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IMA Journal of Management Mathematics
, Volume Advance Article – Aug 29, 2017

31 pages

/lp/ou_press/the-effect-of-the-alliance-between-manufacturer-and-weak-retailer-on-xCjmqG06Yg

- Publisher
- Oxford University Press
- Copyright
- © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
- ISSN
- 1471-678X
- eISSN
- 1471-6798
- D.O.I.
- 10.1093/imaman/dpx005
- Publisher site
- See Article on Publisher Site

Abstract Consider a supply chain consisting of a common manufacturer and two asymmetric retailers, a dominant retailer and a weak one. This article develops a differential game in which the manufacturer sets the wholesale price for the weak retailer and invests advertising to improve goodwill, while both retailers compete for market demand by setting their respective retail prices. We study two market structures: a no-alliance scenario and an alliance scenario to answer a key question that the dominant retailer is facing: Can the dominant retailer benefit from the alliance between the manufacturer and the weak retailer? Our results show that the dominant retailer can take advantage from the alliance when the marginal contribution of goodwill on market demand is relatively high. The weak channel always benefits from alliance because of the strengthened channel power. Under the no-alliance scenario, the dominant retailer with a relatively small market share will set a lower steady-state price than that of the weak retailer, while the opposite situation occurs for the dominant retailer with a relatively large market share. Under the alliance scenario, the dominant retailer’s steady-state price is always higher than that of the weak channel, and both the dominant retailer and the weak channel charge higher steady-state prices compared to the no-alliance scenario when the effect of goodwill on demand is large, while lower prices of the dominant retailer and the weak channel arise for a small effect of goodwill on demand. 1. Introduction In recent decades, a few retailers, such as WalMart, Home Depot and Tesco, are increasingly dominating the retailing market. Research concerning dominant retailers and their impacts on distribution channels has been continuously active during these years. The motivation is crystal: The growing dominance of these powerful retailers has not only altered traditional channel incentives for manufacturers, but also posed threats to other weak (traditional) retailers. It can often be seen in practice that the powerful retailers get favourable wholesale terms via aggressive negotiating tactics with their manufacturers. A study conducted by Shi et al. (2013) reveals that the greater the retailer’s power is, the lower the corresponding wholesale price is. These powerful retailers not only take advantage of their channel power to buy products from their manufacturers at a low price but also employ this competitive price advantage to attack weak retailers and attract more customers. For example, a UBS survey shows that WalMart’s prices for grocery items are as much as 27–39% lower than other weak retailers’ (Koretz, 2002). The low prices of WalMart’s products force down these rival stores’ prices, and hence damage their profits. Additionally, in industries such as music, we observe that some powerful music retailers, such as Best Buy, are able to extract an additional 10% discount vis-a-vis other weak purchasers from the major record companies, which allows them to sell new albums at lower prices, whereby grabbing massive amounts of business away from their competitors (Christman, 2003). These observations reveal that both the manufacturer and the weak retailers are under the threat of powerful retailers. An important question arises that how manufacturers effectively manage asymmetric channel relationships when facing with powerful retailers. On the one hand, when facing a powerful retailer, especially for the giant one, the manufacturer has limited decision-making and bargaining powers. In this case, if the manufacturer makes an alliance with the dominant retailer, he has no advantage in profit redistribution, which may lead the manufacturer to get less profit than expected. On the other hand, compared with the weak retailer, the strong manufacturer has a dominated bargaining power and an absolute advantage over the weak retailer in allocating the allied profit. Therefore, even though the dominant retailer enjoys a large demand, and can run unique feature advertisements, information seminars and trade shows to promote the manufacturer’s products (Amrouche & Yan, 2013), sometimes the manufacturer is not willing to ally with this powerful retailer, but chooses the weak one as the collaborative partner to counter dominant retailer’s oppression and exploitation. For example, in 2004, due to the dispute over the pricing right, Gome1 announced an urgent notice about cleaning the air-conditioning inventory of Gree,2 which ended the collaboration between Gree and Gome (Wang, 2004). Afterwards, Gree adjusts the channel tactics and cooperates with some relatively weak retailers as a community of interests which bears venture and shares benefits together. Another example is that, Haier3 allies with the weak retailers and transforms them into community stores or franchised shops, in order to lessen the dependence on the dominant channel and establish controllable marketing channels. These weak retailers have little bargaining power and their gross margins are only about 3–4%. To form the integrated marketing channels of sales & service, Haier provides many supports for the weak retailers, such as free training, wage subsidy, advertising support, and bears most of the work, such as logistics and after-sale services. In this case, the weak retailers’ final profits are very stable and considerable, which makes possible for this alliance. When the cost of forming an alliance between the manufacturer and the weak retailer is very small, we expect that the weak channel composed by the manufacturer and the weak retailer always gets more profit in the alliance case than that in the no-alliance case. This is because the alliance can eliminate the double marginalization effect and strengthen the weak channels power. However, it is not clear whether this alliance hurts or benefits the dominant retailer. On one hand, the alliance enhances the power of the weak channel, which increases competition between both distribution channels. On the other hand, the dominant retailer gets some positive spillover effects due to additional advertising fees invested by the manufacturer under the alliance. We would like to investigate the following questions: (i) Is such spillover enough to help the dominant retailer to take advantage from the alliance and increase its profit compared to the no-alliance scenario? (ii) Under what conditions will the spillover be helpful for the dominant retailer to counter the threat of the alliance? (iii) How does the dominant retailer change its strategy when facing the alliance between the manufacturer and the weak retailer? In light of the fact that some marketing activities have carry-over effects and the partners usually develop a long-term and evolving relationship, we form a dynamic modelling framework to answer above questions. In this article, we develop a differential game model involving two asymmetric retailers, a dominant retailer and a weak one, and their common manufacturer. We analytically derive the feedback equilibrium pricing and advertising strategies for such a game under two different market structures: The no-alliance structure in which all the channel members make decisions to maximize their individual profits, and the alliance scenario in which the manufacturer allies with the weak retailer to achieve their joint profit maximization. Specifically, in the no-alliance case, the manufacturer is not able to directly influence the dominant retailer’s wholesale price but can set the wholesale price for the weak retailer and invest advertising effort to build goodwill stocks and then enhance marketing sales. Both retailers decide on their respective retail prices. In the alliance scenario, the manufacturer and the weak retailer making up a team maximize the sum of their profits by pricing and investing in advertising. The dominant retailer still determines her own retail price. Finally, we provide the strategies and payoffs comparisons between the two cases. Our analysis leads to many interesting results. First, the dominant retailer gets some positive spillover effect due to additional advertisement offered by the manufacturer, and such spillover may be great enough to help the dominant retailer take advantage from the alliance and increase its profit compared to the no-alliance scenario. When the marginal contribution of goodwill on market demand is significant, the alliance is beneficial to the dominant retailer. Second, the alliance scenario does not necessarily induce the dominant retailer to reduce its retail price. When the marginal contribution of goodwill on market demand is relatively high, the alliance between the manufacturer and the weak retailer leads to higher steady-state retail prices for both distribution channels. Third, a dominant retailer does not necessarily set a lower retail price relative to the weak one, when both retailers use pricing strategy to compete for marketing sales. Under no-alliance scenario, the steady-state price of the dominant retailer with a relatively small market share is lower than that of the weak retailer, while a relatively large market share results in a higher retail price of the dominant retailer. In addition, under alliance scenario, the dominant retailer’s steady-state price is always higher than that of the weak channel. Finally, the alliance always brings about a higher profit to the weak channel. The remainder of the article is organized as follows. We first review the related literature in Section 2 and then introduce a differential game model composed of two asymmetric retailers and a common manufacturer in Section 3. We derive the feedback equilibrium strategies under the no-alliance scenario to establish a baseline for comparison in Section 4 and analyse the alliance scenario in Section 5. In Section 6, the sensitivity analysis and outcome comparisons between the two scenarios are carried out. Meanwhile, we endogenize the dominant retailer’s wholesale price through bargaining to verify the robustness of our main results. Conclusions are finally drawn in Section 7. 2. Literature review This study is most closely related to three streams of literature: Channel power and asymmetric channel relationships, the pricing and advertising and differential game. In the following, we will review these streams of research and present how our work is related to each. This study contributes to the growing stream of literature in channel power and asymmetric channel relationships. Among this stream, most of the literature concerning channel power follows the definition presented by El-Ansary & Stern (1972) that the power of a channel member refers to the ability to control the decision variables. The power structure can be classified into the manufacturer Stackelberg game, the Nash game and the retailer Stackelberg game according to the decision sequences. The manufacturer Stackelberg game has been widely used in the existing literature, e.g., Chutani & Sethi (2012), He et al. (2009) and Xie & Neyret (2009). However, retailers have increased their power equal or even greater than the manufacturers in many industries over the last two decades. Under this condition, Ghadimi et al. (2013) consider the supply chain coordination by cooperative advertising between one manufacturer and two retailers when their relationships are symmetric. Amrouche et al. (2008a) and Jeuland & Shugan (1983) model the interaction between the manufacturer and the retailer as a Nash game where channel members move simultaneously when deciding on their own strategies. Lee & Staelin (1997) assume that the retailer moves first as the Stackelberg leader to set the retail margin. Based on the different channel powers, the issue of asymmetric channel relationships has been widely emphasized in channel research. In the presence of retailer asymmetry, Dukes et al. (2006, 2009, 2014) study the manufacturer’s potential profit, strategic assortment reduction by a dominant retailer and diverging incentives for product quality, respectively. Geylani et al. (2007) examine the asymmetric channels where a dominant retailer and a weak one are supplied by a common manufacturer and present a game-theoretic model to analyse the manufacturer’s strategic response to a dominant retailer. Raju & Zhang (2005) develop a channel model to examine how a manufacturer can best coordinate the channel including a dominant retailer and a competitive fringe. Our article complements these previous works by studying asymmetric channel relationships which consist of two asymmetric retailers and a common manufacturer. Different from these above literatures, this article investigates two scenarios: No-alliance scenario where the three channel members make decisions individually to maximize their own profits, and alliance scenario in which the manufacturer and the weak retailer ally with each other to counter the dominant retailer’s oppression and exploitation. In the no-alliance scenario, the manufacturer plays a Stackelberg game with the two retailers, while the retailers play a Nash game at the retailer level. In the alliance scenario, channel members play a Nash game. We analyse the effect of the manufacturer-weak retailer alliance on channel members’ performances, and especially care about the impact of this alliance on the dominant retailer. Our article is also related to researches on pricing and advertising. Pricing, as an important decision in marketing, has been extensively studied, especially in competitive scenarios. Zhao et al. (2014) consider pricing decisions for two substitutable products in a supply chain with one common retailer and two competitive manufacturers. Amrouche & Yan (2013) and Zhou & Xie (2014) present pricing decisions in a game model with a single supplier and two competing retailers. Notice that the retailers in Zhou & Xie (2014) are symmetric, while the retailers in Amrouche & Yan (2013) are asymmetric. Su & Mukhopadhyay (2012) study pricing decisions in a supply chain populated by a dominant retailer and a number of fringe retailers. They mainly focus on how the manufacturer coordinates such a channel and prevents the grey market activities through contract design. In addition to pricing, manufacturers and retailers also use advertising to promote demands directly, or to build on their goodwill for brands which in turn boosts demand. There are a number of papers studying the direct effect of advertising on demand in a static environment. For example, Karray & Zaccour (2006) and Lu & Liu (2013) consider that only one player (manufacturer or retailer) invests in advertising, while Szmerekovsky & Zhang (2009), Xie & Neyret (2009) and Zaccour (2008) explore both the manufacturer and the retailer invest in advertising by using different function forms concerning demand, price and advertising. In a dynamic environment, advertising investment contributes to the accumulation of goodwill which further increases demand, e.g., Amrouche et al. (2008a), Amrouche et al. (2008b), Karray & Martín-Herrán (2009), Li et al. (2017), Grosset & Viscolani (2008), Karray & Martín-Herrán (2009) and Martín-Herrán et al. (2011). Specifically, Amrouche et al. (2008a) and Amrouche et al. (2008b) consider a marketing channel where a retailer sells its own store brand and the manufacturer’s brand. The demand of products is affected by the price and goodwill stock and both models capture the cross-price effect on demand. Considering a differential game of dual-channel supply chain, Li et al. (2017) investigate the effects of cooperative advertising strategy and the manufacturer’s fairness concern on the channel coordination. Grosset & Viscolani (2008) present a dynamic advertising model which takes the market segmentation into consideration. Karray & Martín-Herrán (2009) also develop a model for pricing competition between national and store brands, but Martín-Herrán et al. (2011) consider pricing competition between different retailers. Similar to these literatures, the demand of product in our article is also affected by self price, cross-price and goodwill stock. However, our research differs from these papers by assuming that the dominant and weak retailers segment the market potential unequally and compete for the terminal market demand by pricing strategies. There has been a sustained interest in differential games of marketing channels. Differential game models have been widely applied to analyse the strategic dynamic interactions between the channel players (Amrouche et al., 2008b). For example, Buratto et al. (2016) consider a finite-horizon differential game and study the strategic interactions between a legal producer and a counterfeiter in a dynamic framework. Nie & Zhang (2017) model a differential game to study the effect of advertising productiveness difference between a manufacturer and a retailer on the distribution channel selection. El Ouardighi et al. (2013) consider a differential game with a single manufacturer who supplies two duopolistic retailers to evaluate how price-competition at the retailing level affects the management way of the upstream and downstream firms. For a good review of the literature on applications of differential games in marketing channels, see Buckdahn et al. (2011), Erickson (1995), He et al. (2009) and Huang et al. (2012). Our work diverges from above studies by incorporating power asymmetry between retailers into the differential games. Considering the wholesale price for the dominant retailer is exogenous, the manufacturer moves first to set the wholesale price for the weak retailer and the advertising effort. Then, reacting to the manufacturer’s decision, both retailers set their respective retail prices simultaneously through a Nash game at the retailing level. Our study is most closely related to Amrouche & Yan (2013). Under the static setting, they study the asymmetric channels where the manufacturer supplies products to a dominant retailer and a weak one. The manufacturer can totally control the wholesale prices for these two retailers and charges them the same price. They analyse the circumstances that make the weak retailer benefit from the manufacturer-dominant retailer alliance. Different from their study, our paper assumes that the manufacturer cannot directly decide the wholesale price for the dominant retailer, but has the ability to set the price for the weak one. Taking advertising’s carry-over effect into account, we model the differential games to investigate the impact of the alliance between the manufacturer and the weak retailer, a strategic response to counter the very powerful retailer, on the strategies of channel members and their performances. 3. Model development We consider a distribution channel consisting of one manufacturer (M) and two asymmetric retailers: A dominant retailer (Retailer 1) and a weak retailer (Retailer 2), in which the manufacturer produces a product and sells it to final consumers through both retailers. In line with Geylani et al. (2007), the distinction between the two asymmetric retailers lies in the control over wholesale prices. For the dominant retailer, the manufacturer is unable to directly influence its wholesale price $$w_{1}$$ which is given exogenously, while the manufacturer is endowed with the ability to offer a take-it-or-leave-it wholesale price $$w_{2}(t)$$ to the weak retailer. Both retailers compete for market demand by setting their individual retail prices $$p_{1}(t)$$ and $$p_{2}(t)$$. In the rest of the article, we refer to the manufacturer as ‘him’ and the retailers as ‘her’. To build the product’s goodwill and ultimately promote demand, the manufacturer tends to invest in advertising. We assume that $$G(t)$$ and $$A(t)$$ denote goodwill and advertising effort, respectively. The goodwill evolution according to the Nerlove & Arrow (1962) goodwill model can be characterized as follows: \begin{equation} \dot{G}(t)=A(t)-\delta G(t), \ G(0)=G_{0}, \end{equation} (1) where $$\delta$$ is a positive coefficient denoting the decay rate of goodwill, and $$G_{0}$$ represents the initial goodwill level. The above specification is the most widely used form for the goodwill functions in marketing channels (see Huang et al., 2012 for a review). The demand $$D_i(t)$$ for retailer $$i$$ ($$i=1, 2$$) is assumed to depend on the retail prices and goodwill in the duopoly market, i.e., \begin{align} D_{1}(t)&=\theta\big(\alpha+\gamma G(t)\big)-p_1(t)+\beta p_2(t), \\ \end{align} (2) \begin{align} D_2(t)&= (1-\theta) \big(\alpha+\gamma G(t)\big) -p_2(t)+\beta p_1(t), \end{align} (3) where $$1/2\leq\theta<1$$ is the market share of the dominant retailer, highlighting the dominant retailer’s advantage over the weak retailer, $$\alpha>0$$ denotes the basic market potential, $$0<\beta<1$$ represents the degree of store substitutability for the product, and $$\gamma>0$$ is the goodwill effectiveness coefficient reflecting the marginal contribution of goodwill on demand. According to equations (2) and (3), the market potential is time-varying and positively influenced by the product goodwill. The demand of each retailer is a decreasing function of her own price, while an increasing function of her rival’s price. Such demand functions have been commonly used in the marketing and operations management literatures to capture the competition between retailers, e.g., Geylani et al. (2007), Ingene & Parry (1995) and Tsay & Agrawal (2000). The advertising cost function is supposed to be convex increasing and quadratic, which can be expressed as \begin{align} &C\big(A(t)\big)= \frac{k}{2} A^2(t), \end{align} (4) where $$k>0$$ is the cost coefficient of the advertising investment. The convex cost function reflects a fact that the advertising effort is subject to diminishing marginal returns. This assumption is in accordance with many of the previous marketing literatures (e.g., Prasad & Sethi, 2004; He et al., 2009; Erickson, 2011; Chutani & Sethi, 2012; Erickson, 2012; Liu et al., 2015). In the following sections, we will present the equilibrium pricing and advertising strategies for no-alliance scenario and alliance scenario, respectively. Then, we will compare the outcomes between these two scenarios to analyse the effects of the manufacturer-weak retailer alliance on the strategies and profits of channel members, and discuss whether the dominant retailer can benefit from this alliance or not. 4. No-alliance scenario In this section, we first establish a performance benchmark by analysing the no-alliance scenario, under which the manufacturer does not cooperate with the weak retailer, i.e., all the supply chain members maximize their individual profits. To concentrate our analysis on research questions, we ignore the production costs, inventory-related costs and supply chain operational costs. Meanwhile, we suppose that the manufacturer’s production quantity is in accord with the retailers’ order quantity, which follows the consumer demand. We use superscript ‘$$N$$’ to signify the no-alliance scenario. Subscripts ‘$$M$$’ and ‘$$i$$’ ($$i=1,2$$) denote the manufacturer, the dominant retailer and the weak retailer, respectively. Thus, assuming an infinite time horizon and a positive discount rate $$\rho$$, the objective functionals of the manufacturer and two retailers are respectively expressed as \begin{align} J_M^{N}&=\displaystyle\int_{0}^{\infty}e^{-\rho t}\left( w_1(t)D_1(t)+w_2(t)D_2(t)-C(A(t))\right)\mathrm{d}t,\\ \end{align} (5) \begin{align} J_{i}^{N}&=\displaystyle\int_{0}^{\infty}e^{-\rho t}\Big(p_i(t)-w_i(t)\Big) D_i(t)\mathrm{d}t,\ i=1, 2. \end{align} (6) We assume that, regardless of whether the manufacturer allies with the weak retailer or not, it remains the same sequence of events. That is, the advertising effort is first announced to attract consumers’ attention and then the retail prices in the dominant and the weak channels are simultaneously determined. Specifically, in the no-alliance case, the manufacturer moves first to set the wholesale price for the weak retailer $$w_{2}(t)$$ and the advertising effort $$A(t)$$. Upon receiving the decisions of the manufacturer, both retailers set their respective retail prices $$p_{i}(t)$$ simultaneously to maximize their individual profits by playing a Nash game at the retailer level. Therefore, taking the dynamic relationships (1)–(6) together, we develop a differential game involving one manufacturer and two differentiated retailers as follows \begin{align} &\max\limits_{w_2(\cdot), A(\cdot)}\displaystyle\int_{0}^{\infty}e^{-\rho t} \left(w_1(t)D_1(t)+w_2(t)D_2(t)-\frac{k}{2} A^2(t)\right) \mathrm{d}t\displaystyle,\notag\\[1ex] &\quad\ \max\limits_{p_i(\cdot)} \displaystyle\displaystyle\int_{0}^{\infty}e^{-\rho t}\Big(p_i(t)-w_i(t)\Big)D_i(t)\mathrm{d}t,\quad i=1, 2, \\ &\quad\mathrm{s}.\mathrm{t}.\notag\\ &\qquad\dot{G}(t)=A(t)-\delta G(t),\ G(0)=G_{0}\geq 0.\notag \end{align} (7) To solve this differential game problem, we employ the backward induction to identify the equilibria of channel members. That is, we start by solving the optimization problems of the two retailers by determining their individual retail prices. Then we substitute the retailers’ response functions into the manufacturer’s problem and recursively derive the manufacturer’s optimal wholesale price and advertising effort. Just as many literatures involving in infinite time horizon differential games (e.g., Amrouche et al., 2008b; De Giovanni, 2011; Erickson, 2011; Erickson, 2012), we also confine our interest to feedback equilibrium strategies, which means that the pricing and advertising equilibrium strategies are dependent on the current goodwill level. Unlike the open-loop concept in which the manufacturer pre-commits to his decisions throughout the game and never departs from his initial plan over time (Chiang, 2012), when the feedback concept is applied, the pricing and advertising decisions is time consistent and made on the basis of the status of the product goodwill at each particular time. Let $$V^N_M$$, $$V^N_1$$ and $$V^N_2$$ denote the value functions for the manufacturer, the dominant retailer and the weak retailer in the no-alliance scenario, respectively. In the following article, wherever there is no confusion, we will omit the function argument $$t$$ for brevity. Thus, according to the differential game (7), the Hamilton–Jacobi–Bellman (HJB) equations for all the players are characterized as follows \begin{align} \rho V^N_M&=\max\limits_{w_{2},A}\left\{w_1 D_1+w_2D_2-\frac{k }{2}A^2+ \frac{\partial V^N_M}{\partial G}(A-\delta G)\right\},\\ \end{align} (8) \begin{align} \rho V^N_1&=\max\limits_{p_{1}}\left\{(p_1-w_1)D_1 + \frac{\partial V^N_1}{\partial G}(A-\delta G)\right\},\\ \end{align} (9) \begin{align} \rho V^N_2&=\max\limits_{p_{2}}\left\{(p_2-w_2)D_2 + \frac{\partial V^N_2}{\partial G}(A-\delta G)\right\}. \end{align} (10) To ensure positive steady-state goodwill levels and globally asymptotically stable equilibria under both the no-alliance scenario and the alliance scenario, we impose the following constraint on the parameters: \begin{align} \frac{\gamma^2}{k}<\frac{\delta(\rho+\delta)(4-\beta^2)^2} {2(2-2\theta+\beta\theta)^2}. \end{align} (11) The following proposition characterizes the feedback equilibrium strategies of all the players and the demands for both retailers in the no-alliance scenario. The proofs for this and all subsequent propositions and corollaries can be found in Appendix. Proposition 1 In the no-alliance scenario, the feedback equilibrium strategies for prices and advertisement of the manufacturer and both retailers are respectively given by \begin{align} &w^N_{2}(G)=\frac{\lambda_4(\alpha+\gamma G)+2\beta w_1}{2(2-\beta^2)},\\ \end{align} (12) \begin{align} &A^N(G)=\frac{k\lambda_1(\rho+2\delta)-\lambda_2}{2k\lambda_1}G +\frac{2\gamma\lambda_1(\beta-\beta\theta+\theta)w_1} {(2-\beta^2)(k\rho\lambda_1+\lambda_2)} +\frac{\alpha\gamma\lambda_4^2}{\lambda_1(k\rho\lambda_1+\lambda_2)},\\ \end{align} (13) \begin{align} &p^N_{1}(G)=\frac{-\lambda_3(\alpha+\gamma G)+2(4-\beta^2)w_1} {2(2-\beta^2)(4-\beta^2)}, \end{align} (14) \begin{align} &p^N_{2}(G)=\frac{\lambda_4(3-\beta^2) (\alpha+\gamma G)+\beta(4-\beta^2) w_1}{(2-\beta^2)(4-\beta^2)} \end{align} (15) and the corresponding demands for both retailers are given by \begin{align} D^{N}_{1}(G)&= \frac{-\lambda_3(\alpha+\gamma G)} {2(2-\beta^2)(4-\beta^2)}-\frac{1-\beta^2}{2-\beta^2}w_1, \\ \end{align} (16) \begin{align} D^{N}_{2}(G)& =\frac{\lambda_4(\alpha+\gamma G)}{2(4-\beta^2)}, \end{align} (17) where \begin{align} \lambda_1&=\sqrt{(2-\beta^2)(4-\beta^2)},\notag\\ \lambda_2&=\sqrt{k^2(\rho+2\delta)^2(2-\beta^2)(4-\beta^2) -2k\gamma^2(2-2\theta+\beta\theta)^2},\notag\\ \lambda_3&=2\beta^3-2\beta^3\theta+6\beta\theta -6\beta+3\beta^2\theta -8\theta,\notag\\ \lambda_4&=2-2\theta+\beta\theta.\notag \end{align} Note that the application of the feedback equilibrium concept makes the pricing and advertising strategies dependent on the state of goodwill. From Proposition 1, we can see that a higher goodwill level means more advertising investment. Additionally, it is easy to verify that the goodwill level has a positive effect on the wholesale price of the weak retailer, and the retail price and demand of each retailer. This result is intuitive because the advertising investment could build up product goodwill among consumers, which, in turn, can lead to an increase in prices and demands of the product. The proposition above also shows that, the weak retailer’s wholesale price and both retailers’ retail prices increase with the dominant retailer’s wholesale price $$w_1$$. Meanwhile, $$\frac{\partial p^N_{1}}{\partial w_1}>\frac{\partial p^N_{2}}{\partial w_1}$$ implies that the effect of the dominant retailer’s wholesale price on her own retail price is greater than the effect on her rival’s retail price, which is consistent with our intuition. Furthermore, it is straightforward to see that, the retail margin of the weak retailer does not depend directly on the dominant retailer’s wholesale price $$w_1$$. The rationale behind this result is the following. From Proposition 1, we can see that the dominant retailer’s wholesale prices $$w_1$$ has two effects on the wholesale and retail prices of the weak retailer: A direct effect and an indirect effect through goodwill $$G$$. Note that $$\frac{\partial w^N_{2}}{\partial w_1}=\frac{\partial p^N_{2}}{\partial w_1}$$, that is to say, the dominant retailer’s wholesale price has the same direct effect on the weak retailer’s wholesale and retail prices. In this case, the retail margin of the weak retailer is affected only by the indirect effect through $$G$$, not by the direct effect of $$w_1$$. Substituting the feedback advertising strategy $$A^N(G)$$ from (13) into state equation (1) yields the following differential equation \begin{align} \dot{G}^N(t)& = \frac{2\gamma\lambda_1 (\beta-\beta\theta+\theta)w_1} {(2-\beta^2)(k\rho\lambda_1+\lambda_2)} +\frac{\alpha\gamma\lambda_4^2} {\lambda_1(k\rho\lambda_1+\lambda_2)} +\frac{k\rho\lambda_1-\lambda_2}{2k\lambda_1}G^N,\ G^{N}(0)=G_{0}. \end{align} (18) At this time, we can derive the time path of the goodwill from (18). Correspondingly, the time paths for the pricing and advertising strategies of the manufacturer and the two retailers in the no-alliance scenario can be solved out, which are summarized in the following proposition. Proposition 2 In the no-alliance scenario, the time path for the goodwill is given by \begin{align} G^N(t)&= (G_{0}-G^N_{\infty})e^{\frac {k\rho\lambda_1-\lambda_2}{2k\lambda_1}t} +G^N_{\infty} \end{align} (19) and accordingly the time paths for the wholesale price of the weak retailer, the advertising effort of the manufacturer and the retail prices of the retailers are respectively given by \begin{align} w^N_2(t)&= \left(\frac{\lambda_4(\alpha+\gamma G_0)+2\beta w_1}{2(2-\beta^2)} -w^N_{2\infty}\right)e^{\frac {k\rho\lambda_1-\lambda_2}{2k\lambda_1}t} +w^N_{2\infty},\\ \end{align} (20) \begin{align} A^N(t)&= \left(\!\frac{\big(k(\rho + 2\delta) \lambda_1 - \lambda_2\big)G_0} {2k\lambda_1} + \frac{2k\lambda_1\gamma w_1(\beta + \theta - \beta\theta)} {k(2 - \beta^2)(k\rho\lambda_1 + \lambda_2)} + \frac{k\alpha\gamma\lambda_4^2} {k\lambda_1(k\rho\lambda_1 + \lambda_2)} - A^N_{\infty}\!\right)e^{\frac {k\rho\lambda_1-\lambda_2}{2k\lambda_1}t}\notag\\ &\quad +A^N_{\infty},\\ \end{align} (21) \begin{align} p^N_1(t)&= \left(\frac{-\lambda_3(\alpha+\gamma G_0) +2w_1(4-\beta^2)}{2\lambda_1^2}-p^N_{1\infty}\right) e^{\frac {k\rho\lambda_1-\lambda_2}{2k\lambda_1}t} +p^N_{1\infty},\\[0.5ex] \end{align} (22) \begin{align} p^N_2(t)&= \left(\frac{\lambda_4(3-\beta^2)(\alpha+\gamma G_0) +\beta w_1(4-\beta^2)}{\lambda_1^2}-p^N_{2\infty}\right) e^{\frac {k\rho\lambda_1-\lambda_2}{2k\lambda_1}t} +p^N_{2\infty}, \end{align} (23) where $$G^N_{\infty}$$, $$w^N_{2\infty}$$, $$A^N_{\infty}$$, $$p^N_{1\infty}$$, $$p^N_{2\infty}$$ are given in Appendix. Note that the subscript‘$$\infty$$’ represents the steady state when $$t\rightarrow+\infty$$. From Proposition 2, we can see that the time path of the goodwill is monotonic overtime. Specifically, when the initial goodwill is higher (lower) than the steady-state goodwill, the product goodwill is initially high (low), but decreases (increases) over the entire planning horizon; while when the initial goodwill is equal to the goodwill at steady state, the goodwill among consumers always keeps at a stable value. In this case, because the prices and the advertisement of the channel members are positively affected by the goodwill as shown in Proposition 1, these equilibrium strategies present the same changing trend as the product goodwill. Put differently, just as the time path of the goodwill is, the time paths of $$w^N_2(t)$$, $$A^N(t)$$, $$p^N_1(t)$$ and $$p^N_2(t)$$ in the no-alliance scenario are also dependent on the gap between the initial product goodwill and the steady-state goodwill. Corollary 1 Under the no-alliance scenario, the manufacturer charges the weak retailer a higher wholesale price at the steady state compared to the wholesale price for the dominant retailer, i.e., $$w^N_{2\infty}>w_{1}$$. Corollary 1 attests to the practical fact discussed previously that the wholesale prices for the dominant retailers are usually lower than those of the weak ones. This corollary reveals insights for a strategic manufacturer’s response to the pricing pressure given by the dominant retailer. Specifically, at the steady state, the manufacturer can respond strategically to the strong pricing power of the dominant retailer about the wholesale price by raising the wholesale price for the weak retailer over that for the dominant retailer. Substituting state trajectory (19) into equilibrium demands (16) and (17) yields the time paths for demands of both retailers: \begin{align} D^N_1(t)&= -\frac{\lambda_3\gamma}{2\lambda_1^2}(G_{0}-G^N_{\infty}) e^{\frac {k\rho\lambda_1-\lambda_2}{2k\lambda_1}t} -\frac{2k\lambda_1^2\lambda_3\alpha\delta(\rho+\delta)(2-\beta^2)} {(2-\beta^2)(\lambda_2^2-k^2\rho^2\lambda_1^2)}\nonumber\\ & \quad{} +\frac{2k(k\lambda_1^2\delta(\rho+\delta)(1-\beta^2) -2\lambda_1^2\lambda_3\gamma^2(\beta+\theta-\beta\theta) -(1-\beta^2)\gamma^2\lambda_4^2)w_1}{(2-\beta^2) (\lambda_2^2-k^2\rho^2\lambda_1^2)},\\ \end{align} (24) \begin{align} D^N_2(t)&= \frac{\gamma\lambda_4}{2(4-\beta^2)}(G_{0}-G^N_{\infty}) e^{\frac {k\rho\lambda_1-\lambda_2}{2k\lambda_1}t}\nonumber\\ &\quad{} +\frac{2k^2\lambda_1^2\lambda_4\alpha\delta (\rho+\delta)(2-\beta^2)+4k\lambda_1^2\lambda_4\gamma^2 (\beta+\theta-\beta\theta)w_1} {2\lambda_1^2(\lambda_2^2-k^2\rho^2\lambda_1^2)}. \end{align} (25) Accordingly, substituting equilibrium strategies (20)–(23) and the demands (24) and (25) into the objective functionals (5) and (6), we can obtain the discounted profits of the manufacturer and both retailers, $$J_{M}^{N}$$, $$J_{1}^{N}$$ and $$J_{2}^{N}$$, in the no-alliance scenario, which are, respectively, given by: \begin{align} J_M^N & = \frac{k(\rho+2\delta)\lambda_1-\lambda_2}{4\lambda_1} G_{0}^2 +\left(\frac{2k\lambda_1\gamma w_1(\beta+\theta-\beta\theta)} {(2-\beta^2)(k\rho\lambda_1+\lambda_2)} +\frac{k\alpha\gamma\lambda_4^2} {\lambda_1(k\rho\lambda_1+\lambda_2)} \right)G_{0}\notag\\[1pt] &\quad{} +\frac{(\beta+\theta-\beta\theta)\alpha w_1-(1-\beta^2)w_1^2}{(2-\beta^2) \rho} + \frac{\alpha^2\lambda_4^2} {4\rho\lambda_1^2} + \frac{1}{2k\rho}\left(\frac{2k\lambda_1\gamma w_1(\beta+\theta-\beta\theta)} {(2-\beta^2)(k\rho\lambda_1+\lambda_2)}\right.\nonumber\\[1pt] &\quad{} \left.+\frac{k\alpha\gamma\lambda_4^2} {\lambda_1(k\rho\lambda_1+\lambda_2)}\right)^2,\\[1pt] \end{align} (26) \begin{align} J_{1}^N& = \frac{k\gamma^2\lambda_3^2}{4\lambda_1^3\lambda_2}G_{0}^2 +\left(\frac{k\gamma^2\lambda_3^2} {\lambda_1^2\lambda_2(k\rho\lambda_1 +\lambda_2)^2}\left(\frac{2k\lambda_1\gamma w_1(\beta+\theta-\beta\theta)}{2-\beta^2} +\frac{k\alpha\gamma\lambda_4^2} {\lambda_1}\right)\right.\notag\\[1pt] &\quad{} \left.+\frac{2k\lambda_3\gamma w_1(1-\beta^2)}{\lambda_1(2-\beta^2)(k\rho\lambda_1+\lambda_2)} +\frac{k\alpha\gamma\lambda_3^2} {\lambda_1^3(k\rho\lambda_1+\lambda_2)} \right)G_{0}+\frac{(1-\beta^2)^2w_1^2} {(2-\beta^2)^2\rho}+\frac{\alpha^2\lambda_3^2} {4\rho\lambda_1^4}\notag\\[1pt] &\quad{} +\frac{(1-\beta^2)\lambda_3\alpha w_1}{(2-\beta^2)\lambda_1^2\rho}+\left(\frac{2k\lambda_1\gamma w_1(\beta+\theta-\beta\theta)} {k\rho(2-\beta^2)(k\rho\lambda_1+\lambda_2)} +\frac{k\alpha\gamma\lambda_4^2} {k\rho\lambda_1(k\rho\lambda_1+\lambda_2)}\right)\nonumber\\[1pt] &\quad{} \times\left(\frac{k\gamma^2\lambda_3^2} {\lambda_1^2\lambda_2(k\rho\lambda_1 +\lambda_2)^2}\left(\frac{2k\lambda_1\gamma w_1(\beta+\theta-\beta\theta)}{2-\beta^2} +\frac{k\alpha\gamma\lambda_4^2}{\lambda_1}\right)\right.\\[1pt] &\quad{} \left.+\frac{2k\lambda_3\gamma w_1(1-\beta^2)}{\lambda_1(2-\beta^2)(k\rho\lambda_1+\lambda_2)} +\frac{k\alpha\gamma\lambda_3^2} {\lambda_1^3(k\rho\lambda_1+\lambda_2)} \right),\notag\\[1pt] \end{align} (27) \begin{align} J_{2}^N & = \frac{k\gamma^2\lambda_1\lambda_4^2} {4\lambda_2(4-\beta^2)^2}G_{0}^2 +\left(\frac{k\gamma^2\lambda_1^2\lambda_4^2} {(4-\beta^2)^2\lambda_2(k\rho\lambda_1+\lambda_2)^2}\left (\frac{2k\lambda_1\gamma w_1(\beta+\theta-\beta\theta)}{2-\beta^2} +\frac{k\alpha\gamma\lambda_4^2} {\lambda_1}\right)\right.\notag\\[1pt] &\quad{} \left. +\frac{k\alpha\gamma\lambda_1\lambda_4^2} {(4-\beta^2)^2(k\rho\lambda_1+\lambda_2)}\right)G_{0} + \left(\frac{2k\lambda_1\gamma w_1(\beta+\theta-\beta\theta)} {k\rho(2-\beta^2)(k\rho\lambda_1+\lambda_2)} +\frac{k\alpha\gamma\lambda_4^2} {k\rho\lambda_1(k\rho\lambda_1+\lambda_2)}\right)\nonumber\\[1pt] &\quad{} \times\left(\frac{k\gamma^2\lambda_1^2\lambda_4^2} {(4-\beta^2)^2\lambda_2(k\rho\lambda_1+\lambda_2)^2} \left(\frac{2k\lambda_1\gamma w_1(\beta+\theta-\beta\theta)}{2-\beta^2} +\frac{k\alpha\gamma\lambda_4^2}{\lambda_1}\right)\right.\nonumber\\[1pt] &\quad{} \left.+\frac{k\alpha\gamma\lambda_1\lambda_4^2} {(4-\beta^2)^2(k\rho\lambda_1+\lambda_2)}\right) +\frac{\alpha^2\lambda_4^2}{4(4-\beta^2)^2\rho}. \end{align} (28) 5. The alliance scenario In the previous section, we have explored the equilibrium strategies and profits of channel members under the no-alliance scenario. In this section, we consider the alliance scenario in which the manufacturer and the weak retailer cooperate with each other, acting as a team, to maximize their joint profits by determining the retail price and advertising effort, and the dominant retailer still sets her own retail price. For simplicity of exposition, from now on we denote the allied team by subscript 2. To focus on our research questions, we do not consider the cost of forming an alliance. Thus, the corresponding differential game can be formulated as follows: \begin{align} &\max\limits_{p_2(\cdot), A(\cdot)}\displaystyle\int_{0}^{\infty}e^{-\rho t} \Big( w_1(t)D_1(t)+p_2(t)D_2(t)-\frac{k}{2} A^2(t)\Big) \mathrm{d}t\displaystyle,\notag\\ &\quad\max\limits_{p_1(\cdot)} \displaystyle\displaystyle\int_{0}^{\infty}e^{-\rho t}\Big(p_1(t)-w_1(t)\Big)D_1(t)\mathrm{d}t,\quad \\ &\quad\mathrm{s}.\mathrm{t}.\notag\\ &\qquad\dot{G}(t)=A(t)-\delta G(t),\ G(0)=G_{0}\geq 0.\notag \end{align} (29) We use the superscript ‘$$C$$’ to refer to the alliance scenario. {Similar to the no-alliance case, in the alliance scenario, the allied team first determines the advertising effort, and then the team and the dominant retailer set their respective retail prices simultaneously to compete in the retail market. Nonetheless, notice from the game problem (29) that, the advertising effort $$A^{C}(t)$$ controlled by the allied team is independent of channel members’ other decisions. Therefore, it will not change channel members’ equilibrium strategies if they set $$A^{C}(t)$$, $$p^{C}_{1}(t)$$ and $$p^{C}_{2}(t)$$ simultaneously. Under such a circumstance, the game between the allied team and the dominant retailer can be viewed as a Nash game. Let $$V^C_2$$ and $$V^C_{1}$$ respectively denote the value functions of the team and the dominant retailer under the alliance case. The HJB equations for the dominant retailer and the team are, respectively, written as \begin{align} \rho V^C_2&= \max\limits_{p_{2},A}\left\{w_1D_1+p_2 D_2-\frac{k}{2} A^2 +\frac{\partial V^C_2}{\partial G}(A-\delta G)\right\}, \\ \end{align} (30) \begin{align} \rho V^C_{1}&= \max\limits_{p_{1}}\left\{(p_{1}-w_{1})D_1+\frac{\partial V^C_{1}}{\partial G}(A-\delta G)\right\}. \end{align} (31) Solving the differential game (29) with the similar method as that in the no-alliance scenario, we can obtain the feedback equilibrium strategies of the dominant retailer and the team under the alliance scenario, which are shown in the following proposition. Proposition 3 In the alliance scenario, the feedback equilibrium strategies of the prices and advertisement for the dominant retailer and the team are, respectively, given by \begin{align} p^C_1(G)&= \frac{-\lambda_6(\alpha+\gamma G) +2w_1+\beta^2 w_1}{4-\beta^2},\\ \end{align} (32) \begin{align} p^C_2(G)&= \frac{\lambda_4(\alpha+\gamma G)+3\beta w_1}{4-\beta^2},\\ \end{align} (33) \begin{align} A^C(G)&= \frac{k(\rho+2\delta)(4-\beta^2)-\lambda_7}{2k(4-\beta^2)}G +\frac{2\gamma(\lambda_5w_1+2\alpha\lambda_4^2)} {(4-\beta^2)(k\rho(4-\beta^2)+\lambda_7)} \end{align} (34) and the corresponding demands for both channel members are given by \begin{align} D^{C}_{1}(G)&= \frac{-\lambda_6 (\alpha+\gamma G)+2\beta^2 w_1-2w_1}{4-\beta^2},\\ \end{align} (35) \begin{align} D^{C}_{2}(G)&= \frac{\lambda_4(\alpha+\gamma G)+\beta^3 w_1-\beta w_1}{4-\beta^2}, \end{align} (36) where \begin{align} \lambda_5&=8\beta-8\beta\theta+\beta^3-\beta^3 \theta+8\theta+\beta^4\theta,\notag\\ \lambda_6&=\beta\theta-\beta-2\theta,\notag\\ \lambda_7&=\sqrt{k^2(\rho+2\delta)^2(4-\beta^2)^2 -8k\gamma^2\lambda_4^2}.\notag \end{align} It is easy to see that the qualitative results of this proposition are similar to those obtained in the no-alliance scenario. The pricing and advertising strategies are linearly dependent on the product goodwill. That is to say, all strategies are state-dependent and the channel members will adjust their strategy trajectories over time according to the current state of the goodwill. Substituting the feedback advertising strategy (34) into the state equation (1), we can obtain the following differential equation: \begin{align} \dot{G}^C(t)=\frac{k \rho(4-\beta^2)-\lambda_7}{2k(4-\beta^2)}G^{C} +\frac{2\gamma(\lambda_5w_1+2\alpha\lambda_4^2)} {(4-\beta^2)(k\rho(4-\beta^2)+\lambda_7)},\ G^{C}(0)=G_{0}. \end{align} (37) At this time, we can derive the time path of the goodwill from (37). Correspondingly, the time paths for the pricing and advertising strategies of the manufacturer, the dominant retailer and the allied team can be obtained in the following proposition. Proposition 4 In the alliance scenario, the time path for the goodwill is given by \begin{align} &G^C(t)=(G_{0}-G^C_{\infty})e^{\frac{k \rho(4-\beta^2)-\lambda_7}{2k(4-\beta^2)}t} +G^C_{\infty} \end{align} (38) and accordingly the time paths for the dominant retailer’s retail price, the team’s retail price and advertising effort are, respectively, given by \begin{align} p^C_1(t)&= \left(\frac{-\lambda_{6}(\alpha+\gamma G_0)+w_1(2+\beta^2) }{4-\beta^2}-p^C_{1\infty}\right)e^{\frac{k \rho(4-\beta^2)-\lambda_7}{2k(4-\beta^2)}t} +p^C_{1\infty},\\ \end{align} (39) \begin{align} p^C_2(t)&= \left(\frac{\lambda_4(\alpha+\gamma G_0)+3\beta w_1}{4-\beta^2}-p^C_{2\infty}\right)e^{\frac{k \rho(4-\beta^2)-\lambda_7}{2k(4-\beta^2)}t} +p^C_{2\infty},\\ \end{align} (40) \begin{align} A^C(t)&= \left(\frac{(k(\rho + 2\delta)(4 - \beta^2) - \lambda_7)G_0} {2k(4-\beta^2)} + \frac{2\gamma(\lambda_5w_1+2\alpha\lambda_4^2)} {(4-\beta^2)(k(4-\beta^2)\rho+\lambda_7)} - A^C_{\infty}\right) e^{\frac{k \rho(4 - \beta^2) - \lambda_7}{2k(4 - \beta^2)}t},\notag\\ &\quad{} +A^C_{\infty}, \end{align} (41) where $$G^C_{\infty}$$, $$p^C_{1\infty}$$, $$p^C_{2\infty}$$ and $$A^C_{\infty}$$ are given in Appendix. It is obviously shown from Proposition 4 that, the time path of the goodwill is monotonic overtime. Meanwhile, similar to the no-alliance case, the monotonicity of the time paths for equilibrium strategies $$p^C_1(t)$$, $$p^C_2(t)$$ and $$A^C(t)$$ in the alliance scenario are also determined by the difference between the initial product goodwill and the steady-state goodwill. Substituting state trajectory (38) into equilibrium demands (35) and (36) yields the time paths for the demands of the dominant retailer and the team as follows: \begin{align} D^C_1(t)&= \frac{\gamma(2\theta+\beta-\beta\theta)} {4-\beta^2}(G_{0}-G^C_{\infty}) e^{\frac{k \rho(4-\beta^2)-\lambda_7}{2k(4-\beta^2)}t}\nonumber\\ &\quad{} - \frac{k\alpha\delta\lambda_6(\rho + \delta) (4 - \beta^2)^2 + (\lambda_5\lambda_6\gamma^2 + 2k\delta (\rho + \delta) (1 - \beta^2)(4 - \beta^2)^2 - \!4(1 - \beta^2) \gamma^2\lambda_4^2)\!w_1} {(4 - \beta^2)(k\delta(\rho + \delta) (4 - \beta^2)^2 - 2\gamma^2\lambda_4^2)},\\ \end{align} (42) \begin{align} D^C_2(t)& = \frac{\gamma(2-2\theta+\beta\theta)} {4-\beta^2}(G_{0}-G^C_{\infty}) e^{\frac{k \rho(4-\beta^2)-\lambda_7}{2k(4-\beta^2)}t} \nonumber\\ &\quad{} + \frac{k\alpha\delta\lambda_4(\rho + \delta) (4 - \beta^2)^2 + (\lambda_5\lambda_4\gamma^2 - k\beta\delta(\rho + \delta)(1 - \beta^2) \!(4 - \beta^2)\!^2 + 2\beta(1 - \beta^2)\!\gamma^2\lambda_4^2)w_1} {(4 - \beta^2)(k\delta(\rho + \delta) (4 - \beta^2)^2 - 2\gamma^2\lambda_4^2)}. \end{align} (43) Accordingly, substituting equilibrium strategies (39)–(41) and the demands (42) and (43) into (29), we can obtain the discounted profits of the dominant retailer and the team, $$J_{1}^{C}$$ and $$J_{2}^{C}$$, in the alliance scenario, which are, respectively, given by: \begin{align} J_{1}^C&= \frac{k\gamma^2\lambda_6^2}{\lambda_7(4-\beta^2)}G_{0}^2 + \left(\frac{8k^2\gamma^3\lambda_6^2 (\lambda_5w_1+2\alpha\lambda_4^2)} {\lambda_7(4 - \beta^2)(k\rho(4 - \beta^2) + \lambda_7)^2} + \frac{4k\gamma\lambda_6(2(1 - \beta^2)w_1+\alpha\lambda_6)} {(4-\beta^2)(k\rho(4-\beta^2)+\lambda_7)}\right)G_{0}\notag\\ &\quad{} + \frac{4(1-\beta^2)^2w_1^2}{(4-\beta^2)^2\rho} + \frac{4\alpha\lambda_6(1-\beta^2)w_1}{(4-\beta^2)^2\rho} + \frac{\alpha^2\lambda_6^2}{(4-\beta^2)^2\rho} + \frac{2\gamma(\lambda_5w_1+2\alpha\lambda_4^2)} {\rho(4-\beta^2)(k(4-\beta^2)\rho+\lambda_7)} \nonumber\\ &\quad{} \times\left(\frac{8k^2\gamma^3\lambda_6^2 (\lambda_5w_1+2\alpha\lambda_4^2)} {\lambda_7(4-\beta^2)(k\rho(4-\beta^2)+\lambda_7)^2} +\frac{4k\gamma\lambda_6(2(1-\beta^2)w_1+\alpha\lambda_6)} {(4-\beta^2)(k\rho(4-\beta^2)+\lambda_7)}\right),\\ \end{align} (44) \begin{align} J_2^C&= \frac{k(\rho+2\delta)(4-\beta^2)-\lambda_7}{4(4-\beta^2)} G_{0}^2+\frac{2k\gamma(\lambda_5w_1+2\alpha\lambda_4^2)} {(4-\beta^2)(k(4-\beta^2)\rho+\lambda_7)}G_{0} \nonumber\\ &\quad{} + \frac{\lambda_5\alpha w_1-(1-\beta^2)(\beta^2+8)w_1^2} {(4-\beta^2)^2\rho} + \frac{\alpha^2\lambda_4^2}{(4-\beta^2)^2\rho} + \frac{4k^2\gamma^2(\lambda_5w_1+2\alpha\lambda_4^2)^2} {2k\rho(4-\beta^2)^2(k(4-\beta^2)\rho+\lambda_7)^2}. \end{align} (45) 6. Results analysis In this section, we give a further analysis on equilibrium strategies and profits under no-alliance and alliance scenarios by exploring the following questions: (i) How do the system parameters affect the strategies, states, demands as well as profits of all the players under each scenario? (ii) What are the differences of the strategies, states as well as profits between the no-alliance and alliance scenarios? (iii) Is it possible for the dominant retailer to benefit from this alliance? If yes, under what conditions? To answer above research questions, in Subsection 6.1, we carry out a series of sensitivity analysis on key system parameters to study their impacts on the strategies and profits of all the players under each scenario. In Subsection 6.2, we compare the strategies and profits obtained under both scenarios to gain more managerial insights. In Subsection 6.3, we endogenize the dominant retailer’s wholesale price through bargaining to verify the robustness of our main results. Note that in an infinite time differential game, the transient phase, no matter how long it looks, is a relatively short duration compared with the steady state period. As a result, consistent with previous studies involving infinite time differential games which always focus on the strategies and channel performance in the steady state, e.g., De Giovanni (2011), Liu et al. (2015) and Zhang et al. (2014), we confine our interest to the long-term behaviors of the players and analyse the steady-state values concerning control and state variables, demands and profits of channel members. Although the analytical solutions are characterized under both scenarios, the complexities of the expressions for the strategies and profits restrain us from analytically making a further analysis with these results. Therefore, we resort to numerical analysis to answer our research questions and gain more managerial insights. As a benchmark case, we set the following parameter values: Demand parameters: $$\alpha=1,$$$$\beta=0.7,$$$$\gamma=0.3$$ and $$\theta=0.7,$$ Cost parameters: $$k=0.8$$ and $$w_1=0.2,$$ Goodwill parameters: $$\delta=0.3$$ and $$G_0=1,$$ Dynamic parameters: $$\rho=0.1.$$ These parameter values are chosen based on previous studies concerning marketing and operations management, e.g., De Giovanni (2011), Liu et al. (2015) and Zhang et al. (2014), which allow for a comprehensive illustration. In addition, these parameters satisfy the constraint condition (11) to guarantee positive control and state paths and globally asymptotically stable equilibria. 6.1 Sensitivity analysis In this subsection, we carry out a sensitivity analysis to study how key system parameters, i.e., the substitutability parameter $$\beta$$, the market share $$\theta$$, the goodwill effectiveness coefficient $$\gamma$$, the goodwill decay rate $$\delta$$, the advertising cost coefficient $$k$$ and the dominant retailer’s wholesale price $$w_{1}$$, affect the steady-state solutions and profits of all the players. As shown in Tables 1 and 2, key system parameters have a similar effect on the steady-state strategies, demands and profits of all members under both no-alliance and alliance scenarios. We observe that the steady-state advertising efforts, prices, product goodwill, demands and profits of all players increase with the substitutability parameter $$\beta$$. That is to say, the manufacturer (or the alliance) invests more in advertising to build up a higher goodwill as competition between retailers increases, which expands the total market potential, and in turn provides an incentive for all the players to choose higher prices. The expanded market potential sizes and the increased prices finally lead to higher profits for all the players. Table 1. Variations in the no-alliance solution Parameter $$w^{N}_{2\infty}$$ $$p^{N}_{1\infty}$$ $$p^{N}_{2\infty}$$ $$A^{N}_{\infty}$$ $$G_{\infty}^{N}$$ $$D_{1\infty}^{N}$$ $$D_{2\infty}^{N}$$ $$J_{M}^{N}$$ $$J_{1}^{N}$$ $$J_{2}^{N}$$ $$J_{T}^{N}$$ Benchmark 0.5416 0.7924 0.7347 0.2437 0.8123 0.5924 0.1931 2.0301 3.6039 0.3819 2.4119 $$\qquad$$ 0.600 0.4459 0.7036 0.6138 0.1985 0.6617 0.5036 0.1679 1.6527 2.6686 0.2947 1.9474 $$\beta\hspace{1.4em}$$0.650 0.4901 0.7444 0.6698 0.2191 0.7303 0.5444 0.1798 1.8254 3.0822 0.3345 2.1599 $$\qquad$$ 0.750 0.6027 0.8499 0.8111 0.2736 0.9118 0.6499 0.2084 2.2759 4.2747 0.4391 2.7149 $$\qquad$$ 0.800 0.6765 0.9198 0.9027 0.3105 1.0349 0.7198 0.2262 2.5757 5.157 0.5096 3.0852 $$\qquad$$ 0.600 0.6088 0.7740 0.8308 0.2774 0.9248 0.5740 0.2220 2.2084 3.3295 0.4975 2.7059 $$\theta\hspace{1.4em}$$0.650 0.5745 0.7830 0.7818 0.2598 0.8659 0.5830 0.2073 2.1150 3.4640 0.4369 2.5519 $$\qquad$$ 0.750 0.5099 0.8022 0.6893 0.2291 0.7635 0.6022 0.1795 1.9531 3.7495 0.3319 2.2850 $$\qquad$$ 0.800 0.4792 0.8122 0.6455 0.2158 0.7194 0.6122 0.1663 1.8838 3.9011 0.2866 2.1704 $$\qquad$$ 0.100 0.4625 0.6761 0.6216 0.0736 0.2452 0.4761 0.1591 1.7112 2.3651 0.2626 1.9738 $$\gamma\hspace{1.4em}$$0.200 0.4903 0.7171 0.6614 0.1525 0.5084 0.5171 0.1711 1.8406 2.8153 0.3063 2.1469 $$\qquad$$ 0.400 0.6257 0.9161 0.8550 0.3576 1.1919 0.7161 0.2293 2.3012 4.9716 0.5111 2.8122 $$\qquad$$ 0.500 0.7624 1.1171 1.0505 0.5133 1.7109 0.9171 0.2881 2.6959 7.4699 0.7432 3.4391 $$\qquad$$ 0.250 0.5798 0.8486 0.7893 0.2912 1.1648 0.6486 0.2095 2.1345 4.1037 0.4293 2.5639 $$\delta\hspace{1.3em}$$0.275 0.5580 0.8165 0.7582 0.2650 0.9637 0.6165 0.2002 2.0767 3.8214 0.4026 2.4793 $$\qquad$$ 0.325 0.5289 0.7738 0.7165 0.2259 0.6950 0.5738 0.1876 1.9916 3.4319 0.3655 2.3571 $$\qquad$$ 0.350 0.5188 0.7590 0.7021 0.2107 0.6020 0.5590 0.1833 1.9595 3.2929 0.3522 2.3117 $$\qquad$$ 0.700 0.5559 0.8134 0.7551 0.2832 0.9441 0.6134 0.1992 2.0647 3.7919 0.3998 2.4645 $$k\hspace{1.3em}$$0.750 0.5482 0.8021 0.7441 0.2620 0.8733 0.6021 0.1959 2.0461 3.6903 0.3901 2.4362 $$\qquad$$ 0.850 0.5359 0.7840 0.7265 0.2278 0.7592 0.5840 0.1906 2.0161 3.5297 0.3748 2.3909 $$\qquad$$ 0.900 0.5308 0.7766 0.7193 0.2138 0.7127 0.5766 0.1885 2.0038 3.4651 0.3686 2.3724 Parameter $$w^{N}_{2\infty}$$ $$p^{N}_{1\infty}$$ $$p^{N}_{2\infty}$$ $$A^{N}_{\infty}$$ $$G_{\infty}^{N}$$ $$D_{1\infty}^{N}$$ $$D_{2\infty}^{N}$$ $$J_{M}^{N}$$ $$J_{1}^{N}$$ $$J_{2}^{N}$$ $$J_{T}^{N}$$ Benchmark 0.5416 0.7924 0.7347 0.2437 0.8123 0.5924 0.1931 2.0301 3.6039 0.3819 2.4119 $$\qquad$$ 0.600 0.4459 0.7036 0.6138 0.1985 0.6617 0.5036 0.1679 1.6527 2.6686 0.2947 1.9474 $$\beta\hspace{1.4em}$$0.650 0.4901 0.7444 0.6698 0.2191 0.7303 0.5444 0.1798 1.8254 3.0822 0.3345 2.1599 $$\qquad$$ 0.750 0.6027 0.8499 0.8111 0.2736 0.9118 0.6499 0.2084 2.2759 4.2747 0.4391 2.7149 $$\qquad$$ 0.800 0.6765 0.9198 0.9027 0.3105 1.0349 0.7198 0.2262 2.5757 5.157 0.5096 3.0852 $$\qquad$$ 0.600 0.6088 0.7740 0.8308 0.2774 0.9248 0.5740 0.2220 2.2084 3.3295 0.4975 2.7059 $$\theta\hspace{1.4em}$$0.650 0.5745 0.7830 0.7818 0.2598 0.8659 0.5830 0.2073 2.1150 3.4640 0.4369 2.5519 $$\qquad$$ 0.750 0.5099 0.8022 0.6893 0.2291 0.7635 0.6022 0.1795 1.9531 3.7495 0.3319 2.2850 $$\qquad$$ 0.800 0.4792 0.8122 0.6455 0.2158 0.7194 0.6122 0.1663 1.8838 3.9011 0.2866 2.1704 $$\qquad$$ 0.100 0.4625 0.6761 0.6216 0.0736 0.2452 0.4761 0.1591 1.7112 2.3651 0.2626 1.9738 $$\gamma\hspace{1.4em}$$0.200 0.4903 0.7171 0.6614 0.1525 0.5084 0.5171 0.1711 1.8406 2.8153 0.3063 2.1469 $$\qquad$$ 0.400 0.6257 0.9161 0.8550 0.3576 1.1919 0.7161 0.2293 2.3012 4.9716 0.5111 2.8122 $$\qquad$$ 0.500 0.7624 1.1171 1.0505 0.5133 1.7109 0.9171 0.2881 2.6959 7.4699 0.7432 3.4391 $$\qquad$$ 0.250 0.5798 0.8486 0.7893 0.2912 1.1648 0.6486 0.2095 2.1345 4.1037 0.4293 2.5639 $$\delta\hspace{1.3em}$$0.275 0.5580 0.8165 0.7582 0.2650 0.9637 0.6165 0.2002 2.0767 3.8214 0.4026 2.4793 $$\qquad$$ 0.325 0.5289 0.7738 0.7165 0.2259 0.6950 0.5738 0.1876 1.9916 3.4319 0.3655 2.3571 $$\qquad$$ 0.350 0.5188 0.7590 0.7021 0.2107 0.6020 0.5590 0.1833 1.9595 3.2929 0.3522 2.3117 $$\qquad$$ 0.700 0.5559 0.8134 0.7551 0.2832 0.9441 0.6134 0.1992 2.0647 3.7919 0.3998 2.4645 $$k\hspace{1.3em}$$0.750 0.5482 0.8021 0.7441 0.2620 0.8733 0.6021 0.1959 2.0461 3.6903 0.3901 2.4362 $$\qquad$$ 0.850 0.5359 0.7840 0.7265 0.2278 0.7592 0.5840 0.1906 2.0161 3.5297 0.3748 2.3909 $$\qquad$$ 0.900 0.5308 0.7766 0.7193 0.2138 0.7127 0.5766 0.1885 2.0038 3.4651 0.3686 2.3724 Note: $$J_{T}^{N}=J_{M}^{N}+J_{2}^{N}$$. Table 1. Variations in the no-alliance solution Parameter $$w^{N}_{2\infty}$$ $$p^{N}_{1\infty}$$ $$p^{N}_{2\infty}$$ $$A^{N}_{\infty}$$ $$G_{\infty}^{N}$$ $$D_{1\infty}^{N}$$ $$D_{2\infty}^{N}$$ $$J_{M}^{N}$$ $$J_{1}^{N}$$ $$J_{2}^{N}$$ $$J_{T}^{N}$$ Benchmark 0.5416 0.7924 0.7347 0.2437 0.8123 0.5924 0.1931 2.0301 3.6039 0.3819 2.4119 $$\qquad$$ 0.600 0.4459 0.7036 0.6138 0.1985 0.6617 0.5036 0.1679 1.6527 2.6686 0.2947 1.9474 $$\beta\hspace{1.4em}$$0.650 0.4901 0.7444 0.6698 0.2191 0.7303 0.5444 0.1798 1.8254 3.0822 0.3345 2.1599 $$\qquad$$ 0.750 0.6027 0.8499 0.8111 0.2736 0.9118 0.6499 0.2084 2.2759 4.2747 0.4391 2.7149 $$\qquad$$ 0.800 0.6765 0.9198 0.9027 0.3105 1.0349 0.7198 0.2262 2.5757 5.157 0.5096 3.0852 $$\qquad$$ 0.600 0.6088 0.7740 0.8308 0.2774 0.9248 0.5740 0.2220 2.2084 3.3295 0.4975 2.7059 $$\theta\hspace{1.4em}$$0.650 0.5745 0.7830 0.7818 0.2598 0.8659 0.5830 0.2073 2.1150 3.4640 0.4369 2.5519 $$\qquad$$ 0.750 0.5099 0.8022 0.6893 0.2291 0.7635 0.6022 0.1795 1.9531 3.7495 0.3319 2.2850 $$\qquad$$ 0.800 0.4792 0.8122 0.6455 0.2158 0.7194 0.6122 0.1663 1.8838 3.9011 0.2866 2.1704 $$\qquad$$ 0.100 0.4625 0.6761 0.6216 0.0736 0.2452 0.4761 0.1591 1.7112 2.3651 0.2626 1.9738 $$\gamma\hspace{1.4em}$$0.200 0.4903 0.7171 0.6614 0.1525 0.5084 0.5171 0.1711 1.8406 2.8153 0.3063 2.1469 $$\qquad$$ 0.400 0.6257 0.9161 0.8550 0.3576 1.1919 0.7161 0.2293 2.3012 4.9716 0.5111 2.8122 $$\qquad$$ 0.500 0.7624 1.1171 1.0505 0.5133 1.7109 0.9171 0.2881 2.6959 7.4699 0.7432 3.4391 $$\qquad$$ 0.250 0.5798 0.8486 0.7893 0.2912 1.1648 0.6486 0.2095 2.1345 4.1037 0.4293 2.5639 $$\delta\hspace{1.3em}$$0.275 0.5580 0.8165 0.7582 0.2650 0.9637 0.6165 0.2002 2.0767 3.8214 0.4026 2.4793 $$\qquad$$ 0.325 0.5289 0.7738 0.7165 0.2259 0.6950 0.5738 0.1876 1.9916 3.4319 0.3655 2.3571 $$\qquad$$ 0.350 0.5188 0.7590 0.7021 0.2107 0.6020 0.5590 0.1833 1.9595 3.2929 0.3522 2.3117 $$\qquad$$ 0.700 0.5559 0.8134 0.7551 0.2832 0.9441 0.6134 0.1992 2.0647 3.7919 0.3998 2.4645 $$k\hspace{1.3em}$$0.750 0.5482 0.8021 0.7441 0.2620 0.8733 0.6021 0.1959 2.0461 3.6903 0.3901 2.4362 $$\qquad$$ 0.850 0.5359 0.7840 0.7265 0.2278 0.7592 0.5840 0.1906 2.0161 3.5297 0.3748 2.3909 $$\qquad$$ 0.900 0.5308 0.7766 0.7193 0.2138 0.7127 0.5766 0.1885 2.0038 3.4651 0.3686 2.3724 Parameter $$w^{N}_{2\infty}$$ $$p^{N}_{1\infty}$$ $$p^{N}_{2\infty}$$ $$A^{N}_{\infty}$$ $$G_{\infty}^{N}$$ $$D_{1\infty}^{N}$$ $$D_{2\infty}^{N}$$ $$J_{M}^{N}$$ $$J_{1}^{N}$$ $$J_{2}^{N}$$ $$J_{T}^{N}$$ Benchmark 0.5416 0.7924 0.7347 0.2437 0.8123 0.5924 0.1931 2.0301 3.6039 0.3819 2.4119 $$\qquad$$ 0.600 0.4459 0.7036 0.6138 0.1985 0.6617 0.5036 0.1679 1.6527 2.6686 0.2947 1.9474 $$\beta\hspace{1.4em}$$0.650 0.4901 0.7444 0.6698 0.2191 0.7303 0.5444 0.1798 1.8254 3.0822 0.3345 2.1599 $$\qquad$$ 0.750 0.6027 0.8499 0.8111 0.2736 0.9118 0.6499 0.2084 2.2759 4.2747 0.4391 2.7149 $$\qquad$$ 0.800 0.6765 0.9198 0.9027 0.3105 1.0349 0.7198 0.2262 2.5757 5.157 0.5096 3.0852 $$\qquad$$ 0.600 0.6088 0.7740 0.8308 0.2774 0.9248 0.5740 0.2220 2.2084 3.3295 0.4975 2.7059 $$\theta\hspace{1.4em}$$0.650 0.5745 0.7830 0.7818 0.2598 0.8659 0.5830 0.2073 2.1150 3.4640 0.4369 2.5519 $$\qquad$$ 0.750 0.5099 0.8022 0.6893 0.2291 0.7635 0.6022 0.1795 1.9531 3.7495 0.3319 2.2850 $$\qquad$$ 0.800 0.4792 0.8122 0.6455 0.2158 0.7194 0.6122 0.1663 1.8838 3.9011 0.2866 2.1704 $$\qquad$$ 0.100 0.4625 0.6761 0.6216 0.0736 0.2452 0.4761 0.1591 1.7112 2.3651 0.2626 1.9738 $$\gamma\hspace{1.4em}$$0.200 0.4903 0.7171 0.6614 0.1525 0.5084 0.5171 0.1711 1.8406 2.8153 0.3063 2.1469 $$\qquad$$ 0.400 0.6257 0.9161 0.8550 0.3576 1.1919 0.7161 0.2293 2.3012 4.9716 0.5111 2.8122 $$\qquad$$ 0.500 0.7624 1.1171 1.0505 0.5133 1.7109 0.9171 0.2881 2.6959 7.4699 0.7432 3.4391 $$\qquad$$ 0.250 0.5798 0.8486 0.7893 0.2912 1.1648 0.6486 0.2095 2.1345 4.1037 0.4293 2.5639 $$\delta\hspace{1.3em}$$0.275 0.5580 0.8165 0.7582 0.2650 0.9637 0.6165 0.2002 2.0767 3.8214 0.4026 2.4793 $$\qquad$$ 0.325 0.5289 0.7738 0.7165 0.2259 0.6950 0.5738 0.1876 1.9916 3.4319 0.3655 2.3571 $$\qquad$$ 0.350 0.5188 0.7590 0.7021 0.2107 0.6020 0.5590 0.1833 1.9595 3.2929 0.3522 2.3117 $$\qquad$$ 0.700 0.5559 0.8134 0.7551 0.2832 0.9441 0.6134 0.1992 2.0647 3.7919 0.3998 2.4645 $$k\hspace{1.3em}$$0.750 0.5482 0.8021 0.7441 0.2620 0.8733 0.6021 0.1959 2.0461 3.6903 0.3901 2.4362 $$\qquad$$ 0.850 0.5359 0.7840 0.7265 0.2278 0.7592 0.5840 0.1906 2.0161 3.5297 0.3748 2.3909 $$\qquad$$ 0.900 0.5308 0.7766 0.7193 0.2138 0.7127 0.5766 0.1885 2.0038 3.4651 0.3686 2.3724 Note: $$J_{T}^{N}=J_{M}^{N}+J_{2}^{N}$$. Table 2. Variations in the alliance solution Parameter $$p^{C}_{1\infty}$$ $$p^{C}_{2\infty}$$ $$A^{C}_{\infty}$$ $$G_{\infty}^{C}$$ $$D_{1\infty}^{C}$$ $$D_{2\infty}^{C}$$ $$J_{1}^{C}$$ $$J_{2}^{C}$$ Benchmark 0.7662 0.5423 0.3610 1.2034 0.5662 0.4023 3.1204 2.7346 $$\qquad$$ 0.600 0.6905 0.4610 0.2921 0.9735 0.4905 0.3410 2.4151 2.2177 $$\beta\hspace{1.4em}$$0.650 0.7256 0.4994 0.3238 1.0793 0.5256 0.3694 2.7330 2.4562 $$\qquad$$ 0.750 0.8133 0.5908 0.4051 1.3504 0.6133 0.4408 3.5981 3.0623 $$\qquad$$ 0.800 0.8688 0.6462 0.4581 1.5268 0.6688 0.4862 4.1955 3.4520 $$\qquad$$ 0.600 0.7478 0.6191 0.4369 1.4564 0.5478 0.4791 2.8295 3.1420 $$\theta\hspace{1.4em}$$0.650 0.7567 0.5793 0.3967 1.3225 0.5567 0.4393 2.9714 2.9275 $$\qquad$$ 0.750 0.7762 0.5078 0.3292 1.0973 0.5762 0.3678 3.2770 2.5612 $$\qquad$$ 0.800 0.7868 0.4755 0.3009 1.0029 0.5868 0.3355 3.4415 2.4056 $$\qquad$$ 0.250 0.7083 0.5031 0.2818 0.9395 0.5083 0.3631 2.6024 2.5396 $$\gamma\hspace{1.4em}$$0.275 0.7350 0.5212 0.3197 1.0655 0.5350 0.3812 2.8384 2.6313 $$\qquad$$ 0.325 0.8027 0.5670 0.4067 1.3556 0.6027 0.4270 3.4597 2.8512 $$\qquad$$ 0.350 0.8455 0.5960 0.4577 1.5255 0.6455 0.4560 3.8712 2.9836 $$\qquad$$ 0.250 0.8480 0.5977 0.4494 1.7977 0.6480 0.4577 3.7645 2.9464 $$\delta\hspace{1.3em}$$0.275 0.8005 0.5655 0.3995 1.4528 0.6005 0.4255 3.3922 2.8273 $$\qquad$$ 0.325 0.7404 0.5249 0.3302 1.0160 0.5404 0.3849 2.9141 2.6603 $$\qquad$$ 0.350 0.7204 0.5114 0.3049 0.8711 0.5204 0.3714 2.7527 2.5996 $$\qquad$$ 0.700 0.7960 0.5625 0.4260 1.4201 0.5960 0.4225 3.3672 2.8103 $$k\hspace{1.3em}$$ 0.750 0.7798 0.5516 0.3908 1.3028 0.5798 0.4116 3.2326 2.7694 $$\qquad$$ 0.850 0.7544 0.5344 0.3354 1.1181 0.5544 0.3944 3.0255 2.7045 $$\qquad$$ 0.900 0.7442 0.5275 0.3132 1.0441 0.5442 0.3875 2.9443 2.6784 Parameter $$p^{C}_{1\infty}$$ $$p^{C}_{2\infty}$$ $$A^{C}_{\infty}$$ $$G_{\infty}^{C}$$ $$D_{1\infty}^{C}$$ $$D_{2\infty}^{C}$$ $$J_{1}^{C}$$ $$J_{2}^{C}$$ Benchmark 0.7662 0.5423 0.3610 1.2034 0.5662 0.4023 3.1204 2.7346 $$\qquad$$ 0.600 0.6905 0.4610 0.2921 0.9735 0.4905 0.3410 2.4151 2.2177 $$\beta\hspace{1.4em}$$0.650 0.7256 0.4994 0.3238 1.0793 0.5256 0.3694 2.7330 2.4562 $$\qquad$$ 0.750 0.8133 0.5908 0.4051 1.3504 0.6133 0.4408 3.5981 3.0623 $$\qquad$$ 0.800 0.8688 0.6462 0.4581 1.5268 0.6688 0.4862 4.1955 3.4520 $$\qquad$$ 0.600 0.7478 0.6191 0.4369 1.4564 0.5478 0.4791 2.8295 3.1420 $$\theta\hspace{1.4em}$$0.650 0.7567 0.5793 0.3967 1.3225 0.5567 0.4393 2.9714 2.9275 $$\qquad$$ 0.750 0.7762 0.5078 0.3292 1.0973 0.5762 0.3678 3.2770 2.5612 $$\qquad$$ 0.800 0.7868 0.4755 0.3009 1.0029 0.5868 0.3355 3.4415 2.4056 $$\qquad$$ 0.250 0.7083 0.5031 0.2818 0.9395 0.5083 0.3631 2.6024 2.5396 $$\gamma\hspace{1.4em}$$0.275 0.7350 0.5212 0.3197 1.0655 0.5350 0.3812 2.8384 2.6313 $$\qquad$$ 0.325 0.8027 0.5670 0.4067 1.3556 0.6027 0.4270 3.4597 2.8512 $$\qquad$$ 0.350 0.8455 0.5960 0.4577 1.5255 0.6455 0.4560 3.8712 2.9836 $$\qquad$$ 0.250 0.8480 0.5977 0.4494 1.7977 0.6480 0.4577 3.7645 2.9464 $$\delta\hspace{1.3em}$$0.275 0.8005 0.5655 0.3995 1.4528 0.6005 0.4255 3.3922 2.8273 $$\qquad$$ 0.325 0.7404 0.5249 0.3302 1.0160 0.5404 0.3849 2.9141 2.6603 $$\qquad$$ 0.350 0.7204 0.5114 0.3049 0.8711 0.5204 0.3714 2.7527 2.5996 $$\qquad$$ 0.700 0.7960 0.5625 0.4260 1.4201 0.5960 0.4225 3.3672 2.8103 $$k\hspace{1.3em}$$ 0.750 0.7798 0.5516 0.3908 1.3028 0.5798 0.4116 3.2326 2.7694 $$\qquad$$ 0.850 0.7544 0.5344 0.3354 1.1181 0.5544 0.3944 3.0255 2.7045 $$\qquad$$ 0.900 0.7442 0.5275 0.3132 1.0441 0.5442 0.3875 2.9443 2.6784 Table 2. Variations in the alliance solution Parameter $$p^{C}_{1\infty}$$ $$p^{C}_{2\infty}$$ $$A^{C}_{\infty}$$ $$G_{\infty}^{C}$$ $$D_{1\infty}^{C}$$ $$D_{2\infty}^{C}$$ $$J_{1}^{C}$$ $$J_{2}^{C}$$ Benchmark 0.7662 0.5423 0.3610 1.2034 0.5662 0.4023 3.1204 2.7346 $$\qquad$$ 0.600 0.6905 0.4610 0.2921 0.9735 0.4905 0.3410 2.4151 2.2177 $$\beta\hspace{1.4em}$$0.650 0.7256 0.4994 0.3238 1.0793 0.5256 0.3694 2.7330 2.4562 $$\qquad$$ 0.750 0.8133 0.5908 0.4051 1.3504 0.6133 0.4408 3.5981 3.0623 $$\qquad$$ 0.800 0.8688 0.6462 0.4581 1.5268 0.6688 0.4862 4.1955 3.4520 $$\qquad$$ 0.600 0.7478 0.6191 0.4369 1.4564 0.5478 0.4791 2.8295 3.1420 $$\theta\hspace{1.4em}$$0.650 0.7567 0.5793 0.3967 1.3225 0.5567 0.4393 2.9714 2.9275 $$\qquad$$ 0.750 0.7762 0.5078 0.3292 1.0973 0.5762 0.3678 3.2770 2.5612 $$\qquad$$ 0.800 0.7868 0.4755 0.3009 1.0029 0.5868 0.3355 3.4415 2.4056 $$\qquad$$ 0.250 0.7083 0.5031 0.2818 0.9395 0.5083 0.3631 2.6024 2.5396 $$\gamma\hspace{1.4em}$$0.275 0.7350 0.5212 0.3197 1.0655 0.5350 0.3812 2.8384 2.6313 $$\qquad$$ 0.325 0.8027 0.5670 0.4067 1.3556 0.6027 0.4270 3.4597 2.8512 $$\qquad$$ 0.350 0.8455 0.5960 0.4577 1.5255 0.6455 0.4560 3.8712 2.9836 $$\qquad$$ 0.250 0.8480 0.5977 0.4494 1.7977 0.6480 0.4577 3.7645 2.9464 $$\delta\hspace{1.3em}$$0.275 0.8005 0.5655 0.3995 1.4528 0.6005 0.4255 3.3922 2.8273 $$\qquad$$ 0.325 0.7404 0.5249 0.3302 1.0160 0.5404 0.3849 2.9141 2.6603 $$\qquad$$ 0.350 0.7204 0.5114 0.3049 0.8711 0.5204 0.3714 2.7527 2.5996 $$\qquad$$ 0.700 0.7960 0.5625 0.4260 1.4201 0.5960 0.4225 3.3672 2.8103 $$k\hspace{1.3em}$$ 0.750 0.7798 0.5516 0.3908 1.3028 0.5798 0.4116 3.2326 2.7694 $$\qquad$$ 0.850 0.7544 0.5344 0.3354 1.1181 0.5544 0.3944 3.0255 2.7045 $$\qquad$$ 0.900 0.7442 0.5275 0.3132 1.0441 0.5442 0.3875 2.9443 2.6784 Parameter $$p^{C}_{1\infty}$$ $$p^{C}_{2\infty}$$ $$A^{C}_{\infty}$$ $$G_{\infty}^{C}$$ $$D_{1\infty}^{C}$$ $$D_{2\infty}^{C}$$ $$J_{1}^{C}$$ $$J_{2}^{C}$$ Benchmark 0.7662 0.5423 0.3610 1.2034 0.5662 0.4023 3.1204 2.7346 $$\qquad$$ 0.600 0.6905 0.4610 0.2921 0.9735 0.4905 0.3410 2.4151 2.2177 $$\beta\hspace{1.4em}$$0.650 0.7256 0.4994 0.3238 1.0793 0.5256 0.3694 2.7330 2.4562 $$\qquad$$ 0.750 0.8133 0.5908 0.4051 1.3504 0.6133 0.4408 3.5981 3.0623 $$\qquad$$ 0.800 0.8688 0.6462 0.4581 1.5268 0.6688 0.4862 4.1955 3.4520 $$\qquad$$ 0.600 0.7478 0.6191 0.4369 1.4564 0.5478 0.4791 2.8295 3.1420 $$\theta\hspace{1.4em}$$0.650 0.7567 0.5793 0.3967 1.3225 0.5567 0.4393 2.9714 2.9275 $$\qquad$$ 0.750 0.7762 0.5078 0.3292 1.0973 0.5762 0.3678 3.2770 2.5612 $$\qquad$$ 0.800 0.7868 0.4755 0.3009 1.0029 0.5868 0.3355 3.4415 2.4056 $$\qquad$$ 0.250 0.7083 0.5031 0.2818 0.9395 0.5083 0.3631 2.6024 2.5396 $$\gamma\hspace{1.4em}$$0.275 0.7350 0.5212 0.3197 1.0655 0.5350 0.3812 2.8384 2.6313 $$\qquad$$ 0.325 0.8027 0.5670 0.4067 1.3556 0.6027 0.4270 3.4597 2.8512 $$\qquad$$ 0.350 0.8455 0.5960 0.4577 1.5255 0.6455 0.4560 3.8712 2.9836 $$\qquad$$ 0.250 0.8480 0.5977 0.4494 1.7977 0.6480 0.4577 3.7645 2.9464 $$\delta\hspace{1.3em}$$0.275 0.8005 0.5655 0.3995 1.4528 0.6005 0.4255 3.3922 2.8273 $$\qquad$$ 0.325 0.7404 0.5249 0.3302 1.0160 0.5404 0.3849 2.9141 2.6603 $$\qquad$$ 0.350 0.7204 0.5114 0.3049 0.8711 0.5204 0.3714 2.7527 2.5996 $$\qquad$$ 0.700 0.7960 0.5625 0.4260 1.4201 0.5960 0.4225 3.3672 2.8103 $$k\hspace{1.3em}$$ 0.750 0.7798 0.5516 0.3908 1.3028 0.5798 0.4116 3.2326 2.7694 $$\qquad$$ 0.850 0.7544 0.5344 0.3354 1.1181 0.5544 0.3944 3.0255 2.7045 $$\qquad$$ 0.900 0.7442 0.5275 0.3132 1.0441 0.5442 0.3875 2.9443 2.6784 An increase in market share $$\theta$$ for the dominant retailer leads to higher retail price, demand and profit of the dominant retailer, while lower wholesale and retail prices, advertising effort, demand and profits in the weak channel under each scenario. With the increase of the dominant retailer’s market share, it is reasonable that the dominant retailer seizes the opportunity to set a high retail price to grab a high profit. However, it is notable from Corollary 1 that, the steady-state wholesale price for the weak retailer is always higher than the exogenous wholesale price for the dominant retailer. The higher profit margin from the weak retailer impels the manufacturer to lower the wholesale price for the weak retailer to maintain a proper demand from the weak channel. The lower profit from the weak channel makes the manufacturer decrease his investment in advertising, which consequently leads to a reduction of the goodwill. Accordingly, the weak retailer determines a lower retail price to stimulate her market demand. Even so, the demand from the weak channel ultimately shrinks due to the increase of the dominant retailer’s market share and the decrease of the goodwill. As a result, the profits of the manufacturer and the weak retailer decrease. Similarly, the prices, demand and profit of the weak retail channel under alliance case are decreasing with the increase of dominant retailer’s market share $$\theta$$. We can also find that the wholesale and retail prices, advertising efforts, demands and profits of all members increase with the marginal contribution of goodwill on demand $$\gamma$$ under both scenarios. As the goodwill effectiveness on market demand strengthens, the manufacturer (or the alliance) has more incentive to invest in advertisement to increase the total market potential size. The manufacturer charges a higher wholesale price for the weak retailer who will transfer the higher price to customers in the no-alliance case, or the alliance charges a higher retail price. On the basis of increased market demand, the dominant retailer also decides a higher retail price to increase profit. Eventually, all members obtain more profits as the goodwill effectiveness on market demand increases. It is intuitive that the decrease of decay coefficient $$\delta$$ or advertising cost coefficient $$k$$ has a similar effect with the increase of the goodwill effectiveness coefficient $$\gamma$$. Next, we’ll examine the effect of the exogenous wholesale price $$w_1$$ on channel members’ profits. Figure 1 characterizes the changes of profits of all players with respect to $$w_1$$ under the no-alliance and alliance cases, respectively. Specifically, Fig. 1(a) shows that in the no-alliance scenario, with the increase of $$w_1$$, the profit of the manufacturer or the weak channel first increases and then decreases, the dominant retailer’s profit decreases monotonically, and the weak retailer’s profit increases monotonically. A higher wholesale price to the dominant retailer induces the manufacturer to invest more in advertising effort which brings higher goodwill, as shown in Fig. 2(a). Correspondingly, the manufacturer asks a higher wholesale price to the weak retailer, and both retailers decide on higher retail prices. For the dominant retailer, the negative effect of the increased retail price on demand outweighs the positive effect of the increased advertisement on demand through goodwill. Thus the demand for the dominant retailer decreases. However, for the weak retailer, the opposite result occurs, i.e., the positive effect of the increased advertisement on demand outweighs the negative effect of the increased retail price on demand, thus the demand for the weak retailer increases with $$w_1$$ in no-alliance case, which are shown in Fig. 2(b). With the increase of $$w_{1}$$, the dominant retailer’s net margin per unit decreases, which further decreases the profit. For the weak retailer, although it is verified from Proposition 1 that $$\frac{\partial w^N_{2}}{\partial w_1}=\frac{\partial p^N_{2}}{\partial w_1}$$ which implies the net margin per unit does not directly depend on the exogenous wholesale price $$w_1$$, it can be calculated that the net margin per unit positively depends on the goodwill which increases with $$w_1$$. Overall, the weak retailer’s net margin per unit increases, which further increases the profit. Fig. 1. View largeDownload slide The profits as $$w_1$$ changes under both scenarios. Fig. 1. View largeDownload slide The profits as $$w_1$$ changes under both scenarios. Fig. 2. View largeDownload slide The impacts of $$w_1$$ on advertising efforts, goodwill levels and demands under both scenarios. Fig. 2. View largeDownload slide The impacts of $$w_1$$ on advertising efforts, goodwill levels and demands under both scenarios. For the manufacturer, when the wholesale price of dominant retailer is relatively small, the positive effects of the increased wholesale prices of both retailers and the weak retailer’s demand on profit occupy the leading position relative to the negative effects of the decreased dominant retailer’s demand and the increased advertising cost. In this case, the profit of the manufacturer is increasing with the dominant retailer’s wholesale price. The situation will be inverse if the wholesale price of dominant retailer is relatively large. Therefore, under the no-alliance scenario, the profit of the manufacturer first increases and then decreases with the increase of $$w_1$$. The change of the weak channel’s profit can be similarly observed and explained as above. As shown in Fig. 1(b) that the changes of profits of the team and the dominant retailer under the alliance scenario have the same trends as those under no-alliance scenario. The difference is that the increased advertisement in the alliance case also expands the demand of the dominant retailer, as depicted in Fig. 2(b). However, we can see that the net margin per unit for the dominant retailer shrinks with the increasing of exogenous wholesale price $$w_1$$, which ultimately reduces her profit. For the alliance, when the wholesale price of dominant retailer is relatively small, the positive effects of the increased demands of both retailers, the weak channel’s retail price and the dominant retailer’s wholesale price on profit take a leading place over the increased advertising cost. In this case, the profit of the alliance is increasing with the dominant retailer’s wholesale price. An opposite result arises when the wholesale price exceeds a certain level. Specifically, when the wholesale price of the dominant retailer is large enough, the negative effect of the greatly increased advertising cost plays a critical role in determining the profitability of the alliance, thus the corresponding profit of the alliance decreases. Next we will compare the difference of steady-state retail prices between the dominant retailer and the weak one. Figure 3 describes the comparisons of steady-state retail prices between the two retailers under no-alliance scenario when the competition intensity between retailers is mild (Fig. 3(a) with $$\beta=0.3$$) or intense (Fig. 3(b) with $$\beta=0.7$$) respectively. Fig. 3. View largeDownload slide Retail price comparisons via $$\gamma$$ and $$\theta$$ with different competition intensities under the no-alliance scenario. Fig. 3. View largeDownload slide Retail price comparisons via $$\gamma$$ and $$\theta$$ with different competition intensities under the no-alliance scenario. We observe from Fig. 3 that, the price comparisons between retailers are mainly affected by the market share of the dominant retailer $$\theta$$. When the dominant retailer holds a large market share, the dominant retailer’s retail price is higher than that of the weak retailer; while for a relatively small market share of the dominant retailer, the dominant retailer will charge a lower price than the weak retailer. This happens because, on the one hand, when the dominant retailer has a relatively low market share, Tab. 1 shows that, the manufacturer invests more in advertisement. In this situation, he tends to push up the wholesale price for the weak retailer to cover his advertising cost, which makes the retail price of the weak retailer increase. On the other hand, the dominant retailer tries to boost consumer demand through setting a low retail price. As a result, with a relatively low $$\theta$$, the retail price of the weak retailer is higher than that of the dominant retailer. In contrast, when facing a relatively large market share $$\theta$$, the dominant retailer sets a higher price to grab super profit from the market, while the manufacturer invests less in advertisement and charges the weak retailer a low wholesale price to maintain a proper demand from the weak channel which then leads to a low retail price of the weak retailer. Therefore, with a relatively large $$\theta$$, the retail price set by the dominant retailer is higher than that of the weak retailer. Notice that, as the value of substitutability parameter $$\beta$$ becomes bigger, the intensity of competition between the two retailers increases. Comparing Fig. 3(a) with Fig. 3(b), we see that, when the market competition becomes more intense, the region where the dominant retailer’s retail price is lower than that of the weak one becomes larger. That is to say, the fiercer the competition is, the more inclined the dominant retailer is to push down her retail price. This offers a possible explanation for why in real economic life the product prices of the dominant retailers like WalMart are always lower, compared with other smaller retailers. We also make a comparison of the retail prices between the dominant retailer and the weak channel under the alliance scenario. The results show that the retail price of the dominant retailer is always higher than that of the weak channel. The explanations are as follows. On one hand, the alliance between the manufacturer and the weak retailer increases the advertising effort and then expands the market potential, which allows the alliance and the dominant retailer in the supply chain to charge high retail prices. On the other hand, the alliance eliminates the double marginalization effect in the weak channel, which greatly reduces the retail price of the weak channel. Overall, the dominant retailer’s price is higher than that of the weak channel. 6.2 Strategies and profits comparison across scenarios In this subsection, we will compare the equilibrium prices, advertising efforts, goodwill levels at the steady state as well as the profits under the no-alliance and alliance scenarios. First, we make a comparison of the steady-state advertising efforts and goodwill levels between the two scenarios. The comparison results are given in the following proposition. Proposition 5 At the steady state, both the equilibrium advertising effort and goodwill level in the alliance scenario are higher than those in the no-alliance case, i.e., $$A_{\infty}^C>A_{\infty}^N$$ and $$G_{\infty}^{C}>G_{\infty}^{N}$$. It is straightforward that the alliance between the manufacturer and the weak retailer strengthens the channel power, which avoids the problem of double marginalization. In this situation, the channel is more efficient as compared to the no-alliance case, and has the ability to invest more in advertising effort to improve the product goodwill among customers. As a result, the goodwill level in the case of alliance is higher than that in the no-alliance case. Next, we compare the pricing strategies and profits of the dominant retailer and the weak retail channel under both settings by numerical analysis. The results are respectively shown in Fig. 4. Fig. 4. View largeDownload slide The retail price and profit comparisons across both scenarios via $$\gamma$$ and $$\theta$$. Fig. 4. View largeDownload slide The retail price and profit comparisons across both scenarios via $$\gamma$$ and $$\theta$$. Generally speaking, there are two factors affecting the pricing policy under the alliance scenario. For one thing, there is a factor promoting the price. Specifically, the alliance enlarges the market potential through the increased advertising effort, which brings more incentives for the alliance and the dominant retailer to raise their retail prices. For another, there is a factor lowering the price. The alliance greatly reduces the retail price of the weak channel by removing the double marginalization effect, which brings a fierce competition to the dominant retailer, and compels the dominant retailer to cut price. Under different system parameters, each factor may take a leading place in determining the results of price comparisons. Figure 4(a) demonstrates the retail price comparisons between the alliance scenario and no-alliance scenario. We observe from Fig. 4(a) that the price comparisons between scenarios are mainly affected by the goodwill effectiveness on market demand, and are not sensitive to the market share of the dominant retailer when the impact of goodwill on demand is relatively small. With a relatively small effect of goodwill on demand, the fierce competition under alliance scenario turns to be a dominant factor which lowers prices of both members. However, when the effect of goodwill on demand is large, the greatly enlarged market potential through the augmented advertising effort in the alliance scenario takes a leading place, which enables the dominant retailer to charge a higher price in the alliance case than that in no-alliance case. When the impact of goodwill on market demand is relatively large, e.g., $$0.4<\gamma<0.8$$, the weak channel’s retail price comparison between scenarios also depends on the market share of the dominant retailer. For a given impact of goodwill on market demand, the weak channel charges a higher price in the alliance case than that in no-alliance case when the market share of the weak channel is relatively large, while the opposite result occurs for a relatively small market share of the weak channel. This is because a large market share of the weak channel further strengthens the motivation of increasing the advertising effort in the alliance scenario, thus brings a higher price. Additionally, for a given market share of the weak channel, a large impact of goodwill on demand implies a higher retail price of the alliance. Similar to the price comparison for the dominant retailer, this is also because the greatly enlarged market potential through the augmented advertising effort in the alliance scenario takes a leading place, which induces the alliance to set a higher price to pursue more profits. However, when both the impact of goodwill on demand and the market share of the weak channel are relatively small, the function of the augmented advertising effort in the alliance scenario to expand demand weakens, so the weak channel will enlarge the market demand through setting a lower price, meanwhile, the absence of double marginalization makes this possible under the alliance case. Moreover, the dominant retailer is more likely to increase the retail price under alliance scenario than the weak channel does. This is due to a relatively large market share taken by the dominant retailer. The augmented advertisement under alliance scenario has a greater effect on the retail pricing of the dominant retailer than on that of the weak channel. Figure 4(b) characterizes the profit comparisons for the dominant retailer and the weak retailer channel under both scenarios. It is shown that the dominant retailer’s profit comparison between scenarios is mainly affected by the goodwill effectiveness on market demand, but is not sensitive to the market share of the dominant retailer. Specifically, the dominant retailer’s profit in no-alliance case is higher than that in alliance case when the effect of goodwill is relatively small. However, when this impact strengthens, e.g., $$\gamma>0.4$$, the dominant retailer can take advantage from the alliance scenario, and get more profit than that under no-alliance scenario. The reason is as follows. For a large goodwill’s effect, the alliance invests more in advertisement which leads to better product goodwill, and the market potential of the dominant retailer expands under alliance scenario. Furthermore, as shown in Fig. 4(a), we can see that in the alliance scenario the price of the dominant retailer is higher when goodwill’s effect is large. Therefore, although facing with a fiercer competition, the dominant retailer gets benefit from the additional advertisement offered by the weak channel through a positive spillover effect in the alliance situation. Such spillover is large enough to help the dominant retailer take advantage from the alliance and increase her profit compared to the no-alliance scenario. However, the dominant retailer’s retail price is lower under alliance scenario when the marginal contribution of goodwill on market demand is weak. In this case, the fiercer competition coming from alliance plays a key role in affecting the profitability of the dominant retailer, thus the dominant retailer is better off in no-alliance case. We can see that, the weak channel always gets more profit in the alliance case than that in no-alliance case for any market share of the dominant retailer and goodwill effectiveness. This is the result of the eliminating of the double marginalization effect and strengthening of the weak channel’s power when competing with the dominant retailer under the alliance scenario. The alliance investing more in advertisement improves the goodwill of product, which combing with the enhanced power of the weak channel ultimately expands the demand, thus resulting in more profit. It should be pointed out that, in our article, we do not take the cost of alliance into consideration when we say that the alliance is always beneficial for the manufacturer and the weak retailer. Actually, in practice, it is only when the gains induced by the alliance are higher than the cost to form an alliance that the manufacturer-weak retailer alliance can be possible proceeded. Additionally, it should be noted that, when the manufacturer and the weak retailer have the incentive to form an alliance, the profit allocation problem arises, which can be settled through the negotiation between the two allied members. Because of the space limitations and the focus of our research, these problems would not be discussed in detail in this study. 6.3 Extensions and robustness The main results of the previous sections rely on the assumption that the wholesale price $$w_{1}$$ is exogenous and cannot be strategically chosen. To illustrate that our results are qualitatively robust, we relax this assumption. To do so, we consider that the wholesale price denoted by $$w_{1}^{N}$$ under no-alliance case is determined through the negotiation between the dominant retailer and the manufacturer, and the wholesale price $$w_{1}^{C}$$ under alliance case is determined through the negotiation between the dominant retailer and the weak channel. The corresponding problem becomes a bargaining game, which is analysed by many researches such as Feng & Lu (2013) and Osborne & Rubinstein (1990). Under no-alliance case, the bargaining problem is a pair ($$J^{N}$$, $$D^{N}$$) where $$J^{N}\in\mathfrak{R}^{2}$$ is the set of all possible profit pairs ($$J_{1}^{N}$$, $$J_{M}^{N}$$) when an agreement is reached. The disagreement point $$D^{N}=(D_{1}^{N}$$, $$D_{M}^{N})\in\mathfrak{R}^{2}$$ represents the profits the two negotiating parties get when they fail to reach an agreement. In our case, we assume that $$D^{N}=(0, 0)$$. Meanwhile, consistent with Balasubramanian & Bhardwaj (2004) and Liu et al. (2015), we assume that the dominant retailer has the same bargaining power as the manufacturer over the wholesale price $$w_{1}^{N}$$. We apply the Nash bargaining model of Nash (1950) to derive the bargaining outcome. Hence, the wholesale price $$w_{1}^{N}$$ agreed upon by the dominant retailer and the manufacturer is determined by solving the following Nash bargaining problem: \begin{align} \max\limits_{w_1^{N}} \Pi^{N}&=\left(J_{1}^{N}(w_{1})-D_{1}^{N}(w_{1})\right)\left(J_{M}^{N}(w_{1})-D_{M}^{N}(w_{1})\right)\notag\\ &=J_{1}^{N}(w_{1})J_{M}^{N}(w_{1}), \end{align} (46) where the term $$\Pi^{N}$$ is the so-called Nash Product. Similarly, under the alliance case, we assume that the profits the two negotiating parties get when they fail to reach an agreement are zero, i.e., $$D_{1}^{C}=D_{2}^{C}=0$$. Then, the wholesale price $$w_{1}^{C}$$ agreed upon by the dominant retailer and the weak channel is determined by solving the following Nash bargaining problem: \begin{align} \max\limits_{w_1^{C}} \Pi^{C}&=\left(J_{1}^{C}(w_{1})-D_{1}^{C}(w_{1})\right)\left(J_{2}^{C}(w_{1})-D_{2}^{C}(w_{1})\right)\notag\\ &=J_{1}^{C}(w_{1})J_{2}^{C}(w_{1}). \end{align} (47) Accordingly, substituting the solutions of above bargaining problems $$w_1^{\ast N}$$ into (26)–(28) and $$w_1^{\ast C}$$ into (44) and (45), respectively, we obtain the profits of channel members under both no-alliance and alliance scenarios in the case of negotiated pricing. Due to the complexity of $$\Pi^{N}$$ in (46) and $$\Pi^{C}$$ in (47), we cannot obtain the wholesale prices of dominant retailers and the profits of channel members under both scenarios analytically. Therefore, we resort to the numerical examples to analyse the comparative results of profits across both scenarios when dominant retailer’s wholesale price is determined through bargaining. Set the base system values as follows: $$\alpha=1$$, $$\beta=0.5$$, $$\theta=0.7$$, $$k=1$$, $$\delta=0.3$$, $$G_0=1$$ and $$\rho=0.1$$. Figure 5 indicates that when endogenizing the dominant retailer’s wholesale price, the main results remain. Specifically, the alliance between the manufacturer and the weak retailer is always beneficial for the weak channel. Moreover, this alliance behaviour hurts the dominant retailer when the margin contribution of goodwill on market demand is relatively low, while the dominant retailer can benefit from the alliance when the effect of goodwill on the demand is high enough. Fig. 5. View largeDownload slide The profit comparisons across both scenarios with $$\gamma$$ when $$w_{1}$$ are determined through bargaining. Fig. 5. View largeDownload slide The profit comparisons across both scenarios with $$\gamma$$ when $$w_{1}$$ are determined through bargaining. 7. Implications and conclusion In this article, we propose a dynamic supply chain model involving one manufacturer and two differentiated retailers. The two retailers are asymmetric in influencing their wholesale prices. The manufacturer is unable to directly control the dominant retailer’s wholesale price which is assumed to be exogenous, but he has the ability to offer a take-it-or-leave-it wholesale price to the weak retailer. Two retailers compete for market demand by setting their individual retail prices. The manufacturer invests in advertisement to improve the product’s goodwill, and then expands the market potential size. The paper considers two different scenarios: The no-alliance scenario in which all the channel members make decisions to maximize their individual profits, and the alliance scenario where the manufacturer allies with the weak retailer to achieve their joint profit maximization. The focus of this article is primarily on whether or not the alliance between the manufacturer and the weak retailer will benefit the dominant retailer. We derive the equilibrium pricing and advertising strategies and the discounted profits of the channel members under the no-alliance and alliance scenarios, respectively. Then we compare the strategies and profits between the two scenarios, which develops several interesting results with important managerial implications for marketing practice as follows. The alliance between the manufacturer and the weak retailer always brings about higher profit for the weak channel. The dominant retailer can also benefit from this alliance when the effect of goodwill on market demand is relatively high. In this case, the manufacturer-weak retailer alliance results in a Pareto improvement for all the channel members. This finding brings some meaningful guidance for channel members that, when distributing the products whose brand image has a great influence on consumers’ purchasing decisions, the dominant retailer will welcome the manufacturer–weak retailer alliance, which causes the manufacturer and weak retailer to ally with each other without scruple. Additionally, when endogenizing the dominant retailer’s wholesale price through the bargaining between the dominant retailer and the manufacturer (or weak channel), these results do not change qualitatively. Comparing the no-alliance case with the alliance case, we find that, with a great marginal contribution of goodwill on market demand, both the dominant retailer and the weak channel will charge higher prices under alliance scenario. However, both the dominant retailer and the weak channel will set lower prices under alliance scenario when the contribution of goodwill on demand is relatively low. Under the no-alliance scenario, the dominant retailer, when facing a large market share, will adopt a higher retail price than that of the weak retailer, while an opposite situation occurs when she has a relatively small market share. In contrast, under the alliance scenario, due to the elimination of the double marginalization effect in the weak channel, the dominant retailer’s price is always higher than that of the weak channel. Both the steady-state advertising effort and goodwill level in the alliance scenario are higher than those in the no-alliance case. That is to say, the manufacturer–weak retailer alliance can induce the advertising effort to a higher level and is better for the brand image of the product. This article has a few limitations despite the importance of the findings and the managerial insights, and some valuable extensions should be noted. We test the findings considering downstream retailers’ competition. Future research can study competition at the upstream level or at both levels. In addition, we consider that only the manufacturer invests in advertising which contributes to the build-up of goodwill. It will be more realistic to relax this assumption and investigate not only the manufacturer but also the retailers can invest in advertising. Funding National Natural Foundation of China No. 61473204 and the Humanity and Social Science Youth Foundation of Ministry of Education of China No. 14YJCZH204. Appendix Proof of Proposition 1. The first-order conditions for the maximization problems of the two retailers in (9) and (10) lead to \begin{align} p^N_1=\frac{1}{2}\big(\theta(\alpha+\gamma G) +\beta p_2+w_1\big), p^N_2=\frac{1}{2}\big((1-\theta)(\alpha+\gamma G\big)+\beta p_1+w_2\big). \notag \end{align} Solving the two equations above leads to \begin{align} p^N_1&= \frac{(2\theta+\beta-\beta\theta)(\alpha+\gamma G) +\beta w_2+2w_1}{4-\beta^2},\\ \end{align} (A1) \begin{align} p^N_2&= \frac{(2-2\theta+\beta\theta)(\alpha+\gamma G) +\beta w_1+2w_2}{4-\beta^2}. \end{align} (A2) Incorporate the reaction functions of both retailers (A1) and (A2) into equation (8). Maximizing the right-hand side of equation (8) by virtue of the first-order conditions, we can obtain the wholesale price for the weak retailer and the advertising effort as follows: \begin{align} w^{N}_{2}&=\frac{(2-2\theta+\beta\theta)(\alpha+\gamma G)+2\beta w_1}{2(2-\beta)^2}, \\ \end{align} (A3) \begin{align} A^{N}&=\frac{1}{k}\frac{\partial V_M^N}{\partial G}. \end{align} (A4) Substituting (A3) back into equations (A1) and (A2) yields the feedback pricing strategies of both retailers shown in (14) and (15). Substituting (14) and (15) into (2) and (3) results in the market demands for both retailers given in (16) and (17). Substituting (A4) into (8) yields \begin{align} \rho V_M^N&= \frac{1-\beta^2}{2-\beta^2} w_1^2+\frac{(\beta-\beta\theta+\theta)(\alpha+\gamma G)}{2-\beta^2}w_1+\frac{(\beta\theta+2-2\theta)^2(\alpha+\gamma G)^2}{4(2-\beta^2)(4-\beta^2)}\notag\\[4pt] &\quad{} +\frac{1}{2k}\left(\frac{\partial V_M^N} {\partial G}\right)^2-\delta G\frac{\partial V_M^N}{\partial G}. \end{align} (A5) Conjecture the following quadratic value function of the manufacturer: \begin{align} V_M^N&= \frac{L_1}{2} G^2+L_2G+L_3, \end{align} (A6) where $$L_1$$, $$L_2$$ and $$L_3$$ are constants to be determined. Substituting this value function as well as its partial derivative with respect to $$G$$ into (A5) and equating the corresponding coefficients, we obtain the following three algebraic Ricatti equations \begin{align} \frac{\rho L_1}{2}&= \frac{\gamma^2(2-2\theta+\beta\theta)^2} {4(4-\beta^2)(2-\beta^2)}+\frac{L_1^2}{2k}-\delta L_1, \\ \end{align} (A7) \begin{align} \rho L_2&= \frac{\alpha\gamma(2-2\theta+\beta\theta)^2} {2(4-\beta^2)(2-\beta^2)}+\frac{(\beta+\theta-\beta\theta)\gamma w_1}{2-\beta^2}+\frac{L_1L_2}{k}-\delta L_2,\\ \end{align} (A8) \begin{align} \rho L_3&= \frac{(\beta+\theta-\beta\theta)\alpha w_1 -(1-\beta^2)w_1^2}{2-\beta^2}+\frac{(2-2\theta+\beta\theta)^2 \alpha^2}{4(2-\beta^2)(4-\beta^2)}+\frac{L_2^2}{2k}. \end{align} (A9) Solving equation (A7), we can obtain \begin{align} L_1&= \frac{k(\rho+2\delta)\lambda_1\pm\lambda_2}{2\lambda_1}.\notag \end{align} The larger root would imply that the goodwill evolution does not converge to the steady-state value, so it is not further considered. As a result, we have \begin{align} L_1&= \frac{k(\rho+2\delta)\lambda_1-\lambda_2}{2\lambda_1},\notag\\ L_2&= \frac{2k\lambda_1\gamma w_1(\beta+\theta-\beta\theta)} {(2-\beta^2)(k\rho\lambda_1+\lambda_2)} +\frac{k\alpha\gamma\lambda_4^2} {\lambda_1(k\rho\lambda_1+\lambda_2)},\notag\\ L_3&= \frac{(\beta+\theta-\beta\theta)\alpha w_1 -(1-\beta^2)w_1^2}{(2-\beta^2) \rho}+\frac{\alpha^2\lambda_4^2} {4\rho\lambda_1^2}\notag\\ &\quad{} +\frac{1}{2k\rho}\left(\frac{2k\lambda_1\gamma w_1(\beta+\theta-\beta\theta)} {(2-\beta^2)(k\rho\lambda_1+\lambda_2)} +\frac{k\alpha\gamma\lambda_4^2} {\lambda_1(k\rho\lambda_1+\lambda_2)}\right)^2.\notag \end{align} Substituting $$L_1$$, $$L_2$$ and $$L_3$$ above into (A6), we can get $$V_M^N$$. Then from (A4), the advertising effort is solved as shown in (13). By the same method as solving the value function of the manufacturer, we can obtain the value functions of both retailers. □ Proof of Proposition 2 Substituting the feedback advertising strategies (13) into state equation (1) yields the following differential equation \begin{align} \dot{G}^N(t)&= \frac{k\rho\lambda_1-\lambda_2}{2k\lambda_1}G^N+ \frac{2\gamma\lambda_1(\beta-\beta\theta+\theta)w_1} {(2-\beta^2)(k\rho\lambda_1+\lambda_2)} +\frac{\alpha\gamma\lambda_4^2} {\lambda_1(k\rho\lambda_1+\lambda_2)}. \end{align} (A10) By solving equation (A10), we obtain \begin{align} G^N(t)&= (G_{0}-G^N_{\infty})e^{\frac{k\rho\lambda_1-\lambda_2}{2k\lambda_1}t} +G^N_{\infty}, \end{align} (A11) where \begin{align} G^N_{\infty}= \frac{\gamma(2(4-\beta^2)(\beta+\theta-\beta\theta)w_1 +\alpha\lambda_4^2)}{ 2k\delta\lambda_1^2(\rho+\delta)-\gamma^2\lambda_4^2}. \end{align} (A12) Substituting (A11) into (12)–(15), we can obtain the time paths of the pricing and advertising strategies (20)–(23), where \begin{align} w^N_{2\infty} = &\frac{k\alpha\delta\lambda_1^2\lambda_4(\rho+\delta) +\big(\lambda_4\gamma^2(4-\beta^2)(\beta+\theta-\beta\theta) +2k\beta\delta\lambda_1^2(\rho+\delta) -\beta\gamma^2\lambda_4^2\big)w_1} {(2-\beta^2)(2k\delta\lambda_1^2(\rho+\delta)-\gamma^2\lambda_4^2)}, \notag\\ p^N_{1\infty} = &\frac{-k\alpha\lambda_3\delta(\rho+\delta)(2-\beta^2) +\big(2k\lambda_1^2\delta(\rho+\delta) -\gamma^2\lambda_4^2-\lambda_3\gamma^2 (\beta+\theta-\beta\theta)\big)w_1}{(2-\beta^2) (2k\delta\lambda_1^2(\rho+\delta)-\gamma^2\lambda_4^2)}, \notag\\ p^N_{2\infty} = &\frac{2k\lambda_4\alpha\delta(\rho + \delta)(2 - \beta^2) (3 - \beta^2) + \big(\!2\lambda_4\gamma^2(3 - \beta^2) (\beta + \theta - \beta\theta) + 2k\lambda_1^2\beta\delta(\rho + \delta) - \beta\gamma^2\lambda_4^2\!\big)w_1} {(2-\beta^2)(2k\delta\lambda_1^2(\rho+\delta)-\gamma^2\lambda_4^2)},\notag\\[0.5ex] A^N_{\infty} = &\frac{\delta\gamma\big(2(4-\beta^2) (\beta+\theta-\beta\theta)w_1+\alpha\lambda_4^2\big)}{ 2k\delta\lambda_1^2(\rho+\delta)-\gamma^2\lambda_4^2}.\notag\\[-40pt] \nonumber \end{align} □ Proof of Corollary 1. Rewrite $$w^N_{2\infty}$$ as follows: \begin{align} w^N_{2\infty}= Aw_{1}+B, \end{align} (A13) where $$A=\frac{\lambda_4\gamma^2(4-\beta^2)(\beta+\theta-\beta\theta) +2k\beta\delta\lambda_1^2(\rho+\delta) -\beta\gamma^2\lambda_4^2}{(2-\beta^2)(2k\delta\lambda_1^2(\rho+\delta)-\gamma^2\lambda_4^2)}$$, and $$B=\frac{k\alpha\delta\lambda_1^2\lambda_4(\rho+\delta)}{(2-\beta^2)(2k\delta\lambda_1^2(\rho+\delta)-\gamma^2\lambda_4^2)}$$. To ensure positive steady-state goodwill level and globally asymptotically stable equilibria under the no-alliance scenario, the following inequality should hold: \begin{align} 2k\delta(2-\beta^2)(4-\beta^2)(\rho+\delta)>\gamma^2(2-2\theta+\beta\theta)^2. \end{align} (A14) Considering (A14), and substituting $$\lambda_1$$, $$\lambda_4$$ into $$A$$, we can get \begin{align} A&=1 + \frac{\begin{array}{c}\gamma^2(\beta\theta - 2\theta + 2)[(4 - \beta^2)(\beta - \beta\theta + \theta) + (\beta\theta - 2\theta + 2)(2 - \beta^2 - \beta)]\\ + 2k\delta(2 - \beta^2)(4 - \beta^2) (\rho + \delta)(\beta + \beta^2 - 2)\end{array}} {(2-\beta^2)[2k\delta(2-\beta^2)(4-\beta^2)(\rho+\delta)-\gamma^2(\beta\theta-2\theta+2)^2]}\notag\\ &>1 + \frac{\begin{array}{c}\gamma^2(\beta\theta - 2\theta+2)[(4 - \beta^2)(\beta - \beta\theta + \theta) + (\beta\theta - 2\theta+2)(2 - \beta^2 - \beta)]\\ + \gamma^2(\beta\theta - 2\theta + 2)^2(\beta + \beta^2 - 2)\end{array}} {(2-\beta^2)[2k\delta(2-\beta^2)(4-\beta^2)(\rho+\delta)-\gamma^2(\beta\theta-2\theta+2)^2]}\notag\\ &=1+\frac{\gamma^2(\beta\theta-2\theta+2)(4-\beta^2)(\beta-\beta\theta+\theta)} {(2-\beta^2)[2k\delta(2-\beta^2)(4-\beta^2)(\rho+\delta)-\gamma^2(\beta\theta-2\theta+2)^2]}\notag\\ &>1. \end{align} (A15) In addition, considering (A14), it can be easily verified that $$B>0$$. Therefore, we have $$w^N_{2\infty}>w_{1}$$. The Corollary (1) holds. □ Proof of Proposition 3. Given the competitor’s retail price, solving the optimization problems (31) and (30) yields the optimal response for the dominant retailer and the alliance, i.e., \begin{align} &p^C_1=\frac{\theta(\alpha+\gamma G)+\beta p_2+w_1}{2},\\ \end{align} (A16) \begin{align} &p^C_2=\frac{(1-\theta)(\alpha+\gamma G)+\beta( p_1+w_1)}{2}, \\ \end{align} (A17) \begin{align} &A^C=\frac{1}{k}\frac{\partial V_2^C}{\partial G}. \end{align} (A18) Combining the two equations (A16) and (A17), we obtain the feedback pricing strategies of both members shown in (32) and (33). Furthermore, substituting (32) and (33) into (2) and (3) yields the corresponding demands characterized in (35) and (36). Substituting (A18) back into (30), we can obtain \begin{align} \rho V_2^C&= \frac{\gamma^2\lambda_8^2 G^2}{(4-\beta^2)^2}+\frac{1}{2k}\left(\frac{\partial V_T^C} {\partial G}\right)^2-\delta G\frac{\partial V_T^C}{\partial G}\notag\\ &\quad{} +(\frac{\gamma(\beta^3-\beta^3 \theta+8\beta-8\beta\theta+8\theta+\beta^4\theta)} {(4-\beta^2)^2}+\frac{2\alpha\gamma\lambda_4^2}{(4-\beta^2)^2})G+\frac{\alpha^2\lambda_4^2}{(4-\beta^2)^2} \nonumber\\ &\quad{} + \frac{(\beta^2-1)(\beta^2+8)w_1^2}{(4-\beta^2)^2}+\frac{(\beta^3-\beta^3 \theta+8\beta-8\beta\theta+8\theta+\beta^4\theta)\alpha w_1}{(4-\beta^2)^2}. \end{align} (A19) Conjecture the following form for the value function of the allied team: $$V_2^C=\frac{M_1}{2} G^2+M_2G+M_3$$, where $$M_1, M_2$$ and $$M_3$$ are constants to be determined. Substituting above value function as well as its partial derivatives into (A19) and equating according coefficients, we can obtain the following three algebraic Ricatti equations \begin{align} \frac{\rho M_1}{2}&= \frac{\gamma^2\lambda_4^2}{(4-\beta^2)^2}+\frac{M_1^2}{2k}-\delta M_1,\\ \end{align} (A20) \begin{align} \rho M_2&= \frac{(\beta^3-\beta^3 \theta+8\beta-8\beta\theta+8\theta+\beta^4\theta)\gamma w_1}{(4-\beta^2)^2}+\frac{2\alpha\gamma\lambda_4^2}{(4-\beta^2)^2}+\frac{M_1M_2}{k}-\delta M_2,\\ \end{align} (A21) \begin{align} \rho M_3&= \frac{(\beta^2-1)(\beta^2+8)w_1^2}{(4-\beta^2)^2}+ \frac{\alpha^2\lambda_4^2}{(4-\beta^2)^2}+\frac{M^2_2}{2k}\notag\\[4pt] &\quad{} +\frac{(\beta^3-\beta^3 \theta+8\beta-8\beta\theta+8\theta+\beta^4\theta)\alpha w_1}{(4-\beta^2)^2}. \end{align} (A22) Solving equation (A20), we can obtain $$M_1=\frac{k(\rho+2\delta)(4-\beta^2)\pm\lambda_7}{2(4-\beta^2)}$$. However, the larger root would imply that goodwill dynamics do not converge to steady-state values, and is not further considered. As a result, we have $$M_1=\frac{k(\rho+2\delta)(4-\beta^2)-\lambda_7}{2(4-\beta^2)}$$, $$M_2=\frac{2k\gamma(\lambda_5 w_1+2\alpha\lambda_4^2)}{(4-\beta^2)(k(4-\beta^2)\rho+\lambda_7)}$$, $$M_3=\frac{\lambda_5\alpha w_1-(1-\beta^2)(\beta^2+8)w_1^2}{(4-\beta^2)^2\rho}+\frac{\alpha^2\lambda_4^2}{(4-\beta^2)^2\rho}+\frac{M^2_2}{2k\rho}$$. Thus, we have $$A^C=\frac{2\gamma(\lambda_5w_1+2\alpha\lambda_4^2)} {(4-\beta^2)(k\rho(4-\beta^2)+\lambda_7)} +\frac{k(\rho+2\delta)(4-\beta^2)-\lambda_7} {2k(4-\beta^2)}G$$. □ Proof of Proposition 4. This proof is similar to that of Proposition 1. The steady-state goodwill and strategies are given as follows: \begin{align} G^C_{\infty} & = \frac{\gamma(\lambda_5w_1+2\alpha\lambda_4^2)} {k\delta(\rho+\delta)(4-\beta^2)^2-2\gamma^2\lambda_4^2}, \notag\\[0.5ex] p^C_{1\infty} & = \frac{-k\alpha\delta\lambda_6(\rho + \delta)(4 - \beta^2)^2 + (k\delta(\rho + \delta)(2 + \beta^2)(4 - \beta^2)^2 - \gamma^2\lambda_5\lambda_6 - 2(2 + \beta^2) \gamma^2\lambda_4^2)w_1} {(4-\beta^2)(k\delta(\rho+\delta)(4-\beta^2)^2 -2\gamma^2\lambda_4^2)},\notag\\[0.5ex] p^C_{2\infty} & = \frac{k\alpha\delta\lambda_4(\rho+\delta)(4-\beta^2)^2 +(\gamma^2\lambda_4\lambda_5 +3k\beta\delta(\rho+\delta)(4-\beta^2)^2 -6\beta\gamma^2\lambda_4^2)w_1}{(4-\beta^2)(k\delta(\rho+\delta) (4-\beta^2)^2-2\gamma^2\lambda_4^2)},\notag\\ A^C_{\infty} & = \frac{\delta\gamma (\lambda_5w_1+2\alpha\lambda_4^2)} {k\delta(\rho+\delta)(4-\beta^2)^2-2\gamma^2\lambda_4^2}.\nonumber\\[-40pt] \nonumber \end{align} □ Proof of Proposition 5. By calculation, we can get \begin{align} &G_{\infty}^{C}-G_{\infty}^{N}\notag\\ &\quad{} = \frac{\gamma(2k\delta\lambda_4\beta^3(1-\beta^2) (\rho+\delta)+\gamma^2\lambda_4^2(\theta(8-\beta^2(4+\beta^2)) +\beta(1-\theta)(8-5\beta^2))w_1)}{(k\delta(\rho+\delta) (4-\beta^2)^2-2\gamma^2(2-2\theta+\beta\theta)^2)(2k\delta(4-\beta^2) (2-\beta^2)(\rho+\delta)-\gamma^2\lambda_4^2)}\notag\\ &\qquad{} +\frac{\alpha \gamma\lambda_4^2(k\delta(\rho+\delta)(4-\beta^2) (4-3\beta^2))}{(k\delta(\rho+\delta)(4-\beta^2)^2 -2\gamma^2(2-2\theta+\beta\theta)^2)(2k\delta(4-\beta^2) (2-\beta^2)(\rho+\delta)-\gamma^2\lambda_4^2)}>0.\notag \end{align} The proof with respect to $$A_{\infty}^{C}>A_{\infty}^{N}$$ is similar to that of $$G_{\infty}^{C}>G_{\infty}^{N}$$. □ References Amrouche, N., Martín-Herrán, G. & Zaccour, G. ( 2015). Pricing and advertising of private and national brands in a dynamic marketing channel. J. Optim. Theory Appl. , 137, 465– 483. 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IMA Journal of Management Mathematics – Oxford University Press

**Published: ** Aug 29, 2017

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