturer in form of a lower wholesale price. This finding contributes a novel rationale for retailer bundling to the bundling literature. Whenever the bundling option causes this concession-extraction effect, the retailer always benefits from the lower wholesale price. The manufacturer, nevertheless, does not necessarily suffer because bundling can lead to a higher consumer demand. We also show that the manufacturer’s marginal production cost plays a critical role in driving the retailer’s bundling decision, concession extraction behavior and consequently the total channel profit. © 2014 New York University. Published by Elsevier Inc. All rights reserved. Keywords: Channel; Bundling; Concession extraction

Introduction Retailers often bundle products from powerful manufacturers who set the wholesale prices with retailers’ own products (or products from fringe manufacturers that have little pricing power). This is the case for retailers who bundle national-brand products with their own private-label products. For example, drug stores such as Walgreen, CVS, and Rite Aid routinely sell bundles of national brand medicines and their own private label drugs: Tylenol is sold in bundles with a private-label decongestant; Sudafed is sold in bundles with a private-label pain reliever (Evans and Salinger 2005). Grocery stores such as Safeway routinely provide discounts on bundles that mix both national brands and private labels (Scaff et al. 2011). Outside of the privatelabel industry, we also observe instances of retailer bundling of products from both powerful and fringe manufacturers. For example, while major gaming console manufacturers (e.g., Nintendo, Microsoft and Sony) often dictate their wholesale prices given their market dominance, manufacturers of games or ∗

Corresponding author at: Owen Graduate School of Management, Vanderbilt University, 401 21st Avenue S, Nashville, TN 37203, United States. Tel.: +1 615 322 1501; fax: +1 615 343 7177. E-mail addresses: [email protected] (Q. Cao), [email protected] (X. Geng), [email protected] (J. Zhang). 1 Post-Doctoral Research Scholar.

gaming accessories often face a commoditized wholesale market and thus cannot raise their wholesale prices even when electronic retailers such as BestBuy and Toys R Us are reaping high margins from games or accessories bundled with gaming consoles (Hills 2007; Sengupta 2013). This paper analyzes a channel consisting of a retailer and a powerful (i.e., wholesale-price-setting) manufacturer when the retailer has a bundling option: the retailer can either sell the manufacturer’s product alone, or bundle it with the retailer’s private-label product (or product from a non-strategic fringe manufacturer at a fixed wholesale price). We focus on the strategic role of retailer bundling and ask the following research questions. First, how does the bundling option affect the interactions between the retailer and the powerful manufacturer, and consequently the wholesale price? Second, how does the bundling option affect retailer profit and manufacturer profit? Third, what role, if any, does the powerful manufacturer’s marginal production cost (“manufacturer cost” hereafter for abbreviation) play in retailer bundling? Our first key finding is that the downstream retailer can use the bundling option as a strategic leverage to extract concessions from the upstream manufacturer in form of a lower wholesale price.2 Such concession extraction can happen when the retailer

2

http://dx.doi.org/10.1016/j.jretai.2014.10.005 0022-4359/© 2014 New York University. Published by Elsevier Inc. All rights reserved.

Hereafter we abbreviate “powerful manufacturer” to “manufacturer.”

Q. Cao et al. / Journal of Retailing 91 (1, 2015) 50–67

bundles (i.e., executes the bundling option) in equilibrium: in this case, it is the retailer’s off-equilibrium threat of unbundling that puts downward pressure on the wholesale price that the manufacturer sets. Concession extraction can also happen when the retailer unbundles (i.e., forfeits the bundling option): the retailer’s off-equilibrium threat of bundling now leads to the concessions. This finding also highlights that it is the option to bundle, rather than the action of bundling in equilibrium, that drives concession extraction. Extant literature highlights two reasons why a firm bundles: bundling results in better price discrimination against consumers (Adams and Yellen 1976; Banciu, Gal-Or, and Mirchandani 2010; Basu and Vitharana 2009; Fang and Norman 2005; Mcafee, Mcmillan, and Whinston 1989; Prasad, Venkatesh, and Mahajan 2010; Schmalensee 1984), and bundling serves as an effective competition tool (Balachander, Ghosh, and Stock 2010; Chen 1997; Ghosh and Balachander 2007; Nalebuff 2004; Whinston 1990). Our first finding complements the above two streams of research by offering a new rationale for product bundling.3 Our second key finding concerns the impact of concession extraction on firm profits. Whenever the retailer extracts concessions from the manufacturer, we show that the retailer always benefits from the bundling option. Interestingly, concession extraction by the retailer does not necessarily hurt the manufacturer. Specifically, when the retailer bundles and extracts concessions from the manufacturer, while the manufacturer faces a reduced margin due to concession extraction, his sales increases because retailer bundling expands the consumer demand.4 This market expansion, and consequently improved total channel profit, is shared by the retailer and the manufacturer. In this case, the bundling option induces both concession extraction and market expansion, thus results in win-win for both channel members. Our third key finding is that the manufacturer’s marginal production cost plays a critical role in driving the retailer’s bundling decision, concession extraction behavior and consequently the total channel profit. When this cost is moderately high, the retailer always extracts concessions from the manufacturer when she bundles. This leads to a lower wholesale price that alleviates the double marginalization problem in the distribution channel, thus results in better coordination between the retailer and manufacturer and improved channel profit. When this cost is low, however, a bundling retailer cannot extract concessions from the manufacturer in most cases because the wholesale price under the no-bundling-option benchmark is already low. In fact, the bundling option actually leads to a higher wholesale price in most cases under a low marginal production cost. This worsens

3 This finding also contributes to the literature on private labels. It is known that retailers can use a private label to obtain a lower wholesale price for a competing national brand due to the demand substitution effect (Mills 1995; Narasimhan and Wilcox 1998). Our work shows that a retailer can use a private label to obtain a wholesale price concession from the manufacturer of a non-competing national brand due to the bundling effect. 4 We use “she” to refer to the retailer and “he” the manufacturer throughout this paper.

51

the double marginalization problem in the distribution channel and reduces total channel profit. This finding suggests that, when studying bundling in a channel context, ignoring marginal product costs can lead to incomplete conclusions. Literature Review Our paper is related to the rich literature on product bundling and tying in marketing and economics. In the context of monopolistic bundling, a number of papers study the optimal bundling strategy in different contexts: when the number of products in a bundle is two (Adams and Yellen 1976; Schmalensee 1984), very large (Armstrong 1999) or finite (Fang and Norman 2005), when component products are complements or substitutes (Venkatesh and Kamakura 2003), when consumers differ in their abilities of assessing the value of a component product (Basu and Vitharana 2009), when the seller auctions off his products (Subramaniam and Venkatesh 2009), when component products are vertically differentiated and production capacity is limited (Banciu, GalOr, and Mirchandani 2010), when the distribution of consumer valuation is heavy-tailed (Ibragimov and Walden 2010), and when products have network externality (Prasad, Venkatesh, and Mahajan 2010). Besides monopolistic bundling, the bundling literature also studies oligopolistic bundling and tying where there are two or more retailers. This stream of literature shows how a firm can use bundling as a competition tool from different perspectives: bundling can leverage a firm’s monopolistic power in one market into another oligopolistic market (Whinston 1990), soften competition (Anderson and Leruth 1993; Balachander, Ghosh, and Stock 2010; Chen 1997; Ghosh and Balachander 2007), or deter entrance (Nalebuff 2004; Wilson, Weiss, and John 1990). See Stremersch and Tellis (2002) and Venkatesh and Mahajan (2009) for comprehensive reviews of the bundling literature. While the above two streams of literature highlight retailer–consumer and retailer–retailer interactions, we focus on retailer–supplier (manufacturer) interactions in this paper. We offer a new reason why a firm bundles: the downstream retailer can use the bundling option as a strategic leverage to extract concessions from the upstream manufacturer in form of a lower wholesale price. Our paper is also related to the broad literature on distribution channel management and coordination. One stream of literature examines various marketing mechanisms for coordinating a decentralized channel (Cui, Raju, and Zhang 2007; Gerstner and Hess 1995; Iyer 1998; Iyer and Villas-Boas 2003; Jeuland and Shugan 1983; Lal 1990; Moorthy 1987; Raju and Zhang 2005). Our paper adds to this literature by showing when and how downstream bundling can improve channel coordination. Another stream of channel literature studies how channel structure affects firms’ marketing decisions and profitability (Bhaskaran and Gilbert 2009; Cai, Dai, and Zhou 2012; Choi 1991; Coughlan 1985; Coughlan and Wernerfelt 1989; Desai, Koenigsberg, and Purohit 2004; Liu and Cui 2010; Liu and Tyagi 2011; Mcguire and Staelin 1983; Shulman, Coughlan, and Savaskan 2010). Our work complements this literature by showing how a decentralized channel structure can distort a downstream firm’s bundling decision. Some recent papers study

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the impact of channel members’ marketing instruments on channel coordination and member profits (Cui, Raju, and Zhang 2008; Desai, Koenigsberg, and Purohit 2010; Dukes and Liu 2010; Guo 2009; Liu and Zhang 2006; Tsay 2001). In contrast, our work highlights how retailer bundling affects channel dynamics and profit distribution. To the best of our knowledge, only two papers examine bundling strategies in a distribution channel. Girju, Prasad, and Ratchford (2012) focuses on an upstream manufacturer’s bundling decision while our paper analyzes a downstream retailer’s bundling decision. Bhargava (2012) and our paper both examine downstream bundling in a channel context. However, our paper differs significantly from Bhargava (2012) in terms of model settings and results. In his model, manufacturer marginal production costs are fixed at zero while we allow for positive costs. These positive costs play an important role in driving our results on profit distribution and channel coordination. Bhargava shows that channel decentralization causes manufacturers to overprice component goods, thus weakening the retailer’s incentive to bundle. In contrast, we show that in the presence of strategic channel interactions, the retailer can use bundling to strengthen her bargaining power and extract concessions from a manufacturer. This can happen even when there is no demand-side factors that favor bundling (e.g., negative valuation correlation across component products). Finally, our work is related to the private label literature. The extant private label literature has identified three main benefits of private labels to a retailer. First, the wholesale prices of competing national brands become lower with an introduction of national brand (Mills 1995; Narasimhan and Wilcox 1998). Second, the gross margin of a private label is higher than that of a competing national brand (Pauwels and Srinivasan 2004). Finally, retailers can use private labels to differentiate from each other (Corstjens and Lal 2000). Our work complements the extant literature by showing that retailers can bundle privatelabel products with branded products to induce lower wholesale prices from manufacturers. Model Consider a distribution channel that consists of an upstream manufacturer selling a product (product 1) to a downstream retailer, which then sells the product to consumers. The manufacturer incurs an exogenous and constant marginal production cost c1 and charges the retailer a wholesale price w1 for product 1. The retailer may choose between two selling strategies for product 1. The retailer can sell product 1 at retail price p1 to consumers, which we refer to as the unbundling strategy.

Manufacturer decides wholesale price w1

Stage 1

Alternatively, the retailer can sell product 1 in two forms simultaneously: standalone product at price p1 and bundled with another product (product 2) at bundle price pB . We refer to this alternative as the mixed-bundling strategy, or the bundling strategy in short. The retailer procures product 2 at a constant procurement cost w2 . The assumption of a constant procurement cost of product 2 for the retailer is reasonable when product 2 is the retailer’s private-label product (Mills 1995; Raju, Sethuraman, and Dhar 1995; Sayman and Raju 2004). This assumption is also reasonable when the manufacturer of product 2 is a price-taker rather than a price-setter, such as when the wholesale market for product 2 is highly competitive and thus no single manufacturer can influence market price in a substantial manner. This assumption also enables us to provide a focused discussion of the pair-wise strategic interactions between the retailer and the manufacturer of product 1 without getting into technical complications of analyzing three-way interactions. For ease of exposition, hereafter by “the manufacturer” we always refer to the strategic manufacturer of product 1. On the demand side, the consumer mass is normalized to 1. For each product i = 1, 2, each consumer has unit demand. For product i, a fraction α of the consumers values the product at l, and the rest values this product at h. We refer to the former (latter) low-valuation (high-valuation) consumers for product i. 0 ≤ α ≤ 1. For simplicity and without loss of generality, we assume that l = 1 and h > 1. The practice of dividing consumers into groups so that consumers are homogeneous within a group and heterogeneous between groups is well adopted in the marketing literature (Chen, Moorthy, and Zhang 2005; Desai et al. 2001; Essegaier, Gupta, and Zhang 2002; Kumar, Rajiv, and Jeuland 2001; Moorthy and Png 1992; Shin and Sudhir 2010). We assume a consumer’s valuation for a bundle equals the sum of her separate valuations for each product. To avoid trivialities, we assume that 0 < c1 < h and 0 < w2 < h. We further assume that consumer valuations of the two products are not correlated. We use a Stackelberg game to model the strategic interactions between the manufacturer and the retailer. In the game, the manufacturer is the Stackelberg leader and moves first to determine the wholesale price of product 1. Then, the retailer makes her bundling and pricing decisions. If the retailer chooses the unbundling strategy, she needs to decide the prices of both products, p1 and p2 , respectively. If the retailer chooses the bundling strategy, she needs to decide all of individual product prices p1 and p2 , and bundle price pB . After the retail prices are determined, demand materializes. A consumer buys a product (either an individual product or a bundle), which yields the maximal utility for her. Fig. 1 summarizes the timing of our model.

Retailer chooses whether to bundle and sets prices p1 , p2 and pB if bundling or p1 and p2 if unbundling.

Stage 2 Fig. 1. Timing of the model.

Demand materializes

Stage 3

Q. Cao et al. / Journal of Retailing 91 (1, 2015) 50–67

Benchmark: Retailer Has No Bundling Option

c1

Before we carry out the main analysis where the retailer can choose between bundling and unbundling, we first characterize a benchmark scenario in which the retailer does not have a bundling option – this benchmark facilitates our later study on how the bundling option affects prices and profits in this channel. For the retailer and without a bundling option, she simply sets p1 and p2 separately. We use superscript “o” to denote the results under no bundling option. For product 1 and given w1 , the table below lists the retailer’s optimal price and profit, and the manufacturer’s profit5 :

If w1 ≤ (1 − (1 − α)h)/α: If w1 > (1 − (1 − α)h)/α:

po1

o πr1

o πm

1 h

1 − w1 (h − w1 )(1 − α)

w1 − c1 (w1 − c1 )(1 − α)

The following lemma characterizes the equilibrium outcome of the Stackelberg game. All proofs are in the Appendix. Lemma 1. When the retailer does not have a bundling option, the equilibrium wholesale price, retail price, retailer profit from product 1, and manufacturer profit, respectively, are:

If c1 ≤ If c1 >

1−(1−α2 )h α2 1−(1−α2 )h α2

po1

o πr1

:

1−(1−α)h α

1

1−

:

h

h

0

o πm 1−(1−α)h α

h H

H

L

LH

1 − (1 − α 2 )h

α2

1 − (1 − α )h

α

h

w2

Fig. 2. Equilibrium retail prices under no bundling option.

The retailer’s profit from product 2 can be derived simio = 1 − w if w ≤ (1 − (1 − α)h)/α, and π o = (h − larly: πr2 2 2 r2 w2 )(1 − α) otherwise. Taking the retailer’s response into consideration, the manufacturer chooses a wholesale price to maximize his profit:  if w1 ≤ (1 − (1 − α)h)/α; w1 − c1 , max πm = (w1 − c1 )(1 − α), if w1 > (1 − (1 − α)h)/α. (1)

wo1

53

1−(1−α)h α

− c1

(h − c1 )(1 − α)

Lemma 1 shows that, when the manufacturer has a cost lower than (1 − (1 − α2 )h)/α2 , he sets a low wholesale price to induce the retailer to serve the whole consumer market. In this situation, a high retail demand benefits the manufacturer more than a high wholesale profit margin. When the manufacturer has a high cost, he instead focuses on gaining a high margin by using a high wholesale price to induce the retailer to serve the high-valuation consumers only.6 Fig. 2 illustrates the equilibrium outcome under no bundling option in the c1 -w2 plane. For example, the top-left region HL represents the case where the retailer prices at high price h for product 1 and at low price 1 for product 2.

5 When a consumer is indifferent between purchasing or not, without loss of generality we assume the former. Similarly, when the retailer is indifferent between purchasing from a manufacturer or not, we assume the former. 6 Note that when h is high and α is low, in the no-bundling equilibrium the firm sets the wholesale price at h even if the marginal costs are very low. The key insights of this paper, however, hold qualitatively.

In the following sections, we first characterize the equilibrium outcome when the retailer has a bundling option, we then compare it with the aforementioned benchmark in order to gauge the impact of the retailer’s bundling option on channel members. Retailer Has a Bundling Option In this section, we consider the scenario in which the retailer has the option to bundle. The first subsection presents and solves the retailer’s problem for any given wholesale price w1 . The second subsection solves the manufacturer’s problem and further characterizes equilibrium outcomes. Retailer’s Problem With a bundling option and for any given wholesale price w1 , the retailer has to make two decisions: whether to offer a product bundle; and how to set prices for individual products, p1 and p2 , and for the bundle (if it is offered), pB , respectively. Note that having a bundling option does not necessarily mean the retailer will execute this option: the retailer will execute the bundling option only when she benefits from doing so. In this paper we allow mixed bundling, under which the retailer not only can offer a product bundle at price pB , but also can offer the two products individually at prices p1 and p2 . Mixed bundling in general is a convoluted problem because of its combinatorial nature. Lemma 2 rules out bundling schemes that can never improve retailer profit (as compared to unbundling) and thus facilitates our further analysis. Lemma 2. The only mixed-bundling scheme that can possibly result in a higher profit for the retailer, as compared to unbundling, is pB = 1 + h, p1 = h and p2 = h. Under this scheme, consumers with low valuations for both products do not purchase; all other consumers purchase the bundle. Under the bundling scheme in Lemma 2, no consumer purchases individual products. Therefore, this mixed-bundling scheme is equivalent to a pure-bundling scheme where the retailer sells the bundle at price 1 + h, and does not offer individual products. Based on Lemma 2, hereafter when considering the retailer’s bundling decision, it is sufficient for us to only consider this pure-bundling strategy with bundle price 1 + h. Another

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Q. Cao et al. / Journal of Retailing 91 (1, 2015) 50–67

Fig. 3. Retailer strategy under bundling option.

important implication of Lemma 2 is that, under bundling, the retailer is able to cover all consumers except for the ones who have low valuations for both products. Note that such market coverage is not feasible under unbundling. Given wholesale price w1 for product 1 and procurement cost w2 for product 2, the next lemma characterizes the sufficient and necessary condition for retailer bundling. For notational convenience, define the following boundary expressions: lb1 (w2 ) = −w2 + 1 + h + (1 − h)/α2 , lb2 (w2 ) = (1 − α)/αw2 + 1 + h − h/α,

to consumers who have low valuations on both products. Second and for product 2, bundling squeezes the retailer’s profit margin (as compared to selling this product separately at high price h). This result that highly lopsided marginal costs of products favor unbundling is consistent with the findings in Schmalensee (1984). The fact that our finding in Lemma 3 is consistent with prior work on bundling is not surprising. Given wholesale prices, the

ub1 (w2 ) = (α/(1 − α))w2 + 1 + h − 1/(1 − α), and

ub2 (w2 ) = −w2 + 1 + h + (1 − h)/α.

(2)

Lemma 3. The retailer will choose to bundle if and only if max{lb1 (w2 ), lb2 (w2 )} < w1 ≤ min{ub1 (w2 ), ub2 (w2 )}.

(3)

Fig. 3 illustrates the retailer’s optimal strategy under various wholesale prices. When inequality (3) holds, the retailer bundles – this corresponds to region B in this figure. Under bundling, the bundle price and the retail demand are 1 + h and 1 − α2 , respectively. In addition, Fig. 3 also illustrates the retailer’s four unbundling regions. The name of each region indicates her optimal unbundling prices in this region. For instance, region HL represents the case where the retailer unbundles, prices high (at h) for product 1 and low (at 1) for product 2. Region B in Fig. 3 reveals a key insight regarding retailer bundling: high or heavily-lopsided wholesale prices are likely to discourage the retailer from bundling. First, when both wholesale prices are high (HH region in Fig. 3), bundling is not optimal for the retailer. Intuitively, when w2 is high enough, choosing a high margin on product 2 (via unbundling and p2 = h) benefits the retailer more than acquiring more market demand on product 2 by bundling. This result that a high marginal cost favors unbundling is consistent with prior work in the bundling literature (Adams and Yellen 1976; Armstrong 1999; Schmalensee 1984). Second, when the wholesale prices are heavily lopsided (as in region HL or LH in Fig. 3), bundling is not optimal for the retailer either. To see the intuition, consider region LH where the retailer’s procurement cost is low for product 1 and high for product 2. By bundling, the retailer may suffer in two ways in this scenario. First and for product 1, the retailer is losing sales

retailer is a monopolistic bundler over consumers – a setup similar to the ones in previous work such as Adams and Yellen (1976) and Schmalensee (1984). As we will show in the next sections, once we consider an endogenously determined wholesale price from the manufacturer of product 1, the strategic interactions between channel partners will lead to significantly new results.

Manufacturer’s Problem In this subsection we analyze how the manufacturer, as the Stackelberg leader, can influence the retailer’s bundling choice and pricing through a wholesale price w1 . First note that, when the wholesale price of product 2 is very low or very high, i.e., w2 < 1 − (1 − α)(h − 1)/α2 or w2 > 1, there does not exist any feasible solution w1 to inequality (3), and thus the retailer will never bundle. Therefore, the manufacturer’s problem boils down to the one when the retailer does not have the bundling option (as in the benchmark scenario). For analytical convenience and without loss of generality, hereafter we impose the following assumption on the wholesale price of product 2: Assumption. w2 ∈ [1 − (1 − α)(h − 1)/α2 , 1]. Note that for comparison purpose we still plot the case of w2 < 1 − (1 − α)(h − 1)/α2 or w2 > 1 in later figures. Given the above assumption, for j = 1, 2, 1 − (1 − α)

Q. Cao et al. / Journal of Retailing 91 (1, 2015) 50–67

55

Fig. 4. Equilibrium under bundling option.

Two other interesting regions are S1 and S2. From (h − 1)/α2 ≤ lbj ≤ (1 − (1 − α)h)/α and (1 − (1 − α)h)/α ≤ Figs. 2 and 4 we know that, in region S1, the retailer never ubj ≤ 1. Therefore, the manufacturer’s problem is: ⎧ w1 − c1 , if 0 ≤ w1 ≤ max{lb1 (w2 ), lb2 (w2 )}; ⎪ ⎪ ⎨ max πm = (w1 − c1 )(1 − α2 ), if max{lb1 (w2 ), lb2 (w2 )} < w1 ≤ min{ub1 (w2 ), ub2 (w2 )}; (4) ⎪ ⎪ ⎩ (w1 − c1 )(1 − α), if min{ub1 (w2 ), ub2 (w2 )} < w1 ≤ h. Proposition 1 characterizes the equilibrium outcome for every pair of c1 and w2 . For ease of exposition, we use the c1 -w2 plane in Fig. 4 to facilitate our discussion of various cases of equilibria. To the right of the c1 -w2 plane are our definitions of four regions, where for notational convenience let j = 1 if w2 ≤ (1 − (1 − α)h)/α and j = 2 otherwise. Consistent with Fig. 2, we label the regions where the bundling option does not matter HL, HH, LL and LH, respectively – results in these regions are the same as the ones in the benchmark scenario. Proposition 1. When the retailer has a bundling option, the equilibrium wholesale price of product 1, retail price, retailer profit from both products, and manufacturer profit, respectively, are:

bundles whether a bundling option exists or not. That said, the bundling option – even when not exercised – can serve as a credible deterrence from retailer side that affects profit margins and market demand in equilibrium. Compared to the equilibrium outcome in region LL in Fig. 2 under no bundling option, in region S1 in Fig. 4 the manufacturer charges a much higher wholesale price. Intuitively, without the bundling option, the manufacturer will expect a wholesale demand of 1 at low price (1 − (1 − α)h)/α; with the bundling option, however, wholesale demand will be reduced to 1 − α2 at this price because of retailer bundling. In region S1, this wholesale demand reduction at a low wholesale price is so severe that the manufacturer finds it better to switch to a high wholesale price strategy.

Regions

wb1

pb1 or pB

πrb

b πm

L’L and L’H B S1, S2

lbj ubj h

1 (unbundling) 1 + h (bundling) h (unbundling)

1 − lbj + (2 − j)(1 − w2 ) + (j − 1)(1 − α)(h − w2 ) (1 − α2 )(1 + h − ubj − w2 ) (2 − j)(1 − w2 ) + (j − 1)(1 − α)(h − w2 )

lbj − c1 (1 − α2 )(ubj − c1 ) (1 − α)(h − c1 )

For the rest regions (HL, HH, LH, and LL), equilibrium outcomes are identical with and without the bundling option. For regions B, L’L, L’H, S1 and S2, Proposition 1 shows that the manufacturer strategically responds to the existence of the retailer bundling option. In region B, it is optimal for the manufacturer to charge a wholesale price of ubj – the highest wholesale price that can induce the retailer to bundle, for j = 1, 2 (recall the retailer bundling region in Fig. 3). The retailer responds by choosing bundling, setting a bundle price 1 + h and selling to all consumers except the ones that have low valuations on both products. Because the manufacturer’s marginal cost c1 is low in regions L’L and L’H, the manufacturer finds it optimal to charge a low enough wholesale price so that the retailer will subsequently unbundle and cover the full market for product 1.

With equilibrium results both when the retailer has a bundling option (Proposition 1) and when she has no bundling option (Lemma 1), we next compare these two set of results and study how the downstream bundling option affects prices and profits. Strategic Role of Retailer Bundling To understand the strategic role of retailer bundling, we need to compare wholesale prices and retailer profits in Lemma 1 and Proposition 1 for all possible pairs of c1 and w2 . This comparison, nevertheless, is quite technical because of the number of possible cases to consider. To facilitate our discussion, we present Fig. 5 which offers a graphical and intuitive representation of all possible cases.

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Q. Cao et al. / Journal of Retailing 91 (1, 2015) 50–67

Downstream Bundling and Strategic Concession Extraction

c1

We first discuss the impact of the downstream bundling option on the upstream manufacturer’s optimal wholesale price. To keep our narratives succinct, hereafter and unless noted otherwise, when we discuss how the bundling option affects firm prices and profits, we always use the case with no bundling option as the benchmark implicitly. For example, in the proposition below, by “a lower wholesale price” we mean “a lower wholesale price as compared to that under no bundling option.”

h HL

HH

B1

1 − (1 − α 2 )h

α2

S1

S2

B2

LH

LL L’L

1−

(1−α)(h −1)

α2

B3

B4

1 − (1 − α )h

α

L’H

1

h

w2

Fig. 5. Regions in c1 –w2 plane for profit comparison.

Fig. 5 is based on a comparison of retailer and manufacturer profits under the bundling option (Fig. 4 and Proposition 1) with the ones under no bundling option (Fig. 2 and Lemma 1). Each labeled region in Fig. 5 represents the set of (c1 , w2 ) values under which we can clearly rank-order retailer (and manufacturer) profits with and without the bundling option – the details of which are discussed in the next several propositions. Note that the division of regions in Fig. 5 largely resembles that in Fig. 4 – see earlier discussion in Manufacturer’s Problem subsection for boundary conditions of each region. The only exception is that we now further divide the bundling region into four smaller ones, B1–B4, as defined below:

Proposition 2. In and only in regions L’L, L’H and B1, the bundling option enables the retailer to extract concessions from the manufacturer in the form of a lower wholesale price.7 One subtle but important implication of Proposition 2 is that not observing bundling in equilibrium does not imply that the bundling option has no impact on the equilibrium. In regions L’L and L’H, the retailer does not bundle in equilibrium, and the retail price and total channel profit remain the same with or without the bundling option. Nevertheless, under the bundling option the wholesale price (lb1 or lb2 ) is lower than that ((1 − (1 − α)h)/α) under no bundling option (see Lemma 1 and Proposition 1). The reason is that bundling serves as an effective off-equilibrium threat that constrains the manufacturer’s wholesale price. Consider region L’L for example. Without the bundling option, the manufacturer can charge a wholesale price as high as (1 − (1 − α)h)/α and still have full market coverage (Lemma 1). When there is a bundling option, however, any wholesale price higher than lb1 will trigger the retailer to adopt bundling and consequently reduce total demand for product 1. The manufacturer thus cannot choose any wholesale price

  1 − (1 − α2 )h (1 + α)ubj − h  B1 = (c1 , w2 )  < c1 ≤ , where j = 1, 2; α α2    1 − (1 − α)h 1 − α2 1 − (1 − α2 )h B2 = (c1 , w2 )  − ub < c ≤ , where j = 1, 2; j 1 α3 α2 α2     1 − (1 − α)h lb1 − (1 − α2 )ub1 1 − α2 1 − (1 − α2 )h (1 + α)ub1 − h 1 − (1 − α)h B3 = (c1 , w2 ) w2 ≤ c > and c ≤ Min , − ub ; and , 1 1 1  α α α2 α2 α3 α2     1 − (1 − α)h lb2 − (1 − α2 )ub2 1 − α2 1 − (1 − α2 )h (1 + α)ub2 − h 1 − (1 − α)h B4 = (c1 , w2 ) w2 > c > and c ≤ Min , − ub . and , 1 1 2  α α α2 α2 α3 α2 

In regions B1–B4 the retailer executes the bundling option in equilibrium, and in all other regions she chooses not to. We next present three sets of findings regarding the strategic role of retailer bundling in a distribution channel. The first set of findings highlight a strategic behavior of the retailer in using downstream bundling or unbundling (as a threat) to extract concessions from the upstream manufacturer in form of a lower wholesale price. The second set of findings show that the concession extraction behavior by the retailer always benefits the retailer profit, yet does not necessarily damage the manufacturer profit. The third set of findings emphasize the critical role that the manufacturer’s marginal production cost plays in driving the retailer’s bundling decision, concession extraction behavior and the consequent total channel profit.

higher than lb1 if he wants to prevent the retailer from choosing bundling. Proposition 2 thus spotlights one new reason – that is unknown in the extant literature on bundling – why a firm bundles: the downstream retailer can use the bundling option to extract concessions (i.e., inducing a lower wholesale price) from the upstream manufacturer. In contrast, prior research on bundling resorts to two other mechanisms to explain the benefit of bundling: price discrimination and bundling as a

7 For completeness of analysis, we also consider all other regions in the proof of this proposition in the Appendix.

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competition tool.8 The literature on bundling for price discrimination highlights the interactions between retailers and consumers, and shows that a firm may better extract consumer surplus through bundling (Adams and Yellen 1976; Banciu, Gal-Or, and Mirchandani 2010; Fang and Norman 2005; Basu and Vitharana 2009; Ibragimov and Walden 2010; Mcafee, Mcmillan, and Whinston 1989; Prasad, Venkatesh, and Mahajan 2010; Schmalensee 1984). The literature on bundling as a competition tool highlights the interactions among peer retailers, and shows that bundling may soften competition (Balachander, Ghosh, and Stock 2010; Chen 1997; Ghosh and Balachander 2007), deter entrance (Nalebuff 2004; Wilson, Weiss, and John 1990), or leverage a firm’s monopolistic power in one market into another oligopolistic market (Whinston 1990). In contrast, our paper highlights the interactions between a downstream retailer and its upstream manufacturer, and offers a new, wholesaleprice-based explanation of retailer bundling. The retailer not only can use bundling as an effective offequilibrium threat (under unbundling equilibrium) to extract concessions, in an analogous manner she can use unbundling as an effective off-equilibrium threat (under bundling equilibrium) to extract concessions as well. The latter is the case in region B1. In this region, the retailer chooses to bundle in equilibrium (rather than unbundling as discussed previously in regions L’L and L’H). Comparing having a bundling option to not having a bundling option in region B1, the wholesale price is lower (ub1 or ub2 ) in the former than that (h) in the latter. This concession over the wholesale price is due to the fact that, if the manufacturer deviates from the equilibrium by charging higher than ub1 or ub2 , the retailer will then switch to unbundling. In other words, the retailer’s threat to unbundle effectively sets an upper-bound on the wholesale price.9 The phenomenon of concession extraction applies only in regions L’L, L’H and B1. In other regions, the bundling option either has no impact on the wholesale price (regions HL, HH, LL and LH), or actually results in a higher wholesale price (regions S1, S2, B2, B3 and B4). In particular, in regions S1 and S2 the manufacturer charges a higher wholesale price in order to induce the retailer to serve only the high-valuation consumers. In regions B2, B3 and B4 the manufacturer charges a higher wholesale price both to extract a higher share of the bundling profit from the retailer and to make the unbundling alternative unattractive to the retailer.

8 Another obvious reason for bundling is that bundling leads to significant savings on assembling costs. For example, Nalebuff (2004, p. 161) comments that: “A car is a bundle of seats, engine, steering wheel, gas pedal, cup holders, and much more. An obvious explanation for many bundles is that the company can integrate the products better than its customers can.” This assembling-costsaving argument received little attention in theoretical expositions perhaps due to its straightforwardness. 9 In region B1 two mechanisms apply simultaneously: on manufacturer– retailer interaction side, the bundling option constrains how high the wholesale price can go; on retailer–consumer interaction side, bundling enables the retailer to price discriminate consumers better than unbundling. The latter price discrimination effect, nevertheless, does not apply in regions L’L and L’H where neither retail price or retail demand is affected by the bundling option.

57

Firm Profits We now discuss how the aforementioned concession extraction phenomenon (enabled by the retailer’s bundling option) affects the profits of the retailer and the manufacturer. For this purpose, we limit our discussion to regions L’L, L’H and B1 where concession extraction happens.10 Corollary 1. Whenever the retailer is able to use the bundling option to extract concessions from the manufacturer, the bundling option always increases retailer profit. In regions L’L and L’H, the bundling option does not affect the retailer’s bundling decision and the retail price, thus the total profit in the channel (i.e., retailer profit plus manufacturer profit) does not change. The only effect the bundling option has is the concession extraction effect, which increases the margin the retailer gets from product 1. Consequently, the bundling option improves retailer profit. In region B1 the concession extraction effect also exists, thus the retailer saves on the wholesale price. Furthermore (and different from regions L’L and L’H), in this region the retailer bundles and thus receives a higher revenue due to demand expansion (than possible under no bundling option). Therefore, both the cost reduction on the wholesale side and the revenue boost on the retail side contribute to the increase in retailer profit. While concession extraction always benefits the retailer, it does not necessarily leads to a lower manufacturer profit as shown in the next proposition. Proposition 3. In region B1, the bundling option increases manufacturer profit. In regions L’L, L’H, the bundling option decreases manufacturer profit. The above result regarding manufacturer profit in region B1 is somewhat surprising: from Proposition 2 we know that the manufacturer is losing on the wholesale margin in this region due to the concession it offers the retailer, yet Proposition 3 says that the manufacturer still receives a higher profit (than that under no bundling option). To understand the intuition, compare region B1 in Fig. 5 to region HL or HH in Fig. 2: while under no bundling option product 1 is only sold to consumers who have high valuation over this product, under bundling product 1 is sold (through the bundle) to all consumers except for the ones who have low valuations on both products. In order words, the manufacturer captures a demand of 1 − α under no bundling option, and captures a larger demand of 1 − α2 under bundling. Proposition 3 shows that, in region B1 and for the manufacturer, the gain from demand expansion dominates the loss from a lower wholesale price. This proposition thus provides a new rationale for a manufacturer to sell his product through a bundling retailer (than through a retailer with no bundling option): when the manufacturer cost is moderately high, selling through a bundling retailer enables a manufacturer to benefit from the increased demand caused by retailer bundling.

10 For completeness of analysis, we present profit results for all regions in the Appendix.

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Impact of the Manufacturer’s Marginal Production Cost on Channel Coordination In this subsection we highlight the critical impact of the manufacturer’s marginal production cost (manufacturer cost for abbreviation) on retailer bundling and channel profit.11 Recall that Fig. 5 already illustrates that both the manufacturer cost and the procurement cost of product 2 affect firm profits (as the bounds of the regions in this figure are determined by these two costs). Below we present a focused discussion on the relationship between the manufacturer cost c1 and the channel profit, and offer clear intuitions on the role of cost in our model without getting into the technical details of the many boundaries (among all regions) in Fig. 5. For notational convenience, define cˆ 1 ≡ (1 − (1 − α2 )h)/α2 . Proposition 4. The bundling option increases channel profit only if the manufacturer cost c1 is moderately higher than cˆ 1 . The bundling option decreases channel profit only if the manufacturer cost c1 is lower than cˆ 1 . Proposition 4 shows that the impact of the bundling option on the channel profit depends critically on the manufacturer cost. When c1 > cˆ 1 , the bundling option increases channel profit in region B1 (as compared to that under no bundling option).12 To see the intuition, first notice that in this region the retailer uses the bundling option to extract concessions from the manufacturer (recall Proposition 2). The moderately high manufacturer cost plays a critical role in driving this concession extraction phenomenon: with no bundling option, under a high marginal production cost the manufacturer prefers a high margin to a high demand, and therefore will charge the highest possible wholesale price h to the retailer (Lemma 1); with the bundling option, and if the marginal production cost is not extremely high (i.e., it is moderately high), retailer bundling expands the demand and thus offers the optimal balance between margin and demand for the manufacturer. However, retailer bundling will not happen when the manufacturer charges the extremely high wholesale price of h, thus the manufacturer is willing to offer a concession (by cutting the wholesale price) to induce retailer bundling. The resulting lower wholesale price (ub1 or ub2 as compared to h) then alleviates the double marginalization problem in the distribution channel, thus results in better coordination between the retailer and manufacturer and improved channel profit. When c1 < cˆ 1 , the bundling option decreases channel profit in regions S1, S2, B2, B3 and B4. From Lemma 1 we know that the wholesale price under no bundling option is already low ((1 − (1 − α)h)/α) when the manufacturer cost is low (as the manufacturer prefers demand to margin). Therefore, the bundling option does not incentivize the manufacturer to offer concessions to the retailer (by cutting his already-low

11 By “channel profit” we mean the retailer profit and the manufacturer profit combined. 12 For all other regions under c > c ˆ 1 , the bundling option does not affect the 1 channel profit. See the proof of Proposition 4 for the analysis of channel profit in all regions.

wholesale price) in all of regions S1, S2, B2, B3 and B4.13 In fact, the manufacturer even increases the wholesale price in these five regions under the bundling option. The higher wholesale price in these regions then exacerbates the double marginalization problem in the distribution channel, thus results in worse coordination between the retailer and manufacturer and reduced channel profit. Model Extensions In this section, we consider three model extensions. In the first extension, the manufacturer decides whether or not to allow retailer bundling. In the second extension, the consumer valuation for each product follows a uniform distribution. In the third extension, we consider an alternative model where the manufacturer sells directly to consumers. Manufacturer Decides Whether to Allow Bundling In our base model, we assume that the retailer has the power to decide on whether or not to bundle after the manufacturer announces the wholesale price. In this extension, we consider an alternative model in which the manufacturer decides on whether the retailer can bundle her product with another product. Specifically, in Stage 1 the manufacturer decides whether to allow retailer bundling, as well as the wholesale price w1 . In Stage 2, if the manufacturer allows retailer bundling, the retailer decides her bundling strategy; otherwise, the retailer sells products 1 and 2 separately. Lemma 3 characterizes the equilibrium outcome. Lemma 3. When the manufacturer decides whether or not allow retailer bundling, the equilibrium manufacturer and retailer actions are: Regions

Manufacturer allows bundling?

Retailer executes bundling?

B1, B2 B3, B4, S1, S2, L’L, L’H HL, HH, LH, LL

Y N Indifferent

Y N/A N

For the regions in which the manufacturer allows retailer bundling (B1 and B2), the equilibrium wholesale price, retail price, retailer profit from both products, and manufacturer profit are identical to those under a retailer bundling option (Proposition 1). For the regions in which the manufacturer does not allow retailer bundling or is indifferent (all other regions), equilibrium prices and profits are identical to those under no bundling option (Lemma 1). Proposition 5 characterizes the impacts of retailer bundling option on the performance of different channel members.

13 The manufacturer does offer concessions in regions L’L and L’H. Nevertheless, the concessions in these two regions are not strong enough to induce any change in the consumer market (either in terms of retail bundling decision or retail price), thus do not affect channel profit.

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Proposition 5. Suppose that the manufacturer can decide whether or not to allow retailer bundling.

w1 h

(i) In region B1, retailer bundling decreases the wholesale price and improves both firm profits; (ii) In region B2, retailer bundling increases the wholesale price, improves manufacturer profit and lowers retailer profit; (iii) In all other regions, the bundling option does not affect either firm profit. Proposition 5 shows that the key insights obtained from the base model continue to hold. The retailer can still obtain a lower wholesale price from the manufacturer even when the manufacturer can decide whether his product can be bundled. Different from the base model, the manufacturer is never worse off with retailer bundling in this extension. Uniform Valuation Distribution In this extension we assume that consumer valuation for either product follows a uniform distribution with support [1, h]. It is straightforward to verify that, without a bundling option, the equilibrium wholesale price, retail price, retailer profit from product 1 and manufacturer profit, respectively, are14 : wo1 =

(c1 + h) , 2

po1 =

o and πm =

(3h + c1 ) , 4

o πr1 =

(h − c1 )2 , 16(h − 1)

(h − c1 )2 . 8(h − 1)

59

Unbundle

Bundle

o

h Fig. 6. Retailer strategy under bundling option.

the retailer  bundles with bundle price pb2 =

2 2 4 + w1 + w2 + (w1 + w2 − 2) + 6(h − 1) /3. Otherwise, the retailer does not bundle and prices product 1 at (h + w1 )/2. Lemma 4, extending McCardle et al. (2007), presents a full characterization of retailer bundling decision. Fig. 6 illustrates the retailer’s strategy: in the circle area the retailer bundles; otherwise, she does not bundle. A comparison of Figs. 6 and 2 (retailer strategy under two-point valuation distribution) reveals that retailer bundling strategies under two-point and uniform distributions are analogous. The two figures differ only in the shapes of the bundling region: one’s border is smooth and the other’s border has kinks. The key intuition under the two-point case – that the retailer bundles only when its procurement costs are not too high or lop-sided – continues to hold under the uniform case. Given the retailer’s response, the manufacturer sets his wholesale price to maximize his profit:

In the case with a bundling option, we restrict our attention to pure bundling.15 The lemma below characterizes the retailer’s best response (bundling decision and corresponding retail prices) for any given wholesale price pair (w1 , w2 ). ⎧ 2(2h − w1 − w2 )2 2(2h − w1 − w2 )3 (h − w1 )2 + (h − w2 )2 h+3 ⎪ ⎪ (w1 − c1 ) if ≥ and w1 + w2 ≥ ; ⎪ 2 2 ⎪ 4 2 9(h − 1) 27(h − 1) ⎪ ⎪



 ⎨ 2 2 2 2 Max πm =

w1 >c1

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(w1 − c1 ) 1 −

(w1 − c1 )

(pb2 − 2) 2(h − 1)

(h − w1 ) 2(h − 1)

2

w2

if (pb2 − w1 − w2 ) 1 −

(pb2 − 2)

2(h − 1)2

≥

(h − w1 ) + (h − w2 ) 4

and w1 + w2 <

h+3 ; 2

otherwise.

Lemma 4. If (2(2h − w1 − w2 )3 )/(27(h − 1)2 ) ≥ ((h − w1 )2 + (h − w2 )2 )/4 and w1 + w2 ≥ (h + 3)/2, the retailer bundles with bundle price 2(h + w1 + w2 )/3. 2 If (pb2 − w1 − w2 )[1 − (pb2 − 2) /(2(h − 1)2 )] ≥ 2 2 and w1 + w2 < (h + 3)/2, ((h − w1 ) + (h − w2 ) )/4 14 To keep the narrative succinct, we assume c > 4 − 3h in this extension to 1 avoid corner solutions. Including the corner solutions does not qualitatively change the results. 15 It is a common practice in the bundling literature with continuous valuations to consider only pure bundling. See, for example, Fang and Norman (2005) and McCardle et al. (2007).

The manufacturer’s problem turns out to be analytically convoluted. We therefore conduct a computational study to examine how a bundling option affects manufacturer and retailer profits. This computational study is facilitated by the fact that the above profit function of the manufacturer is unimodal in each segment (see the end of Appendix for proof). We generate two problem instances by setting h to 1.5 and 2, respectively. Given each problem instance and for every possible pair of c1 and w2 , we compute the equilibrium wholesale price using a Fibonacci search (where the unimodality result improves the search efficiency). We then compute and compare the equilibrium outcomes of the bundling-option and no-bundling-option games, and obtain results qualitatively analogous to those under the two-point valuation distribution. Specifically, the retailer can

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Q. Cao et al. / Journal of Retailing 91 (1, 2015) 50–67 2.0

1.5

c1

No Impact 1.0

Win 0.5

Lose 0.0 0.0

0.5

1.0

1.5

2.0

w2 Fig. 7. Impact of bundling on retailer profitability with uniform consumer valuations.

still be better off with bundling, as illustrated in Fig. 7 (when h = 2). Manufacturer Sells to Consumers Directly In this extension, we consider an alternative model – the direct-selling model – where the manufacturer sells directly to consumers. The goal of this analysis is to determine whether it is possible for the manufacturer to profit more under our base model (where the bundling retailer serves as an intermediary between the manufacturer and consumers) than under this alternative direct-selling model. Lemma 5 characterizes the manufacturer’s profit in the direct-selling model. Lemma 5. In the direct-selling model, the manufacturer’s optimal profit is  if c1 ≤ (1 − (1 − α)h)/α; 1 − c1 , d πm = (h − c1 )(1 − α), if c1 > (1 − (1 − α)h)/α. Lemma 5 shows that, under the direct-selling model, the manufacturer either aims for high market demand at a low margin (when c1 ≤ (1 − (1 − α)h)/α), or for high margin at a low demand (when c1 > (1 − (1 − α)h)/α). We next compare the manufacturer’s profit in our base model with that in the direct-selling model, the result of which is presented in Proposition 6. Proposition 6. The manufacturer’s profit under the base model is higher than that under the direct-selling model if and only if his cost satisfies 1 − (1 − α2 )ubj (1 + α)ubj − h ≤ c1 ≤ , α2 α

where j = 1, 2.

Proposition 6 shows that a manufacturer can benefit from selling his product through a bundling retailer than selling it directly to consumers. This happens when the manufacturer cost is medium, i.e., (1 − (1 − α2 )ubj )/α2 ≤ c1 ≤ ((1 + α)ubj − h)/α. Intuitively, under a medium marginal production cost of the manufacturer, bundling enables the retailer to balance between margin and demand and thus to extract the maximum possible surplus from consumers. A large enough share of this benefit is then passed over to the manufacturer – either in the form of a better margin when (1 − (1 − α2 )ubj )/α2 < c1 ≤ (1 − (1 + α)h)/α or in the form of a larger demand when (1 − (1 − α)h)/α < c1 ≤ ((1 + α)ubj − h)/α, as compared to the direct-selling model. As a result, the manufacturer benefits more from retailer bundling under the base model than from the elimination of double marginalization under the direct-selling model. Therefore, bundling provides a new rationale for a manufacturer to sell his product through a retailer instead of selling it directly. Note that the cost condition (1 − (1 − α2 )ubj )/α2 ≤ c1 ≤ ((1 + α)ubj − h)/α is tighter than that of region B1, i.e., (1 − (1 − α2 )h)/α2 ≤ c1 ≤ ((1 + α)ubj − h)/α. Therefore, this region (in which the manufacturer profits more than he does under direct selling) is enclosed by region B1. This is not surprising: In region B1, the manufacturer’s profit is higher as compared to his profit in a decentralized distribution channel without retailer bundling; for the manufacturer to outperform one in a centralized system, bundling has to yield an even better balance between the manufacturer’s margin and demand. Managerial Implications In this section we summarize the implications of our research to the managers of both a downstream retailer and an upstream manufacturer. The first and foremost managerial message of our research is that, in a channel structure, the manager of a retailer should not look at her bundling decision in isolation. Rather, the downstream bundling option affects the strategic dynamics in a channel. Particular to our model, the downstream bundling option can (but not always) enable a retailer to impose downward pressure on a manufacturer’s wholesale price decision, which we refer to as the concession extraction role of retailer bundling in this paper. For the managers of a retailer, our research provides guidelines on when such a concession extraction strategy is possible, the details of which are summarized in Table 1. Table 1 summarizes Propositions 2–5, and provides a bird’s eye view of the key implications of our research for managers. In Table 1, the first column on regions corresponds to the regions in Fig. 5. Columns 2 and 3 list the cost conditions that apply to each region. Here, “product 1” refers to the product of the strategic manufacturer, and “product 2” refers to either the retailer’s own private label product or a product procured from a fringe manufacturer that is a price taker (e.g., due to a commoditized wholesale market). For each set of cost conditions, column 4 lists the retailer’s optimal bundling strategy, and column 5 highlights whether the retailer is able to extract concessions (in the form of a lower wholesale price) from a manufacturer.

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61

Table 1 Managerial implications. Region

B1

Cost structure

Retailer bundling strategy

Production cost of product 1

Procurement cost of product 2

Moderately high

Medium

Bundle

Concession extraction

Yes

Profit implications Retailer profit

Manufacturer Channel profit profit

Higher

Higher

Higher Lower

B2, B3, B4

Low

Medium

Bundle

No

Lower

a

L’L, L’H

Low

Moderately low or moderately high

Unbundle

Yes

Higher

Lower

Same

S1, S2

Moderately low

Moderately low or moderately high

Unbundle

No

Lower

Lower

Lower

Other regions a

The retailer bundling option has no impact on firm strategy or profit.

The manufacturer profit is higher in region B2 and lower in regions B3 and B4.

The no-bundling-option benchmark is used for all results in this table (e.g., by saying retailer profit is higher, we mean it is higher than that under the no-bundling-option benchmark). Table 1 shows that the manager of a retailer can leverage the downstream bundling option to extract concessions from her upstream manufacturer in two cases. The first case is when the marginal production cost of product 1 is moderately high and the procurement cost of product 2 is medium (region B1). Under this case, the bundling option (which is executed in equilibrium) benefits the retailer on two fronts: At the retail side, bundling enables the retailer to better extract surplus from consumers (the classical argument for bundling in the literature). At the wholesale side, the bundling option enables the retailer to extract concessions from the manufacturer (the new rationale uncovered in our model). An important message in this case for the manager of a retailer is the following: it is the ex ante option to bundle, rather than the ex post action of bundling, that drives the concession extraction phenomenon. In other words, the manufacturer is willing to propose a lower wholesale price only because, if the wholesale price is any higher, the retailer has the freedom to (and indeed will) switch to unbundling which is not preferred by the manufacturer under this case. The second case of concession extraction by a retailer is when the marginal production cost of product 1 is low and the procurement cost of product 2 is either moderately low or moderately high (regions L’L and L’H). In this case, the retailer does not execute the bundling option in equilibrium, thus the bundling option does not benefit the retailer at the retail side. Nevertheless and on the wholesale side, the retailer’s off-equilibrium threat of bundling forces a lower wholesale price from the manufacturer. One takeaway for the managers of a retailer in this case is that, even if bundling is not observed in equilibrium and thus does not affect the retail market, a retailer can still use the bundling option (in particular, the off-equilibrium threat to bundle) as a strategic tool to gain leverage from manufacturers.

For the manager of a manufacturer, our research shows that concession extraction by a retailer is not necessarily detrimental to the bottom line of the manufacturer. If the manufacturer’s marginal production cost is moderately high and the retailer’s private label product carries a medium marginal procurement cost (region B1), retailer bundling leads to both a squeeze of wholesale margin and an expansion of consumer demand. It turns out the benefit the manufacturer receives due to the expanded demand outweighs the loss he suffers due to the tighter margin. Therefore overall retailer bundling is beneficial to the manufacturer. In cases where the bundling option hurts the manufacturer (cost structures are reflected by regions L’L and L’H), the manager of a manufacturer may consider limiting the retailer’s bundling strategy space if this is feasible.16 Specifically, in regions L’L and L’H, if possible the manufacturer should contractually forbid the retailer from bundling his product with other products. Doing so ensures that the manufacturer never loses from downstream bundling. Concluding Remarks This paper offers a new rationale for retailer bundling in a distribution channel: By bundling a product from a powerful manufacturer with a private-label product (or a product from a price-taking fringe manufacturer), a retailer can force a lower wholesale price from the powerful manufacturer. The retailer can obtain such a concession even when the powerful manufacturer can specify whether its product can be bundled. While concession extraction always benefits the retailer, it does not necessarily hurt the manufacturer. This is because retailer bundling expands the market – the gain from increased demand can outweigh the loss of margin for the manufacturer.

16 This discussion corresponds to the model extension in Manufacturer Decides Whether to Allow Bundling subsection.

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Therefore, the manufacturer can be better off selling through a bundling retailer (than selling through a no-bundling retailer). We find that the manufacturer’s marginal production cost plays a critical role in driving the retailer’s concession extraction behavior and consequently the total channel profit. When this cost is moderately high, retailer bundling results in a lower wholesale price that in turn alleviates the double marginalization problem in the distribution channel, thus results in better channel efficiency as measured by total channel profit. A broader message from this paper is that, when studying the impact of bundling on retailer performance, ignoring the channel context it is embedded in can sometimes result in inaccurate conclusions. Our findings provide clear and strong evidence on the importance of pushing research at the intersections of the bundling and channel management literatures. Acknowledgements The authors thank the Editor, Shankar Ganesan, and three anonymous referees for their constructive suggestions that have improved the paper significantly. Appendix. Proof of Lemma 1. The manufacturer’s profit function (1) is monotonically increasing within each segment of the wholesale price. Therefore, the manufacturer should charge either w1 = (1 − (1 − α)h)/α or w1 = h. Accordingly his profit o = (1 − (1 − α)h)/α − c or π o = (h − c )(1 − is either πm 1 1 m α). The former profit dominates the latter if any only if c1 ≤ (1 − (1 − α2 )h)/α2 . Retail price and retailer profit immeo and π o . diately follow from the expressions of po1 , πr1 r2 Proof of Lemma 2. Under unbundling, each product is priced and sold separately. If the price for product i (i = 1, 2) is 1, then demand for product i is 1. If the price is h, then demand is 1 − α. Therefore, the four possible unbundling strategies and their corresponding profits are: p1 = 1, p2 = 1, and the retailer’s profit is πLL = 1 − w1 + 1 − w2 ; p1 = 1, p2 = h, and πLH = 1 − w1 + (1 − α)(h − w2 ); p1 = h, p2 = 1, and πHL = (1 − α)(h − w1 ) + 1 − w2 ; p1 = h, p2 = h, and πHH = (1 − α)(h − w1 ) + (1 − α)(h − w2 ). Subscript LL denotes the strategy that the retailer uses lowpricing strategies in both markets. If the retailer pure bundles, possible strategies and the retailer’s corresponding profits are: • pB = 1 + 1. All consumers buy the bundle. Therefore, retailer profit is πB = 2 − w1 − w2 = πLL . • pB = 1 + h. Only consumers with low valuations for both products do not buy. Therefore, demand for the bundle is 1 − α2 . Retailer profit is πB = (1 − α2 )(1 + h − w1 − w2 ). This strategy can possibly give the retailer a higher profit than those unbundling strategies.

• pB = h + h. Only consumers with high valuations for both products buy. Demand is (1 − α)2 and profit is πB = (1 − α)2 (h + h − w1 − w2 ) = πHH . Now we consider mixed-bundling. We only need to consider mixed-bundling strategies where the bundle price is lower than the sum of individual product prices. Such strategies are: • p1 = h, p2 = h, pB = 1 + h. Under this strategy, no consumer buys any individual product. Demand for the bundle is 1 − α2 . Retailer profit is πB = (1 − α2 )(1 + h − w1 − w2 ). This strategy is identical to the pure-bundling one where pB = 1 + h. • p1 = h(1), p2 = (1)h, pB = 1 + 1. All consumers buy the bundle, which yields profit πB = 1 − w1 + 1 − w2 . This strategy is equivalent to the unbundling one where p1 = h and p2 = 1. To summarize, the only bundling scheme that can possibly yield a higher profit for the retailer than unbundling is pB = 1 + h, which is also equivalent to the strategy p1 = h, p2 = h, pB = 1 + h. Proof of Lemma 3. First consider pure bundling with bundle price pB = 1 + h. The demand for the bundle is 1 − α2 and retail profit margin is 1+ h − w1 − w2 , which yields a profit πB = (1 − α2 )(1 + h − w1 − w2 ). The retailer bundles if and only if πB ≥ max{πLL , πLH , πHL , πHH }, where πLL , πLH , πHL , πHH are defined in the proof of Lemma 2. We examine two cases below. Case 1: w2 < (1 − (1 − α)h)/α. In this case, when w1 ≥ (1 − (1 − α)h)/α, πHL = max{πLL , πLH , πHL , πHH }. Therefore, with a bundling option, the retailer bundles if and only if πB ≥ πHL . When w1 < (1 − (1 − α)h)/α, πLL = max{πLL , πLH , πHL , πHH }, thus the retailer bundles if and only if πB ≥ πLL . Combining these two cases of w1 together, the retailer bundles if and only if −w2 + 1 + h + (1 − h)/α2 ≤ w1 ≤ (α/(1 − α))w2 + 1 + h − 1/(1 − α). That is, lb1 (w2 ) ≤ w1 ≤ ub1 (w2 ). Case 2: (1 − (1 − α)h)/α ≤ w2 ≤ h. Similarly, the retailer bundles if and only if πB ≥ πHH and πB ≥ πLH . That is, ((1 − α)/α)w2 + 1 + h − h/α ≤ w1 ≤ −w2 + 1 + h + (1 − h)/α, or lb2 (w2 ) ≤ w1 ≤ ub2 (w2 ). Note that when w2 < (1 − (1 − α)h)/α, we have ub1 − ub2 < 0 and lb1 − lb2 > 0. Similarly, when (1 − (1 − α)h)/α ≤ c2 ≤ h, we have ub2 ≤ ub1 and lb2 ≥ lb1 . Therefore, we can rewrite the retailer’s bundling condition as max{lb1 , lb2 } < w1 ≤ min{ub1 , ub2 }. Proof of Proposition 1. Proof is similar to that of Lemma 1. In the rest proofs, all retailer and manufacturer profits under the no-bundling-option and bundling-option equilibria are from Lemma 1 and Proposition 1. To facilitate the rest proofs, we replace ubj and lbj (j = 1, 2) as expressions of α and h in the

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definitions of regions B1 through B4. Using (2), we can rewrite Consider the left part. Retailer profit in the bundling equilibrium these definitions as: is      1 − (1 − α2 )h 1+α h − 1 − αh − α 1+α (1 + α)2 + (α2 − α − 1)h  ; B1 = (c1 , w2 )  ≤ c1 ≤ Min w2 + ,− w2 +  α2 1−α 1−α α α2     1+α 1 + α2 + α3 + α3 h 1 − α2 α 3 + α2 + α + α 3 h − α2 h 1 − (1 − α2 )h  B2 = (c1 , w2 ) Max − < c1 ≤ ; w2 + , w2 + α α3 α2 α3 α2    1 − (1 − α)h 1 + α + α2 1 + α2 + α3 + α4 − (1 − α)(1 + α + α2 + α3 )h B3 = (c1 , w2 ) c2 ≤ w + and c1 > − 2 α α2 α4  2 2 3 3 1 − (1 − α )h 1+α 1+α +α +α h 1+α h − 1 − αh − α and c1 ≤ Min ,− , w2 + w2 + ; α2 α α3 1−α 1−α    1 − (1 − α)h 1 − α3 α2 + α3 − 1 − (1 − α)α2 h B4 = (c1 , w2 ) w2 > and c1 > w + 2 α α3 α3   1 − (1 − α2 )h 1 − α2 α 3 + α2 + α + α3 h − α2 h 1+α (1 + α)2 + (α2 − α − 1)h . and c1 ≤ Min , w2 + ,− w2 + α2 α2 α3 α α2 To be analytically rigorous and complete, we provide extended versions of Propositions 2 and 3 and Corollary 1 in this appendix. The extended versions of Propositions 2 and 3 and Corollary 1 characterize the impact of a bundling option for all problem parameters, i.e., in all regions of the c1 –w2 plane.

πrb = (1 − α2 )(1 + h − ub1 − w2 ) = (h − ub1 )(1 − α) + 1 − w2 ,

Proposition 2. (i) In regions L’L, L’H and B1, the bundling option enables the retailer to extract concessions from the manufacturer in the form of a lower wholesale price. (ii) In regions B2, B3, B4, S1, and S2, the bundling option increases the wholesale price. (iii) In regions HL, HH, LH, and LL, the bundling option does not change the wholesale price. Proof of Proposition 2. It is easy to verify that lbj ≤ (1 − (1 − α)h)/α and (1 − (1 − α)h)/α ≤ ubj ≤ h. Therefore, wb1 ≤ wo1 in regions L’L, L’H, and B1, and wb1 ≥ wo1 in regions B2, B3, and B4. It follows from a straightforward comparison of Lemma 1 and Proposition 1 that wb1 ≥ wo1 in regions S1 and S2, and wb1 = wo1 in regions HL, HH, LH, and LL. This completes the proof. Corollary 1. (i) In regions L’L, L’H and B1, the bundling option improves retailer profit. (ii) In regions B2, B3, B4, S1, and S2, the bundling option lowers retailer profit. (iii) In regions HL, HH, LH, and LL, the bundling option does not change retailer profit. Proof of Corollary 1. Consider part (i). In region L’L, retailer profit in the bundling-option equilibrium is πrb = 1 − lb1 ≥ 1 − (1 − (1 − α)h)/α = πro , so the retailer is better off in region L’L. Similarly, the retailer is better off in region L’H. Now consider region B1, which we partition into two parts: left where w2 ≤ (1 − (1 − α)h)/α, and right where w2 > (1 − (1 − α)h)/α.

where the second equality follows from the fact that ub1 is the highest wholesale price the manufacturer can charge such that the retailer is indifferent between unbundling and bundling. In the no-bundling-option benchmark, the retailer’s profit is πro = 1 − w2 in this region. It follows immediately that πrb > πro . Thus the retailer is better off. The same conclusion holds in the right part of region B1. Now we show part (ii). Consider region B3 first. Retailer profit in the bundling equilibrium is πrb = (1 − α2 )(1 + h − ub1 − w2 ) = (h − ub1 )(1 − α) + 1 − w2 , where the last equality follows from the fact that ub1 is the highest wholesale price the manufacturer can charge such that the retailer is indifferent between unbundling and bundling. The retailer’s profit under no-bundling-option benchmark is πro = 1 − (1 − (1 − α)h)/α + 1 − w2 . The profit difference is πrb − πro = (h − ub1 )(1 − α) − 1 + (1 − (1 − α)h)/α ≤ [h − (1 − (1 − α)h)/α](1 − α) − 1 + (1 − (1 − α)h)/α = 0. Therefore, the retailer is worse off in region B3. Similarly, she is worse off in regions B2 and B4. In region S1 and for the retailer, the difference between her profits in the bundling-option and no-bundling-option equilibria is πrb − πro = 0 + 1 − w2 −

1 − (1 − (1 − α)h) < 0. α + 1 − w2

Therefore, the retailer is also worse off in region S1. Similarly, we can show that the same conclusion holds in region S2. Part (iii) follows directly from the fact that, in regions HL, HH, LL, and LH, the wholesale prices and the retailer strategies are the same as the ones in the two equilibria.

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Proposition 3. (i) In regions B1 and B2, the bundling option improves manufacturer profit. (ii) In regions B3, B4, S1, and S2, L’L, and L’H, the bundling option lowers manufacturer profit. (iii) In regions HL, HH, LH, and LL, the bundling option does not change manufacturer profit. Proof of Proposition 3. We prove part (i) first. Consider the left part of region B1, where c1 and w2 are such that c1 > (1 − (1 − α2 )h)/α2 , w2 ≤ (1 − (1 − α)h)/α and c1 ≤ (1 + α)w2 /(1 − α) + (h − 1 − αh − α)/(1 − α). By Lemma 1, the manufacturer’s profits in the no-bundling-option o = (1 − and bundling-option equilibria, respectively, are πm b 2 α)(h − c1 ) and πm = (1 − α )(ub1 − c1 ). From (2) we have b − πo πm m

o πm + πro =

1 − (1 − α)h − c1 α 1 − (1 − (1 − α)h) + + 1 − w2 . α

(5)

With the bundling option, the joint profit is

= −α(1 − α)c1 + α(1 + α)w2 − α2 − hα2 − α + αh   . 1+α h − 1 − αh − α ≥ −α(1 − α) w2 + + α(1 + α)w2 − α2 − hα2 − α + αh = 0 1−α 1−α

Similarly, in the right part of region B1 and region B2, we can b − π o ≥ 0. show that πm m We now prove part (ii). In region B3, the manufacturer’s o = (1 − (1 − α)h)/α − c profit in those two equilibria are πm 1 b 2 and πm = (1 − α )(ub1 − c1 ). From (2) we have b − πo πm m

increases, following from part (i) of Corollary 1 and part (i) of Proposition 3. In HL and HH, however, the channel profit does not change following from part (iii) of Corollary 1 and Proposition 3. c1 < cˆ 1 implies regions B2, B3, B4, S1, S2, L’L, L’H, LL, and LH. Now we show that channel profit does not increases in those regions. We now show that channel profit decreases in B2. Consider the left part of region B2 where w2 ≤ (1 − (1 − α)h)/α. In the no-bundling-option benchmark, the sum of manufacturer 1 and retailer profits is

b πm + πrb = (ub1 − c1 )(1 − α2 ) + (1 + h − ub1 − w2 )(1 − α2 )

= (ub1 − c1 )(1 − α2 ) + (h − ub1 )(1 − α) + 1 − w2 , (6)

= α2 c1 + α(1 + α)w2 + (h − 1 − α2 − α3 − α3 h)/α   , 1+α 1 + α 2 + α3 + α3 h 1 − (1 − α)h ≤ α2 − + α(1 + α)w2 + h − α − w2 + =0 3 α α α

where the inequality follows from the fact that c1 ≤ −(1 + α)w2 /α + (1 + α2 + α3 + α3 h)/α3 in region B3. Therefore, the bundling option lowers the manufacturer profit. Similar conclusion holds in region B4. In region S1, manufacturer profit in the bundling-option b = (1 − α)(h − c ). In the no-bundling-option equilibrium is πm 1 o = (1 − (1 − α)h)/α − c when equilibrium his profit is πm 1 c1 ≤ (1 − (1 − α2 )h)/α2 . It follows that b o πm − πm = α(c1 − (1 − (1 − α2 )h)/α2 ) ≤ 0.

Therefore, the manufacturer is worse off. In region L’L, manb = ufacturer profit in the bundling-option equilibrium is πm 2 b ub1 − c1 . Given w2 ∈ [1 − (1 − α)(h − 1)/α , 1], πm ≤ (1 − (1 − α)h)/α − c1 , which is the manufacturer profit in the nobundling-option benchmark. Therefore, the manufacturer is worse off in region L’L. The proofs for region S2 and L’H are similar. Part (iii) follows directly from the fact that, in regions HL, HH, LL, and LH, the wholesale prices and the retailer strategies are the same as the ones in the two equilibria. Proof of Proposition 4. When c1 ≥ cˆ 1 , there are the following regions: B1, HL, and HH. In region B1, the channel profit

where the last equality follows from the fact that the manufacturer extracts the retailer’s bundling margin by setting a high wholesale price such that the retailer is indifferent between bundling and unbundling. The joint profit increases if and only if the difference between (6) and (5) are non-negative, i.e., c1 ≥ −w2 +

(h − 1) + h + 1. α2

(7)

Further, we show that no pair of (c1 , w2 ) satisfies (7) in region B2. It is sufficient to show that the corner point of region B2, i.e., ((1 − (1 − α)h)/α, (1 − (1 − α2 )h)/α2 ), is not in the region enclosed by (7). It follows that 1 − (1 − α)h 1 − (1 − α2 )h + α α2 h−1 h−1 <1+h− 2 < 0, − 2 α −h−1 α −h−1 which indicates that no pair of (c1 , w2 ) satisfies (7) in region B2. Therefore, the joint profit decreases in the left of region B2. Similarly, we can show that the joint profit decreases in the right of region B2. In regions B3, B4, S1, and S2, channel profit decreases, which follows immediately from part (ii) of Corollary 1 and Proposition 3, as both manufacturer and retailer are worse off. In all other

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regions, it is easy to verify that channel profit does not change. This completes the proof. Proof of Lemma 3. Consider the manufacturer’s decision on whether to allow retailer bundling in Stage 1. If the manufacturer does not allow retailer bundling, the problem boils down to the one in which the retailer does not have a bundling option. Therefore, the equilibrium of this subgame is identical to that under no bundling option (Lemma 1). When the manufacturer allows retailer bundling, the problem boils down to the one in which the retailer has a bundling option. Hence, the subgame outcome is identical to that under a bundling option (Proposition 1). The manufacturer allows retailer bundling only if he is better off under a retailer bundling option, i.e., in regions B1 and B2. In regions HL, HH, LH, and LL, the manufacturer’s profits are identical in the two subgames and therefore he is indifferent between allowing bundling or not. In all other regions where the manufacturer is worse off, he does not allow retailer bundling. This completes the proof. Proof of Proposition 5. The results in regions B1 and B2 follow from Corollary 1 and Proposition 3. Results in all other regions follow as the equilibrium outcomes are identical to those in the benchmark scenario. This completes the proof. Proof of Lemma 4. The retail demand with respect to any given retail price is D = 1 − (p − 2)2 /(2(h − 1)2 ) if p < h + 1, and D = (2h − p)2 /(2(h − 1)2 ) if p ≥ h + 1. For any given w1 and w2 , the retailer’s profit is ⎧

 ⎪ (p − 2)2 ⎪ ⎪ (p if 2 ≤ p ≤ h + 1; − w − w ) 1 − 1 2 ⎨ 2(h − 1)2 πr = ⎪ (2h − p)2 ⎪ ⎪ if h + 1 < p ≤ 2h; ⎩ (p − w1 − w2 ) 2(h − 1)2 Consider the first part of the retailer’s profit function. Taking first derivative and setting it to 0 yields 1−

(p − 2)2 p−2 − (p − w1 − w2 ) = 0, 2(h − 1)2 (h − 1)2

with two roots    1 2 2 b1 p = 4 + w1 + w2 − (w1 + w2 − 2) + 6(h − 1) , 3    1 pb2 = 4 + w1 + w2 + (w1 + w2 − 2)2 + 6(h − 1)2 . 3 It follows immediately that (p − w1 − w2 )(1 − (p − 2)2 / 2(h − 1)2 ) as a function of p decreases on (− ∞ , pb1 ], increases on (pb1 , pb2 ], and decreases on (pb2 , −∞). Note that pb1 ≤ 2 and pb2 ≥ 2. Therefore, the retailer’s maximal profit in the interval [2, h + 1] is achieved at p∗1 = min{pb2 , h + 1}. Now consider the second part of the retailer’s profit function, the first derivative of which is (2h − p)(2h − 3p + 2w1 + 2w2 ) . 2(h − 1)2 It follows that (p − w1 − w2 )(2h − p)2 /2(h − 1)2 as a function of p increases on (−∞, 2(h + w1 + w2 )/3], decreases on (2(h + w1 + w2 )/3, 2h], and increases on (2h, −∞). Consequently, the

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retailer’s maximal profit in the interval [h + 1, 2h] is achieved at p∗2 = Max{2(h + w1 + w2 )/3, h + 1}. It is easy to verify that, for p = h + 1, the left-side and the right-side derivatives of the manufacturer’s profit are identical. Therefore, the retailer’s profit function is differentiable at p = h + 1. When w1 + w2 ≤ (h + 3)/2, p∗1 = pb2 and p∗2 = h + 1. Therefore, the retailer charges pb2 to achieve the maximal profit associated with bundling. The retailer bundles if and only if (pb2 − w1 − w2 )[1 − 2 (pb2 − 2) /(2(h − 1)2 )] ≥ ((h − w1 )2 + (h − w2 )2 )/4. When w1 + w2 > (h + 3)/2, p∗1 = h + 1 and p∗2 = 2(h + w1 + w2 )/3. The retailer charges 2(h + w1 + w2 )/3 to achieve the maximal bundling profit. The retailer bundles if and only if (2(2h − w1 − w2 )3 )/(27(h − 1)2 ) ≥ ((h − w1 )2 + (h − w2 )2 )/4. This completes the proof. Unimodality of the manufacturer’s profit function under uniform distribution Next, we show that the manufacturer’s profit function is unimodal in w1 in each segment. Using the basic properties of polynomial functions, we can easily show that (w1 − c1 )2(2h − w1 − w2 )2 /9(h − 1)2 is unimodal in w1 on the support (0, 2h). Here, we only show that (w1 − c1 )[1 − 2 (pb2 − 2) /2(h − 1)2 ] is unimodal in w1 . 2 It is sufficient to show that (w1 − c1 )[(pb2 − 2) / 2(h − 1)2 − 1] is quasiconvex in w1 . Note that the first and second derivatives of pb2 (w1 ) with respect to w1 are   pb2  (w) =

1 3

1+





w1 + w2 − 2

(w1 + w2 − 2)2 + 6(h − 1)2 −

pb2  (w) =

1 3

> 0,

(w1 + w2 − 2)2 + 6(h − 1)2

(w1 + w2 − 2)2



(w1 + w2 − 2)2 + 6(h − 1)2

(w1 + w2 − 2)2 + 6(h − 1)2

> 0.

Therefore, pb2 (w1 ) is increasing and convex in w1 . Taking 2 the first derivative of [(pb2 − 2) /2(h − 1)2 − 1] with respect b2 b2 to w1 yields (p − 2)p (w1 )/(h − 1)2 . Because pb2 − 2 and 2 pb2 (w1 ) are increasing and positive, [(pb2 − 2) /2(h − 1)2 − 1] is convex (and non-positive) in w1 . Also, w1 –c1 is concave and non-negative. It is well known that the product of a non-negative concave function and a non-positive convex function is quasiconvex; see, for example, Bazaraa, Sherali, and Shetty (2006, p. 157, 3.62). It then follows that (w1 − 2 c1 )[(pb2 − 2) /2(h − 1)2 − 1] is quasiconvex in w1 . This completes the proof. Proof of Lemma 5. The proof is similar to that of Lemma 1. Proof of Proposition 6. First note that manufacturer profit d ) is higher than the sum of manufacturer under direct selling (πm o + π o ). and retailer profits in the no-bundling benchmark (πm r Therefore, to derive for when the manufacturer benefits from selling to a bundling retailer (as compared to selling directly), it is sufficient for us to only consider region B1. For brevity, assume w2 ≤ (1 − (1 − α)h)/α, while the proof for the other case is simib = lar. In region B1, manufacturer profit in our base model is πm

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(1 − α2 )(ub1 − c1 ). When c1 ≤ (1 − (1 − α)h)/α, manufacturer d = 1 − c . The manufacturer benprofit under direct selling πm 1 b − πd ≥ efits from selling to a bundling retailer if and only if πm m 2 2 0, i.e., c1 ≥ (1 − (1 − α )ub1 )/α . When c1 > (1 − (1 − α)h)/α, d = (1 − α2 )(h − c ) ≤ π b , which follows from the definiπm 1 m tion of region B1, i.e., c1 ≤ ((1 + α)ub1 − h)/α. This completes the proof.

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