https://orcid.org/0000-0001-7032-0112
Abstract
The promotion of water-saving products is one of the vital ways to implement water conservation action, and advertising is a significant way to promote water-saving products. Taking the two-level Supply Chain consisting of a leading manufacturer and a retailer as an example and considering the advertising cost-sharing ratio of the two, as well as the government's R&D subsidies to manufacturers and product subsidies to consumers, this study establishes differential game models in three cases, that is, non-cooperative contract without cost sharing, cooperative contract with cost sharing, and collaborative cooperation contract. Also, numerical simulation is adopted to analyze the sensitivity of important parameters. The results show that the product goodwill and market demand for water-saving products can achieve Pareto optimality under the collaborative cooperation contract. In addition, the cooperative contract with cost sharing can realize Pareto improvement of the optimal benefit of the Supply Chain under certain conditions. Moreover, in the absence of the government's R&D subsidies, the overall benefits can achieve Pareto optimality under the collaborative cooperation contract. This study provides theoretical guidance and reference for the advertising cooperation strategy for the main bodies in the Supply Chain.
HIGHLIGHTS
Considering the influence of government subsidies to manufacturers and product subsidies to consumers, this study constructs three contracts to optimize the advertising cooperation strategy of the main bodies in the Supply Chain in the promotion of water-saving products.
This study adopts numerical simulation analysis to verify the long-term benefits of the main bodies of the Supply Chain under different contracts.
INTRODUCTION
According to ‘the United Nations World Water Development Report 2018’, there are 17 countries in the dilemma of ‘extreme water scarcity’ currently, that is, a quarter of the world's total population suffers from water shortages. The demand for water resources is growing at an annual rate of 1% and will continue to accelerate over the next 20 years (WWAP, 2018). Moreover, two-thirds of the population is expected to face water scarcity by 2025 due to climate change (Ungureanu et al., 2020). For example, the Middle East, especially Iran, has suffered from several droughts in the past 40 years, and the problem of water shortage is quite severe (Zobeidi et al., 2022). The promotion of water-saving products is one of the important ways to implement water conservation action and drive high-quality economic and social development. Many countries and regions are committed to the research and development of water-saving products and related technologies, along with numerous water policies, to ensure the sustainable use of water resources (Getie et al., 2020; Alhejji et al., 2021; Cipolletta et al., 2021). In 2012, China issued the ‘national guidelines on Strengthening the Quality Improvement and Promotion of Water-Saving Products’. It clearly pointed out that water-using techniques, equipment, and products that do not meet water-saving standards should be eliminated as soon as possible, and water-saving products for domestic use should be vigorously promoted. At present, low water-consumption products are particularly favored by environmentalists (Gómez-Llanos et al., 2020), and more people tend to buy water-saving products (Rasoulkhani et al., 2018; Tian & Chen, 2022). More than a dozen water-saving products have been issued with water efficiency labeling, including washing machines, dishwashers, toilets, and shower heads (Liang et al., 2022). However, there are several problems in the promotion of water-saving products, mainly due to insufficient publicity, difficulties in technological breakthroughs, false information labels, and weak consumers' intention to purchase, which is a hot topic in the current. Moreover, few scholars have carried out detailed research on the promotion of water-saving products from the perspective of the water-saving product Supply Chain.
To encourage innovation in water-saving technologies, Australia, Europe, the United States, Canada, and China have begun to implement water efficiency labeling systems. China has also actively implemented a series of highly targeted policies and measures (Zhang et al., 2021b). For example, governments set up some major special projects for technology R&D and provide financial support. In addition, some cities in China, such as Beijing, have implemented energy-saving subsidies for home appliances since 2017, that is, those who purchase water-saving showers and toilets with water efficiency labeling will receive government subsidies of 20% of the sales price. Meanwhile, some manufacturers also work with retailers to strengthen advertising cooperation, improve product goodwill, and enhance users’ water-saving awareness. For example, Hengjie Sanitary Ware, one of the manufacturers that focus on water-saving products, launched a super water-saving toilet in 2012, and formed strategic alliances with retailers such as JD and Suning. Hengjie Sanitary Ware specializes in production, and JD and Suning are responsible for advertising so that these water-saving products can be quickly recognized and accepted by consumers.
Many scholars have emphasized the advantages of cooperative advertising, such as achieving Pareto improvement, improving performance (Chenavaz & Eynan, 2021; Zhang et al., 2021a), stimulating advertising efforts and sales, controlling inventory, and reducing costs (De Giovanni et al., 2019; Xiao et al., 2019). However, cooperative advertising can only be achieved under the condition that the profit and cost distribution ratios are satisfactory to all parties in the Supply Chain. It all depends on the negotiation ability of both parties, so it is also a disadvantage of cooperative advertising (Aust & Buscher, 2014). Currently, many scholars have carried out systematic research on cooperative advertising of the Green Supply Chain (GSC). Most of these studies take the advertising cost-sharing ratio as the only influencing factor, and seldom consider the government's R&D subsidies to manufacturers and product subsidies to consumers. In addition, there are very few studies on cooperative advertising strategies in the promotion of water-saving products. Therefore, this study takes the promotion of water-saving products as the research object and is committed to the research on the long-term collaborative cooperation between manufacturers and retailers.
LITERATURE REVIEW
In this section, we will sort out the literature from two perspectives: influencing factors and policies for the promotion of water-saving products, and the application of cooperative advertising strategy in the Supply Chain.
Many scholars have carried out research on the influencing factors of water-saving products (such as small-capacity/dual-flush toilets, water-type washing machines, throttling faucets and rainwater collection tanks in the water supply of small-capacity toilets, etc.) in different regions with different economic levels by questionnaire surveys and empirical analysis. The study found that there are four main influencing factors, including attitude characteristics, environmental problems, socio-economic characteristics and attribute characteristics of water-saving products (Millock & Nauges, 2010; Grafton et al., 2011; Thiam et al., 2021). In terms of promotion policies for water-saving products, as early as the 1990s, the Goleta area of California adopted ultra-low water-consumption flush toilets to replace old-style toilets, and distributed 35,000 low-water-consuming showers to the local people for free to prolong the service life of the water supply system (Dunham et al., 1995). Later, California introduced a series of incentives to mitigate the threat of drought in the region (LADWP, 2015), such as implementing a low-flow toilet rebate program and distributing free home plumbing retrofit kits. In 2005, WELS of Australia formulated the minimum water-saving efficiency of water-saving products and determined its water efficiency labeling (Chong et al., 2008). Beijing promoted 100,000 sets of high-efficiency water-saving products in China in 2018 and provided subsidies for some products. In addition, it also promotes the standardization of water-saving products by prohibiting the sale of unlabeled water-saving products. To sum up, most scholars mainly focus on the factors and policies that affect the promotion of water-saving products, and less on how to use cooperative advertising strategy to effectively promote water-saving products.
Meanwhile, there is much research on the application of cooperative advertising strategy in the Supply Chain. Cooperative advertising strategies generally refer to advertising cost-sharing mechanisms adopted by manufacturers and retailers in the Supply Chain, that is, the manufacturer pays the retailer part of the advertising bill to compensate the latter for advertising their products (Jørgensen et al., 2000). Since the early 1990s, the cooperative advertising strategy has been widely used in the Supply Chain in the United States (Yue et al., 2006), of which the cooperative advertising expenditure in 2000 has reached 15 billion dollars (Chutani & Sethi, 2012). Dant and Berger say that 25–40% of a retailer's promotional advertising costs are financed by manufacturers (Dant & Berger, 1996). Many scholars have also studied the application of cooperative advertising strategies in the GSC such as water saving, low carbon, and coal industries (Pan et al., 2012; Yan et al., 2016; Chutani & Sethi, 2018). In the GSC, scholars mostly take the secondary Supply Chain consisting of a single manufacturer and a single retailer as the research object, and build a static model of cooperative advertising (Abolfazl & Zohreh, 2016; Chaab & Rasti, 2016). There are also a few scholars to build a dynamic model of cooperative advertising with a single manufacturer and two retailers, or two manufacturers and one retailer (Karray et al., 2017; Xu et al., 2019), and study the impact of cooperative advertising on the performance of the Supply Chain. The research on cooperative advertising under the dynamic model is closer to reality, but most scholars only consider the impact of advertising investment on product demand and rarely involve the issue that product demand is also affected by government subsidies. Based on the literature (Mahdiraji et al., 2021), this paper selects a single manufacturer and a single retailer as the research objects. The closer the cooperation between the main bodies (such as single manufacturer and single retailer, single supplier and single manufacturer), the higher the level of green technology, and the higher the market demand for green products (Mohsin et al., 2021). In addition, considering consumer preferences and social preferences, some scholars have studied the promotion of green products and found that a cooperative advertising strategy can change consumer preferences and the marginal profit of the main bodies in the Supply Chain (Xia et al., 2020). However, the implementation of the cooperative advertising strategy is also conditional. Manufacturers will be willing to share the retailer's advertising costs if and only if the retailer advertises below the stated price (He et al., 2011). Finally, the scholars analyzed the cooperative advertising strategy of the GSC in different situations (such as competitive Supply Chain, sudden Supply Chain breakage, etc.), and further confirmed that cooperative advertising strategy is an effective incentive for promoting green products. It can not only significantly increase the market demand, but also increase the benefits of each main body of the Supply Chain (Aust & Buscher, 2012; Karray & Amin, 2015; Ma et al., 2019).
MODELING AND SOLUTION
This paper is devoted to the study of the Supply Chain consisting of a manufacturer and a retailer over a long period of time. The manufacturer plays the leader, and the retailer is the follower. The technical level of the manufacturer will affect the product goodwill and further affect the market demand. In addition, the publicity level of retailers has played a certain role in promoting the sales of water-saving products. Finally, manufacturers as leaders incentivize retailers to promote water-saving products through certain advertising cost-sharing measures. Based on the above analysis, this study proposes 5 hypotheses as follows.
of the manufacturer and the input cost 
of the retailer at time t are calculated in Equations (1) and (2).
represent the technical effort level of the manufacturer and the publicity effort level of the retailer at the time t, respectively. Also, 
represent the input cost coefficient of the manufacturer and retailer, respectively. It can be known from the equations that the input cost of manufacturer and retailer at time t increase with the increase of their effort level, respectively, and the increase rate shows an upward trend. The use of quadratic functions to describe input costs is a form commonly used in the literature (Jørgensen & Zaccour, 2014; Chen & Wang, 2020).Hypothesis 2 Suppose the input cost of manufacturer is 
, the selling price of manufacturer for water-saving products is 
, and the selling price of retailer for water-saving products is 
. Based on the literature (Choi, 1991), it can obtain 
.
is affected by the technical effort level 
of manufacturers, the publicity effort level 
of retailers, and the government's subsidy intensity 
for water-saving products.
represents the natural decay coefficient of product goodwill over time. 
and 
represent the influence coefficient of the technical effort level of manufacturer and the publicity effort level of retailer, respectively. 
indicates the government's subsidy intensity for water-saving products.
is affected by the selling price e of retailer, the product goodwill W and the government's subsidy intensity 
for water-saving products.
represents the potential market demand for water-saving products, 
represents the influence coefficient of the selling price on the market demand, 
represents the influence coefficient of the government's subsidy intensity, and 
represents the influence coefficient of product goodwill.Hypothesis 5 The advertising cost-sharing ratio provided by the manufacturer to the retailer is 
, and 
. Government R&D subsidy drives the technical effort level of manufacturer. The government's R&D subsidy is based on the input cost of the manufacturer, assuming that the government's incentive ratio to the manufacturer is 
, and 
.
Model N: Non-cooperative contract without cost sharing
, and then the retailer determines their publicity effort level 
for water-saving products. Assuming both bodies make decisions in an infinite time interval and the discount factor is 
, the objective functions of manufacturer and retailer are 
and 
, respectively. Then, Equations (5) and (6) can be obtained as follows.
can be obtained from Equation (7).
of the manufacturer and the optimal profit value function 
of the retailer can be obtained by the following equations, respectively.
and 
, are transformed into 
and 
. Also, the optimal control problem of both bodies satisfies the following Hamiltonian–Jacobi–Bellman (HJB) equation.
Proposition 1 In the above model, the equilibrium result under a non-cooperative contract without cost sharing is as follows.
- The optimal effort level
of manufacturers, the optimal effort level 
of retailers, and the government's R&D subsidy ratio 
to manufacturers are in the following.(14)
- The optimal benefit function
of the manufacturer, the optimal benefit function 
of the retailer, and the optimal benefit function 
of the Supply Chain are as follows.(15)
(16)
(17)
. First, conduct the first-order derivative of 
in Equation (13). Then, let Equation (13) be 0 and solve the equation, and substitute the result of 
into Equation (12). Equation (12) is a concave function of 
. Apply the above steps to Equation (12), then the optimal effort level of manufacturer and retailer can be obtained respectively, that is, 
and 
.
According to the structural characteristics of Equations (12) and (13), it can be inferred that the linear equation of W is the solution of the HJB equation.
and 
, the values of 
and 
can be obtained as follows.
Then, substituting the above conclusions into Equations (19) and (20), the optimal benefit functions of the manufacturer, the retailer and the Supply Chain can be obtained respectively as shown in Equations (15)–(17).
. According to the optimal control theory, 
satisfies the following HJB equation for any 
.
is satisfied, and the value of 
can be obtained later.
Corollary 1 In the non-cooperative contract without cost sharing, the technical effort level 
of the manufacturer is negatively correlated with their input cost coefficient 
, discount rate 
and product goodwill decay rate 
; and it is positively correlated with the influence coefficient 
of water-saving technology on product goodwill, their own marginal revenue 
, the influence coefficient 
of product goodwill on market demand, and the government subsidy ratio 
to manufacturers.
Corollary 2 In the non-cooperative contract without cost sharing, the publicity effort level 
of retailers is negatively correlated with their input cost coefficient 
, discount rate 
and product goodwill decay rate 
; and it is positively correlated with the influence coefficient 
of publicity level on product goodwill, their own revenue 
and the influence coefficient 
of product goodwill on market demand.
Corollary 3 The subsidy ratio 
of government to manufacturers is determined by the ratio of the manufacturer's profit margin 
to the sum 
of the manufacturer's and retailer's profit margins.
Model S: cooperative contract with cost sharing
Generally speaking, in order to motivate retailers, the manufacturer will share a part of the retailer's publicity cost, and the sharing rate is denoted by 
. Therefore, the manufacturer-led Supply Chain of water-saving products, that is, the Stackelberg master–slave differential game model, is formed. In this model, manufacturers and retailers make decisions to maximize their own interests, and the decision-making process is divided into two stages. In the first stage, the manufacturer determines their own technical effort level and the ratio of the publicity cost to share; and in the second stage, retailers determine their own publicity effort level based on the technical effort level and the publicity cost-sharing ratio determined by the manufacturer.
, 
and 
, are as follows, respectively.
Proposition 2 In the above model, the equilibrium result under the cooperative contract with cost sharing is as follows.
- The technical effort level
of manufacturers, the publicity effort level 
of retailers, the publicity cost-sharing ratio 
of manufacturers and the ratio 
of government subsidies to manufacturers are as follows, respectively.(32)
- The optimal benefit function of the manufacturer, the retailer, and the Supply Chain, namely
, 
, and 
, are as follows, respectively.(33)
(34)
(35)
. First, conduct the first-order derivative of Equation (31). Then, let it be 0 and solve the equation, and substitute the result of 
into Equation (30). Equation (30) is a concave function of 
and 
. Apply the above steps to Equation (30), then the optimal effort level of manufacturer and retailer (i.e., 
and 
) can be obtained, as well as the optimal cost-sharing ratios for manufacturers (
).
According to the structural characteristics of Equations (30) and (31), it is certain that the linear equation of W is the solution of the HJB equation.
and 
, the values of 
and 
can be obtained as follows.
Substitute Equation (39) into Equation (36) to obtain the values of 
, 
and 
as shown in Equation (32). Substitute the above conclusions into Equations (30) and (31), the optimal benefit function of the manufacturers, retailers and the Supply Chain can be obtained as shown in Equations (33)–(35).
. According to the optimal control theory, 
satisfies the HJB equation for any 
.
, and the value of 
can be obtained.
of government subsidies to manufacturers can be obtained as follows.
, the manufacturer will share part of the retailer's advertising cost, and the manufacturer's optimal sharing ratio 
is proportional to their marginal revenue 
and inversely proportional to the retailer's marginal revenue 
.Proof: according to Equation (24), we can get 
, then Corollary 3 can be proved.
Corollary 5 When the manufacturer shares a certain proportion of the retailer's advertising cost, the optimal publicity effort level of the retailer is positively related to the marginal revenue of itself and the manufacturer. In addition, the impact of the manufacturer's marginal revenue on the publicity effort level of the retailer is greater than that of the retailer's own marginal revenue on itself.
Corollary 6 When the manufacturer shares a certain proportion of the retailer's advertising cost, the retailer's publicity effort level is greater than that under no cost-sharing contract.
Proof: 
, then Corollary 6 is proved.
Model C: collaborative cooperation contract
of the Supply Chain and the objective function 
of the government are as follows.
Proposition 3 In the above model, the equilibrium result under the cooperative contract is as follows.
- The technical effort level
of manufacturers, the publicity effort level 
of retailers, and the ratio 
of government subsidies to manufacturers are as follows.(48)
- The optimal function
of the overall benefit of the Supply Chain is as follows.(49)
and 
respectively and setting it to 0, we can obtain equations in the following.
, and the value of 
can be obtained as follows.
Substitute Equation (52) into Equation (50) to get the values of 
and 
as shown in Equation (48). Substituting the above conclusions into Equation (51), the objective function of the Supply Chain can be obtained as shown in Equation (49).
Substituting the above conclusions into Equation (51), the objective function of the Supply Chain can be obtained as shown in Equation (49).
. According to the optimal control theory, 
satisfies the HJB equation for any 
.
, and the value 
of can be obtained later.
Corollary 7 In the collaborative cooperation contract, the optimal effort level of the manufacturer and the retailer (i.e., 
and 
) and the overall optimal benefit 
of the Supply Chain are positively related to the sum 
of the marginal benefit of the manufacturer and the retailer, and the influence coefficient 
of product goodwill on market demand. Meanwhile, it is negatively correlated with the input cost coefficients of manufacturers and retailers (i.e., 
and 
), and the natural decay coefficient 
of product goodwill.
Corollary 8 In the collaborative cooperation contract, the government does not need to subsidize the manufacturers for technical R&D, but only needs to subsidize water-saving products. Also, the more the government subsidizes the product, the greater the market demand, and the greater the optimal benefit of the Supply Chain.
COMPARATIVE ANALYSIS
In the three cases, a comparative study on the optimal effort strategy and optimal benefit of manufacturers and retailers, government R&D subsidy to manufacturers and product subsidy to consumers is carried out, and the following conclusions are obtained.
Theorem 1 In the collaborative cooperation contract, the technical effort level 
of the manufacturer and the publicity effort level 
of the retailer are both the highest.
Proof: For manufacturers of water-saving products, 
can be obtained from Equations (14), (32), and (48).
Hence, 
, the following equations can be derived from Equations (14), (32), and (48):
Hence, 
In one word, when 
, the technical effort level of the manufacturer remains unchanged under the three contracts. Compared with non-cooperative contract without cost-sharing, the publicity effort level of the retailer has increased under cost-sharing contracts. While under the collaborative cooperation contract, both the technical effort level of the manufacturer and the publicity effort level of the retailer are the highest.
Theorem 2 Compared with the non-cooperative contract without cost sharing, the benefits of manufacturers and retailers (i.e., 
and 
) have achieved Pareto improvement under the cooperative contract with cost sharing.
Hence, 
.
Hence, 
.
When 
, the benefits of both the manufacturer and the retailer in the cooperative contract with cost sharing are greater than those without cost sharing. This shows that the benefits of both the manufacturer and the retailer have achieved Pareto improvement through advertising cost-sharing approach.
Theorem 3 In the absence of government subsidy, the overall optimal value of the Supply Chain under the collaborative cooperation contract is greater than that in the other two cases.
Proof: Based on Theorem 2, it is known that 
, and 
Hence, 
.
Therefore, when 
and 
, the overall benefit of the Supply Chain under the cooperative contract of cost sharing is greater than the benefit of that under the non-cooperative contract without cost sharing. Meanwhile, the overall benefit of the Supply Chain under the collaborative cooperation contract is greater than that under the cooperative contract with cost sharing. However, it is worth noting that only when the individual benefits of the manufacturer and the retailer under the collaborative cooperation contract are greater than that under the cooperative contract with cost sharing, both bodies are likely to adopt the collaborative strategy. However, how the two bodies distribute benefits under the collaborative cooperation contract depends on the negotiating ability of the two. That is, if the final scheme is feasible, the benefits of both manufacturers and retailers can achieve Pareto optimality.
Theorem 4 Under the collaborative cooperation contract, manufacturers are the least dependent on government R&D subsidies.
Proof: From Equations (26), (44), and (57), it can be known that 
and 
, then 
.
In other words, when the government does not provide the manufacturer with technical R&D subsidies under the collaborative cooperation contract, the manufacturer will determine their technical effort level according to the principle of optimal output to achieve the optimal output. In addition, the proportion of government subsidies to manufacturers is determined by that of manufacturers’ profits in the Supply Chain under the cooperative contract with cost sharing. The proportions of government subsidies to manufacturers are equal to the two cases of the non-cooperative contract and the cooperative contract with cost sharing. With the closer cooperation between manufacturers and retailers, the optimal proportion of government subsidies to manufacturers decreases. Finally, under the collaborative cooperation contract, the proportion of government subsidies to manufacturers is 0.
Theorem 5 The higher the government's subsidy ratio 
for water-saving products, the greater the product goodwill 
and market demand Q.
Proof: Conduct the first derivative of Equations (3) and (4), respectively, and let it be 0, then 
, and 
.
According to Theorem 5, the higher the government's product subsidy ratio to consumers, the higher the product goodwill and the greater the market demand. Combined with the conclusions in Model C, it can be seen that the government can realize the Pareto optimal of the overall benefit of the Supply Chain only by providing product subsidies to consumers. Also, the higher the government's subsidy ratio to consumers, the greater the overall benefit of the Supply Chain.
NUMERICAL SIMULATION
In cooperative and non-cooperative games, the optimal decisions and benefits of manufacturers and retailers, as well as the total benefit of the Supply Chain, depend on the choice of parameters in the model. The data used in this paper were obtained from actual cases and were processed to meet these hypotheses. In order to explore the impact of the changes of each parameter on manufacturers and retailers and based on the literature (Gao et al., 2016; Chen et al., 2020), this study assumes that the values of the parameters are as follows: 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
. It should be noted that these parameters are set to meet the conditions mentioned in the manufacturer's cost sharing.
The technical effort level of manufacturer and the publicity effort level of retailer in the three contracts are shown in Table 1.
The technical effort level of manufacturer and the publicity effort level of retailer in the three contracts.
| Model | Non-cooperative contract without cost sharing (N) | Cooperative contract with cost sharing (S) | Collaborative cooperation contract (C) | Results comparison |
|---|---|---|---|---|
 | 16 | 16 | 16 |  |
 | 12.5 | 13.75 | 20 |  |
 | 0.625 | 0.625 | 0 |  |
 | – | 0.09 | – | – |
 | 475.7665 | 496.6 | 600.7665 |  |
| Manufacturer's profit | 5,936.499 | 6,077.125 | – |  |
| Retailer's profit | 10,331.665 | 10,696.25 | – |  |
| Supply Chain profit | 16,268.164 | 16,773.375 | 16,830.664 |  |
| Model | Non-cooperative contract without cost sharing (N) | Cooperative contract with cost sharing (S) | Collaborative cooperation contract (C) | Results comparison |
|---|---|---|---|---|
| | 16 | 16 | 16 | |
| | 12.5 | 13.75 | 20 | |
| | 0.625 | 0.625 | 0 | |
| | – | 0.09 | – | – |
| | 475.7665 | 496.6 | 600.7665 | |
| Manufacturer's profit | 5,936.499 | 6,077.125 | – | |
| Retailer's profit | 10,331.665 | 10,696.25 | – | |
| Supply Chain profit | 16,268.164 | 16,773.375 | 16,830.664 | |
It can be seen from Table 1 that: (1) The technical effort level of the manufacturer is equal under the three contracts, while the publicity effort level of the retailer increases with the deepening of the cooperation between the two, and reaches the maximum when the cooperation is coordinated. (2) Under the non-cooperative contract without cost sharing, the government subsidy intensity to manufacturers is fixed. With the highest level of cooperation between manufacturers and retailers, the government's subsidy will drop to zero. Hence, manufacturers and retailers can achieve the maximum overall benefit based on the optimal output ratio. (3) The market demand for water-saving products is the highest under the collaborative cooperation contract, the next one under the cooperative contract with cost sharing, and the smallest under the non-cooperative contract without cost sharing. (4) Compared with the non-cooperative contract without cost sharing, the profits of both the manufacturer and the retailer under the cooperative contract with cost sharing have achieved Pareto improvement. (5) The overall benefit of the Supply Chain reaches the maximum value under the collaborative cooperation contract, and the overall benefit under the cooperative contract with cost sharing reaches the second best.
When the main bodies of the Supply Chain implement the non-cooperative contract without cost sharing, the goodwill of water-saving products is 
; the market demand for water-saving products is 
; the manufacturer's optimal benefit is 
; and the retailer's optimal benefit is 
. When the main bodies of the Supply Chain implement the cooperative contract with cost sharing, the goodwill of the water-saving product is 
. The market demand for water-saving products is 
; the manufacturer's optimal benefit is 
; and the retailer's optimal benefit is 
. When the main bodies of the Supply Chain implement the collaborative cooperation contract, the product goodwill is 
; the market demand for water-saving products is 
; and the overall optimal benefit of the Supply Chain is 
.
on the Pareto improvement results under the cooperative contract with cost sharing, Figure 6 is the impact of the change of parameter 
on the retailer's publicity effort level under the cooperative contract with cost sharing, Figure 7 is the impact of the change of parameter 
on the overall interests of the Supply Chain under the cooperative contract with cost sharing, and Figure 8 is the overall benefit of the Supply Chain under the three contracts. It can be seen from Figure 1 that the product goodwill W increases and tends to be stable over time. The product goodwill in Model C is the largest, followed by Model S, and that in Model N is the smallest. The results show that, compared with the non-cost sharing contract, the cost sharing contract can improve the goodwill of water-saving products. Also, when the manufacturer and the retailer cooperate, the goodwill can be further improved significantly.
According to Figure 2, the market demand Q increases and tends to be stable over time. The market demand in Model C is the largest, while that in Model S is second, and that in Model N is the smallest. The results show that, compared with non-cooperative contract without cost sharing, if the manufacturer shares the retailer's advertising cost, the market demand for water-saving products can be increased.
Figure 3 shows that the cooperative contract with cost sharing can achieve Pareto improvement of the manufacturer and the retailer, and the improvement result for the retailer's benefit is better than that for the manufacturer. Since the manufacturer shares part of the retailer's publicity cost, the retailer reduces their publicity input cost. The retailer will put more effort into promoting the water-saving product to improve their own revenue and product goodwill, and the increase in sales will ultimately improve the profitability of the manufacturer and the retailer, resulting in a Pareto improvement for the two.
Figure 4 shows that under the cooperative contract with cost sharing, that is,
, as s increases, the manufacturer's cost-sharing ratio also increases gradually. Since 
does not change, that is, as the manufacturer's profit margin 
increases, the proportion of the retailer's advertising cost that the manufacturer is willing to share also gradually increases. In the same market, when the manufacturer's profit increases, the manufacturer expects to occupy a larger market share. Meanwhile, the publicity of retailers can improve the product goodwill, increase its sales volume, thereby increasing the profits of manufacturers, and finally achieve a win-win situation for the two. The result also validates Corollary 4.Figure 5 shows that the Pareto improvement of the cooperative contract with cost sharing on the benefits of the manufacturer and the retailer increases with the decrease of the retailer's input cost coefficient. Also, the benefit improvement effect of the retailer is better than that of the manufacturer. The smaller the retailer's input cost and input cost coefficient, the higher their benefit added value. Meanwhile, the higher the benefit added value of the manufacturer, the stronger the manufacturer's willingness to share the cost; and vice versa. It is consistent with the reality. When the retailer's input cost is lower, the publicity effect achieves better economic benefits, and further manufacturers also obtain higher economic benefits. Therefore, manufacturers will be more willing to cooperate with retailers.
As can be seen from Figure 6, under the contracts with cost sharing, with the increase of the retailer's publicity effort cost
, the retailer's effort level gradually decreases. In addition, with the increase of the value 
, the publicity effort level 
of the retailer under the collaborative cooperation contract is always greater than the publicity effort level 
of the retailer under the cooperative contract with cost sharing. Therefore, compared with the cooperative contract with cost sharing, the collaborative cooperation contract can improve the retailer's publicity effort level, and their publicity effort cost is negatively correlated with the effort level.As can be seen from Figure 7, under the contracts with cost sharing, the greater the retailer's publicity effort cost
, the smaller the overall benefit of the Supply Chain over time. Moreover, with the increase of the value 
, the overall benefit 
of the Supply Chain under the collaborative cooperation contract is always greater than the overall benefit 
of that under the cooperative contract with cost sharing. In other words, under the contracts with cost sharing, the collaborative cooperation contract is not only conducive to the cooperation and communication of the Supply Chain, but also to improve its overall revenue.Figure 8 shows that the overall benefits of the Supply Chain have increased and stabilized over time. Also, it is clear that the overall benefit of the Supply Chain in Model C is the largest, followed by that in Model S, and the smallest in Model N. Compared with the non-cooperative contract without cost sharing, the overall benefit of the Supply Chain under the contracts with cost sharing is much greater than that of the non-cooperative cooperation, which also verifies Theorem 3. This conclusion implies that cooperative decision-making is superior to non-cooperative decision-making, which can provide a theoretical basis and reference for manufacturers and retailers on advertising and promotion of water-saving products.
Product goodwill under three contracts.
Market demand under three contracts.
The optimal benefits of the manufacturer and the retailer in Model N and Model S.
The impact of the change of parameter s on the cost sharing ratio.
The effect of the change of parameter 
on the Pareto improvement results under the cooperative contract with cost sharing.
The effect of the change of parameter on the Pareto improvement results under the cooperative contract with cost sharing.
The impact of the change of parameter 
on the retailer's publicity effort level under the cooperative contract with cost sharing.
The impact of the change of parameter on the retailer's publicity effort level under the cooperative contract with cost sharing.
The impact of the change of parameter 
on the overall benefit of the Supply Chain under the cooperative contract with cost sharing.
The impact of the change of parameter on the overall benefit of the Supply Chain under the cooperative contract with cost sharing.
The overall benefit of the Supply Chain under the three contracts.
CONCLUSION AND DISCUSSION
Taking a two-level Supply Chain consisting of a manufacturer and a retailer as the research object, based on a differential game theory and taking into account government R&D subsidies and product subsidies for manufacturers, this study establishes three game models of water-saving products, namely the non-cooperative contract without cost-sharing, the cooperative game with cost sharing, and the collaborative cooperation game. A comparative analysis was also carried out. The results show the following: (1) Compared with the non-cooperative contract without cost sharing, the product goodwill and market demand for water-saving products under the cooperative contract with cost sharing have achieved Pareto improvement, and finally reached Pareto optimality. (2) Under the cooperative contract with cost sharing, if the manufacturer shares the retailer's publicity cost, the Pareto improvement of the benefits of both bodies can be achieved respectively, and the condition of the cost-sharing contract is 
. (3) When transitioning from the non-cooperative contract without cost sharing to the cooperative contract with cost sharing, the degree of change in the effort level of the manufacturer and the retailer is different. As the manufacturer shares some of the advertising costs, the retailer will increase its effort level, whereas the manufacturer's effort level will remain the same. (4) Under the cooperative contract with cost sharing, as the retailer's publicity effort cost increases, their publicity effort level and the overall benefit of the Supply Chain decrease. Moreover, retailers’ publicity effort level and the overall benefits of the Supply Chain under the collaborative cooperation contract are consistently better than that under the cooperative contract with cost sharing. (5) When manufacturers and retailers cooperate, the benefits of the two and the overall benefits of the Supply Chain are optimal, which also provides a theoretical basis and decision-making reference for the main bodies in advertising cooperation.
In addition, the research has positive guiding significance for the decision-making of the members of the Supply Chain and the policy-making of the government under the water scarcity scenario. The purpose of advertising is to maximize profits. Since the fact that manufacturers share the advertising costs of retailers can significantly improve the revenue of both and the overall revenue of the Supply Chain, the manufacturer should choose the appropriate advertising method according to the product profit. When the manufacturer's profit is higher, it is the best choice for the manufacturer to choose the collaborative cooperation contract. The cooperation among Supply Chain members has become the key to the decision-making process of enterprise management. However, the conclusion of the collaborative cooperation contract not only needs to weigh the profitability and cost and revenue distribution ratio of both parties, but also depends on the negotiating ability of the two, which is also a thorny issue faced by the members of the Supply Chain members. In addition, when manufacturers share the publicity cost with retailers, retailers should improve their publicity effort level to increase the exposure of water-saving products. In addition, the government should carry out education on national water conditions, raise public awareness of water conservation, and guide consumers to choose water-saving products; moreover, the government can issue incentive policies for water-saving technology R&D and strengthen international cooperation to promote international water-saving technology transfer, thereby reducing R&D costs and improving the conversion rate of water-saving products.
To sum up, the collaborative cooperation contract is an ideal state in the Supply Chain. The cost-sharing ratios and benefit distribution ratios among all parties in the Supply Chain play a leading role in the signing and execution of collaborative cooperation contract, but how to determine these ratios is still in the black box. In addition, the competition between the two manufacturers is also a hot spot. In the future, we will focus on the conflict of cooperative advertising under the collaborative cooperation contract of the Supply Chain.
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.
CONFLICT OF INTEREST
The authors declare there is no conflict.
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