# Analysis on the Water Supply Chain Model under Revenue-Sharing Contract considering Marketing Effort, Water Purity

Authors: Lijia Huang; Deshan Tang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

A two-tier water supply chain including a manufacturer and a retailer under revenue-sharing contract is constructed. And the contribution of the model is that marketing effort and water purity has been considered. First, four models including the centralized model (model B) and decentralized models (models BM, I, and II) are established and analyzed. Second, the Stackelberg game model is used to discuss the pricing strategy of water supply chain members in centralized and decentralized scenarios. The comparison results show that revenue-sharing contract is beneficial to improve the level of product greening, the profit of supply chain members, and the overall profit of the water supply chain compared with model BM. However, it leads to the decrease of retailers’ green marketing efforts and the wholesale price of water. In addition, revenue-sharing contract through bargaining makes bigger influence than revenue-sharing contract. Marketing can stimulate the increase of the green product’s market demand on one hand, and on the other hand, it generates the amount of marketing cost. In this study, the profit is that marketing produces cannot offset the cost that it brings. Thus, it will be important to take some measures to make up the loss that marketing generated.


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## Body

## 1. Introduction

As the blood of the earth, the source of life, and the cradle of civilization, water has important resource function, ecological function, and economic function. Due to the scarcity of water resources and the discharge of a larger number of domestic sewage and industrial wastewater, the manufacturing process of drinking water is critical. With the modern day acceleration of industrialization and urbanization, the demand for water increased rapidly. Moreover, water pollution incidents that happened occasionally also had promoted the demand for water, especially clean water. Nowadays, plenty of enterprises such as NONGFU SPRING, China’s drinking water industry benchmarking enterprise, is focusing on research and development and promotion of mineral water.Recently, low-carbon life is a hot topic, and a lot of companies are devoting themselves to green products. Also, consumers are shifting their preferences to more environment friendly brands. The green brand image and green brand value can be transformed into the customer’s brand loyalty when customers have the concept of green consumption [1]. Therefore, retailers end up going green marketing [2].Collaboration between partners is beneficial to bring both environment and economy improvement to water supply partners [3]. To motivate water supply chain member’s cooperation, it is a common practice in the world that leading enterprises propose a series of green manufacturing requirements, including energy conservation, environmental protection, energy efficiency, and environmental emission, in the procurement process [4]. Only the water suppliers who meet the requirements can become the suppliers of brand enterprises. In addition, in the financial sector, green finance is used to guide funds to green industries to control and reduce pollution-related investment, so as to promote green production in the whole society.In this study, two-tier water supply chain models under different Stackelberg scenarios are introduced. Revenue-sharing contract is studied by considering both water marketing effort and water purity. Thus, several questions should be addressed. (1) How do water supply chain members’ decisions and profitability are impacted when considering water purity and marketing efforts under revenue-sharing contracts? (2) What are the optimal decision strategies for water supply chain members under different scenarios? Therefore, our study has the following contributes.(1)
Water supply chain models under both centralized scenario and decentralized scenario are constructed. Decision-strategy for supply chain members is analyzed and compared in different models.(2)
Under decentralized scenario, the imparts of revenue-sharing contract on different members and the whole supply chain have been analyzedThere are 4 sections concluded in this study. The 2nd section introduces the previous researchers. In Section 3, models of water supply chain considering water purity and marketing effort are studied. Section 4 presents numerical analysis. We summarize the research and come up with management insights in Section 5.


## 2. Literature Review

Water supply chain has been studied academically for decades. Since about 2004, supply chains have been widely used in water management. As an instance, research [5] applied supply chain in water resources allocation and dispatch. In 2005, [6] built a hybrid agent-based model to estimate residential water demand by simulating the residential water supply chain. Research [7] demonstrated that the related supply chain management theory and methodology can be applied in water resources allocation and dispatching. The bullwhip effect was studied in the water resource supply chain information management by using stochastic control theory [8]. The optimization of water supply chain management was studied in literature [9], and they came up with a systematic engineering method to reduce the total water demand. In 2007, the study [10] established an optimal multiobjective model of water resource allocation. In 2008, the studies [11, 12] researched joint pricing of water on the basis of cooperative and noncooperative game theories in a two-tier water supply chain. Study [13] developed a spatial water distribution plan that can save cost and time. In 2010, Kogan and Tapiero [14] discussed the economic benefits of two-stage water supply chain by constructing a zero-sum stochastic differential game model. Elala et al. [15] talked about the safety of water supply chain and provided practical suggestions for communities, enterprises, and local authorities to realize water supply chain risk management. Zhou et al. [16] also discussed the water supply chain risk management experiences based on the case study in Sweden. Portable water supply chain from water collecting to water distribution by taking advantage of the theory of product life cycle chain has been studied in research [17]. In [18], water supply chain transformation has been discussed by constructing a multiperiod mixed integer program model, and they found that comprehensive integration of water supply chain networks and coordination of water production will bring economic and environmental benefits if they are considered within the planned time frame. For water supply-demand with uncertainty, the study [19] utilized the artificial neural network and stationary chain to predict water demand. In 2015, by updating the artificial fish swarm algorithm and the supply chain management theory, the problem of water resources scheduling was solved [20]. Water Data Warehouse (a software) was designed to share the water supply demand information in the whole water supply chain [21]. In [3], taking the reliability of water supply into consideration, they constructed a multiperiod-mixed integer linear programming model to achieve the water supply reliability maximization, and it was found that water demand has a huge effect on the reliability of water supply chain.Water supply chain coordination is a branch of supply chain management. There are also many research studies on water supply chain coordination. For example, on the basis of communication and coordination and the principle-agent theory, water resource supply chain contract was constructed [5, 7]. Research [22] designed a multilevel, Pareto optimization decision-making model to deal with the multilevel cooperative decision-making problem of the South-to-North Water Transfer Project. In 2006, Dai et al. [23] established a decision-making model of supply chain agent under the contract effect by applying group gaming decision theory on the basis of water resources supply chain. A study [24] observed water inventory coordination by using inventory control theory and proposed transfer payment coordination strategy of water inventory management under VMI theory. In order to derive water resource, Kondili and Kaldellis [25] developed a decision support system (DSS) to coordinate the conflicts between customers’ demand in water supply chain. In 2009, a research [14] analyzed the optimal, respectively, in the minimum commitment and flexibility contract based on a two-stage water supply chain. In 2012, a study [26] demonstrated the optimization modeling approach for the pricing and coordinating schemes of SNWD-ER project. Chen [27] considered the water suppliers’ profitability and distributers’ profitability under the perspective of social responsibility and economic benefit. In [28–33], the pricing strategies of a competitive two-tier water supply chain including one supplier and two distributers under two-part pricing contract and wholesale price contract was studied, and the findings reveal many practical management insights. In 2018, a study [3] investigated the decision-making strategies of water supply chain members by considering who played the leader and follower, and water resources management insights were provided. In 2019, water supply chain equilibrium and coordination were studied in [31]; in addition, fairness factors are considered to compare the supply chain performances, social welfare, and consumer surplus under different equilibrium strategies and coordination strategies. In 2020, literature [34] revealed that revenue and cost sharing contract could effectively coordinate and improve the water supply chain performance (Figure 1 and Table 1).Figure 1
Two-tier water supply chain.Table 1
Parameters and their meanings.
ParameterMeaningaTotal market potential anda>0qActual demand of the marketkPrice elasticity of demand andk>0eThe retailer’s green marketing effort to promote the green productγThe impact of the marketing effort on demandηMarketing cost coefficientηe2/2The total cost of the green marketing effortcThe manufacturer’s cost of purifying waterwThe wholesale price of productspThe retail price of green productsθWater purity and0<θ≤1αThe sensitivity of consumer to water purity improvementβThe cost rate of research and development (R&D)βθ2The R&D investment of manufactureφRevenue-sharing ratio


## 3. Model Description

### 3.1. Assumptions and Demand Function

(1)
We assume that in the centralized decision-making system, the manufacturers determine the green marketing efforte, while in the decentralized system, the retailers decide the green marketing effort. Based on [35–37], the total green marketing effort cost is ηe2/2, where η stands for the marketing cost coefficient.(2)
In this study, the demand function is of retail price, green marketing effort, and demand. Retailers’ investment is in green marketing in order to promote the green market, and the greater the marketing efforts, the greater the effect on demand.(3)
Manufacturers’ green R&D mainly means that green manufacturing can use fewer resources to produce the same product and improve resource utilization rate. Water purityθ0<θ≤1 improvement indicates that manufactures should increase research and development costs, and we assume that all the costs were born by the manufacturers. Referring to [38], the R&D investment of manufacture is βθ2, where β means the cost rate of research and development.(4)
Parameterα represents the sensitivity of consumer to water purity improvement. The bigger the manufacturers invest in R&D, the more αθ impact to demand.(5)
The parameters and their meanings used in this study are given in Table1(6)
The manufacture purified ground water and the retailer buys it at wholesale price and sells it to the market at retail price (Figure1). We consider that the actual demand of the market q is a linear function depending on retail price, green marketing effort, and water purity. The demand function is shown in the following equation.(1)qp,e,θ=a−kp+γe+αθ.


### 3.2. Stackelberg Equilibrium and Profitability Analysis

#### 3.2.1. Profit Analysis under Centralized Decision Making (Model B)

Centralized decision-making refers to a commodity produced and sold by the manufacturer, that is to say, a manufacturer integrates production and sales (vertical integration); more specifically, there is only one interest subject in the whole supply chain except customers, which is called integrated manufacturer. In the centralized model, we suppose that a central decision-maker is responsible for the retail price, the green marketing effort, and water purity.The water supply chain’s profit function is shown as(2)ΠSCB=p−ca−kp+γe+αθ−βθ2−ηe22.After solving the first-order derivative ofΠSCB with respect to p, θ, and e, we get(3)∂ΠSCB∂p=a−2kp+γe+αθ+ck,(4)∂ΠSCB∂θ=αp−cα−2βθ,(5)∂ΠSCB∂e=γp−cγ−ηe.Afterwards, we set all the equations (3)–(5), all equal to zero to obtain equilibrium values of model B. The equilibrium values of the equation set are as follows:(6)pSC∗B=−2ckβη−cα2η+2γ2β+2aβηα2η+2γ2β−4kηβ,(7)θSC∗B=kcηα−ηaαα2η+2γ2β−4kηβ,(8)eSC∗B=2kβcγ−2βaγα2η+2γ2β−4kηβ.Substitute (6)–(8) into functions (1) and (2) to get the optimal actual market demand qSC∗B and the profit of supply chain ΠSC∗B.(9)qSC∗B=2k2ηβc−2kηβaα2η+2γ2β−4kηβ,ΠSC∗B=2βηack−βηa2−βηc2k2α2η+2γ2β−4kηβ.Lemma 1.
If it satisfies the conditions thatα2−4kβ<0 and α2η+2γ2β−4βηk<0, the Hessian matrix of the profit function of the centralized supply chain is negative definite matrix. There exists a maximum value in the point p∗B,θ∗B,e∗B, and the profit function of ΠSCB is strictly concave in the retail price p, the green marketing effort e, and water purity θ.Proof (Appendix).Proposition 1.
The Stackelberg equilibrium of the centralized supply chain is(10)pSC∗B=−2ckβη−cα2η+2γ2β+2aβηα2η+2γ2β−4kηβ,θSC∗B=kcηα−ηaαα2η+2γ2β−4kηβ,eSC∗B=2kβcγ−2βaγα2η+2γ2β−4kηβ,qSC∗B=2k2ηβc−2kηβaα2η+2γ2β−4kηβ,ΠSC∗B=2βηack−βηa2−βηc2k2α2η+2γ2β−4kηβ.To ensure that all the equilibrium values are meaningful, the necessary condition iskc−a<0 and γ2−2kη<0. Moreover, when β∗B≥ηαkc−a−α/γ2−2kη, it guaranteed that 0<θSC∗B≤1.Proof (Appendix).


#### 3.2.2. Profit Analysis under Decentralized Decision Making (Model BM)

The decentralized decision-making game model means that the manufacturer and the retailer are two independent interest subjects to maximize their profit functions. In this study, we assume the Stackelberg game relationship between manufacturer and seller; manufacturer is the leader and decides the wholesale price and the greening degree of a product; the retailer is a follower who determines the retail price of the product and the level of the green marketing effort according to the manufacturer’s decision. In this case, the profit function of the manufacturer and retailer is expressed as follows, respectively:(11)ΠRBM=p−wa−kp+γe+αθ−ηe22,(12)ΠMBM=w−ca−kp+γe+αθ−βθ2.According to the backward algorithm, in the first stage, based on the anticipation of retailer’s retail price and marketing effort, the manufacture sets the wholesale price and purity of drinking water. In the second stage, concerning the wholesale price and marketing effort, the retailer decides the retail price and marketing effort. According to the backward algorithm, the retail price and marketing effort are obtained through the first derivative of the retailer’s profit function with respect to the retail price and marketing effort. Afterwards, substitute them into the profit function of the manufacturer to get its first derivative with respect to the wholesale price and water purity.The first-derivative result ofΠRB with respect to p and e is shown as(13)∂ΠRBM∂p=a−kp−kp−w+αθ+γe=0,∂ΠRBM∂e=γp−w−ηe=0.Next, we can obtain the optimal retail price and marketing effort of the retailer as follows:(14)pRBM=−−wγ2+ηa+ηkw+αηθγ2−2ηk,(15)eRBM=−γa−kw+αθγ2−2ηk.PutpRBM and eRBM into the profit function of manufacturer ΠMBM. Then, we get(16)ΠMBM=−βγ2θ2+2βηθ2k−αηθkw+αηcθk+ηk2w2−ηakw+ηackγ2−2ηk.After taking the first derivative ofΠMB with respect to w and θ, we get(17)∂ΠMBM∂w=−ηak+ηck2−2ηk2w+αηθkγ2−2ηk=0,(18)∂ΠMBM∂θ=−2βθγ2−4βηθk−αηck+αηkwγ2−2ηk=0.By combining the equations (17) and (18), we derive that(19)wM∗BM=ηcα2k+2βcγ2k+2βaγ2−4βηck2−4βηakkηα2+4βγ2−8βηk,(20)θM∗BM=αηck−αηaηα2+4βγ2−8βηk.Submit (19) and (20) into the equations (14) and (15), and the equilibrium value of pRBM and eRBM is shown as(21)pR∗BM=ηcα2k+2βcγ2k+2βar2−2βηck2−6βηakkηα2+4βγ2−8βηk.(22)eR∗BM=−2βγa−ckηα2+4βγ2−8βηk.Next, we submit (19)–(22) into equations (1), (11), and (12) and obtain the optimal actual market demand qM∗BM and the maximized profits of the retailer ΠR∗BM and manufacturer ΠM∗BM.Lemma 2.
Under Lemma1 and Proposition 1, the profit function ΠMBM and ΠRBM is strictly concave in w, θ and p, e, respectively.Proof (Appendix).Proposition 2.
In this model, the manufacturer’s optimal whole pricewM∗BM, water purity θM∗BM, and maximized profit ΠM∗BM; the retailer’s retail price pR∗BM, marketing effort eR∗BM, and maximized profit ΠR∗BM; the actual market demand qM∗BM, and the profit of the supply chain ΠSC∗BM are obtained as(23)wM∗BM=ηcα2k+2βcγ2k+2βaγ2−4βηck2−4βηakkηα2+4βγ2−8βηk,θM∗BM=αηck−αηaηα2+4βγ2−8βηk,pR∗BM=ηcα2k+2βcγ2k+2βar2−2βηck2−6βηakkηα2+4βγ2−8βηk,eR∗BM=−2βγa−ckηα2+4βγ2−8βηk,ΠM∗BM=2βηack−βηa2−βηc2k2ηα2+4βγ2−8βηk,ΠR∗BM=−2β2ηa−ck2γ2−2ηkηα2+4βγ2−8βηk2,ΠSC∗BM=−βηa−ck2α2η+6βγ2−12βηkηα2+4βγ2−8βηk2,qM∗BM=2βηkck−aηα2+4βγ2−8βηk.Based on Lemma1 and Proposition 1, the equilibrium values make sense. Moreover, to ensure that 0<θM∗BM≤1, we get β∗BM≥ηαkc−a−α/4γ2−2kη.Proof (Appendix).


#### 3.2.3. Revenue-Sharing Contract (Model I)

A revenue-sharing model (model I) is where manufacturers play a leading role, and a revenue-sharing contract is through the bargaining model (model II). In these models above, the proportion of income sharing on the premise of maximizing their own interests is decided by manufacturers and the bargaining between manufacturers and retailers, respectively.In the manufacturer-led revenue-sharing model, the revenue-sharing ratioφ0<φ<1 is determined by manufacturers, which means that the proportion of retailer gains from the final revenue is φ, and the manufacture shares the remaining percentage 1−φ. Therefore, the profit function of the manufacturer and retailer is obtained as follows:(24)ΠMI=w−ca−kp+γe+αθ−βθ2+1−φp−wa−kp+γe+αθ,(25)ΠRI=φp−wa−kp+γe+αθ−ηe22.Similar to the decision-making process of model BM, we get(26)wM∗Iφ=α2ηck+2βγ2φck+2βγ2φa−4βηck2−4βηφakkα2η−4βηφk−4βηk+4βγ2φ,(27)θM∗Iφ=αηck−aα2η−4βηφk−4βηk+4βγ2φ,(28)pR∗Iφ=−2βηak+2βγ2φa+α2ηck−2βηck2+2βγ2φck−4βηφakkα2η−4βηφk−4βηk+4βγ2φ,(29)eR∗Iφ=−2βγφa−ckα2η−4βηφk−4βηk+4βγ2φ.Substituting equations (26)–(29) into equations (1), (24), and (25), we get the optimal market demand qM∗BM and the optimal profit of the manufacturer ΠM∗I and retailer ΠR∗I to get(30)qM∗Iφ=−2βηka−ckα2η−4βηφk−4βηk+4βγ2φ,(31)ΠM∗Iφ=−βηa2−2βηack+βηc2k2α2η−4βηφk−4βηk+4βγ2φ,(32)ΠR∗Iφ=−2β2ηφa−ck2γ2φ−2ηkα2η−4βηφk−4βηk+4βγ2φ2,(33)ΠSC∗Iφ=−βηa−ck2α2η+2βγ2φ2+4βγ2φ−8βηφk−4βηkα2η−4βηφk−4βηk+4βγ2φ2.Lemma 3.
Whenγ2<ηk, α2−4βk<0, and ck−a<0, the profit function ΠMI is strictly concave in w and θ, and ΠRI is strictly concave in p and e. Moreover, the equations (26)–(33) are meaningful. And to ensure 0<θM∗Iφ≤1, we get β∗I≥αα2γ2−4βηk2α+a−ck/4k2α2η−8βηk+4βγ2.Proof (Appendix).
Since the first and second derivatives ofΠR∗I with respect to φ are(34)∂ΠR∗I∂ϕ=4β2η2a−ck2α2ηk−4βηk2+4βηφk2−α2γ2φα2η−4βηφk−4βηk+4βγ2φ3,∂2ΠR∗I∂ϕ2<0.
The optimalφ∗I=4βηk2−α2ηk/4βηk2−α2γ2 is next put it into equations (26)–(33) to get:Proposition 3.
In model I, the manufacturer’s optimal whole pricewM∗I, water purity θM∗I, maximized profit ΠM∗I, the retailer’s retail price pR∗I, marketing effort eR∗I, maximized profit ΠR∗I, the actual market demand qM∗I, and the profit of the supply chain ΠSC∗I are obtained as(35)wM∗I=cα2γ2+2βcγ2k+2βaγ2−4βηck2−4βηakα2γ2+4βγ2k−8βηk2,θM∗I=−αα2γ2−4βηk2a−ckα2−4βkα2γ2+4βγ2k−8βηk2,pR∗I=cα4γ2−4ηcα2βk2−4ηaα2βk−8cβ2γ2k2−8aβ2γ2k+8ηcβ2k3+24ηaβ2k2α2−4βkα2γ2+4βγ2k−8βηk2,eR∗I=−2βγka−ckα2γ2+4βγ2k−8βηk2,ΠM∗I=−βα2γ2−4βηk2a−ck2α2−4βkα2γ2+4βγ2k−8βηk2,ΠR∗I=2β2ηk2a−ck2α2−4βkα2γ2+4βγ2k−8βηk2,ΠSC∗I=−βα2γ2−6βηk2a−ck2α2−4βkα2γ2+4βγ2k−8βηk2,qM∗I=−2βkα2γ2−4βηk2a−ckα2−4βkα2γ2+4βγ2k−8βηk2.Proof (Appendix).


#### 3.2.4. Revenue-Sharing Contract through Bargaining (Model II)

Revenue-sharing contract through bargaining means that the revenue-sharing ratio is determined by the manufacturers and retailers through bargaining. To study the impact of bargaining on revenue-sharing contract, we adopt the vertical Nash game process. The decision-making sequence is as follows: in the first stage, the manufacture and the retailer bargain on the revenue-sharing ratio,φ, which means the retailer takes possession of φ proportion of the total revenue while the manufacture shares the remaining. In the second stage, the manufacture decides the wholesale price and water purity of drinking water by anticipating the retailer’s retail price and the marketing effort. Finally, the retailer decides the retail price and marketing effort taking the manufacturer’s whole price and water purity into consideration.We construct the bargaining process between the supply chain members by using Nash bargain game. By substituting equations (31) and (32) into the function that ΠIIφ=ΠM∗IϕΠR∗Iφ, we can obtain(36)ΠII=ΠM∗IΠR∗I=2β3η2φa−ck4γ2φ−2ηkα2η+4βγ2φ−4βηk−4βηφk3.Lemma 4.
(1)
Whenφ∗II≤3−1, there does not exists a solution to the Nash bargaining problem(2)
The equilibrium valueφ to this Nash bargaining problem is given as(37)φ∗II=ηα2γ2+σ−8βηk2+4βγ2k4βγ2γ2−ηk,whereσ=α4γ4−8α2βηγ2k2+64β2η2k4−96β2ηγ2k3+48β2γ4k2According to Cauchy inequality, we getφ∗II≤3−1ηk/γ2Proposition 4.
The optimal strategies of wholesale price, water purity, retail price, marketing effort, product demand, the manufacturer’s profit, retailer’s profit, and the water supply chain’s profit are displayed as(38)wM∗II=γ2a+γ2ck−2ηakα2γ2+σ+2γ2−ηkα2γ2ck−8βηk2a+4βγ4ka−ck2kγ2−ηk2α2γ2+σ−8βηk2,θM∗II=−γ2αa−ck2α2γ2+σ−8βηk2,pR∗II=γ2a+γ2ck−2ηakα2γ2+σ−4βηk2+2γ2−ηkα2γ2ck−4βηk2a2kγ2−ηk2α2γ2+σ−8βηk2,eR∗II=−γa−ckα2γ2+σ−8βηk2+4βγ2k2γ2−ηk2α2γ2+σ−8βηk2,ΠM∗II=−βγ2a−ck22α2γ2+σ−8βηk2,ΠR∗II=−ηγ2a−ck2α2γ2+σ−4βγ4kα2γ2+σ−8βηk2+4βγ4k8γ2−ηk22α2γ2+σ−8βηk22,ΠSC∗II=−γ2a−ck22γ2α2η+4βγ4−8βηkα2γ2+σ+8βγ2γ2−ηkα2γ2−4βηk28γ2−ηk22α2γ2+σ−8βηk22,qM∗II=−2βγ2ka−ck2α2γ2+σ−8βηk2.


### 3.3. Comparative Analysis

Proposition 5.
The water purity, retail price, wholesale price, and the retailer’s marketing effort under the three models satisfy the following relationships which are derived through algebraic comparison:(1)
θSC∗B>θM∗II>θM∗I>θM∗BM(2)
eSC∗B>eR∗BM>eR∗I>eR∗II(3)
wM∗BM>wM∗I>wM∗II(4)
qSC∗B>qM∗II>qM∗I>qM∗BM
Proposition5 distributes the differences among the strategies in terms of the product’s water purity, the retailer’s green marketing effort, the product’s wholesale price and retail price, and product demand of the four models in this study.It is natural that the product’s water purity for model B (centralized model) is highest since it is the fully coordinated scenario and lowest under model BM (decentralized model without revenue-sharing contract). Moreover, product’s water purity for revenue-sharing contract through bargaining is higher than that under revenue-sharing contract (i.e.,θM∗II>θM∗I). Therefore, revenue-sharing contract is beneficial to the improvement of product’s water purity and cooperation can contribute to it.As for the comparison of product’s water purity, naturally, it is highest in the fully coordinated scenario, while lowest under revenue-sharing contract situation (i.e.,eR∗BM>eR∗I>eR∗II). According to the revenue-sharing contract, the retailer returns a percentage of its sales to the manufacturer and maintains profit through decreasing the marketing cost. Therefore, revenue-sharing contract leads to the decrease of the marketing effort and revenue-sharing contract though bargaining makes it worse.The wholesale price is lowest in revenue-sharing contract through bargaining, followed by under revenue-sharing contract and highest in decentralized condition without any communication between supply chain members (i.e.,wM∗BM>wM∗I>wM∗II). It is probably that the manufacturer sells products to the retailer at a wholesale price below the products’ manufacturing cost. As a result, effective communication gives rise to the decline of the wholesale price.Proof (Appendix).Proposition 6.
The water supply chain’s profit under the four models meets the relationship:(1)
ΠM∗II>ΠM∗I>ΠM∗BM and ΠR∗I>ΠR∗BM(2)
ΠSC∗B>ΠSC∗II>ΠSC∗I>ΠSC∗BM
Proposition6 shows the profit differences of the manufacturer and the retailer among the models. The manufacturer benefits from the revenue-sharing contract due to it as the Stackelberg leader, while the retail as the follower fails to benefit from the contract; only if φ=1, the retailer earns profit equal to that under the decentralized model without revenue-sharing contract. The comparison of green supply chain’s profit indicates that there exists the double marginalization effect. It is very likely to be caused by the goals of supply chain members that perhaps conflict with each other. As a result, revenue-sharing contract are conducive to the boosting of the profit; furthermore, the profit has been increased even more under the revenue-sharing contract (i.e., ΠSC∗II>ΠSC∗I>ΠSC∗BM). Therefore, the goal of constructing water supply chain is to realize integrated management, promote cooperation between upstream and downstream enterprises, and finally achieve higher overall performance.Proof (Appendix).Proposition 7.
In models B, BM, I, and II, we have(1)
The increase of consumer sensitivity to water purity improvementα induces the increase of the optimal wholesale price, water purity, retail price, marketing effort, manufacturer’s profit, and retailer’s profit(2)
The increase of the cost rate of R&Dβ leads to the decrease of the optimal wholesale price, water purity, retail price, marketing effort, manufacturer’s profit, and retailer’s profitProof (Appendix).


## 3.1. Assumptions and Demand Function

(1)
We assume that in the centralized decision-making system, the manufacturers determine the green marketing efforte, while in the decentralized system, the retailers decide the green marketing effort. Based on [35–37], the total green marketing effort cost is ηe2/2, where η stands for the marketing cost coefficient.(2)
In this study, the demand function is of retail price, green marketing effort, and demand. Retailers’ investment is in green marketing in order to promote the green market, and the greater the marketing efforts, the greater the effect on demand.(3)
Manufacturers’ green R&D mainly means that green manufacturing can use fewer resources to produce the same product and improve resource utilization rate. Water purityθ0<θ≤1 improvement indicates that manufactures should increase research and development costs, and we assume that all the costs were born by the manufacturers. Referring to [38], the R&D investment of manufacture is βθ2, where β means the cost rate of research and development.(4)
Parameterα represents the sensitivity of consumer to water purity improvement. The bigger the manufacturers invest in R&D, the more αθ impact to demand.(5)
The parameters and their meanings used in this study are given in Table1(6)
The manufacture purified ground water and the retailer buys it at wholesale price and sells it to the market at retail price (Figure1). We consider that the actual demand of the market q is a linear function depending on retail price, green marketing effort, and water purity. The demand function is shown in the following equation.(1)qp,e,θ=a−kp+γe+αθ.


## 3.2. Stackelberg Equilibrium and Profitability Analysis

### 3.2.1. Profit Analysis under Centralized Decision Making (Model B)

Centralized decision-making refers to a commodity produced and sold by the manufacturer, that is to say, a manufacturer integrates production and sales (vertical integration); more specifically, there is only one interest subject in the whole supply chain except customers, which is called integrated manufacturer. In the centralized model, we suppose that a central decision-maker is responsible for the retail price, the green marketing effort, and water purity.The water supply chain’s profit function is shown as(2)ΠSCB=p−ca−kp+γe+αθ−βθ2−ηe22.After solving the first-order derivative ofΠSCB with respect to p, θ, and e, we get(3)∂ΠSCB∂p=a−2kp+γe+αθ+ck,(4)∂ΠSCB∂θ=αp−cα−2βθ,(5)∂ΠSCB∂e=γp−cγ−ηe.Afterwards, we set all the equations (3)–(5), all equal to zero to obtain equilibrium values of model B. The equilibrium values of the equation set are as follows:(6)pSC∗B=−2ckβη−cα2η+2γ2β+2aβηα2η+2γ2β−4kηβ,(7)θSC∗B=kcηα−ηaαα2η+2γ2β−4kηβ,(8)eSC∗B=2kβcγ−2βaγα2η+2γ2β−4kηβ.Substitute (6)–(8) into functions (1) and (2) to get the optimal actual market demand qSC∗B and the profit of supply chain ΠSC∗B.(9)qSC∗B=2k2ηβc−2kηβaα2η+2γ2β−4kηβ,ΠSC∗B=2βηack−βηa2−βηc2k2α2η+2γ2β−4kηβ.Lemma 1.
If it satisfies the conditions thatα2−4kβ<0 and α2η+2γ2β−4βηk<0, the Hessian matrix of the profit function of the centralized supply chain is negative definite matrix. There exists a maximum value in the point p∗B,θ∗B,e∗B, and the profit function of ΠSCB is strictly concave in the retail price p, the green marketing effort e, and water purity θ.Proof (Appendix).Proposition 1.
The Stackelberg equilibrium of the centralized supply chain is(10)pSC∗B=−2ckβη−cα2η+2γ2β+2aβηα2η+2γ2β−4kηβ,θSC∗B=kcηα−ηaαα2η+2γ2β−4kηβ,eSC∗B=2kβcγ−2βaγα2η+2γ2β−4kηβ,qSC∗B=2k2ηβc−2kηβaα2η+2γ2β−4kηβ,ΠSC∗B=2βηack−βηa2−βηc2k2α2η+2γ2β−4kηβ.To ensure that all the equilibrium values are meaningful, the necessary condition iskc−a<0 and γ2−2kη<0. Moreover, when β∗B≥ηαkc−a−α/γ2−2kη, it guaranteed that 0<θSC∗B≤1.Proof (Appendix).


### 3.2.2. Profit Analysis under Decentralized Decision Making (Model BM)

The decentralized decision-making game model means that the manufacturer and the retailer are two independent interest subjects to maximize their profit functions. In this study, we assume the Stackelberg game relationship between manufacturer and seller; manufacturer is the leader and decides the wholesale price and the greening degree of a product; the retailer is a follower who determines the retail price of the product and the level of the green marketing effort according to the manufacturer’s decision. In this case, the profit function of the manufacturer and retailer is expressed as follows, respectively:(11)ΠRBM=p−wa−kp+γe+αθ−ηe22,(12)ΠMBM=w−ca−kp+γe+αθ−βθ2.According to the backward algorithm, in the first stage, based on the anticipation of retailer’s retail price and marketing effort, the manufacture sets the wholesale price and purity of drinking water. In the second stage, concerning the wholesale price and marketing effort, the retailer decides the retail price and marketing effort. According to the backward algorithm, the retail price and marketing effort are obtained through the first derivative of the retailer’s profit function with respect to the retail price and marketing effort. Afterwards, substitute them into the profit function of the manufacturer to get its first derivative with respect to the wholesale price and water purity.The first-derivative result ofΠRB with respect to p and e is shown as(13)∂ΠRBM∂p=a−kp−kp−w+αθ+γe=0,∂ΠRBM∂e=γp−w−ηe=0.Next, we can obtain the optimal retail price and marketing effort of the retailer as follows:(14)pRBM=−−wγ2+ηa+ηkw+αηθγ2−2ηk,(15)eRBM=−γa−kw+αθγ2−2ηk.PutpRBM and eRBM into the profit function of manufacturer ΠMBM. Then, we get(16)ΠMBM=−βγ2θ2+2βηθ2k−αηθkw+αηcθk+ηk2w2−ηakw+ηackγ2−2ηk.After taking the first derivative ofΠMB with respect to w and θ, we get(17)∂ΠMBM∂w=−ηak+ηck2−2ηk2w+αηθkγ2−2ηk=0,(18)∂ΠMBM∂θ=−2βθγ2−4βηθk−αηck+αηkwγ2−2ηk=0.By combining the equations (17) and (18), we derive that(19)wM∗BM=ηcα2k+2βcγ2k+2βaγ2−4βηck2−4βηakkηα2+4βγ2−8βηk,(20)θM∗BM=αηck−αηaηα2+4βγ2−8βηk.Submit (19) and (20) into the equations (14) and (15), and the equilibrium value of pRBM and eRBM is shown as(21)pR∗BM=ηcα2k+2βcγ2k+2βar2−2βηck2−6βηakkηα2+4βγ2−8βηk.(22)eR∗BM=−2βγa−ckηα2+4βγ2−8βηk.Next, we submit (19)–(22) into equations (1), (11), and (12) and obtain the optimal actual market demand qM∗BM and the maximized profits of the retailer ΠR∗BM and manufacturer ΠM∗BM.Lemma 2.
Under Lemma1 and Proposition 1, the profit function ΠMBM and ΠRBM is strictly concave in w, θ and p, e, respectively.Proof (Appendix).Proposition 2.
In this model, the manufacturer’s optimal whole pricewM∗BM, water purity θM∗BM, and maximized profit ΠM∗BM; the retailer’s retail price pR∗BM, marketing effort eR∗BM, and maximized profit ΠR∗BM; the actual market demand qM∗BM, and the profit of the supply chain ΠSC∗BM are obtained as(23)wM∗BM=ηcα2k+2βcγ2k+2βaγ2−4βηck2−4βηakkηα2+4βγ2−8βηk,θM∗BM=αηck−αηaηα2+4βγ2−8βηk,pR∗BM=ηcα2k+2βcγ2k+2βar2−2βηck2−6βηakkηα2+4βγ2−8βηk,eR∗BM=−2βγa−ckηα2+4βγ2−8βηk,ΠM∗BM=2βηack−βηa2−βηc2k2ηα2+4βγ2−8βηk,ΠR∗BM=−2β2ηa−ck2γ2−2ηkηα2+4βγ2−8βηk2,ΠSC∗BM=−βηa−ck2α2η+6βγ2−12βηkηα2+4βγ2−8βηk2,qM∗BM=2βηkck−aηα2+4βγ2−8βηk.Based on Lemma1 and Proposition 1, the equilibrium values make sense. Moreover, to ensure that 0<θM∗BM≤1, we get β∗BM≥ηαkc−a−α/4γ2−2kη.Proof (Appendix).


### 3.2.3. Revenue-Sharing Contract (Model I)

A revenue-sharing model (model I) is where manufacturers play a leading role, and a revenue-sharing contract is through the bargaining model (model II). In these models above, the proportion of income sharing on the premise of maximizing their own interests is decided by manufacturers and the bargaining between manufacturers and retailers, respectively.In the manufacturer-led revenue-sharing model, the revenue-sharing ratioφ0<φ<1 is determined by manufacturers, which means that the proportion of retailer gains from the final revenue is φ, and the manufacture shares the remaining percentage 1−φ. Therefore, the profit function of the manufacturer and retailer is obtained as follows:(24)ΠMI=w−ca−kp+γe+αθ−βθ2+1−φp−wa−kp+γe+αθ,(25)ΠRI=φp−wa−kp+γe+αθ−ηe22.Similar to the decision-making process of model BM, we get(26)wM∗Iφ=α2ηck+2βγ2φck+2βγ2φa−4βηck2−4βηφakkα2η−4βηφk−4βηk+4βγ2φ,(27)θM∗Iφ=αηck−aα2η−4βηφk−4βηk+4βγ2φ,(28)pR∗Iφ=−2βηak+2βγ2φa+α2ηck−2βηck2+2βγ2φck−4βηφakkα2η−4βηφk−4βηk+4βγ2φ,(29)eR∗Iφ=−2βγφa−ckα2η−4βηφk−4βηk+4βγ2φ.Substituting equations (26)–(29) into equations (1), (24), and (25), we get the optimal market demand qM∗BM and the optimal profit of the manufacturer ΠM∗I and retailer ΠR∗I to get(30)qM∗Iφ=−2βηka−ckα2η−4βηφk−4βηk+4βγ2φ,(31)ΠM∗Iφ=−βηa2−2βηack+βηc2k2α2η−4βηφk−4βηk+4βγ2φ,(32)ΠR∗Iφ=−2β2ηφa−ck2γ2φ−2ηkα2η−4βηφk−4βηk+4βγ2φ2,(33)ΠSC∗Iφ=−βηa−ck2α2η+2βγ2φ2+4βγ2φ−8βηφk−4βηkα2η−4βηφk−4βηk+4βγ2φ2.Lemma 3.
Whenγ2<ηk, α2−4βk<0, and ck−a<0, the profit function ΠMI is strictly concave in w and θ, and ΠRI is strictly concave in p and e. Moreover, the equations (26)–(33) are meaningful. And to ensure 0<θM∗Iφ≤1, we get β∗I≥αα2γ2−4βηk2α+a−ck/4k2α2η−8βηk+4βγ2.Proof (Appendix).
Since the first and second derivatives ofΠR∗I with respect to φ are(34)∂ΠR∗I∂ϕ=4β2η2a−ck2α2ηk−4βηk2+4βηφk2−α2γ2φα2η−4βηφk−4βηk+4βγ2φ3,∂2ΠR∗I∂ϕ2<0.
The optimalφ∗I=4βηk2−α2ηk/4βηk2−α2γ2 is next put it into equations (26)–(33) to get:Proposition 3.
In model I, the manufacturer’s optimal whole pricewM∗I, water purity θM∗I, maximized profit ΠM∗I, the retailer’s retail price pR∗I, marketing effort eR∗I, maximized profit ΠR∗I, the actual market demand qM∗I, and the profit of the supply chain ΠSC∗I are obtained as(35)wM∗I=cα2γ2+2βcγ2k+2βaγ2−4βηck2−4βηakα2γ2+4βγ2k−8βηk2,θM∗I=−αα2γ2−4βηk2a−ckα2−4βkα2γ2+4βγ2k−8βηk2,pR∗I=cα4γ2−4ηcα2βk2−4ηaα2βk−8cβ2γ2k2−8aβ2γ2k+8ηcβ2k3+24ηaβ2k2α2−4βkα2γ2+4βγ2k−8βηk2,eR∗I=−2βγka−ckα2γ2+4βγ2k−8βηk2,ΠM∗I=−βα2γ2−4βηk2a−ck2α2−4βkα2γ2+4βγ2k−8βηk2,ΠR∗I=2β2ηk2a−ck2α2−4βkα2γ2+4βγ2k−8βηk2,ΠSC∗I=−βα2γ2−6βηk2a−ck2α2−4βkα2γ2+4βγ2k−8βηk2,qM∗I=−2βkα2γ2−4βηk2a−ckα2−4βkα2γ2+4βγ2k−8βηk2.Proof (Appendix).


### 3.2.4. Revenue-Sharing Contract through Bargaining (Model II)

Revenue-sharing contract through bargaining means that the revenue-sharing ratio is determined by the manufacturers and retailers through bargaining. To study the impact of bargaining on revenue-sharing contract, we adopt the vertical Nash game process. The decision-making sequence is as follows: in the first stage, the manufacture and the retailer bargain on the revenue-sharing ratio,φ, which means the retailer takes possession of φ proportion of the total revenue while the manufacture shares the remaining. In the second stage, the manufacture decides the wholesale price and water purity of drinking water by anticipating the retailer’s retail price and the marketing effort. Finally, the retailer decides the retail price and marketing effort taking the manufacturer’s whole price and water purity into consideration.We construct the bargaining process between the supply chain members by using Nash bargain game. By substituting equations (31) and (32) into the function that ΠIIφ=ΠM∗IϕΠR∗Iφ, we can obtain(36)ΠII=ΠM∗IΠR∗I=2β3η2φa−ck4γ2φ−2ηkα2η+4βγ2φ−4βηk−4βηφk3.Lemma 4.
(1)
Whenφ∗II≤3−1, there does not exists a solution to the Nash bargaining problem(2)
The equilibrium valueφ to this Nash bargaining problem is given as(37)φ∗II=ηα2γ2+σ−8βηk2+4βγ2k4βγ2γ2−ηk,whereσ=α4γ4−8α2βηγ2k2+64β2η2k4−96β2ηγ2k3+48β2γ4k2According to Cauchy inequality, we getφ∗II≤3−1ηk/γ2Proposition 4.
The optimal strategies of wholesale price, water purity, retail price, marketing effort, product demand, the manufacturer’s profit, retailer’s profit, and the water supply chain’s profit are displayed as(38)wM∗II=γ2a+γ2ck−2ηakα2γ2+σ+2γ2−ηkα2γ2ck−8βηk2a+4βγ4ka−ck2kγ2−ηk2α2γ2+σ−8βηk2,θM∗II=−γ2αa−ck2α2γ2+σ−8βηk2,pR∗II=γ2a+γ2ck−2ηakα2γ2+σ−4βηk2+2γ2−ηkα2γ2ck−4βηk2a2kγ2−ηk2α2γ2+σ−8βηk2,eR∗II=−γa−ckα2γ2+σ−8βηk2+4βγ2k2γ2−ηk2α2γ2+σ−8βηk2,ΠM∗II=−βγ2a−ck22α2γ2+σ−8βηk2,ΠR∗II=−ηγ2a−ck2α2γ2+σ−4βγ4kα2γ2+σ−8βηk2+4βγ4k8γ2−ηk22α2γ2+σ−8βηk22,ΠSC∗II=−γ2a−ck22γ2α2η+4βγ4−8βηkα2γ2+σ+8βγ2γ2−ηkα2γ2−4βηk28γ2−ηk22α2γ2+σ−8βηk22,qM∗II=−2βγ2ka−ck2α2γ2+σ−8βηk2.


## 3.2.1. Profit Analysis under Centralized Decision Making (Model B)

Centralized decision-making refers to a commodity produced and sold by the manufacturer, that is to say, a manufacturer integrates production and sales (vertical integration); more specifically, there is only one interest subject in the whole supply chain except customers, which is called integrated manufacturer. In the centralized model, we suppose that a central decision-maker is responsible for the retail price, the green marketing effort, and water purity.The water supply chain’s profit function is shown as(2)ΠSCB=p−ca−kp+γe+αθ−βθ2−ηe22.After solving the first-order derivative ofΠSCB with respect to p, θ, and e, we get(3)∂ΠSCB∂p=a−2kp+γe+αθ+ck,(4)∂ΠSCB∂θ=αp−cα−2βθ,(5)∂ΠSCB∂e=γp−cγ−ηe.Afterwards, we set all the equations (3)–(5), all equal to zero to obtain equilibrium values of model B. The equilibrium values of the equation set are as follows:(6)pSC∗B=−2ckβη−cα2η+2γ2β+2aβηα2η+2γ2β−4kηβ,(7)θSC∗B=kcηα−ηaαα2η+2γ2β−4kηβ,(8)eSC∗B=2kβcγ−2βaγα2η+2γ2β−4kηβ.Substitute (6)–(8) into functions (1) and (2) to get the optimal actual market demand qSC∗B and the profit of supply chain ΠSC∗B.(9)qSC∗B=2k2ηβc−2kηβaα2η+2γ2β−4kηβ,ΠSC∗B=2βηack−βηa2−βηc2k2α2η+2γ2β−4kηβ.Lemma 1.
If it satisfies the conditions thatα2−4kβ<0 and α2η+2γ2β−4βηk<0, the Hessian matrix of the profit function of the centralized supply chain is negative definite matrix. There exists a maximum value in the point p∗B,θ∗B,e∗B, and the profit function of ΠSCB is strictly concave in the retail price p, the green marketing effort e, and water purity θ.Proof (Appendix).Proposition 1.
The Stackelberg equilibrium of the centralized supply chain is(10)pSC∗B=−2ckβη−cα2η+2γ2β+2aβηα2η+2γ2β−4kηβ,θSC∗B=kcηα−ηaαα2η+2γ2β−4kηβ,eSC∗B=2kβcγ−2βaγα2η+2γ2β−4kηβ,qSC∗B=2k2ηβc−2kηβaα2η+2γ2β−4kηβ,ΠSC∗B=2βηack−βηa2−βηc2k2α2η+2γ2β−4kηβ.To ensure that all the equilibrium values are meaningful, the necessary condition iskc−a<0 and γ2−2kη<0. Moreover, when β∗B≥ηαkc−a−α/γ2−2kη, it guaranteed that 0<θSC∗B≤1.Proof (Appendix).


## 3.2.2. Profit Analysis under Decentralized Decision Making (Model BM)

The decentralized decision-making game model means that the manufacturer and the retailer are two independent interest subjects to maximize their profit functions. In this study, we assume the Stackelberg game relationship between manufacturer and seller; manufacturer is the leader and decides the wholesale price and the greening degree of a product; the retailer is a follower who determines the retail price of the product and the level of the green marketing effort according to the manufacturer’s decision. In this case, the profit function of the manufacturer and retailer is expressed as follows, respectively:(11)ΠRBM=p−wa−kp+γe+αθ−ηe22,(12)ΠMBM=w−ca−kp+γe+αθ−βθ2.According to the backward algorithm, in the first stage, based on the anticipation of retailer’s retail price and marketing effort, the manufacture sets the wholesale price and purity of drinking water. In the second stage, concerning the wholesale price and marketing effort, the retailer decides the retail price and marketing effort. According to the backward algorithm, the retail price and marketing effort are obtained through the first derivative of the retailer’s profit function with respect to the retail price and marketing effort. Afterwards, substitute them into the profit function of the manufacturer to get its first derivative with respect to the wholesale price and water purity.The first-derivative result ofΠRB with respect to p and e is shown as(13)∂ΠRBM∂p=a−kp−kp−w+αθ+γe=0,∂ΠRBM∂e=γp−w−ηe=0.Next, we can obtain the optimal retail price and marketing effort of the retailer as follows:(14)pRBM=−−wγ2+ηa+ηkw+αηθγ2−2ηk,(15)eRBM=−γa−kw+αθγ2−2ηk.PutpRBM and eRBM into the profit function of manufacturer ΠMBM. Then, we get(16)ΠMBM=−βγ2θ2+2βηθ2k−αηθkw+αηcθk+ηk2w2−ηakw+ηackγ2−2ηk.After taking the first derivative ofΠMB with respect to w and θ, we get(17)∂ΠMBM∂w=−ηak+ηck2−2ηk2w+αηθkγ2−2ηk=0,(18)∂ΠMBM∂θ=−2βθγ2−4βηθk−αηck+αηkwγ2−2ηk=0.By combining the equations (17) and (18), we derive that(19)wM∗BM=ηcα2k+2βcγ2k+2βaγ2−4βηck2−4βηakkηα2+4βγ2−8βηk,(20)θM∗BM=αηck−αηaηα2+4βγ2−8βηk.Submit (19) and (20) into the equations (14) and (15), and the equilibrium value of pRBM and eRBM is shown as(21)pR∗BM=ηcα2k+2βcγ2k+2βar2−2βηck2−6βηakkηα2+4βγ2−8βηk.(22)eR∗BM=−2βγa−ckηα2+4βγ2−8βηk.Next, we submit (19)–(22) into equations (1), (11), and (12) and obtain the optimal actual market demand qM∗BM and the maximized profits of the retailer ΠR∗BM and manufacturer ΠM∗BM.Lemma 2.
Under Lemma1 and Proposition 1, the profit function ΠMBM and ΠRBM is strictly concave in w, θ and p, e, respectively.Proof (Appendix).Proposition 2.
In this model, the manufacturer’s optimal whole pricewM∗BM, water purity θM∗BM, and maximized profit ΠM∗BM; the retailer’s retail price pR∗BM, marketing effort eR∗BM, and maximized profit ΠR∗BM; the actual market demand qM∗BM, and the profit of the supply chain ΠSC∗BM are obtained as(23)wM∗BM=ηcα2k+2βcγ2k+2βaγ2−4βηck2−4βηakkηα2+4βγ2−8βηk,θM∗BM=αηck−αηaηα2+4βγ2−8βηk,pR∗BM=ηcα2k+2βcγ2k+2βar2−2βηck2−6βηakkηα2+4βγ2−8βηk,eR∗BM=−2βγa−ckηα2+4βγ2−8βηk,ΠM∗BM=2βηack−βηa2−βηc2k2ηα2+4βγ2−8βηk,ΠR∗BM=−2β2ηa−ck2γ2−2ηkηα2+4βγ2−8βηk2,ΠSC∗BM=−βηa−ck2α2η+6βγ2−12βηkηα2+4βγ2−8βηk2,qM∗BM=2βηkck−aηα2+4βγ2−8βηk.Based on Lemma1 and Proposition 1, the equilibrium values make sense. Moreover, to ensure that 0<θM∗BM≤1, we get β∗BM≥ηαkc−a−α/4γ2−2kη.Proof (Appendix).


## 3.2.3. Revenue-Sharing Contract (Model I)

A revenue-sharing model (model I) is where manufacturers play a leading role, and a revenue-sharing contract is through the bargaining model (model II). In these models above, the proportion of income sharing on the premise of maximizing their own interests is decided by manufacturers and the bargaining between manufacturers and retailers, respectively.In the manufacturer-led revenue-sharing model, the revenue-sharing ratioφ0<φ<1 is determined by manufacturers, which means that the proportion of retailer gains from the final revenue is φ, and the manufacture shares the remaining percentage 1−φ. Therefore, the profit function of the manufacturer and retailer is obtained as follows:(24)ΠMI=w−ca−kp+γe+αθ−βθ2+1−φp−wa−kp+γe+αθ,(25)ΠRI=φp−wa−kp+γe+αθ−ηe22.Similar to the decision-making process of model BM, we get(26)wM∗Iφ=α2ηck+2βγ2φck+2βγ2φa−4βηck2−4βηφakkα2η−4βηφk−4βηk+4βγ2φ,(27)θM∗Iφ=αηck−aα2η−4βηφk−4βηk+4βγ2φ,(28)pR∗Iφ=−2βηak+2βγ2φa+α2ηck−2βηck2+2βγ2φck−4βηφakkα2η−4βηφk−4βηk+4βγ2φ,(29)eR∗Iφ=−2βγφa−ckα2η−4βηφk−4βηk+4βγ2φ.Substituting equations (26)–(29) into equations (1), (24), and (25), we get the optimal market demand qM∗BM and the optimal profit of the manufacturer ΠM∗I and retailer ΠR∗I to get(30)qM∗Iφ=−2βηka−ckα2η−4βηφk−4βηk+4βγ2φ,(31)ΠM∗Iφ=−βηa2−2βηack+βηc2k2α2η−4βηφk−4βηk+4βγ2φ,(32)ΠR∗Iφ=−2β2ηφa−ck2γ2φ−2ηkα2η−4βηφk−4βηk+4βγ2φ2,(33)ΠSC∗Iφ=−βηa−ck2α2η+2βγ2φ2+4βγ2φ−8βηφk−4βηkα2η−4βηφk−4βηk+4βγ2φ2.Lemma 3.
Whenγ2<ηk, α2−4βk<0, and ck−a<0, the profit function ΠMI is strictly concave in w and θ, and ΠRI is strictly concave in p and e. Moreover, the equations (26)–(33) are meaningful. And to ensure 0<θM∗Iφ≤1, we get β∗I≥αα2γ2−4βηk2α+a−ck/4k2α2η−8βηk+4βγ2.Proof (Appendix).
Since the first and second derivatives ofΠR∗I with respect to φ are(34)∂ΠR∗I∂ϕ=4β2η2a−ck2α2ηk−4βηk2+4βηφk2−α2γ2φα2η−4βηφk−4βηk+4βγ2φ3,∂2ΠR∗I∂ϕ2<0.
The optimalφ∗I=4βηk2−α2ηk/4βηk2−α2γ2 is next put it into equations (26)–(33) to get:Proposition 3.
In model I, the manufacturer’s optimal whole pricewM∗I, water purity θM∗I, maximized profit ΠM∗I, the retailer’s retail price pR∗I, marketing effort eR∗I, maximized profit ΠR∗I, the actual market demand qM∗I, and the profit of the supply chain ΠSC∗I are obtained as(35)wM∗I=cα2γ2+2βcγ2k+2βaγ2−4βηck2−4βηakα2γ2+4βγ2k−8βηk2,θM∗I=−αα2γ2−4βηk2a−ckα2−4βkα2γ2+4βγ2k−8βηk2,pR∗I=cα4γ2−4ηcα2βk2−4ηaα2βk−8cβ2γ2k2−8aβ2γ2k+8ηcβ2k3+24ηaβ2k2α2−4βkα2γ2+4βγ2k−8βηk2,eR∗I=−2βγka−ckα2γ2+4βγ2k−8βηk2,ΠM∗I=−βα2γ2−4βηk2a−ck2α2−4βkα2γ2+4βγ2k−8βηk2,ΠR∗I=2β2ηk2a−ck2α2−4βkα2γ2+4βγ2k−8βηk2,ΠSC∗I=−βα2γ2−6βηk2a−ck2α2−4βkα2γ2+4βγ2k−8βηk2,qM∗I=−2βkα2γ2−4βηk2a−ckα2−4βkα2γ2+4βγ2k−8βηk2.Proof (Appendix).


## 3.2.4. Revenue-Sharing Contract through Bargaining (Model II)

Revenue-sharing contract through bargaining means that the revenue-sharing ratio is determined by the manufacturers and retailers through bargaining. To study the impact of bargaining on revenue-sharing contract, we adopt the vertical Nash game process. The decision-making sequence is as follows: in the first stage, the manufacture and the retailer bargain on the revenue-sharing ratio,φ, which means the retailer takes possession of φ proportion of the total revenue while the manufacture shares the remaining. In the second stage, the manufacture decides the wholesale price and water purity of drinking water by anticipating the retailer’s retail price and the marketing effort. Finally, the retailer decides the retail price and marketing effort taking the manufacturer’s whole price and water purity into consideration.We construct the bargaining process between the supply chain members by using Nash bargain game. By substituting equations (31) and (32) into the function that ΠIIφ=ΠM∗IϕΠR∗Iφ, we can obtain(36)ΠII=ΠM∗IΠR∗I=2β3η2φa−ck4γ2φ−2ηkα2η+4βγ2φ−4βηk−4βηφk3.Lemma 4.
(1)
Whenφ∗II≤3−1, there does not exists a solution to the Nash bargaining problem(2)
The equilibrium valueφ to this Nash bargaining problem is given as(37)φ∗II=ηα2γ2+σ−8βηk2+4βγ2k4βγ2γ2−ηk,whereσ=α4γ4−8α2βηγ2k2+64β2η2k4−96β2ηγ2k3+48β2γ4k2According to Cauchy inequality, we getφ∗II≤3−1ηk/γ2Proposition 4.
The optimal strategies of wholesale price, water purity, retail price, marketing effort, product demand, the manufacturer’s profit, retailer’s profit, and the water supply chain’s profit are displayed as(38)wM∗II=γ2a+γ2ck−2ηakα2γ2+σ+2γ2−ηkα2γ2ck−8βηk2a+4βγ4ka−ck2kγ2−ηk2α2γ2+σ−8βηk2,θM∗II=−γ2αa−ck2α2γ2+σ−8βηk2,pR∗II=γ2a+γ2ck−2ηakα2γ2+σ−4βηk2+2γ2−ηkα2γ2ck−4βηk2a2kγ2−ηk2α2γ2+σ−8βηk2,eR∗II=−γa−ckα2γ2+σ−8βηk2+4βγ2k2γ2−ηk2α2γ2+σ−8βηk2,ΠM∗II=−βγ2a−ck22α2γ2+σ−8βηk2,ΠR∗II=−ηγ2a−ck2α2γ2+σ−4βγ4kα2γ2+σ−8βηk2+4βγ4k8γ2−ηk22α2γ2+σ−8βηk22,ΠSC∗II=−γ2a−ck22γ2α2η+4βγ4−8βηkα2γ2+σ+8βγ2γ2−ηkα2γ2−4βηk28γ2−ηk22α2γ2+σ−8βηk22,qM∗II=−2βγ2ka−ck2α2γ2+σ−8βηk2.


## 3.3. Comparative Analysis

Proposition 5.
The water purity, retail price, wholesale price, and the retailer’s marketing effort under the three models satisfy the following relationships which are derived through algebraic comparison:(1)
θSC∗B>θM∗II>θM∗I>θM∗BM(2)
eSC∗B>eR∗BM>eR∗I>eR∗II(3)
wM∗BM>wM∗I>wM∗II(4)
qSC∗B>qM∗II>qM∗I>qM∗BM
Proposition5 distributes the differences among the strategies in terms of the product’s water purity, the retailer’s green marketing effort, the product’s wholesale price and retail price, and product demand of the four models in this study.It is natural that the product’s water purity for model B (centralized model) is highest since it is the fully coordinated scenario and lowest under model BM (decentralized model without revenue-sharing contract). Moreover, product’s water purity for revenue-sharing contract through bargaining is higher than that under revenue-sharing contract (i.e.,θM∗II>θM∗I). Therefore, revenue-sharing contract is beneficial to the improvement of product’s water purity and cooperation can contribute to it.As for the comparison of product’s water purity, naturally, it is highest in the fully coordinated scenario, while lowest under revenue-sharing contract situation (i.e.,eR∗BM>eR∗I>eR∗II). According to the revenue-sharing contract, the retailer returns a percentage of its sales to the manufacturer and maintains profit through decreasing the marketing cost. Therefore, revenue-sharing contract leads to the decrease of the marketing effort and revenue-sharing contract though bargaining makes it worse.The wholesale price is lowest in revenue-sharing contract through bargaining, followed by under revenue-sharing contract and highest in decentralized condition without any communication between supply chain members (i.e.,wM∗BM>wM∗I>wM∗II). It is probably that the manufacturer sells products to the retailer at a wholesale price below the products’ manufacturing cost. As a result, effective communication gives rise to the decline of the wholesale price.Proof (Appendix).Proposition 6.
The water supply chain’s profit under the four models meets the relationship:(1)
ΠM∗II>ΠM∗I>ΠM∗BM and ΠR∗I>ΠR∗BM(2)
ΠSC∗B>ΠSC∗II>ΠSC∗I>ΠSC∗BM
Proposition6 shows the profit differences of the manufacturer and the retailer among the models. The manufacturer benefits from the revenue-sharing contract due to it as the Stackelberg leader, while the retail as the follower fails to benefit from the contract; only if φ=1, the retailer earns profit equal to that under the decentralized model without revenue-sharing contract. The comparison of green supply chain’s profit indicates that there exists the double marginalization effect. It is very likely to be caused by the goals of supply chain members that perhaps conflict with each other. As a result, revenue-sharing contract are conducive to the boosting of the profit; furthermore, the profit has been increased even more under the revenue-sharing contract (i.e., ΠSC∗II>ΠSC∗I>ΠSC∗BM). Therefore, the goal of constructing water supply chain is to realize integrated management, promote cooperation between upstream and downstream enterprises, and finally achieve higher overall performance.Proof (Appendix).Proposition 7.
In models B, BM, I, and II, we have(1)
The increase of consumer sensitivity to water purity improvementα induces the increase of the optimal wholesale price, water purity, retail price, marketing effort, manufacturer’s profit, and retailer’s profit(2)
The increase of the cost rate of R&Dβ leads to the decrease of the optimal wholesale price, water purity, retail price, marketing effort, manufacturer’s profit, and retailer’s profitProof (Appendix).


## 4. Numerical Analysis

In Section4, we compare seven groups of parameters in the four models (B, BM, I, and II), including the water purity of the manufacturer, the green marketing level of the retailer, the wholesale price of the product, the actual market demand quantity of the product, the profit of the manufacturer, the profit of the retailer, and the total profit of the supply chain. In order to explain the effect of the optimal values, we set values to parameters as the following: a = 20, k = 5, c = 3, η=4, γ=2, and α=4. Seen from the four models, we get three different value ranges of the cost rate of research and development (R&D) β, and(39)β≥maxηαkc−a−αγ2−2kη,ηαkc−a−α4γ2−2kη,αα2γ2−4βηk2α+a−ck4k2α2η−8βηk+4βγ2≥4.In addition,β decides the water purity directly; generally speaking, when the manufacturer increases the investment in research and development, the water purity will increase. Moreover, higher water purity usually possesses higher value, and the retailer is willing to make more effort to the marketing of the green product. Therefore, we assume that β is the independent variable.The comparison results are shown in Figures2–8 by taking advantage of Matlab 2013. As shown in Figure 2, the revenue-sharing contract model is superior to model BM (without any contract) in improving the water purity. And furthermore, the revenue-sharing contract through the bargaining model is better than the revenue-sharing contract model. Figure 3 demonstrates that the retailer makes the least marketing effort in model II to save cost, since the retailer has to share its revenue with the manufacturer. In Figure 4, there is no wholesale price in centralized condition as results of supply chain members are considered as a whole. Contract (revenue-sharing contract or revenue-sharing contract through bargaining) leads to the decrease of the product’s wholesale price. Product with higher water purity is more likely to be accepted by consumers; as consequence, the sequence of product’s actual market demand among the four models can be seen from Figure 5. In Figures 6 and 7, both the profit of manufacturer and the profit of retailer can be promoted through the contract model. Therefore, both of them will adopt revenue-sharing contract. In Figure 8, contracts (revenue-sharing contract and revenue-sharing contract through bargaining) contribute to the increase of the profit of the whole supply chain, and furthermore, the profit level of the latter contract is higher than that of the former contract.Figure 2
The comparison of optimal water purity.Figure 3
The comparison of the optimal marketing effort.Figure 4
The comparison of the optimal wholesale price.Figure 5
The comparison of the optimal product market demand.Figure 6
The comparison of the optimal manufacturer’s profit.Figure 7
The comparison of the optimal retailer’s profit.Figure 8
The comparison of the optimal supply chain’s profit.


## 5. Conclusion

From the comparison above, the result shows that revenue-sharing contract is beneficial to the increase of water purity, the water supply chain members’ profit, and the overall profit of water supply chain, while giving rise to the decrease of the retailer’s marketing effort and the product’s wholesale price compared with the decentralized model. Moreover, revenue-sharing contract through bargaining exert a more significant influence. In a word, revenue-sharing contract through bargaining eliminate the double marginal effect among the water supply chain more effectively and is an effective way to promote the cooperation among the water supply chain members. It is worth mentioning that revenue-sharing contract brings negative impact to the retailer’s marketing effort. Marketing can stimulate the increase of the green product’s market demand on the one hand, and on the other hand, it generates amount of marketing cost. In this study, the profit that marketing produces cannot offset the cost that it brings. Thus, it will be important to take some measures to make up the loss that marketing generated.In this study, we have made a little innovation, which is that the marketing effort and water purity have been considered in the demand function. However, there are some shortcomings of this study. For example, first, we only consider a two-tier water supply chain; second, although the demand function has been revised, it is still a simple linear function concerning the marketing effort, water purity, and retail price, which makes the application of this model limited in a sense. In our further study, we will pay attention to two streams. One is that considering the supply chain that has more than one manufacturer and retailer. The other is to study the difference among revenue-sharing contract, costing-sharing contract, and mixed-sharing contract.Management insights: by signing revenue-sharing contracts, manufacturers, wholesalers, and retailers can form a community of interests, always maintain the real-time interaction among them, get firsthand customer feedback and demand in a timely manner, and then receive orders according to customer demand. In this way, customized products should be manufactured to ensure that provided products are according to customer requirements. Overall, a virtuous loop should be established to realize value creation. Second, because revenue-sharing contracts weaken retailers’ incentives to market green products, manufacturers can use other methods, such as sales discounts, exchange promises, and extended payback periods to motivate retailers. Third, the future market competition is the competition between the supply chains. Therefore, it is more and more important for the development of the supply chain to form a stable win-win partnership among all partners in the supply chain.



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*Source: 5593463-2021-04-13.xml*