- Email: [email protected]
Pricing decisions in a dual channels system with different power structures
Economic Modelling 29 (2012) 523–533
Contents lists available at SciVerse ScienceDirect
Economic Modelling journal homepage: www.elsevier.com/locate/ecmod
Pricing decisions in a dual channels system with different power structures Rong Zhang, Bin Liu ⁎, Wenliang Wang Department of Management Science, Henan Agricultural University, Zhengzhou, 450002, China
a r t i c l e
i n f o
Article history: Accepted 31 August 2011 Keyword: Dual channels Pricing decisions Asymmetric distribution channel Game theory Equilibrium
a b s t r a c t This paper investigates the effect of product substitutability and relative channel status on pricing decisions under different power structures of a dual exclusive channel system where each manufacturer distributes its goods through a single exclusive retailer but two goods are substitute. A linear demand based on the utility function of a representative consumer is assumed, and three game scenarios(Manufacturer Stackelberg, Retailer Stackelbeg and Vertical Nash) are examined under symmetric and asymmetric related channel status. It is shown that no power structure is always the best for the entire supply chain though all members on supply chain have incentive to lead the Stackelberg game. Meanwhile, the vertical Nash game is an equilibrium for the members, however, a Prisoner's dilemma necessarily incurs for the entire supply chain because the Retailer Stackelberg or the Manufacturer Stackelberg can gain the better performance than that in vertical Nash for the entire supply chain when the product substitutability is moderate or higher and the asymmetric relative channel status is moderate, while consumers always get the most welfare from the vertical Nash game. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Pricing decisions has significantly reshaped supply chain behavior in the last two decades, especially for dual or multiple supply chain system. Due to supply chains are generally comprised of individual members who are often guided by conflicting objective functions, there are always two topics revolving of dual channel system. One is who, retailer and manufacturer, should be the leadership in Stackelberg game in each channel, and the other is how the competition between both channels affects pricing decision. Actually, the first topic concerns the power structure, which refers to a member's relative ability to control the decision making process in the supply chain. For example, Wal-Mart usually is considered to be the Stackelberg leader (RetailerStackelberg) and his manufacturers are followers (Almehdawe and Mantin (2010)), Apple is also viewed as to be the Stackelberg leader(Manufacturer-Stackelberg) (Lee and Ren (2011)), however, Wal-Mart and Procter & Gamble (P&G) are almost equal in bargaining(Vertical Nash)(Almehdawe and Mantin (2010)). Specifically, the more powerful firm moves first in a Stackelberg game. The second topic investigates the competition of dual channels with different relative channel status, such as symmetric channel (Doraiswamy et al. (1979)) and asymmetric channel (Choi (1996)). This paper considers a two-echelon distribution system that consists of two manufacturer and two retailers, and each manufacturer sells a product exclusively through a franchised retailer. However,
⁎ Corresponding author. E-mail address: [email protected] (B. Liu). 0264-9993/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econmod.2011.08.024
two products have similar function with a higher substitutability, and the demand from customers is met through the retailer. Such a system is called a dual exclusive channel distribution system. The objective of this paper is not only to analyze the pricing decisions of dual exclusive channel system but also to concern the power balance between retailers and manufacturers. E.g. which power structure is the best for the retailers, manufacturers, entire supply chain and customers, respectively, under pricing competition? And how does the product substitutability and asymmetric relative channel power affect the optimal power balance? Several studies have been done on analyzing the interactions between manufacturers and retailers in literature. Different objectives of channels members, however, create conflicts within a channel. Numerous scholars have focused on vertical coordinations among channel members through such measures as various transfer pricing schemes or formal agreements (such as, Zusman and Etgar (1981), McGuire and Staelin (1983), Jeuland and Shugan (1983)), implicit understanding(such as, Shugan (1985)), and formation of conjectures(such as, Jeuland and Shugan (1988)) to achieve maximum channel profit. Many of these studies have considered only one manufacturer (monopoly) and its channel intermediaries, and the analysis of competition and cooperation is confined to members in the same channel. An area often neglected in the previous studies is the market structure in which retailers retain more power than manufacturer do in the retail price decision. While most channel studies have focused on the manufacturer dominance over retailers, an intermediary that deals with a number of competing products is often a large retailer that can influence the market substantially. Therefore, this paper also focuses on the power balance between the channel members
524
R. Zhang et al. / Economic Modelling 29 (2012) 523–533
and its effects on equilibrium prices and profits. Consequently, besides the two most popular power structures in the previous studies (i.e. the Manufacturer Stackelberg and the Nash Game) a third game is also considered in this paper: the Retailer Stackelberg. The interested reader can refer these game settings to Choi (1991) and Huang and Li (2001). Manufacturer Stackelberg (MS). Each manufacturer chooses the wholesale price using the response function of the retailer. Each retailer determines the retail price of each product so as to maximize his profits given the respective wholesale price. Vertical Nash (VN). Each manufacturer chooses its wholesale price conditional on the retailers' selling prices of dual channels. The retailers determine their optimal selling prices conditional on respective wholesale prices. Retailer Stackelberg (RS). Each manufacturer chooses its wholesale price conditional on both retailers selling prices. The retailers set up their selling prices using the reaction functions of both manufacturers in terms of respective wholesale prices. On the power structures, most results in literature is the member always benefits from playing the leader's role in Stackelberg game, such as McGuire and Staelin (1983) for a single channel system, Choi (1991) and Choi (1996) for dual channel with common retailer and so on. Differ from the previous works, this paper investigates the same issues but for an exclusive dual channel with symmetric and asymmetric relative channel status. Another area related to this paper is channel competition of dual channel supply chain system. On dual channels, Paralar and Wang (1993) develop a two-firm competitive newsboy model where the firms face independent random demands. In a paper closely related to Paralar's, Karjalainen (1992) also analyzes the case where independent firm demands are aggregated to form industry demand. Then, Choi (1996) deals with a channel structure in which there are duopoly manufacturers and duopoly common retailers, and Lippmand and Macardle (1997) considers a competitive version of the classical newsboy problem in which a firm must choose an inventory or production level for a perishable good with random demand. The optimal solution is a fractal of the demand distribution and to investigate the impact of competition upon industry inventory. Cai et al. (2009) evaluates the impact of price discount contracts and pricing schemes on the dual-channel supply chain competition, meanwhile, from supplier-Stackelberg, retailerStackelberg, and Nash game theoretic perspectives, also show that the scenarios with price discount contracts can outperform the non-contract scenarios. Cai (2010) investigates the influence of channel structures and channel coordination on the supplier, the retailer, and the entire supply chain in the context of two singlechannel and two dual-channel supply chains. Lu et al. (2011) highlights the importance of service from manufacturers in the interactions between two competing manufacturers and their common retailer, facing end consumers who are sensitive to both retail price and manufacturer service, and a game-theoretic framework is applied to obtain the equilibrium solutions for every entity. Note that the logic underlying our model is distinctly different from that used by Paralar and Wang (1993), Karjalainen (1992) or Lippmand and Macardle (1997) in their models the firm demands are independent, whereas in our model they are dependent on selling prices of both channels and are substitutes.To differ from Choi (1996), our paper investigates a dual exclusive dual channels, and to differ from Cai (2010),our paper concerns the pricing decisions under different power structures. Based on the above literature, this paper investigates the price competition of an exclusive dual channel system with symmetric and asymmetric relative channel status, and applies three game
models to analyze the influences of pricing decisions on product substitutability and channel power with the view of manufacturers, retailers, the entire supply chain and customers respectively. To differ from the previous literature, this paper shows no power structure is always the best for the entire supply chain though all members on supply chain have incentive to lead the Stackelberg game. Meanwhile, the vertical Nash game is an equilibrium for the members, however, a Prisoner's dilemma necessarily incurs for the entire supply chain when the product substitutability is moderate or higher and the asymmetric relative channel status is moderate, while consumers always get the most welfare from the vertical Nash game. The remainder of this paper is organized as follows. We first describe the model in Section 2, and establish the price competition models in Section 3. Then, we analyze the equilibrium entire supply chain efficient, consumers welfare, and others effects in Section 4. Lastly, we summarize the results and provide some further research topics related to the dual channels system in Section 5. 2. The model We mainly explore the price competition in the dual exclusive channel system. We use Di, i = 1, 2, to represent the demand to product/channel i. 1 Correspondingly, the retail prices are denoted by pi, i = 1, 2, and the wholesale prices are wi, i = 1, 2.We assume the initial base demand for product i is Ai. To obtain logically consistent demand functions in different channel structures, we adopt the framework established by Ingene and Parry (2004) (Chapter 11) and Ingene and Parry (2007) and employ a utility function of a representative consumer as follows: 2 2 U≡ ∑ Ai Di −Di =2 −θD1 D2 − ∑ pi Di ; i¼1;2
i¼1;2
ð1Þ
where θ (0 ≤ θ b 1) denotes the channel substitutability. When θ = 0, the products are purely monopolistic; while θ goes to 1, the products converge to purely substitutable. The channels are demand interdependent (unless θ = 0). The aggregate demand decreases in θ. Maximization of Eq. (1) yields the demand for product i as follows. Di ¼
Ai −θA3−i −pi þ θp3−i ; i ¼ 1; 2: 1−θ2
ð2Þ
To facilitate our discussion, we define Ω≡
A1 A2
as the relative channel status of product 1 over product 2 in the dual exclusive channel. If Ω N 1, product 1 is superior to product 2 in terms of the initial base demand, and vice versa. Ω = 1 implies the channel system is symmetric in the relative channel status, and Ω ≠ 1 means the channel system is asymmetric.
1 The terms of product and channel are used interchangeably for the scenarios of dual exclusive, such as Cai (2010), and monopoly common retailer channels, such as Choi (1991). The impact of different retailers on the demand in the dual exclusive channel is assumed away in order to compare the impact of the above two channel structures, but is restored in the duopoly common retailer channel. 2 As stated by Ingene and Parry (2004), a system of linear demand curves must have the form of the underlying utility function.
R. Zhang et al. / Economic Modelling 29 (2012) 523–533
The term Πmi denotes the manufacturer i's profit, Πri is Retailer i's profit, Πi is the channel i's profit, and Π is the total profit for the entire supply chain including all manufacturers and retailers. Meanwhile, the superscript MS represents for the Manufacturer Stackelberg scenario, RS represents for the Retailer Stackelberg scenario, VN represents for the Vertical Nash scenario, and ⁎ represents the optimization. We specify each profit function and the corresponding feasible domain when the need arises. Product i's cost is normalized to zero. For the parsimony purpose, we also normalize the channel operational costs to zero. 3 In the dual exclusive channel case, each manufacturer sells its product through its own downstream retailer. The manufacturers' and the retailers' profit functions can be written as Πmi ¼ wi Di ;
ð3Þ
Πri ¼ mi Di ;
ð4Þ
525
where mi = pi − wi and i = 1, 2, the retail margin on product i. Further, the supply chain i's profit function is Πi ¼ Πmi þ Πri ;
ð5Þ
and the total profit profit for the entire supply chain including all manufacturers and retailers is 2
Π ¼ ∑ Πi :
ð6Þ
i¼1
As in many previous studies, each member of the channel is assumed to seek for maximizing its own profit and and no cooperation is assumed between the members. This is the most common and legal institutional arrangement of the channel structure under consideration. We now derive analytical equilibrium solutions for this base model under each power structure scenario.
3. Price competition under different power structures 3.1. Manufacturer Stackelberg (MS) Under the MS, the manufacturers take their retailer's reaction function into consideration for their respective price decisions. Retailer i's reaction function given wholesale prices wi can be derived from the First Order Conditions (FOC) of Eq. (4).
∂Πri Ai −θA3−i −2pi þ θp3−i þ wi ¼ ¼ 0; i ¼ 1; 2; ∂pi 1−θ2
ð7Þ 2
2
2
2
2
2
∂ Πr1 2 ∂ Πr1 ∂ Πr2 ∂ Πr1 ∂ Πr2 4−θ ¼− b 0 and − ¼ 2 N 0 for any θ ∈ [0, 1), ∂p1 ∂p2 ∂p2 ∂p1 1−θ2 ∂p21 ∂p21 ∂p22 1−θ2 satisfying the second order condition for a maximum. From Eq. (7), retailer i's reaction functions can be derived: for which the Hessian matrix is negative due to
pi ðwi ; w3−i Þ ¼
2ðAi þ wi Þ−θðA3−i þ θAi −w3−i Þ ; i ¼ 1; 2; 4−θ2
ð8Þ
which is linear in both wholesale prices. These functions imply that the retail prices depend on not only the wholesale price of own channel but also that of rival channel. By the reaction function (8), the manufacturer i's Nash equilibrium wholesale prices can be reached from the following FOC of the manufacturer i's profit maximization problems: 2 2 2−θ Ai −θA3−i −2 2−θ wi þ θw3−i ∂ Π ðw ; w Þ ¼ ; i ¼ 1; 2: ∂wi mi i 3−i 4−5θ2 þ θ4
ð9Þ
Solving Eq. (9)= 0 results into the following whole prices MS⁎ wi
¼
8−9θ2 þ 2θ4 Ai −θ 2−θ2 A3−i 16−17θ2 þ 4θ4
; i ¼ 1; 2;
ð10Þ
and the corresponding retail prices can obtain from Eq. (8):
MS⁎ pi
¼
8−9θ2 þ 2θ4 Ai −θ 2−θ2 A3−i 2 3−θ2 64−84θ2 þ 33θ4 −4θ6
; i ¼ 1; 2:
ð11Þ
Certainly, the nonnegative retail prices and wholesale prices require 2 4 2 8−9θ þ 2θ Ai ≥ θ 2−θ A3−i ; i ¼ 1; 2;
ð12Þ
3 We have also analyzed the cases with asymmetric non-zero operational costs and found that all qualitative results hold. In fact, with non-zero operational costs, we can redefine 1 Ω≡ AA12 −c −c2 , where ci denotes the operational cost in channel i, and the corresponding discussion falls through similarly.
526
R. Zhang et al. / Economic Modelling 29 (2012) 523–533
which implies θ ∈ [0, 1) if Ai = A3 − i. So, we yield the common feasible area for θ is θ ∈ [0, 1), and will be narrower because of more asymmetry if Ai ≠ A3 − i. Then, the Retailer i's margin profit can be reached MS⁎ mi
¼
2
2−θ
2 4 2 8−9θ þ 2θ Ai −θ 2−θ A3−i 64−84θ2 þ 33θ4 −4θ6
; i ¼ 1; 2:
ð13Þ
Furthermore, it is easy to get the optimal profits of manufacture i, Retailer i, channel i and entire supply chain as follows.
ΠMS⁎ mi ΠMS⁎ ri ΠMS⁎ i ΠMS⁎
2 8−9θ2 þ 2θ4 Ai −θ 2−θ2 A3−i 2−θ2 ¼ ; 2 4−5θ2 þ θ4 16−17θ2 þ 4θ4 2 2 2−θ2 8−9θ2 þ 2θ4 Ai −θ 2−θ2 A3−i ¼ ; 2 1−θ2 64−84θ2 þ 33θ4 −4θ6 2 8−9θ2 þ 2θ4 Ai −θ 2−θ2 A3−i 2 6−5θ2 þ θ4 ¼ ; 2 1−θ2 64−84θ2 þ 33θ4 −4θ6 64−140θ2 þ 109θ4 −35θ6 þ 4θ8 A21 þ A22 −4θ 2−θ2 8−9θ2 þ 2θ4 A1 A2 2 6−5θ2 þ θ4 ¼ : 2 1−θ2 64−84θ2 þ 33θ4 −4θ6
Especially, when the related channel powers are assumed equal(A1 = A2 = 1), the problem becomes symmetric, which is much easier to solve. This symmetry is meaningful because it presents the most drastic competition market. Under the symmetry assumption, all optimal wholesale prices, retail prices, manufacturer i's profit, Retailer i's profit, channel i's profit, entire supply chain's profit and Retailer i's margin profit are summarized in Table 1. 3.2. Vertical Nash (VN) The assumption VN implies a Nash equilibrium among all members. Here each manufacturer takes as given the competing retail price and the retailer's margin on its own product, whereas the retailer conditions its margins on wholesale prices. The FOC for this equilibrium involves the above equation of retailer's profits maximization condition (7) and the following manufacturer i's profit maximization condition: ∂Πmi Ai −pi −wi −θðA3−i −p3−i Þ ¼ ¼ 0; i ¼ 1; 2: ∂wi 1−θ2
ð14Þ
∂2 Πm1 1 ¼− b 0 and It is easily established that the second order Hessian matrix of this problem is negative definite due to 1−θ2 ∂w21 2 2 2 2 ∂ Πm1 ∂ Πm2 ∂ Πm1 ∂ Πm2 1 − ¼ N 0 for any θ ∈ [0, 1), implying a solution for the Eqs. (7) and (14) is a Nash equilibrium. The 2 ∂w1 ∂w2 ∂w2 ∂w1 ∂w21 ∂w22 ð1−θ2 Þ resulting equilibrium retail and wholesale prices are
VN⁎ wi
¼
3−2θ2 Ai −θA3−i 9−4θ2
;
ð15Þ
Table 1 Equilibrium prices and profits under different power structures given A1 = A2 = 1. Equilibrium
MS
VN
RS
w⁎i
2−θ−θ2 4−θ−2θ2 2ð1−θÞ 3−θ2 8−6θ−3θ2 þ 2θ3
1−θ 3−2θ
1−θ 4−3θ
2ð1−θÞ 3−2θ
3ð1−θÞ 4−3θ
1 3 þ θ−2θ2
1 4 þ θ−3θ2
p⁎i D⁎i ΠMi⁎ ΠRi⁎ Π⁎i Π⁎ m⁎i
2−θ2 8 þ 2θ−9θ2 −θ3 þ 2θ4 2−θ2 2−θ−θ2 ð2 þ θ−θ2 Þð4−θ−2θ2 Þ 2 ð1−θÞ 2−θ2
2
2
ð1 þ θÞð8−6θ−3θ2 þ 2θ3 Þ 2 6−6θ−5θ2 þ 5θ3 þ θ4 −θ5
1−θ ð4−3θÞ2 ð1 þ θÞ
1−θ
2ð1−θÞ
ð3−2θÞ2 ð1 þ θÞ
ð4−3θÞ2 ð1 þ θÞ
2ð1−θÞ
3ð1−θÞ
2
ð3−2θÞ2 ð1 þ θÞ
ð4−3θÞ2 ð1 þ θÞ
4ð1−θÞ
6ð1−θÞ
2
ð3−2θÞ2 ð1 þ θÞ
ð4−3θÞ2 ð1 þ θÞ
1−θ 3−2θ
2ð1−θÞ 4−3θ
ð2−θÞ2 ð1 þ θÞð4−θ−2θ2 Þ 4 6−6θ−5θ2 þ 5θ3 þ θ4 −θ5 ð2−θÞ2 ð1 þ θÞð4−θ−2θ2 Þ ð1−θÞ 2−θ2 2 8−6θ−3θ þ 2θ3
1−θ ð3−2θÞ2 ð1 þ θÞ
R. Zhang et al. / Economic Modelling 29 (2012) 523–533
VN⁎ pi
¼
VN⁎ 2wi
¼
2 3−2θ2 Ai −2θA3−i 9−4θ2
:
527
ð16Þ
for i = 1, 2. Certainly, the nonnegative retail and wholesale prices require 2 3−2θ Ai ≥ θA3−i ; i ¼ 1; 2;
ð17Þ
which implies θ ∈ [0, 1) if Ai = A3 − i. Then, the Retailer i's margin profit can be reached VN⁎
¼
mi
3−2θ2 Ai −θA3−i 9−4θ2
; i ¼ 1; 2:
ð18Þ
Furthermore, the optimal profits of manufacture i, Retailer i, channel i and entire supply chain are listed as follows.
ΠVN⁎ mi ΠVN⁎ i ΠVN⁎
2 3−2θ2 Ai −θA3−i ¼ ΠVN⁎ ¼ 2 ; ri 9−4θ2 1−θ2 2 2 3−2θ2 Ai −θA3−i VN⁎ ; ¼ 2Πmi ¼ 2 9−4θ2 1−θ2 2 9−11θ2 þ 4θ4 A2i þ A23−i −4θ 3−2θ2 Ai A3−i : ¼ 2 9−4θ2 1−θ2
Especially, under the symmetry assumption(A1 = A2 = 1), all optimal wholesale prices, retail prices, manufacturer i's profit, Retailer i's profit, channel i's profit, entire supply chain's profit and Retailer i's margin profit are summarized in Table 1. 3.3. Retailer Stackelberg (RS) Under the assumption RS, the retailers become the leader and the manufacturers are the followers. In this market, the leaders take the followers' reaction functions into account for their own retail prices decisions. Manufacturer i conditions her wholesale price on the retail margin mi and the rival's price p3 − i. The manufacturer i's reaction function can be derived from the following FOC: ∂Πmi Ai −mi −2wi −θðA3−i −p3−i Þ ¼ ¼ 0; i ¼ 1; 2: ∂wi 1−θ2
ð19Þ
Notice Eq. (19) is similar to the condition (14), but its retail price is the function of retail margin. It can be easily seen that the second order ∂2 Πs1 2 ∂2 Πs1 ∂2 Πs2 ∂2 Πs1 ∂2 Πs2 4 ¼− b 0 and − ¼ N 0 for any θ ∈ [0, 1), implyHession matrix is negative definite because 2 2 2 2 2 1−θ ∂w1 ∂w2 ∂w2 ∂w1 ∂w1 ∂w1 ∂w2 ð1−θ2 Þ ing that a solution of Eq. (19) is a Nash equilibrium between both manufacturers. The resulting reaction functions are wi ðmi ; p3−i Þ ¼
1 ðA −mi −θðA3−i −p3−i ÞÞ; i ¼ 1; 2 2 i
or equivalently, wi ðmi ; p3−i Þ ¼ Ai −pi −θðA3−i −p3−i Þ; i ¼ 1; 2:
ð20Þ
That is, the wholesale price of channel i is positively related to its rival's retail price, while it is negatively related to own Retailer's margin(or price). Maximizing the Retailer i's profit with manufacturer i's reaction function in Eq. (20), we can get the optimal retail prices by
RS⁎ pi
¼
3
4−3θ2 Ai −θA3−i 16−9θ2
; i ¼ 1; 2;
ð21Þ
and from the manufacturers' reaction functions in Eq. (20) we can obtain the following equilibrium wholesale prices.
RS⁎ wi
1 RS⁎ ¼ pi ¼ 3
4−3θ2 Ai −θA3−i 16−9θ2
; i ¼ 1; 2:
ð22Þ
It is easy to find the nonnegative retail and wholesale prices require 2 4−3θ Ai ≥ θA3−i ; i ¼ 1; 2;
ð23Þ
528
R. Zhang et al. / Economic Modelling 29 (2012) 523–533
which implies θ ∈ [0, 1) if Ai = A3 − i. Then, the corresponding retail margins can be derived from mi = pi − wi: RS⁎
mi
¼
2 4−3θ Ai −θA3−i 16−9θ2
; i ¼ 1; 2:
ð24Þ
Furthermore, the optimal profits of Retailer i, manufacturer i, channel i's profit and entire supply chain are listed as follows.
ΠRS⁎ ri ΠRS⁎ i ΠRS⁎
2 4−3θ2 Ai −θA3−i ¼ 2ΠRS⁎ 2 ; mi ¼ 16−9θ2 1−θ2 2 3 4−3θ2 Ai −θA3−i RS⁎ ¼ 3Πmi ¼ ; 2 2 2 16−9θ 1−θ 3 16−23θ2 þ 9θ4 A2i þ A23−i −4θ 4−3θ2 Ai A3−i : ¼ 2 16−9θ2 1−θ2 2
As the previous subsections, with the assumption of identical relative channel status(A1 = A2 = 1), these results can be simplified in Table 1. In the next section, we will discuss the equilibrium prices and profit with the views of entire supply chain, manufacturers, retailers and customers, and try to find out which power structure is the best for the above members and how the products substitutability and relative channel status affect the power structure? 4. Equilibrium analysis on pricing decision This section discusses several implications of the results derived in the previous sections. In particular, we focus on which power structure will be the best for the manufacturers, retailers, channels and entire supply chain under symmetric and asymmetric channel power? Also, we compare the results from our dual exclusive channel system with those of single channel. In the following, we first compare these profits and profits among different power structures under symmetric relative channel status, that is Ω=1. Then, we will discuss the similar problems under asymmetric relative channel status, that is Ω≠1. Finally, these results are compared with that in a previous study, in which single channel system or the dual channel but with common retailer or manufacturer were considered.
The retail margins of the respective power structure are simply the differences between retail and wholesale prices, and they are in the reverse order from that of wholesale prices (shown in Fig. 3). Associated with retail prices, the realized demand levels are very close in the two Stackelberg games and larger in the VN because of the lower equilibrium retail prices. VN⁎
RS⁎
≥ Di
Di
MS⁎
≥ Di
;
which is shown in Fig. 4. Therefore, when only two Stackelberg games are considered, consumers are almost indifferent as to who is the market leader. Then, we compare the suppliers' and retailers' profits in different power structures. a manufacturer makes the largest profit in MS, and the smallest in RS (shown in Fig. 5) as expected:
4.1. Equilibrium analysis with symmetric related channel power MS⁎
Due to symmetry, here we only consider some characters with the view of Retailer i, manufacturer i, and channel i. From the Table 1, we can get some results as follows. Lemma 1. If both channels have the same related channel powers (i.e., A1 = A2 = 1),an individual member has necessarily an incentive to play the leader's role, while consumers always benefit from lower prices and realize the most demand when there is no market leadership. However, the supply chain as a whole cannot always benefit from any power structure, for example, VN is the best if and only if θ ∈ [0, 0.592), RS is the best if and only if θ ∈ [0.592, 0.682), and MS is the best if and only if θ ∈ [0.682, 1). Firstly, we compare the wholesale prices and retail prices under different power structure. The wholesale price is the highest in the MS followed by the VN and then RS (shown in Fig. 1):
VN⁎
RS⁎
Πmi ≥ Πmi ≥ Πmi : Again, equalities hold only when the wholesale price is zero, and therefore profit is zero. On the other hand, the retailer profits are the reverse order (shown in Fig. 6): RS⁎
VN⁎
Πri ≥ Πri
MS⁎
≥ Πri :
Interestingly, the differences of all members' profit under different power structure become smaller as the product substitutability grows.
Price 0.5
MS
0.4 MS⁎
wi
VN⁎
≥ wi
RS⁎
≥ wi
On the other hand, equilibrium retail prices are most indistinguishable (shown in Fig. 2). Analytically, the VN has the lowest retail price, and this insight also implies the customers wish get more welfare from the channel system without leadership. Interestingly, in the MS and RS, the retail prices are very close, which have a common starting point and end point(when θ = 0, pMS⁎ ¼ pRS⁎ ¼ 34, and when i i θ → 1, piMS ⁎ = piRS ⁎ → 0), and MS⁎
pi
RS⁎
≥ pi
VN⁎
≥ pi
:
VN
0.3
RS 0.2 0.1
0.2
0.4
0.6
0.8
1.0
θ
Fig. 1. Comparison of wholesale price wi among MS, VN and RS given A1 = A2 = 1.
R. Zhang et al. / Economic Modelling 29 (2012) 523–533
529
Profit
Price 0.7
0.12
0.6
0.10
0.5 VN
RS
MS
MS VN
0.08
0.4
RS
0.06
0.3 0.04
0.2
0.02
0.1 0.2
0.4
0.6
0.8
1.0
θ
Fig. 2. Comparison of retail prices pi among MS, VN and RS given A1 = A2 = 1.
Profit 0.5
0.2
0.4
0.6
0.8
1.0
θ
Fig. 5. Comparison of manufacturer i's profits among MS, VN and RS given A1 = A2 = 1.
Profit 0.12 RS
0.4 0.3
0.10
RS VN
0.08
VN
MS
0.06
MS 0.2
0.04 0.1
0.02 0.2
0.4
0.6
0.8
1.0
θ
0.2
0.4
0.6
θ 1.0
0.8
Fig. 3. Comparison of Retailer i's margin mi among MS, VN and RS given A1 = A2 = 1.
Fig. 6. Comparison of retailer i's profits among MS, VN and RS given A1 = A2 = 1.
Finally, we compare the profits of supply chain as a whole under different power structure. We can find that any power structure can not always ensure the supply chain getting the best performance (shown in Fig. 7), and
leadership but negatively affected by its competitor. While a new leadership will make new market for each member, it also encroaches the other member in existing market and thus intensifies the competition between them. This leaves the other members no choice but to become the leader to counteract its competitor's market encroachment. Therefore, the leadership is a dominant strategy for both members.
ΠVN⁎ ≥ ΠRS⁎ ΠRS⁎ ≥ ΠVN⁎ ΠRS⁎ ≥ ΠMS⁎ ΠMS⁎ ≥ ΠRS⁎
≥ ΠMS⁎ ; ≥ ΠMS⁎ ; ≥ ΠVN⁎ ; ≥ ΠVN⁎ ;
for for for for
θ ∈ ½0; 0:592Þ; θ ∈ ½0:592; 0:605Þ; θ ∈ ½0:605; 0:682Þ; θ ∈ ½0:682; 1Þ:
Lemma 1 is rather straightforward. Only consider the competition among retailer and manufacture in a common supply chain, it echoes the conventional wisdom that a member is rewarded with its own
Theorem 2. In the scenario of symmetric relative channel status, it is a dominant strategy and equilibrium for both members and the entire supply chain to choice the vertical Nash game. However, the entire supply chain and both members encounter a prisoner's dilemma if the product substitutability is sufficiently high (e.g., for the entire supply chain, 0.592 ≤ θ b 1 when Ω = 1).
Profit
Demand 0.50
0.4
VN
0.45 0.3 0.40
MS 0.2
VN
0.35
RS RS
0.30
MS
0.1
0.25 0.0
0.2
0.4
0.6
0.8
1.0
θ
Fig. 4. Comparison of demands in channel i, Di,among MS, VN and RS given A1 = A2 = 1.
0.2
0.4
0.6
0.8
1.0
θ
Fig. 7. Comparison of entire supply chain coefficients among MS, VN and RS given A1 = A2 = 1.
530
R. Zhang et al. / Economic Modelling 29 (2012) 523–533
Theorem 2 suggests that the VN could make the entire supply chain worsen than others, especially under higher substitutability, although it is beneficial for entire supply chain to do so when product substitutability is lower. We graphically presented this insight in Fig. 7. The rationale is explained as follows. If the channel substitutability is low, both channels behave relatively more monopolistic, and thus all players and the entire supply chain can benefit from VN game. As the product substitutability grows, the competition between channels becomes more drastic, and each player always pursue for taking the role of leader to avoid the loss of own profit. Thus, the retail prices and the wholesale prices will increase more than that in VN, and the consumers welfare descends. The Theorem 2 also suggests all players should adopt the VN game when the product substitutability is lower, adopt the RS game when it is moderate, and adopt the MS when it is higher. Furthermore, due to all members can benefit from playing the leader in games, the VN game, which implies both or no leader in the game, is an equilibrium for all members. However, the entire supply chain encounters a Prisoner's dilemma when the substitutability is higher, and all members always encounter the Prisoner's dilemma because they always prefer the Stackelberg game as a leader to that as a follower. 4.2. Equilibrium analysis with asymmetric related channel power Under the dual channel system with asymmetric relative channel power, i.e., Ω ≠ 1, we will investigate the joint effect of product substitutability and asymmetric channel power in different power structure. Through the performances comparisons among different power structure, the result can be reached as follows. Lemma 3. Under the asymmetric relative channel status(i.e.,Ω ≠ 1), an individual members has necessarily an incentive to play the leader's role MS PS ≠ MS if all members are the possible leaderships. That is, Πmi N Πmi and ΠriRS N ΠriPS ≠ RS, where i = 1, 2 and PS = MS, RS, VN in the common feasible area. Furthermore, consumers always benefit from VN. Lemma 1 is rather intuitive. Similar to the dual channel with symmetric relative channel status, all members have an incentive to play the leader's role whether their relative channel power is stronger or weaker. Meanwhile, the consumer always benefit from VN game because the intensified competition let them get a lower price and then a higher demand. Actually, the VN game is necessarily the outcome from the phenomena under which all members try to lead the channel. Furthermore, the asymmetric relative channel power does not affect the decision behaviors of all members because the related channel power depicts the situation of own channel in competition between dual channels, while the leadership depicts the power of members in own channels. An interesting question is whether the asymmetric related channel power affects the equilibrium for all members, even the entire supply chain? The following Theorem 4 will give us the answer.
members can get more profit from own-leadership Stackelberg game than that from vertical Nash game, and the entire supply chain also encounters a Prisoner's dilemma when the channel substitutability is higher. Interestingly, the Theorem 4 suggests that the channel system should choice VN, RS and MS in turn with the increase the channel substitutability when the relative channel status is symmetric (or fixed), while the channel system should also choice VN, RS and MS in turn with the increase of relative channel status when the channel substitutability is fixed (shown in Fig. 8). Meanwhile, the insight tells us a dominance situation for entire supply chain (shown in Fig. 9), the VN game is the best for the entire supply chain when the channel substitutability is lower or the relative channel status is stronger respectively, the RS game is the best when the channel substitutability and the relative channel status are moderate, the MS game is the best when the channel substitutability is higher, and otherwise, that is, the channel substitutability is higher and the relative channel status is weaker or stronger, probably it is a nonevalue area for all members. Similar to the results of previous works on single channel and dual channel with a common retailer, all members in exclusive dual channel always have an incentive to play the leader's role, however, this paper also identify the equilibrium areas in scenarios of symmetric and asymmetric relative channel status. To differ from previous works for a single channel, this paper shows that none power structure always be the best for the entire supply chain though all members on supply chain have incentive to lead the Stackelberg game. Meanwhile, the vertical Nash is an equilibrium for the members to choice their power structure, however, a Prisoner's dilemma necessarily incur for the entire supply chain when the product substitutability is moderate or higher and the asymmetric related channel power is moderate, while consumers always get the most welfare from the vertical Nash. 5. Conclusion and further research Pricing competition is a predominance aspect of the consumer goods industry. This paper investigates the effect of product substitutability and relative channel status on pricing competition under different power structure of a dual exclusive channel system where each manufacturer distributes its goods through an exclusive retailer but two goods have substitutability in demand. A linear demand based on the utility function of a representative consumer is assumed, and three game scenarios(Manufacturer Stackelberg, Retailer Stackelbeg and Vertical Nash) on power structures are examined under symmetric and asymmetric related channel powers. This paper evaluates the efficacy of pricing competition under three game power structures through an asymmetric dual exclusive channel. We identify the equilibrium areas in scenarios of symmetric and asymmetric relative channel status. To differ from previous works for a single channel, our analysis demonstrates that none power structure always be
Theorem 4. In the scenario of asymmetric relative channel status, it is a dominant strategy and equilibrium for both members and the entire supply chain to choice the vertical Nash game. However, the channel system encounters a prisoner's dilemma if the product substitutability is higher. RS−VN
ˆ For example, the RS is the best when Ω i MS−VN
ˆ MS is the best when θ N Ω i
MS−VN
ˆ ðθÞ b θ b Ω i
ðθÞ, and the
ðθÞ, where i = 1, 2.
The Theorem 4 is similar to the Theorem 2, but it depicts that the channel substitutability and asymmetric channel power how to influence the decision behaviors of all members. Due to that all members necessarily have an incentive to lead the Stackelberg game, the vertical Nash game, which implies no leadership in game and also implies all player are leaders, is an equilibrium for all members. However, all members always encounter a Prisoner's dilemma because all
VN
RS
MS
Fig. 8. Comparison of entire supply chain efficiencies among MS, VN and RS.
R. Zhang et al. / Economic Modelling 29 (2012) 523–533
531
Proof of Lemma 3. Due to similarity, here we just investigate the profit comparison of Retailer 1 and Manufacturer 1, respectively, under different power structure PS. From the conditions (12), (17) and (23) to ensure nonnegative retail prices and wholesale prices, we can get the Common Feasible Area(CFA) is: Ω Z
Fig. 9. Dominance situation for entire supply chain among MS, VN and RS.
the best for the entire supply chain though all members on supply chain have incentive to lead the Stackelberg game. Meanwhile, the vertical Nash could be an equilibrium for the members and the entire supply chain to choice their power structure, however, a Prisoner's dilemma necessarily incur for the entire supply chain because with modertae or higher substitutability and moderate asymmetric relative channel power the Retailer Stackelberg or the manufacturer Stackelberg can gain the better performance than that in vertical Nash for the entire supply chain, while consumers always get the most welfare from the vertical Nash. While our work provides a general framework for future study on pricing competition on dual channel system, there are some future research priorities. First, due to limited space, this paper has been focused on pricing competition of the asymmetric dual exclusive channel. In a separate work, we have been investigating the effort effect and pricing competition in similar channel structures, and considered retailer effort, manufacturer effort and hybrid effort respectively. Secondly, this paper inherits the Stackelberg game setting from Doraiswamy et al. (1979), McGuire and Staelin (1983) and many others. Nevertheless, it is well known that a different game setting might alter some of our results especially quantitatively (see Choi (1991) and Xie and Neyret (2009). It is certainly interesting to explore the disparity among different game settings under different decision mode, such as quantity competition. Finally, other asymmetric factors can be considered. Such as, the analysis of asymmetric operational costs together with asymmetric initial base demand can be linearly transformed into a new relative channel status variable and all qualitative results hold.
CFA
ðθÞ ¼
θ 3−2θ2
2
and
¯ CFA ðθÞ ¼ 3−2θ : Ω θ
θ 2−θ2 θ θ because ≥ ≥ and they equate each other 3−2θ2 4−3θ2 8−9θ2 þ 2θ4 when θ = 0 and θ = 1. In the following, we compare Manufacturer 1's profits in different cases. MS − VN To compare case MS and VN, we temporarily define ΔΠm1 as the Manufacturer 1's profits in MS minus the one in VN. Reorganizing A1 MS − VN and solving ΔΠm1 = 0 yields two roots: A1 and A2 into Ω ¼ A2 ˆ MS−VN ðθÞ ¼ Ω m1−1 ˆ MS−VN Ω m1−2 ðθÞ ¼
K1 þ K2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8−6θ2 þ θ4
K3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi K1 −K2 8−6θ2 þ θ4 K3
;
;
where þ 766θ3 −940θ5 þ 549θ7 −152θ9 þ 16θ11 ; K1 ¼ −240θ K2 ¼ θ 144−361θ2 þ 321θ4 −120θ6 þ 16θ8 ; K3 ¼ 576−1776θ2 þ 2087θ4 −1168θ6 þ 312θ8 −32θ10 : MS−VN
MS−VN
CFA ˆ ˆ Due to complexity, we can get Ω ðθÞ N Ω Z m1−1 ðθÞ N Ωm1−2 ðθÞ with countour plots, which is a tool of Mathematica. Thus, the Manufacturer 1 always prefers MS to VN. MS − RS To compare case MS and RS, we can define ΔΠm1 as the Manufacturer 1's profits in MS minus the one in RS, and get two roots MS − RS which meet ΔΠm1 = 0.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8−6θ2 þ θ4 ¼ ; L3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L1 −L2 8−6θ2 þ θ4 ˆ MS−RS ; Ω m1−2 ðθÞ ¼ L3 L1 þ L2
ˆ MS−RS ðθÞ Ω m1−1
where Acknowledgements The authors are grateful to Editors and anonymous referee for their very valuable comments and suggestions, which have significantly helped improving the quality of this paper. The author gratefully acknowledges support from the National Science Foundation of China through grants 70871035 and 71171074, the Program for Science and Technology Innovation Talents in the Universities of Henan Province through grant 2010HASTIT030, and Nature Science Fund of Henan Education Committee through grants 2008A630020 and 2009A630022. Appendix A
Proof of Lemma 1. It is straightforward, though a little tedious, to get these results from Table 1. Here we omit these process, but they are illustrated by Figs. 5–7.□. Proof of Theorem 2. The Theorem 2 is a direct result from Lemma 1.□.
3
5
7
9
11
L1 ¼ 4096θ−13824θ þ 19344θ −14356θ þ 5937θ −1289θ 13
þ 114θ ; L2 ¼ θ3 256−672θ2 þ 633θ4 −253θ6 þ 36θ8 ; L3 ¼ 16384−63488θ2 þ 103488θ4 −92080θ6 þ 48342θ8 −14976θ10 þ 2532θ12 −180θ14 : MS−RS
MS−RS
CFA ˆ ˆ We can also prove Ω ðθÞN Ω Z m1−1 ðθÞN Ωm1−2 ðθÞ with countour plots. Thus, the Manufacturer 1 prefers MS to RS. In summary, the Manufacturer 1 always has an incentive to play the leader's role. In the following, we compare Retailer 1's profits in different cases. RS − VN To compare case RS and VN, we temporarily define ΔΠr1 as the Retailer 1's profits in RS minus the one in VN. Solving RS − VN ΔΠr1 = 0 yields two roots:
pffiffiffi M1 þ M2 2 ; M3 pffiffiffi M −M2 2 RS−VN ˆ r1−2 ðθÞ ¼ 1 ; Ω M3
ˆ RS−VN ðθÞ ¼ Ω r1−1
532
R. Zhang et al. / Economic Modelling 29 (2012) 523–533
where 3
5
7
9
11
M1 ¼ 16384θ−49152θ þ 57152θ −30968θ þ 6348θ þ 848θ 13
15
−565θ þ 66θ ; M2 ¼ θ3 2048−6912θ2 þ 9352θ4 −6494θ6 þ 2439θ8 −469θ10 þ 36θ12 ;
Proof of Theorem 4. Using the similar method in Lemma 3, here we compare the entire supply chain efficiency among MS, RS and VN. To compare the VN with MS. We temporarily define ΔΠ VN − MS as the channel system's profits in VN minus the one in MS. Reorganizing A1 and A2 into Ω ¼ AA12 and solving ΔΠ VN − MS = 0 yields two roots:
M3 ¼ 65536−253952θ2 þ 410880θ4 −357440θ6 þ 178620θ8 −50108θ10 þ 6769θ12 −156θ14 −36θ16 : RS−VN
RS−VN
ˆ ˆ Due to complexity, we can get Ω ðθÞ N Ω Z r1−1 ðθÞ N Ωr1−2 ðθÞ. Thus, the Retailer 1 always prefers RS to VN. RS − MS To compare case RS and MS, we can define ΔΠs1 as the Retailer 1's profits in RS minus the one in MS, and get two roots which meet RS − MS ΔΠr1 = 0. CFA
pffiffiffi N1 −N2 2 ; N3 pffiffiffi N þ N2 2 RS−MS ˆ r1−2 ðθÞ ¼ 1 ; Ω N3
RS−MS
RS−MS
ˆ ˆ We can also prove Ω ðθÞN Ω Z r1−1 ðθÞN Ωr1−2 ðθÞ with countour plots. Thus, the Retailer 1 prefers RS to MS. In summary, the Retailer 1 always has an incentive to play the leader's role. Due to symmetry, we can prove the Retailer 2 and Manufacture 2 have an incentive to play his(her) leader's roles, respectively. To consider the consumers welfare. To compare case VN and MS, we can define ΔU VN − MS as the utility in VN minus the one in MS.
Þ¼
O3
Actually, O1(A12 + A22) ≥ 2O1A1A2, and it is easy to find 2O1 + O2 ≥ 0 4 with countour plot. So, consumers always prefer VN to MS. Similarly, we can define ΔU VN − RS as the utility in VN minus the one in RS.
2o3
4
6
8
10
12
o1 ¼ 16384−58368θ þ 83648θ −61468θ þ 24401θ −4973θ þ 408θ ; 2 4 6 8 10 12 o2 ¼ 2θ 4096−17920θ þ 29456θ −23740θ þ 10027θ −2131θ þ 180θ ; 2 o3 ¼ 1−θ2 1024−1920θ2 þ 1284θ4 −361θ6 þ 36θ8 :
Actually, o1(A12 + A22) ≥ 2o1A1A2, and it is easy to find 2o1 + o2 ≥ 0 5 with countour plot. So, consumers always prefer VN to RS. In summary, consumers always prefer VN to others power structure. □. 2O1 + O2 = 0 only when θ = 0 2o1 + o2 = 0 only when θ = 0.
¼
ˆ RS−VN ðθÞ ¼ Ω 2
n1 þ n2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 256−880θ2 þ 841θ4 −300θ6 þ 36θ8
n3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 n1 −n2 − 256−880θ þ 841θ4 −300θ6 þ 36θ8 n3
;
;
where n1 ¼ 20480θ−68096θ3 þ 91600θ5 −63512θ7 þ 23882θ9 −4620θ11
RS−MS
ˆ We have Ω 1
RS−MS
ˆ ðθÞ ≥ Ω 2
VN−MS
ðθÞ and only equates if Ω = 1. Mean-
RS−MS
VN−MS
RS−MS
ˆ ˆ ˆ ˆ ðθÞ N Ω ðθÞ and Ω ðθÞ b Ω ðθÞ. while, Ω 1 1 1 1 Therefore, for the entire supply chain system the VN dominate VN−MS
ˆ others when Ω N Ω 1 others when
VN−MS
ˆ Ω i
RS−MS
ˆ ðθÞ b Ω b Ω 1
RS−MS
ˆ others when Ω b Ω 1
VN−MS
ˆ ðθÞ and Ω b Ω 2
ðθÞ, and the MS dominate
RS−MS
ˆ ðθÞ and Ω N Ω 2
ðθÞ, the RS dominate
ðθÞ. □
References
θ2 o1 A21 þ A22 þ o2 A1 A2
where
5
ˆ RS−MS ðθÞ Ω 1
þ 2982θ12 −216θ14 :
2 O1 ¼ 4032−13844θ þ 19139θ4 −13562θ6 þ 5185θ8 −1016θ10 þ 80θ12 ; 2 O2 ¼ 2θ 480−2728θ þ 4993θ4 −4217θ6 þ 1810θ8 −384θ10 þ 32θ12 ; 2 O3 ¼ 1−θ2 576−1012θ2 þ 633θ4 −168θ6 þ 16θ8 :
2
VN−MS
þ 360θ13 ; n2 ¼ 1024−2944θ2 þ 3204θ4 −1645θ6 þ 397θ8 −36θ10 ; n3 ¼ 16384−62464θ2 þ 103616θ4 −95980θ6 þ 52865θ8 −17093θ10
2 O1 A21 þ A22 þ O2 A1 A2
where
¼
;
where
VN−MS
CFA
4
k3
;
ˆ ˆ We have Ω ðθÞ ≥ Ω ðθÞ and only equates if Ω = 1. 1 2 To compare the RS with MS. We temporarily define ΔΠ RS − MS as the channel system's profits in RS minus the one in MS. Solving ΔΠ RS − MS = 0 yields two roots:
N3 ¼ −288 þ 528θ2 −217θ4 −60θ6 þ 36θ8 :
VN−MS
k3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k1 −k2 − 100−396θ þ 385θ4 −136θ6 þ 16θ8
þ 880θ12 −64θ14 :
3 N1 ¼ 20θ−314θ þ 259θ5 −66θ7 ; N2 ¼ θ 144−289θ2 þ 181θ4 −36θ6 ;
ΔU
ˆ MS−VN ðθÞ ¼ Ω 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 100−396θ2 þ 385θ4 −136θ6 þ 16θ8
þ 128θ13 ; k2 ¼ 576−1588θ2 þ 1645θ4 −801θ6 þ 184θ8 −16θ10 ; k3 ¼ 5760−20216θ2 þ 31654θ4 −28441θ6 þ 15509θ8 −5020θ10
where
VN−MS
¼
k1 þ k2
k1 ¼ 9024θ−28840θ3 þ 37344θ5 −24962θ7 þ 9064θ9 −1696θ11
ˆ RS−MS ðθÞ ¼ Ω r1−1
ΔU
ˆ VN−MS ðθÞ Ω 1
Almehdawe, E., Mantin, B., 2010. Vendor managed inventory with a capacitated manufacturer and multiple retailers: retailer versus manufacturer leadership. International Journal of Production Economics 128 (1), 292–302. Cai, G., 2010. Channel selection and coordination in dual-channel supply chains. Journal of Retailing 86 (1), 22–36. Cai, G.G., Zhang, Z., Zhang, M., 2009. Game theoretical perspectives on dual-channel supply chain competition with price discounts and pricing schemes. International Journal of Production Economics 117 (1), 80–96. Choi, S.C., 1991. Pricing competition in a channel structure with a common retailer. Marketing Science 10, 271–296. Choi, S.C., 1996. Pricing competition in a duopoly common retailer channel. Journal of Retailing 72 (2), 117–134. Doraiswamy, K., McGuire, T.W., Staelin, R., 1979. An analysis of alternative advertising strategies in a competitive franchise framework. In: N.B., et al. (Ed.), 1979 Educators' Conference Proceedings. American Marketing Associtation, Chicago, pp. 463–467. Huang, Z., Li, S.X., 2001. Co-op advertising models in a manufacturer–retailer supply chain: a game theory approach. European Journal of Operational Research 135 (3), 527–544. Ingene, C.A., Parry, M.E., 2004. Mathematical Models of Distribution Channels. Springer.
R. Zhang et al. / Economic Modelling 29 (2012) 523–533 Ingene, C., Parry, M., 2007. Bilateral monopoly, identical competitors/distributors, and game-theoretic analyses of distribution channels. Journal of the Academy of Marketing Science 35 (4), 586–602. Jeuland, A.P., Shugan, S.M., 1983. Managing channel profits. Marketing Science 2 (3), 239–272. Jeuland, A.P., Shugan, S.M., 1988. Channel of distribution profits when channel members form conjectures. Marketing Science 7 (2), 202–210. Karjalainen, R., 1992. The Newsboy Game. University of Pennsylvania working paper. Lee, J.-Y., Ren, L., 2011. Vendor-managed inventory in a global environment with exchange rate uncertainty. International Journal of Production Economics 130 (2), 169–174. Lippmand, S., Macardle, K., 1997. The competitive newsboy. Operations Research 45 (1), 54–65.
533
Lu, J.-C., Tsao, Y.-C., Charoensiriwath, C., 2011. Competition under manufacturer service and retail price. Economic Modelling 28 (3), 1256–1264. McGuire, T., Staelin, R., 1983. An industry equilibrium analysis of downstream vertical integration. Marketing Science 2 (2), 161–191. Paralar, M., Wang, D., 1993. Diversification under yield randomness in inventory models. European Journal of Operational Research 66 (1), 52–64. Shugan, S.M., 1985. Implicit understanding in channels of distribution. Management Science 31 (4), 435–460. Xie, J., Neyret, A., 2009. Co-op advertising and pricing models in manufacturercretailer supply chains. Computers and Industrial Engineering 56, 1375–1385. Zusman, P., Etgar, M., 1981. The marketing channel as an equilibrium set of contracts. Management Science 27 (2), 284–302.
Contents lists available at SciVerse ScienceDirect
Economic Modelling journal homepage: www.elsevier.com/locate/ecmod
Pricing decisions in a dual channels system with different power structures Rong Zhang, Bin Liu ⁎, Wenliang Wang Department of Management Science, Henan Agricultural University, Zhengzhou, 450002, China
a r t i c l e
i n f o
Article history: Accepted 31 August 2011 Keyword: Dual channels Pricing decisions Asymmetric distribution channel Game theory Equilibrium
a b s t r a c t This paper investigates the effect of product substitutability and relative channel status on pricing decisions under different power structures of a dual exclusive channel system where each manufacturer distributes its goods through a single exclusive retailer but two goods are substitute. A linear demand based on the utility function of a representative consumer is assumed, and three game scenarios(Manufacturer Stackelberg, Retailer Stackelbeg and Vertical Nash) are examined under symmetric and asymmetric related channel status. It is shown that no power structure is always the best for the entire supply chain though all members on supply chain have incentive to lead the Stackelberg game. Meanwhile, the vertical Nash game is an equilibrium for the members, however, a Prisoner's dilemma necessarily incurs for the entire supply chain because the Retailer Stackelberg or the Manufacturer Stackelberg can gain the better performance than that in vertical Nash for the entire supply chain when the product substitutability is moderate or higher and the asymmetric relative channel status is moderate, while consumers always get the most welfare from the vertical Nash game. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Pricing decisions has significantly reshaped supply chain behavior in the last two decades, especially for dual or multiple supply chain system. Due to supply chains are generally comprised of individual members who are often guided by conflicting objective functions, there are always two topics revolving of dual channel system. One is who, retailer and manufacturer, should be the leadership in Stackelberg game in each channel, and the other is how the competition between both channels affects pricing decision. Actually, the first topic concerns the power structure, which refers to a member's relative ability to control the decision making process in the supply chain. For example, Wal-Mart usually is considered to be the Stackelberg leader (RetailerStackelberg) and his manufacturers are followers (Almehdawe and Mantin (2010)), Apple is also viewed as to be the Stackelberg leader(Manufacturer-Stackelberg) (Lee and Ren (2011)), however, Wal-Mart and Procter & Gamble (P&G) are almost equal in bargaining(Vertical Nash)(Almehdawe and Mantin (2010)). Specifically, the more powerful firm moves first in a Stackelberg game. The second topic investigates the competition of dual channels with different relative channel status, such as symmetric channel (Doraiswamy et al. (1979)) and asymmetric channel (Choi (1996)). This paper considers a two-echelon distribution system that consists of two manufacturer and two retailers, and each manufacturer sells a product exclusively through a franchised retailer. However,
⁎ Corresponding author. E-mail address: [email protected] (B. Liu). 0264-9993/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econmod.2011.08.024
two products have similar function with a higher substitutability, and the demand from customers is met through the retailer. Such a system is called a dual exclusive channel distribution system. The objective of this paper is not only to analyze the pricing decisions of dual exclusive channel system but also to concern the power balance between retailers and manufacturers. E.g. which power structure is the best for the retailers, manufacturers, entire supply chain and customers, respectively, under pricing competition? And how does the product substitutability and asymmetric relative channel power affect the optimal power balance? Several studies have been done on analyzing the interactions between manufacturers and retailers in literature. Different objectives of channels members, however, create conflicts within a channel. Numerous scholars have focused on vertical coordinations among channel members through such measures as various transfer pricing schemes or formal agreements (such as, Zusman and Etgar (1981), McGuire and Staelin (1983), Jeuland and Shugan (1983)), implicit understanding(such as, Shugan (1985)), and formation of conjectures(such as, Jeuland and Shugan (1988)) to achieve maximum channel profit. Many of these studies have considered only one manufacturer (monopoly) and its channel intermediaries, and the analysis of competition and cooperation is confined to members in the same channel. An area often neglected in the previous studies is the market structure in which retailers retain more power than manufacturer do in the retail price decision. While most channel studies have focused on the manufacturer dominance over retailers, an intermediary that deals with a number of competing products is often a large retailer that can influence the market substantially. Therefore, this paper also focuses on the power balance between the channel members
524
R. Zhang et al. / Economic Modelling 29 (2012) 523–533
and its effects on equilibrium prices and profits. Consequently, besides the two most popular power structures in the previous studies (i.e. the Manufacturer Stackelberg and the Nash Game) a third game is also considered in this paper: the Retailer Stackelberg. The interested reader can refer these game settings to Choi (1991) and Huang and Li (2001). Manufacturer Stackelberg (MS). Each manufacturer chooses the wholesale price using the response function of the retailer. Each retailer determines the retail price of each product so as to maximize his profits given the respective wholesale price. Vertical Nash (VN). Each manufacturer chooses its wholesale price conditional on the retailers' selling prices of dual channels. The retailers determine their optimal selling prices conditional on respective wholesale prices. Retailer Stackelberg (RS). Each manufacturer chooses its wholesale price conditional on both retailers selling prices. The retailers set up their selling prices using the reaction functions of both manufacturers in terms of respective wholesale prices. On the power structures, most results in literature is the member always benefits from playing the leader's role in Stackelberg game, such as McGuire and Staelin (1983) for a single channel system, Choi (1991) and Choi (1996) for dual channel with common retailer and so on. Differ from the previous works, this paper investigates the same issues but for an exclusive dual channel with symmetric and asymmetric relative channel status. Another area related to this paper is channel competition of dual channel supply chain system. On dual channels, Paralar and Wang (1993) develop a two-firm competitive newsboy model where the firms face independent random demands. In a paper closely related to Paralar's, Karjalainen (1992) also analyzes the case where independent firm demands are aggregated to form industry demand. Then, Choi (1996) deals with a channel structure in which there are duopoly manufacturers and duopoly common retailers, and Lippmand and Macardle (1997) considers a competitive version of the classical newsboy problem in which a firm must choose an inventory or production level for a perishable good with random demand. The optimal solution is a fractal of the demand distribution and to investigate the impact of competition upon industry inventory. Cai et al. (2009) evaluates the impact of price discount contracts and pricing schemes on the dual-channel supply chain competition, meanwhile, from supplier-Stackelberg, retailerStackelberg, and Nash game theoretic perspectives, also show that the scenarios with price discount contracts can outperform the non-contract scenarios. Cai (2010) investigates the influence of channel structures and channel coordination on the supplier, the retailer, and the entire supply chain in the context of two singlechannel and two dual-channel supply chains. Lu et al. (2011) highlights the importance of service from manufacturers in the interactions between two competing manufacturers and their common retailer, facing end consumers who are sensitive to both retail price and manufacturer service, and a game-theoretic framework is applied to obtain the equilibrium solutions for every entity. Note that the logic underlying our model is distinctly different from that used by Paralar and Wang (1993), Karjalainen (1992) or Lippmand and Macardle (1997) in their models the firm demands are independent, whereas in our model they are dependent on selling prices of both channels and are substitutes.To differ from Choi (1996), our paper investigates a dual exclusive dual channels, and to differ from Cai (2010),our paper concerns the pricing decisions under different power structures. Based on the above literature, this paper investigates the price competition of an exclusive dual channel system with symmetric and asymmetric relative channel status, and applies three game
models to analyze the influences of pricing decisions on product substitutability and channel power with the view of manufacturers, retailers, the entire supply chain and customers respectively. To differ from the previous literature, this paper shows no power structure is always the best for the entire supply chain though all members on supply chain have incentive to lead the Stackelberg game. Meanwhile, the vertical Nash game is an equilibrium for the members, however, a Prisoner's dilemma necessarily incurs for the entire supply chain when the product substitutability is moderate or higher and the asymmetric relative channel status is moderate, while consumers always get the most welfare from the vertical Nash game. The remainder of this paper is organized as follows. We first describe the model in Section 2, and establish the price competition models in Section 3. Then, we analyze the equilibrium entire supply chain efficient, consumers welfare, and others effects in Section 4. Lastly, we summarize the results and provide some further research topics related to the dual channels system in Section 5. 2. The model We mainly explore the price competition in the dual exclusive channel system. We use Di, i = 1, 2, to represent the demand to product/channel i. 1 Correspondingly, the retail prices are denoted by pi, i = 1, 2, and the wholesale prices are wi, i = 1, 2.We assume the initial base demand for product i is Ai. To obtain logically consistent demand functions in different channel structures, we adopt the framework established by Ingene and Parry (2004) (Chapter 11) and Ingene and Parry (2007) and employ a utility function of a representative consumer as follows: 2 2 U≡ ∑ Ai Di −Di =2 −θD1 D2 − ∑ pi Di ; i¼1;2
i¼1;2
ð1Þ
where θ (0 ≤ θ b 1) denotes the channel substitutability. When θ = 0, the products are purely monopolistic; while θ goes to 1, the products converge to purely substitutable. The channels are demand interdependent (unless θ = 0). The aggregate demand decreases in θ. Maximization of Eq. (1) yields the demand for product i as follows. Di ¼
Ai −θA3−i −pi þ θp3−i ; i ¼ 1; 2: 1−θ2
ð2Þ
To facilitate our discussion, we define Ω≡
A1 A2
as the relative channel status of product 1 over product 2 in the dual exclusive channel. If Ω N 1, product 1 is superior to product 2 in terms of the initial base demand, and vice versa. Ω = 1 implies the channel system is symmetric in the relative channel status, and Ω ≠ 1 means the channel system is asymmetric.
1 The terms of product and channel are used interchangeably for the scenarios of dual exclusive, such as Cai (2010), and monopoly common retailer channels, such as Choi (1991). The impact of different retailers on the demand in the dual exclusive channel is assumed away in order to compare the impact of the above two channel structures, but is restored in the duopoly common retailer channel. 2 As stated by Ingene and Parry (2004), a system of linear demand curves must have the form of the underlying utility function.
R. Zhang et al. / Economic Modelling 29 (2012) 523–533
The term Πmi denotes the manufacturer i's profit, Πri is Retailer i's profit, Πi is the channel i's profit, and Π is the total profit for the entire supply chain including all manufacturers and retailers. Meanwhile, the superscript MS represents for the Manufacturer Stackelberg scenario, RS represents for the Retailer Stackelberg scenario, VN represents for the Vertical Nash scenario, and ⁎ represents the optimization. We specify each profit function and the corresponding feasible domain when the need arises. Product i's cost is normalized to zero. For the parsimony purpose, we also normalize the channel operational costs to zero. 3 In the dual exclusive channel case, each manufacturer sells its product through its own downstream retailer. The manufacturers' and the retailers' profit functions can be written as Πmi ¼ wi Di ;
ð3Þ
Πri ¼ mi Di ;
ð4Þ
525
where mi = pi − wi and i = 1, 2, the retail margin on product i. Further, the supply chain i's profit function is Πi ¼ Πmi þ Πri ;
ð5Þ
and the total profit profit for the entire supply chain including all manufacturers and retailers is 2
Π ¼ ∑ Πi :
ð6Þ
i¼1
As in many previous studies, each member of the channel is assumed to seek for maximizing its own profit and and no cooperation is assumed between the members. This is the most common and legal institutional arrangement of the channel structure under consideration. We now derive analytical equilibrium solutions for this base model under each power structure scenario.
3. Price competition under different power structures 3.1. Manufacturer Stackelberg (MS) Under the MS, the manufacturers take their retailer's reaction function into consideration for their respective price decisions. Retailer i's reaction function given wholesale prices wi can be derived from the First Order Conditions (FOC) of Eq. (4).
∂Πri Ai −θA3−i −2pi þ θp3−i þ wi ¼ ¼ 0; i ¼ 1; 2; ∂pi 1−θ2
ð7Þ 2
2
2
2
2
2
∂ Πr1 2 ∂ Πr1 ∂ Πr2 ∂ Πr1 ∂ Πr2 4−θ ¼− b 0 and − ¼ 2 N 0 for any θ ∈ [0, 1), ∂p1 ∂p2 ∂p2 ∂p1 1−θ2 ∂p21 ∂p21 ∂p22 1−θ2 satisfying the second order condition for a maximum. From Eq. (7), retailer i's reaction functions can be derived: for which the Hessian matrix is negative due to
pi ðwi ; w3−i Þ ¼
2ðAi þ wi Þ−θðA3−i þ θAi −w3−i Þ ; i ¼ 1; 2; 4−θ2
ð8Þ
which is linear in both wholesale prices. These functions imply that the retail prices depend on not only the wholesale price of own channel but also that of rival channel. By the reaction function (8), the manufacturer i's Nash equilibrium wholesale prices can be reached from the following FOC of the manufacturer i's profit maximization problems: 2 2 2−θ Ai −θA3−i −2 2−θ wi þ θw3−i ∂ Π ðw ; w Þ ¼ ; i ¼ 1; 2: ∂wi mi i 3−i 4−5θ2 þ θ4
ð9Þ
Solving Eq. (9)= 0 results into the following whole prices MS⁎ wi
¼
8−9θ2 þ 2θ4 Ai −θ 2−θ2 A3−i 16−17θ2 þ 4θ4
; i ¼ 1; 2;
ð10Þ
and the corresponding retail prices can obtain from Eq. (8):
MS⁎ pi
¼
8−9θ2 þ 2θ4 Ai −θ 2−θ2 A3−i 2 3−θ2 64−84θ2 þ 33θ4 −4θ6
; i ¼ 1; 2:
ð11Þ
Certainly, the nonnegative retail prices and wholesale prices require 2 4 2 8−9θ þ 2θ Ai ≥ θ 2−θ A3−i ; i ¼ 1; 2;
ð12Þ
3 We have also analyzed the cases with asymmetric non-zero operational costs and found that all qualitative results hold. In fact, with non-zero operational costs, we can redefine 1 Ω≡ AA12 −c −c2 , where ci denotes the operational cost in channel i, and the corresponding discussion falls through similarly.
526
R. Zhang et al. / Economic Modelling 29 (2012) 523–533
which implies θ ∈ [0, 1) if Ai = A3 − i. So, we yield the common feasible area for θ is θ ∈ [0, 1), and will be narrower because of more asymmetry if Ai ≠ A3 − i. Then, the Retailer i's margin profit can be reached MS⁎ mi
¼
2
2−θ
2 4 2 8−9θ þ 2θ Ai −θ 2−θ A3−i 64−84θ2 þ 33θ4 −4θ6
; i ¼ 1; 2:
ð13Þ
Furthermore, it is easy to get the optimal profits of manufacture i, Retailer i, channel i and entire supply chain as follows.
ΠMS⁎ mi ΠMS⁎ ri ΠMS⁎ i ΠMS⁎
2 8−9θ2 þ 2θ4 Ai −θ 2−θ2 A3−i 2−θ2 ¼ ; 2 4−5θ2 þ θ4 16−17θ2 þ 4θ4 2 2 2−θ2 8−9θ2 þ 2θ4 Ai −θ 2−θ2 A3−i ¼ ; 2 1−θ2 64−84θ2 þ 33θ4 −4θ6 2 8−9θ2 þ 2θ4 Ai −θ 2−θ2 A3−i 2 6−5θ2 þ θ4 ¼ ; 2 1−θ2 64−84θ2 þ 33θ4 −4θ6 64−140θ2 þ 109θ4 −35θ6 þ 4θ8 A21 þ A22 −4θ 2−θ2 8−9θ2 þ 2θ4 A1 A2 2 6−5θ2 þ θ4 ¼ : 2 1−θ2 64−84θ2 þ 33θ4 −4θ6
Especially, when the related channel powers are assumed equal(A1 = A2 = 1), the problem becomes symmetric, which is much easier to solve. This symmetry is meaningful because it presents the most drastic competition market. Under the symmetry assumption, all optimal wholesale prices, retail prices, manufacturer i's profit, Retailer i's profit, channel i's profit, entire supply chain's profit and Retailer i's margin profit are summarized in Table 1. 3.2. Vertical Nash (VN) The assumption VN implies a Nash equilibrium among all members. Here each manufacturer takes as given the competing retail price and the retailer's margin on its own product, whereas the retailer conditions its margins on wholesale prices. The FOC for this equilibrium involves the above equation of retailer's profits maximization condition (7) and the following manufacturer i's profit maximization condition: ∂Πmi Ai −pi −wi −θðA3−i −p3−i Þ ¼ ¼ 0; i ¼ 1; 2: ∂wi 1−θ2
ð14Þ
∂2 Πm1 1 ¼− b 0 and It is easily established that the second order Hessian matrix of this problem is negative definite due to 1−θ2 ∂w21 2 2 2 2 ∂ Πm1 ∂ Πm2 ∂ Πm1 ∂ Πm2 1 − ¼ N 0 for any θ ∈ [0, 1), implying a solution for the Eqs. (7) and (14) is a Nash equilibrium. The 2 ∂w1 ∂w2 ∂w2 ∂w1 ∂w21 ∂w22 ð1−θ2 Þ resulting equilibrium retail and wholesale prices are
VN⁎ wi
¼
3−2θ2 Ai −θA3−i 9−4θ2
;
ð15Þ
Table 1 Equilibrium prices and profits under different power structures given A1 = A2 = 1. Equilibrium
MS
VN
RS
w⁎i
2−θ−θ2 4−θ−2θ2 2ð1−θÞ 3−θ2 8−6θ−3θ2 þ 2θ3
1−θ 3−2θ
1−θ 4−3θ
2ð1−θÞ 3−2θ
3ð1−θÞ 4−3θ
1 3 þ θ−2θ2
1 4 þ θ−3θ2
p⁎i D⁎i ΠMi⁎ ΠRi⁎ Π⁎i Π⁎ m⁎i
2−θ2 8 þ 2θ−9θ2 −θ3 þ 2θ4 2−θ2 2−θ−θ2 ð2 þ θ−θ2 Þð4−θ−2θ2 Þ 2 ð1−θÞ 2−θ2
2
2
ð1 þ θÞð8−6θ−3θ2 þ 2θ3 Þ 2 6−6θ−5θ2 þ 5θ3 þ θ4 −θ5
1−θ ð4−3θÞ2 ð1 þ θÞ
1−θ
2ð1−θÞ
ð3−2θÞ2 ð1 þ θÞ
ð4−3θÞ2 ð1 þ θÞ
2ð1−θÞ
3ð1−θÞ
2
ð3−2θÞ2 ð1 þ θÞ
ð4−3θÞ2 ð1 þ θÞ
4ð1−θÞ
6ð1−θÞ
2
ð3−2θÞ2 ð1 þ θÞ
ð4−3θÞ2 ð1 þ θÞ
1−θ 3−2θ
2ð1−θÞ 4−3θ
ð2−θÞ2 ð1 þ θÞð4−θ−2θ2 Þ 4 6−6θ−5θ2 þ 5θ3 þ θ4 −θ5 ð2−θÞ2 ð1 þ θÞð4−θ−2θ2 Þ ð1−θÞ 2−θ2 2 8−6θ−3θ þ 2θ3
1−θ ð3−2θÞ2 ð1 þ θÞ
R. Zhang et al. / Economic Modelling 29 (2012) 523–533
VN⁎ pi
¼
VN⁎ 2wi
¼
2 3−2θ2 Ai −2θA3−i 9−4θ2
:
527
ð16Þ
for i = 1, 2. Certainly, the nonnegative retail and wholesale prices require 2 3−2θ Ai ≥ θA3−i ; i ¼ 1; 2;
ð17Þ
which implies θ ∈ [0, 1) if Ai = A3 − i. Then, the Retailer i's margin profit can be reached VN⁎
¼
mi
3−2θ2 Ai −θA3−i 9−4θ2
; i ¼ 1; 2:
ð18Þ
Furthermore, the optimal profits of manufacture i, Retailer i, channel i and entire supply chain are listed as follows.
ΠVN⁎ mi ΠVN⁎ i ΠVN⁎
2 3−2θ2 Ai −θA3−i ¼ ΠVN⁎ ¼ 2 ; ri 9−4θ2 1−θ2 2 2 3−2θ2 Ai −θA3−i VN⁎ ; ¼ 2Πmi ¼ 2 9−4θ2 1−θ2 2 9−11θ2 þ 4θ4 A2i þ A23−i −4θ 3−2θ2 Ai A3−i : ¼ 2 9−4θ2 1−θ2
Especially, under the symmetry assumption(A1 = A2 = 1), all optimal wholesale prices, retail prices, manufacturer i's profit, Retailer i's profit, channel i's profit, entire supply chain's profit and Retailer i's margin profit are summarized in Table 1. 3.3. Retailer Stackelberg (RS) Under the assumption RS, the retailers become the leader and the manufacturers are the followers. In this market, the leaders take the followers' reaction functions into account for their own retail prices decisions. Manufacturer i conditions her wholesale price on the retail margin mi and the rival's price p3 − i. The manufacturer i's reaction function can be derived from the following FOC: ∂Πmi Ai −mi −2wi −θðA3−i −p3−i Þ ¼ ¼ 0; i ¼ 1; 2: ∂wi 1−θ2
ð19Þ
Notice Eq. (19) is similar to the condition (14), but its retail price is the function of retail margin. It can be easily seen that the second order ∂2 Πs1 2 ∂2 Πs1 ∂2 Πs2 ∂2 Πs1 ∂2 Πs2 4 ¼− b 0 and − ¼ N 0 for any θ ∈ [0, 1), implyHession matrix is negative definite because 2 2 2 2 2 1−θ ∂w1 ∂w2 ∂w2 ∂w1 ∂w1 ∂w1 ∂w2 ð1−θ2 Þ ing that a solution of Eq. (19) is a Nash equilibrium between both manufacturers. The resulting reaction functions are wi ðmi ; p3−i Þ ¼
1 ðA −mi −θðA3−i −p3−i ÞÞ; i ¼ 1; 2 2 i
or equivalently, wi ðmi ; p3−i Þ ¼ Ai −pi −θðA3−i −p3−i Þ; i ¼ 1; 2:
ð20Þ
That is, the wholesale price of channel i is positively related to its rival's retail price, while it is negatively related to own Retailer's margin(or price). Maximizing the Retailer i's profit with manufacturer i's reaction function in Eq. (20), we can get the optimal retail prices by
RS⁎ pi
¼
3
4−3θ2 Ai −θA3−i 16−9θ2
; i ¼ 1; 2;
ð21Þ
and from the manufacturers' reaction functions in Eq. (20) we can obtain the following equilibrium wholesale prices.
RS⁎ wi
1 RS⁎ ¼ pi ¼ 3
4−3θ2 Ai −θA3−i 16−9θ2
; i ¼ 1; 2:
ð22Þ
It is easy to find the nonnegative retail and wholesale prices require 2 4−3θ Ai ≥ θA3−i ; i ¼ 1; 2;
ð23Þ
528
R. Zhang et al. / Economic Modelling 29 (2012) 523–533
which implies θ ∈ [0, 1) if Ai = A3 − i. Then, the corresponding retail margins can be derived from mi = pi − wi: RS⁎
mi
¼
2 4−3θ Ai −θA3−i 16−9θ2
; i ¼ 1; 2:
ð24Þ
Furthermore, the optimal profits of Retailer i, manufacturer i, channel i's profit and entire supply chain are listed as follows.
ΠRS⁎ ri ΠRS⁎ i ΠRS⁎
2 4−3θ2 Ai −θA3−i ¼ 2ΠRS⁎ 2 ; mi ¼ 16−9θ2 1−θ2 2 3 4−3θ2 Ai −θA3−i RS⁎ ¼ 3Πmi ¼ ; 2 2 2 16−9θ 1−θ 3 16−23θ2 þ 9θ4 A2i þ A23−i −4θ 4−3θ2 Ai A3−i : ¼ 2 16−9θ2 1−θ2 2
As the previous subsections, with the assumption of identical relative channel status(A1 = A2 = 1), these results can be simplified in Table 1. In the next section, we will discuss the equilibrium prices and profit with the views of entire supply chain, manufacturers, retailers and customers, and try to find out which power structure is the best for the above members and how the products substitutability and relative channel status affect the power structure? 4. Equilibrium analysis on pricing decision This section discusses several implications of the results derived in the previous sections. In particular, we focus on which power structure will be the best for the manufacturers, retailers, channels and entire supply chain under symmetric and asymmetric channel power? Also, we compare the results from our dual exclusive channel system with those of single channel. In the following, we first compare these profits and profits among different power structures under symmetric relative channel status, that is Ω=1. Then, we will discuss the similar problems under asymmetric relative channel status, that is Ω≠1. Finally, these results are compared with that in a previous study, in which single channel system or the dual channel but with common retailer or manufacturer were considered.
The retail margins of the respective power structure are simply the differences between retail and wholesale prices, and they are in the reverse order from that of wholesale prices (shown in Fig. 3). Associated with retail prices, the realized demand levels are very close in the two Stackelberg games and larger in the VN because of the lower equilibrium retail prices. VN⁎
RS⁎
≥ Di
Di
MS⁎
≥ Di
;
which is shown in Fig. 4. Therefore, when only two Stackelberg games are considered, consumers are almost indifferent as to who is the market leader. Then, we compare the suppliers' and retailers' profits in different power structures. a manufacturer makes the largest profit in MS, and the smallest in RS (shown in Fig. 5) as expected:
4.1. Equilibrium analysis with symmetric related channel power MS⁎
Due to symmetry, here we only consider some characters with the view of Retailer i, manufacturer i, and channel i. From the Table 1, we can get some results as follows. Lemma 1. If both channels have the same related channel powers (i.e., A1 = A2 = 1),an individual member has necessarily an incentive to play the leader's role, while consumers always benefit from lower prices and realize the most demand when there is no market leadership. However, the supply chain as a whole cannot always benefit from any power structure, for example, VN is the best if and only if θ ∈ [0, 0.592), RS is the best if and only if θ ∈ [0.592, 0.682), and MS is the best if and only if θ ∈ [0.682, 1). Firstly, we compare the wholesale prices and retail prices under different power structure. The wholesale price is the highest in the MS followed by the VN and then RS (shown in Fig. 1):
VN⁎
RS⁎
Πmi ≥ Πmi ≥ Πmi : Again, equalities hold only when the wholesale price is zero, and therefore profit is zero. On the other hand, the retailer profits are the reverse order (shown in Fig. 6): RS⁎
VN⁎
Πri ≥ Πri
MS⁎
≥ Πri :
Interestingly, the differences of all members' profit under different power structure become smaller as the product substitutability grows.
Price 0.5
MS
0.4 MS⁎
wi
VN⁎
≥ wi
RS⁎
≥ wi
On the other hand, equilibrium retail prices are most indistinguishable (shown in Fig. 2). Analytically, the VN has the lowest retail price, and this insight also implies the customers wish get more welfare from the channel system without leadership. Interestingly, in the MS and RS, the retail prices are very close, which have a common starting point and end point(when θ = 0, pMS⁎ ¼ pRS⁎ ¼ 34, and when i i θ → 1, piMS ⁎ = piRS ⁎ → 0), and MS⁎
pi
RS⁎
≥ pi
VN⁎
≥ pi
:
VN
0.3
RS 0.2 0.1
0.2
0.4
0.6
0.8
1.0
θ
Fig. 1. Comparison of wholesale price wi among MS, VN and RS given A1 = A2 = 1.
R. Zhang et al. / Economic Modelling 29 (2012) 523–533
529
Profit
Price 0.7
0.12
0.6
0.10
0.5 VN
RS
MS
MS VN
0.08
0.4
RS
0.06
0.3 0.04
0.2
0.02
0.1 0.2
0.4
0.6
0.8
1.0
θ
Fig. 2. Comparison of retail prices pi among MS, VN and RS given A1 = A2 = 1.
Profit 0.5
0.2
0.4
0.6
0.8
1.0
θ
Fig. 5. Comparison of manufacturer i's profits among MS, VN and RS given A1 = A2 = 1.
Profit 0.12 RS
0.4 0.3
0.10
RS VN
0.08
VN
MS
0.06
MS 0.2
0.04 0.1
0.02 0.2
0.4
0.6
0.8
1.0
θ
0.2
0.4
0.6
θ 1.0
0.8
Fig. 3. Comparison of Retailer i's margin mi among MS, VN and RS given A1 = A2 = 1.
Fig. 6. Comparison of retailer i's profits among MS, VN and RS given A1 = A2 = 1.
Finally, we compare the profits of supply chain as a whole under different power structure. We can find that any power structure can not always ensure the supply chain getting the best performance (shown in Fig. 7), and
leadership but negatively affected by its competitor. While a new leadership will make new market for each member, it also encroaches the other member in existing market and thus intensifies the competition between them. This leaves the other members no choice but to become the leader to counteract its competitor's market encroachment. Therefore, the leadership is a dominant strategy for both members.
ΠVN⁎ ≥ ΠRS⁎ ΠRS⁎ ≥ ΠVN⁎ ΠRS⁎ ≥ ΠMS⁎ ΠMS⁎ ≥ ΠRS⁎
≥ ΠMS⁎ ; ≥ ΠMS⁎ ; ≥ ΠVN⁎ ; ≥ ΠVN⁎ ;
for for for for
θ ∈ ½0; 0:592Þ; θ ∈ ½0:592; 0:605Þ; θ ∈ ½0:605; 0:682Þ; θ ∈ ½0:682; 1Þ:
Lemma 1 is rather straightforward. Only consider the competition among retailer and manufacture in a common supply chain, it echoes the conventional wisdom that a member is rewarded with its own
Theorem 2. In the scenario of symmetric relative channel status, it is a dominant strategy and equilibrium for both members and the entire supply chain to choice the vertical Nash game. However, the entire supply chain and both members encounter a prisoner's dilemma if the product substitutability is sufficiently high (e.g., for the entire supply chain, 0.592 ≤ θ b 1 when Ω = 1).
Profit
Demand 0.50
0.4
VN
0.45 0.3 0.40
MS 0.2
VN
0.35
RS RS
0.30
MS
0.1
0.25 0.0
0.2
0.4
0.6
0.8
1.0
θ
Fig. 4. Comparison of demands in channel i, Di,among MS, VN and RS given A1 = A2 = 1.
0.2
0.4
0.6
0.8
1.0
θ
Fig. 7. Comparison of entire supply chain coefficients among MS, VN and RS given A1 = A2 = 1.
530
R. Zhang et al. / Economic Modelling 29 (2012) 523–533
Theorem 2 suggests that the VN could make the entire supply chain worsen than others, especially under higher substitutability, although it is beneficial for entire supply chain to do so when product substitutability is lower. We graphically presented this insight in Fig. 7. The rationale is explained as follows. If the channel substitutability is low, both channels behave relatively more monopolistic, and thus all players and the entire supply chain can benefit from VN game. As the product substitutability grows, the competition between channels becomes more drastic, and each player always pursue for taking the role of leader to avoid the loss of own profit. Thus, the retail prices and the wholesale prices will increase more than that in VN, and the consumers welfare descends. The Theorem 2 also suggests all players should adopt the VN game when the product substitutability is lower, adopt the RS game when it is moderate, and adopt the MS when it is higher. Furthermore, due to all members can benefit from playing the leader in games, the VN game, which implies both or no leader in the game, is an equilibrium for all members. However, the entire supply chain encounters a Prisoner's dilemma when the substitutability is higher, and all members always encounter the Prisoner's dilemma because they always prefer the Stackelberg game as a leader to that as a follower. 4.2. Equilibrium analysis with asymmetric related channel power Under the dual channel system with asymmetric relative channel power, i.e., Ω ≠ 1, we will investigate the joint effect of product substitutability and asymmetric channel power in different power structure. Through the performances comparisons among different power structure, the result can be reached as follows. Lemma 3. Under the asymmetric relative channel status(i.e.,Ω ≠ 1), an individual members has necessarily an incentive to play the leader's role MS PS ≠ MS if all members are the possible leaderships. That is, Πmi N Πmi and ΠriRS N ΠriPS ≠ RS, where i = 1, 2 and PS = MS, RS, VN in the common feasible area. Furthermore, consumers always benefit from VN. Lemma 1 is rather intuitive. Similar to the dual channel with symmetric relative channel status, all members have an incentive to play the leader's role whether their relative channel power is stronger or weaker. Meanwhile, the consumer always benefit from VN game because the intensified competition let them get a lower price and then a higher demand. Actually, the VN game is necessarily the outcome from the phenomena under which all members try to lead the channel. Furthermore, the asymmetric relative channel power does not affect the decision behaviors of all members because the related channel power depicts the situation of own channel in competition between dual channels, while the leadership depicts the power of members in own channels. An interesting question is whether the asymmetric related channel power affects the equilibrium for all members, even the entire supply chain? The following Theorem 4 will give us the answer.
members can get more profit from own-leadership Stackelberg game than that from vertical Nash game, and the entire supply chain also encounters a Prisoner's dilemma when the channel substitutability is higher. Interestingly, the Theorem 4 suggests that the channel system should choice VN, RS and MS in turn with the increase the channel substitutability when the relative channel status is symmetric (or fixed), while the channel system should also choice VN, RS and MS in turn with the increase of relative channel status when the channel substitutability is fixed (shown in Fig. 8). Meanwhile, the insight tells us a dominance situation for entire supply chain (shown in Fig. 9), the VN game is the best for the entire supply chain when the channel substitutability is lower or the relative channel status is stronger respectively, the RS game is the best when the channel substitutability and the relative channel status are moderate, the MS game is the best when the channel substitutability is higher, and otherwise, that is, the channel substitutability is higher and the relative channel status is weaker or stronger, probably it is a nonevalue area for all members. Similar to the results of previous works on single channel and dual channel with a common retailer, all members in exclusive dual channel always have an incentive to play the leader's role, however, this paper also identify the equilibrium areas in scenarios of symmetric and asymmetric relative channel status. To differ from previous works for a single channel, this paper shows that none power structure always be the best for the entire supply chain though all members on supply chain have incentive to lead the Stackelberg game. Meanwhile, the vertical Nash is an equilibrium for the members to choice their power structure, however, a Prisoner's dilemma necessarily incur for the entire supply chain when the product substitutability is moderate or higher and the asymmetric related channel power is moderate, while consumers always get the most welfare from the vertical Nash. 5. Conclusion and further research Pricing competition is a predominance aspect of the consumer goods industry. This paper investigates the effect of product substitutability and relative channel status on pricing competition under different power structure of a dual exclusive channel system where each manufacturer distributes its goods through an exclusive retailer but two goods have substitutability in demand. A linear demand based on the utility function of a representative consumer is assumed, and three game scenarios(Manufacturer Stackelberg, Retailer Stackelbeg and Vertical Nash) on power structures are examined under symmetric and asymmetric related channel powers. This paper evaluates the efficacy of pricing competition under three game power structures through an asymmetric dual exclusive channel. We identify the equilibrium areas in scenarios of symmetric and asymmetric relative channel status. To differ from previous works for a single channel, our analysis demonstrates that none power structure always be
Theorem 4. In the scenario of asymmetric relative channel status, it is a dominant strategy and equilibrium for both members and the entire supply chain to choice the vertical Nash game. However, the channel system encounters a prisoner's dilemma if the product substitutability is higher. RS−VN
ˆ For example, the RS is the best when Ω i MS−VN
ˆ MS is the best when θ N Ω i
MS−VN
ˆ ðθÞ b θ b Ω i
ðθÞ, and the
ðθÞ, where i = 1, 2.
The Theorem 4 is similar to the Theorem 2, but it depicts that the channel substitutability and asymmetric channel power how to influence the decision behaviors of all members. Due to that all members necessarily have an incentive to lead the Stackelberg game, the vertical Nash game, which implies no leadership in game and also implies all player are leaders, is an equilibrium for all members. However, all members always encounter a Prisoner's dilemma because all
VN
RS
MS
Fig. 8. Comparison of entire supply chain efficiencies among MS, VN and RS.
R. Zhang et al. / Economic Modelling 29 (2012) 523–533
531
Proof of Lemma 3. Due to similarity, here we just investigate the profit comparison of Retailer 1 and Manufacturer 1, respectively, under different power structure PS. From the conditions (12), (17) and (23) to ensure nonnegative retail prices and wholesale prices, we can get the Common Feasible Area(CFA) is: Ω Z
Fig. 9. Dominance situation for entire supply chain among MS, VN and RS.
the best for the entire supply chain though all members on supply chain have incentive to lead the Stackelberg game. Meanwhile, the vertical Nash could be an equilibrium for the members and the entire supply chain to choice their power structure, however, a Prisoner's dilemma necessarily incur for the entire supply chain because with modertae or higher substitutability and moderate asymmetric relative channel power the Retailer Stackelberg or the manufacturer Stackelberg can gain the better performance than that in vertical Nash for the entire supply chain, while consumers always get the most welfare from the vertical Nash. While our work provides a general framework for future study on pricing competition on dual channel system, there are some future research priorities. First, due to limited space, this paper has been focused on pricing competition of the asymmetric dual exclusive channel. In a separate work, we have been investigating the effort effect and pricing competition in similar channel structures, and considered retailer effort, manufacturer effort and hybrid effort respectively. Secondly, this paper inherits the Stackelberg game setting from Doraiswamy et al. (1979), McGuire and Staelin (1983) and many others. Nevertheless, it is well known that a different game setting might alter some of our results especially quantitatively (see Choi (1991) and Xie and Neyret (2009). It is certainly interesting to explore the disparity among different game settings under different decision mode, such as quantity competition. Finally, other asymmetric factors can be considered. Such as, the analysis of asymmetric operational costs together with asymmetric initial base demand can be linearly transformed into a new relative channel status variable and all qualitative results hold.
CFA
ðθÞ ¼
θ 3−2θ2
2
and
¯ CFA ðθÞ ¼ 3−2θ : Ω θ
θ 2−θ2 θ θ because ≥ ≥ and they equate each other 3−2θ2 4−3θ2 8−9θ2 þ 2θ4 when θ = 0 and θ = 1. In the following, we compare Manufacturer 1's profits in different cases. MS − VN To compare case MS and VN, we temporarily define ΔΠm1 as the Manufacturer 1's profits in MS minus the one in VN. Reorganizing A1 MS − VN and solving ΔΠm1 = 0 yields two roots: A1 and A2 into Ω ¼ A2 ˆ MS−VN ðθÞ ¼ Ω m1−1 ˆ MS−VN Ω m1−2 ðθÞ ¼
K1 þ K2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8−6θ2 þ θ4
K3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi K1 −K2 8−6θ2 þ θ4 K3
;
;
where þ 766θ3 −940θ5 þ 549θ7 −152θ9 þ 16θ11 ; K1 ¼ −240θ K2 ¼ θ 144−361θ2 þ 321θ4 −120θ6 þ 16θ8 ; K3 ¼ 576−1776θ2 þ 2087θ4 −1168θ6 þ 312θ8 −32θ10 : MS−VN
MS−VN
CFA ˆ ˆ Due to complexity, we can get Ω ðθÞ N Ω Z m1−1 ðθÞ N Ωm1−2 ðθÞ with countour plots, which is a tool of Mathematica. Thus, the Manufacturer 1 always prefers MS to VN. MS − RS To compare case MS and RS, we can define ΔΠm1 as the Manufacturer 1's profits in MS minus the one in RS, and get two roots MS − RS which meet ΔΠm1 = 0.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8−6θ2 þ θ4 ¼ ; L3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L1 −L2 8−6θ2 þ θ4 ˆ MS−RS ; Ω m1−2 ðθÞ ¼ L3 L1 þ L2
ˆ MS−RS ðθÞ Ω m1−1
where Acknowledgements The authors are grateful to Editors and anonymous referee for their very valuable comments and suggestions, which have significantly helped improving the quality of this paper. The author gratefully acknowledges support from the National Science Foundation of China through grants 70871035 and 71171074, the Program for Science and Technology Innovation Talents in the Universities of Henan Province through grant 2010HASTIT030, and Nature Science Fund of Henan Education Committee through grants 2008A630020 and 2009A630022. Appendix A
Proof of Lemma 1. It is straightforward, though a little tedious, to get these results from Table 1. Here we omit these process, but they are illustrated by Figs. 5–7.□. Proof of Theorem 2. The Theorem 2 is a direct result from Lemma 1.□.
3
5
7
9
11
L1 ¼ 4096θ−13824θ þ 19344θ −14356θ þ 5937θ −1289θ 13
þ 114θ ; L2 ¼ θ3 256−672θ2 þ 633θ4 −253θ6 þ 36θ8 ; L3 ¼ 16384−63488θ2 þ 103488θ4 −92080θ6 þ 48342θ8 −14976θ10 þ 2532θ12 −180θ14 : MS−RS
MS−RS
CFA ˆ ˆ We can also prove Ω ðθÞN Ω Z m1−1 ðθÞN Ωm1−2 ðθÞ with countour plots. Thus, the Manufacturer 1 prefers MS to RS. In summary, the Manufacturer 1 always has an incentive to play the leader's role. In the following, we compare Retailer 1's profits in different cases. RS − VN To compare case RS and VN, we temporarily define ΔΠr1 as the Retailer 1's profits in RS minus the one in VN. Solving RS − VN ΔΠr1 = 0 yields two roots:
pffiffiffi M1 þ M2 2 ; M3 pffiffiffi M −M2 2 RS−VN ˆ r1−2 ðθÞ ¼ 1 ; Ω M3
ˆ RS−VN ðθÞ ¼ Ω r1−1
532
R. Zhang et al. / Economic Modelling 29 (2012) 523–533
where 3
5
7
9
11
M1 ¼ 16384θ−49152θ þ 57152θ −30968θ þ 6348θ þ 848θ 13
15
−565θ þ 66θ ; M2 ¼ θ3 2048−6912θ2 þ 9352θ4 −6494θ6 þ 2439θ8 −469θ10 þ 36θ12 ;
Proof of Theorem 4. Using the similar method in Lemma 3, here we compare the entire supply chain efficiency among MS, RS and VN. To compare the VN with MS. We temporarily define ΔΠ VN − MS as the channel system's profits in VN minus the one in MS. Reorganizing A1 and A2 into Ω ¼ AA12 and solving ΔΠ VN − MS = 0 yields two roots:
M3 ¼ 65536−253952θ2 þ 410880θ4 −357440θ6 þ 178620θ8 −50108θ10 þ 6769θ12 −156θ14 −36θ16 : RS−VN
RS−VN
ˆ ˆ Due to complexity, we can get Ω ðθÞ N Ω Z r1−1 ðθÞ N Ωr1−2 ðθÞ. Thus, the Retailer 1 always prefers RS to VN. RS − MS To compare case RS and MS, we can define ΔΠs1 as the Retailer 1's profits in RS minus the one in MS, and get two roots which meet RS − MS ΔΠr1 = 0. CFA
pffiffiffi N1 −N2 2 ; N3 pffiffiffi N þ N2 2 RS−MS ˆ r1−2 ðθÞ ¼ 1 ; Ω N3
RS−MS
RS−MS
ˆ ˆ We can also prove Ω ðθÞN Ω Z r1−1 ðθÞN Ωr1−2 ðθÞ with countour plots. Thus, the Retailer 1 prefers RS to MS. In summary, the Retailer 1 always has an incentive to play the leader's role. Due to symmetry, we can prove the Retailer 2 and Manufacture 2 have an incentive to play his(her) leader's roles, respectively. To consider the consumers welfare. To compare case VN and MS, we can define ΔU VN − MS as the utility in VN minus the one in MS.
Þ¼
O3
Actually, O1(A12 + A22) ≥ 2O1A1A2, and it is easy to find 2O1 + O2 ≥ 0 4 with countour plot. So, consumers always prefer VN to MS. Similarly, we can define ΔU VN − RS as the utility in VN minus the one in RS.
2o3
4
6
8
10
12
o1 ¼ 16384−58368θ þ 83648θ −61468θ þ 24401θ −4973θ þ 408θ ; 2 4 6 8 10 12 o2 ¼ 2θ 4096−17920θ þ 29456θ −23740θ þ 10027θ −2131θ þ 180θ ; 2 o3 ¼ 1−θ2 1024−1920θ2 þ 1284θ4 −361θ6 þ 36θ8 :
Actually, o1(A12 + A22) ≥ 2o1A1A2, and it is easy to find 2o1 + o2 ≥ 0 5 with countour plot. So, consumers always prefer VN to RS. In summary, consumers always prefer VN to others power structure. □. 2O1 + O2 = 0 only when θ = 0 2o1 + o2 = 0 only when θ = 0.
¼
ˆ RS−VN ðθÞ ¼ Ω 2
n1 þ n2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 256−880θ2 þ 841θ4 −300θ6 þ 36θ8
n3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 n1 −n2 − 256−880θ þ 841θ4 −300θ6 þ 36θ8 n3
;
;
where n1 ¼ 20480θ−68096θ3 þ 91600θ5 −63512θ7 þ 23882θ9 −4620θ11
RS−MS
ˆ We have Ω 1
RS−MS
ˆ ðθÞ ≥ Ω 2
VN−MS
ðθÞ and only equates if Ω = 1. Mean-
RS−MS
VN−MS
RS−MS
ˆ ˆ ˆ ˆ ðθÞ N Ω ðθÞ and Ω ðθÞ b Ω ðθÞ. while, Ω 1 1 1 1 Therefore, for the entire supply chain system the VN dominate VN−MS
ˆ others when Ω N Ω 1 others when
VN−MS
ˆ Ω i
RS−MS
ˆ ðθÞ b Ω b Ω 1
RS−MS
ˆ others when Ω b Ω 1
VN−MS
ˆ ðθÞ and Ω b Ω 2
ðθÞ, and the MS dominate
RS−MS
ˆ ðθÞ and Ω N Ω 2
ðθÞ, the RS dominate
ðθÞ. □
References
θ2 o1 A21 þ A22 þ o2 A1 A2
where
5
ˆ RS−MS ðθÞ Ω 1
þ 2982θ12 −216θ14 :
2 O1 ¼ 4032−13844θ þ 19139θ4 −13562θ6 þ 5185θ8 −1016θ10 þ 80θ12 ; 2 O2 ¼ 2θ 480−2728θ þ 4993θ4 −4217θ6 þ 1810θ8 −384θ10 þ 32θ12 ; 2 O3 ¼ 1−θ2 576−1012θ2 þ 633θ4 −168θ6 þ 16θ8 :
2
VN−MS
þ 360θ13 ; n2 ¼ 1024−2944θ2 þ 3204θ4 −1645θ6 þ 397θ8 −36θ10 ; n3 ¼ 16384−62464θ2 þ 103616θ4 −95980θ6 þ 52865θ8 −17093θ10
2 O1 A21 þ A22 þ O2 A1 A2
where
¼
;
where
VN−MS
CFA
4
k3
;
ˆ ˆ We have Ω ðθÞ ≥ Ω ðθÞ and only equates if Ω = 1. 1 2 To compare the RS with MS. We temporarily define ΔΠ RS − MS as the channel system's profits in RS minus the one in MS. Solving ΔΠ RS − MS = 0 yields two roots:
N3 ¼ −288 þ 528θ2 −217θ4 −60θ6 þ 36θ8 :
VN−MS
k3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k1 −k2 − 100−396θ þ 385θ4 −136θ6 þ 16θ8
þ 880θ12 −64θ14 :
3 N1 ¼ 20θ−314θ þ 259θ5 −66θ7 ; N2 ¼ θ 144−289θ2 þ 181θ4 −36θ6 ;
ΔU
ˆ MS−VN ðθÞ ¼ Ω 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 100−396θ2 þ 385θ4 −136θ6 þ 16θ8
þ 128θ13 ; k2 ¼ 576−1588θ2 þ 1645θ4 −801θ6 þ 184θ8 −16θ10 ; k3 ¼ 5760−20216θ2 þ 31654θ4 −28441θ6 þ 15509θ8 −5020θ10
where
VN−MS
¼
k1 þ k2
k1 ¼ 9024θ−28840θ3 þ 37344θ5 −24962θ7 þ 9064θ9 −1696θ11
ˆ RS−MS ðθÞ ¼ Ω r1−1
ΔU
ˆ VN−MS ðθÞ Ω 1
Almehdawe, E., Mantin, B., 2010. Vendor managed inventory with a capacitated manufacturer and multiple retailers: retailer versus manufacturer leadership. International Journal of Production Economics 128 (1), 292–302. Cai, G., 2010. Channel selection and coordination in dual-channel supply chains. Journal of Retailing 86 (1), 22–36. Cai, G.G., Zhang, Z., Zhang, M., 2009. Game theoretical perspectives on dual-channel supply chain competition with price discounts and pricing schemes. International Journal of Production Economics 117 (1), 80–96. Choi, S.C., 1991. Pricing competition in a channel structure with a common retailer. Marketing Science 10, 271–296. Choi, S.C., 1996. Pricing competition in a duopoly common retailer channel. Journal of Retailing 72 (2), 117–134. Doraiswamy, K., McGuire, T.W., Staelin, R., 1979. An analysis of alternative advertising strategies in a competitive franchise framework. In: N.B., et al. (Ed.), 1979 Educators' Conference Proceedings. American Marketing Associtation, Chicago, pp. 463–467. Huang, Z., Li, S.X., 2001. Co-op advertising models in a manufacturer–retailer supply chain: a game theory approach. European Journal of Operational Research 135 (3), 527–544. Ingene, C.A., Parry, M.E., 2004. Mathematical Models of Distribution Channels. Springer.
R. Zhang et al. / Economic Modelling 29 (2012) 523–533 Ingene, C., Parry, M., 2007. Bilateral monopoly, identical competitors/distributors, and game-theoretic analyses of distribution channels. Journal of the Academy of Marketing Science 35 (4), 586–602. Jeuland, A.P., Shugan, S.M., 1983. Managing channel profits. Marketing Science 2 (3), 239–272. Jeuland, A.P., Shugan, S.M., 1988. Channel of distribution profits when channel members form conjectures. Marketing Science 7 (2), 202–210. Karjalainen, R., 1992. The Newsboy Game. University of Pennsylvania working paper. Lee, J.-Y., Ren, L., 2011. Vendor-managed inventory in a global environment with exchange rate uncertainty. International Journal of Production Economics 130 (2), 169–174. Lippmand, S., Macardle, K., 1997. The competitive newsboy. Operations Research 45 (1), 54–65.
533
Lu, J.-C., Tsao, Y.-C., Charoensiriwath, C., 2011. Competition under manufacturer service and retail price. Economic Modelling 28 (3), 1256–1264. McGuire, T., Staelin, R., 1983. An industry equilibrium analysis of downstream vertical integration. Marketing Science 2 (2), 161–191. Paralar, M., Wang, D., 1993. Diversification under yield randomness in inventory models. European Journal of Operational Research 66 (1), 52–64. Shugan, S.M., 1985. Implicit understanding in channels of distribution. Management Science 31 (4), 435–460. Xie, J., Neyret, A., 2009. Co-op advertising and pricing models in manufacturercretailer supply chains. Computers and Industrial Engineering 56, 1375–1385. Zusman, P., Etgar, M., 1981. The marketing channel as an equilibrium set of contracts. Management Science 27 (2), 284–302.