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Supplier encroachment under nonlinear pricing with imperfect substitutes: Bargaining power versus revenue-sharing
Accepted Manuscript
Supplier encroachment under nonlinear pricing with imperfect substitutes: Bargaining power versus revenue-sharing Huixiao Yang , Jianwen Luo , Qinhong Zhang PII: DOI: Reference:
S0377-2217(17)31160-8 10.1016/j.ejor.2017.12.027 EOR 14886
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
17 May 2017 11 December 2017 18 December 2017
Please cite this article as: Huixiao Yang , Jianwen Luo , Qinhong Zhang , Supplier encroachment under nonlinear pricing with imperfect substitutes: Bargaining power versus revenue-sharing, European Journal of Operational Research (2017), doi: 10.1016/j.ejor.2017.12.027
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Highlights Nonlinear pricing coupled with Revenue-sharing can coordinate the chain. Further increase of direct sales cost and product substitution can profit both firms. An encroaching supplier always hurts the retailer, even without direct sales. A less powerful supplier is more likely to benefit from encroachment. A powerful and relatively efficient supplier hurts from the threat of encroaching.
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* Corresponding author: Jianwen Luo. Tel.: +862152302053. E-mail addresses: [email protected] (H. Yang); [email protected] (J. Luo); [email protected] (Q. Zhang).
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Supplier encroachment under nonlinear pricing with imperfect substitutes: Bargaining power versus revenue-sharing Huixiao Yang a,b, Jianwen Luo b,*, Qinhong Zhang c School of Management, Jinan University, Guangzhou 510632, China
b
Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai 200030, China
c
Sino-US Global Logistics Institute, Shanghai Jiao Tong University, Shanghai 200030, China
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a
Abstract: We explore the impact of nonlinear pricing (NP) on supplier encroachment in a supply chain, consisting of a retailer (he) and a supplier (she), who can sell through either the retailer, her direct channel, or both. The two channels’ products are imperfect substitutes. The firms bilaterally negotiate over the
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wholesale price and the quantity using NP. To better align their behaviors, they can implement revenue-sharing (RS) to share the retailer’s sales revenue. Our analysis shows that NP coupled with RS coordinates the supply chain under encroachment when the supplier seizes all the retailer’s sales revenue. In what follows, the supplier subsidizes the retailer’s acquisition of product through a negative wholesale
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price. Surprisingly, further increase of the supplier’s direct selling cost and product substitution degree can benefit both firms. Contrary to the prior research, the retailer always hurts from the supplier’s ability to
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encroach, even when an inactive direct channel is introduced. Furthermore, a supplier with weak power is more likely to benefit from encroachment, but one with strong power hurts from initiating an inactive direct
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channel when her retail disadvantage is limited. Our main results are robust even after altering some
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assumptions in the basic model.
Keywords: E-commerce; Supplier encroachment; Bargaining power; Nonlinear price contracts;
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Revenue-sharing contracts
1. Introduction
With the rapid development of E-commerce and the growth of customer’s acceptance of a direct channel, upstream suppliers/manufacturers might introduce their own direct channels in addition to relying on third-party resellers/retailers, thereby competing with their resellers in the retail market. Such competition is often referred to as “supplier encroachment” in the literature, which has taken various forms. These include catalog sales, factory outlets and online stores (Nair and Pleasance 2005). Such examples can be observed in various industries, for instance, electronic industry (e.g., Apple, Dell, Sony,
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and Samsung), apparel and fashion accessory industry (e.g., Adidas, Coach, Estee Lauder and Nike), beverage and food industry (e.g., Budweiser Beer, Campbell Soup and Coca-Cola), travel industry, like hotels and airlines (e.g., American Airlines, Hilton), etc. It is widely acknowledged that the participation of a supplier in the retail market can pose a serious threat to existing retailers. Contrary to the conventional wisdom that supplier encroachment mostly
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hurts retailers, current research indicates that supplier encroachment may benefit incumbent retailers, because such an encroachment can induce demand-enhancing activity (Tsay and Agrawal 2004), lower double marginalization effect (Arya et al. 2007, Cattani et al. 2006, Chiang et al. 2003), promotion of brand awareness (Blair and Lafontaine 2005), and stronger incentive to reduce production cost (Yoon 2016). To reach such a conclusion, most papers assume that the upstream supplier acts as a Stackelberg
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leader and firms adopt a wholesale price-only contract. Despite its wide use, the supplier-led assumption that a supplier offers a “take-it-or-leave-it” contract to a retailer is inconsistent with various empirical evidences, which indicate the wide occurrence of bilateral bargaining in channel relationships (Draganska et al. 2010). Examples span raw material markets, manufacturing, service industries, etc.
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(Bajari et al. 2009). As Lovejoy (2010) indicated, a bargaining model would be more suitable in many supply chain circumstances, because the solutions resulting from the Stackelberg framework can be
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extremely impractical for various issues. Models of bilateral bargaining, though better revealing the power structure in supply chain, have sporadic adoption in the literature. While much has been known
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not so transparent.
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about supplier encroachment in the supplier-led Stackelberg game, the result in the bargaining game is
When the products sold through the direct and indirect channels are perfect substitutes and the
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supplier can implement nonlinear pricing (NP) to coordinate the channel, however, the benefit from supplier encroachment disappears (Li et al. 2015), which makes an interesting contrast to previous research. Nonlinear price contracts are widespread in practice. For example, many suppliers offer a menu of contract specifying the wholesale price and quantity to their retailers (Ha 2001, Li et al. 2015). Yet, consumers may view the retail products of direct and indirect channels as imperfect substitutes. Say, some consumers might particularly appreciate the convenience of in-home shopping offered by a supplier’s online store, while others may have a strong bias towards personal touch delivered by a brick-and-mortar retailer. In fact, product substitutions can considerably influence the model outcome.
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For example, Matsui (2011) shows that the degree of substitution among products significantly affects the negative effects of transfer prices on social welfare under price competition. However, the effect of product differentiation on firms’ behaviors and profits is still unclear under encroachment with nonlinear pricing. Building upon the modeling framework of Li et al. (2015), we seek to further understand the
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implications of supplier encroachment on contractual agreements and firms’ profitability under imperfect substitutes by considering two related managerial factors: (i) The firms’ bargaining power and (ii) the additional incentives schemes (e.g., revenue-sharing contracts) that the firms may adopt. To mitigate channel conflict and better align their behaviors, the firms between the upstream and downstream can implement revenue-sharing (RS) to share the retailer’s sales revenue. RS now is
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commonly used in different industries, e.g., agriculture, manufacturing, pharmaceuticals, etc. (Heese and Kemahlıoğlu-Ziya 2014). It is well known that RS coordinates the supply chain and arbitrarily allocates the supply chain’s profit, but it limits a risk neutral retailer’s incentive to exert costly effort (Cachon and Lariviere 2005). Also, it reduces the supplier’s incentive to leak the incumbent retailer’s
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private information about demand to the entrant retailer (Kong et al. 2013), but coordinates supply chain decision on innovation investment (Wang and Shin 2015). Although the effect of RS on coordinating a
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supply chain and firm’s incentives in various contexts has been generally investigated, its effect on aligning firms’ behaviors under encroachment is not so perspicuous.
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The popularity of supplier encroachment, bilateral bargaining, NP, imperfect substitutes, and RS in
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practice brings to the forefront a few research questions: When two firms bilaterally negotiate over NP with a RS clause and the two channels’ products are imperfect substitutes, (1) How do the supplier’s
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retail disadvantage and product substitution affect the two firms’ profitability? (2) Can NP coupled with RS coordinate the supply chain system? (3) Can Pareto gains continue to arise from supplier encroachment? (4) Does the threat of selling directly remain to be a boon for the supplier? To answer the above questions, we use a stylized model of a supply chain, consisting of a retailer and a supplier, who can sell through a retailer, or her direct channel, or both, where the two channels’ products are imperfect substitutes. The two firms engage in bilateral negotiation using NP coupled with RS. Our paper contributes to the literature in four ways. First, we prove that further increase of the
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supplier’s direct selling cost and product substitutions can benefit both firms. Second, we demonstrate that NP coupled with RS can coordinate the system under encroachment when the supplier squeezes all the retailer’s sales revenue. Third, we verify that the retailer always gets hurt from the supplier’s ability to encroach, even when an inactive direct channel is introduced. Finally, we establish that a supplier with weak power is more likely to benefit from encroachment, but one with strong power hurts from
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initiating an inactive direct channel when her retail disadvantage is limited. The remainder of this paper is organized as follows. Section 2 reviews the related literature. Section 3 introduces the model in detail. Section 4 provides the analysis of the model. Section 5 delivers two extensions of our model. Finally, section 6 summarizes our insights and limitations. The proofs of all the
2. Literature Review
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formal results appear in the appendix.
In this section, we review the related literature to corroborate the originality and importance of our study. Our paper is mainly related to three streams of literature: Supplier encroachment, bilateral negotiation, and revenue-sharing contracts.
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This work serves as a commentary on the papers on supplier encroachment. Encroachment arises
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when the upstream supplier launches her own direct channel to sell directly to the final consumers. Contrast to conventional wisdom, recent analytical research shows that supplier encroachment can
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benefit both firms, e.g., Arya et al. (2007), Cai (2010), Cattani et al. (2006), Chiang et al. (2003), Dumrongsiri et al. (2008), Tsay and Agrawal (2004), Yoon (2016), etc. However, Li et al. (2014, 2015)
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extend Arya et al. (2007) to incorporate asymmetric information about demand and find that supplier encroachment could lead to win-win, lose-lose, win-lose, or lose-win outcomes. Ha et al. (2016) show
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that supplier encroachment always hurts the retailer when endogenous product quality and heterogeneous customers arise. Hence, the effect of supplier encroachment on firms’ profitability is still unclear. Therefore, enormous papers seek to evaluate the effect of supplier encroachment on firms’ profitability, e.g., Caldieraro (2016), Hsiao and Chen (2014), Xiao et al. (2014). Furthermore, the prevalence of supplier encroachment drives many researchers to study the dual-channel system, where a supplier can sell through a retail channel, or a direct channel, or both, e.g., Chen et al. (2017), Feng et al. (2017), Lu et al. (2018), Matsui (2016, 2017), Qing et al. (2017), Yan et al. (2015), Yu et al. (2017), etc. Our work differs from the above papers along two important dimensions. First, we consider the two firms
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engage in bilateral negotiation game, rather than a Stackelberg game adopted by most existing research. Second, these papers assume that a wholesale price-only contract is used, while our work consider the two firms reach an agreement via NP coupled with RS. Finally, our work is closely related to Li et al. (2015), which shows that when the demand information is asymmetric and the supplier can use NP, the supplier’s ability to encroach can either hurt or benefit both firms. However, they derive the result in a
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Stackelberg game and perfect substitutes, while we consider the case in bilateral negotiation game and imperfect substitutes. Furthermore, they focus on the effect of asymmetric information, while we consider the impact of RS.
Our work also relates to the growing stream of literature on bargaining game. Models of bilateral bargaining game have received increasing concerns recently. Nagarajan and Bassok (2008) use a
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bargaining game framework to study the component suppliers’ decision to form a coalition. Hereafter, many papers seek to further understand the implications of bargaining power on firms’ decisions and profits, e.g., Aydin and Heese (2015), Feng and Lu (2012, 2013a, 2013b), Feng et al. (2015), Guo and Iyer (2013), Heese (2015), Leng et al. (2016), Sheu and Gao (2014), Wang et al. (2013), Zhong et al. (2016). Our
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study is most closely related to Yoon (2016), which examines the spillover effect from a manufacturer’s cost-reducing investment when the manufacturer can encroach into retail space and the firms negotiate
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over the wholesale price and quantity under imperfect substitutes in the extensions. Our study differs from Yoon (2016) along three important aspects: First, Yoon (2016) assumes that both firms possess equal
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bargaining power, which is a special case of ours; second, Yoon (2016) considers the spillover effect,
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whereas we examine the supplier’s retail disadvantage; finally, unlike our paper, Yoon (2016) does not survey the impact of RS. Our study is also closely related to Qing et al. (2017), which considers a
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supplier’s capacity-allocation under bargaining when she can allocate one element through dual-channel. However, the contract to be negotiated in their works is a linear price contract, while it is a nonlinear price contract in ours. Besides, they consider the supplier’s capacity-allocation problem, while we study the impact of RS. Finally, they seek to derive the conditions of the supplier’s equilibrium channel choices, but we strive to explore the effect of the supplier’s encroaching ability on firms’ decisions and profitability. Finally, our paper relates to the papers on revenue sharing contracts. Revenue sharing contracts have been recognized as an effective tool to better align firms’ incentives by inducing the firms to act in the
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best interest of the whole supply chain. Cachon and Lariviere (2005) show that revenue sharing contracts equal to buy-backs in various contexts and it can coordinate a supply chain and arbitrarily allocate the channel’s profit. Hereafter, massive papers concentrate on the coordinating revenue-sharing contract design issues in various settings. For example, Arani et al. (2016), Avinadav et al. (2015), Cai et al. (2017), Chakraborty et al. (2015), Heese and Kemahlıoğlu-Ziya (2014), Hu et al. (2016), Lim et al. (2015), Xie et al.
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(2017), Xu et al. (2014), etc. Besides, many researchers seek to understand the effect of RS on firms’ incentives to exert costly efforts, e.g., Bhaskaran and Krishnan (2009), Hsueh (2014), Heese and Kemahlıoğlu-Ziya (2016), Krishnan et al. (2004), etc. Furthermore, some researchers make a comparison between revenue-sharing contracts and the other contracts, e.g., Jin et al. (2015), Kouvelis and Zhao (2016), Palsule-Desai (2013), Wang and Shin (2015), Yang and Chen (2017), Yenipazarli (2017), Zhang et
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al. (2016), etc. Finally, many researchers attempt to study the effect of RS on aligning firms’ strategical behaviors, Cai et al. (2012), Kong et al. (2013), Yang et al, (2017), etc. To the best of our knowledge, our paper is the first to employ RS to coordinate firms’ behaviors under supplier encroachment.
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3. The basic model
We consider a supply chain system, consisting of a single retailer (labeled r) and a single supplier
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(labeled s), who can sell either through the retailer, her direct channel (e.g., her own online store or retail store), or both. Similarly to Arya et al. (2007) and Yoon (2016), we assume that the products sold through
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the retailer and direct channel are symmetric substitutes, i.e., the substitution degree of the retailer’s product over the supplier’s equals that of the latter over the former. The case of asymmetric substitutes
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will be examined in §5.1. Then, the consumer demand for the product can be captured by a linear and downward-sloping inverse demand function, pk
a ql
qk , k , l
r , s , and k
l , where pk and q k
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represent the market clearing price and the quantity of the product for firm-k, respectively, and a is the market size. The parameter
(0,1] measures the degree of substitution between the two channels’
products, thus the products become more homogenous and the retail market is more competitive as θ increases. Note that the two retail products turn out to be perfect substitutes when independent when function to be
approaches 1, and
approaches 0. Note that we have normalized the slope of this inverse demand
1 . This function has been widely used in the modeling literature on supplier
encroachment (e.g., Arya et al. 2007, Li et al. 2014, 2015, Yoon 2016). The supplier manufactures the product at a constant unit cost, which is normalized to zero.
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Compared with the supplier, the retailer typically enjoys an advantage in the sales process, which can stem from more direct contact with customers, superior knowledge of customer preferences, the cost advantage of bulk shipping, and economies of scope with other retailing activities. To model this advantage, we follow Arya et al. (2007) to normalize the retailer’s per-unit selling cost to zero, and denote the supplier’s per-unit direct selling cost as 𝑐 ∈ (0, 𝑎). Note that this assumption is consistent
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with some business practices in real world. For example, Kevin Mansell, Kohl’s Chief Executive, said online operating margins at Kohl’s are nearly 4%, less than half the 10% operating margins in its nearly 1,200 physical stores, and Target Corp. says its profit margins will fall as its online sales grow (Kapner 2014). Besides, this assumption has also been employed by recent research on supplier encroachment, e.g., Li et al. (2014, 2015), and Ha et al. (2016). This suggests that the retailer’s distribution channel is
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more efficient than the supplier’s direct channel. As a result, it may be in both firms’ favor to fulfill more demand by the retailer channel. To this end, the two firms can implement some incentive schemes to better align their behaviors, e.g., using revenue-sharing (RS) to share the sales revenue. As an endeavor to build a more realistic model, we use a bilateral bargaining framework to capture the strategic
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interactions between the supplier and retailer, where their interactions are modeled as a two stages game, as illustrated in Fig. 1.
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, then Two firms negotiate over Supplier determines her two firms’ profits are realized. for a given .
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s=1
s=2
Fig. 1. The sequence of events.
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The timeline is as below. (1) In the first stage, s
1 , the supplier negotiates bilaterally with her
retailer over ( w, qr ) using nonlinear pricing (NP), where w is the wholesale price that the supplier
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charges for each unit purchased, qr stands for the quantity that the retailer sells to the end customers, represents the rate of revenue that the retailer shares with his supplier per-unit sold. For ease of comparison, we follow Kong et al. (2013) to assume that
is exogenous. Nonetheless, we will analyze
the effect of RS on firms’ decisions and profits through some sensitivity analysis (see Proposition 1 and Theorem 1). The assumption that the two firms negotiate over w and qr is based on the concept of Nash bargaining solution (Nash 1950, 1953), which is analogous to Yoon (2016). If the negotiation breaks down, the retailer earns zero profit, and so does the supplier without encroachment, otherwise, she
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gains the monopolistic profit by selling only through her direct channel. (2) In the second stage, s
2,
the supplier chooses the quantity ( qs ) sold directly to the end consumers if a direct channel is introduced, and then both firms bring their entire order quantities to the market and their profits are realized. Note that the assumption that the retailer orders before the supplier is consistent with the fact that the supplier has no means to credibly commit to not adjust her own order quantity after receiving the
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retailer’s order. Nevertheless, we will discuss the alternative case where the supplier can make a credible commitment and place an order before the retailer in §5.2.
Furthermore, we assume that the two firms’ bargaining power under encroachment is identical to that under no-encroachment for the results to be comparable. Following Arya et al. (2007), we assume
paper are common knowledge to both firms.
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that the encroachment does not affect retail demand function. Finally, we assume all parameters in our
The notational scheme used in this work is as follows. In general, qxj *,k , wxj * , and
j* x ,k
, denote the
output, wholesale price and profit in equilibrium, respectively: x can be n (no-encroachment) or e (encroachment); k can be r (retailer), s (supplier), or t (total system); j can be N (without revenue-sharing) or RS * e ,r
denotes the retailer’s optimal profit with revenue-sharing
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RS (with revenue-sharing). For example,
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under encroachment. Additional notations will be introduced later.
4. Analysis
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In this section, we first consider a benchmark case without direct channel, then discuss the case in which the supplier sells directly to consumers. Finally, we compare the main results with the benchmark case.
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4.1. The no-encroachment case
In this subsection, we consider the no-encroachment case, that is, the firms can only reach consumers
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through the retailer’s channel, which serves as a benchmark. For convenience, we index this case by subscript n . We first examine a centralized system wherein a single decision-maker jointly manufactures and sells the products to the final customers, which serves as a first-best benchmark. We index this case by superscript C. In the centralized system under no-encroachment, a centralized firm chooses its output qnC to maximize the system’s profit max
which is maximized at qnC *
C n
(qnC )
C n
:
( a qnC )qnC ,
a 2 . Thus, the system’s profit under no-encroachment is
(1) C* n
a2 4 .
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We next study the decentralized supply chain system in which the two firms individually make decisions. Thus, the firms’ profits with revenue-sharing ( j j n ,r
)( a qnj ,r )qnj ,r
(1
j n ,s
RS ) or without ( j
( a qnj ,r )qnj ,r
N ) can be formulated as:
wnj qnj ,r ,
(2)
wnj qnj ,r ,
(3)
Consider the negotiation between the two firms over ( wnj , qnj ) for an exogenously given
. Then
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wnj and qnj , r can be derived by a generalized Nash bargaining (GNB) scheme. The GNB scheme serves
as a common methodology for deriving negotiation outcomes in bilateral monopoly settings, and it has been widely used in the modeling literature nowadays (e.g., Feng and Lu 2013a, 2013b, Leng et al. 2016, [0,1] denote the supplier’s bargaining power vis-à-vis her retailer, where
Wang et al. 2013, etc.). Let
1 ) indicates the retailer (the supplier) totally holds the bargaining power, whereas
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0 (
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represents the case of equal bargaining power. As a result, the cases where the supplier or the retailer acts as a Stackelberg leader, and both firms possess equal bargaining power are special cases of our work. With the assumption that both firms have zero reservation profit, the GNB scheme in this case is defined to solve the next optimization problem.
is the so-called Nash product.
(
j n ,s
) (
j 1 n ,r
)
,
(4)
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where
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max ( wnj , qnj ,r )
Substituting Eqs. (2) and (3) into Eq. (4), and performing the optimization in Eq. (4) provide the )a 2 . Then we can obtain the optimal quantity, qnj *,r
(
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optimal wholesale price wnj * two firms’ maximal profits,
j* n ,r
(1
)a2 4 and
a 2 , and the
a2 4 .
j* n ,s
j* n ,t
j* n ,r
j* n ,s
. Thus,
a2 4 . Interestingly, the relationships between the two firms’ profits suggest that the
j* n ,t
is allocated proportionally based on the players’ bargaining power. Besides, it is
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trade surplus
j* n ,t
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Denote the decentralized supply chain system’s profit under no-encroachment as
simple to see that qnj *,r
qnC * and
j* n ,t
C* n
, which implies that under bargaining, NP coupled with RS
can coordinate the supply chain under no-encroachment. To offer further insights, we explore the impact of key model inputs (i.e.,
and
) on the main
equilibrium outcomes, as the next proposition shows. Proposition 1. Under no-encroachment: (i) wnRS *
0,
qnRS,r*
(ii) wnj *
0,
qnj *,r
0, 0,
qnRS,t * qnj *,t
0, 0,
RS * n ,r j* n ,r
RS * n ,s
0, 0,
j* n ,s
RS * n ,t
0 , and 0 , and
j* n ,t
0; 0.
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Proposition 1 states that under no-encroachment: With a larger
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, the supplier charges less, which
implies that RS can alleviate double marginalization effect under no-encroachment. However, the retailer’s output and the firms’ profits are independent of
, which suggests that RS does not affect the
retailer’s output and two firms’ profits under no-encroachment. This is because the two firms can always rebalance their profits via adjusting the wholesale price according to
and
(see the expression of
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wnj * ). Besides, when the supplier is more powerful, the supplier charges more and earns more, but the
retailer earns less, which means that a more powerful supplier will charge more and earns more, while the retailer gains less. However, the retailer’s output is independent of
, which implies that
does
not affect the retailer’s output. Because they can reallocate the profit according to their respective bargaining power, which induces them to order at the first-best level. Finally, the decentralized system’s and
, as the system is coordinated
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output and profit under no-encroachment are independent of under this setting.
4.2. The encroachment case
Now, we consider the encroachment case where the supplier sells directly to the final consumers after
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supplying the wholesale product to her retailer. For convenience, we index this case by subscript e.
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We first survey a centralized system in which a centralized firm jointly manufactures and virtually decides on the outputs of the direct ( qCe, d ) and indirect ( qCe, i ) channels sold to the customers, which serves
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as a first-best benchmark. We index this case by superscript C. To facilitate comparison, we assume that the centralized firm’s direct and indirect sales costs are the same as those of the decentralized system.
C e
qCe , d
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Solving
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And then, the centralized firm’s problem under encroachment can be established by: max
0 and
C e
C e
(qCe ,d , qCe ,i )
qCe ,i
( a qCe ,i
qCe ,d )qCe ,i
(a
qCe ,i
qCe ,d
c )qCe ,d ,
(5)
0 provides the optimal outputs of the direct and indirect
channels for the centralized firm, then we obtain the centralized firm’s profit immediately. Lemma 1. In a centralized system: (i) When c qCe,*d
[(1
ceC (or, equivalently, )a c ] 2(1
system’s profit is (ii) When c
C* e
2
) and qCe,*i
[2(1
C e
), the centralized firm sells directly, then the optimal outputs are
[(1
)a
)( a c )a c 2 ] 4(1
ceC (or, equivalently,
C e
c ] 2(1 2
2
) , the total output is qCe,t*
C ) , where ce
(1
)a ,
C e
(2 a c ) 2(1
) , and the
1 c a;
), the centralized firm does not sell directly, and the optimal
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outcomes are identical to those of with no-encroachment case. We then survey the decentralized supply chain system under encroachment in which the two partners individually make their decisions. The equilibrium outcomes are determined by backward induction following the timeline described in §3. In the last stage, the supplier’s optimization problem (given ( wej , qej ,r ) and (qej,s )
( a qej,r
qej,s )qej,r
(a
qej,r
qej,s
c )qej,s
wej qej,r .
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j e ,s
max
) is: (6)
The first term in Eq. (6) is the supplier’s profit share of the retailer’s sales revenue, the second term is the supplier’s profit from her direct channel, and the last term is the supplier’s payments from selling the wholesale products. Performing the optimization in Eq. (6) delivers qej,s (qej,r ) [ a c (1
) .
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supplier’s best response output sold directly to final consumers, where
qej ,r ] 2 , the
In the first stage, anticipating the supplier’s response output qej ,s (qej ,r ) , the two firms adopt nonlinear pricing (NP) and negotiate over ( wej , qej ,r ) for a given max ( wej , qej ,r ) j e ,s
o s
is given in Eq. (6),
o s
ED j e ,r
) (
j 1 e ,r
)
,
(7)
is the supplier’s reservation profit if the negotiation breaks down, and
is given in the next equation:
j e ,r
j e ,s
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where
(
to maximize the Nash product, solving:
(1
qej ,r
)( a
wej qej ,r .
qej ,s )qej ,r
(8)
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When the negotiation breaks down, the supplier sells only through her direct channel and incurs marginal selling cost c, but the retailer is expelled from the market and makes no profit. If this is the case, o s
( a qso c )qso , which is maximized at qso
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the supplier’s profit becomes
reservation profit with the ability to encroach is
o s
( a c ) 2 . Thus, the supplier’s
( a c )2 4 .
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Performing the optimization in Eq. (7), we can obtain the main equilibrium outcomes when the supplier holds the ability to encroach. Lemma 2. In a decentralized system: (i) When c wej *
(1
qej ,*r
2[(1
cej (or, equivalently, 2 ) (a c) 2 )a
profits are
j* e ,r
3
2
2
c] (1
.
j
e
j
), the supplier sells directly, and her optimal wholesale price is
)( a qej ,*r ) (1
(
, and qej,*s
{[4 (1
)[(1
c ]2
)a
j
2
and
)[1
) ] 2 qej ,*r 4 , the outputs of two firms are
(3
) 2 ]( a c ) 2 a} 2 j* e ,s
[(1
)a
j
, respectively, and the two firms’ maximal c ]2
j
o s
, respectively, where
j
4
2
,
ACCEPTED MANUSCRIPT Submitted to European Journal of Operational Research (2017) 1–32
(ii) When cej
cnj (or, equivalently,
c
e
j
j n
), the supplier opens an inactive direct channel and sells
nothing directly, her optimal wholesale price is wej * output is qej ,*r j* e ,s
j* e ,r
(1
o s
)
[(1
)a
j* e ,r
) ]( a c) , and the retailer’s
) 4 ( (1
)( a c )[4 a (4
2
)( a c )] 4
2
and
.
cnj (or, equivalently,
(iii) When c
(
, the two firms’ maximal profits are
(a c)
13
j n
), the supplier will not encroach, and the optimal outcomes are
CR IP T
identical to those of with no-encroachment case. The above result reveals that the supplier will directly sell the product to consumers (i.e., qej ,*s when her direct selling cost (c) is sufficiently small or the degree of product substitution ( sufficiently low. When c or
) is
is intermediate, the supplier opens an inactive direct channel (i.e.,
0 ) and no direct sales occur. Finally, the supplier does not encroach when c or
is sufficiently
AN US
qej ,*s
0)
high. This is because as c increases, the supplier’s retail disadvantage increases, which makes her direct channel become much inefficient, thus inducing her to diminish the use of direct channel. Particularly, when c is intermediate, the supplier refers all customers to the retailer channel for demand fulfillment. When c is sufficiently great, the direct channel is not a viable option for the supplier. On the other hand, increases, the retail market competition becomes more intense, which makes the supplier’s retail
M
as
ED
disadvantage become more significant, thus leading to the result identical to that of an increased c. Furthermore, we can observe from the expression of
j* e ,s
that the supplier owning the encroaching 1 and c
0.
PT
ability earns more when using dual-channel as compared to the direct channel, unless
This observation indicates that a supplier with the ability to encroach would not expel her retailer from
CE
the retail market, unless the two channels’ products are perfect substitutes and the supplier is as efficient as the retailer in the sales process.
qej,*t
AC
Denote the supply chain system’s output and profit under NP in encroachment as q ej ,*t and {[4(1
surplus (
) (1
j* e ,t
o s
)2
2
]( a c ) 2(2
)a} 2
j
, and
j* e ,t
[(1
c ]2
)a
j
o s
j* e ,t
. Thus,
. Similarly, the net trade
) is allocated proportionally based on the players’ bargaining power.
The next theorem gives more insights about the effect of key model inputs on the main equilibrium results under encroachment. Theorem 1. Under encroachment ( c (0, cej ) ): (i) qej,*r j* e ,s
c c
0;
qej ,*s
c
0;
0
if 0
c
cIj ,
j* e ,r j* e ,s
c c
0;
when
0
if cIj
1
c
,
cej ;
j* e ,s
c
0
for all c (0, cej ) ; when
1
1,
ACCEPTED MANUSCRIPT Submitted to European Journal of Operational Research (2017) 1–32
(ii) qej,*r
0,
(iii) qeRS,r * 0
0,
qej ,*s
qeRS,s *
0 , 2( 2
1)
j* e ,r
,
0
c1j :
(iv) When c
qej,*r
0 and
j* e ,r j r ,1
e
and
j
, but
(0,
j t ,1
e
j
e
j
(0,
e
j
qej,*s
) , but
(0,
0 for all
2( 2
0 ; however, when
RS * e ,t
1)
1
,
* qeRS ,t
if
0
0
,
1
0;
0 for all
qej ,*s
0 , and
RS * e ,s
; when
ceRS
0 for all j* e ,s
0 if
j* e ,s
0 ,
for all
0 otherwise. Furthermore,
qeRS,t *
0;
j* e ,s
RS * e ,r
0 ,
* qeRS ,t
and
0
e
j
) ; when c
),
c1j : 0 and
j* e ,r
cej
. Furthermore,
j s ,1
0 if
0.
,
qej ,*s 0 if
qej ,*r
j s ,1
0 if j r ,1
,
0 if
j* e ,s
j t ,1
,
e
j
,
0 if
qej ,*r
CR IP T
14
j* e ,r
0
Theorem 1 declares that under bargaining with encroachment using NP coupled with RS, the retailer’s
AN US
profit increases in c, because an increase in c can enlarge the retailer’s output and reduce the supplier’s reservation profit, both of which promote the net trade surplus that can improve the retailer’s profit. Surprisingly, the supplier’s profit increases in c when
is large and c is intermediate. This can be
explained as follow. An increase in c exerts two opposite effects on the supplier’s profit (see the j* e ,s
expression of
j* e ,t
in Lemma 2): On the one hand, it increases the net trade surplus (as indicated by the o s
), as they will expand the employment of the retailer channel and reduce the use
M
expression of
ED
of the direct channel when c enlarges (see Theorem 1 (i)); on the other hand, it decreases the supplier’s reservation profit, as the expression of
o s
indicated. The first effect dominates the latter when
is
PT
large and c is intermediate, the situation reverses otherwise, thus resulting in the above results. These findings suggest that the further increase of the supplier’s retail disadvantage can improve both firms’
CE
profits when her disadvantage is intermediate and her bargaining power is strong. We illustrate these results in Fig. 2.
AC
120 j* e,k
13
j* e ,s
q eRS,t *
80
12
40
11 j* e ,r
0 0
cI j
2
Fig. 2. Impact of ( , ,
on ,
j 4 ce
and
c
10 0.0
0.5
1
1.0
Fig. 3. Impact of
.
)
(
,
on
,
.
)
ACCEPTED MANUSCRIPT Submitted to European Journal of Operational Research (2017) 1–32
15
Furthermore, the two firms’ outputs under encroachment are independent of
. Because it is in both
firms’ interests to enlarge the overall pie that can be divided according to their bargaining power. As expected, the retailer’s profit decreases in firms’ profits increase in
, while the supplier’s profit increases in
. Because the two firms’ interests are better aligned as
. Moreover, both increases, which
can induce them to shift more demand to a more efficient channel (retailer channel), thereby yielding a
CR IP T
larger net trade surplus that improves both firms’ profits. This finding suggests that each firm will set a as large as possible (up to 1 in our context) when retailer. Yet, the total output decreases in decreases in
can be determined by either the supplier or
when
is small; otherwise, it first increases then
. An explanation is that the supplier becomes cooperative when the retail market
competition is mild, which prompts her to cut down her direct quantity as
increases; otherwise, as is
AN US
increases, the supplier would act aggressively to slightly lower down her direct output when
small, but to rapidly reduce her output otherwise. We illustrate these results in Fig.3. Moreover, ceRS decreases in 7
, which indicates that RS reduces the supplier’s incentive to sell directly. 7
q ej ,*s
5
M
q ej ,*k
0.8
PT
3 0.85
j s ,1
0.9
(
AC
Fig. 4. Impact of
90
j* e,k
6
5
4
1
0.8
0.85
0.9
(b)
)
CE
(a)
0.95
q ej ,*s
q ej ,*r
ED
q ej ,*r
q ej ,*k
on q ej ,*r and q ej ,*s . ( a
j* e ,s
70
20 ,
0.95
(
0.6 ,
1
)
0.5 )
j* e ,s
j* e,k
60
40 30 j* e ,r
j* e ,r
0
10 0.8
0.9
(a)
(
1
)
j t ,1
0.8
(b)
0.9
e
(
)
j
1
ACCEPTED MANUSCRIPT 16
Submitted to European Journal of Operational Research (2017) 1–32
Fig. 5. Impact of
on
j* e ,r
and
j* e ,s
.(a
What’s more, both firms’ profits always decrease in c is high and
20 ,
0.5 ,
0.2 )
when c is low, but they increase in
when
is intermediate. This can be explained as below. When the retail market competition is
mild (with a small
), both firms reduce their channels’ outputs to restrict over-competition as
increases (see Fig. 4); otherwise, as
increases, they would reduce the retailer channel’s output to
CR IP T
prevent excessive competition if the supplier turns out to be an efficient competitor, which in turn leaves a broad market space for the supplier, thus stimulating the supplier to rise her output (see Fig. 4); however, if the supplier’s retail disadvantage is marked, the supplier would cut down the direct sales quantity in order not to diminish unduly the efficient (retailer) channel’s demand, which can induce the
AN US
retailer to enlarge his output (see Fig. 4), thereby enhancing the net trade surplus that can secure Pareto improvement for the two firms. These unexpected results suggest that, when the supplier’s retail disadvantage is moderate, further increase of the product substitution can benefit both firms when the product substitution is intermediate. We illustrate the above results in Fig. 5.
M
Unsurprisingly, a great product substitution dulls the supplier’s incentive to encroach.
encroachment immediately.
ED
From Theorem 1(iii), we can see the impact of RS on the two firms’ decisions and profits under
Corollary 1. In a decentralized system under encroachment: qeRS,r * N* e ,t
; however, when
2( 2
qeN,t* otherwise; furthermore, ceRS
RS *
1) , qe ,t
qeN,t* for all
; when
qeN,s* , 2( 2
RS * e ,r
N* e ,r
RS *
1) , qe ,t
,
RS * e ,s
q eN,t* if
N* e ,s
2
ceN .
CE
qeRS,t *
RS * e ,t
PT
and
qeN,r* , qeRS,s *
Corollary 1 says that under encroachment with bargaining using NP, RS induces the two firms to
AC
expand the use of the efficient indirect channel but contract the employment of the direct channel, and it improves the two firms’ and the total system’s profits. However, when the retail market competition is mild, RS always reduces the total system’s output; otherwise, RS enlarges it when reduces it when
is small, RS
is large. Finally, RS limits the supplier’s incentive to encroach, which highlights the
key role of RS in better aligning different players’ behaviors. Comparing the results under encroachment between the decentralized and centralized systems provides the next result. Theorem 2. Under encroachment: cej
ceC , qej ,*r
qCe ,*i , qej ,*s
qCe ,*d , qej ,*t
qCe ,t* , and
j* e ,t
C* e ,t
, if and only if
,
ACCEPTED MANUSCRIPT Submitted to European Journal of Operational Research (2017) 1–32
1 , the above equations hold, wherein we
RS *
(1
)[(1
)a
c] 2
17
0.
The above result uncovers that when the two firms participate in negotiation using NP under encroachment, the system is perfectly coordinated when the supplier owns all the retailer’s sales revenue. Because in this case, the two firms’ interests are perfectly aligned, therefore they would order at the first-best levels. Note that when
1 , the negotiated wholesale price is negative, which implies that
CR IP T
the supplier subsidizes the retailer’s acquisition of product (according to his bargaining power) through a negative wholesale price. This finding implies that NP coupled with RS can perfectly coordinate the supply chain under encroachment when the supplier seizes all the retailer’s sales revenue. This novel result contributes to the growing literature on supply chain coordination.
AN US
4.3. No-encroachment vs. encroachment
In §4.1 and §4.2, we have explored the profitability of the two firms under no-encroachment and encroachment, respectively. In this subsection, we will investigate the impact of supplier encroachment on two firms’ and the system’s profit. To facilitate the comparison, we assume that
and
in the case
of encroachment are identical to those of no-encroachment.
M
Following the previous academic research, one might expect that the retailer remains better off from
ED
supplier encroachment when the supplier’s downstream disadvantage is pronounced in our context. This logic, however, misses a key point: The supplier can refrain from encroachment when her
PT
disadvantage is sufficiently marked. We justify this conjecture in the next theorem. cej ):
Theorem 3. Under encroachment ( 0 c qnj *,r when 0
CE
(i) For the retailer: qej ,*r
(ii) For the supplier: (a) When c
csj,1
or csj,2
AC
0
j* e ,s
j* n ,s
if csj,1
c
c
cej ,
c 2
j* e ,s
c1 , qej ,*r
,
j* e ,s
j* n ,s
if csj,1
cej . Especially, when
j* n ,s
qnj *,r when c1
cej ;
c
j* e ,r
j* n ,r
for all c (0, cej ) ; (b) when c
1 and
csj,2 ; (c) when
1 , csj,1
0 and
3
,
j* e ,s
for all c (0, cej ) ;
2 j* e ,s j* n ,s
3 j* n ,s
,
j* e ,s
if 0
j* n ,s
c
if
csj,1 ,
for all c (0, cej ) .
Theorem 3 declares that under bargaining using NP coupled with RS, the retailer orders more under encroachment than under no-encroachment when c is sufficiently large, the situation is reversed when c is sufficiently small. Surprisingly, the retailer always hurts when the supplier encroaches, regardless of both his bargaining power and the revenue share. This is because on the one hand, the supplier’s ability to encroach on the reselling channel endows her with a substantial reservation profit when the negotiation
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Submitted to European Journal of Operational Research (2017) 1–32
breaks down, which diminishes the net trade surplus, thereby ultimately hurting the retailer;1 on the other hand, the supplier’s incentive to encroach is weaker when both firms participate in bilateral negotiate when using NP as compared to the wholesale price contract (see Aray et al. 2007).2 This finding makes an amazing contrast to previous finding based on the supplier-led Stackelberg game, which declares that supplier encroachment mostly can also make the retailer better off (e.g., Arya et al.
CR IP T
2007, Cattani 2006, Chiang et al. 2003, Tsay and Agrawal 2004). Yet, the case for the supplier depends on her bargaining power: When always benefits from encroachment; when her direct selling cost is small or large; when
is small, the supplier
is intermediate, she is better off from encroachment when is large, she gains from encroachment only when her
direct selling cost is small. These findings can be explained as follows. The supplier’s profit under
AN US
encroachment is composed of her gain from the net trade surplus and her reservation profit. When her bargaining power is small, her gain from the net trade surplus under both encroachment case and no-encroachment case are limited, which can be dominated by her reservation profit. Hence, her profit under encroachment mainly depends on her reservation profit when her bargaining power is small. Yet,
M
when her bargaining power is large, her gain from the net trade surplus and her reservation profit are very comparable, both of which are oppositely affected by an increased c, as discussed in Theorem 1(i). These
ED
results reveal that the supplier is always better off from opening a direct channel when her bargaining power is small, while she benefits from it when her direct selling cost is sufficiently limited or
PT
pronounced when her bargaining power is intermediate, whereas she gains from it only if her direct
CE
selling cost is sufficiently small when her bargaining power is large. In short, a less powerful supplier is more likely to benefit from encroachment. Particularly, when the supplier holds all the bargaining
AC
power and the two channels’ products are perfect substitutes, the supplier always hurts from encroachment, because in this case, the supplier can fully control the retailer channel, including the wholesale price, output and the resulting retail price. As a result, the supplier cannot earn any additional profit from the introduction of a direct channel when the two channels’ products are perfect substitutes.
1
One can use the methodology employed in the preceding subsections to verify that the retailer can gain from encroachment when the supplier’s direct selling cost is large if the supplier also has zero reservation profit when the negotiation breaks down.
2
When the two firms negotiate over a wholesale price by GNB, Aray et al. (2007) show that encroachment can produce Pareto gains whenever the supplier’s bargaining power is sufficiently pronounced.
ACCEPTED MANUSCRIPT Submitted to European Journal of Operational Research (2017) 1–32
19
This result is akin to Li et al. (2015). The above findings indicate that supplier encroachment can lead to a win-lose or a lose-lose outcome for the supplier and retailer when the two firms engage in negotiation using NP coupled with RS. These results make an interesting contrast to that derived by Yoon (2016), who use the similar framework to show that encroachment can still create Pareto gains when the product substitution is sufficiently small
CR IP T
if the supplier can make a cost-reducing investment. Intuitively, the threat of launching a direct channel seems to be a boon for the supplier. However, this threat can induce the retailer to significantly upward distort his order quantity, which can in turn make the supplier worse off. We justify this guesswork in the next theorem. Theorem 4. When the supplier opens an inactive direct channel ( cej qnj *,r and
(ii) when csj,3
j* e ,t
,
j* n ,r
j* e ,s
j* n ,s
for all c [ cej , cnj ] ;
for all c [ cej , cnj ] ; when
cnj ;
c
0 and
j* e ,t
j* n ,t
.
4
,
j* e ,s
j* n ,s
if cej
c
csj,3 ,
j* e ,s
j* n ,s
if
M
(iii)
c
4
j* e ,r
cnj ):
AN US
(i) qej ,*r
c
The above results tell that when the supplier launches an inactive direct channel, the retailer orders
ED
more but earns less than under the no-encroachment case. This is because the threat of selling directly can force the retailer to order more, and the ability to encroach endows the supplier with a substantial
PT
reservation profit, which reduces the retailer’s profitability in the negotiation. This finding shows that the retailer always suffers from an inactive direct channel developed by the supplier. However, the supplier
CE
always benefits from introducing an inactive direct channel when her bargaining power is weak; otherwise, she hurts from it when her cost is relatively small. This is because, as discussed in Theorem 3,
AC
the supplier’s profit includes two parts: One from the net trade surplus and the other from her reservation profit. As c increases, her threat to sell directly is reduced, which mitigates the retailer’s order quantity distortion that can enhance system profit and augment the net trade surplus; on the other hand, the supplier’s reservation profit decreases in c. The opposite effects of the two gains on the supplier’s profitability resemble those of the case under encroachment, as discussed in Theorem 3. This finding uncovers that a less powerful supplier with the ability to encroach always gains from the threat to sell directly to the customers, but the situation for a powerful supplier is reversed when her retail disadvantage is not too pronounced. Furthermore, the system’s profit is smaller with an inactive direct
ACCEPTED MANUSCRIPT 20
Submitted to European Journal of Operational Research (2017) 1–32
channel than that under no-encroachment, because the supplier’s threat to sell directly can prompt the retailer to overorder, which can reduce the system’s performance. This finding indicates that an inactive direct channel launched by the supplier can reduce the system’s profit. Although the retailer always hurts from supplier encroachment, encroachment can enhance the system’s output and profit, as the next proposition confirms.
when cbj
c
cej or
qnj *,t for all c j b
j e
[0, cnj ) ; (ii)
. Particularly, cbj
j* e ,t
when 0 c
j* n ,t
0 when
1.
cbj or 0
j b
; and
CR IP T
Proposition 2. (i) qej ,*t
j* e ,t
j* n ,t
As expected, the above proposition verifies that the supplier’s ability to encroach can enlarge the system’s total output, even when she introduces an inactive direct channel. Besides, the system benefits cbj ) or the
AN US
from supplier encroachment when the supplier’s direct selling cost is sufficiently small ( c degree of product substitution is sufficiently low (
j
b
). Moreover, the threshold of supplier’s selling
cost ( cbj ) from which the system gains more from encroachment equals zero when the two channels’ products are perfect substitutes. This result implies that when the two firms use NP coupled with RS in
M
negotiation, the system’s benefit from encroachment disappears once the two channels’ products are perfect substitutes. This finding is like Li et al. (2015), but we expand their result deriving from a
ED
supplier-led Stackelberg game to a negotiation game with RS.
5. Extensions
PT
In this section, we consider two extensions of the basic model analyzed in §4 by altering some assumptions to further assess the robustness of the main results of the basic model. The analysis in this
CE
section confirms that the retailer always hurts from supplier encroachment under bargaining when the two firms use NP coupled with RS, even when consumers view the retailer’s products as superior to
AC
those of the direct channel, or when the supplier can credibly precommit to her quantity sold directly before the negotiation.
5.1. Asymmetric substitutes In the basic model, we assumed that the products sold through the retailer and direct channel are symmetric substitutes. However, a direct marketer typically provides the final consumers with only a virtual description of the product, but lacking of the use of smell, taste, touch, which can lead to evaluation mistakes by shoppers. Although the product can be returned after a mistaken purchase, the
ACCEPTED MANUSCRIPT Submitted to European Journal of Operational Research (2017) 1–32
21
return process can be troublesome, thereby reducing the consumers’ expectation of consumption value. Besides, the consumer will typically be required to wait several days for delivery when purchasing the product from a direct channel. Furthermore, the postsale service from a direct channel is less convenient than that from a retailer’s store, as the direct seller is usually located at a distance. Therefore, the products sold through the retailer’s store can be regarded as superior to those of a direct channel. To
CR IP T
reflect this element, we assume that the formers are perfect substitutes for the latters, but the reverse is not true. Then, the two firms’ profits under encroachment are: j e ,r
)( a qej,r
(1
j e ,s
( a qej,r
wej qej,r .
qej,s )qej,r
qej,s )qej,r
( a qej,r
qej,s
wej qej,r .
c )qej,s
AN US
where the “→” reflects the asymmetric substitutes setting.
(9) (10)
Employing the methodology developed in §4.2, we can obtain the next lemma. Lemma 3. In a decentralized system under asymmetric substitutes:
(1
(or, equivalently,
)( a c ) 2
of two firms are qej,*r
)[2 a ( a c qej,*r ) ] 2 (1
(
[(1
maximal profits are
)a (1
j* e ,r
(1
respectively,
j
4( a
)c ]
)[(1
j
)a
, and qej,*s
2c ) { [ a c
(a
c ) ]2
8(1
4 (1
)
,
(ii) When cej
c
cnj (or, equivalently,
CE
AC j* e ,r
j n
(1
( a 2c)
(iii) When c
j
and
j* e ,s
) ( a c )( a
2c )
)[4c
2
c (1
a c
(a
j
(1
)
2
2
]qej,*r 4 , the outputs
, respectively, and the two firms’
[(1
)a
a 2 a
c) } ,
(1
j
)c ]2 4
( a c )2 4 ,
j
j
, 4
(1
)2
(1
)2
2
,
.
1
output
1 3
cej
j e
j n
), the supplier opens an inactive direct channel and sells
nothing directly, her optimal wholesale price is wej * retailer’s
)
[ j ( a c ) 2 a] 2
)c]2 4
(1
2
)[2(2
where
PT
j e
), the supplier sells directly, her wholesale price is
j e
M
wej *
cej
c
ED
(i) When
is
qej,*r
,
(a c)
)2 a]( a c ) 4
2
and
the j* e ,s
(
)a
two j* e ,r
(1
[(1
) 4 (
firms’
) ]( a c) , and the
maximal
) ( a c )2 4 , where cnj
profits (1
are )a 2 ,
a.
cnj (or, equivalently,
j n
), the supplier will not encroach, and the optimal outcomes are
identical to those of with no-encroachment case. The above result resembles that under the case of symmetric substitutes. Theorem 5. Under asymmetric substitutes:
ACCEPTED MANUSCRIPT 22
Submitted to European Journal of Operational Research (2017) 1–32
j* e ,r
for all c (0, cej ) ;
j* n ,r
(ii) For the supplier: (a) When 0 j* e ,s
csj,1
or csj,2
csj,1
c
if
j* n ,s
a [2(1
)2
[(1
) 2
cej ,
c csj,1
j* e ,s
a[
)2 ]
)(1
,
j* e ,s
where j
for all c (0, cej ) ; (b) when
j* n ,s
if csj,1
j* n ,s
,
cej
c
]
(1
4
(1
4
,
)2 ]
(1
csj,2 ; (c) when
c
. Especially, when
j
,
a [2(1
)
1 and
(0, cej ) .
c
)2
2 ) (1 csj,2
5
4
,
5
j* e ,s
j* e ,s
if 0
j* n ,s
]a [ 0 and
j
(1
j* n ,s
2
]
j* n ,s
, ,
)2 ]
j* e ,s
if
csj,1 ,
c
)2
4 [4 (1
5
1 , csj,1
,
for all
CR IP T
(i) For the retailer:
The above result confirms that the retailer always hurts from supplier encroachment even when consumers view the retailer’s products as superior to those of the direct channel. Besides, the supplier is small. However, when
AN US
always benefits from encroachment when
encroachment profits her when her direct selling cost is small or large. When
is intermediate,
is large, encroachment
benefits her only when her direct selling cost is small. These findings are consistent with those under symmetric substitutes.
j
j* e ,r
j* n ,r
for all c [cej , cnj ] ;
(ii) When csj,3
c
6
cnj ,
,
where
j* e ,s
for all c [cej , cnj ] ; when
j* n ,s
6
2
ED
(i)
M
Theorem 6. When the supplier opens an inactive direct channel under asymmetric substitutes ( ce
(4
2
) , csj,3
a
[2
6
(1
,
j* e ,s
) ] a [4
j* n ,s
(1
if cej )
2
c
csj,3 ,
c
j* e ,s
cnj ):
j* n ,s
if
].
PT
The above theorem affirms that when the supplier’s direct selling cost is intermediate, the retailer
CE
always hurts even if the supplier launches an inactive direct channel. In addition, the supplier always benefits from introducing an inactive direct channel when her bargaining power is weak; otherwise, she
AC
benefits from it when her selling cost is sufficiently large, she hurts from it when her cost is relatively small. These findings also like those under symmetric substitutes.
5.2. Supplier ordering first In this extension, we discuss a new timeline wherein the supplier can credibly precommit to her own output quantity sold directly, thereby choosing her direct channel’s output before the negotiation. We use “^” to reflect the setting where the supplier orders first. In this setting, the original sequence of events is replaced by the following: First, the supplier ˆ ej , qˆ ej ,r ) with an announces her direct channel’s output, qˆ ej ,s ; second, the two firms negotiate over ( w
ACCEPTED MANUSCRIPT Submitted to European Journal of Operational Research (2017) 1–32
exogenously given
23
. Using the methodology employed in §4.2, we can get the next lemma.
Lemma 4. In a decentralized system when the supplier orders first: 1:
ˆ ej * w
[(
[(1
)a c ] 2(1
ˆ
j* e ,s
) ( a c )] 2 , the two firms’ outputs are qˆ ej,*r
)a (1
[(1
)a
ˆ j , the supplier sells directly, and her optimal wholesale price is e
cˆej or
(a) if c
2
) , and the two firms’ maximal profits are
c ]2 4(1
2
)
(a
c )2 4 ; (b) if c
cˆej * or
ˆ
j* e ,s
(2 a
(ii) When
c )c 4
(a
(1
)[(1
(1
)a
2
) and qˆ ej ,*s
c ]2 4(1
2
) and
)a 2 , and the
)( a c )2 2 a (
a 2 , the two firms’ maximal profits are
c )2 4 .
c ] 2(1
ˆ
j* e ,r
(1
)(2 a
c )c 4
and
1 , the supplier also opens an inactive direct channel and sells nothing directly, and the
AN US
ˆ
qˆ ej ,*r
j* e ,r
)a
ˆ j * , the supplier opens an inactive direct e
channel and sells nothing directly, her optimal wholesale price is wˆ ej * retailer’s output is
[(1
CR IP T
(i) When 0
equilibrium outcomes are identical to those of (i)(b).
The above results are like those of the original model. Theorem 7. ˆ ej *,r
j* n ,r
and ˆ ej *,s
j* n ,s
for all c (0, a) and all
(0,1) .
M
The above theorem states that when the supplier orders first, the retailer always hurts when the
ED
supplier owns the ability to encroach, even when the supplier introduces an inactive direct channel. This result is identical to that of the original model. However, the supplier always benefits from her ability to sell direct, even when she launches an inactive direct channel. This finding makes an interesting contrast
PT
to that of the original model where the supplier places an order after the retailer, because the option to
CE
order first endows the supplier with a priority to move, which can translate into a benefit to her.
6. Conclusion
AC
In this paper, we attempt to further understand the impact of nonlinear pricing (NP) on supplier encroachment in a linear demand setting. We consider a supply chain system, consisting of a single retailer and a single supplier, who can sell a product either through the retailer, her direct channel, or both. The two channels’ products are imperfect substitutes. To reach a trade agreement, the two firms engage in bilaterally negotiation to bargain over the wholesale price and the quantity using NP. To better align their behaviors, they can implement revenue-sharing (RS) to share the retailer’s sales revenue. Based on our analytical results, we get the following novel insights:
ACCEPTED MANUSCRIPT 24
Submitted to European Journal of Operational Research (2017) 1–32
Further increase of the supplier’s direct selling cost can improve both firms’ profits when the cost is intermediate and her bargaining power is strong, and further increase of substitution degree can benefit both firms when the degree is intermediate and the selling cost is large. NP coupled with RS coordinates the supply chain when the supplier extracts all the retailer’s sales revenue. In what follows, the supplier subsidizes the retailer’s acquisition of product through a
CR IP T
negative wholesale price. This novel result contributes to the literature on supply chain coordination. Contrary to previous research, the retailer always hurts from the supplier’s ability to encroach, even when an inactive direct channel is introduced.
A less powerful supplier is more likely to benefit from encroachment, but a powerful supplier is worse off from adding an inactive direct channel when her retail disadvantage is limited.
AN US
In addition, the key insights of our paper remain valid even after altering some assumptions. There are some limitations to our study. First, throughout this paper, there is only a monopolist supplier in the upstream market. Yet, it is likely to observe horizontal competitions between suppliers. Second, as in Yoon (2016), we do not consider the additional investment cost to make the two channels’
M
products become imperfect substitutes. Third, the supplier can offer products of different quality levels or being horizontally differentiated for the two channels (see Ha et al. 2016). Finally, we assume that all
ED
the information is perfectly symmetric between the firms in our study, but in practice, the supplier can be more knowledgeable about her direct selling cost (see Çakanyıldırım et al. 2012), whereas the retailer
PT
is likely more informed about the demand (see Li et al. 2014, 2015). These complexities will irrefutably
CE
affect the interactions between the two firms and thereby altering the role played by NP, bargaining power and RS in our current setting. It would be interesting to incorporate these factors into our model
AC
in the future.
Acknowledgments The authors thank Robert Graham Dyson (the Editor), and two anonymous referees for their most diligent and constructive suggestions and comments that have considerably improved the presentation and contents of the paper. The authors also gratefully acknowledge the second author, Jianwen Luo, as the corresponding author of the paper. The authors are partially supported by the Natural Science Foundation of China [71372107, 71772122, 71771146 and 71431004].
ACCEPTED MANUSCRIPT Submitted to European Journal of Operational Research (2017) 1–32
25
Appendix: Proofs The proofs of Proposition 1 and Lemma 1 are straightforward, and omit the others for brevity. Proof of Lemma 2: qej ,r ) 2
(a c
condition (FOC) jointly yields qej,*r
j e ,s
2 a
. To ensure qej,*r
0 , where
4 [(1
)a
j
4
]
2
(1
)
(a c) 2
j
c]
and wej *
x( )
4( a c ) [ 2
4(1
cej
, we need c
. Note that
or
)2 a2
(1
)a
(1
2 ) (a
c) 2
(
(1
)( a c )2 , thus qej ,*r
(a
a
j
2 a
, or x( ) [4 (1
0 , then let x( ) )a]
0
c ) (1
)
(2) We then consider qej ,s (qej ,r ) 0 , which implies that qej,r
if
e
(a c)
j
j e ,r
2
0 , we obtain
(which
is
(a c)
( wej )
4 (
)
values, i.e.,
Proof of Theorem 1: 2
0 ; Let
cIj
[
j* e ,s
j
(c )
4(1
AC
c
j
(a) cIj
cej
cIj
(b) 0
0;
qej ,*s
CE
c
cej
wej
(1
j n
2)a , or
e
j
j n
j
c
2
(a c)
]a (4
1
, where
j
2
1
)
j
0,
j
where
4
)
(1
)2
j* e ,s
2
1,
1
(c )
]a (4 1
j* e ,s
)2
(1
) 2 , then, when
1 , thus when
1
(a c)
where
a 2,
qnj *,r
)a .
wej *
(
)a
cej ,
c
or for
.
[4(1
(1
j
; then substituting q ej ,s and q ej ,*r into Eq.
and solving the FOC yields
be a quadratic function of c , then )
e
2( a c ) (1
]( a c ) , the above results are hold for the rest range of c values, i.e., cnj
the rest range of
(1) qej,*r
with respect to
PT
)
which implies that qej,*r
) 2 ]( a c )
.
ED
(7) and differentiating [(1
0,
M
(3) Finally, we consider qej ,s (qej ,r )
and
. In this case, the encroachment case does not
cnj
, which can be rewritten as c
0
rejected),
exist, thus the supplier’s optimization problem is the same as the no-encroachment case, then qej ,r thus, we have a 2
qej ,*r )
)( a
qej ,*r 4 , the other two group solutions are rejected as both can lead to
) ] c )2 4
(a
2[(1
AN US
(3
, then substituting qej ,s (qej ,r )
(a c)
( wej , qej ,r ) with respect to wej and q ej ,r , respectively, and solving the first-order
into Eq. (7) and differentiating
[1
0 , which implies that qej ,r
CR IP T
(1) We first consider qej,s (qej,r )
c
2
j* e ,r
;
c
2(1
) [(1
)a
j
c]
goes upwards and its symmetrical axis is
2
j
,
j* e ,s
0
)
0 , thus, c
if 0
for all c (0, cej ) ;
0
c
cIj ;
j* e ,s
c
0
if cIj
cej .
c
(2) The proof of part (ii) is easy and hence is omitted. (3) qeRS,r * (1
)(1
Moreover,
4(1
)[(1
)[(1
)a
qeRS,t *
a quadratic function of
)a
c ]2 [(1
2
c]
( j )2 )a
2
(
j 2
)
0, c ] z( ) (
0
,
RS * e ,s RS 2
qeRS,s *
2
(4 RS * e ,r
) , where z( )
)[(1
(1 2
) 2
c ] ( j )2
)a
RS * e ,t
0, 2(2
)
, then z( ) goes upwards and its symmetrical axis is
RS * e ,r
0 , RS * e ,r
2
4(1 0
(1
2 )
0.
) . Let z( ) be
(2
)
1 , thus
ACCEPTED MANUSCRIPT 26
Submitted to European Journal of Operational Research (2017) 1–32
[0,1) . Note that z(
(a) when z( (b) when z( 0
0 , i.e., 0
0)
1
2( 2
0 , i.e., 2( 2
0)
, z( ) 0 when
(the other root of z( )
0
ceRS
(4) (a) qej,*r
a
j r ,1
4( a
c) ( a
otherwise, j r ,1
e
it
j
(b) qej,*s qej,*s
[8( a
c)
2
(4
* qeRS ,t
)
)
roots,
)a] (
j 2
)
)
which
[ a2
]2
)a 2
c) ( a
qej ,*s
. Let
c )2
2(1 j t ,1
where
)[(1
)a
j* e ,s
2[4 (1
i.e.,
[2
1
c1j , where c1j
j t ,1
j* e ,r
2
qej ,*r
2 2(1
)]
e
j
for
c1j , and its roots are
j s ,1
e
j
j t ,1
(0,
all
e
j
a2
)
4( a c )2 ]
.
Note
;
that
4( a c )2 .
, then it goes downwards, and
j s ,1
2 a [2( a
c)
]
j s ,2
and
a2 ; otherwise, it has no real roots, which means
( a)2
c )2
c1j ,
[ a2
[4( a
c )2
)a 2
(1
c ) ]2
2( a
2( a
c)
a2 . j* e ,r
e
j
0 if
j t ,1
j* e ,r
,
0
j t ,1
if
,
c1j .
if c
, as their expressions reveals, thus is omitted for brevity. j
) 2] a
if c
0
, then it
2)a , and its roots are
a , which is always hold when
4( a c )] ( j )2 , then
a
(1
be a quadratic function of
M
) . Note that
c ][
like
CE
cej
Finally,
j
4( a c ) a . Note that
(d) The case of
such that z( ) 0 when
(which is rejected), where
means
)a2 ] , which is always hold when 4( a
(1
j* e ,r
e
PT
[4( a
(0,
0 , Thus,
[0,1) ; 1
0,
)
be a quadratic function of
4( a c )2
1 (which is rejected), where
0 for all
1
4( a c )2 ]
ED
qej ,*s
qej ,*r
. Let
0 has real roots if and only if c ]
solves z( )
1
0. j 2
c )} (
4( a
real
(4
0 for all
2
4(1
holds.
RS 2
) a(
)( a
j r ,2
no 2
0)
z(
0 has real roots if and only if c
and
has [(2
2 a [2( a c )
(c)
)
2
2
qej,*r
goes downwards, and
1)
is rejected as it is larger than 1).
2(4
2{2
4 , z(
1 , there exists a unique
1 , where
1
4
1) , then z( )
1)
Then we can easily see the part about Furthermore,
2
0)
CR IP T
0 for all
AN US
z( )
0.
AC
Proof of Corollary 1:
We only provide the proofs of the relationship between qeRS,t * and qeN,t* when 2( 2
obtained from Theorem 1 readily. When thus qeRS,t *
qeN,t*
if
2
, qeRS,t *
q eN,t*
1) , qeRS,t *
otherwise, where
2
2( 2
qeN,t*
[(1
[(2
)2
8] (2
)
0 ; qej ,*s
)a
1) ,
)2
c ][(2
the other parts can be
8 (2
)
]
(4
3 2)
RS
,
) .
Proof of Theorem 2： cej
ceC 2(1 2
4(1
(1 2
)
)(2 j
2
0 )
j
; 0.
) a
j
qej ,*t
qCe ,t*
0 ; qCe,*i (1
qej ,*r )(2
(1
)2
)[(1
)a
2
qej ,*r 4(1 c ] 2(1
2
)
j
0
;
qCe ,*d C* e ,t
(1
)(2 j* e ,t
(1
)[(1
)a
c]
)2 [(1
)a
c ]2
ACCEPTED MANUSCRIPT Submitted to European Journal of Operational Research (2017) 1–32
27
Proof of Theorem 3: (1) For the retailer: qnj *,r 2
(1
[4c (4 j
)
j* e ,r
(ii) As
j
a 4
0,
c
j* n ,r
qej ,*r
thus
j* e ,r
) , then
j* e ,r
, then qej ,*r
qnj *,r when 0
qnj *,r when c1
j* n ,r
j
)a2 [4
(1
c1 , where c1
c
j* e ,r
(0, cej ) ,
( j )2 ] 4( j )2
(1
j* n ,s
j
g(c ) 4
4)a . Note that cej
, where g(c )
(4
)[(1
2
)
(2) For the supplier: j* e ,s
(1 2 2
)[(1
for all c (0, cej ) .
When c (0, cej ) , let
(1
c1
cej .
c
0 (see Theorem 1(i)), then, when c
j 2
(a
j
)a] 2
cej ]2
)a
j
(1
)2 ] 2 a2 4( j )2
4(1
j
)
0 , thus
CR IP T
(i) qej,*r
j
2
)( a c )2
j
[(4
)a 2
8 ( a c)a] . Let
g(c ) be a quadratic function of c . Obviously, g(c ) goes upwards. Thus,
.
[4
2
(4
then, 2
] a (4
)
j
a [(4
)
2
2
j
2
,
4]
. Moreover, g(c
j
2
(b) When
j* n ,s
3
4 [4 (1
c
csj,2 ;
if csj,1
CE
j* e ,s
2
3
cs
a2 [4(1
a2 ( j ) . Then,
(a) When
cej )
, g(c
csj,1
0
a [4
,
0 , Note that 2(2
0)
j* e ,s
(4
)
j
j* n ,s
)(1
2
)2
0 , thus
2
] , g(c
j* e ,s
j* n ,s
cej
and
1
)
cej )
2
(1
) (1
cs
c
2
]
j* e ,s
j* n ,s
j* n ,s
4
a
1
j
) 2
0 , g(c
j* e ,s
csj,1 ,
cs
2(2
j
2
0 when
)2
)(1
0 , thus
if 0
)
for all c (0, cej ) when
] a (4
) . Note that the symmetrical axis of g(c ) is c
PT
2
j
)
0 has real roots, i.e., c
when
2 j
, g(c )
M
(ii) When
(1
0 for all c (0, cej ) , which means that
ED
2
4 , g(c )
2
AN US
(i) When
if csj,1
c
a [(4
2
csj,1
a
2
)
)
{4 [4
c
csj,2
a [(4
4] , j
4]
cej when
, thus cs
cej )
if 0
) and c
)2
(1
or csj,2
c
c
) 2
2
] }
cej ,
cej .
AC
Proof of Theorem 4:
(1) For the retailer: When c [ cej , cnj ] , [3a (1
)2 a (4
2
)c ]
(1
c
) a 4 , thus,
(2) For the supplier: when c [ cej , cnj ] , [(4( a c )
qej ,*r
a] . Note that m( cej ) [4(1
j* e ,r
j* e ,s
) (1
0 , thus qej ,*r
)a2 4 (1
(1 j* n ,s
)2
implies c0
2
(4
2
2
) , m(c ) goes upwards and cnj
cnj , thus m(c )
m( cnj )
2
m(c ) 4
a quadratic function of c , then its symmetrical axis is c0 (i) When
qej ,*r ( cnj )
]a2 4 a c0
a2
) 2 a2 16
qnj *,r ; j* n ,r
, m( cnj )
2
a [(1 (1
)(4
0 for all c [ cej , cnj ] , which means
(1 )
2
2
j* e ,s
(1
2
.
, where m(c ) [(1
j
j* e ,r
)
) 2 a2 16
2
4 ]( a c )2
a
0 . Let m(c ) be
4 ]. ) a 2[4 j* n ,s
(1
)
2
]
0 , which
for all c [ cej , cnj ] .
ACCEPTED MANUSCRIPT 28
Submitted to European Journal of Operational Research (2017) 1–32
2
(4
(a) m( cej ) 0 2
2
(4
)
4
cej
j* e ,t j* e ,t
2
)2 ] , c
, cej
csj,3
j* e ,r
j* e ,s
cnj )
(c
] ,
j* e ,s
thus
csj,4
a [2
cnj , but csj,3
( a c )[c j* n ,t
a2 4
(1
) ] a [4
(1
cnj . Therefore, when
(1
2
)a]
j* e ,t
, then
4
c
( )a
2
) (1
( )c
( )cej
( )a
(2) When c [ cej , cnj ] , qej ,*t j* e ,t
(3)
j* n ,t
h(c ) 4
2
)
.
Note 2
(1
)
( ( )a
that j
2
qej ,*r , and qej ,*r
, where h(c )
j
qnj *,t
j 2
c
( )c ) 2
[2( a c )
j
,
( )
when j
j* e ,s
,
all
csj,3
a [2
j* n ,s
j* e ,t
2[4(1
) (1
2 2
(1
0 when cbj
ED
j
a ( j )2
)
4 2
c
cej , where c
0 ,
( )a
c ( cnj )
CE
h( ) goes upwards. Let h( )
) and
AC j b
e
(1
j
j* e ,s
j* n ,s
)(2
)
( )c
0 ;
when
if
j* e ,t
0 , thus
2(1
qnj *,t for all c
)2
2
]ac
4(1
0 , Thus, there exists a unique c cbj solves h(c )
(ii) Rewriting h(c ) as a quadratic function of
(a)
0 , thus
0
and
( )
0 ,
[0, cej ] .
)2 a2 , where
j
0 , i.e., c
0)
cbj such that h(c )
cbj
a [4
(1
4
)
)2
(1
] a
.
)2 a2
4(1
0 when 0 j
2
j
0 ,
cbj ,
c
(the other
0 is rejected as it is larger than cej ).
root of h(c )
2( a c ) (2 a
) ]
qnj *,r , as confirmed in Theorem 4(i).
PT
h(c )
cej )
cnj ,
c
(i)Let h(c ) be a quadratic function of c , then h(c ) goes upwards. Note that h(c h(c
when
(1
( )
where
0 . Hence, qej ,*t
if csj,3
2
a]
AN US
4(1
qej ,*t
,
M
( )
[ cej , cnj ]
c
for
)2 ] , recall that m( cnj )
)(1
.
[0, cej )
c
When
j* n ,s
0 , we can obtain its two roots, i.e., c
Proof of Proposition 2: (1)
cnj . Note that:
0 , which implies c0
csj,3 .
c
(3)
)2
4 [4 (1
4
. Let m(c )
)(1
(1 4
where
c0
.
4
when
,
4
(b) m( cej ) 0 a [4
) , m(c ) goes downwards, and cnj
CR IP T
2
(ii) When
)a
0 be a quadratic equation of 2( a c ) (2 a
j 1
4a
)2 ( a c )2
, then h( ) [(1
2
2
[(3
) , where 2 ]2
)a
a2 ]
2
8( a c ) a 4( a c )2 , thus
, the equation’s two roots are
(1
) (2 a c )c .
(3
)a (2 a c )c
(1
2(2 a c )c
j b
)a2 , which
always holds. (b) When (1 (1
)
)a
that (1
)
2c
(2 a c )c ,
4a
2
2c
(2 a c )c )
2
j 1
[(3
1 , then
)a 2 ]2
j 1
should be rejected; otherwise,
4(2 a c )c (1
4( a c )(3a c )c 2 (2 a c )c
)2 a2
0
4(2 a c )c
0 ,which always holds. Thus
j 1
1 , then
4a2 c 2 (2 a c )c j 1
j 1
e
0 (note
is rejected.
Proof of Lemma 3: Employing the same process developed in the proof of Lemma 2, we can easily show that Lemma 3 holds.
j
ACCEPTED MANUSCRIPT Submitted to European Journal of Operational Research (2017) 1–32
29
Proof of Theorem 5: Using the same process developed in the proof of Theorem 3, we can easily show that Theorem 5 holds. Proof of Theorem 6: Following the same process established in the proof of Theorem 4, we can easily show that Theorem 6 holds. Proof of Lemma 4:
jointly yields wˆ ej *
qˆ ej ,*r
)[( a c )2 4 ( a c
(1
qˆ ej ,s )qˆ ej ,s ] qˆ ej ,*r
(
the other two group solutions are rejected as both can lead to ˆ ej ,r
CR IP T
( wˆ ej , qˆ ej ,r ) with respect to wˆ ej and qˆ ej ,r , respectively and solving the FOC
(1) In the second stage, differentiating
)( a
qˆ ej ,*r
qˆ ej ,s ) , and qˆ ej ,*r
0 and ˆ ej ,s
qˆ ej ,s ,
a2
( a c )2 4 .
(2) In the first stage, substituting wˆ ej * and qˆ ej ,*r into Eq. (10) and taking the second derivative of ˆ ej ,s with
(i) When 0 ˆj
ˆ j* e
2
1,
qˆ ej ,s
e ,s
0
2
ˆj
2
ˆj
e ,s
(qˆ ej ,s )
e ,s
2
(qˆ ej ,s )
qˆ ej ,*s
yields
1 c a ; (b) qˆ ej ,*s
[(1
0 if c
2
2 (
1) , then:
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respect to qˆ ej ,s yields
0 , which means that there exists a unique qˆ ej ,s maximizing ˆ ej , s . Solving 2
)a c ] 2(1
2
2
ˆj
(qˆ ej ,s )
e ,s
0 , thus
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1,
Note
that
qˆ ej ,*s
(a)
ˆj
qˆ ej ,s
e ,s
c
(1
2
)(1
)a2 4
cˆej * or
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(2) When c (1
)( a c )2 4
j* n ,r
; (b) ˆ ej *,s
j* n ,s
(1
)( a
c )2 4
c
cˆej *
(1
)a
0 . Thus, we can readily
[(1
)a
ˆ j*
e ,r
c ]2 4(1
0 , then ˆ ej *,r
c 2
)
ˆ j * (c e ,r
j* n ,s
(1
)( a c )2 4
cˆej * )
0.
ˆ j * , the supplier opens an inactive direct channel and sells nothing. (a) ˆ j * e e ,r
0 ; (b) ˆ ej *,s
or
0 . Then, we can readily obtain the
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ˆ j * , the supplier sells directly. (a) Note that e
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cˆej * or
(1) When c
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Proof of Theorem 7:
if
0 , which means that the supplier’s profit decreases
in qˆ ej ,s , hence she will sell nothing in her direct channel, i.e., she sets qˆ ej ,*s equilibrium outcomes.
0
ˆ j * , then the supplier will set qˆ j * e e ,s
cˆej * or
obtain the equilibrium outcomes. (ii) When
) .
j* n ,r
0.
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Supplier encroachment under nonlinear pricing with imperfect substitutes: Bargaining power versus revenue-sharing Huixiao Yang , Jianwen Luo , Qinhong Zhang PII: DOI: Reference:
S0377-2217(17)31160-8 10.1016/j.ejor.2017.12.027 EOR 14886
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
17 May 2017 11 December 2017 18 December 2017
Please cite this article as: Huixiao Yang , Jianwen Luo , Qinhong Zhang , Supplier encroachment under nonlinear pricing with imperfect substitutes: Bargaining power versus revenue-sharing, European Journal of Operational Research (2017), doi: 10.1016/j.ejor.2017.12.027
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Submitted to European Journal of Operational Research (2017) 1–32
Highlights Nonlinear pricing coupled with Revenue-sharing can coordinate the chain. Further increase of direct sales cost and product substitution can profit both firms. An encroaching supplier always hurts the retailer, even without direct sales. A less powerful supplier is more likely to benefit from encroachment. A powerful and relatively efficient supplier hurts from the threat of encroaching.
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* Corresponding author: Jianwen Luo. Tel.: +862152302053. E-mail addresses: [email protected] (H. Yang); [email protected] (J. Luo); [email protected] (Q. Zhang).
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Supplier encroachment under nonlinear pricing with imperfect substitutes: Bargaining power versus revenue-sharing Huixiao Yang a,b, Jianwen Luo b,*, Qinhong Zhang c School of Management, Jinan University, Guangzhou 510632, China
b
Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai 200030, China
c
Sino-US Global Logistics Institute, Shanghai Jiao Tong University, Shanghai 200030, China
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a
Abstract: We explore the impact of nonlinear pricing (NP) on supplier encroachment in a supply chain, consisting of a retailer (he) and a supplier (she), who can sell through either the retailer, her direct channel, or both. The two channels’ products are imperfect substitutes. The firms bilaterally negotiate over the
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wholesale price and the quantity using NP. To better align their behaviors, they can implement revenue-sharing (RS) to share the retailer’s sales revenue. Our analysis shows that NP coupled with RS coordinates the supply chain under encroachment when the supplier seizes all the retailer’s sales revenue. In what follows, the supplier subsidizes the retailer’s acquisition of product through a negative wholesale
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price. Surprisingly, further increase of the supplier’s direct selling cost and product substitution degree can benefit both firms. Contrary to the prior research, the retailer always hurts from the supplier’s ability to
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encroach, even when an inactive direct channel is introduced. Furthermore, a supplier with weak power is more likely to benefit from encroachment, but one with strong power hurts from initiating an inactive direct
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channel when her retail disadvantage is limited. Our main results are robust even after altering some
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assumptions in the basic model.
Keywords: E-commerce; Supplier encroachment; Bargaining power; Nonlinear price contracts;
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Revenue-sharing contracts
1. Introduction
With the rapid development of E-commerce and the growth of customer’s acceptance of a direct channel, upstream suppliers/manufacturers might introduce their own direct channels in addition to relying on third-party resellers/retailers, thereby competing with their resellers in the retail market. Such competition is often referred to as “supplier encroachment” in the literature, which has taken various forms. These include catalog sales, factory outlets and online stores (Nair and Pleasance 2005). Such examples can be observed in various industries, for instance, electronic industry (e.g., Apple, Dell, Sony,
ACCEPTED MANUSCRIPT Submitted to European Journal of Operational Research (2017) 1–32
3
and Samsung), apparel and fashion accessory industry (e.g., Adidas, Coach, Estee Lauder and Nike), beverage and food industry (e.g., Budweiser Beer, Campbell Soup and Coca-Cola), travel industry, like hotels and airlines (e.g., American Airlines, Hilton), etc. It is widely acknowledged that the participation of a supplier in the retail market can pose a serious threat to existing retailers. Contrary to the conventional wisdom that supplier encroachment mostly
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hurts retailers, current research indicates that supplier encroachment may benefit incumbent retailers, because such an encroachment can induce demand-enhancing activity (Tsay and Agrawal 2004), lower double marginalization effect (Arya et al. 2007, Cattani et al. 2006, Chiang et al. 2003), promotion of brand awareness (Blair and Lafontaine 2005), and stronger incentive to reduce production cost (Yoon 2016). To reach such a conclusion, most papers assume that the upstream supplier acts as a Stackelberg
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leader and firms adopt a wholesale price-only contract. Despite its wide use, the supplier-led assumption that a supplier offers a “take-it-or-leave-it” contract to a retailer is inconsistent with various empirical evidences, which indicate the wide occurrence of bilateral bargaining in channel relationships (Draganska et al. 2010). Examples span raw material markets, manufacturing, service industries, etc.
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(Bajari et al. 2009). As Lovejoy (2010) indicated, a bargaining model would be more suitable in many supply chain circumstances, because the solutions resulting from the Stackelberg framework can be
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extremely impractical for various issues. Models of bilateral bargaining, though better revealing the power structure in supply chain, have sporadic adoption in the literature. While much has been known
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not so transparent.
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about supplier encroachment in the supplier-led Stackelberg game, the result in the bargaining game is
When the products sold through the direct and indirect channels are perfect substitutes and the
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supplier can implement nonlinear pricing (NP) to coordinate the channel, however, the benefit from supplier encroachment disappears (Li et al. 2015), which makes an interesting contrast to previous research. Nonlinear price contracts are widespread in practice. For example, many suppliers offer a menu of contract specifying the wholesale price and quantity to their retailers (Ha 2001, Li et al. 2015). Yet, consumers may view the retail products of direct and indirect channels as imperfect substitutes. Say, some consumers might particularly appreciate the convenience of in-home shopping offered by a supplier’s online store, while others may have a strong bias towards personal touch delivered by a brick-and-mortar retailer. In fact, product substitutions can considerably influence the model outcome.
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For example, Matsui (2011) shows that the degree of substitution among products significantly affects the negative effects of transfer prices on social welfare under price competition. However, the effect of product differentiation on firms’ behaviors and profits is still unclear under encroachment with nonlinear pricing. Building upon the modeling framework of Li et al. (2015), we seek to further understand the
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implications of supplier encroachment on contractual agreements and firms’ profitability under imperfect substitutes by considering two related managerial factors: (i) The firms’ bargaining power and (ii) the additional incentives schemes (e.g., revenue-sharing contracts) that the firms may adopt. To mitigate channel conflict and better align their behaviors, the firms between the upstream and downstream can implement revenue-sharing (RS) to share the retailer’s sales revenue. RS now is
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commonly used in different industries, e.g., agriculture, manufacturing, pharmaceuticals, etc. (Heese and Kemahlıoğlu-Ziya 2014). It is well known that RS coordinates the supply chain and arbitrarily allocates the supply chain’s profit, but it limits a risk neutral retailer’s incentive to exert costly effort (Cachon and Lariviere 2005). Also, it reduces the supplier’s incentive to leak the incumbent retailer’s
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private information about demand to the entrant retailer (Kong et al. 2013), but coordinates supply chain decision on innovation investment (Wang and Shin 2015). Although the effect of RS on coordinating a
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supply chain and firm’s incentives in various contexts has been generally investigated, its effect on aligning firms’ behaviors under encroachment is not so perspicuous.
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The popularity of supplier encroachment, bilateral bargaining, NP, imperfect substitutes, and RS in
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practice brings to the forefront a few research questions: When two firms bilaterally negotiate over NP with a RS clause and the two channels’ products are imperfect substitutes, (1) How do the supplier’s
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retail disadvantage and product substitution affect the two firms’ profitability? (2) Can NP coupled with RS coordinate the supply chain system? (3) Can Pareto gains continue to arise from supplier encroachment? (4) Does the threat of selling directly remain to be a boon for the supplier? To answer the above questions, we use a stylized model of a supply chain, consisting of a retailer and a supplier, who can sell through a retailer, or her direct channel, or both, where the two channels’ products are imperfect substitutes. The two firms engage in bilateral negotiation using NP coupled with RS. Our paper contributes to the literature in four ways. First, we prove that further increase of the
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supplier’s direct selling cost and product substitutions can benefit both firms. Second, we demonstrate that NP coupled with RS can coordinate the system under encroachment when the supplier squeezes all the retailer’s sales revenue. Third, we verify that the retailer always gets hurt from the supplier’s ability to encroach, even when an inactive direct channel is introduced. Finally, we establish that a supplier with weak power is more likely to benefit from encroachment, but one with strong power hurts from
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initiating an inactive direct channel when her retail disadvantage is limited. The remainder of this paper is organized as follows. Section 2 reviews the related literature. Section 3 introduces the model in detail. Section 4 provides the analysis of the model. Section 5 delivers two extensions of our model. Finally, section 6 summarizes our insights and limitations. The proofs of all the
2. Literature Review
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formal results appear in the appendix.
In this section, we review the related literature to corroborate the originality and importance of our study. Our paper is mainly related to three streams of literature: Supplier encroachment, bilateral negotiation, and revenue-sharing contracts.
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This work serves as a commentary on the papers on supplier encroachment. Encroachment arises
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when the upstream supplier launches her own direct channel to sell directly to the final consumers. Contrast to conventional wisdom, recent analytical research shows that supplier encroachment can
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benefit both firms, e.g., Arya et al. (2007), Cai (2010), Cattani et al. (2006), Chiang et al. (2003), Dumrongsiri et al. (2008), Tsay and Agrawal (2004), Yoon (2016), etc. However, Li et al. (2014, 2015)
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extend Arya et al. (2007) to incorporate asymmetric information about demand and find that supplier encroachment could lead to win-win, lose-lose, win-lose, or lose-win outcomes. Ha et al. (2016) show
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that supplier encroachment always hurts the retailer when endogenous product quality and heterogeneous customers arise. Hence, the effect of supplier encroachment on firms’ profitability is still unclear. Therefore, enormous papers seek to evaluate the effect of supplier encroachment on firms’ profitability, e.g., Caldieraro (2016), Hsiao and Chen (2014), Xiao et al. (2014). Furthermore, the prevalence of supplier encroachment drives many researchers to study the dual-channel system, where a supplier can sell through a retail channel, or a direct channel, or both, e.g., Chen et al. (2017), Feng et al. (2017), Lu et al. (2018), Matsui (2016, 2017), Qing et al. (2017), Yan et al. (2015), Yu et al. (2017), etc. Our work differs from the above papers along two important dimensions. First, we consider the two firms
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engage in bilateral negotiation game, rather than a Stackelberg game adopted by most existing research. Second, these papers assume that a wholesale price-only contract is used, while our work consider the two firms reach an agreement via NP coupled with RS. Finally, our work is closely related to Li et al. (2015), which shows that when the demand information is asymmetric and the supplier can use NP, the supplier’s ability to encroach can either hurt or benefit both firms. However, they derive the result in a
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Stackelberg game and perfect substitutes, while we consider the case in bilateral negotiation game and imperfect substitutes. Furthermore, they focus on the effect of asymmetric information, while we consider the impact of RS.
Our work also relates to the growing stream of literature on bargaining game. Models of bilateral bargaining game have received increasing concerns recently. Nagarajan and Bassok (2008) use a
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bargaining game framework to study the component suppliers’ decision to form a coalition. Hereafter, many papers seek to further understand the implications of bargaining power on firms’ decisions and profits, e.g., Aydin and Heese (2015), Feng and Lu (2012, 2013a, 2013b), Feng et al. (2015), Guo and Iyer (2013), Heese (2015), Leng et al. (2016), Sheu and Gao (2014), Wang et al. (2013), Zhong et al. (2016). Our
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study is most closely related to Yoon (2016), which examines the spillover effect from a manufacturer’s cost-reducing investment when the manufacturer can encroach into retail space and the firms negotiate
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over the wholesale price and quantity under imperfect substitutes in the extensions. Our study differs from Yoon (2016) along three important aspects: First, Yoon (2016) assumes that both firms possess equal
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bargaining power, which is a special case of ours; second, Yoon (2016) considers the spillover effect,
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whereas we examine the supplier’s retail disadvantage; finally, unlike our paper, Yoon (2016) does not survey the impact of RS. Our study is also closely related to Qing et al. (2017), which considers a
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supplier’s capacity-allocation under bargaining when she can allocate one element through dual-channel. However, the contract to be negotiated in their works is a linear price contract, while it is a nonlinear price contract in ours. Besides, they consider the supplier’s capacity-allocation problem, while we study the impact of RS. Finally, they seek to derive the conditions of the supplier’s equilibrium channel choices, but we strive to explore the effect of the supplier’s encroaching ability on firms’ decisions and profitability. Finally, our paper relates to the papers on revenue sharing contracts. Revenue sharing contracts have been recognized as an effective tool to better align firms’ incentives by inducing the firms to act in the
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best interest of the whole supply chain. Cachon and Lariviere (2005) show that revenue sharing contracts equal to buy-backs in various contexts and it can coordinate a supply chain and arbitrarily allocate the channel’s profit. Hereafter, massive papers concentrate on the coordinating revenue-sharing contract design issues in various settings. For example, Arani et al. (2016), Avinadav et al. (2015), Cai et al. (2017), Chakraborty et al. (2015), Heese and Kemahlıoğlu-Ziya (2014), Hu et al. (2016), Lim et al. (2015), Xie et al.
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(2017), Xu et al. (2014), etc. Besides, many researchers seek to understand the effect of RS on firms’ incentives to exert costly efforts, e.g., Bhaskaran and Krishnan (2009), Hsueh (2014), Heese and Kemahlıoğlu-Ziya (2016), Krishnan et al. (2004), etc. Furthermore, some researchers make a comparison between revenue-sharing contracts and the other contracts, e.g., Jin et al. (2015), Kouvelis and Zhao (2016), Palsule-Desai (2013), Wang and Shin (2015), Yang and Chen (2017), Yenipazarli (2017), Zhang et
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al. (2016), etc. Finally, many researchers attempt to study the effect of RS on aligning firms’ strategical behaviors, Cai et al. (2012), Kong et al. (2013), Yang et al, (2017), etc. To the best of our knowledge, our paper is the first to employ RS to coordinate firms’ behaviors under supplier encroachment.
M
3. The basic model
We consider a supply chain system, consisting of a single retailer (labeled r) and a single supplier
ED
(labeled s), who can sell either through the retailer, her direct channel (e.g., her own online store or retail store), or both. Similarly to Arya et al. (2007) and Yoon (2016), we assume that the products sold through
PT
the retailer and direct channel are symmetric substitutes, i.e., the substitution degree of the retailer’s product over the supplier’s equals that of the latter over the former. The case of asymmetric substitutes
CE
will be examined in §5.1. Then, the consumer demand for the product can be captured by a linear and downward-sloping inverse demand function, pk
a ql
qk , k , l
r , s , and k
l , where pk and q k
AC
represent the market clearing price and the quantity of the product for firm-k, respectively, and a is the market size. The parameter
(0,1] measures the degree of substitution between the two channels’
products, thus the products become more homogenous and the retail market is more competitive as θ increases. Note that the two retail products turn out to be perfect substitutes when independent when function to be
approaches 1, and
approaches 0. Note that we have normalized the slope of this inverse demand
1 . This function has been widely used in the modeling literature on supplier
encroachment (e.g., Arya et al. 2007, Li et al. 2014, 2015, Yoon 2016). The supplier manufactures the product at a constant unit cost, which is normalized to zero.
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Compared with the supplier, the retailer typically enjoys an advantage in the sales process, which can stem from more direct contact with customers, superior knowledge of customer preferences, the cost advantage of bulk shipping, and economies of scope with other retailing activities. To model this advantage, we follow Arya et al. (2007) to normalize the retailer’s per-unit selling cost to zero, and denote the supplier’s per-unit direct selling cost as 𝑐 ∈ (0, 𝑎). Note that this assumption is consistent
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with some business practices in real world. For example, Kevin Mansell, Kohl’s Chief Executive, said online operating margins at Kohl’s are nearly 4%, less than half the 10% operating margins in its nearly 1,200 physical stores, and Target Corp. says its profit margins will fall as its online sales grow (Kapner 2014). Besides, this assumption has also been employed by recent research on supplier encroachment, e.g., Li et al. (2014, 2015), and Ha et al. (2016). This suggests that the retailer’s distribution channel is
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more efficient than the supplier’s direct channel. As a result, it may be in both firms’ favor to fulfill more demand by the retailer channel. To this end, the two firms can implement some incentive schemes to better align their behaviors, e.g., using revenue-sharing (RS) to share the sales revenue. As an endeavor to build a more realistic model, we use a bilateral bargaining framework to capture the strategic
M
interactions between the supplier and retailer, where their interactions are modeled as a two stages game, as illustrated in Fig. 1.
ED
, then Two firms negotiate over Supplier determines her two firms’ profits are realized. for a given .
PT
s=1
s=2
Fig. 1. The sequence of events.
CE
The timeline is as below. (1) In the first stage, s
1 , the supplier negotiates bilaterally with her
retailer over ( w, qr ) using nonlinear pricing (NP), where w is the wholesale price that the supplier
AC
charges for each unit purchased, qr stands for the quantity that the retailer sells to the end customers, represents the rate of revenue that the retailer shares with his supplier per-unit sold. For ease of comparison, we follow Kong et al. (2013) to assume that
is exogenous. Nonetheless, we will analyze
the effect of RS on firms’ decisions and profits through some sensitivity analysis (see Proposition 1 and Theorem 1). The assumption that the two firms negotiate over w and qr is based on the concept of Nash bargaining solution (Nash 1950, 1953), which is analogous to Yoon (2016). If the negotiation breaks down, the retailer earns zero profit, and so does the supplier without encroachment, otherwise, she
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9
gains the monopolistic profit by selling only through her direct channel. (2) In the second stage, s
2,
the supplier chooses the quantity ( qs ) sold directly to the end consumers if a direct channel is introduced, and then both firms bring their entire order quantities to the market and their profits are realized. Note that the assumption that the retailer orders before the supplier is consistent with the fact that the supplier has no means to credibly commit to not adjust her own order quantity after receiving the
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retailer’s order. Nevertheless, we will discuss the alternative case where the supplier can make a credible commitment and place an order before the retailer in §5.2.
Furthermore, we assume that the two firms’ bargaining power under encroachment is identical to that under no-encroachment for the results to be comparable. Following Arya et al. (2007), we assume
paper are common knowledge to both firms.
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that the encroachment does not affect retail demand function. Finally, we assume all parameters in our
The notational scheme used in this work is as follows. In general, qxj *,k , wxj * , and
j* x ,k
, denote the
output, wholesale price and profit in equilibrium, respectively: x can be n (no-encroachment) or e (encroachment); k can be r (retailer), s (supplier), or t (total system); j can be N (without revenue-sharing) or RS * e ,r
denotes the retailer’s optimal profit with revenue-sharing
M
RS (with revenue-sharing). For example,
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under encroachment. Additional notations will be introduced later.
4. Analysis
PT
In this section, we first consider a benchmark case without direct channel, then discuss the case in which the supplier sells directly to consumers. Finally, we compare the main results with the benchmark case.
CE
4.1. The no-encroachment case
In this subsection, we consider the no-encroachment case, that is, the firms can only reach consumers
AC
through the retailer’s channel, which serves as a benchmark. For convenience, we index this case by subscript n . We first examine a centralized system wherein a single decision-maker jointly manufactures and sells the products to the final customers, which serves as a first-best benchmark. We index this case by superscript C. In the centralized system under no-encroachment, a centralized firm chooses its output qnC to maximize the system’s profit max
which is maximized at qnC *
C n
(qnC )
C n
:
( a qnC )qnC ,
a 2 . Thus, the system’s profit under no-encroachment is
(1) C* n
a2 4 .
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We next study the decentralized supply chain system in which the two firms individually make decisions. Thus, the firms’ profits with revenue-sharing ( j j n ,r
)( a qnj ,r )qnj ,r
(1
j n ,s
RS ) or without ( j
( a qnj ,r )qnj ,r
N ) can be formulated as:
wnj qnj ,r ,
(2)
wnj qnj ,r ,
(3)
Consider the negotiation between the two firms over ( wnj , qnj ) for an exogenously given
. Then
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wnj and qnj , r can be derived by a generalized Nash bargaining (GNB) scheme. The GNB scheme serves
as a common methodology for deriving negotiation outcomes in bilateral monopoly settings, and it has been widely used in the modeling literature nowadays (e.g., Feng and Lu 2013a, 2013b, Leng et al. 2016, [0,1] denote the supplier’s bargaining power vis-à-vis her retailer, where
Wang et al. 2013, etc.). Let
1 ) indicates the retailer (the supplier) totally holds the bargaining power, whereas
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0 (
12
represents the case of equal bargaining power. As a result, the cases where the supplier or the retailer acts as a Stackelberg leader, and both firms possess equal bargaining power are special cases of our work. With the assumption that both firms have zero reservation profit, the GNB scheme in this case is defined to solve the next optimization problem.
is the so-called Nash product.
(
j n ,s
) (
j 1 n ,r
)
,
(4)
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where
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max ( wnj , qnj ,r )
Substituting Eqs. (2) and (3) into Eq. (4), and performing the optimization in Eq. (4) provide the )a 2 . Then we can obtain the optimal quantity, qnj *,r
(
PT
optimal wholesale price wnj * two firms’ maximal profits,
j* n ,r
(1
)a2 4 and
a 2 , and the
a2 4 .
j* n ,s
j* n ,t
j* n ,r
j* n ,s
. Thus,
a2 4 . Interestingly, the relationships between the two firms’ profits suggest that the
j* n ,t
is allocated proportionally based on the players’ bargaining power. Besides, it is
AC
trade surplus
j* n ,t
CE
Denote the decentralized supply chain system’s profit under no-encroachment as
simple to see that qnj *,r
qnC * and
j* n ,t
C* n
, which implies that under bargaining, NP coupled with RS
can coordinate the supply chain under no-encroachment. To offer further insights, we explore the impact of key model inputs (i.e.,
and
) on the main
equilibrium outcomes, as the next proposition shows. Proposition 1. Under no-encroachment: (i) wnRS *
0,
qnRS,r*
(ii) wnj *
0,
qnj *,r
0, 0,
qnRS,t * qnj *,t
0, 0,
RS * n ,r j* n ,r
RS * n ,s
0, 0,
j* n ,s
RS * n ,t
0 , and 0 , and
j* n ,t
0; 0.
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Proposition 1 states that under no-encroachment: With a larger
11
, the supplier charges less, which
implies that RS can alleviate double marginalization effect under no-encroachment. However, the retailer’s output and the firms’ profits are independent of
, which suggests that RS does not affect the
retailer’s output and two firms’ profits under no-encroachment. This is because the two firms can always rebalance their profits via adjusting the wholesale price according to
and
(see the expression of
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wnj * ). Besides, when the supplier is more powerful, the supplier charges more and earns more, but the
retailer earns less, which means that a more powerful supplier will charge more and earns more, while the retailer gains less. However, the retailer’s output is independent of
, which implies that
does
not affect the retailer’s output. Because they can reallocate the profit according to their respective bargaining power, which induces them to order at the first-best level. Finally, the decentralized system’s and
, as the system is coordinated
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output and profit under no-encroachment are independent of under this setting.
4.2. The encroachment case
Now, we consider the encroachment case where the supplier sells directly to the final consumers after
M
supplying the wholesale product to her retailer. For convenience, we index this case by subscript e.
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We first survey a centralized system in which a centralized firm jointly manufactures and virtually decides on the outputs of the direct ( qCe, d ) and indirect ( qCe, i ) channels sold to the customers, which serves
PT
as a first-best benchmark. We index this case by superscript C. To facilitate comparison, we assume that the centralized firm’s direct and indirect sales costs are the same as those of the decentralized system.
C e
qCe , d
AC
Solving
CE
And then, the centralized firm’s problem under encroachment can be established by: max
0 and
C e
C e
(qCe ,d , qCe ,i )
qCe ,i
( a qCe ,i
qCe ,d )qCe ,i
(a
qCe ,i
qCe ,d
c )qCe ,d ,
(5)
0 provides the optimal outputs of the direct and indirect
channels for the centralized firm, then we obtain the centralized firm’s profit immediately. Lemma 1. In a centralized system: (i) When c qCe,*d
[(1
ceC (or, equivalently, )a c ] 2(1
system’s profit is (ii) When c
C* e
2
) and qCe,*i
[2(1
C e
), the centralized firm sells directly, then the optimal outputs are
[(1
)a
)( a c )a c 2 ] 4(1
ceC (or, equivalently,
C e
c ] 2(1 2
2
) , the total output is qCe,t*
C ) , where ce
(1
)a ,
C e
(2 a c ) 2(1
) , and the
1 c a;
), the centralized firm does not sell directly, and the optimal
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outcomes are identical to those of with no-encroachment case. We then survey the decentralized supply chain system under encroachment in which the two partners individually make their decisions. The equilibrium outcomes are determined by backward induction following the timeline described in §3. In the last stage, the supplier’s optimization problem (given ( wej , qej ,r ) and (qej,s )
( a qej,r
qej,s )qej,r
(a
qej,r
qej,s
c )qej,s
wej qej,r .
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j e ,s
max
) is: (6)
The first term in Eq. (6) is the supplier’s profit share of the retailer’s sales revenue, the second term is the supplier’s profit from her direct channel, and the last term is the supplier’s payments from selling the wholesale products. Performing the optimization in Eq. (6) delivers qej,s (qej,r ) [ a c (1
) .
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supplier’s best response output sold directly to final consumers, where
qej ,r ] 2 , the
In the first stage, anticipating the supplier’s response output qej ,s (qej ,r ) , the two firms adopt nonlinear pricing (NP) and negotiate over ( wej , qej ,r ) for a given max ( wej , qej ,r ) j e ,s
o s
is given in Eq. (6),
o s
ED j e ,r
) (
j 1 e ,r
)
,
(7)
is the supplier’s reservation profit if the negotiation breaks down, and
is given in the next equation:
j e ,r
j e ,s
M
where
(
to maximize the Nash product, solving:
(1
qej ,r
)( a
wej qej ,r .
qej ,s )qej ,r
(8)
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When the negotiation breaks down, the supplier sells only through her direct channel and incurs marginal selling cost c, but the retailer is expelled from the market and makes no profit. If this is the case, o s
( a qso c )qso , which is maximized at qso
CE
the supplier’s profit becomes
reservation profit with the ability to encroach is
o s
( a c ) 2 . Thus, the supplier’s
( a c )2 4 .
AC
Performing the optimization in Eq. (7), we can obtain the main equilibrium outcomes when the supplier holds the ability to encroach. Lemma 2. In a decentralized system: (i) When c wej *
(1
qej ,*r
2[(1
cej (or, equivalently, 2 ) (a c) 2 )a
profits are
j* e ,r
3
2
2
c] (1
.
j
e
j
), the supplier sells directly, and her optimal wholesale price is
)( a qej ,*r ) (1
(
, and qej,*s
{[4 (1
)[(1
c ]2
)a
j
2
and
)[1
) ] 2 qej ,*r 4 , the outputs of two firms are
(3
) 2 ]( a c ) 2 a} 2 j* e ,s
[(1
)a
j
, respectively, and the two firms’ maximal c ]2
j
o s
, respectively, where
j
4
2
,
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(ii) When cej
cnj (or, equivalently,
c
e
j
j n
), the supplier opens an inactive direct channel and sells
nothing directly, her optimal wholesale price is wej * output is qej ,*r j* e ,s
j* e ,r
(1
o s
)
[(1
)a
j* e ,r
) ]( a c) , and the retailer’s
) 4 ( (1
)( a c )[4 a (4
2
)( a c )] 4
2
and
.
cnj (or, equivalently,
(iii) When c
(
, the two firms’ maximal profits are
(a c)
13
j n
), the supplier will not encroach, and the optimal outcomes are
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identical to those of with no-encroachment case. The above result reveals that the supplier will directly sell the product to consumers (i.e., qej ,*s when her direct selling cost (c) is sufficiently small or the degree of product substitution ( sufficiently low. When c or
) is
is intermediate, the supplier opens an inactive direct channel (i.e.,
0 ) and no direct sales occur. Finally, the supplier does not encroach when c or
is sufficiently
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qej ,*s
0)
high. This is because as c increases, the supplier’s retail disadvantage increases, which makes her direct channel become much inefficient, thus inducing her to diminish the use of direct channel. Particularly, when c is intermediate, the supplier refers all customers to the retailer channel for demand fulfillment. When c is sufficiently great, the direct channel is not a viable option for the supplier. On the other hand, increases, the retail market competition becomes more intense, which makes the supplier’s retail
M
as
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disadvantage become more significant, thus leading to the result identical to that of an increased c. Furthermore, we can observe from the expression of
j* e ,s
that the supplier owning the encroaching 1 and c
0.
PT
ability earns more when using dual-channel as compared to the direct channel, unless
This observation indicates that a supplier with the ability to encroach would not expel her retailer from
CE
the retail market, unless the two channels’ products are perfect substitutes and the supplier is as efficient as the retailer in the sales process.
qej,*t
AC
Denote the supply chain system’s output and profit under NP in encroachment as q ej ,*t and {[4(1
surplus (
) (1
j* e ,t
o s
)2
2
]( a c ) 2(2
)a} 2
j
, and
j* e ,t
[(1
c ]2
)a
j
o s
j* e ,t
. Thus,
. Similarly, the net trade
) is allocated proportionally based on the players’ bargaining power.
The next theorem gives more insights about the effect of key model inputs on the main equilibrium results under encroachment. Theorem 1. Under encroachment ( c (0, cej ) ): (i) qej,*r j* e ,s
c c
0;
qej ,*s
c
0;
0
if 0
c
cIj ,
j* e ,r j* e ,s
c c
0;
when
0
if cIj
1
c
,
cej ;
j* e ,s
c
0
for all c (0, cej ) ; when
1
1,
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(ii) qej,*r
0,
(iii) qeRS,r * 0
0,
qej ,*s
qeRS,s *
0 , 2( 2
1)
j* e ,r
,
0
c1j :
(iv) When c
qej,*r
0 and
j* e ,r j r ,1
e
and
j
, but
(0,
j t ,1
e
j
e
j
(0,
e
j
qej,*s
) , but
(0,
0 for all
2( 2
0 ; however, when
RS * e ,t
1)
1
,
* qeRS ,t
if
0
0
,
1
0;
0 for all
qej ,*s
0 , and
RS * e ,s
; when
ceRS
0 for all j* e ,s
0 if
j* e ,s
0 ,
for all
0 otherwise. Furthermore,
qeRS,t *
0;
j* e ,s
RS * e ,r
0 ,
* qeRS ,t
and
0
e
j
) ; when c
),
c1j : 0 and
j* e ,r
cej
. Furthermore,
j s ,1
0 if
0.
,
qej ,*s 0 if
qej ,*r
j s ,1
0 if j r ,1
,
0 if
j* e ,s
j t ,1
,
e
j
,
0 if
qej ,*r
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14
j* e ,r
0
Theorem 1 declares that under bargaining with encroachment using NP coupled with RS, the retailer’s
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profit increases in c, because an increase in c can enlarge the retailer’s output and reduce the supplier’s reservation profit, both of which promote the net trade surplus that can improve the retailer’s profit. Surprisingly, the supplier’s profit increases in c when
is large and c is intermediate. This can be
explained as follow. An increase in c exerts two opposite effects on the supplier’s profit (see the j* e ,s
expression of
j* e ,t
in Lemma 2): On the one hand, it increases the net trade surplus (as indicated by the o s
), as they will expand the employment of the retailer channel and reduce the use
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expression of
ED
of the direct channel when c enlarges (see Theorem 1 (i)); on the other hand, it decreases the supplier’s reservation profit, as the expression of
o s
indicated. The first effect dominates the latter when
is
PT
large and c is intermediate, the situation reverses otherwise, thus resulting in the above results. These findings suggest that the further increase of the supplier’s retail disadvantage can improve both firms’
CE
profits when her disadvantage is intermediate and her bargaining power is strong. We illustrate these results in Fig. 2.
AC
120 j* e,k
13
j* e ,s
q eRS,t *
80
12
40
11 j* e ,r
0 0
cI j
2
Fig. 2. Impact of ( , ,
on ,
j 4 ce
and
c
10 0.0
0.5
1
1.0
Fig. 3. Impact of
.
)
(
,
on
,
.
)
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15
Furthermore, the two firms’ outputs under encroachment are independent of
. Because it is in both
firms’ interests to enlarge the overall pie that can be divided according to their bargaining power. As expected, the retailer’s profit decreases in firms’ profits increase in
, while the supplier’s profit increases in
. Because the two firms’ interests are better aligned as
. Moreover, both increases, which
can induce them to shift more demand to a more efficient channel (retailer channel), thereby yielding a
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larger net trade surplus that improves both firms’ profits. This finding suggests that each firm will set a as large as possible (up to 1 in our context) when retailer. Yet, the total output decreases in decreases in
can be determined by either the supplier or
when
is small; otherwise, it first increases then
. An explanation is that the supplier becomes cooperative when the retail market
competition is mild, which prompts her to cut down her direct quantity as
increases; otherwise, as is
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increases, the supplier would act aggressively to slightly lower down her direct output when
small, but to rapidly reduce her output otherwise. We illustrate these results in Fig.3. Moreover, ceRS decreases in 7
, which indicates that RS reduces the supplier’s incentive to sell directly. 7
q ej ,*s
5
M
q ej ,*k
0.8
PT
3 0.85
j s ,1
0.9
(
AC
Fig. 4. Impact of
90
j* e,k
6
5
4
1
0.8
0.85
0.9
(b)
)
CE
(a)
0.95
q ej ,*s
q ej ,*r
ED
q ej ,*r
q ej ,*k
on q ej ,*r and q ej ,*s . ( a
j* e ,s
70
20 ,
0.95
(
0.6 ,
1
)
0.5 )
j* e ,s
j* e,k
60
40 30 j* e ,r
j* e ,r
0
10 0.8
0.9
(a)
(
1
)
j t ,1
0.8
(b)
0.9
e
(
)
j
1
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Submitted to European Journal of Operational Research (2017) 1–32
Fig. 5. Impact of
on
j* e ,r
and
j* e ,s
.(a
What’s more, both firms’ profits always decrease in c is high and
20 ,
0.5 ,
0.2 )
when c is low, but they increase in
when
is intermediate. This can be explained as below. When the retail market competition is
mild (with a small
), both firms reduce their channels’ outputs to restrict over-competition as
increases (see Fig. 4); otherwise, as
increases, they would reduce the retailer channel’s output to
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prevent excessive competition if the supplier turns out to be an efficient competitor, which in turn leaves a broad market space for the supplier, thus stimulating the supplier to rise her output (see Fig. 4); however, if the supplier’s retail disadvantage is marked, the supplier would cut down the direct sales quantity in order not to diminish unduly the efficient (retailer) channel’s demand, which can induce the
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retailer to enlarge his output (see Fig. 4), thereby enhancing the net trade surplus that can secure Pareto improvement for the two firms. These unexpected results suggest that, when the supplier’s retail disadvantage is moderate, further increase of the product substitution can benefit both firms when the product substitution is intermediate. We illustrate the above results in Fig. 5.
M
Unsurprisingly, a great product substitution dulls the supplier’s incentive to encroach.
encroachment immediately.
ED
From Theorem 1(iii), we can see the impact of RS on the two firms’ decisions and profits under
Corollary 1. In a decentralized system under encroachment: qeRS,r * N* e ,t
; however, when
2( 2
qeN,t* otherwise; furthermore, ceRS
RS *
1) , qe ,t
qeN,t* for all
; when
qeN,s* , 2( 2
RS * e ,r
N* e ,r
RS *
1) , qe ,t
,
RS * e ,s
q eN,t* if
N* e ,s
2
ceN .
CE
qeRS,t *
RS * e ,t
PT
and
qeN,r* , qeRS,s *
Corollary 1 says that under encroachment with bargaining using NP, RS induces the two firms to
AC
expand the use of the efficient indirect channel but contract the employment of the direct channel, and it improves the two firms’ and the total system’s profits. However, when the retail market competition is mild, RS always reduces the total system’s output; otherwise, RS enlarges it when reduces it when
is small, RS
is large. Finally, RS limits the supplier’s incentive to encroach, which highlights the
key role of RS in better aligning different players’ behaviors. Comparing the results under encroachment between the decentralized and centralized systems provides the next result. Theorem 2. Under encroachment: cej
ceC , qej ,*r
qCe ,*i , qej ,*s
qCe ,*d , qej ,*t
qCe ,t* , and
j* e ,t
C* e ,t
, if and only if
,
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1 , the above equations hold, wherein we
RS *
(1
)[(1
)a
c] 2
17
0.
The above result uncovers that when the two firms participate in negotiation using NP under encroachment, the system is perfectly coordinated when the supplier owns all the retailer’s sales revenue. Because in this case, the two firms’ interests are perfectly aligned, therefore they would order at the first-best levels. Note that when
1 , the negotiated wholesale price is negative, which implies that
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the supplier subsidizes the retailer’s acquisition of product (according to his bargaining power) through a negative wholesale price. This finding implies that NP coupled with RS can perfectly coordinate the supply chain under encroachment when the supplier seizes all the retailer’s sales revenue. This novel result contributes to the growing literature on supply chain coordination.
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4.3. No-encroachment vs. encroachment
In §4.1 and §4.2, we have explored the profitability of the two firms under no-encroachment and encroachment, respectively. In this subsection, we will investigate the impact of supplier encroachment on two firms’ and the system’s profit. To facilitate the comparison, we assume that
and
in the case
of encroachment are identical to those of no-encroachment.
M
Following the previous academic research, one might expect that the retailer remains better off from
ED
supplier encroachment when the supplier’s downstream disadvantage is pronounced in our context. This logic, however, misses a key point: The supplier can refrain from encroachment when her
PT
disadvantage is sufficiently marked. We justify this conjecture in the next theorem. cej ):
Theorem 3. Under encroachment ( 0 c qnj *,r when 0
CE
(i) For the retailer: qej ,*r
(ii) For the supplier: (a) When c
csj,1
or csj,2
AC
0
j* e ,s
j* n ,s
if csj,1
c
c
cej ,
c 2
j* e ,s
c1 , qej ,*r
,
j* e ,s
j* n ,s
if csj,1
cej . Especially, when
j* n ,s
qnj *,r when c1
cej ;
c
j* e ,r
j* n ,r
for all c (0, cej ) ; (b) when c
1 and
csj,2 ; (c) when
1 , csj,1
0 and
3
,
j* e ,s
for all c (0, cej ) ;
2 j* e ,s j* n ,s
3 j* n ,s
,
j* e ,s
if 0
j* n ,s
c
if
csj,1 ,
for all c (0, cej ) .
Theorem 3 declares that under bargaining using NP coupled with RS, the retailer orders more under encroachment than under no-encroachment when c is sufficiently large, the situation is reversed when c is sufficiently small. Surprisingly, the retailer always hurts when the supplier encroaches, regardless of both his bargaining power and the revenue share. This is because on the one hand, the supplier’s ability to encroach on the reselling channel endows her with a substantial reservation profit when the negotiation
ACCEPTED MANUSCRIPT 18
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breaks down, which diminishes the net trade surplus, thereby ultimately hurting the retailer;1 on the other hand, the supplier’s incentive to encroach is weaker when both firms participate in bilateral negotiate when using NP as compared to the wholesale price contract (see Aray et al. 2007).2 This finding makes an amazing contrast to previous finding based on the supplier-led Stackelberg game, which declares that supplier encroachment mostly can also make the retailer better off (e.g., Arya et al.
CR IP T
2007, Cattani 2006, Chiang et al. 2003, Tsay and Agrawal 2004). Yet, the case for the supplier depends on her bargaining power: When always benefits from encroachment; when her direct selling cost is small or large; when
is small, the supplier
is intermediate, she is better off from encroachment when is large, she gains from encroachment only when her
direct selling cost is small. These findings can be explained as follows. The supplier’s profit under
AN US
encroachment is composed of her gain from the net trade surplus and her reservation profit. When her bargaining power is small, her gain from the net trade surplus under both encroachment case and no-encroachment case are limited, which can be dominated by her reservation profit. Hence, her profit under encroachment mainly depends on her reservation profit when her bargaining power is small. Yet,
M
when her bargaining power is large, her gain from the net trade surplus and her reservation profit are very comparable, both of which are oppositely affected by an increased c, as discussed in Theorem 1(i). These
ED
results reveal that the supplier is always better off from opening a direct channel when her bargaining power is small, while she benefits from it when her direct selling cost is sufficiently limited or
PT
pronounced when her bargaining power is intermediate, whereas she gains from it only if her direct
CE
selling cost is sufficiently small when her bargaining power is large. In short, a less powerful supplier is more likely to benefit from encroachment. Particularly, when the supplier holds all the bargaining
AC
power and the two channels’ products are perfect substitutes, the supplier always hurts from encroachment, because in this case, the supplier can fully control the retailer channel, including the wholesale price, output and the resulting retail price. As a result, the supplier cannot earn any additional profit from the introduction of a direct channel when the two channels’ products are perfect substitutes.
1
One can use the methodology employed in the preceding subsections to verify that the retailer can gain from encroachment when the supplier’s direct selling cost is large if the supplier also has zero reservation profit when the negotiation breaks down.
2
When the two firms negotiate over a wholesale price by GNB, Aray et al. (2007) show that encroachment can produce Pareto gains whenever the supplier’s bargaining power is sufficiently pronounced.
ACCEPTED MANUSCRIPT Submitted to European Journal of Operational Research (2017) 1–32
19
This result is akin to Li et al. (2015). The above findings indicate that supplier encroachment can lead to a win-lose or a lose-lose outcome for the supplier and retailer when the two firms engage in negotiation using NP coupled with RS. These results make an interesting contrast to that derived by Yoon (2016), who use the similar framework to show that encroachment can still create Pareto gains when the product substitution is sufficiently small
CR IP T
if the supplier can make a cost-reducing investment. Intuitively, the threat of launching a direct channel seems to be a boon for the supplier. However, this threat can induce the retailer to significantly upward distort his order quantity, which can in turn make the supplier worse off. We justify this guesswork in the next theorem. Theorem 4. When the supplier opens an inactive direct channel ( cej qnj *,r and
(ii) when csj,3
j* e ,t
,
j* n ,r
j* e ,s
j* n ,s
for all c [ cej , cnj ] ;
for all c [ cej , cnj ] ; when
cnj ;
c
0 and
j* e ,t
j* n ,t
.
4
,
j* e ,s
j* n ,s
if cej
c
csj,3 ,
j* e ,s
j* n ,s
if
M
(iii)
c
4
j* e ,r
cnj ):
AN US
(i) qej ,*r
c
The above results tell that when the supplier launches an inactive direct channel, the retailer orders
ED
more but earns less than under the no-encroachment case. This is because the threat of selling directly can force the retailer to order more, and the ability to encroach endows the supplier with a substantial
PT
reservation profit, which reduces the retailer’s profitability in the negotiation. This finding shows that the retailer always suffers from an inactive direct channel developed by the supplier. However, the supplier
CE
always benefits from introducing an inactive direct channel when her bargaining power is weak; otherwise, she hurts from it when her cost is relatively small. This is because, as discussed in Theorem 3,
AC
the supplier’s profit includes two parts: One from the net trade surplus and the other from her reservation profit. As c increases, her threat to sell directly is reduced, which mitigates the retailer’s order quantity distortion that can enhance system profit and augment the net trade surplus; on the other hand, the supplier’s reservation profit decreases in c. The opposite effects of the two gains on the supplier’s profitability resemble those of the case under encroachment, as discussed in Theorem 3. This finding uncovers that a less powerful supplier with the ability to encroach always gains from the threat to sell directly to the customers, but the situation for a powerful supplier is reversed when her retail disadvantage is not too pronounced. Furthermore, the system’s profit is smaller with an inactive direct
ACCEPTED MANUSCRIPT 20
Submitted to European Journal of Operational Research (2017) 1–32
channel than that under no-encroachment, because the supplier’s threat to sell directly can prompt the retailer to overorder, which can reduce the system’s performance. This finding indicates that an inactive direct channel launched by the supplier can reduce the system’s profit. Although the retailer always hurts from supplier encroachment, encroachment can enhance the system’s output and profit, as the next proposition confirms.
when cbj
c
cej or
qnj *,t for all c j b
j e
[0, cnj ) ; (ii)
. Particularly, cbj
j* e ,t
when 0 c
j* n ,t
0 when
1.
cbj or 0
j b
; and
CR IP T
Proposition 2. (i) qej ,*t
j* e ,t
j* n ,t
As expected, the above proposition verifies that the supplier’s ability to encroach can enlarge the system’s total output, even when she introduces an inactive direct channel. Besides, the system benefits cbj ) or the
AN US
from supplier encroachment when the supplier’s direct selling cost is sufficiently small ( c degree of product substitution is sufficiently low (
j
b
). Moreover, the threshold of supplier’s selling
cost ( cbj ) from which the system gains more from encroachment equals zero when the two channels’ products are perfect substitutes. This result implies that when the two firms use NP coupled with RS in
M
negotiation, the system’s benefit from encroachment disappears once the two channels’ products are perfect substitutes. This finding is like Li et al. (2015), but we expand their result deriving from a
ED
supplier-led Stackelberg game to a negotiation game with RS.
5. Extensions
PT
In this section, we consider two extensions of the basic model analyzed in §4 by altering some assumptions to further assess the robustness of the main results of the basic model. The analysis in this
CE
section confirms that the retailer always hurts from supplier encroachment under bargaining when the two firms use NP coupled with RS, even when consumers view the retailer’s products as superior to
AC
those of the direct channel, or when the supplier can credibly precommit to her quantity sold directly before the negotiation.
5.1. Asymmetric substitutes In the basic model, we assumed that the products sold through the retailer and direct channel are symmetric substitutes. However, a direct marketer typically provides the final consumers with only a virtual description of the product, but lacking of the use of smell, taste, touch, which can lead to evaluation mistakes by shoppers. Although the product can be returned after a mistaken purchase, the
ACCEPTED MANUSCRIPT Submitted to European Journal of Operational Research (2017) 1–32
21
return process can be troublesome, thereby reducing the consumers’ expectation of consumption value. Besides, the consumer will typically be required to wait several days for delivery when purchasing the product from a direct channel. Furthermore, the postsale service from a direct channel is less convenient than that from a retailer’s store, as the direct seller is usually located at a distance. Therefore, the products sold through the retailer’s store can be regarded as superior to those of a direct channel. To
CR IP T
reflect this element, we assume that the formers are perfect substitutes for the latters, but the reverse is not true. Then, the two firms’ profits under encroachment are: j e ,r
)( a qej,r
(1
j e ,s
( a qej,r
wej qej,r .
qej,s )qej,r
qej,s )qej,r
( a qej,r
qej,s
wej qej,r .
c )qej,s
AN US
where the “→” reflects the asymmetric substitutes setting.
(9) (10)
Employing the methodology developed in §4.2, we can obtain the next lemma. Lemma 3. In a decentralized system under asymmetric substitutes:
(1
(or, equivalently,
)( a c ) 2
of two firms are qej,*r
)[2 a ( a c qej,*r ) ] 2 (1
(
[(1
maximal profits are
)a (1
j* e ,r
(1
respectively,
j
4( a
)c ]
)[(1
j
)a
, and qej,*s
2c ) { [ a c
(a
c ) ]2
8(1
4 (1
)
,
(ii) When cej
c
cnj (or, equivalently,
CE
AC j* e ,r
j n
(1
( a 2c)
(iii) When c
j
and
j* e ,s
) ( a c )( a
2c )
)[4c
2
c (1
a c
(a
j
(1
)
2
2
]qej,*r 4 , the outputs
, respectively, and the two firms’
[(1
)a
a 2 a
c) } ,
(1
j
)c ]2 4
( a c )2 4 ,
j
j
, 4
(1
)2
(1
)2
2
,
.
1
output
1 3
cej
j e
j n
), the supplier opens an inactive direct channel and sells
nothing directly, her optimal wholesale price is wej * retailer’s
)
[ j ( a c ) 2 a] 2
)c]2 4
(1
2
)[2(2
where
PT
j e
), the supplier sells directly, her wholesale price is
j e
M
wej *
cej
c
ED
(i) When
is
qej,*r
,
(a c)
)2 a]( a c ) 4
2
and
the j* e ,s
(
)a
two j* e ,r
(1
[(1
) 4 (
firms’
) ]( a c) , and the
maximal
) ( a c )2 4 , where cnj
profits (1
are )a 2 ,
a.
cnj (or, equivalently,
j n
), the supplier will not encroach, and the optimal outcomes are
identical to those of with no-encroachment case. The above result resembles that under the case of symmetric substitutes. Theorem 5. Under asymmetric substitutes:
ACCEPTED MANUSCRIPT 22
Submitted to European Journal of Operational Research (2017) 1–32
j* e ,r
for all c (0, cej ) ;
j* n ,r
(ii) For the supplier: (a) When 0 j* e ,s
csj,1
or csj,2
csj,1
c
if
j* n ,s
a [2(1
)2
[(1
) 2
cej ,
c csj,1
j* e ,s
a[
)2 ]
)(1
,
j* e ,s
where j
for all c (0, cej ) ; (b) when
j* n ,s
if csj,1
j* n ,s
,
cej
c
]
(1
4
(1
4
,
)2 ]
(1
csj,2 ; (c) when
c
. Especially, when
j
,
a [2(1
)
1 and
(0, cej ) .
c
)2
2 ) (1 csj,2
5
4
,
5
j* e ,s
j* e ,s
if 0
j* n ,s
]a [ 0 and
j
(1
j* n ,s
2
]
j* n ,s
, ,
)2 ]
j* e ,s
if
csj,1 ,
c
)2
4 [4 (1
5
1 , csj,1
,
for all
CR IP T
(i) For the retailer:
The above result confirms that the retailer always hurts from supplier encroachment even when consumers view the retailer’s products as superior to those of the direct channel. Besides, the supplier is small. However, when
AN US
always benefits from encroachment when
encroachment profits her when her direct selling cost is small or large. When
is intermediate,
is large, encroachment
benefits her only when her direct selling cost is small. These findings are consistent with those under symmetric substitutes.
j
j* e ,r
j* n ,r
for all c [cej , cnj ] ;
(ii) When csj,3
c
6
cnj ,
,
where
j* e ,s
for all c [cej , cnj ] ; when
j* n ,s
6
2
ED
(i)
M
Theorem 6. When the supplier opens an inactive direct channel under asymmetric substitutes ( ce
(4
2
) , csj,3
a
[2
6
(1
,
j* e ,s
) ] a [4
j* n ,s
(1
if cej )
2
c
csj,3 ,
c
j* e ,s
cnj ):
j* n ,s
if
].
PT
The above theorem affirms that when the supplier’s direct selling cost is intermediate, the retailer
CE
always hurts even if the supplier launches an inactive direct channel. In addition, the supplier always benefits from introducing an inactive direct channel when her bargaining power is weak; otherwise, she
AC
benefits from it when her selling cost is sufficiently large, she hurts from it when her cost is relatively small. These findings also like those under symmetric substitutes.
5.2. Supplier ordering first In this extension, we discuss a new timeline wherein the supplier can credibly precommit to her own output quantity sold directly, thereby choosing her direct channel’s output before the negotiation. We use “^” to reflect the setting where the supplier orders first. In this setting, the original sequence of events is replaced by the following: First, the supplier ˆ ej , qˆ ej ,r ) with an announces her direct channel’s output, qˆ ej ,s ; second, the two firms negotiate over ( w
ACCEPTED MANUSCRIPT Submitted to European Journal of Operational Research (2017) 1–32
exogenously given
23
. Using the methodology employed in §4.2, we can get the next lemma.
Lemma 4. In a decentralized system when the supplier orders first: 1:
ˆ ej * w
[(
[(1
)a c ] 2(1
ˆ
j* e ,s
) ( a c )] 2 , the two firms’ outputs are qˆ ej,*r
)a (1
[(1
)a
ˆ j , the supplier sells directly, and her optimal wholesale price is e
cˆej or
(a) if c
2
) , and the two firms’ maximal profits are
c ]2 4(1
2
)
(a
c )2 4 ; (b) if c
cˆej * or
ˆ
j* e ,s
(2 a
(ii) When
c )c 4
(a
(1
)[(1
(1
)a
2
) and qˆ ej ,*s
c ]2 4(1
2
) and
)a 2 , and the
)( a c )2 2 a (
a 2 , the two firms’ maximal profits are
c )2 4 .
c ] 2(1
ˆ
j* e ,r
(1
)(2 a
c )c 4
and
1 , the supplier also opens an inactive direct channel and sells nothing directly, and the
AN US
ˆ
qˆ ej ,*r
j* e ,r
)a
ˆ j * , the supplier opens an inactive direct e
channel and sells nothing directly, her optimal wholesale price is wˆ ej * retailer’s output is
[(1
CR IP T
(i) When 0
equilibrium outcomes are identical to those of (i)(b).
The above results are like those of the original model. Theorem 7. ˆ ej *,r
j* n ,r
and ˆ ej *,s
j* n ,s
for all c (0, a) and all
(0,1) .
M
The above theorem states that when the supplier orders first, the retailer always hurts when the
ED
supplier owns the ability to encroach, even when the supplier introduces an inactive direct channel. This result is identical to that of the original model. However, the supplier always benefits from her ability to sell direct, even when she launches an inactive direct channel. This finding makes an interesting contrast
PT
to that of the original model where the supplier places an order after the retailer, because the option to
CE
order first endows the supplier with a priority to move, which can translate into a benefit to her.
6. Conclusion
AC
In this paper, we attempt to further understand the impact of nonlinear pricing (NP) on supplier encroachment in a linear demand setting. We consider a supply chain system, consisting of a single retailer and a single supplier, who can sell a product either through the retailer, her direct channel, or both. The two channels’ products are imperfect substitutes. To reach a trade agreement, the two firms engage in bilaterally negotiation to bargain over the wholesale price and the quantity using NP. To better align their behaviors, they can implement revenue-sharing (RS) to share the retailer’s sales revenue. Based on our analytical results, we get the following novel insights:
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Submitted to European Journal of Operational Research (2017) 1–32
Further increase of the supplier’s direct selling cost can improve both firms’ profits when the cost is intermediate and her bargaining power is strong, and further increase of substitution degree can benefit both firms when the degree is intermediate and the selling cost is large. NP coupled with RS coordinates the supply chain when the supplier extracts all the retailer’s sales revenue. In what follows, the supplier subsidizes the retailer’s acquisition of product through a
CR IP T
negative wholesale price. This novel result contributes to the literature on supply chain coordination. Contrary to previous research, the retailer always hurts from the supplier’s ability to encroach, even when an inactive direct channel is introduced.
A less powerful supplier is more likely to benefit from encroachment, but a powerful supplier is worse off from adding an inactive direct channel when her retail disadvantage is limited.
AN US
In addition, the key insights of our paper remain valid even after altering some assumptions. There are some limitations to our study. First, throughout this paper, there is only a monopolist supplier in the upstream market. Yet, it is likely to observe horizontal competitions between suppliers. Second, as in Yoon (2016), we do not consider the additional investment cost to make the two channels’
M
products become imperfect substitutes. Third, the supplier can offer products of different quality levels or being horizontally differentiated for the two channels (see Ha et al. 2016). Finally, we assume that all
ED
the information is perfectly symmetric between the firms in our study, but in practice, the supplier can be more knowledgeable about her direct selling cost (see Çakanyıldırım et al. 2012), whereas the retailer
PT
is likely more informed about the demand (see Li et al. 2014, 2015). These complexities will irrefutably
CE
affect the interactions between the two firms and thereby altering the role played by NP, bargaining power and RS in our current setting. It would be interesting to incorporate these factors into our model
AC
in the future.
Acknowledgments The authors thank Robert Graham Dyson (the Editor), and two anonymous referees for their most diligent and constructive suggestions and comments that have considerably improved the presentation and contents of the paper. The authors also gratefully acknowledge the second author, Jianwen Luo, as the corresponding author of the paper. The authors are partially supported by the Natural Science Foundation of China [71372107, 71772122, 71771146 and 71431004].
ACCEPTED MANUSCRIPT Submitted to European Journal of Operational Research (2017) 1–32
25
Appendix: Proofs The proofs of Proposition 1 and Lemma 1 are straightforward, and omit the others for brevity. Proof of Lemma 2: qej ,r ) 2
(a c
condition (FOC) jointly yields qej,*r
j e ,s
2 a
. To ensure qej,*r
0 , where
4 [(1
)a
j
4
]
2
(1
)
(a c) 2
j
c]
and wej *
x( )
4( a c ) [ 2
4(1
cej
, we need c
. Note that
or
)2 a2
(1
)a
(1
2 ) (a
c) 2
(
(1
)( a c )2 , thus qej ,*r
(a
a
j
2 a
, or x( ) [4 (1
0 , then let x( ) )a]
0
c ) (1
)
(2) We then consider qej ,s (qej ,r ) 0 , which implies that qej,r
if
e
(a c)
j
j e ,r
2
0 , we obtain
(which
is
(a c)
( wej )
4 (
)
values, i.e.,
Proof of Theorem 1: 2
0 ; Let
cIj
[
j* e ,s
j
(c )
4(1
AC
c
j
(a) cIj
cej
cIj
(b) 0
0;
qej ,*s
CE
c
cej
wej
(1
j n
2)a , or
e
j
j n
j
c
2
(a c)
]a (4
1
, where
j
2
1
)
j
0,
j
where
4
)
(1
)2
j* e ,s
2
1,
1
(c )
]a (4 1
j* e ,s
)2
(1
) 2 , then, when
1 , thus when
1
(a c)
where
a 2,
qnj *,r
)a .
wej *
(
)a
cej ,
c
or for
.
[4(1
(1
j
; then substituting q ej ,s and q ej ,*r into Eq.
and solving the FOC yields
be a quadratic function of c , then )
e
2( a c ) (1
]( a c ) , the above results are hold for the rest range of c values, i.e., cnj
the rest range of
(1) qej,*r
with respect to
PT
)
which implies that qej,*r
) 2 ]( a c )
.
ED
(7) and differentiating [(1
0,
M
(3) Finally, we consider qej ,s (qej ,r )
and
. In this case, the encroachment case does not
cnj
, which can be rewritten as c
0
rejected),
exist, thus the supplier’s optimization problem is the same as the no-encroachment case, then qej ,r thus, we have a 2
qej ,*r )
)( a
qej ,*r 4 , the other two group solutions are rejected as both can lead to
) ] c )2 4
(a
2[(1
AN US
(3
, then substituting qej ,s (qej ,r )
(a c)
( wej , qej ,r ) with respect to wej and q ej ,r , respectively, and solving the first-order
into Eq. (7) and differentiating
[1
0 , which implies that qej ,r
CR IP T
(1) We first consider qej,s (qej,r )
c
2
j* e ,r
;
c
2(1
) [(1
)a
j
c]
goes upwards and its symmetrical axis is
2
j
,
j* e ,s
0
)
0 , thus, c
if 0
for all c (0, cej ) ;
0
c
cIj ;
j* e ,s
c
0
if cIj
cej .
c
(2) The proof of part (ii) is easy and hence is omitted. (3) qeRS,r * (1
)(1
Moreover,
4(1
)[(1
)[(1
)a
qeRS,t *
a quadratic function of
)a
c ]2 [(1
2
c]
( j )2 )a
2
(
j 2
)
0, c ] z( ) (
0
,
RS * e ,s RS 2
qeRS,s *
2
(4 RS * e ,r
) , where z( )
)[(1
(1 2
) 2
c ] ( j )2
)a
RS * e ,t
0, 2(2
)
, then z( ) goes upwards and its symmetrical axis is
RS * e ,r
0 , RS * e ,r
2
4(1 0
(1
2 )
0.
) . Let z( ) be
(2
)
1 , thus
ACCEPTED MANUSCRIPT 26
Submitted to European Journal of Operational Research (2017) 1–32
[0,1) . Note that z(
(a) when z( (b) when z( 0
0 , i.e., 0
0)
1
2( 2
0 , i.e., 2( 2
0)
, z( ) 0 when
(the other root of z( )
0
ceRS
(4) (a) qej,*r
a
j r ,1
4( a
c) ( a
otherwise, j r ,1
e
it
j
(b) qej,*s qej,*s
[8( a
c)
2
(4
* qeRS ,t
)
)
roots,
)a] (
j 2
)
)
which
[ a2
]2
)a 2
c) ( a
qej ,*s
. Let
c )2
2(1 j t ,1
where
)[(1
)a
j* e ,s
2[4 (1
i.e.,
[2
1
c1j , where c1j
j t ,1
j* e ,r
2
qej ,*r
2 2(1
)]
e
j
for
c1j , and its roots are
j s ,1
e
j
j t ,1
(0,
all
e
j
a2
)
4( a c )2 ]
.
Note
;
that
4( a c )2 .
, then it goes downwards, and
j s ,1
2 a [2( a
c)
]
j s ,2
and
a2 ; otherwise, it has no real roots, which means
( a)2
c )2
c1j ,
[ a2
[4( a
c )2
)a 2
(1
c ) ]2
2( a
2( a
c)
a2 . j* e ,r
e
j
0 if
j t ,1
j* e ,r
,
0
j t ,1
if
,
c1j .
if c
, as their expressions reveals, thus is omitted for brevity. j
) 2] a
if c
0
, then it
2)a , and its roots are
a , which is always hold when
4( a c )] ( j )2 , then
a
(1
be a quadratic function of
M
) . Note that
c ][
like
CE
cej
Finally,
j
4( a c ) a . Note that
(d) The case of
such that z( ) 0 when
(which is rejected), where
means
)a2 ] , which is always hold when 4( a
(1
j* e ,r
e
PT
[4( a
(0,
0 , Thus,
[0,1) ; 1
0,
)
be a quadratic function of
4( a c )2
1 (which is rejected), where
0 for all
1
4( a c )2 ]
ED
qej ,*s
qej ,*r
. Let
0 has real roots if and only if c ]
solves z( )
1
0. j 2
c )} (
4( a
real
(4
0 for all
2
4(1
holds.
RS 2
) a(
)( a
j r ,2
no 2
0)
z(
0 has real roots if and only if c
and
has [(2
2 a [2( a c )
(c)
)
2
2
qej,*r
goes downwards, and
1)
is rejected as it is larger than 1).
2(4
2{2
4 , z(
1 , there exists a unique
1 , where
1
4
1) , then z( )
1)
Then we can easily see the part about Furthermore,
2
0)
CR IP T
0 for all
AN US
z( )
0.
AC
Proof of Corollary 1:
We only provide the proofs of the relationship between qeRS,t * and qeN,t* when 2( 2
obtained from Theorem 1 readily. When thus qeRS,t *
qeN,t*
if
2
, qeRS,t *
q eN,t*
1) , qeRS,t *
otherwise, where
2
2( 2
qeN,t*
[(1
[(2
)2
8] (2
)
0 ; qej ,*s
)a
1) ,
)2
c ][(2
the other parts can be
8 (2
)
]
(4
3 2)
RS
,
) .
Proof of Theorem 2： cej
ceC 2(1 2
4(1
(1 2
)
)(2 j
2
0 )
j
; 0.
) a
j
qej ,*t
qCe ,t*
0 ; qCe,*i (1
qej ,*r )(2
(1
)2
)[(1
)a
2
qej ,*r 4(1 c ] 2(1
2
)
j
0
;
qCe ,*d C* e ,t
(1
)(2 j* e ,t
(1
)[(1
)a
c]
)2 [(1
)a
c ]2
ACCEPTED MANUSCRIPT Submitted to European Journal of Operational Research (2017) 1–32
27
Proof of Theorem 3: (1) For the retailer: qnj *,r 2
(1
[4c (4 j
)
j* e ,r
(ii) As
j
a 4
0,
c
j* n ,r
qej ,*r
thus
j* e ,r
) , then
j* e ,r
, then qej ,*r
qnj *,r when 0
qnj *,r when c1
j* n ,r
j
)a2 [4
(1
c1 , where c1
c
j* e ,r
(0, cej ) ,
( j )2 ] 4( j )2
(1
j* n ,s
j
g(c ) 4
4)a . Note that cej
, where g(c )
(4
)[(1
2
)
(2) For the supplier: j* e ,s
(1 2 2
)[(1
for all c (0, cej ) .
When c (0, cej ) , let
(1
c1
cej .
c
0 (see Theorem 1(i)), then, when c
j 2
(a
j
)a] 2
cej ]2
)a
j
(1
)2 ] 2 a2 4( j )2
4(1
j
)
0 , thus
CR IP T
(i) qej,*r
j
2
)( a c )2
j
[(4
)a 2
8 ( a c)a] . Let
g(c ) be a quadratic function of c . Obviously, g(c ) goes upwards. Thus,
.
[4
2
(4
then, 2
] a (4
)
j
a [(4
)
2
2
j
2
,
4]
. Moreover, g(c
j
2
(b) When
j* n ,s
3
4 [4 (1
c
csj,2 ;
if csj,1
CE
j* e ,s
2
3
cs
a2 [4(1
a2 ( j ) . Then,
(a) When
cej )
, g(c
csj,1
0
a [4
,
0 , Note that 2(2
0)
j* e ,s
(4
)
j
j* n ,s
)(1
2
)2
0 , thus
2
] , g(c
j* e ,s
j* n ,s
cej
and
1
)
cej )
2
(1
) (1
cs
c
2
]
j* e ,s
j* n ,s
j* n ,s
4
a
1
j
) 2
0 , g(c
j* e ,s
csj,1 ,
cs
2(2
j
2
0 when
)2
)(1
0 , thus
if 0
)
for all c (0, cej ) when
] a (4
) . Note that the symmetrical axis of g(c ) is c
PT
2
j
)
0 has real roots, i.e., c
when
2 j
, g(c )
M
(ii) When
(1
0 for all c (0, cej ) , which means that
ED
2
4 , g(c )
2
AN US
(i) When
if csj,1
c
a [(4
2
csj,1
a
2
)
)
{4 [4
c
csj,2
a [(4
4] , j
4]
cej when
, thus cs
cej )
if 0
) and c
)2
(1
or csj,2
c
c
) 2
2
] }
cej ,
cej .
AC
Proof of Theorem 4:
(1) For the retailer: When c [ cej , cnj ] , [3a (1
)2 a (4
2
)c ]
(1
c
) a 4 , thus,
(2) For the supplier: when c [ cej , cnj ] , [(4( a c )
qej ,*r
a] . Note that m( cej ) [4(1
j* e ,r
j* e ,s
) (1
0 , thus qej ,*r
)a2 4 (1
(1 j* n ,s
)2
implies c0
2
(4
2
2
) , m(c ) goes upwards and cnj
cnj , thus m(c )
m( cnj )
2
m(c ) 4
a quadratic function of c , then its symmetrical axis is c0 (i) When
qej ,*r ( cnj )
]a2 4 a c0
a2
) 2 a2 16
qnj *,r ; j* n ,r
, m( cnj )
2
a [(1 (1
)(4
0 for all c [ cej , cnj ] , which means
(1 )
2
2
j* e ,s
(1
2
.
, where m(c ) [(1
j
j* e ,r
)
) 2 a2 16
2
4 ]( a c )2
a
0 . Let m(c ) be
4 ]. ) a 2[4 j* n ,s
(1
)
2
]
0 , which
for all c [ cej , cnj ] .
ACCEPTED MANUSCRIPT 28
Submitted to European Journal of Operational Research (2017) 1–32
2
(4
(a) m( cej ) 0 2
2
(4
)
4
cej
j* e ,t j* e ,t
2
)2 ] , c
, cej
csj,3
j* e ,r
j* e ,s
cnj )
(c
] ,
j* e ,s
thus
csj,4
a [2
cnj , but csj,3
( a c )[c j* n ,t
a2 4
(1
) ] a [4
(1
cnj . Therefore, when
(1
2
)a]
j* e ,t
, then
4
c
( )a
2
) (1
( )c
( )cej
( )a
(2) When c [ cej , cnj ] , qej ,*t j* e ,t
(3)
j* n ,t
h(c ) 4
2
)
.
Note 2
(1
)
( ( )a
that j
2
qej ,*r , and qej ,*r
, where h(c )
j
qnj *,t
j 2
c
( )c ) 2
[2( a c )
j
,
( )
when j
j* e ,s
,
all
csj,3
a [2
j* n ,s
j* e ,t
2[4(1
) (1
2 2
(1
0 when cbj
ED
j
a ( j )2
)
4 2
c
cej , where c
0 ,
( )a
c ( cnj )
CE
h( ) goes upwards. Let h( )
) and
AC j b
e
(1
j
j* e ,s
j* n ,s
)(2
)
( )c
0 ;
when
if
j* e ,t
0 , thus
2(1
qnj *,t for all c
)2
2
]ac
4(1
0 , Thus, there exists a unique c cbj solves h(c )
(ii) Rewriting h(c ) as a quadratic function of
(a)
0 , thus
0
and
( )
0 ,
[0, cej ] .
)2 a2 , where
j
0 , i.e., c
0)
cbj such that h(c )
cbj
a [4
(1
4
)
)2
(1
] a
.
)2 a2
4(1
0 when 0 j
2
j
0 ,
cbj ,
c
(the other
0 is rejected as it is larger than cej ).
root of h(c )
2( a c ) (2 a
) ]
qnj *,r , as confirmed in Theorem 4(i).
PT
h(c )
cej )
cnj ,
c
(i)Let h(c ) be a quadratic function of c , then h(c ) goes upwards. Note that h(c h(c
when
(1
( )
where
0 . Hence, qej ,*t
if csj,3
2
a]
AN US
4(1
qej ,*t
,
M
( )
[ cej , cnj ]
c
for
)2 ] , recall that m( cnj )
)(1
.
[0, cej )
c
When
j* n ,s
0 , we can obtain its two roots, i.e., c
Proof of Proposition 2: (1)
cnj . Note that:
0 , which implies c0
csj,3 .
c
(3)
)2
4 [4 (1
4
. Let m(c )
)(1
(1 4
where
c0
.
4
when
,
4
(b) m( cej ) 0 a [4
) , m(c ) goes downwards, and cnj
CR IP T
2
(ii) When
)a
0 be a quadratic equation of 2( a c ) (2 a
j 1
4a
)2 ( a c )2
, then h( ) [(1
2
2
[(3
) , where 2 ]2
)a
a2 ]
2
8( a c ) a 4( a c )2 , thus
, the equation’s two roots are
(1
) (2 a c )c .
(3
)a (2 a c )c
(1
2(2 a c )c
j b
)a2 , which
always holds. (b) When (1 (1
)
)a
that (1
)
2c
(2 a c )c ,
4a
2
2c
(2 a c )c )
2
j 1
[(3
1 , then
)a 2 ]2
j 1
should be rejected; otherwise,
4(2 a c )c (1
4( a c )(3a c )c 2 (2 a c )c
)2 a2
0
4(2 a c )c
0 ,which always holds. Thus
j 1
1 , then
4a2 c 2 (2 a c )c j 1
j 1
e
0 (note
is rejected.
Proof of Lemma 3: Employing the same process developed in the proof of Lemma 2, we can easily show that Lemma 3 holds.
j
ACCEPTED MANUSCRIPT Submitted to European Journal of Operational Research (2017) 1–32
29
Proof of Theorem 5: Using the same process developed in the proof of Theorem 3, we can easily show that Theorem 5 holds. Proof of Theorem 6: Following the same process established in the proof of Theorem 4, we can easily show that Theorem 6 holds. Proof of Lemma 4:
jointly yields wˆ ej *
qˆ ej ,*r
)[( a c )2 4 ( a c
(1
qˆ ej ,s )qˆ ej ,s ] qˆ ej ,*r
(
the other two group solutions are rejected as both can lead to ˆ ej ,r
CR IP T
( wˆ ej , qˆ ej ,r ) with respect to wˆ ej and qˆ ej ,r , respectively and solving the FOC
(1) In the second stage, differentiating
)( a
qˆ ej ,*r
qˆ ej ,s ) , and qˆ ej ,*r
0 and ˆ ej ,s
qˆ ej ,s ,
a2
( a c )2 4 .
(2) In the first stage, substituting wˆ ej * and qˆ ej ,*r into Eq. (10) and taking the second derivative of ˆ ej ,s with
(i) When 0 ˆj
ˆ j* e
2
1,
qˆ ej ,s
e ,s
0
2
ˆj
2
ˆj
e ,s
(qˆ ej ,s )
e ,s
2
(qˆ ej ,s )
qˆ ej ,*s
yields
1 c a ; (b) qˆ ej ,*s
[(1
0 if c
2
2 (
1) , then:
AN US
respect to qˆ ej ,s yields
0 , which means that there exists a unique qˆ ej ,s maximizing ˆ ej , s . Solving 2
)a c ] 2(1
2
2
ˆj
(qˆ ej ,s )
e ,s
0 , thus
M
1,
Note
that
qˆ ej ,*s
(a)
ˆj
qˆ ej ,s
e ,s
c
(1
2
)(1
)a2 4
cˆej * or
AC
(2) When c (1
)( a c )2 4
j* n ,r
; (b) ˆ ej *,s
j* n ,s
(1
)( a
c )2 4
c
cˆej *
(1
)a
0 . Thus, we can readily
[(1
)a
ˆ j*
e ,r
c ]2 4(1
0 , then ˆ ej *,r
c 2
)
ˆ j * (c e ,r
j* n ,s
(1
)( a c )2 4
cˆej * )
0.
ˆ j * , the supplier opens an inactive direct channel and sells nothing. (a) ˆ j * e e ,r
0 ; (b) ˆ ej *,s
or
0 . Then, we can readily obtain the
ED
ˆ j * , the supplier sells directly. (a) Note that e
CE
cˆej * or
(1) When c
PT
Proof of Theorem 7:
if
0 , which means that the supplier’s profit decreases
in qˆ ej ,s , hence she will sell nothing in her direct channel, i.e., she sets qˆ ej ,*s equilibrium outcomes.
0
ˆ j * , then the supplier will set qˆ j * e e ,s
cˆej * or
obtain the equilibrium outcomes. (ii) When
) .
j* n ,r
0.
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