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Manufacturer's pricing strategy in a two-level supply chain with competing retailers and advertising cost dependent demand
Economic Modelling 38 (2014) 102–111
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Economic Modelling journal homepage: www.elsevier.com/locate/ecmod
Manufacturer's pricing strategy in a two-level supply chain with competing retailers and advertising cost dependent demand B.C. Giri ⁎, S. Sharma Department of Mathematics, Jadavpur University, Kolkata 700032, India
a r t i c l e
i n f o
Article history: Accepted 1 November 2013 Available online 21 January 2014 Keywords: Supply chain Advertising cost dependent demand Pricing strategy
a b s t r a c t The paper studies a two-echelon supply chain comprising one manufacturer and two competing retailers with advertising cost dependent demand. The manufacturer acts as the Stackelberg leader who specifies wholesale price for each retailer. The two retailers compete with each other in advertising and they have different sales costs. The manufacturer uses one of the following two pricing strategies: (i) setting the same wholesale price for both the retailers irrespective of the difference in their sales costs; (ii) setting different wholesale prices for the retailers depending on their sales costs. Two models are developed. In the first model, the manufacturer shares a fraction of each retailer's advertising cost while in the second model, the manufacturer does not share any retailer's advertising expenses. In both the models, we derive the retailers' and manufacturer's optimal strategies. A numerical example is given to illustrate the theoretical results developed in each model. Computational results show that it is always beneficial for the manufacturer to adopt different wholesale pricing strategy for the retailers. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Rapid technological advancement, increasing expectation of customer and shortened product life-cycle have intensified competition and introduced new challenges in the market. To overcome the situation, companies should have good flexibility to respond and capability to adopt new strategies to utilize greater share of the market demand. More and more companies are resorted to depend on service and product quality in order to avoid traditional competition which focuses solely on price. Advertising is one way of promotional campaign used by many firms to provide customers the brand knowledge of their products and services and other specialties of their organizations. Depending upon the product type and market demand, either the manufacturer or the retailer or both participate in advertising. Two-tier advertising or co-operative (co-op) advertising is an interactive scheme in a manufacturer–retailer system in which the manufacturer pays a part of the advertising expenditure incurred by the retailer in local advertisement. For example, Small World Toys offers a 2% advertising on total net purchases (Small World Toys, 2007). In personal computer industry, IBM offers a 50–50 split of advertising costs with retailer while Apple Computer pays 75% of the media cost (Brennan, 1988). Manufacturer uses co-op advertising to strengthen the image of the brand and motivate immediate sales at the retail level while retailer uses local advertising to bring potential customers to the stage of buying. The sharing of local advertising cost by the manufacturer is intended to influence ⁎ Corresponding author. E-mail addresses: [email protected] (B.C. Giri), [email protected] (S. Sharma). 0264-9993/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econmod.2013.11.005
retailers to spend more in local advertisement which in turn generates greater sales volume. Berger (1972) was probably the first who discussed co-op advertising in a manufacturer–retailer channel. His work was subsequently extended by many authors (Berger and Magliozzi, 1992; Fulop, 1988; Somers et al., 1990) under different co-op advertising settings. Roslow et al. (1993) discussed co-op advertising under supply chain framework and demonstrated that cooperation in advertising investment could increase the profit of the whole supply chain. Khouja and Robbins (2003) studied the effect of advertising on the variance of demand under newsboy framework. However, the above studies are based on the assumption that demand at the retailer's end depends explicitly on retailer's local advertisement. Taking manufacturer's national level advertisement investment in addition to retailer's local advertising into consideration, Huang et al. (2002) and Li et al. (2002) independently developed manufacturer–retailer supply chain models under co-op advertising setting. Using game theory, Huang and Li (2001) further discussed the issue of co-op advertisement for supply chain with one manufacturer and one retailer. In their model, they showed that if the ratio of the manufacturer's and retailer's marginal profits is relatively high, the manufacturer offers a positive amount of advertising allowance to the retailer. In case the marginal profit ratio is relatively low, the manufacturer is reluctant to offer any advertising allowance. Szmerekovsky and Zhang (2009), Xie and Wei (2009) extended the model of Huang et al. (2002) assuming that customer demand is dependent on retail price as well as advertising efforts of channel members. Yue et al. (2006) also extended Huang et al. (2002) model assuming that the manufacturer directly offers a price discount to customers. To increase profits of the parties involved in the supply chain, they
B.C. Giri, S. Sharma / Economic Modelling 38 (2014) 102–111
recommended that coordination in local and co-operative advertising with a partnership relation between manufacturer and retailer is the best solution. In the context of co-operative advertising, Kunter (2012) analyzed a royalty payment contract for coordination of a manufacturer–retailer channel. Consumer demand was assumed to be simultaneously affected by the retail price and marketing efforts of manufacturer and retailer. Aust and Buscher (2012) addressed the optimal pricing and advertising decisions in a manufacturer–retailer supply chain where the consumer demand depends on the retail price as well as the channel members' advertising expenditures. Additionally, they considered a co-operative advertising program where the manufacturer can bear a fraction of retailer's advertising costs. Zhang et al. (2013) proposed a dynamic cooperative advertising model for a manufacturer–retailer supply chain taking into account the impact of advertising on the reference price and analyzed how the reference price would influence the decisions of all the channel members. Chen (2011) developed a model to study the combined effect of co-operative advertising, return policy and channel coordination for supply chain. Recently, Aust and Buscher (2014) made a comprehensive updated review of literature on co-operative advertising in supply chain management. From the perspective of customer buying behavior, it is seen that besides price other factors such as promotional activity through advertising also influence customers' preferences and their purchasing decisions and hence market demand. Advertising is the most effective method of promotion and hence advertising investment plays a key role in a firm's marketing decisions. Through advertising the retailer gives the customers brand knowledge of the product, detailed description of the product as well as review of the product in the market. All these useful guidelines stimulate customers' purchasing decisions and thus bring potential customers to the stage of buying. Such attribute of market demand can be seen in insurance industries, departmental stores, supermarkets, newsvendor products like fashion apparel, personal computers or supply chains with fixed retail price. Wang et al. (2010), Yan (2010) and Wang (2011) considered demand dependent on investment on advertising. For more literature concerning advertising sensitive demand, readers can be referred to articles contributed by Jorgensen et al. (2000), Karray and Zaccour (2006), Xie and Neyret (2009). The primary interest of the present study is focused on the competitive behavior of manufacturer and retailers where mode of competition shifts from strategies that consider price differentiation to strategies that include differentiation in terms of advertising efforts. We consider a supply chain with one manufacturer and two competing retailers who face advertising cost dependent demand. In the proposed model, the effect of price is ignored; price remains fundamental as a basis for competition. However, it may not be the only means of determining competition. For instance, when different firms offer the same retail price for a product, the one which promotes the product more definitely has a better edge to consume greater market demand than the others through advertising done by the retailer. Viscolani and Zaccour (2009) examined a duopoly problem where each firm's current sales is proportional to its goods in stock, which is related to the firm's own advertising effort and negatively related to that of its competitor. The idea that one player's advertising effort may hamper competitor's sales can also be seen in the articles provided by Anderson and Renault (2009), Barigozzi et al. (2009). Viscolani (2012) proposed a model for two competing manufacturers selling substitutable products in a homogeneous market. Demand for each manufacturer is positively correlated with his own advertising effort and negatively correlated with advertising effort of his rival. Lu et al. (2011) modeled a game-theoretic problem considering two competing manufacturers and one common retailer, facing end consumers who are sensitive to both retail price and manufacturer service. We investigate the effect of manufacturer's pricing strategy in the supply chain. The manufacturer who acts as Stackelberg leader specifies
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wholesale prices to retailers. The manufacturer uses one of the following pricing strategies: (i) a common wholesale price for both the retailers while disregarding the difference of their sales costs (strategy 1) and (ii) different wholesale prices for two retailers according to their different sales costs (strategy 2). We determine optimal strategies of the manufacturer and retailers in response to manufacturer's different pricing strategies, effects of manufacturer's different pricing strategies on each member's performance and the whole supply chain's performance, the pricing strategy which is best suited to the manufacturer in response to different sales cost scenarios. Lau and Lau (1999) analyzed the pricing and return policies of a monopolistic manufacturer for single-period commodities. In our present study, we assume in Model I that the manufacturer takes part in co-op advertising and provides the retailers a fraction of local advertising costs incurred by the retailers while in Model II the manufacturer does not participate in co-op advertising. Both the models are developed under the framework of two pricing strategies set by the manufacturer. We investigate which strategy is beneficial for channel members as well as for the whole channel. Among the existing literature on co-operative advertising, very few articles discussed a channel where a single manufacturer sells a product through two or more competing retailers. Related articles in this issue were presented by Wang et al. (2011) and He et al. (2011). This channel structure, however, represents numerous markets including those made up of specialty stores (e.g., consumer electronics, sporting goods, automobile parts, to name a few), departmental stores and supermarkets. The main contributions of our paper are the following. First, our paper extends the current literature on co-operative advertising to account for a supply chain with multiple retailers. Most of the previous research on co-operative advertising was done under the traditional setting of a bilateral monopoly model where one manufacturer sells through one retailer. Second, it takes into account the competitive behavior of two retailers in terms of advertising efforts, i.e. demand of each retailer is not only related to his own advertising investment but also on the other retailer's advertising expenditure. Third, it studies the effects of the manufacturer's pricing strategies in conjunction with retailers' gaming interaction, and discusses the related impact on supply chain's decisions and performance when all the channel members are affected by advertising investment dependent demand. This makes our contribution unique because no such analysis in connection with co-operative advertising has been done before. The rest of the paper is organized as follows: Model description and assumptions are presented in the next section. Model I is formulated in Section 3 with co-operative advertising under different pricing strategies of the manufacturer. Section 4 describes Model II when the manufacturer does not participate in co-operative advertising. Section 5 is devoted to the discussion related to the effects of manufacturer's pricing strategies in Model II on retailers' and manufacturer's decision making. In Section 6, theoretical results are verified through a numerical example. Finally, the paper is concluded and scope of future research is suggested in Section 7. 2. Assumptions and model description We consider a two-echelon supply chain comprising one manufacturer and two competing retailers for trading a single product. The manufacturer acts as the Stackelberg leader and sets wholesale price to two retailers. Suppose that the unit retail price of the product is p and unit production cost is c. The two retailers have different sales efficiencies and hence have different sales costs. Let the retailer i's unit sales cost be si (i = 1,2). Without any loss of generalization, we assume that s1 b s2. We assume that the two retailers compete with each other on advertising and the market demand depends on their advertising expenses. Further, the price differentiation is negligible to the customers at the time of purchase and the demand is mainly influenced by the
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advertising costs incurred by the two competing retailers. For each retailer, the demand increases with own advertising while it decreases with the advertising done by his rival. We define the demand at the retailer i as (Choi, 1991, 1996; Wu et al., 2012; Xie and Wei, 2009; Zhang and Xie, 2012): DðI i Þ ¼ α þ β
qffiffiffiffiffi pffiffiffi Ii −γ I j ; i ¼ 1; 2 and j ¼ 3−i:
Here, Ii denotes retailer i's advertising expenditure, α denotes the primary demand of retailer i, β denotes the demand sensitivity of the retailer i on its own advertising (Padmanabhan and Png, 1997) while γ denotes the demand sensitivity of retailer i on its rival's advertising. The linear and symmetrical demand model represents a situation in which two retailers have equal competing powers in a duopolistic market place. We assume that γ b β so that the response functions are negatively sloped which, in turn, ensures the existence of the Nash equilibrium. This appears reasonable since sales are relatively more sensitive to retailer's own advertising than the advertising done by the competing retailer. This demand pattern is widely used in marketing research (see Desiraju and Moorthy, 1997; Ingene and Parry, 1995) and in some economics research (see Singh and Vives, 1984; Vives, 1984, 1985).
and the retailer 2's optimal profit as " # 2 β ðp−w1 −s2 Þ βγðp−w1 −s1 Þ r TP 12 I12 ¼ ðp−w1 −s2 Þ α þ − : 4ð1−t 2 Þ 2ð1−t 1 Þ
3.1.2. The manufacturer's problem The manufacturer faces the problem to determine the wholesale price which maximizes his/her profit. The manufacturer's profit is given by m
TP 1 ðw1 Þ ¼
2 h X
qffiffiffiffiffi qffiffiffiffiffiffii ðw1 −cÞ α þ β I1i −γ I1 j −t 1 I11 −t 2 I12 ; j ¼ 3−i:
i¼1
ð2Þ Substituting the values of I∗11 and I∗12 given in Proposition 1, the manufacturer's profit can be written as
β p−w1 −s1 p−w1 −s2 m þ TP 1 ðw1 Þ ¼ ðw1 −cÞ 2α þ ðβ−γÞ 2 1−t 1 1−t 2 −
t 1 ðp−w1 −s1 Þ2 β2 t 2 ðp−w1 −s2 Þ2 β2 − : 4ð1−t 1 Þ2 4ð1−t 2 Þ2
ð3Þ
3. Model I: Manufacturer share a fraction of retailers' advertising expenditures Proposition 2. For the pricing strategy 1, the manufacturer's optimal wholesale price is given by
3.1. Case1: Manufacturer's pricing strategy 1 In this section, we assume that manufacturer declares beforehand that she/he would share a fraction ti of retailer's advertising expenditure prior to the start of selling season. The manufacturer then sets a common wholesale price w1 irrespective of retailers' sales costs si's, i = 1,2. In response, the retailers observe w1 and follow Nash equilibrium to determine their optimal advertising expenditures. 3.1.1. The retailer's problem The retailer i faces the problem of determining advertising expenditure I1i which maximizes his/her profit. Retailer i's profit is given by
i ¼ 1; 2 and j ¼ 3−i:
ð1Þ Proposition 1. For the pricing strategy 1 of the manufacturer, the retailers 2
1 and 2's optimal advertising expenditures are given by I11 ¼ ðp−w1 −s1 Þ2 β
2
2
¼−
ðp−w1 −si Þβ 3=2
4I1i
b0; r
2
2
2
1i
1 −s1 Þ β 1 −s2 Þ β and I 12 ¼ ðp−w . This completes the proof. I11 ¼ ðp−w 4ð1−t Þ2 4ð1−t Þ2 1
d2 TP m 1 1 β2 t1 t2 1 ¼ −β ð β−γ Þ þ þ − b0: ð5Þ 1−t 1 1−t 2 2 ð1−t 1 Þ2 ð1−t 2 Þ2 dw21 dTP m 1 dw1
¼ 0, we obtain after
4α ðp−s1 Þt 1 β ðp−s2 Þt 2 β p þ c−s1 p þ c−s2 þ þ þ ð β−γ Þ þ β 1−t 1 1−t 2 ð1−t 1 Þ2 ð1−t 2 Þ2 w1 ¼ : t1 t2 1 1 þ þ þ 2 ð β−γ Þ β 1−t 1 1−t 2 ð1−t 1 Þ2 ð1−t 2 Þ2
implying that TPr1i(I1i) is concave in I1i. solving dT PdI1i ðI1i Þ ¼ 0; we obtain 2
ð4Þ
some calculations
Proof. From Eq. (1), we have d
t 1 ðp−w1 −s1 Þβ 2 t 2 ðp−w1 −s2 Þβ2 þ 2ð1−t 1 Þ2 2ð1−t 2 Þ2
Therefore, TPm 1 (w1) is concave in w1. Solving
4ð1−t 2 Þ
TP r1i ðI1i Þ d2 I 1i
dTP m β ðβ−γÞ p−w1 −s1 p−w1 −s2 1 1 1 ¼ 2α þ þ −ðw1 −cÞ þ dw1 1−t 1 1−t 2 2 1−t 1 1−t 2
4ð1−t 1 Þ
and I12 ¼ ðp−w1 −s2 Þ2 β ,respectively.
2
Proof. From Eq. (3), we have
þ
qffiffiffiffiffiffi pffiffiffiffiffi r TP 1i ðI1i Þ ¼ ðp−w1 −si Þ α þ β I1i −γ I 1 j −ð1−t i ÞI1i ;
2
4α ðp−s1 Þt 1 β ðp−s2 Þt 2 β p þ c−s1 p þ c−s2 þ þ þ ð β−γ Þ þ β 1−t 1 1−t 2 ð1−t 1 Þ2 ð1−t 2 Þ2 w1 ¼ : t1 t2 1 1 þ þ þ 2ðβ−γÞ β 2 2 1−t 1 1−t 2 ð1−t 1 Þ ð1−t 2 Þ
2
Proposition 1 indicates that the retailers' optimal advertising expenditures increase as manufacturer's participation rate for subsidizing retailers' advertising costs increases. Substituting the values of I∗11 and I∗12 in Eq. (1), we get the retailer 1's optimal profit as " # β2 ðp−w1 −s1 Þ βγðp−w1 −s2 Þ r − TP 11 I11 ¼ ðp−w1 −s1 Þ α þ 4ð1−t 1 Þ 2ð1−t 2 Þ
This completes the proof. From Eqs. (1) and (2) and from Propositions 1 and 2, the profit of the entire supply chain is given by w r r m TP 1 ¼ TP 11 I11 þ TP 12 I 12 þ TP 1 w1 qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi ¼ ðp−c−s1 Þ α þ β I 11 −γ I 12 þ ðp−c−s2 Þ α þ β I12 −γ I 11 :
−I11 −I12
ð6Þ
B.C. Giri, S. Sharma / Economic Modelling 38 (2014) 102–111
3.2. Case 2: Manufacturer's pricing strategy 2
and
With pricing strategy 2, the manufacturer first sets wholesale prices w21 and w22 for retailers 1 and 2, respectively according to their different sales costs s1 and s2. The retailers then follow Nash equilibrium to determine their optimal advertising expenditures. 3.2.1. The retailer's problem The retailer's problem is to determine the advertising cost that maximizes his/her profit. The retailer's profit is given by r TP 2i ðI 2i Þ
qffiffiffiffiffiffi pffiffiffiffiffi ¼ ðp−w2i −si Þ α þ β I 2i −γ I 2 j −ð1−t i ÞI 2i ;
105
i ¼ 1; 2 and j ¼ 3−i:
ð7Þ
"
! # β2 t 1 β2 βγ 1 1 B þ þ A þ 1−t 1 2ð1−t 1 Þ2 2 1−t 1 1−t 2 ! ! 2 β2 t1 β2 β2 t2 β2 β 2 γ2 1 1 þ þ þ − 1−t 1 2ð1−t 1 Þ2 1−t 2 2ð1−t 2 Þ2 4 1−t 1 1−t 2
w22 ¼
ð13Þ where,
2 2 β t2 β βγ p−s1 c ðp þ c−s2 Þ þ ð p−s Þ− þ 2 2 1−t 1 1−t 2 2ð1−t 2 Þ 2ð1−t 2 Þ2
: 2 2 β t1 β βγ p−s2 c B¼αþ ðp þ c−s1 Þ þ ð p−s Þ− þ 1 2 1−t 2 1−t 1 2ð1−t 1 Þ 2ð1−t 1 Þ2
A¼αþ
Proposition 3. For the pricing strategy 2 of the manufacturer, the retailers 2
1 and 2's optimal advertising expenditures are I21 ¼ ðp−w21 −s12Þ 2
ðp−w22 −s2 Þ β 4ð1−t 2 Þ2
2
β2
4ð1−t 1 Þ
and I 22 ¼
,respectively.
Proof. The proof is similar to Proposition 1. Substituting the values of I∗21 and I∗22 in Eq. (7), the retailer 1's optimal profit can be derived as "
# 2 β ðp−w21 −s1 Þ βγðp−w22 −s2 Þ r − TP 21 I21 ¼ ðp−w21 −s1 Þ α þ ð8Þ 4ð1−t 1 Þ 2ð1−t 2 Þ
Proof. The proof is given in Appendix A. From Eqs. (7) and (10), and from Propositions 3 and 4, the profit of the entire supply chain is given by w r r m TP 2 ¼ TP 21 I21 þ TP 22 I 22 þ TP 2 w21 ; w22 qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi ¼ ðp−c−s1 Þ α þ β I 21 −γ I22 þ ðp−c−s2 Þ α þ β I 22 −γ I21 :
−I21 −I22
ð14Þ
and the retailer 2's optimal profit as " # β2 ðp−w22 −s2 Þ βγðp−w21 −s1 Þ r − : ð9Þ TP 22 I22 ¼ ðp−w22 −s2 Þ α þ 4ð1−t 2 Þ 2ð1−t 1 Þ
4. Model II: Manufacturer does not share retailers' advertising expenditures 4.1. Case 1: Manufacturer's pricing strategy 1
3.2.2. The manufacturer's problem The manufacturer's problem is to select different wholesale prices w2is for two retailers that maximize his/her profit. The manufacturer's profit is given by m
TP 2 ðw21 ; w22 Þ ¼
2 X
qffiffiffiffiffi qffiffiffiffiffiffi ðw2i −cÞ α þ β I 2i −γ I2 j −t 1 I 21 −t 2 I22 ; j ¼ 3−i:
i¼1
ð10Þ Substituting the values of
I∗21
" m
TP 2 ðw21 ; w22 Þ ¼ ðw21 −cÞ α þ "
and
I∗22
from Proposition 3, we obtain
# β2 ðp−w21 −s1 Þ βγðp−w22 −s2 Þ − 2ð1−t 1 Þ 2ð1−t 2 Þ
þðw22 −cÞ α þ
# β2 ðp−w22 −s2 Þ βγðp−w21 −s1 Þ − 2ð1−t 2 Þ 2ð1−t 1 Þ
t ðp−w21 −s1 Þ2 β2 t 2 ðp−w22 −s2 Þ2 β2 − : − 1 4ð1−t 1 Þ2 4ð1−t 2 Þ2
w21
# 2 2 2 2 β t2 β β t1 β βγ 1 1 B Aþ þ þ þ 2 2 1−t 2 2ð1−t 2 Þ 1−t 1 2ð1−t 1 Þ 2 1−t 1 1−t 2 ! ! ¼ " 2 # βγ 1 1 β2 t 1 β2 β2 t 2 β2 β2 γ2 1 1 − þ þ þ þ 1−t 1 2ð1−t 1 Þ2 1−t 2 2ð1−t 2 Þ2 4 2 1−t 1 1−t 2 1−t 1 1−t 2 −
A βγ 1 1 þ 2 1−t 1 1−t 2
4.1.1. The retailer's problem The retailer i's problem is to determine advertising expenditure which maximizes his/her profit. The profit of retailer i is given by qffiffiffiffiffiffi pffiffiffiffiffi r TP 3i ðI3i Þ ¼ ðp−w3 −si Þ α þ β I3i −γ I 3 j −I 3i ; i ¼ 1; 2 and j ¼ 3−i:
ð15Þ From Proposition 1, by putting t1 = t2 = 0, we find that for manufacturer's pricing strategy 1, retailers 1 and 2's optimal advertising expenditures are
ð11Þ
Proposition 4. For the pricing strategy 2, the manufacturer's optimal wholesale prices are !"
In this section, we assume that manufacturer does not share any advertising expenditure incurred by the retailers and he(she) sets the same wholesale price w3 to each retailer. The sequence of events is as follows: The manufacturer first sets a common wholesale price w3 irrespective of retailers' sales costs si, i = 1,2. In response, the retailers observe w3 and follow Nash's equilibrium to determine their optimal advertising expenditures.
!
ð12Þ
I 31 ¼
ðp−w3 −s1 Þ2 β2 4
and
I32 ¼
ðp−w3 −s2 Þ2 β2 ; 4
ð16Þ
respectively. From (16), we have I∗31 N I∗32 since s1 b s2. Therefore, retailer 1 spends more than retailer 2 on advertising. Substituting the values of I∗31 and I∗32 in Eq. (15), we get the retailer 1's optimal profit as " # β2 ðp−w3 −s1 Þ βγðp−w3 −s2 Þ r − TP 31 I31 ¼ ðp−w3 −s1 Þ α þ 4 2
ð17Þ
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B.C. Giri, S. Sharma / Economic Modelling 38 (2014) 102–111
Substituting t1 = 0 and t2 = 0 in Proposition 3, we find that for pricing strategy 2 of the manufacturer, retailers 1 and 2's optimal advertising expenditures are given by
and the retailer 2's optimal profit as " # β2 ðp−w3 −s2 Þ βγðp−w3 −s1 Þ r − TP 32 I32 ¼ ðp−w3 −s2 Þ α þ : 4 2
ð18Þ
Proposition 5. For the manufacturer's pricing strategy 1, retailer 1 gains more profit than retailer 2. Proof. From Eqs. (17) and (18), we have r r TP 31 I31 −TP 32 I32 ¼ α ½ðp−w3 −s1 Þ−ðp−w3 −s2 Þ i β2 h 2 2 ðp−w3 −s1 Þ − ðp−w3 −s2 Þ þ 4 " # 2 β ððp−w3 −s1 Þ þ ðp−w3 −s2 ÞÞ ¼ ðs2 −s1 Þ α þ 4
N0
I 41 ¼
2 2 β ðp−w41 −s1 Þ 4
and
I 42 ¼
2 2 β ðp−w42 −s2 Þ ; 4
ð24Þ
respectively. Substituting the values of I∗41 and I∗42 in Eq. (23), the optimal profits of retailers 1 and 2 can be obtained as " # β2 ðp−w41 −s1 Þ βγðp−w42 −s2 Þ r − TP 41 I41 ¼ ðp−w41 −s1 Þ α þ ð25Þ 4 2 " # and β2 ðp−w42 −s2 Þ βγðp−w41 −s1 Þ r − ; TP 42 I42 ¼ ðp−w42 −s2 Þ α þ 4 2 respectively. ð26Þ
since s 2 N s 1 and p − w 3 − s 1 N 0 and p − w 3 − s 2 N 0. Thus, TPr31(I∗31) N TPr32(I∗32) which means that for the manufacturer's pricing strategy 1, retailer 1 gains higher profit than retailer 2. This completes the proof.
4.2.2. The manufacturer's problem The manufacturer's problem is to determine the wholesale prices for retailers that maximize his/her profit. The manufacturer's profit is given by
4.1.2. The manufacturer's problem The manufacturer's problem is to determine the wholesale price that maximizes his/her profit. The manufacturer's profit is given by
TP 4 ðw41 ; w42 Þ ¼
m
TP 3 ðw3 Þ ¼
2 h X
qffiffiffiffiffi qffiffiffiffiffiffii ðw3 −cÞ α þ β I 3i −γ I3 j ; j ¼ 3−i:
ð19Þ
i¼1
m
w3 ¼
α p þ c s1 þ s2 þ − : βðβ−γÞ 2 4
Substituting the values of I∗41 and I∗42, we have "
m TP 4 ðw41 ; w42 Þ
# β2 ðp−w41 −s1 Þ βγðp−w42 −s2 Þ − ¼ ðw41 −cÞ α þ 2 2 " #: 2 β ðp−w42 −s2 Þ βγðp−w41 −s1 Þ − þ ðw42 −cÞ α þ 2 2 ð28Þ
Substituting t1 = 0 and t2 = 0 in Proposition 4, we find that for pricing strategy 2, the manufacturer's optimal wholesale prices for retailers 1 and 2 are
ð21Þ
ð27Þ
i¼1
Substituting the values of I∗31 and I∗32 from Eq. (16), we have h s þ s2 i m ð20Þ TP 3 ðw3 Þ ¼ ðw3 −cÞ 2α þ βðβ−γ Þ p−w3 − 1 2 Substituting t1 = 0 and t2 = 0 in Proposition 2, the manufacturer's optimal wholesale price for pricing strategy 1 is given by
qffiffiffiffiffi 2 qffiffiffiffiffiffi X ðw4i −cÞ α þ β I4i −γ I 4 j ; j ¼ 3−i:
w41 ¼
α p þ c s1 þ − βðβ−γÞ 2 2
ð29Þ
α p þ c s2 þ − : βðβ−γ Þ 2 2
ð30Þ
and From Eqs. (15),(16),(19) and (21), the total profit of the entire supply chain is given by w r r m TP 3 ¼ TP 31 I 31 þ TP 32 I 32 þ TP 3 w3 qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi ¼ ðp−c−s1 Þ α þ β I 31 −γ I32 þ ðp−c−s2 Þ α þ β I 32 −γ I 31 :
−I31 −I32
ð22Þ 4.2. Case 2: Manufacturer's pricing strategy 2 In this strategy, the manufacturer first sets different wholesale prices w41 and w42 to retailers 1 and 2, respectively according to their different sales costs s1 and s2. Then the retailers follow Nash's equilibrium to determine their optimal advertising expenditures. 4.2.1. The retailer's problem The retailer i faces the problem to determine advertising expenditure that maximizes his/her profit. The profit of retailer i is given by qffiffiffiffiffiffi pffiffiffiffiffi r TP 4i ðI4i Þ ¼ ðp−w4i −si Þ α þ β I4i −γ I 4 j −I 4i ; i ¼ 1; 2 and j ¼ 3−i:
ð23Þ
w42 ¼
∗ ∗ From Eqs. (29) and (30), we get w41 −w42 ¼ s −s 2 N0, i.e. w41 N w42. This implies that the manufacturer will charge a greater wholesale price to retailer 1 than retailer 2. 2
1
Proposition 6. For the pricing strategy 2 of the manufacturer, retailer 1 gains higher profit than retailer 2. Proof. The proof is given in Appendix A. From Propositions 5 and 6, we find that the retailer 1's profit is higher than the retailer 2's profit under both the pricing strategies of the manufacturer. From Eqs. (23), (24), (27), (29) and (30), the total profit of the entire supply chain is given by w r r m TP 4 ¼ TP 41 I41 þ TP 42 I 42 þ TP 4 w41 ; w42 qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi ¼ ðp−c−s1 Þ α þ β I 41 −γ I42 þ ðp−c−s2 Þ α þ β I 42 −γ I41 :
−I41 −I42 :
ð31Þ
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5. Discussion In this section, we investigate the effects of manufacturer's different pricing strategies in Model II on the retailers' optimal advertising expenditures and their profits, manufacturer's optimal wholesale prices and his/her profit, and also profit of the whole supply chain. Proposition 7. w∗41 N w∗3 N w∗42 and w3 ¼ w
þw42 41
2
.
Proof. From Eqs. (21), (29) and (30), we have w41 −w3 ¼ s −s 4 N0 , ∗ which implies that w∗41 N w∗3. Also, w3 −w42 ¼ s −s N0. Thus, w N w∗42. 3 4 w þw ∗ ∗ ∗ ∗ Since w41 − w3 = w3 − w42, we have w3 ¼ 2 . The above results indicate that with pricing strategy 2, manufacturer sets a higher wholesale price for retailer 1 and lower wholesale price for retailer 2 compared to those with pricing strategy 1. With regard to retailer's optimal advertising expenditure, now we have the following result. pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi Proposition 8. I∗41 b I∗31, I32⁎ b I42⁎ and I31 þ I32 ¼ I 41 þ I42 . 2
2
41
have w∗41 N w∗3 − s1)2
1
1
107
The above Proposition suggests that for manufacturer's pricing strategy 1, the profit of the supply chain as a whole is higher than that for pricing strategy 2. 6. Numerical study In this section, we illustrate the developed models for a numerical example. The parameter values are taken as: α = 5, β = 2, γ = 0.2, p = $12, c = $4, s1 = $1, s2 = $2, t1 = 0.1, t2 = 0.2. The optimal results of the developed models are given below:
42
w∗3
w∗42
b w∗3. Therew∗42 − s2)2 N
and Proof. From Proposition 7, we fore, (p − w∗41 − s1)2 b (p − and (p − (p − w∗3 − s2)2, and hence from Eqs. (16) and (24), wepffiffiffiffiffiffi have pffiffiffiffiffiffi ∗ ∗ ∗ I∗41 b I Eqs. (16) and (24), we have I − I 31 and I42 N I32. Also, from 31 41 ¼ ffiffiffiffiffiffi p p ffiffiffiffiffiffi β β w −w w −w N0 and I N0 . Thus from − I ¼ 2 2 42pffiffiffiffiffiffi 32 pffiffiffiffiffiffi 41 1 3 pffiffiffiffiffiffi 42 pffiffiffiffiffiffi Proposition have pffiffiffiffiffiffi I31 − I41 ¼ I42 − I32 , and hence we have pffiffiffiffiffiffi pffiffiffiffiffiffi7 we pffiffiffiffiffiffi I31 þ I32 ¼ I41 þ I42 . Therefore, with respect to the manufacturer's pricing strategy 2, retailer 2 has a higher advertising expenditure and retailer 1 has a lower advertising expenditure compared to those with respect to the pricing strategy 1. m m Proposition 9. TP 4 w41 ; w42 NTP 3 w3 : Proof. See Appendix A. We are now in a position to state that the manufacturer obtains a higher profit from his pricing strategy 2 than strategy 1. With regard to retailers' profits, we have the result stated in the following Proposition:
Model I. For the pricing strategy 1, the optimal results are obtained as w∗1 = $ 8.59, I∗11 = $ 7.20, I∗12 = $ 3.12, TPr11 = $ 17.70, TPr12 = $ 8.81, w TPm 1 = $ 81.38 and TP1 = $ 107.75. For the pricing strategy 2, the optimal results are obtained as w∗21 = $ 8.87, w∗22 = $ 8.35, I∗21 = $ 5.61, I∗22 = $ 4.23, TPr21 = $ 14.83, w TPr22 = $ 10.83, TPm 2 = $ 81.62 and TP2 = $ 107.28. Model II. For the model with manufacturer's pricing strategy 1, the optimal results are obtained as w∗3 = $ 8.64, I∗31 = $ 5.57, I∗32 = $ 1.85, w TPr31 = $ 16.74, TPr32 = $ 8.02, TPm 3 = $ 77.47 and TP3 = $ 102.22. For the pricing strategy 2 of the manufacturer, the optimal results are obtained as w∗41 = $ 8.89, w∗42 = $ 8.39, I∗41 = $ 4.46, I∗42 = $ 2.60, w TPr41 = $ 14.33, TPr42 = $ 9.97, TPm 4 = $ 77.74 and TP4 = $ 102.05. From the above numerical results of two models we see that retailer 1 obtains a higher profit than retailer 2 under both the pricing strategies of the manufacturer. Also, advertising expenditure of retailer 1 is greater than that of retailer 2 irrespective of the pricing strategies set by the manufacturer. With respect to pricing strategy 2 set by the manufacturer, retailer 2 has a higher advertising expenditure while retailer 1 has a lower advertising expenditure compared to those with pricing strategy 1. The manufacturer always obtains a higher profit with pricing strategy 2 than that with pricing strategy 1. The total profit of the supply chain is higher in pricing strategy 1 than that in pricing strategy 2. All the observations described above corresponding to numerical results are consistent with the theoretical results obtained in two models. From the results of Models I and II, we also find that cooperative advertising increases the profit not only for the manufacturer and the retailers but also for the whole supply chain.
Proposition 10. TPr41(I∗41) b TPr31(I∗31) and TPr42(I∗42) N TPr32(I∗32). Proof. See Appendix A. From the above Proposition, we find that retailer 1 gains a lower profit from manufacturer's pricing strategy 2 than pricing strategy 1, and retailer 2 gains a higher profit from pricing strategy 2 than pricing strategy 1. Now, with respect to the profit of the entire supply chain, we have the following Proposition: Proposition 11. The profit of the supply chain is higher for manufacturer's pricing strategy 1 than that for pricing strategy 2 i.e. w TPw 3 N TP4 . Proof. From Propositions 7 and 8 and from Eqs. (16), (22), (24) and (31), we have qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi w w TP 3 −TP 4 ¼ ðp−c−s1 Þ α þ β I 31 −γ I32 þ ðp−c−s2 Þ α þ β I 32 −γ I 31 − q q ffiffiffiffiffiffi q qffiffiffiffiffiffi ffiffiffiffiffiffi ffiffiffiffiffiffi −ðp−c−s1 Þ α þ β I41 −γ I 42 −ðp−c−s2 Þ α þ β I 42 −γ I 41
þ I 41 þ I42 −I 31 −I 32 2 qffiffiffiffiffiffi qffiffiffiffiffiffi 2 2 i : β h ¼ ðβ þ γÞ I 31 − I 41 ðs2 −s1 Þ þ p−w42 −s2 − p−w3 −s2 4 2 2 2 i β h p−w3 −s1 − p−w41 −s1 − 4 βðβ þ 4γ Þðs2 −s1 Þ2 N0 ¼ 32 w Therefore, TPw 3 N TP4 . This completes the proof.
6.1. Effect of competition In this subsection, we examine the effect of competition on optimal advertising expenditures of the retailers, the profits of the manufacturer, retailers, and the entire supply chain in Model II. The effect of competition is illustrated in Figs. 1–3. Fig. 1 shows that retailer 1 obtains a higher profit than retailer 2 under each of the manufacturer's pricing strategy and for the pricing strategy 1, retailer 1's profit is higher than that for pricing strategy 2. Also, for the pricing strategy 1, retailer 2's profit is lower than that for pricing strategy 2. Fig. 2 shows that, for pricing strategy 1, manufacturer obtains lower profit than that for pricing strategy 2. In case of profit of the entire supply chain, pricing strategy 1 yields a higher profit than pricing strategy 2. From Figs. 1 and 2, we find that increase in competition decreases the profit of the retailers, manufacturer and the entire supply chain irrespective of manufacturer's pricing strategy. Thus increase in competition between retailers is detrimental for the retailers, the manufacturer and also for the entire supply chain. Fig. 3 shows that, with increase in competition, retailers' optimal advertising expenditures decrease. 6.2. Sensitivity analysis In this subsection, we examine the impact of unit retail price p and manufacturer's unit production cost c on profits of the retailers, manufacturer and the entire supply chain, and on the retailers' optimal advertising expenditures. We perform the sensitivity analysis on parameters
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18
6
TPr
TP r
31
32
TP r
41
*
TP r
*
I31
42
*
I32
*
I41
I42
16 5
advertising expenditure
profit
14
12
10
8
6
4 0.2
4
3
2
1
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0
γ
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
γ Fig. 1. Impact of competition on retailers' profits. Fig. 3. Impact of competition on retailers' advertising expenditures.
based on Model II under each of the manufacturer's pricing strategies. The results are reflected in Figs. 4–9 which show that an increase in unit retail price p increases the profits of the retailers, the manufacturer and the entire supply chain under each pricing strategy of the manufacturer. Also, we find that each retailer's optimal advertising expenditure increases with increase in unit retail price under both the pricing strategies of the manufacturer. In Figs. 7–9, we find that the profits of the retailers, manufacturer and the entire supply chain all decrease with increase in manufacturer's unit production cost c under each pricing strategy adopted by the manufacturer. Also, each retailer's optimal advertising expenditure decreases with increase in the manufacturer's unit production cost. These observations can be explained on the basis of the fact that, in each pricing strategy, an increase in unit retail price results in an increase in the unit sales profit and therefore, the retailers tend to invest more on advertising to acquire greater demand and hence order more quantities from the manufacturer. Thus, retailers' optimal advertising expenditures and the profits of the retailers, manufacturer and the entire supply chain all increase with an increase in unit retail
price. However, with an increase in unit manufacturing cost, the manufacturer charges a higher wholesale price which leads to a decrease in unit sales profit of the retailers. Then, the retailers order less quantities from the manufacturer and invest less in advertising. Thus, the retailers' optimal advertising expenditures and the profits of the retailers, manufacturer and the entire supply chain decrease with an increase in unit manufacturing cost. 7. Conclusions In this paper, we study a two-echelon supply chain comprising one manufacturer and two competing retailers whose demand depends on advertising. The manufacturer acts as the Stackelberg leader and the two retailers are the followers. To set wholesale price, the manufacturer takes one of the two strategies: (1) negotiate with both the retailers simultaneously and set a wholesale price that applies to both of them (strategy 1); and (2) negotiate with two retailers differently and set different wholesale prices to two retailers according to their different sales 300
105 m 3 m TP 4 w TP 3 w TP 4
r 31 r TP 32 m TP 3 w TP 3
TP
TP 100
200
90
profit
profit
95
250
150
85 100 80 50 75
70 0.2
0 0.25
0.3
0.35
γ
0.4
0.45
0.5
0.55
0.6
Fig. 2. Impact of competition on manufacturer's and the entire supply chain's profits.
12
13
14
15
16
17
unit retail price (p) Fig. 4. Unit retail price vs. profit under pricing strategy 1.
18
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109
120
300
r 31 r TP 32 m TP 3 w TP 3
TP r 41 r TP 42 m TP 4 w TP 4
TP 250
80
150
profit
profit
200
100
60
100
40
50
20
0
0 12
13
14
15
16
17
18
4
4.5
5
5.5
6
unit production cost (c)
unit retail price (p) Fig. 5. Unit retail price vs. profit under pricing strategy 2.
Fig. 7. Unit production cost vs. profit under pricing strategy 1.
costs (strategy 2). Two models are developed. The first model is developed under the assumption that the manufacturer shares a fraction of retailer's advertising expense while in the second model the manufacturer does not share advertising costs incurred by the retailers. In our model, it is assumed that manufacturer declares a fixed participation rate for each retailer before the start of the selling season. In both models, we derive retailers' optimal advertising expenditures and their optimal profits, and the manufacturer's optimal wholesale price and profit under each pricing strategy of the manufacturer. The study based on Model II generates several meaningful insights: (i) Retailer 1 always obtains a higher profit than retailer 2 irrespective of the pricing strategy adopted by the manufacturer. This is simply because retailer 1 has lower sales cost than retailer 2. For pricing strategy 1, retailer 1 obtains higher profit than for pricing strategy 2 while retailer 2's profit is higher under pricing strategy 2 than that under pricing strategy 1. (ii) With pricing strategy 2, the manufacturer sets a higher wholesale price to retailer 1 and lower wholesale price to retailer 2
compared to that under pricing strategy 1. Consequently, retailer 1 invests less in advertising with pricing strategy 2 as compared to pricing strategy 1 while retailer 2 invests more in advertising with pricing strategy 2 than with pricing strategy 1. (iii) With pricing strategy 2, manufacturer obtains higher profit than with pricing strategy 1 and hence the manufacturer ideally wants to adopt pricing strategy 2. (iv) The total profit of the entire supply chain is higher with pricing strategy 1 than that with pricing strategy 2. (v) An increase in unit retail price increases the retailers' optimal advertising expenditures, the profits of the retailers, manufacturer and the entire supply chain but the scenario is reversed with an increase in unit manufacturing cost. (vi) A tougher competition between retailers decreases profits of the retailers, manufacturer and the entire supply chain. The study based on model I shows that, with co-operative advertising, the profits of the retailers, manufacturer and entire supply chain always increase compared to the case of non-cooperative advertising. Various insights gained in Model II also hold good for model I.
30
120 r 41 r TP 42 m TP 4 w TP 4
TP *
I31
25
100
*
I41
20
80
* 42
I 15
profit
advertising expenditure
I*32
60
10
40
5
20
0 12
0
13
14
15
16
17
unit retail price (p) Fig. 6. Unit retail price vs. retailers' optimal advertising expenditures.
18
4
4.5
5
5.5
6
6.5
unit production cost (c) Fig. 8. Unit production cost vs. profit under pricing strategy 2.
7
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B.C. Giri, S. Sharma / Economic Modelling 38 (2014) 102–111
6
The Hessian matrix is
*
I 31
0
I*
5
32 *
H1 ¼
advertising expenditure
I 41
0
*
I 42
4
A
∂2 TP m 2 ∂w222
β2 ð2−t Þ
2
3
βγ 1 2 1−t 1
1−t 1
þ
1 1−t 2
β 2 ð2−t 2 Þ − 2ð1−t 2 Þ2
1−t 2
1 C C: A
Therefore, 2 β4 ð2−t 1 Þð2−t 2 Þ β2 γ2 1 1 − þ 2 2 1−t"1 1−t 2 4 ð1−t 1 Þ ð1−t 2 Þ 4 #: 4 2 2 2 2 β ð4−2ðt 1 þ t 2 Þ þ t 1 t 2 Þ β γ ð1−t 2 Þ þ ð1−t 1 Þ þ 2ð1−t 1 Þð1−t 2 Þ ¼ − 4 4 ð1−t 1 Þ2 ð1−t 2 Þ2 ð1−t 1 Þ2 ð1−t 2 Þ2
2
jH 1 j ¼
Since β N γ, to show that |H1| N 0, it is sufficient to show that
0 4
4.5
5
5.5
6
6.5
2 2 4−2ðt 1 þ t 2 Þ þ t 1 t 2 − ð1−t 1 Þ þ ð1−t 2 Þ þ 2ð1−t 1 Þð1−t 2 Þ N0:
7
unit production cost (c) Fig. 9. Unit production cost vs. retailers' optimal advertising expenditures.
Now,
Our model can be extended in several ways. One immediate extension is inclusion of stochastic demand instead of deterministic demand. Secondly, one can generalize the model considering two or more manufacturers. Thirdly, retailers' competition in price as well as in advertising would be another worthy extension of our paper. These are few among various research directions based on our work, some of which will be considered in our future research efforts. Acknowledgment
2 2 4−2ðt 1 þ t 2 Þ þ t 1 t 2 − ð1−t 1 Þ þ ð1−t 2 Þ þ 2ð1−t 1 Þð1−t 2 Þ 2
2
¼ 2ðt 1 þ t 2 Þ−t 1 t 2 −t 1 −t 2 ¼ ðt 1 þ t 2 Þ½2−ðt 1 þ t 2 Þ þ t 1 t 2 N0
:
since 0bt1 ; t 2 ≤1
Thus, we have |H1| N 0, i.e. the Hessian matrix H1 is negative definite and hence TPm 2 (w21,w22) is concave in w21 and w22. From Eqs. (32) and (33), we obtain, after some tedious calculations, the values of w∗21 and w∗22 given in Eqs. (12) and (13), respectively. This completes the proof. Proof of Proposition 6. From Eqs. (25) and (26), we have
The first author gratefully acknowledges the financial support provided by the University Grants Commission, New Delhi, under UGC-DRS Programme (2012–2017). The research work of the second author was funded by State Govt. Fellowship Scheme, Jadavpur University. Appendix A
r r TP 41 I 41 −TP 42 I 42 ¼ α p−w41 −s1 − p−w42 −s2 þ 2 i β 2 h 2 p−w41 −s1 − p−w42 −s2 4 " β 2 ¼ p−w41 −s1 − p−w42 −s2 αþ p−w41 −s1 þ 4 : # p−w42 −s2 Þ " # β 2 ¼ w42 −w41 þ s2 −s1 α þ p−w41 −s1 þ p−w42 −s2 4 Þ N0 . From Eqs. (29) and (30), we have, w42 −w41 þ s2 −s1 ¼ ðs −s 2 Also, (p − w∗41 − s1) N 0 and (p − w∗42 − s2) N 0. Hence, TPr41(I∗41) N TPr42(I∗42). This completes the proof.
Proof of Proposition 4. From Eq. (11), we have
2
∂TP m β2 t ðp−w21 −s1 Þβ2 ðw21 −cÞβ2 2 ¼αþ ðp−w21 −s1 Þ þ 1 − 2ð1−t 1 Þ 2ð1−t 1 Þ ∂w21 2ð1−t 1 Þ2 ðp−w22 −s2 Þβγ ðw22 −cÞβγ þ − 2ð1−t 2 Þ 2ð1−t 1 Þ ð32Þ m
1
∂2 TP m 2 ∂w21 ∂w22 C C
1 − 2 B 2ð1−t 1 Þ ¼B @ βγ 1 1 þ
1
2
∂2 TP m 2 B ∂w2 B 2 21m @ ∂ TP 2 ∂w22 ∂w21
1
Proof of Proposition 9. From Eqs. (19) and (27), we have qffiffiffiffiffiffi qffiffiffiffiffiffi m m TP 4 w41 ; w42 −TP 3 w3 ¼ w41 −c α þ β I 41 −γ I 42 þ w42 −c ffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi q qffiffiffiffiffiffi α þ β I 42 −γ I 41 − w3 −c α þ β I 31 −γ I 32 − q q ffiffiffiffiffiffi ffiffiffiffiffiffi w3 −c α þ β I 32 −γ I 31 :
2
∂ TP 2 β ð2−t 1 Þ ¼− b0 2ð1−t 1 Þ2 ∂w221
∂2 TP m βγ 1 1 2 ¼ þ N0 2 1−t 1 1−t 2 ∂w22 ∂w21 m 2 ∂TP 2 β t ðp−w22 −s2 Þβ2 ðw22 −cÞβ2 : ¼αþ ðp−w22 −s2 Þ þ 2 − 2ð1−t 2 Þ 2ð1−t 2 Þ ∂w22 2ð1−t 2 Þ2 ðp−w21 −s1 Þβγ ðw21 −cÞβγ þ − 2ð1−t 1 Þ 2ð1−t 2 Þ 2 m 2 ∂ TP 2 β ð2−t 2 Þ ¼− b0 2ð1−t 2 Þ2 ∂w222 ð33Þ
∗ ∗ ∗ FrompPropositions pffiffiffiffiffiffi ffiffiffiffiffiffi pffiffiffiffiffiffi 7 and 8, we have w41 + w42 = 2w3 and I32 ¼ I41 þ I42 , respectively and so,
pffiffiffiffiffiffi I31 þ
qffiffiffiffiffiffi qffiffiffiffiffiffi m m I41 − I42 : TP 4 w41 ; w42 −TP 3 w3 ¼ ðβ þ γÞ w41 −w3 from Eq. (24), From p Eqs. (21) and (29), we have w41 −w3 ¼ s −s 4 and ffiffiffiffiffiffi pffiffiffiffiffiffi m m β we have I − I 42 ¼ 4 ðs2 −s1 Þ. Thus, we obtain TP 4 w41 ; w42 −TP 3 βðβþγÞ41 ðs −s Þ2 w3 ¼ N0 since s2 N s1. This completes the proof of the 16 Proposition. 2
2
1
1
B.C. Giri, S. Sharma / Economic Modelling 38 (2014) 102–111
Proof of Proposition 10. From Eqs. (15) and (23), we have qffiffiffiffiffiffi qffiffiffiffiffiffi r r TP 31 I 31 −TP 41 I 41 ¼ p−w3 −s1 α þ β I 31 −γ I 32 − p−w41 −s1 qffiffiffiffiffiffi qffiffiffiffiffiffi : α þ β I 41 −γ I 42 þ I 41 −I 31
From Eqs. (21) and (29), we have w41 ¼ w3 þ s −s 4 . Then from Eqs. (16) and (24) and from Proposition 8, we have 2
1
qffiffiffiffiffiffi qffiffiffiffiffiffi r r TP 31 I 31 −TP 41 I 41 ¼ p−w3 −s1 α þ β I 31 −γ I32 − p−w41 −s1 qffiffiffiffiffiffi qffiffiffiffiffiffi β 2 h 2 i 2 α þ β I41 −γ I42 þ p−w −s − p−w3 −s1 4 ffiffiffiffiffiffi 41 1 qffiffiffiffiffiffi q s −s ¼ p−w3 −s1 α þ β I31 −γ I 32 − p−w3 − 2 1 −s1 4 qffiffiffiffiffiffi qffiffiffiffiffiffi β 2 α þ β I41 −γ I42 þ 2p−w3 −w41 −2s1 w3 −w41 qffiffiffiffiffiffi 4 qffiffiffiffiffiffi : N p−w3 −s1 α þ β I 31 −γ I32 − p−w3 −s1 qffiffiffiffiffiffi qffiffiffiffiffiffi β 2 s −s α þ β I41 −γ I42 − w41 −w3 2p−w3 −w3 − 2 1 −2s1 4 4 2 h s −s i β ðβ þ γÞ β ¼ w41 −w3 2 p−w3 −s1 − 2 1 p−w3 −s1 w41 −w3 − 4 4 2 βγðp−w3 −s1 Þðs2 −s1 Þ β 2 ðs2 −s1 Þ2 ¼ N0 þ 64 8
Therefore, TPr31(I∗31) N TPr41(I∗41). From Eqs. (21) and (30), we have w3 ¼ w42 þ s −s 4 . Then from Eqs. (15), (16), (23) and (24) and from Proposition 8, we obtain 2
1
qffiffiffiffiffiffi qffiffiffiffiffiffi r r TP 42 I 42 −TP 32 I 32 ¼ p−w42 −s2 α þ β I 42 −γ I 41 − p−w3 −s2 qffiffiffiffiffiffi qffiffiffiffiffiffi α þ β I 32 −γ I 31 þ I 32 −I 42 qffiffiffiffiffiffi qffiffiffiffiffiffi s −s ¼ p−w42 −s2 α þ β I 42 −γ I 41 − p−w42 − 2 1 −s2 4 qffiffiffiffiffiffi qffiffiffiffiffiffi α þ β I 32 −γ I 31 þ I 32 −I 42 : qffiffiffiffiffiffi qffiffiffiffiffiffi N p−w42 −s2 ðβ þ γÞ I 42 − I32 þ I32 −I 42 2 2 i β ðβ þ γÞ β 2 h ¼ p − w3 − s2 − p − w42 − s2 p−w42 −s2 w3 −w42 þ 4 2 2 2 βγðp−w42 −s2 Þðs2 −s1 Þ β ðs2 −s1 Þ N0 þ ¼ 64 8
Hence, TPr42(I42⁎) N TP32⁎(I32⁎). This completes the proof. References Anderson, S.P., Renault, R., 2009. Comparative advertising: disclosing horizontal match information. RAND J. Econ. 40 (3), 558–581. Aust, G., Buscher, U., 2012. Vertical cooperative advertising and pricing decisions in a manufacturer–retailer supply chain: a game-theoretic approach. Eur. J. Oper. Res. 223, 473–482. Aust, G., Buscher, U., 2014. Cooperative advertising models in supply chain management: a review. Eur. J. Oper. Res. 234 (1), 1–14. Barigozzi, F., Garella, P.G., Peitz, M., 2009. With a little help from my enemy: comparative advertising as a signal of quality. J. Econ. Manag. Strateg. 18 (4), 1071–1094. Berger, P.D., 1972. Vertical cooperative advertising ventures. J. Mark. Res. 9, 309–312. Berger, P.D., Magliozzi, T., 1992. Optimal co-op advertising decisions in direct-mail operations. J. Oper. Res. Soc. 43, 1079–1086. Brennan, L., 1988. How retailers are putting it all together? Sales Mark. Manag. 62–65 (May). Chen, T.H., 2011. Coordinating the ordering and advertising policies for a single-period commodity in a two-level supply chain. Comput. Ind. Eng. 61, 1268–1274. Choi, S., 1991. Price competition in a channel structure with a common retailer. Mark. Sci. 10 (4), 271–296. Choi, S., 1996. Price competition in a duopoly common retailer channel. J. Retail. 72 (2), 117–134.
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Desiraju, D., Moorthy, S., 1997. Managing a distribution channel under asymmetric information with performance requirements. Manag. Sci. 43 (12), 1644–1682. Fulop, C., 1988. The role of advertising in the retail marketing mix. Int. J. Advert. 7, 99–117. He, X., Krishnamoorthy, A., Prasad, A., Sethi, S.P., 2011. Retail competition and cooperative advertising. Oper. Res. Lett. 39, 11–16. Huang, Z., Li, S.X., 2001. Co-op advertising models in manufacturer–retailer supply chains: a game theory approach. Eur. J. Oper. Res. 135 (3), 527–544. Huang, Z., Li, S.X., Mahajan, V., 2002. An analysis of manufacturer–retailer supply chain coordination in cooperative advertising. Decis. Sci. 33 (3), 469–494. Ingene, C.A., Parry, M.E., 1995. Channel coordination when retailers compete. Mark. Sci. 14 (4), 360–377. Jorgensen, S., Sigue, S.P., Zaccour, G., 2000. Dynamic cooperative advertising in a channel. J. Retail. 76 (1), 71–92. Karray, S., Zaccour, G., 2006. Could co-op advertising be a manufacturer's counter strategy to store brands? J. Bus. Res. 59 (9), 1008–1015. Khouja, M., Robbins, S.S., 2003. Linking advertising and quantity decisions in the singleperiod inventory model. Int. J. Prod. Econ. 86, 93–105. Kunter, M., 2012. Coordination via cost and revenue sharing in manufacturer–retailer channels. Eur. J. Oper. Res. 216, 477–486. Lau, H.S., Lau, A.H.L., 1999. Manufacturer's pricing strategy and return policy for a single period commodity. Eur. J. Oper. Res. 116, 291–304. Li, S.X., Huang, Z., Zhu, J., Chau, P.Y.K., 2002. Cooperative advertising, game theory and manufacturer–retailer supply chains. Omega 30, 347–357. Lu, J.C., Tsao, Y.C., Charoensiriwath, C., 2011. Competition under manufacturer service and retail price. Econ. Model. 28, 1256–1264. Padmanabhan, V., Png, I.P.L., 1997. Manufacturers returns policies and retail competition. Mark. Sci. 16 (1), 81–94. Roslow, S., Laskey, H.A., Nicholls, J.A.F., 1993. The enigma of cooperative advertising. J. Bus. Ind. Mark. 8, 70–79. Singh, N., Vives, X., 1984. Price and quantity competition in a differentiated duopoly. RAND J. Econ. 15 (4), 546–554. Small World Toys, 2007. http://www.smallworldtoys.com/coop.asp. Somers, T.M., Gupta, Y.P., Herriot, S.R., 1990. Analysis of cooperative advertising expenditures: a transfer-function modeling approach. J. Advert. Res. 35–45 (OctoberNovember). Szmerekovsky, J.G., Zhang, J., 2009. Pricing and two-tier advertising with one manufacturer and one retailer. Eur. J. Oper. Res. 192, 904–917. Viscolani, B., 2012. Pure-strategy Nash equilibria in an advertising game with interference. Eur. J. Oper. Res. 216, 605–612. Viscolani, B., Zaccour, G., 2009. Advertising strategies in a differential game with negative competitor's interference. J. Optim. Theory Appl. 140 (1), 153–170. Vives, X., 1984. Duopoly information equilibrium: cournot and Bertrand. J. Econ. Theory 34, 71–94. Vives, X., 1985. On the efficiency of Bertrand and cournot equilibria with product differentiation. J. Econ. Theory 36, 166–175. Wang, T., 2011. Sensitivity analysis for newsvendor model with general advertisementsensitive demand. Procedia Eng. 15, 1209–1213. Wang, S., Zhou, Y., Wang, J., 2010. Supply chain coordination with two production modes and random demand depending on advertising expenditure and selling price. Int. J. Syst. Sci. 41, 1257–1272. Wang, S.D., Zhou, Y.W., Min, J., Zhong, Y.G., 2011. Coordination of cooperative advertising models in a one-manufacturer two-retailer supply chain system. Comput. Ind. Eng. 61, 1053–1071. Wu, C.H., Chen, C.W., Hsieh, C.C., 2012. Competitive pricing decisions in a two-echelon supply chain with horizontal and vertical competition. Int. J. Prod. Econ. 135 (1), 265–274. Xie, J.X., Neyret, A., 2009. Co-op advertising and pricing models in manufacturer–retailer supply chains. Comput. Ind. Eng. 56, 1375–1385. Xie, J., Wei, J.C., 2009. Coordinating advertising and pricing in a manufacturer–retailer channel. Eur. J. Oper. Res. 197, 785–791. Yan, R., 2010. Cooperative advertising, pricing strategy and firm performance in the emarketing age. J. Acad. Mark. Sci. 38 (4), 510–519. Yue, J., Austin, J., Wang, M.C., Huang, Z., 2006. Coordination of cooperative advertising in a two-level supply chain when manufacturer offers discount. Eur. J. Oper. Res. 168, 65–85. Zhang, J., Xie, J., 2012. A game theoretical study of cooperative advertising with multiple retailers in a distribution channel. J. Syst. Sci. Syst. Eng. 21 (1), 37–55. Zhang, J., Gou, Q., Liang, L., Huang, Z., 2013. Supply chain coordination through cooperative advertising with reference price effect. Omega 41, 345–353.
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Economic Modelling journal homepage: www.elsevier.com/locate/ecmod
Manufacturer's pricing strategy in a two-level supply chain with competing retailers and advertising cost dependent demand B.C. Giri ⁎, S. Sharma Department of Mathematics, Jadavpur University, Kolkata 700032, India
a r t i c l e
i n f o
Article history: Accepted 1 November 2013 Available online 21 January 2014 Keywords: Supply chain Advertising cost dependent demand Pricing strategy
a b s t r a c t The paper studies a two-echelon supply chain comprising one manufacturer and two competing retailers with advertising cost dependent demand. The manufacturer acts as the Stackelberg leader who specifies wholesale price for each retailer. The two retailers compete with each other in advertising and they have different sales costs. The manufacturer uses one of the following two pricing strategies: (i) setting the same wholesale price for both the retailers irrespective of the difference in their sales costs; (ii) setting different wholesale prices for the retailers depending on their sales costs. Two models are developed. In the first model, the manufacturer shares a fraction of each retailer's advertising cost while in the second model, the manufacturer does not share any retailer's advertising expenses. In both the models, we derive the retailers' and manufacturer's optimal strategies. A numerical example is given to illustrate the theoretical results developed in each model. Computational results show that it is always beneficial for the manufacturer to adopt different wholesale pricing strategy for the retailers. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Rapid technological advancement, increasing expectation of customer and shortened product life-cycle have intensified competition and introduced new challenges in the market. To overcome the situation, companies should have good flexibility to respond and capability to adopt new strategies to utilize greater share of the market demand. More and more companies are resorted to depend on service and product quality in order to avoid traditional competition which focuses solely on price. Advertising is one way of promotional campaign used by many firms to provide customers the brand knowledge of their products and services and other specialties of their organizations. Depending upon the product type and market demand, either the manufacturer or the retailer or both participate in advertising. Two-tier advertising or co-operative (co-op) advertising is an interactive scheme in a manufacturer–retailer system in which the manufacturer pays a part of the advertising expenditure incurred by the retailer in local advertisement. For example, Small World Toys offers a 2% advertising on total net purchases (Small World Toys, 2007). In personal computer industry, IBM offers a 50–50 split of advertising costs with retailer while Apple Computer pays 75% of the media cost (Brennan, 1988). Manufacturer uses co-op advertising to strengthen the image of the brand and motivate immediate sales at the retail level while retailer uses local advertising to bring potential customers to the stage of buying. The sharing of local advertising cost by the manufacturer is intended to influence ⁎ Corresponding author. E-mail addresses: [email protected] (B.C. Giri), [email protected] (S. Sharma). 0264-9993/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econmod.2013.11.005
retailers to spend more in local advertisement which in turn generates greater sales volume. Berger (1972) was probably the first who discussed co-op advertising in a manufacturer–retailer channel. His work was subsequently extended by many authors (Berger and Magliozzi, 1992; Fulop, 1988; Somers et al., 1990) under different co-op advertising settings. Roslow et al. (1993) discussed co-op advertising under supply chain framework and demonstrated that cooperation in advertising investment could increase the profit of the whole supply chain. Khouja and Robbins (2003) studied the effect of advertising on the variance of demand under newsboy framework. However, the above studies are based on the assumption that demand at the retailer's end depends explicitly on retailer's local advertisement. Taking manufacturer's national level advertisement investment in addition to retailer's local advertising into consideration, Huang et al. (2002) and Li et al. (2002) independently developed manufacturer–retailer supply chain models under co-op advertising setting. Using game theory, Huang and Li (2001) further discussed the issue of co-op advertisement for supply chain with one manufacturer and one retailer. In their model, they showed that if the ratio of the manufacturer's and retailer's marginal profits is relatively high, the manufacturer offers a positive amount of advertising allowance to the retailer. In case the marginal profit ratio is relatively low, the manufacturer is reluctant to offer any advertising allowance. Szmerekovsky and Zhang (2009), Xie and Wei (2009) extended the model of Huang et al. (2002) assuming that customer demand is dependent on retail price as well as advertising efforts of channel members. Yue et al. (2006) also extended Huang et al. (2002) model assuming that the manufacturer directly offers a price discount to customers. To increase profits of the parties involved in the supply chain, they
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recommended that coordination in local and co-operative advertising with a partnership relation between manufacturer and retailer is the best solution. In the context of co-operative advertising, Kunter (2012) analyzed a royalty payment contract for coordination of a manufacturer–retailer channel. Consumer demand was assumed to be simultaneously affected by the retail price and marketing efforts of manufacturer and retailer. Aust and Buscher (2012) addressed the optimal pricing and advertising decisions in a manufacturer–retailer supply chain where the consumer demand depends on the retail price as well as the channel members' advertising expenditures. Additionally, they considered a co-operative advertising program where the manufacturer can bear a fraction of retailer's advertising costs. Zhang et al. (2013) proposed a dynamic cooperative advertising model for a manufacturer–retailer supply chain taking into account the impact of advertising on the reference price and analyzed how the reference price would influence the decisions of all the channel members. Chen (2011) developed a model to study the combined effect of co-operative advertising, return policy and channel coordination for supply chain. Recently, Aust and Buscher (2014) made a comprehensive updated review of literature on co-operative advertising in supply chain management. From the perspective of customer buying behavior, it is seen that besides price other factors such as promotional activity through advertising also influence customers' preferences and their purchasing decisions and hence market demand. Advertising is the most effective method of promotion and hence advertising investment plays a key role in a firm's marketing decisions. Through advertising the retailer gives the customers brand knowledge of the product, detailed description of the product as well as review of the product in the market. All these useful guidelines stimulate customers' purchasing decisions and thus bring potential customers to the stage of buying. Such attribute of market demand can be seen in insurance industries, departmental stores, supermarkets, newsvendor products like fashion apparel, personal computers or supply chains with fixed retail price. Wang et al. (2010), Yan (2010) and Wang (2011) considered demand dependent on investment on advertising. For more literature concerning advertising sensitive demand, readers can be referred to articles contributed by Jorgensen et al. (2000), Karray and Zaccour (2006), Xie and Neyret (2009). The primary interest of the present study is focused on the competitive behavior of manufacturer and retailers where mode of competition shifts from strategies that consider price differentiation to strategies that include differentiation in terms of advertising efforts. We consider a supply chain with one manufacturer and two competing retailers who face advertising cost dependent demand. In the proposed model, the effect of price is ignored; price remains fundamental as a basis for competition. However, it may not be the only means of determining competition. For instance, when different firms offer the same retail price for a product, the one which promotes the product more definitely has a better edge to consume greater market demand than the others through advertising done by the retailer. Viscolani and Zaccour (2009) examined a duopoly problem where each firm's current sales is proportional to its goods in stock, which is related to the firm's own advertising effort and negatively related to that of its competitor. The idea that one player's advertising effort may hamper competitor's sales can also be seen in the articles provided by Anderson and Renault (2009), Barigozzi et al. (2009). Viscolani (2012) proposed a model for two competing manufacturers selling substitutable products in a homogeneous market. Demand for each manufacturer is positively correlated with his own advertising effort and negatively correlated with advertising effort of his rival. Lu et al. (2011) modeled a game-theoretic problem considering two competing manufacturers and one common retailer, facing end consumers who are sensitive to both retail price and manufacturer service. We investigate the effect of manufacturer's pricing strategy in the supply chain. The manufacturer who acts as Stackelberg leader specifies
103
wholesale prices to retailers. The manufacturer uses one of the following pricing strategies: (i) a common wholesale price for both the retailers while disregarding the difference of their sales costs (strategy 1) and (ii) different wholesale prices for two retailers according to their different sales costs (strategy 2). We determine optimal strategies of the manufacturer and retailers in response to manufacturer's different pricing strategies, effects of manufacturer's different pricing strategies on each member's performance and the whole supply chain's performance, the pricing strategy which is best suited to the manufacturer in response to different sales cost scenarios. Lau and Lau (1999) analyzed the pricing and return policies of a monopolistic manufacturer for single-period commodities. In our present study, we assume in Model I that the manufacturer takes part in co-op advertising and provides the retailers a fraction of local advertising costs incurred by the retailers while in Model II the manufacturer does not participate in co-op advertising. Both the models are developed under the framework of two pricing strategies set by the manufacturer. We investigate which strategy is beneficial for channel members as well as for the whole channel. Among the existing literature on co-operative advertising, very few articles discussed a channel where a single manufacturer sells a product through two or more competing retailers. Related articles in this issue were presented by Wang et al. (2011) and He et al. (2011). This channel structure, however, represents numerous markets including those made up of specialty stores (e.g., consumer electronics, sporting goods, automobile parts, to name a few), departmental stores and supermarkets. The main contributions of our paper are the following. First, our paper extends the current literature on co-operative advertising to account for a supply chain with multiple retailers. Most of the previous research on co-operative advertising was done under the traditional setting of a bilateral monopoly model where one manufacturer sells through one retailer. Second, it takes into account the competitive behavior of two retailers in terms of advertising efforts, i.e. demand of each retailer is not only related to his own advertising investment but also on the other retailer's advertising expenditure. Third, it studies the effects of the manufacturer's pricing strategies in conjunction with retailers' gaming interaction, and discusses the related impact on supply chain's decisions and performance when all the channel members are affected by advertising investment dependent demand. This makes our contribution unique because no such analysis in connection with co-operative advertising has been done before. The rest of the paper is organized as follows: Model description and assumptions are presented in the next section. Model I is formulated in Section 3 with co-operative advertising under different pricing strategies of the manufacturer. Section 4 describes Model II when the manufacturer does not participate in co-operative advertising. Section 5 is devoted to the discussion related to the effects of manufacturer's pricing strategies in Model II on retailers' and manufacturer's decision making. In Section 6, theoretical results are verified through a numerical example. Finally, the paper is concluded and scope of future research is suggested in Section 7. 2. Assumptions and model description We consider a two-echelon supply chain comprising one manufacturer and two competing retailers for trading a single product. The manufacturer acts as the Stackelberg leader and sets wholesale price to two retailers. Suppose that the unit retail price of the product is p and unit production cost is c. The two retailers have different sales efficiencies and hence have different sales costs. Let the retailer i's unit sales cost be si (i = 1,2). Without any loss of generalization, we assume that s1 b s2. We assume that the two retailers compete with each other on advertising and the market demand depends on their advertising expenses. Further, the price differentiation is negligible to the customers at the time of purchase and the demand is mainly influenced by the
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advertising costs incurred by the two competing retailers. For each retailer, the demand increases with own advertising while it decreases with the advertising done by his rival. We define the demand at the retailer i as (Choi, 1991, 1996; Wu et al., 2012; Xie and Wei, 2009; Zhang and Xie, 2012): DðI i Þ ¼ α þ β
qffiffiffiffiffi pffiffiffi Ii −γ I j ; i ¼ 1; 2 and j ¼ 3−i:
Here, Ii denotes retailer i's advertising expenditure, α denotes the primary demand of retailer i, β denotes the demand sensitivity of the retailer i on its own advertising (Padmanabhan and Png, 1997) while γ denotes the demand sensitivity of retailer i on its rival's advertising. The linear and symmetrical demand model represents a situation in which two retailers have equal competing powers in a duopolistic market place. We assume that γ b β so that the response functions are negatively sloped which, in turn, ensures the existence of the Nash equilibrium. This appears reasonable since sales are relatively more sensitive to retailer's own advertising than the advertising done by the competing retailer. This demand pattern is widely used in marketing research (see Desiraju and Moorthy, 1997; Ingene and Parry, 1995) and in some economics research (see Singh and Vives, 1984; Vives, 1984, 1985).
and the retailer 2's optimal profit as " # 2 β ðp−w1 −s2 Þ βγðp−w1 −s1 Þ r TP 12 I12 ¼ ðp−w1 −s2 Þ α þ − : 4ð1−t 2 Þ 2ð1−t 1 Þ
3.1.2. The manufacturer's problem The manufacturer faces the problem to determine the wholesale price which maximizes his/her profit. The manufacturer's profit is given by m
TP 1 ðw1 Þ ¼
2 h X
qffiffiffiffiffi qffiffiffiffiffiffii ðw1 −cÞ α þ β I1i −γ I1 j −t 1 I11 −t 2 I12 ; j ¼ 3−i:
i¼1
ð2Þ Substituting the values of I∗11 and I∗12 given in Proposition 1, the manufacturer's profit can be written as
β p−w1 −s1 p−w1 −s2 m þ TP 1 ðw1 Þ ¼ ðw1 −cÞ 2α þ ðβ−γÞ 2 1−t 1 1−t 2 −
t 1 ðp−w1 −s1 Þ2 β2 t 2 ðp−w1 −s2 Þ2 β2 − : 4ð1−t 1 Þ2 4ð1−t 2 Þ2
ð3Þ
3. Model I: Manufacturer share a fraction of retailers' advertising expenditures Proposition 2. For the pricing strategy 1, the manufacturer's optimal wholesale price is given by
3.1. Case1: Manufacturer's pricing strategy 1 In this section, we assume that manufacturer declares beforehand that she/he would share a fraction ti of retailer's advertising expenditure prior to the start of selling season. The manufacturer then sets a common wholesale price w1 irrespective of retailers' sales costs si's, i = 1,2. In response, the retailers observe w1 and follow Nash equilibrium to determine their optimal advertising expenditures. 3.1.1. The retailer's problem The retailer i faces the problem of determining advertising expenditure I1i which maximizes his/her profit. Retailer i's profit is given by
i ¼ 1; 2 and j ¼ 3−i:
ð1Þ Proposition 1. For the pricing strategy 1 of the manufacturer, the retailers 2
1 and 2's optimal advertising expenditures are given by I11 ¼ ðp−w1 −s1 Þ2 β
2
2
¼−
ðp−w1 −si Þβ 3=2
4I1i
b0; r
2
2
2
1i
1 −s1 Þ β 1 −s2 Þ β and I 12 ¼ ðp−w . This completes the proof. I11 ¼ ðp−w 4ð1−t Þ2 4ð1−t Þ2 1
d2 TP m 1 1 β2 t1 t2 1 ¼ −β ð β−γ Þ þ þ − b0: ð5Þ 1−t 1 1−t 2 2 ð1−t 1 Þ2 ð1−t 2 Þ2 dw21 dTP m 1 dw1
¼ 0, we obtain after
4α ðp−s1 Þt 1 β ðp−s2 Þt 2 β p þ c−s1 p þ c−s2 þ þ þ ð β−γ Þ þ β 1−t 1 1−t 2 ð1−t 1 Þ2 ð1−t 2 Þ2 w1 ¼ : t1 t2 1 1 þ þ þ 2 ð β−γ Þ β 1−t 1 1−t 2 ð1−t 1 Þ2 ð1−t 2 Þ2
implying that TPr1i(I1i) is concave in I1i. solving dT PdI1i ðI1i Þ ¼ 0; we obtain 2
ð4Þ
some calculations
Proof. From Eq. (1), we have d
t 1 ðp−w1 −s1 Þβ 2 t 2 ðp−w1 −s2 Þβ2 þ 2ð1−t 1 Þ2 2ð1−t 2 Þ2
Therefore, TPm 1 (w1) is concave in w1. Solving
4ð1−t 2 Þ
TP r1i ðI1i Þ d2 I 1i
dTP m β ðβ−γÞ p−w1 −s1 p−w1 −s2 1 1 1 ¼ 2α þ þ −ðw1 −cÞ þ dw1 1−t 1 1−t 2 2 1−t 1 1−t 2
4ð1−t 1 Þ
and I12 ¼ ðp−w1 −s2 Þ2 β ,respectively.
2
Proof. From Eq. (3), we have
þ
qffiffiffiffiffiffi pffiffiffiffiffi r TP 1i ðI1i Þ ¼ ðp−w1 −si Þ α þ β I1i −γ I 1 j −ð1−t i ÞI1i ;
2
4α ðp−s1 Þt 1 β ðp−s2 Þt 2 β p þ c−s1 p þ c−s2 þ þ þ ð β−γ Þ þ β 1−t 1 1−t 2 ð1−t 1 Þ2 ð1−t 2 Þ2 w1 ¼ : t1 t2 1 1 þ þ þ 2ðβ−γÞ β 2 2 1−t 1 1−t 2 ð1−t 1 Þ ð1−t 2 Þ
2
Proposition 1 indicates that the retailers' optimal advertising expenditures increase as manufacturer's participation rate for subsidizing retailers' advertising costs increases. Substituting the values of I∗11 and I∗12 in Eq. (1), we get the retailer 1's optimal profit as " # β2 ðp−w1 −s1 Þ βγðp−w1 −s2 Þ r − TP 11 I11 ¼ ðp−w1 −s1 Þ α þ 4ð1−t 1 Þ 2ð1−t 2 Þ
This completes the proof. From Eqs. (1) and (2) and from Propositions 1 and 2, the profit of the entire supply chain is given by w r r m TP 1 ¼ TP 11 I11 þ TP 12 I 12 þ TP 1 w1 qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi ¼ ðp−c−s1 Þ α þ β I 11 −γ I 12 þ ðp−c−s2 Þ α þ β I12 −γ I 11 :
−I11 −I12
ð6Þ
B.C. Giri, S. Sharma / Economic Modelling 38 (2014) 102–111
3.2. Case 2: Manufacturer's pricing strategy 2
and
With pricing strategy 2, the manufacturer first sets wholesale prices w21 and w22 for retailers 1 and 2, respectively according to their different sales costs s1 and s2. The retailers then follow Nash equilibrium to determine their optimal advertising expenditures. 3.2.1. The retailer's problem The retailer's problem is to determine the advertising cost that maximizes his/her profit. The retailer's profit is given by r TP 2i ðI 2i Þ
qffiffiffiffiffiffi pffiffiffiffiffi ¼ ðp−w2i −si Þ α þ β I 2i −γ I 2 j −ð1−t i ÞI 2i ;
105
i ¼ 1; 2 and j ¼ 3−i:
ð7Þ
"
! # β2 t 1 β2 βγ 1 1 B þ þ A þ 1−t 1 2ð1−t 1 Þ2 2 1−t 1 1−t 2 ! ! 2 β2 t1 β2 β2 t2 β2 β 2 γ2 1 1 þ þ þ − 1−t 1 2ð1−t 1 Þ2 1−t 2 2ð1−t 2 Þ2 4 1−t 1 1−t 2
w22 ¼
ð13Þ where,
2 2 β t2 β βγ p−s1 c ðp þ c−s2 Þ þ ð p−s Þ− þ 2 2 1−t 1 1−t 2 2ð1−t 2 Þ 2ð1−t 2 Þ2
: 2 2 β t1 β βγ p−s2 c B¼αþ ðp þ c−s1 Þ þ ð p−s Þ− þ 1 2 1−t 2 1−t 1 2ð1−t 1 Þ 2ð1−t 1 Þ2
A¼αþ
Proposition 3. For the pricing strategy 2 of the manufacturer, the retailers 2
1 and 2's optimal advertising expenditures are I21 ¼ ðp−w21 −s12Þ 2
ðp−w22 −s2 Þ β 4ð1−t 2 Þ2
2
β2
4ð1−t 1 Þ
and I 22 ¼
,respectively.
Proof. The proof is similar to Proposition 1. Substituting the values of I∗21 and I∗22 in Eq. (7), the retailer 1's optimal profit can be derived as "
# 2 β ðp−w21 −s1 Þ βγðp−w22 −s2 Þ r − TP 21 I21 ¼ ðp−w21 −s1 Þ α þ ð8Þ 4ð1−t 1 Þ 2ð1−t 2 Þ
Proof. The proof is given in Appendix A. From Eqs. (7) and (10), and from Propositions 3 and 4, the profit of the entire supply chain is given by w r r m TP 2 ¼ TP 21 I21 þ TP 22 I 22 þ TP 2 w21 ; w22 qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi ¼ ðp−c−s1 Þ α þ β I 21 −γ I22 þ ðp−c−s2 Þ α þ β I 22 −γ I21 :
−I21 −I22
ð14Þ
and the retailer 2's optimal profit as " # β2 ðp−w22 −s2 Þ βγðp−w21 −s1 Þ r − : ð9Þ TP 22 I22 ¼ ðp−w22 −s2 Þ α þ 4ð1−t 2 Þ 2ð1−t 1 Þ
4. Model II: Manufacturer does not share retailers' advertising expenditures 4.1. Case 1: Manufacturer's pricing strategy 1
3.2.2. The manufacturer's problem The manufacturer's problem is to select different wholesale prices w2is for two retailers that maximize his/her profit. The manufacturer's profit is given by m
TP 2 ðw21 ; w22 Þ ¼
2 X
qffiffiffiffiffi qffiffiffiffiffiffi ðw2i −cÞ α þ β I 2i −γ I2 j −t 1 I 21 −t 2 I22 ; j ¼ 3−i:
i¼1
ð10Þ Substituting the values of
I∗21
" m
TP 2 ðw21 ; w22 Þ ¼ ðw21 −cÞ α þ "
and
I∗22
from Proposition 3, we obtain
# β2 ðp−w21 −s1 Þ βγðp−w22 −s2 Þ − 2ð1−t 1 Þ 2ð1−t 2 Þ
þðw22 −cÞ α þ
# β2 ðp−w22 −s2 Þ βγðp−w21 −s1 Þ − 2ð1−t 2 Þ 2ð1−t 1 Þ
t ðp−w21 −s1 Þ2 β2 t 2 ðp−w22 −s2 Þ2 β2 − : − 1 4ð1−t 1 Þ2 4ð1−t 2 Þ2
w21
# 2 2 2 2 β t2 β β t1 β βγ 1 1 B Aþ þ þ þ 2 2 1−t 2 2ð1−t 2 Þ 1−t 1 2ð1−t 1 Þ 2 1−t 1 1−t 2 ! ! ¼ " 2 # βγ 1 1 β2 t 1 β2 β2 t 2 β2 β2 γ2 1 1 − þ þ þ þ 1−t 1 2ð1−t 1 Þ2 1−t 2 2ð1−t 2 Þ2 4 2 1−t 1 1−t 2 1−t 1 1−t 2 −
A βγ 1 1 þ 2 1−t 1 1−t 2
4.1.1. The retailer's problem The retailer i's problem is to determine advertising expenditure which maximizes his/her profit. The profit of retailer i is given by qffiffiffiffiffiffi pffiffiffiffiffi r TP 3i ðI3i Þ ¼ ðp−w3 −si Þ α þ β I3i −γ I 3 j −I 3i ; i ¼ 1; 2 and j ¼ 3−i:
ð15Þ From Proposition 1, by putting t1 = t2 = 0, we find that for manufacturer's pricing strategy 1, retailers 1 and 2's optimal advertising expenditures are
ð11Þ
Proposition 4. For the pricing strategy 2, the manufacturer's optimal wholesale prices are !"
In this section, we assume that manufacturer does not share any advertising expenditure incurred by the retailers and he(she) sets the same wholesale price w3 to each retailer. The sequence of events is as follows: The manufacturer first sets a common wholesale price w3 irrespective of retailers' sales costs si, i = 1,2. In response, the retailers observe w3 and follow Nash's equilibrium to determine their optimal advertising expenditures.
!
ð12Þ
I 31 ¼
ðp−w3 −s1 Þ2 β2 4
and
I32 ¼
ðp−w3 −s2 Þ2 β2 ; 4
ð16Þ
respectively. From (16), we have I∗31 N I∗32 since s1 b s2. Therefore, retailer 1 spends more than retailer 2 on advertising. Substituting the values of I∗31 and I∗32 in Eq. (15), we get the retailer 1's optimal profit as " # β2 ðp−w3 −s1 Þ βγðp−w3 −s2 Þ r − TP 31 I31 ¼ ðp−w3 −s1 Þ α þ 4 2
ð17Þ
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B.C. Giri, S. Sharma / Economic Modelling 38 (2014) 102–111
Substituting t1 = 0 and t2 = 0 in Proposition 3, we find that for pricing strategy 2 of the manufacturer, retailers 1 and 2's optimal advertising expenditures are given by
and the retailer 2's optimal profit as " # β2 ðp−w3 −s2 Þ βγðp−w3 −s1 Þ r − TP 32 I32 ¼ ðp−w3 −s2 Þ α þ : 4 2
ð18Þ
Proposition 5. For the manufacturer's pricing strategy 1, retailer 1 gains more profit than retailer 2. Proof. From Eqs. (17) and (18), we have r r TP 31 I31 −TP 32 I32 ¼ α ½ðp−w3 −s1 Þ−ðp−w3 −s2 Þ i β2 h 2 2 ðp−w3 −s1 Þ − ðp−w3 −s2 Þ þ 4 " # 2 β ððp−w3 −s1 Þ þ ðp−w3 −s2 ÞÞ ¼ ðs2 −s1 Þ α þ 4
N0
I 41 ¼
2 2 β ðp−w41 −s1 Þ 4
and
I 42 ¼
2 2 β ðp−w42 −s2 Þ ; 4
ð24Þ
respectively. Substituting the values of I∗41 and I∗42 in Eq. (23), the optimal profits of retailers 1 and 2 can be obtained as " # β2 ðp−w41 −s1 Þ βγðp−w42 −s2 Þ r − TP 41 I41 ¼ ðp−w41 −s1 Þ α þ ð25Þ 4 2 " # and β2 ðp−w42 −s2 Þ βγðp−w41 −s1 Þ r − ; TP 42 I42 ¼ ðp−w42 −s2 Þ α þ 4 2 respectively. ð26Þ
since s 2 N s 1 and p − w 3 − s 1 N 0 and p − w 3 − s 2 N 0. Thus, TPr31(I∗31) N TPr32(I∗32) which means that for the manufacturer's pricing strategy 1, retailer 1 gains higher profit than retailer 2. This completes the proof.
4.2.2. The manufacturer's problem The manufacturer's problem is to determine the wholesale prices for retailers that maximize his/her profit. The manufacturer's profit is given by
4.1.2. The manufacturer's problem The manufacturer's problem is to determine the wholesale price that maximizes his/her profit. The manufacturer's profit is given by
TP 4 ðw41 ; w42 Þ ¼
m
TP 3 ðw3 Þ ¼
2 h X
qffiffiffiffiffi qffiffiffiffiffiffii ðw3 −cÞ α þ β I 3i −γ I3 j ; j ¼ 3−i:
ð19Þ
i¼1
m
w3 ¼
α p þ c s1 þ s2 þ − : βðβ−γÞ 2 4
Substituting the values of I∗41 and I∗42, we have "
m TP 4 ðw41 ; w42 Þ
# β2 ðp−w41 −s1 Þ βγðp−w42 −s2 Þ − ¼ ðw41 −cÞ α þ 2 2 " #: 2 β ðp−w42 −s2 Þ βγðp−w41 −s1 Þ − þ ðw42 −cÞ α þ 2 2 ð28Þ
Substituting t1 = 0 and t2 = 0 in Proposition 4, we find that for pricing strategy 2, the manufacturer's optimal wholesale prices for retailers 1 and 2 are
ð21Þ
ð27Þ
i¼1
Substituting the values of I∗31 and I∗32 from Eq. (16), we have h s þ s2 i m ð20Þ TP 3 ðw3 Þ ¼ ðw3 −cÞ 2α þ βðβ−γ Þ p−w3 − 1 2 Substituting t1 = 0 and t2 = 0 in Proposition 2, the manufacturer's optimal wholesale price for pricing strategy 1 is given by
qffiffiffiffiffi 2 qffiffiffiffiffiffi X ðw4i −cÞ α þ β I4i −γ I 4 j ; j ¼ 3−i:
w41 ¼
α p þ c s1 þ − βðβ−γÞ 2 2
ð29Þ
α p þ c s2 þ − : βðβ−γ Þ 2 2
ð30Þ
and From Eqs. (15),(16),(19) and (21), the total profit of the entire supply chain is given by w r r m TP 3 ¼ TP 31 I 31 þ TP 32 I 32 þ TP 3 w3 qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi ¼ ðp−c−s1 Þ α þ β I 31 −γ I32 þ ðp−c−s2 Þ α þ β I 32 −γ I 31 :
−I31 −I32
ð22Þ 4.2. Case 2: Manufacturer's pricing strategy 2 In this strategy, the manufacturer first sets different wholesale prices w41 and w42 to retailers 1 and 2, respectively according to their different sales costs s1 and s2. Then the retailers follow Nash's equilibrium to determine their optimal advertising expenditures. 4.2.1. The retailer's problem The retailer i faces the problem to determine advertising expenditure that maximizes his/her profit. The profit of retailer i is given by qffiffiffiffiffiffi pffiffiffiffiffi r TP 4i ðI4i Þ ¼ ðp−w4i −si Þ α þ β I4i −γ I 4 j −I 4i ; i ¼ 1; 2 and j ¼ 3−i:
ð23Þ
w42 ¼
∗ ∗ From Eqs. (29) and (30), we get w41 −w42 ¼ s −s 2 N0, i.e. w41 N w42. This implies that the manufacturer will charge a greater wholesale price to retailer 1 than retailer 2. 2
1
Proposition 6. For the pricing strategy 2 of the manufacturer, retailer 1 gains higher profit than retailer 2. Proof. The proof is given in Appendix A. From Propositions 5 and 6, we find that the retailer 1's profit is higher than the retailer 2's profit under both the pricing strategies of the manufacturer. From Eqs. (23), (24), (27), (29) and (30), the total profit of the entire supply chain is given by w r r m TP 4 ¼ TP 41 I41 þ TP 42 I 42 þ TP 4 w41 ; w42 qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi ¼ ðp−c−s1 Þ α þ β I 41 −γ I42 þ ðp−c−s2 Þ α þ β I 42 −γ I41 :
−I41 −I42 :
ð31Þ
B.C. Giri, S. Sharma / Economic Modelling 38 (2014) 102–111
5. Discussion In this section, we investigate the effects of manufacturer's different pricing strategies in Model II on the retailers' optimal advertising expenditures and their profits, manufacturer's optimal wholesale prices and his/her profit, and also profit of the whole supply chain. Proposition 7. w∗41 N w∗3 N w∗42 and w3 ¼ w
þw42 41
2
.
Proof. From Eqs. (21), (29) and (30), we have w41 −w3 ¼ s −s 4 N0 , ∗ which implies that w∗41 N w∗3. Also, w3 −w42 ¼ s −s N0. Thus, w N w∗42. 3 4 w þw ∗ ∗ ∗ ∗ Since w41 − w3 = w3 − w42, we have w3 ¼ 2 . The above results indicate that with pricing strategy 2, manufacturer sets a higher wholesale price for retailer 1 and lower wholesale price for retailer 2 compared to those with pricing strategy 1. With regard to retailer's optimal advertising expenditure, now we have the following result. pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi Proposition 8. I∗41 b I∗31, I32⁎ b I42⁎ and I31 þ I32 ¼ I 41 þ I42 . 2
2
41
have w∗41 N w∗3 − s1)2
1
1
107
The above Proposition suggests that for manufacturer's pricing strategy 1, the profit of the supply chain as a whole is higher than that for pricing strategy 2. 6. Numerical study In this section, we illustrate the developed models for a numerical example. The parameter values are taken as: α = 5, β = 2, γ = 0.2, p = $12, c = $4, s1 = $1, s2 = $2, t1 = 0.1, t2 = 0.2. The optimal results of the developed models are given below:
42
w∗3
w∗42
b w∗3. Therew∗42 − s2)2 N
and Proof. From Proposition 7, we fore, (p − w∗41 − s1)2 b (p − and (p − (p − w∗3 − s2)2, and hence from Eqs. (16) and (24), wepffiffiffiffiffiffi have pffiffiffiffiffiffi ∗ ∗ ∗ I∗41 b I Eqs. (16) and (24), we have I − I 31 and I42 N I32. Also, from 31 41 ¼ ffiffiffiffiffiffi p p ffiffiffiffiffiffi β β w −w w −w N0 and I N0 . Thus from − I ¼ 2 2 42pffiffiffiffiffiffi 32 pffiffiffiffiffiffi 41 1 3 pffiffiffiffiffiffi 42 pffiffiffiffiffiffi Proposition have pffiffiffiffiffiffi I31 − I41 ¼ I42 − I32 , and hence we have pffiffiffiffiffiffi pffiffiffiffiffiffi7 we pffiffiffiffiffiffi I31 þ I32 ¼ I41 þ I42 . Therefore, with respect to the manufacturer's pricing strategy 2, retailer 2 has a higher advertising expenditure and retailer 1 has a lower advertising expenditure compared to those with respect to the pricing strategy 1. m m Proposition 9. TP 4 w41 ; w42 NTP 3 w3 : Proof. See Appendix A. We are now in a position to state that the manufacturer obtains a higher profit from his pricing strategy 2 than strategy 1. With regard to retailers' profits, we have the result stated in the following Proposition:
Model I. For the pricing strategy 1, the optimal results are obtained as w∗1 = $ 8.59, I∗11 = $ 7.20, I∗12 = $ 3.12, TPr11 = $ 17.70, TPr12 = $ 8.81, w TPm 1 = $ 81.38 and TP1 = $ 107.75. For the pricing strategy 2, the optimal results are obtained as w∗21 = $ 8.87, w∗22 = $ 8.35, I∗21 = $ 5.61, I∗22 = $ 4.23, TPr21 = $ 14.83, w TPr22 = $ 10.83, TPm 2 = $ 81.62 and TP2 = $ 107.28. Model II. For the model with manufacturer's pricing strategy 1, the optimal results are obtained as w∗3 = $ 8.64, I∗31 = $ 5.57, I∗32 = $ 1.85, w TPr31 = $ 16.74, TPr32 = $ 8.02, TPm 3 = $ 77.47 and TP3 = $ 102.22. For the pricing strategy 2 of the manufacturer, the optimal results are obtained as w∗41 = $ 8.89, w∗42 = $ 8.39, I∗41 = $ 4.46, I∗42 = $ 2.60, w TPr41 = $ 14.33, TPr42 = $ 9.97, TPm 4 = $ 77.74 and TP4 = $ 102.05. From the above numerical results of two models we see that retailer 1 obtains a higher profit than retailer 2 under both the pricing strategies of the manufacturer. Also, advertising expenditure of retailer 1 is greater than that of retailer 2 irrespective of the pricing strategies set by the manufacturer. With respect to pricing strategy 2 set by the manufacturer, retailer 2 has a higher advertising expenditure while retailer 1 has a lower advertising expenditure compared to those with pricing strategy 1. The manufacturer always obtains a higher profit with pricing strategy 2 than that with pricing strategy 1. The total profit of the supply chain is higher in pricing strategy 1 than that in pricing strategy 2. All the observations described above corresponding to numerical results are consistent with the theoretical results obtained in two models. From the results of Models I and II, we also find that cooperative advertising increases the profit not only for the manufacturer and the retailers but also for the whole supply chain.
Proposition 10. TPr41(I∗41) b TPr31(I∗31) and TPr42(I∗42) N TPr32(I∗32). Proof. See Appendix A. From the above Proposition, we find that retailer 1 gains a lower profit from manufacturer's pricing strategy 2 than pricing strategy 1, and retailer 2 gains a higher profit from pricing strategy 2 than pricing strategy 1. Now, with respect to the profit of the entire supply chain, we have the following Proposition: Proposition 11. The profit of the supply chain is higher for manufacturer's pricing strategy 1 than that for pricing strategy 2 i.e. w TPw 3 N TP4 . Proof. From Propositions 7 and 8 and from Eqs. (16), (22), (24) and (31), we have qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi w w TP 3 −TP 4 ¼ ðp−c−s1 Þ α þ β I 31 −γ I32 þ ðp−c−s2 Þ α þ β I 32 −γ I 31 − q q ffiffiffiffiffiffi q qffiffiffiffiffiffi ffiffiffiffiffiffi ffiffiffiffiffiffi −ðp−c−s1 Þ α þ β I41 −γ I 42 −ðp−c−s2 Þ α þ β I 42 −γ I 41
þ I 41 þ I42 −I 31 −I 32 2 qffiffiffiffiffiffi qffiffiffiffiffiffi 2 2 i : β h ¼ ðβ þ γÞ I 31 − I 41 ðs2 −s1 Þ þ p−w42 −s2 − p−w3 −s2 4 2 2 2 i β h p−w3 −s1 − p−w41 −s1 − 4 βðβ þ 4γ Þðs2 −s1 Þ2 N0 ¼ 32 w Therefore, TPw 3 N TP4 . This completes the proof.
6.1. Effect of competition In this subsection, we examine the effect of competition on optimal advertising expenditures of the retailers, the profits of the manufacturer, retailers, and the entire supply chain in Model II. The effect of competition is illustrated in Figs. 1–3. Fig. 1 shows that retailer 1 obtains a higher profit than retailer 2 under each of the manufacturer's pricing strategy and for the pricing strategy 1, retailer 1's profit is higher than that for pricing strategy 2. Also, for the pricing strategy 1, retailer 2's profit is lower than that for pricing strategy 2. Fig. 2 shows that, for pricing strategy 1, manufacturer obtains lower profit than that for pricing strategy 2. In case of profit of the entire supply chain, pricing strategy 1 yields a higher profit than pricing strategy 2. From Figs. 1 and 2, we find that increase in competition decreases the profit of the retailers, manufacturer and the entire supply chain irrespective of manufacturer's pricing strategy. Thus increase in competition between retailers is detrimental for the retailers, the manufacturer and also for the entire supply chain. Fig. 3 shows that, with increase in competition, retailers' optimal advertising expenditures decrease. 6.2. Sensitivity analysis In this subsection, we examine the impact of unit retail price p and manufacturer's unit production cost c on profits of the retailers, manufacturer and the entire supply chain, and on the retailers' optimal advertising expenditures. We perform the sensitivity analysis on parameters
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18
6
TPr
TP r
31
32
TP r
41
*
TP r
*
I31
42
*
I32
*
I41
I42
16 5
advertising expenditure
profit
14
12
10
8
6
4 0.2
4
3
2
1
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0
γ
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
γ Fig. 1. Impact of competition on retailers' profits. Fig. 3. Impact of competition on retailers' advertising expenditures.
based on Model II under each of the manufacturer's pricing strategies. The results are reflected in Figs. 4–9 which show that an increase in unit retail price p increases the profits of the retailers, the manufacturer and the entire supply chain under each pricing strategy of the manufacturer. Also, we find that each retailer's optimal advertising expenditure increases with increase in unit retail price under both the pricing strategies of the manufacturer. In Figs. 7–9, we find that the profits of the retailers, manufacturer and the entire supply chain all decrease with increase in manufacturer's unit production cost c under each pricing strategy adopted by the manufacturer. Also, each retailer's optimal advertising expenditure decreases with increase in the manufacturer's unit production cost. These observations can be explained on the basis of the fact that, in each pricing strategy, an increase in unit retail price results in an increase in the unit sales profit and therefore, the retailers tend to invest more on advertising to acquire greater demand and hence order more quantities from the manufacturer. Thus, retailers' optimal advertising expenditures and the profits of the retailers, manufacturer and the entire supply chain all increase with an increase in unit retail
price. However, with an increase in unit manufacturing cost, the manufacturer charges a higher wholesale price which leads to a decrease in unit sales profit of the retailers. Then, the retailers order less quantities from the manufacturer and invest less in advertising. Thus, the retailers' optimal advertising expenditures and the profits of the retailers, manufacturer and the entire supply chain decrease with an increase in unit manufacturing cost. 7. Conclusions In this paper, we study a two-echelon supply chain comprising one manufacturer and two competing retailers whose demand depends on advertising. The manufacturer acts as the Stackelberg leader and the two retailers are the followers. To set wholesale price, the manufacturer takes one of the two strategies: (1) negotiate with both the retailers simultaneously and set a wholesale price that applies to both of them (strategy 1); and (2) negotiate with two retailers differently and set different wholesale prices to two retailers according to their different sales 300
105 m 3 m TP 4 w TP 3 w TP 4
r 31 r TP 32 m TP 3 w TP 3
TP
TP 100
200
90
profit
profit
95
250
150
85 100 80 50 75
70 0.2
0 0.25
0.3
0.35
γ
0.4
0.45
0.5
0.55
0.6
Fig. 2. Impact of competition on manufacturer's and the entire supply chain's profits.
12
13
14
15
16
17
unit retail price (p) Fig. 4. Unit retail price vs. profit under pricing strategy 1.
18
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109
120
300
r 31 r TP 32 m TP 3 w TP 3
TP r 41 r TP 42 m TP 4 w TP 4
TP 250
80
150
profit
profit
200
100
60
100
40
50
20
0
0 12
13
14
15
16
17
18
4
4.5
5
5.5
6
unit production cost (c)
unit retail price (p) Fig. 5. Unit retail price vs. profit under pricing strategy 2.
Fig. 7. Unit production cost vs. profit under pricing strategy 1.
costs (strategy 2). Two models are developed. The first model is developed under the assumption that the manufacturer shares a fraction of retailer's advertising expense while in the second model the manufacturer does not share advertising costs incurred by the retailers. In our model, it is assumed that manufacturer declares a fixed participation rate for each retailer before the start of the selling season. In both models, we derive retailers' optimal advertising expenditures and their optimal profits, and the manufacturer's optimal wholesale price and profit under each pricing strategy of the manufacturer. The study based on Model II generates several meaningful insights: (i) Retailer 1 always obtains a higher profit than retailer 2 irrespective of the pricing strategy adopted by the manufacturer. This is simply because retailer 1 has lower sales cost than retailer 2. For pricing strategy 1, retailer 1 obtains higher profit than for pricing strategy 2 while retailer 2's profit is higher under pricing strategy 2 than that under pricing strategy 1. (ii) With pricing strategy 2, the manufacturer sets a higher wholesale price to retailer 1 and lower wholesale price to retailer 2
compared to that under pricing strategy 1. Consequently, retailer 1 invests less in advertising with pricing strategy 2 as compared to pricing strategy 1 while retailer 2 invests more in advertising with pricing strategy 2 than with pricing strategy 1. (iii) With pricing strategy 2, manufacturer obtains higher profit than with pricing strategy 1 and hence the manufacturer ideally wants to adopt pricing strategy 2. (iv) The total profit of the entire supply chain is higher with pricing strategy 1 than that with pricing strategy 2. (v) An increase in unit retail price increases the retailers' optimal advertising expenditures, the profits of the retailers, manufacturer and the entire supply chain but the scenario is reversed with an increase in unit manufacturing cost. (vi) A tougher competition between retailers decreases profits of the retailers, manufacturer and the entire supply chain. The study based on model I shows that, with co-operative advertising, the profits of the retailers, manufacturer and entire supply chain always increase compared to the case of non-cooperative advertising. Various insights gained in Model II also hold good for model I.
30
120 r 41 r TP 42 m TP 4 w TP 4
TP *
I31
25
100
*
I41
20
80
* 42
I 15
profit
advertising expenditure
I*32
60
10
40
5
20
0 12
0
13
14
15
16
17
unit retail price (p) Fig. 6. Unit retail price vs. retailers' optimal advertising expenditures.
18
4
4.5
5
5.5
6
6.5
unit production cost (c) Fig. 8. Unit production cost vs. profit under pricing strategy 2.
7
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B.C. Giri, S. Sharma / Economic Modelling 38 (2014) 102–111
6
The Hessian matrix is
*
I 31
0
I*
5
32 *
H1 ¼
advertising expenditure
I 41
0
*
I 42
4
A
∂2 TP m 2 ∂w222
β2 ð2−t Þ
2
3
βγ 1 2 1−t 1
1−t 1
þ
1 1−t 2
β 2 ð2−t 2 Þ − 2ð1−t 2 Þ2
1−t 2
1 C C: A
Therefore, 2 β4 ð2−t 1 Þð2−t 2 Þ β2 γ2 1 1 − þ 2 2 1−t"1 1−t 2 4 ð1−t 1 Þ ð1−t 2 Þ 4 #: 4 2 2 2 2 β ð4−2ðt 1 þ t 2 Þ þ t 1 t 2 Þ β γ ð1−t 2 Þ þ ð1−t 1 Þ þ 2ð1−t 1 Þð1−t 2 Þ ¼ − 4 4 ð1−t 1 Þ2 ð1−t 2 Þ2 ð1−t 1 Þ2 ð1−t 2 Þ2
2
jH 1 j ¼
Since β N γ, to show that |H1| N 0, it is sufficient to show that
0 4
4.5
5
5.5
6
6.5
2 2 4−2ðt 1 þ t 2 Þ þ t 1 t 2 − ð1−t 1 Þ þ ð1−t 2 Þ þ 2ð1−t 1 Þð1−t 2 Þ N0:
7
unit production cost (c) Fig. 9. Unit production cost vs. retailers' optimal advertising expenditures.
Now,
Our model can be extended in several ways. One immediate extension is inclusion of stochastic demand instead of deterministic demand. Secondly, one can generalize the model considering two or more manufacturers. Thirdly, retailers' competition in price as well as in advertising would be another worthy extension of our paper. These are few among various research directions based on our work, some of which will be considered in our future research efforts. Acknowledgment
2 2 4−2ðt 1 þ t 2 Þ þ t 1 t 2 − ð1−t 1 Þ þ ð1−t 2 Þ þ 2ð1−t 1 Þð1−t 2 Þ 2
2
¼ 2ðt 1 þ t 2 Þ−t 1 t 2 −t 1 −t 2 ¼ ðt 1 þ t 2 Þ½2−ðt 1 þ t 2 Þ þ t 1 t 2 N0
:
since 0bt1 ; t 2 ≤1
Thus, we have |H1| N 0, i.e. the Hessian matrix H1 is negative definite and hence TPm 2 (w21,w22) is concave in w21 and w22. From Eqs. (32) and (33), we obtain, after some tedious calculations, the values of w∗21 and w∗22 given in Eqs. (12) and (13), respectively. This completes the proof. Proof of Proposition 6. From Eqs. (25) and (26), we have
The first author gratefully acknowledges the financial support provided by the University Grants Commission, New Delhi, under UGC-DRS Programme (2012–2017). The research work of the second author was funded by State Govt. Fellowship Scheme, Jadavpur University. Appendix A
r r TP 41 I 41 −TP 42 I 42 ¼ α p−w41 −s1 − p−w42 −s2 þ 2 i β 2 h 2 p−w41 −s1 − p−w42 −s2 4 " β 2 ¼ p−w41 −s1 − p−w42 −s2 αþ p−w41 −s1 þ 4 : # p−w42 −s2 Þ " # β 2 ¼ w42 −w41 þ s2 −s1 α þ p−w41 −s1 þ p−w42 −s2 4 Þ N0 . From Eqs. (29) and (30), we have, w42 −w41 þ s2 −s1 ¼ ðs −s 2 Also, (p − w∗41 − s1) N 0 and (p − w∗42 − s2) N 0. Hence, TPr41(I∗41) N TPr42(I∗42). This completes the proof.
Proof of Proposition 4. From Eq. (11), we have
2
∂TP m β2 t ðp−w21 −s1 Þβ2 ðw21 −cÞβ2 2 ¼αþ ðp−w21 −s1 Þ þ 1 − 2ð1−t 1 Þ 2ð1−t 1 Þ ∂w21 2ð1−t 1 Þ2 ðp−w22 −s2 Þβγ ðw22 −cÞβγ þ − 2ð1−t 2 Þ 2ð1−t 1 Þ ð32Þ m
1
∂2 TP m 2 ∂w21 ∂w22 C C
1 − 2 B 2ð1−t 1 Þ ¼B @ βγ 1 1 þ
1
2
∂2 TP m 2 B ∂w2 B 2 21m @ ∂ TP 2 ∂w22 ∂w21
1
Proof of Proposition 9. From Eqs. (19) and (27), we have qffiffiffiffiffiffi qffiffiffiffiffiffi m m TP 4 w41 ; w42 −TP 3 w3 ¼ w41 −c α þ β I 41 −γ I 42 þ w42 −c ffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi q qffiffiffiffiffiffi α þ β I 42 −γ I 41 − w3 −c α þ β I 31 −γ I 32 − q q ffiffiffiffiffiffi ffiffiffiffiffiffi w3 −c α þ β I 32 −γ I 31 :
2
∂ TP 2 β ð2−t 1 Þ ¼− b0 2ð1−t 1 Þ2 ∂w221
∂2 TP m βγ 1 1 2 ¼ þ N0 2 1−t 1 1−t 2 ∂w22 ∂w21 m 2 ∂TP 2 β t ðp−w22 −s2 Þβ2 ðw22 −cÞβ2 : ¼αþ ðp−w22 −s2 Þ þ 2 − 2ð1−t 2 Þ 2ð1−t 2 Þ ∂w22 2ð1−t 2 Þ2 ðp−w21 −s1 Þβγ ðw21 −cÞβγ þ − 2ð1−t 1 Þ 2ð1−t 2 Þ 2 m 2 ∂ TP 2 β ð2−t 2 Þ ¼− b0 2ð1−t 2 Þ2 ∂w222 ð33Þ
∗ ∗ ∗ FrompPropositions pffiffiffiffiffiffi ffiffiffiffiffiffi pffiffiffiffiffiffi 7 and 8, we have w41 + w42 = 2w3 and I32 ¼ I41 þ I42 , respectively and so,
pffiffiffiffiffiffi I31 þ
qffiffiffiffiffiffi qffiffiffiffiffiffi m m I41 − I42 : TP 4 w41 ; w42 −TP 3 w3 ¼ ðβ þ γÞ w41 −w3 from Eq. (24), From p Eqs. (21) and (29), we have w41 −w3 ¼ s −s 4 and ffiffiffiffiffiffi pffiffiffiffiffiffi m m β we have I − I 42 ¼ 4 ðs2 −s1 Þ. Thus, we obtain TP 4 w41 ; w42 −TP 3 βðβþγÞ41 ðs −s Þ2 w3 ¼ N0 since s2 N s1. This completes the proof of the 16 Proposition. 2
2
1
1
B.C. Giri, S. Sharma / Economic Modelling 38 (2014) 102–111
Proof of Proposition 10. From Eqs. (15) and (23), we have qffiffiffiffiffiffi qffiffiffiffiffiffi r r TP 31 I 31 −TP 41 I 41 ¼ p−w3 −s1 α þ β I 31 −γ I 32 − p−w41 −s1 qffiffiffiffiffiffi qffiffiffiffiffiffi : α þ β I 41 −γ I 42 þ I 41 −I 31
From Eqs. (21) and (29), we have w41 ¼ w3 þ s −s 4 . Then from Eqs. (16) and (24) and from Proposition 8, we have 2
1
qffiffiffiffiffiffi qffiffiffiffiffiffi r r TP 31 I 31 −TP 41 I 41 ¼ p−w3 −s1 α þ β I 31 −γ I32 − p−w41 −s1 qffiffiffiffiffiffi qffiffiffiffiffiffi β 2 h 2 i 2 α þ β I41 −γ I42 þ p−w −s − p−w3 −s1 4 ffiffiffiffiffiffi 41 1 qffiffiffiffiffiffi q s −s ¼ p−w3 −s1 α þ β I31 −γ I 32 − p−w3 − 2 1 −s1 4 qffiffiffiffiffiffi qffiffiffiffiffiffi β 2 α þ β I41 −γ I42 þ 2p−w3 −w41 −2s1 w3 −w41 qffiffiffiffiffiffi 4 qffiffiffiffiffiffi : N p−w3 −s1 α þ β I 31 −γ I32 − p−w3 −s1 qffiffiffiffiffiffi qffiffiffiffiffiffi β 2 s −s α þ β I41 −γ I42 − w41 −w3 2p−w3 −w3 − 2 1 −2s1 4 4 2 h s −s i β ðβ þ γÞ β ¼ w41 −w3 2 p−w3 −s1 − 2 1 p−w3 −s1 w41 −w3 − 4 4 2 βγðp−w3 −s1 Þðs2 −s1 Þ β 2 ðs2 −s1 Þ2 ¼ N0 þ 64 8
Therefore, TPr31(I∗31) N TPr41(I∗41). From Eqs. (21) and (30), we have w3 ¼ w42 þ s −s 4 . Then from Eqs. (15), (16), (23) and (24) and from Proposition 8, we obtain 2
1
qffiffiffiffiffiffi qffiffiffiffiffiffi r r TP 42 I 42 −TP 32 I 32 ¼ p−w42 −s2 α þ β I 42 −γ I 41 − p−w3 −s2 qffiffiffiffiffiffi qffiffiffiffiffiffi α þ β I 32 −γ I 31 þ I 32 −I 42 qffiffiffiffiffiffi qffiffiffiffiffiffi s −s ¼ p−w42 −s2 α þ β I 42 −γ I 41 − p−w42 − 2 1 −s2 4 qffiffiffiffiffiffi qffiffiffiffiffiffi α þ β I 32 −γ I 31 þ I 32 −I 42 : qffiffiffiffiffiffi qffiffiffiffiffiffi N p−w42 −s2 ðβ þ γÞ I 42 − I32 þ I32 −I 42 2 2 i β ðβ þ γÞ β 2 h ¼ p − w3 − s2 − p − w42 − s2 p−w42 −s2 w3 −w42 þ 4 2 2 2 βγðp−w42 −s2 Þðs2 −s1 Þ β ðs2 −s1 Þ N0 þ ¼ 64 8
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