Pricing decisions for substitutable products in a two-echelon supply chain with firms׳ different channel powers

Int. J. Production Economics 153 (2014) 243–252

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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe

Pricing decisions for substitutable products in a two-echelon supply chain with firms' different channel powers Jing Zhao a, Jie Wei b,n, Yongjian Li c a

School of Science, Tianjin Polytechnic University, Tianjin 300387, PR China General Courses Department, Military Transportation University, Tianjin 300161, PR China c Business School, Nankai University, Tianjin 300071, PR China b

art ic l e i nf o

a b s t r a c t

Article history: Received 25 October 2012 Accepted 3 March 2014 Available online 15 March 2014

Pricing decisions for two substitutable products in a supply chain with one common retailer and two competitive manufacturers are considered in this paper. The purpose of this paper is to analyze the effects of the two manufacturers' different competitive strategies and the channel members' different power structures on the optimal pricing decisions. One centralized pricing model and seven decentralized pricing models are developed, and the corresponding analytical equilibrium solutions are obtained by using game-theoretic approach. Finally, numerical studies are presented to illustrate the effectiveness of the theoretical results, and to gain additional managerial insights. & 2014 Elsevier B.V. All rights reserved.

Keywords: Pricing Channel power Competitive strategies Game theory

1. Introduction Today, with the rapid development of retail industry, it is common that a retailer retails differential but substitutable products of an item procured from different manufacturers. For example, Wal-Mart sells substitute brands of detergents like, Unilever brands (Surf, Wisk) and P & G brands (Tide, Gain, Cheer). Although the manufacturers who can supply substitutable products to the common retailer compete on many levels (for example, product quality, customer service, technology support, etc.), price competition between manufacturers is inevitable and critical (Xia, 2011). Manufacturers must determine their products' prices based on their own cost structures, and act or react to their competitors' attempts to earn maximum profits. Moreover, the manufacturers' pricing strategy must never be an aimless price fight but a rational action adjusted to specific competitive situations. In a marketing channel, both the manufacturer and the retailer have “power”, defined by EI-Ansary and Stern (1972) as “the ability of one channel member to control the decision variables in the marketing strategy of another member in a given channel at a different level of distribution” (Pan et al., 2010). Different power structures appear in real-world supply chains. For example, powerful manufacturers, such as Microsoft and Intel, play a more dominant role than downstream members in some chains. In contrast, powerful retailers (e.g., Walmart and Carrefour) play a more dominant role than upstream members in other chains, who can influence each product's sales by lowering price while maintaining their profit margins on sales by squeezing profit

n

Corresponding author. Tel.: þ 86 22 84657651. E-mail address: [email protected] (J. Wei).

http://dx.doi.org/10.1016/j.ijpe.2014.03.005 0925-5273/& 2014 Elsevier B.V. All rights reserved.

from their suppliers (manufacturers). Moreover, firms at the same stage of a supply chain may consider their decisions (e.g., on sale pricing strategy, facility investment, R&D expenditures, etc.) as strategic moves in timing, which will lead to different power structures between them. A real-world example is that Tesco announced to follow Wal-Mart to enter India grocery market (Wu et al., 2012). In the present study, we consider a two-echelon supply chain with two competitive manufacturers and one common retailer. The common retailer orders two substitutable products from the two manufacturers, and sells them in the same market. The retailer determines the sale prices to the market, and the two manufacturers determine the wholesale prices to the retailer. In the following discussion, “he” represents one of the two manufacturers, and “she” represents the common retailer. Based on different channel power structures, there are different competitive strategies between the two manufacturers, e.g., Bertrand competition, cooperation and Stackelberg competition. When the two manufacturers implement Bertrand competition, each manufacturer independently makes his decision by assuming his rival's decision as a parameter. When the two manufacturers adopt cooperative pricing, i.e., the two manufacturers are willing to make their decisions jointly to maximize the total profit of the upstream market. When the two manufacturers implement Stackelberg competition, they make their pricing decisions successively. An important issue has arisen in the management of such a supply chain, i.e., how should the common retailer and the two manufacturers make the pricing decisions to maximize their own profits facing different channel power structures? Our main interests are to investigate how the two manufacturers and the common retailer make their own pricing decisions when facing different market power structures

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J. Zhao et al. / Int. J. Production Economics 153 (2014) 243–252

and to analyze the effects of the two manufacturers' different competitive strategies and the channel members' different power structures on the optimal pricing decisions. To the best of our knowledge, no research has studied the problem in a two-echelon supply chain. A significant amount of research has been done about substitutable products supply chain (e.g., Karakul and Chan, 2008, Liu et al., 2013, Nagarajan and Rajagopalan, 2008 and Zhang et al., 2013). Much of it has focused on joint production/inventory and pricing problems. For related surveys, see Chan et al. (2004) and Yano and Gilbert (2003). In some recent studies, for example, Tang and Yin (2007) developed a base model with deterministic demand to examine how a retailer determines the order quantity and the retail prices of two substitutable products jointly under the fixed and variable pricing strategies. Nagarajan and Rajagopalan (2008) solved the centralized newsboy type inventory problem with two substitutable products whose demands are negatively correlated. Karakul and Chan (2008) considered a single period joint pricing and procurement problem of two one-way substitutable products. Pasternack and Drezner (1991) considered a centralized two-product inventory model with full substitution. They proved that the total expected profit function is concave and derived the analytical expressions of the optimal inventory levels. Stavrulaki (2011) studied a retailer's inventory strategy for two products, which are substitutable and have inventory dependent demand. Some other research papers studied channel coordination of substitutable products supply chain, e.g., Yao et al. (2008), Hsieh and Wu (2009) and Sinha and Sarmah (2010). There has been some work on the pricing of substitutable products (e.g., Li et al., 2013, Choi, 1996, Xia, 2011 andChen and Chang, 2013). Choi (1996) considered two manufacturers using two common buyers that price-compete with each other. Trivedi (1998) considered two manufacturers and two common retailers for pricing decisions, and showed that the presence of competitive effects at both retailer and manufacturer levels of distribution has a significant impact on prices. Chen and Chang (2013) considered the dynamic pricing for two products (one is new and the other is remanufactured products). They assumed that market demand is price-dependent and partially substitutable between the new and remanufactured products. Xia (2011) studied competition between two coexisting suppliers in a two-echelon supply chain, where each supplier offers one type of the two substitutable products to multiple buyers. Zhao et al. (2012a,b) studied the pricing problem of substitutable products in two-echelon fuzzy supply chain. Choi (1991) analyzed different power structures in a supply chain with two manufactures selling substitutable products through a common retailer, and developed manufacturer Stackelberg, retailer Stackelberg, and vertical Nash pricing models. Different from the above studies, this paper focuses on the effects of two competitive manufacturers' different pricing strategies and the channel members' different power structures on the optimal pricing decisions of two substitutable products. The main contributions of this paper can be listed as follows. First, eight pricing models are established by considering the two manufacturers' different pricing strategies and the channel members' different power structures, which extends the current literature related to the pricing of substitutable products. Second, by using the game-theoretical approach, analytical equilibrium solutions are obtained in each of the eight pricing models. Third, the effect of retail substitutability on the equilibrium decisions and profits of the chain members in these pricing models are investigated. Fourth, through numerical studies, some valuable insights are obtained: (1) when the two manufacturers adopt Bertrand competition or cooperation strategy, the leadership between the two manufacturers and the common retailer does not affect the maximum profit of the whole supply chain, namely, the maximum profit of the whole supply chain in MSC model equals that in RSC

model and the maximum profit of the whole supply chain in MSB model equals that in RSB model. (2) Both of the two manufacturers cannot benefit from their cooperation strategy simultaneously, for example, the manufacturer 2's profit in MSC model is lower than that in MSS model. This insight is helpful for a manufacturer who is the follower in MS games, because it tells the manufacturer that a suitable profit-split should be negotiated with his rival before agreeing to act in union. The rest of the paper is organized as follows. Section 2 gives the problem description and notations. Section 3 details our key analytical results. In Section 4, some valuable insights are obtained by using numerical studies. Some concluding comments are presented in Section 5.

2. Problem description Consider a supply chain with one common retailer and two competitive manufacturers (labeled manufacturer 1 and manufacturer 2). The manufacturer i produces a product i with unit manufacturing cost ci and wholesales the product to the common retailer with unit wholesale price wi ; i ¼ 1; 2. The common retailer then sells the product i to the end consumer with unit retail price pi. Products 1 and 2 are substitutable with each other and all activities occur in a single period. The common retailer and the two manufacturers make their price decisions in order to achieve their own maximal profits and behave as if they have perfect information of the demands and the cost structures of other channel members. We assume that the demand Di for product i is a function of the retail prices p1 and p2, which is commonly adopted in the literature (Choi, 1991, 1996; Lee and Staelin, 1997; McGuire and Staelin, 1983; Trivedi, 1998). The general demand functions are given as follows: Di ¼ ai  βpi þ γpj ;

i ¼ 1; 2; j ¼ 3  i;

ð1Þ

where parameter ai denotes the primary demand for product i, parameters β and γ denote the measures of the responsiveness of each product's market demand to its own price and its competitive product's price, respectively. We assume the parameters β and γ satisfy β 4 γ 4 0, which means that the demand for a product is more sensitive to the changes in its own price than to the changes in the price of the other competitive product. The profits of the common retailer and the manufacturers 1 and 2 are, respectively, represented as follows: 2

π r ðp1 ; p2 Þ ¼ ∑ ½ðpi  wi Þðai  βpi þ γpj Þ;

ð2Þ

π m1 ðw1 Þ ¼ ðw1  c1 Þða1 βp1 þ γp2 Þ;

ð3Þ

π m2 ðw2 Þ ¼ ðw2  c2 Þða2 βp2 þ γp1 Þ:

ð4Þ

i¼1

3. Analytical results 3.1. Centralized pricing model (CD model) As a benchmark to evaluate pricing decisions under different decision cases, we first examine the centralized pricing model, namely, there is one entity that aims to optimize the whole supply chain performance. Both the two manufacturers' and the common retailer's decisions are fully coordinated in this case. The wholesale prices w1 and w2 are regarded as inner transfer prices, which only influences the profits of each participant, but not on the profit of the whole supply chain. The total profit of the whole supply chain is determined by manufacturing costs and retail prices, and the two manufacturers' decisions for wholesale prices will be an

J. Zhao et al. / Int. J. Production Economics 153 (2014) 243–252

effective way to coordinate the relationships of participants in this supply chain. Let π c ðp1 ; p2 Þ be the total profit of the centralized supply chain, which can be given as 2

π c ðp1 ; p2 Þ ¼ ∑ ½ðpi  ci Þðai  βpi þ γpj Þ;

j ¼ 3  i:

ð5Þ

and p2, as follows: pn1 ðw1 ; w2 Þ ¼

w 1 a1 β þ a2 γ þ ; 2 2ðβ2  γ 2 Þ

ð9Þ

pn2 ðw1 ; w2 Þ ¼

w2 a1 γ þ a2 β þ : 2 2ðβ2  γ 2 Þ

ð10Þ

i¼1

According to the above description, we know that the objective of the centralized supply chain is to maximize the total profit π c ðp1 ; p2 Þ, which can be denoted as " # CD model :

max π c ðp1 ; p2 Þ ¼ max

ðp1 ;p2 Þ

2

∑ ½ðpi  ci Þðai  βpi þ γpj Þ :

ðp1 ;p2 Þ i ¼ 1

ð6Þ n

In the CD model, we obtain the optimal retail prices p1 and pn2 , by solving ∂π c ðp1 ; p2 Þ=∂p1 ¼ 0 and ∂π c ðp1 ; p2 Þ=∂p2 ¼ 0 simultaneously, as

245

Having the information about the decisions of the common retailer, the two manufacturers would then use them to maximize their profits simultaneously. Substituting pn1 ðw1 ; w2 Þ and pn2 ðw1 ; w2 Þ in Eqs. (9) and (10) into the two manufacturers' profits in Eqs. (3) and (4), respectively, and applying the first-order conditions to the resulting profits in terms of the wholesale prices w1 and w2 gives the two manufacturers' wholesale prices wn1 and wn2 at equilibrium as follows: wn1 ¼

2a1 β þ2c1 β2 þ a2 γ þ c2 βγ ; 4β2  γ 2

ð11Þ

2a2 β þ2c2 β2 þ a1 γ þ c1 βγ : 4β2  γ 2

ð12Þ

pn1 ¼

c1 a1 β þ a2 γ þ ; 2 2ðβ2 γ 2 Þ

ð7Þ

wn2 ¼

pn2 ¼

c2 a1 γ þ a2 β þ : 2 2ðβ2 γ 2 Þ

ð8Þ

Hence, with Eqs. (9)–(12), we can see that the common retailer's equilibrium retail prices pn1 and pn2 are

Eqs. (7) and (8) indicate that, in the CD model, the optimal retail price pni of product i increases as the unit manufacturing cost ci increases and only half the change in the unit manufacturing cost ci is reflected in the optimal retail price pni , whereas the optimal retail price pni is independent of the unit manufacturing cost cj, i ¼ 1; 2 and j ¼ 3  i. 3.2. Pricing models in MS game In this subsection, we assume that the two manufacturers act as the Stackelberg leaders and the common retailer acts as the Stackelberg follower. The game-theoretical approach is used to analyze the pricing decision. The leader in every decision scenario makes his decision to maximize his profit, conditioned on the follower's response. The problem is solved backwards. Namely, the decision of the follower is solved first, given that the leader's decision has been observed. Based on different competitive strategies between the two manufacturers, e.g., Bertrand competition, cooperation and Stackelberg competition, we will establish three pricing models in this subsection, namely, the pricing model when the two manufacturers implement the Bertrand competition (MSB model), the pricing model when the two manufacturers act in cooperation (MSC model), and the pricing model when the two manufacturers implement Stackelberg competition (MSS model).

3.2.1. The MSB model In the MSB model, the two manufacturers first announce wholesale prices of their products, the common retailer observes the wholesale prices and then decides the retail prices she is going to charge for two products. The MSB model is formulated as 88 π m1 ðw1 ; pn1 ðw1 ; w2 Þ; pn2 ðw1 ; w2 ÞÞ > < max > w1 > > > > > π m2 ðw2 ; pn1 ðw1 ; w2 Þ; pn2 ðw1 ; w2 ÞÞ < : max w2

pn1 ¼

wn1 a1 β þ a2 γ þ ; 2 2ðβ2  γ 2 Þ

ð13Þ

pn2 ¼

wn2 a1 γ þa2 β þ ; 2 2ðβ2  γ 2 Þ

ð14Þ

where wn1 and wn2 are defined as in Eqs. (11) and (12). 3.2.2. The MSC model When the two manufacturers recognize their interdependence and agree to act in union in order to maximize the total profit of the upstream production market, they adopt the cooperation solution in MS game. In this decision case, the two manufacturers first announce their wholesale prices of the two substitutable products with the objective to maximize their total profit, the retailer observes the wholesale prices and then decides the retail prices she is going to charge for the two products with the intent to maximize her profit. The MSC model is formulated as 8 max ½π m1 ðw1 ; pn1 ðw1 ; w2 Þ; pn2 ðw1 ; w2 ÞÞ þ π m2 ðw2 ; pn1 ðw1 ; w2 Þ; pn2 ðw1 ; w2 ÞÞ > > > < ðw1 ;w2 Þ pn1 ðw1 ; w2 Þ; pn2 ðw1 ; w2 Þ are derived from solving the following problem > > > max π r ðp1 ; p2 Þ : ðp1 ;p2 Þ

Given earlier decisions w1 and w2 made by the two manufacturers, we can have the retailer's best response functions as in Eqs. (9) and (10). Having the information about the decisions of the common retailer, the two manufacturers would then use them to maximize their total profit. Substituting pn1 ðw1 ; w2 Þ and pn2 ðw1 ; w2 Þ in Eqs. (9) and (10), respectively, into the two manufacturers' total profit π m1 ðw1 Þ þ π m2 ðw2 Þ, and applying the first-order conditions to the resulting profits in terms of the wholesale prices w1 and w2 gives the two manufacturers' wholesale prices, denoted as wn1 and wn2 respectively, at equilibrium as follows: wn1 ¼

ð15Þ

> pn1 ðw1 ; w2 Þ; pn2 ðw1 ; w2 Þ are derived from solving the following problem > > > > > max π r ðp ; p Þ > 1 2 : ðp ;p Þ

c1 a1 β þ a2 γ þ ; 2 2ðβ2 γ 2 Þ

wn2 ¼

c2 a2 β þ a1 γ þ : 2 2ðβ2 γ 2 Þ

ð16Þ

We first derive the common retailer's decisions as follows. Given earlier decisions w1 and w2 made by the two manufacturers, we can derive the retailer's best response functions, by setting ∂π r ðp1 ; p2 Þ=∂p1 and ∂π r ðp1 ; p2 Þ=∂p2 equal to zero and solving for p1

Based on the results in Eqs. (7), (8), (15), and (16), we can see that the optimal retail prices in the CD model equal the optimal wholesale prices in the MSB model. This is consistent with our intuitions.

1

2

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J. Zhao et al. / Int. J. Production Economics 153 (2014) 243–252

With these equilibrium values in Eqs. (7), (8), (15), and (16), the common retailer's equilibrium retail prices in the MSC model, denoted as pn1 and pn2 respectively, can be obtained as pn1 ¼

c1 3ða1 β þ a2 γÞ þ ; 4 4ðβ2  γ 2 Þ

ð17Þ

pn2 ¼

c2 3ða1 γ þa2 βÞ þ : 4 4ðβ2  γ 2 Þ

ð18Þ

3.2.3. The MSS model When the two manufacturers play Stackelberg game, we assume that the manufacturer 1 acts as a Stackelberg leader and the manufacturer 2 acts as a Stackelberg follower. So, the manufacturer 1 first announces the wholesale price w1, and the manufacturer 2 then decides the wholesale price w2 to maximize his profit, the retailer finally decides the retail prices when knowing the two manufacturers' decisions. The MSS model is formulated as follows: 8 max π m1 ðw1 ; pn1 ðw1 ; wn2 ðw1 ÞÞ; pn2 ðw1 ; wn2 ðw1 ÞÞÞ > > > w1 > > > > wn2 ðw1 Þ is derived from solving the following problem > > > <8 π m2 ðw2 ; pn1 ðw1 ; w2 Þ; pn2 ðw1 ; w2 ÞÞ > max > < w2 >> > n > p1 ðw1 ; w2 Þ; pn2 ðw1 ; w2 Þ are derived from solving the following problem > > > >> > > > max π r ðp1 ; p2 Þ > > : : ðp1 ;p2 Þ

Similar to the MSB model, given earlier decisions w1 and w2 made by the two manufacturers respectively, we can have the retailer's best response functions as in Eqs. (9) and (10). Having the information about the common retailer's decisions, the manufacturer 2 would then use them to maximize his profit π m2 ðw2 ; pn1 ðw1 ; w2 Þ; pn2 ðw1 ; w2 ÞÞ for given wholesale price w1. Substituting pn1 ðw1 ; w2 Þ and pn2 ðw1 ; w2 Þ in Eqs. (9) and (10), respectively, into the manufacturer 2's profit in Eq. (4), and applying the firstorder condition to the resulting profit in terms of the wholesale price w2 gives the manufacturer 2's wholesale price, denoted as w2 ðw1 Þ, at equilibrium as follows: w2 ðw1 Þ ¼

γw1 þ a2 þ c2 β : 2β

ð19Þ

Substituting pn1 ðw1 ; w2 Þ and pn2 ðw1 ; w2 Þ in Eqs. (9) and (10), respectively, and the manufacturer 2's wholesale price in Eq. (19) into the manufacturer 1's profit in (3), and applying the first-order condition to the resulting profit in terms of the wholesale price w1 gives the manufacturer 1's optimal wholesale price wn1 at equilibrium as follows: 2a1 β þ a2 γ þ c2 βγ c1 γ 2 þ2c1 β2

3.3. Pricing models in RS game In this subsection, we assume that the common retailer acts as the Stackelberg leader and the two manufacturers act as the Stackelberg followers. Similar to the pricing models in MS game, we will also establish three pricing models in this subsection, namely, the pricing model when the two manufacturers implement the Bertrand competition (RSB model), the pricing model when the two manufacturers act in cooperation (RSC model), and the pricing model when the two manufacturers implement Stackelberg competition (RSS model). 3.3.1. The RSB model In this decision case when the two manufacturers pursue the Bertrand competition in RS game, the common retailer first announces the retail prices she is going to charge for the two substitutable products, the two manufacturers observe the retail prices and then decide the wholesale prices they are going to charge for the two products simultaneously. The RSB model is formulated as

8 max π r ðp1 ; p2 ; wn1 ðp1 ; p2 Þ; wn2 ðp1 ; p2 ÞÞ > > > ðp1 ;p2 Þ > > > > < wn1 ðp1 ; p2 Þ; wn2 ðp1 ; p2 Þ are derived from solving the following problem 8 π m1 ðw1 Þ < max > w1 > > > > > max π m2 ðw2 Þ > :: w 2

We first derive the two manufacturers' decisions as follows. Without loss of generality, let mi be the margin of product i enjoyed by the retailer, namely, pi ¼ wi þ mi ;

where mi 4 0; i ¼ 1; 2. Given earlier decisions p1 and p2 made by the retailer, we can derive the two manufacturers' best response functions as follows: wn1 ðp1 ; p2 Þ ¼  p1 þ

a1 þ c 1 β γ þ p2 ; β β

ð25Þ

wn2 ðp1 ; p2 Þ ¼  p2 þ

a2 þ c 2 β γ þ p1 : β β

ð26Þ

Having the information about the decisions of the two manufacturers, the common retailer would then use them to maximize her profit. Substituting wn1 ðp1 ; p2 Þ and wn2 ðp1 ; p2 Þ in Eqs. (25) and (26), respectively, into the common retailer's profit in Eq. (2), and applying the first-order conditions to the resulting profit in terms of the retail prices p1 and p2 gives the common retailer's retail prices pn1 and pn2 at equilibrium as follows:

ð20Þ

pn1 ¼

With Eqs. (19) and (20), one can easily have the manufacturer 2's optimal wholesale price in the MSS model, denoted as wn2 , at equilibrium as follows:

pn2 ¼

wn1 ¼

wn2 ¼

3β2  2γ 2

:

γð2a1 β þ a2 γ þ c2 βγ  c1 γ 2 þ 2c1 β2 Þ a2 þc2 β : þ 2β 2βð3β2  2γ 2 Þ

ð24Þ

A1 A3 þ6A2 γ A23  36γ 2 A2 A3 þ6A1 γ A23  36γ 2

;

ð27Þ

;

ð28Þ

where ð21Þ A1 ¼ 3a1 þ c1 β 

Similarly, using Eqs. (20), (21), (9), and (10), one can have the common retailer's optimal retail prices in the MSS model, denoted as pn1 and pn2 , at equilibrium as follows:

2a2 γ þ c2 βγ ; β

A2 ¼ 3a2 þ c2 β 

2a1 γ þ c1 βγ ; β

A3 ¼ 4β þ

2γ 2 : β

Hence, with Eqs. (25)–(28), we can see that the two manufacturers' equilibrium wholesale prices wn1 and wn2 in the RSB model are

pn1 ¼

wn1 a1 β þ a2 γ þ ; 2 2ðβ2  γ 2 Þ

ð22Þ

wn1 ¼  pn1 þ

a1 þ c1 β γ n þ p2 ; β β

ð29Þ

pn2 ¼

γwn1 þ a2 þ c2 β a1 γ þ a2 β þ ; 4β 2ðβ2  γ 2 Þ

ð23Þ

wn2 ¼  pn2 þ

a2 þ c2 β γ n þ p1 ; β β

ð30Þ

where wn1 is defined as in Eq. (20).

where pn1 and pn2 are defined as in Eqs. (27) and (28).

J. Zhao et al. / Int. J. Production Economics 153 (2014) 243–252

3.3.2. The RSC model In this decision case when the two manufacturers adopt the cooperation strategy in RS game, we assume that the two manufacturers recognize their interdependence and agree to act in union in order to maximize their total profit. The common retailer first announces the retail prices she is going to charge for the two substitutable products, and the two manufacturers observe the retail prices and then decide the wholesale prices simultaneously with the objective to maximize their total profit. The RSC model is formulated as follows: 8 max π r ðp1 ; p2 ; wn1 ðp1 ; p2 Þ; wn2 ðp1 ; p2 ÞÞ > > > < ðp1 ;p2 Þ wn1 ðp1 ; p2 Þ; wn2 ðp1 ; p2 Þ are derived from solving the following problem > > > : max ðπ m1 ðw1 Þ þ π m2 ðw2 ÞÞ ðw1 ;w2 Þ

We first need to derive the two manufacturers' decisions. Given earlier decisions p1 and p2 made by the retailer, we can derive the two manufacturers' best response functions, by applying the first-order conditions to the two manufacturers' total profit π m1 ðw1 Þ þπ m2 ðw2 Þ in terms of the wholesale prices w1 and w2, as follows: a1 β þ a2 γ wn1 ðp1 ; p2 Þ ¼  p1 þ c1 þ 2 ; β  γ2

ð31Þ

a2 β þ a1 γ : β2  γ 2

ð32Þ

wn2 ðp1 ; p2 Þ ¼  p2 þ c2 þ

Having the information about the decisions of the two manufacturers, the common retailer would then use them to maximize her profit. Substituting wn1 ðp1 ; p2 Þ and wn2 ðp1 ; p2 Þ in Eqs. (31) and (32), respectively, into the common retailer's profit in Eq. (2), and applying the first-order conditions to the resulting profit in terms of the retail prices p1 and p2 gives the retailer's retail prices pn1 and pn2 , respectively, at equilibrium as follows: 3ða1 β þ a2 γÞ c1 þ ; 4 4ðβ2  γ 2 Þ

ð33Þ

3ða1 γ þa2 βÞ c2 p2 ¼ þ : 4 4ðβ2  γ 2 Þ

ð34Þ

pn1 ¼ n

Based on Eqs. (31)–(34), we can see that the two manufacturers' equilibrium wholesale prices wn1 and wn2 in the RSC model are wn1 ¼

3c1 a1 β þ a2 γ þ ; 4 4ðβ2 γ 2 Þ

ð35Þ

wn2 ¼

3c2 a2 β þ a1 γ þ : 4 4ðβ2 γ 2 Þ

ð36Þ

3.3.3. The RSS model In this decision case when the two manufacturers play Stackelberg game, we assume that the manufacturer 1 acts as a Stackelberg leader and the manufacturer 2 acts as a Stackelberg follower. The common retailer first announces the retail prices, manufacturer 1 then decides the wholesale price w1 to maximize his profit, and, finally, manufacturer 2 decides the wholesale price w2 he is going to charge. The RSS model is formulated as 8 max π r ðp1 ; p2 ; wn1 ðp1 ; p2 Þ; wn2 ðp1 ; p2 ÞÞ > > ðp1 ;p2 Þ > > > > n > > w1 ðp1 ; p2 Þ is derived from solving the following problem > > < 8 max π ðw Þ m1 1 > w > > < 1 > > wn ðp ; p Þ is derived from solving the following problem > 2 > 2 1 > > > max π ðw Þ >> > > m2 2 : w > > 2 :

247

Given earlier decisions w1, p1 and p2 made by the manufacturer 1 and the common retailer, we can derive the manufacturer 2's best response function as follows: γ a2 þ c 2 β wn2 ðp1 ; p2 Þ ¼  p2 þ p1 þ : β β

ð37Þ

Substituting wn2 ðp1 ; p2 Þ in Eq. (37) into the manufacturer 1's profit in Eq. (3) and solving the first-order condition of the resulting profit for w1 gives manufacturer 1's best response function: wn1 ðp1 ; p2 Þ ¼

β ða1  βp1 þ γp2 Þ þ c1 : β2  γ 2

ð38Þ

Substituting wn1 ðp1 ; p2 Þ and wn2 ðp1 ; p2 Þ in Eqs. (38) and (37), respectively, into the common retailer's profit in Eq. (2), and applying the first-order conditions to the resulting profit in terms of the retail prices p1 and p2 gives the common retailer's equilibrium retail prices pn1 and pn2 as follows: pn1 ¼

M1 M5 þ M3 M4 ; M2 M5  M3 M6

ð39Þ

pn2 ¼

M1 M6 þ M2 M4 ; M2 M5  M3 M6

ð40Þ

where a1 ð3β2  γ 2 Þ a2 γ 4β3  2βγ 2 2γ 2  c2 γ; M 2 ¼ ; þ c1 β  2 þ 2 2 β β β2  γ 2 β γ 3β2 γ  γ 3 a1 βγ γða2 þ c2 βÞ ; M3 ¼ 2 þ 3γ; M 4 ¼ 3a2 þ c2 β  2  β β  γ2 β  γ2 M1 ¼

M 5 ¼ 4β  γ þ

βγ 2 ; β  γ2 2

M 6 ¼ 4γ þ

β2 γ γ2  : 2 β β γ 2

With Eqs. (37)–(40), we can see that the two manufacturers' equilibrium wholesale prices wn1 and wn2 in the RSS model are wn1 ¼

β ða1  βpn1 þ γpn2 Þ þ c1 ; β2  γ 2

γ a2 þ c2 β : wn2 ¼  pn2 þ pn1 þ β β

ð41Þ

ð42Þ

where pn1 and pn2 are defined as in Eqs. (39) and (40), respectively. 3.4. The vertical nash model (NG) In NG model, we assume that every firm has equal bargaining power and thus makes their decisions simultaneously. The NG model can be formulated as follows: 8 max π r ðp1 ; p2 Þ > > > ðp1 ;p2 Þ > < max π m1 ðw1 Þ w1 > > > > max : w2 π m2 ðw2 Þ: This scenario arises in a market in which there are relatively small to medium-sized manufacturers and retailers. Since a manufacturer cannot dominate the market over the retailer, his price decision is conditioned on how the retailer prices the product. On the other hand, the retailer must also condition her retail price decision on the wholesale prices. Consider that the decisions of the two manufacturers and the common retailer are already derived in the MSB model and the RSB model. Based on pn1 ðw1 ; w2 Þ and pn2 ðw1 ; w2 Þ in Eqs. (9) and (10), and wn1 ðp1 ; p2 Þ and wn2 ðp1 ; p2 Þ in Eqs. (25) and (26), we can obtain the equilibrium values of the chain members' decisions, denoted as pn1 , pn2 , wn1 and wn2 , in the NG model as follows: pn1 ¼

3βB1 þ γB2 ; 9β2  γ 2

ð43Þ

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J. Zhao et al. / Int. J. Production Economics 153 (2014) 243–252

pn2 ¼

3βB2 þγB1 ; 9β2 γ 2

wn1 ¼  n

w2 ¼ 

ð44Þ

3βB1 þ γB2 a1 þ c1 β γð3βB2 þ γB1 Þ þ ; þ β 9β2  γ 2 βð9β2 γ 2 Þ 3βB2 þ γB1 9β2  γ 2

ð45Þ

a2 þ c2 β γð3βB1 þ γB2 Þ þ þ ; β βð9β2 γ 2 Þ

ð46Þ

where B1 ¼ a1 þc1 β þ

βða1 β þa2 γÞ ; β2  γ 2

B2 ¼ a2 þ c2 β þ

separate the effects of different power structures in the pricing models from the effects of differences in each parameter. The following theorem summarizes the results when two substitutable products are symmetric. Theorem. When two substitutable products have symmetric parameters, the equilibrium solutions for the eight pricing models can be summarized as shown in Tables 1 and 2, where a ¼ a1 ¼ a2 and c ¼ c 1 ¼ c2 . Tables 1 and 2 indicate that the two substitutable products achieve equal wholesale prices and equal retail prices except in the MSS and RSS models. From this, the following Insight 1 can be obtained.

βða1 γ þ a2 βÞ : β2  γ 2

4. Comparisons and managerial implications In this section, we first give the equilibrium solutions for the eight pricing models when two substitutable products are symmetric to compare and analyze them. Second, due to the complicated form of the analytic results, we carry out sensitivity analyses through numerical studies of the key parameters for examining their influences on the equilibrium solutions. Third, we compare the analytical results obtained using numerical approach and study the behavior of firms facing the changing decision environment. On the basis of comparison and analysis, some managerial insights are derived. 4.1. Comparison and analysis of the equilibrium solutions In this subsection, we consider the special case that the parameters in eight pricing models established above are all symmetric, that is, a1 ¼ a2 and c1 ¼ c2 . The reason for restricting the two substitutable products to have identical parameter values is to enable a comparison of the eight pricing models and to gain more insights into the managerial implication. The asymmetry between the products creates problems during comparison of decision models (Tsay and Agrawal, 2000; Mishra and Raghunathan, 2004). Thus, a comparison using simplified parameter structures can

Insight 1. The two substitutable products achieve equal wholesale prices and equal retail prices in CD, MSB, MSC, NG, RSB, and RSC models. This means that the two manufacturers' different competitive strategies and the differences of channel powers between the two echelons do not make the two substitutable products achieve different wholesale prices and different retail prices in CD, MSB, MSC, NG, RSB, and RSC models. Discussion 1. Sensitivity analyses of parameters a and c. With some simple algebraic manipulations, it is possible to verify that ∂pni =∂a 4 0; ∂wni =∂a 4 0; ∂pni =∂c 4 0 and ∂wni =∂c 4 0; i ¼ 1; 2 are valid in CD, MSB, MSC, MSS, NG, and RSC models. Moreover, one can see that ∂pni =∂c ¼ 12 in CD model, ∂wni =∂c ¼ 12 ; ∂pni =∂c ¼ 14 in MSC model, and ∂wni =∂c ¼ 34 ; ∂pni =∂c ¼ 14 ; i ¼ 1; 2 in RSC model. However, it is intractable to analyze the changes of optimal decisions with parameters a and c in RSB and RSS models because the equilibrium solutions are complicated. Numerical studies will be used to illustrate the effects of parameters a and c on the optimal decisions in RSB and RSS models. See Figs. 1 and 2, where default values of parameters are β ¼ 0:8; γ ¼ 0:5; a ¼ 90; and c A f7; 8; 9; 10; 11g. As illustrated in Figs. 1 and 2, in RSB and RSS models, an increase in parameter c increases the optimal retail prices (pn1 ; pn2 )

Table 1 Equilibrium solutions in CD and MS models when two substitutable products are symmetric. Price

CD model

MSB model

MSC model

MSS model

n

w1

–

a þ βc 2β  γ

c a þ 2 2ðβ  γÞ

wn2

–

a þ βc 2β  γ

c a þ 2 2ðβ  γÞ

pn1

c a þ 2 2ðβ  γÞ

a þ βc a þ 2ð2β  γÞ 2ðβ  γÞ

c 3a þ 4 4ðβ  γÞ

pn2

c a þ 2 2ðβ  γÞ

a þ βc a þ 2ð2β  γÞ 2ðβ  γÞ

c 3a þ 4 4ðβ  γÞ

ð2β þ γÞa þ ð2β2  γ 2 þ βγÞc 3β2  2γ 2 ð2β þ γÞaγ þ ð2β2  γ 2 þ βγÞc a þ βc þ 2β 2βð3β2  2γ 2 Þ 2 2 ð2β þ γÞa þ ð2β  γ þ βγÞc a þ 2ðβ  γÞ 2ð3β2  2γ 2 Þ ð2β þ γÞaγ þ ð2β2  γ 2 þ βγÞc a a þ βc þ þ 2ðβ  γÞ 4β 4βð3β2  2γ 2 Þ

Table 2 Equilibrium solutions in RS and NG models when two substitutable products are symmetric. Price

NG model

RSB model

RSC model

RSS model

wn1

ðβ þ γÞ½βaþ ðaþ βcÞðβ  γÞ a þ βc þ βðβ  γÞð3β  γÞ β ðβ þ γÞ½βaþ ðaþ βcÞðβ  γÞ a þ βc þ βðβ  γÞð3β  γÞ β aþ βc βa þ 3β  γ ð3β  γÞðβ  γÞ aþ βc βa þ 3β  γ ð3β  γÞðβ  γÞ

aþ βc ðγ  βÞδ1 þ β β aþ βc ðγ  βÞδ1 þ β β δ1

a 4ðβ  γÞ a 4ðβ  γÞ 3a 4ðβ  γÞ 3a 4ðβ  γÞ

  β δ2 δ6 þ δ4 δ5 δ 2 δ 6 þ δ 4 δ5 aβ þγ þc δ3 δ6  δ4 δ7 δ 3 δ 6  δ 4 δ7 β2  γ 2 ðδ2 δ6 þ δ4 δ5 Þγ δ2 δ7 þ δ3 δ5 a2 þ c2 β  þ ðδ3 δ6  δ4 δ7 Þβ δ3 δ6  δ4 δ7 β δ2 δ6 þ δ4 δ5 δ3 δ6  δ4 δ7 δ2 δ7 þ δ3 δ5 δ3 δ6  δ4 δ7

wn2 pn1 pn2

δ1

3c 4 3c þ 4 c þ 4 c þ 4 þ

Note: δ1 ¼ ð½ð3a þ βcÞβ  ð2a þ βcÞγð2β2 þ γ 2 þ 3βγÞÞ=ð2ð2β2 þ γ 2 Þ2  18γ 2 Þ; δ2 ¼ að3β2  γ 2 Þ=ðβ2  γ 2 Þþ cðβ  γÞ  2ðaγ=βÞ; δ3 ¼ ð4β3  2βγ 2 Þ=ðβ2  γ 2 Þ þ 2γ 2 =β; δ4 ¼ ð3β2 γ  γ 3 Þ=ðβ2  γ 2 Þ þ 3γ; δ5 ¼ 3aþ cβ  aβγ=ðβ2  γ 2 Þ ðγða þ cβÞÞ=β; δ6 ¼ 4β  γ þ βγ 2 =ðβ2  γ 2 Þ; δ7 ¼ 4γ þ β2 γ=ðβ2  γ 2 Þ  γ 2 =β.

J. Zhao et al. / Int. J. Production Economics 153 (2014) 243–252

200

249

1200 *

180 160

is

w*1

is

* w 2

is p1

is w*

2

800

1

is p*2

price

Price

*

is w1

is p*

140

*

is p2

1000

120 100

600

400

80 200 60 40

7

7.5

8

8.5

9

9.5

10

10.5

0 0.25

11

0.3

0.35

0.45

Fig. 3. Change of optimal prices with β in MSB model.

Fig. 1. Changes of optimal prices with c in the RSB model.

7

220

x 104 * r * πm1 π* m2

is π

200

6

*

is w1

180 160

is

140

is

is is

is w*

5

2 p* 1 * p 2

profit

Price

0.4

β

c

120 100

4 3 2

80 1

60 40

7

7.5

8

8.5

9

9.5

10

10.5

0 0.25

11

0.3

c

Consider that it is intractable to analyze the changes of optimal decisions with parameters β and γ in MSS and RSS models because the equilibrium solutions are complicated. We will use numerical studies to illustrate the effects of parameters β and γ on the optimal decisions and maximum profits of channel members. First, we explore how the optimal retail prices, wholesale prices, and the maximum profits are affected by changes in the self-price sensitivity β. Figs. 3–6 present the changes of the optimal prices and the maximum profits with the parameter β in the MSB and MSS models, where the default values of the parameters are a1 ¼ a2 ¼ 100; γ ¼ 0:2; c1 ¼ c2 ¼ 7; and β A f0:25; 0:3; 0:35; 0:4; 0:45g.

0.45

Fig. 4. Changes of maximum profits with β in MSB model.

and optimal wholesale prices (wn1 ; wn2 ) of the two substitutable products. Moreover, the same results can be obtained for parameter a in RSB and RSS models. From the above analysis, the following Insight 2 can be derived.

Discussion 2. Sensitivity analyses of parameters β and γ.

0.4

β

Fig. 2. Changes of optimal prices with c in the RSS model.

1400 is p*1 is p*

1200

2 *

is w1

1000

price

Insight 2. The optimal prices will increase as the parameter c increases. This is independent of the two manufacturers' different competitive strategies and the differences of channel powers between the two echelons. The same results are also valid for parameter a. Moreover, the optimal prices have the fixed growth rates as parameter c increases in CD, MSC, and RSC models.

0.35

*

is w2

800 600 400 200 0 0.25

0.3

0.35

0.4

0.45

β Fig. 5. Change of optimal prices with β in MSS model.

It follows from Figs. 3 to 6 that the optimal wholesale prices, the optimal retail prices, and the corresponding maximum profits of both the two manufacturers and the common retailer, all decrease with the self-price sensitivity β in the MSB and MSS

250

J. Zhao et al. / Int. J. Production Economics 153 (2014) 243–252

models. The same effect of the changes in β on the optimal prices and the maximum profits can be derived for the other six pricing models, and the details are, therefore, not discussed in this paper.

6

x 104 *

is πr

* m1 * πm2

is π

5

is

profit

4

3

2

1

0 0.25

0.3

0.35

0.4

0.45

β Fig. 6. Changes of maximum profits with β in MSS model.

Next, the mechanism by which the optimal retail prices, wholesale prices, and the maximum profits are affected by the changes in the cross-price sensitivity γ is investigated. Figs. 7–10 present the changes of the optimal prices and the maximum profits with parameter γ in the MSB and MSS models, where the default values of the parameters are a1 ¼ a2 ¼ 100; β ¼ 0:8; c1 ¼ c2 ¼ 7; and γ A f0:35; 0:4; 0:45; 0:5; 0:55g. It follows from Figs. 7 to 10 that the optimal wholesale prices, the optimal retail prices, and the corresponding maximum profits of both the two manufacturers and the retailer all increase with the cross-price sensitivity γ in the MSB and MSS models. The same effect of the changes in γ on the optimal prices and the maximum profits can be derived for the other six pricing models, and the details are also, therefore, not discussed in this paper. Moreover, using numerical studies, we also have the following results. For brevity, these numerical graphs are not presented in this paper. (2.1) In both MS and RS pricing cases, the optimal retail prices and optimal wholesale prices are most sensitive to the changes of the parameter β when the two manufacturers adopt the cooperation strategy, followed by the Stackelberg competition strategy, then the Bertrand competition strategy. (2.2) In both MS and RS pricing cases, the manufacturer 1's maximum profit is most sensitive to the changes of parameter β when the two manufacturers adopt the cooperation strategy, followed by the Stackelberg competition strategy, then the Bertrand

260

300 *

240

is p1

*

is p2

220

is

200

is

*

is p2

250

w*1 * w2

*

is w1 *

is w2

200

180

price

price

*

is p1

160

150

140 120

100

100 80

0.35

0.4

0.45

0.5

50

0.55

0.35

0.4

Fig. 7. Change of optimal prices with γ in MSB model.

0.55

9000

11000

is π

10000

is

9000

is

* r * πm1 * πm2

8000 7000

*

is πr

is π*m1 *

is πm2

6000

profit

8000

profit

0.5

Fig. 9. Change of optimal prices with γ in MSS model.

12000

7000 6000

5000 4000

5000

3000

4000

2000

3000 2000

0.45

γ

γ

0.35

0.4

0.45

0.5

γ Fig. 8. Changes of maximum profits with γ in MSB model.

0.55

1000

0.35

0.4

0.45

0.5

γ Fig. 10. Changes of maximum profits with γ in MSS model.

0.55

J. Zhao et al. / Int. J. Production Economics 153 (2014) 243–252

competition strategy. However, the common retailer's maximum profit is most sensitive to the changes of parameter β when the two manufacturers adopt the Bertrand competition strategy, followed by the Stackelberg competition strategy, then the cooperation strategy. Moreover, the manufacturer 2's maximum profit is most sensitive to the changes of parameter β when the two manufacturers adopt the Stackelberg competition strategy, followed by the cooperation strategy, then the Bertrand competition strategy. (2.3) The optimal retail prices, optimal wholesale prices and channel members' maximum profits all decrease with increasing values of β in eight pricing models. Which means that, regardless of the channel members' bargain powers and pricing strategies, the optimal prices and the maximum profits decrease with decreasing values of self-price sensitivity β. (2.4) In both MS and RS pricing cases, the optimal retail prices and optimal wholesale prices are most sensitive to the changes of parameter γ when the two manufacturers adopt cooperation strategy, followed by the Stackelberg competition strategy, then the Bertrand competition strategy. (2.5) In both MS and RS pricing cases, the manufacturer 1's maximum profit is most sensitive to the changes of parameter γ when the two manufacturers adopt cooperation strategy, followed by the Bertrand competition strategy, then the Stackelberg competition strategy. However, the common retailer's maximum profit is most sensitive to the changes of parameter γ when the two manufacturers adopt the Bertrand competition strategy, followed by the Stackelberg competition strategy, then the cooperation strategy. Moreover, the manufacturer 2's maximum profit is most sensitive to the changes of parameter γ when the two manufacturers adopt the Stackelberg competition strategy, followed by the cooperation strategy, then the Bertrand competition strategy. (2.6) The optimal retail prices, optimal wholesale prices and channel members' maximum profits all increase with increasing values of γ in eight pricing models, which means that, regardless of the channel members' bargain powers and pricing strategies, the optimal prices and the maximum profits increase with increasing values of cross-price sensitivity γ. 4.2. A numerical example Due to the complicated form of the analytic results, in this subsection, we compare the analytical results obtained from the above eight pricing models using numerical approach and study the behavior of firms facing the changing decision environment. We assume that the parameters take the values as follows: the manufacturing costs c1 ¼ 7 and c2 ¼ 9, the market bases a1 ¼ 100 and a2 ¼ 90, and price elasticities β ¼ 0:8 and γ ¼ 0:5. The corresponding results are shown as in Tables 3 and 4. Discussion 3. From Tables 3 and 4, we can have the following results. (3.1) The maximum profit of the whole supply chain in CD model is higher than that in decentralized pricing models. On the Table 3 Maximum profit of total system and every firm under different pricing models. Scenario

π nc

π nm1

π nm2

π nr

CD MSB MSC MSS RSB RSC RSS NG

14,307 13,238 10,730 11,730 13,238 10,730 12,548 13,947

– 3040 3876 2359 1520 1938 1470 2073

– 2636 3278 3940 1318 1639 1445 1735

– 7561 3577 5432 10,399 7153 9633 10,139

251

Table 4 Optimal prices under different pricing models. Scenario

pn1

wn1

pn1  wn1

pn2

wn2

pn2  wn2

CD MSB MSC MSS RSB RSC RSS NG

163.7564 207.3473 242.1346 236.2458 207.3473 242.1346 222.0069 189.2110

– 94.1818 163.7564 151.9789 50.5909 85.3782 61.9136 57.9093

– 113.1655 78.3782 84.2670 156.7564 156.7564 160.0933 131.3018

160.9103 201.5012 236.8654 210.5320 201.5012 236.8654 208.7518 184.1925

– 90.1818 160.9103 108.2434 49.5909 84.9551 51.5025 55.5644

– 111.3193 75.9551 102.2886 151.9103 151.9103 157.2493 128.6280

other hand, the maximum profit of the whole supply chain in NG model is the biggest one among the decentralized pricing models. (3.2) The maximum profit of the whole supply chain in MSC model equals that in RSC model and the maximum profit of the whole supply chain in MSB model equals that in RSB model, which tells us that the leadership between the two manufacturers and the common retailer does not affect the maximum profit of the whole supply chain when the two manufacturers' adopt Bertrand competition or cooperation pricing strategies. However, the leadership between the two manufacturers and the common retailer affects the maximum profit of the whole supply chain when the two manufacturers' play Stackelberg game. (3.3) One can observe directly from Table 3 that the two manufacturers' different pricing strategies affect the maximum profits of the two manufacturers and the common retailer. The retailer achieves her highest profit in RSB model, followed by NG model, and achieves her lowest profit in MSS model, which implies that the common retailer who acts as the leader does not always have the superiority of gaining profit. Therefore, the common retailer, as a Stackelberg leader in RS games, should find a way to induce the two manufacturers to implement the Bertrand competition strategy. (3.4) The manufacturer 1 achieves his highest profit in MSC model, followed by MSB model, and achieves his lowest profit in RSS model, and the manufacturer 2 achieves his highest profit in MSS model, followed by MSC model, and achieves his lowest profit in RSB model. One can see that the leader does not have the advantage to achieve higher profit in MS game cases, which seems counter-intuitive. This insight is helpful for the leader of the two manufacturers, because the best action strategy for whom is to adopt cooperation and is not to adopt the Stackelberg competition in MS game. Moreover, the manufacturer 1 benefits more from their cooperation action than the manufacturer 2. The retailer can gain the highest profit when the two manufacturers pursue Bertrand competition strategy in both MS game and RS pricing cases. (3.5) From Table 3, we can also see that cooperation strategy does not always benefit two manufacturers, for example, the manufacturer 2's profit in MSC model is lower than that in MSS model. This insight is helpful for a manufacturer who is the follower in MS games, because *which tells the manufacturer that a suitable profit-split should be negotiated with his rival before agreeing to act in union. (3.6) One can observe directly from Table 4 that the two manufacturers' cooperation action will result in the lowest profit margins for the retailer in both RS and MS pricing cases. Both two substitutable products achieve the highest wholesale prices in MSC model and achieve the lowest wholesale prices in RSB model. (3.7) Table 4 tells us that two substitutable products achieve the lowest retail prices in CD model and achieve the highest retail prices in RSC model, and retail prices of the two substitutable products in MSB model equal to those in RSB model respectively regardless of the leadership between the two manufacturers and the common retailer. The same result occurs in MSC and

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J. Zhao et al. / Int. J. Production Economics 153 (2014) 243–252

RSC models. On the other hand, comparing among the decentralized decision cases, the retail prices of the two substitutable products obtain the lowest values in NG model, which means that the consumers can get the highest profits among the decentralized decision cases when there is no channel leadership. (3.8) Comparing among the decentralized decision cases, the profit margins of the two substitutable products obtain the highest values in RSS model and achieve the lowest values in MSC model. Moreover, it is interesting to note that the profit margins of the two substitutable products in RSB model equal to those in RSC model regardless of the two manufacturers' competitive pricing actions. 5. Some concluding comments We analyze two competitive manufacturers' and one common retailer's pricing decisions for two substitutable products by considering the two manufacturers' three kinds of competitive strategies: Bertrand competition, cooperation and Stackelberg competition. As a benchmark to evaluate channel decision in different decision cases, we first develop the centralized pricing model and derive the optimal retail prices. We then establish the pricing models in decentralized decision cases (e.g., the MS and RS pricing models) through considering the two manufacturers' three kinds of competitive strategies, and the NG pricing model. Finally, we provide comparison of the maximum profits and optimal decisions of the whole supply chain and every supply chain members. The analytical and numerical results reveal some insights into the economic behavior of firms. However, our results are based upon some assumptions about the pricing models of substitutable products. Thus, several extensions to the analysis in this paper are possible. First, as opposed to the risk neutral supply chain members considered in this paper, one could study the case where the supply chain members with different attitudes towards risk could also examine the influence of their attitudes towards risk on individual profits and the profit of the whole supply chain. Second, this paper considers a case with two-products, deterministic-linear-demand function in a singleperiod, one can study the case with multiple-products and other forms of demand function in stochastic/fuzzy multiple periods environment. Third, we assume that both the two manufacturers and the common retailer have symmetric information about costs and demands. So, an extension would be to consider the supply chain with information asymmetry, such as asymmetry in cost information and demand information. Finally, one can also consider the coordination of the supply chain under linear or isoelastic demand with symmetric and asymmetric information. Acknowledgments The authors wish to express their sincerest thanks to the editors and anonymous referees for their constructive comments and suggestions on the paper. We gratefully acknowledge the support of (i) National Natural Science Foundation of China, No. 71301116, for J. Zhao; (ii) National Natural Science Foundation of

China (NSFC), Research Fund Nos. 71371186 and 71001106, for J. Wei; (iii) The Major Program of the National Social Science Fund of China (Grant no.13&ZD147), and National Natural Science Foundation of China, No. 71372100, for Y.J. Li.

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