Cooperative Advertising Strategy Selection Problem for considering Pricing and Advertising Decisions in a Two-Period Online Supply Chain

Abstract

This article studies the cooperative advertising problem of a two-period online supply chain consisting of a manufacturer and an online retail platform. The manufacturer provides national advertising in the first period to build the brand image and increase the awareness of the product. And the online retail platform provides platform advertising for selling the product to consumers on its platform during two periods. The manufacturer and the online retail platform may choose different cooperative advertising strategies for national advertising and platform advertising, which are one-way subsidy strategy, two-way subsidy strategy, and revenue-share strategy. We formulate a Stackelberg game model to study the cooperative advertising problem by taking price and advertising effect into account and analyze how the profit is influenced in different cooperative advertising strategies. We find that under the revenue-share strategy, the manufacturer provides a higher subsidy rate for the online retail platform advertising than that in other cooperative advertising strategies. Interestingly, there are conditions where, while just the manufacturer contributes a percentage of the platform advertising and the online retail platform has no effort on the national advertising, the total profit would be better than that in revenue-share strategy even in revenue-share strategy, the cooperative relationship is closer between the manufacture and the online retail platform.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Cooperative advertising is a powerful tool commonly used in marketing channels where one party in the channel agrees to subsidize the advertising expenditure of the other [1]. Nagler [2] found that in 2000, the total expenditures on cooperative advertising in the USA were estimated at $15 billion, and about $50 billion was spent on cooperative advertising in 2009 [3]. The practice of cooperative advertising has seen a significant increase in recent years [1, 4]. In a typical arrangement, manufacturers agree to share part of the local advertising cost for retailers, which is called a one-way subsidy. While manufacturers are willing to bear part of the local advertising cost for retailers, retailers are also willing to bear part of the cost of manufacturers’ national advertising, which is called a two-way subsidy [5]. In this case, manufacturers and retailers bear part of each other’s advertising expenses, commonly known as the subsidy rate.

In the era of online retail, the cooperative advertising relationship is reflected in the cooperation process between national advertising and platform advertising. Generally, platform advertising (such as “Tianhe Plan” of Alibaba and “Donglian Plan” of JD.com) is managed by online retail platforms because online retail platforms usually better understand the effectiveness of platform advertising media, platform preferences, and the interests of demographic consumers. As a powerful tool commonly used in marketing channels, cooperative advertising not only enables manufacturers and retailers to share part of the advertising costs for each other but also enables them to benefit from this alliance by increasing sales [6]. For example, in China, “The Media Resource Replacement Program” is currently a collaborative marketing strategy widely adopted by online retail platforms and manufacturers. In this plan, manufacturers would reflect the brand visual elements of online retail platforms (platform name, logo, etc.) in their national advertising, and online retailer platforms provide a corresponding proportion of platform advertising resource replacement in return [7]. Manufacturers intend to build a brand image by national advertising whereas online retail platforms aim at increasing sales by platform advertising [8, 9]. Advertising cooperation between manufacturers and online retail platforms can effectively improve the profit of the participating parties [10]. For example, since 2015, Tmall (B2C platform of Alibaba) has established advertising cooperation with more than 200 brands, such as Maserati, Beats, and Swarovski, and helps brands build the double harvest of brand and sales on “Super Brand Day.” And Amazon announced a new advertising program for its retail business to compete with Google and Facebook in 2018. Apart from cooperative advertising, manufacturers and online retail platforms also have other methods to strengthen the cooperation relationship, such as revenue-share strategy [11]. For example, the Apple App Store and Google Play use uniform revenue-share strategies to engage with all developers whose apps they sell; through Amazon Marketplace, the seller pays Amazon a certain percentage of the retail price and a fixed fee [12]. Under the revenue-share strategy, manufacturers could be motive to spend more on adverting and reduce the wholesale price because they could get a portion of online retail platforms’ sales profit, and online retail platforms may benefit from the advertising and the lower wholesale price even if they must share their revenue with manufacturers in the retail market.

Based on these observations, we are interested in the following questions. (1) What is the impact of cooperative advertising on firms’ advertising decisions and price decisions during a two-period supply chain? (2) How do the profits for the manufacturer and the online retail platform change in different cooperative advertising strategies? (3) Whether and when can the firms benefit from cooperative advertising?

We attempt to address these issues in this paper; we develop a two-period online supply chain system consisting of a manufacturer and an online retail platform. Different from previous models, this paper considers that the impact of national advertising, platform advertising, and product price on demand is changing over time, and this naturally divides the selling season into two periods. The manufacturer sells his product through the online retail platform and chooses different levels of national advertising. The online retail platform sells the product to consumers and chooses different levels of platform advertising. The manufacturer and the online retail platform may choose different cooperative advertising strategies for national advertising and platform advertising, which are one-way subsidy strategy, two-way subsidy strategy, and revenue-share strategy. We formulate a Stackelberg game model to study the cooperative advertising problem by taking price and advertising effect into account and analyze how the profit is influenced in different cooperative advertising strategies.

The findings reveal the impacts of the cooperation strategies on cooperative advertising. First, under the revenue-share strategy, the manufacturer provides a higher subsidy rate for platform advertising than that in other strategies. Second, in the revenue-share strategy, there is a special region. With the increase of the revenue-share rate, the online retail platform obtains the manufacturer’s advertising subsidy support by transferring more revenue to the manufacturer. Thrid, by identifying the value of the cooperative advertising, the manufacturer and the online retail platform just taking a two-way subsidy strategy is the best choice for the online supply chain system.

This paper contributes to the literature in the following aspects. First, it considers the cooperative advertising problem in a multi-period supply chain, while most articles consider the cooperative advertising problem in a single-period supply chain [10, 13, 14]. Second, this paper considers the impact of price and advertising on market demand simultaneously and considers the different effects of national advertising and platform advertising on consumers in different periods. Third, different from Zhang et al. [15] who consider the cooperative advertising problem under one- and two-way subsidy strategies, this paper adds a revenue-share strategy to enrich the cooperative advertising choice for enterprises.

We organize the remainder of this paper as follows. We discuss the relevant literature in Section 2. The model and the analysis of results are presented in Sections 3 and 4. We provide a numerical analysis in Section 5. In Section 6, we conclude the paper and point out further research for future work. The derivation of equilibrium and the proof of propositions are provided in the Appendix.

2. Literature Review

Cooperative advertising was first used in the United States in the 1990s. As cooperative advertising programs are widespread in practice, many practitioners and researchers pay more attention to the effect of cooperative advertising on demand and profit [16]. Chen [17] studied the profit model of manufacturers and retailers from the perspective of traditional retail channels and online direct sales channels. On the basis of Chen [17], Chen et al. [13] studied the problem of cooperative advertising under the simultaneous price and advertising competition between manufacturers’ online channels and traditional retail channels. Chen et al. [13] pointed out that the increase of price discounts will reduce the enthusiasm of retailers’ advertising investment. From the perspective of the free-rider effect, Tao and Li [14] considered the pricing strategies of members in the O2O channel supply chain and the impact of the free-rider effect on channel profits and analyzed the optimal pricing strategy under centralized and decentralized. Different from Tao and Li [14], Wang and Shu [18] considered the cooperative advertising strategy of the O2O supply chain, studied three game theory models, and deduced the optimal decision of advertising effort and participation rate among supply chain members. On this basis, Shu and Wang [7] further studied the impact of channel power structure on the cooperative advertising decision-making of O2O supply chain channel members. They found that the manufacturer’s national advertising investment and the overall profit of the supply chain in the centralized decision-making model were the largest.

In the following papers, some researchers have extended the cooperative advertising model and explored the cooperative advertising strategy in the O2O supply chain composed of sellers and online platform agents under the “Internet plus” environment [19]. Li et al. [19] established a mathematical model of cooperative advertising in O2O supply chain composed of sellers and network platform agents, proposed three cooperative advertising modes, and deduced the optimal decision-making of advertising level and participation rate among supply chain members by using quantitative modeling method for cooperative advertising in the supply chain. On the basis of Li et al. [19], Li et al. [20] conducted further extended research. They constructed three different game decision models for a dual-channel manufacturer supply chain system carrying out online direct sales business and analyzed and compared the optimal decisions of dual-channel commodity pricing and advertising investment in different situations. When the manufacturer carries out an online direct sales channel through the third-party online retail platform, the manufacturer should actively choose to carry out a cooperative advertising strategy with the online retail platform and traditional retailers at the same time [10]. In the cooperative advertising practice between online retail platforms and manufacturers, the platform cross-sales effect is the key factor affecting the formulation and implementation of a cooperative advertising plan. When the platform cross-sales effect is large, the profit of online retail platform is greater than that of manufacturers [21].

Taking into account the impact of advertising on the reference price, Zhang et al. [15] propose a dynamic cooperative advertising model for a manufacturer-retailer supply chain. Assuming that consumers’ goodwill and product reference price are affected by advertising, they use differential dynamics equation to model and consider how the reference price effect affects the decision-making of all channel members under the advertising cooperation strategy of one- and two-way subsidies. Different from them, our paper extends the understanding of the research by considering the case where there is a two-period operation in the online supply chain system, and the manufacturer and the online retail platform could use a two-way subsidy strategy or revenue-share strategy to enhance the cooperative degree based on the result in Qi and Xie [22]. The demand function is affected by price and advertising simultaneously, and the national advertising and the platform advertising have different effects on consumers during the two periods.

Table 1 reveals that some publications considered a two-period supply chain, while others considered one or two aspects of cooperative advertising. The studies rarely considered different cooperative advertising strategies, the effects of advertising, and price simultaneously, so we will address this gap in our study.

Table 1

The summary of literature and factors addressed by our study.

Reference	Online cooperative advertising	Two period supply chain	One-way Subsidy Contract	Two-way Subsidy Contract	Revenue-Share Contract	Advertising and Price
Zhang et al [15]			✓	✓		✓
He et al [23]		✓	✓	✓	✓
Chen [17]	✓		✓		✓	✓
Chen et al [13]	✓		✓		✓	✓
Tao and Li [14]			✓
Li et al [19]	✓		✓	✓
Yan and he [24]		✓	✓	✓	✓	✓
Shu and Wang [7]	✓		✓	✓
Shu and Wang [21]	✓			✓	✓
Gao et al [10]	✓		✓
Li et al [20]	✓		✓			✓
Our study	✓	✓	✓	✓	✓	✓

3. Model Framework

In this section, we consider an online supply chain consisting of a single manufacturer and a single online retail platform in a two-period setting, and the online retail platform sells the product to consumers by its retail platform. Both the manufacturer and the online retail platform would provide the advertising for the product. The manufacturer provides national advertising in the first period to build the brand image and increase the awareness of the product, and the online retail platform provides platform advertising to promote consumers to buy the product on its platform during two periods [23]. For example, Apple would make the national advertising on Google and other browsers to build Apple’s image. At the same time, Amazon Marketplace would provide the platform advertising for Apple to attract consumers to buy these products on his online platform. The manufacturer and the online retail platform choose different strategies for the cooperation between national advertising and platform advertising, which are one-way subsidy strategy, two-way subsidy strategy, and revenue-share strategy. In this paper, the manufacturer is the leader, and the online retail platform is the follower. The manufacturer decides the national advertising level $A$ and the subsidy rate ${\varphi }_1$ and ${\varphi }_2$ for platform advertising [24]. And the online retail platform decides the platform advertising level $e_i$ and the retail price $p_i$ during two periods ($i\in \mathrm{1,2}$). Especially, the wholesale price $w$ is an exogenous variable and $w<p_i$. The specific structure is shown in Figure 1.

[figure omitted; refer to PDF]

By researching the prevalent assumption in the literature, we can assume the consumer demand as the following form:$$\begin{array}{}\text{(1)}& D{p}_i,e_i,A=g{p}_i\times h{e}_i,A.\end{array}$$

The demand function is similar to Xie and Wei [25], SeyedEsfahani et al. [26], Aust and Buscher [27], and He et al. [28], where $g{p}_i$ reflects the impact of the retail price on the demand and $h{e}_i,A$ reflects the impact of the advertising expenditures on the demand. The demand expression for the first period is as follows:$$\begin{array}{}\text{(2)}& D_1=1-\delta p_1{a}_1+\beta A+\lambda e_1,\end{array}$$where $\delta$, $\beta$, and $\lambda$ are positive constants and $\beta$ and $\lambda$ reflect the efficacy of each type of advertising in generating sales. $a_1>0$ is the potential demand for the first period. Because $1-\delta p_1>0$, we get $0<p_1<1/\delta$ and $w<p_1<1/\delta$.

Generally, manufacturers usually do national advertising when the product first appears in the market. Because manufacturers in the market are normally good at R & D (research and development) and producing but not marketing. We assume that the manufacturer only does the national advertising in the first period.

Therefore, the demand function in the second period would be as follows:$$\begin{array}{}\text{(3)}& D_2=1-\delta p_2{a}_2+\theta \lambda e_2,\end{array}$$where $a_2>0$ is the potential demand for the second period and $0<\theta \le 1$ means the platform advertising effect would weaken in the second period. For the same reason, we can get $0<p_2<1/\delta$ and $w<p_2<1/\delta$.

And the cost functions for the national advertising and the platform advertising with respect to the advertising level are as follows:$$\begin{array}{c}\text{(4)}& CA=\frac{1}{2}{A}^2,C{e}_1=\frac{1}{2}{e}_1^2,\\ C{e}_2=\frac{1}{2}{e}_2^2.\end{array}$$

The descriptions of the other parameters are shown in Table 2.

Table 2

Descriptions of parameters.

Parameter symbol	Symbol description
$i\in \mathrm{1,2}$	The first period and the second period
$j\in o,t,r$	$o$ – one-way subsidy strategy, $t$ – two-way subsidy strategy, and $r$ – revenue-share strategy
$p_i^j$	Retail prices of the products under the strategy $j$ in the $i$ period
$e_i^j$	Platform advertising level of the product under the strategy $j$ in the $i$ period
${\varphi }_i^j$	The subsidy rate of the platform advertising under the strategy $j$ in the $i$ period
$A^j$	National advertising level of the product under the strategy $j$
$\psi$	The subsidy rate of the national advertising by the online retail platform
$\eta$	The revenue-share rate
${\pi }_R^j$	Profit of the online retail platform under the strategy $j$
${\pi }_M^j$	Profit of the manufacturer under the strategy $j$

3.1. One-Way Subsidy Strategy

In this situation, only the manufacturer shares part of the platform advertising cost for the online retail platform. And the game sequence is as follows: first, the manufacturer determines the subsidy rate for the platform advertising. Second, the manufacturer determines the national advertising level. Third, the online retail platform determines the retail price and the platform advertising level for the first period. Finally, the online retail platform determines the above two variables for the second period. The profit functions for the manufacturer and the online retail platform are given by$$\begin{array}{c}\text{(5)}& {\pi }_M^o=w\times D_1+w\times D_2-\frac{1}{2}{A}^{o2}-\frac{1}{2}{\varphi }_1^o{e}_1^{o2}-\frac{1}{2}{\varphi }_2^o{e}_2^{o2},{\pi }_R^o=p_1^o-w{D}_1+p_2^o-w{D}_2-\frac{1}{2}1-{\varphi }_1^o{e}_1^{o2}-\frac{1}{2}1-{\varphi }_2^o{e}_2^{o2},\end{array}$$where ${\varphi }_i^o$ is the manufacturer’s subsidy rate for the platform advertising and $i\in \mathrm{1,2}$.

Using backward induction, we obtain the Stackelberg equilibrium results.

Proposition 1.

The optimal advertising level, subsidy rate, and retail price are as follows:$$\begin{array}{c}\text{(6)}& A^o=\frac{1}{2}w\beta 1-w\delta ,{\varphi }_i^o=\frac{5w\delta -1}{1+3w\delta },\\ e_1^o=\frac{1-w\delta 1+3w\delta \lambda }{8\delta },\\ e_2^o=\frac{1-w\delta 1+3w\delta \theta \lambda }{8\delta },\\ p_i^o=\frac{1+w\delta }{2\delta },\end{array}$$where $i\in \mathrm{1,2}$. The profits for the manufacturer and the online retail platform are as follows:$$\begin{array}{c}\text{(7)}& {\pi }_M^o=\frac{-1+w\delta -16w^2{\beta }^2{\delta }^2+16w^3{\beta }^2{\delta }^2-{\lambda }^2-5w\delta {\lambda }^2-3w^2{\delta }^2{\lambda }^2+9w^3{\delta }^3{\lambda }^2-{\theta }^2{\lambda }^2-5w\delta {\theta }^2{\lambda }^2-3w^2{\delta }^2{\theta }^2{\lambda }^2+9w^3{\delta }^3{\theta }^2{\lambda }^2-64w{\delta }^2{a}_1-64w{\delta }^2{a}_2}{128{\delta }^2},{\pi }_R^o=\frac{{-1+w\delta }^2-8w{\beta }^2\delta +8w^2{\beta }^2{\delta }^2-{\lambda }^2-2w\delta {\lambda }^2+3w^2{\delta }^2{\lambda }^2-{\theta }^2{\lambda }^2-2w\delta {\theta }^2{\lambda }^2+3w^2{\delta }^2{\theta }^2{\lambda }^2-16\delta a_1-16\delta a_2}{-64{\delta }^2}.\end{array}$$

Proposition 1 shows that as the wholesale price $w$ varies, the optimal subsidy rate ${\varphi }_i^o$ falls into two different regimes. When the wholesale price is high ($1/5\delta \le w<1/\delta$), the manufacturer chooses to invest optimal platform advertising level, and when the wholesale price is low ($0<w<1/5\delta$), the manufacturer chooses to invest nothing in the platform advertising.

In fact, if the wholesale price is lower in the wholesale market, manufacturers would not get enough money to invest in the cooperative advertising strategy. The profit of manufacturers in the wholesale market will affect their willingness and decision on sharing the advertising cost of retailers.

Then, differentiating $A^o$ and $e_i^o$ with respect to $w$, we have Proposition 2.

Proposition 2.

Studying the sensitivity analysis with respect to $w$.

(a) $\partial A^o/\partial w>0$ and $\partial e_i^o/\partial w>0$, when $0<w<1/3\delta$

(b) $\partial A^o/\partial w>0$ and $\partial e_i^o/\partial w<0$, when $1/3\delta <w<1/2\delta$

(c) $\partial A^o/\partial w<0$ and $\partial e_i^o/\partial w<0$, when $1/2\delta <w<1/\delta$

In this research, we consider the national and platform advertising levels are affected by the wholesale price in the market and get the different changing trends for $A^o$ and $e_i^o$ presented in Figure 2 and express them mathematically as follows.

From Figure 2, we know that when the wholesale price is very low in region I ($0<w<1/3\delta$), with the wholesale price increasing, the manufacturer and the online retail platform could pay more effort on the advertising. The manufacturer could get more profits in the wholesale market with the wholesale price increasing, and the online retail platform hopes to incentive more consumers to buy the product with the platform advertising. When the wholesale price is moderate in region II ($1/3\delta <w<1/2\delta$), with the wholesale price increasing, the manufacturer could pay more effort on the national advertising, but the online retail platform reduces the effort on the platform advertising. The online retail platform could pay more money in the wholesale market with the increasing wholesale price. When the wholesale price is very high in region III ($1/2\delta <w<1/\delta$), with the wholesale price increasing, the manufacturer and the online retail platform could pay less effort on the advertising. The manufacturer could get a lot of profits in the wholesale market and have no incentive to make great efforts on the national advertising.

In reality, when the wholesale price of products in the wholesale market is higher, manufacturers have no motivation to do advertising for products. Higher wholesale price may be due to the higher value of the product itself or no alternative product in the market. So manufacturers do not need advertising to increase the market demand for products.

In this section, we characterize that just the manufacturer contributes some efforts for platform advertising. Next section, we characterize the two cases: “two-way subsidy strategy” and “revenue-share strategy” for cooperation advertising between manufacturer and the online retail platform.

[figure omitted; refer to PDF]

From Figure 4, we find that in a two-way subsidy strategy, the total profit of the supply chain is always better than that in other cooperation strategies ($S^t>S^r$ and $S^t>S^o$). In a two-way subsidy strategy, the manufacturer and the online retail platform bear each other’s advertising investment costs and share the advertising investment risks. A two-way subsidy strategy encourages them to pursue more corporate profits by appropriately increasing advertising investment and brings higher total profits of the supply chain. At the same time, we also find that when the wholesale price of the product takes a special value, the total profit of the supply chain in a one-way subsidy strategy is the same as that in the revenue-share strategy. Set this specific value to $w\prime$ here. When $w<w\prime$, the closer cooperation between the manufacturer and the online retail platform in revenue-share strategy could make the total profit better than that in one-way subsidy strategy ($S^r>S^o$). Because when the wholesale price of the product is low, the online retail platform has greater power and ability to invest in advertising to promote the product. The total profit of the whole supply chain will increase due to the improvement of the product popularity and competitiveness. When $w>w\prime$, the closer cooperation between the manufacturer and the online retail platform in revenue-share strategy could make the total profit worse than that in one-way subsidy strategy ($S^r<S^o$). Because when the wholesale price of the product is high, the online retail platform needs to invest more purchase costs in the wholesale market. At the same time, the online retail platform also needs to share its sales revenue with the manufacturer in revenue-share strategy. It would have no power and ability to invest in advertising, which will affect the market sales of the product and reduce revenue.

In practical terms, starting from the overall interests of the supply chain, supply chain participants should actively explore reasonable cooperative alliances to increase the total profit of the supply chain.

We compare the profit of the online retail platform under three strategies and get Figure 5 as follows.

[figure omitted; refer to PDF]

From Figure 5, we find that the profit of the online retail platform under a two-way subsidy strategy is always better than that in one-way subsidy strategy and revenue-share strategy even under two-way subsidy strategy, the online retail platform must provide some money for the manufacturer’s national advertising. In particular, in the revenue-share strategy, the online retail platform needs to transfer part of the product sales revenue to the manufacturer in exchange for cooperation so that his revenue is far less than the profit in other cooperation strategies. In fact, retailers should actively strengthen the cooperation between retailers and manufacturers and improve their profit by signing a two-way subsidy strategy.

Furthermore, we compare the manufacturer’s profit under three strategies and also observe the interesting conclusion (see Figure 6).

[figure omitted; refer to PDF]

Because, under the revenue-share strategy, the manufacturer would get a portion of the online retail platform’s selling product profit, we consider that the profit of the manufacturer under the revenue-share strategy is better than that in other strategies. Figure 6 shows that when the wholesale price is high, the manufacturer’s profit is worse than that in other strategies even if he would get the income from the wholesale market and the retail market. In revenue-share strategy, the online retail platform needs to share the product sales revenue of its own retail market with the manufacturer. With the increase of product wholesale price, the online retail platform will also increase product retail price to strive for more profits. Too high product retail prices will hurt consumers’ enthusiasm to buy the product. When the manufacturer’s revenue in the wholesale market decreases, his share of sales revenue will also decrease. Therefore, when the wholesale price of the product is too high, the manufacturer’s profit may be lower than that in the other two strategies. Actually, manufacturers can choose the cooperation strategy with retailers according to the changes of product wholesale price, rather than invariable.

6. Conclusion

As an important consideration when consumers decide whether to buy a product or not, the retail price and the advertising can both influence consumers’ willingness [15]. This paper investigates the cooperative advertising problem by taking price and advertising effect into account in the two-period online supply chain. Utilizing the Stackelberg game, the manufacturer’s and the online retail platform’s optimal level on national and platform advertising are calculated in three different cooperation adverting strategies. As Zhang et al. [15], in a two-way subsidy strategy, both the manufacturer and the online retail platform share the other’s adverting costs. Especially, in a revenue-share strategy, the manufacturer both share the online retail platform’s adverting costs and the retailing income.

The main results of this paper we found as follows: first, under revenue-share cooperative advertising strategy, the manufacturer provides a higher subsidy rate for the online retail platform advertising than that in other cooperative advertising strategies. However, the online retailer platform does not blindly invest more platform advertising costs in revenue-share strategy than that in other strategies because of the high platform advertising subsidy rate. The level of advertising is affected by the wholesale price of products. Second, in the revenue-share strategy, there is a special region. With the increase of the revenue-share rate, the online retail platform obtains the manufacturer’s advertising subsidy support by transferring more revenue to the manufacturer. When the online retail platform increases its investment on platform advertising, the manufacturer will also increase his subsidy rate for platform advertising. Thrid, with identifying the value of the cooperative advertising, it is the best choice for the online supply chain system that the manufacturer and the online retail platform just take two-way subsidy strategy. Interestingly, there are conditions where, while just the manufacturer contributes a percentage of the online retail platform’s platform advertising in a one-way subsidy strategy, the total profit of the online supply chain would be better than that in revenue-share strategy even if the cooperation relationship is closer.

In future research, some valuable extensions of this paper could include the following. First, the characteristics of the network market as bilateral network externalities can be considered. It may be more interesting if the impact of the definition on advertising can be positive or negative. Second, it may be interesting that assume a dynamic model for a two-period supply chain, in which the impact ability of cooperative advertising in the latter stage is based on the impact of the previous stage. Finally, there are multiple manufacturers and multiple online retail platforms, and there is competition between them. The level of competition has an impact on the best subsidy rate of all manufacturers and the best advertising level of all online retail platforms [29].

Acknowledgments

This work was supported by the National Science Foundation of China (grant no. 72031009), the National Social Foundation of China (grant no. 20BJL135), and Henan Province Soft Science Research (grant no. 222400410440).

Appendix

Appendix

Proposition 10.

Using backward induction and solving the first-order condition of ${\pi }_R$, we get$$\begin{array}{c}\text{(A.1)}& p_2^o=\frac{1+w\delta }{2\delta },e_2^o=-\frac{{-1+w\delta }^2\theta \lambda }{4\delta -1+{\varphi }_2^o}.\end{array}$$

Substituting $p_2$ and $e_2$ into ${\pi }_r$ and solving the first-order condition of ${\pi }_R$, we have$$\begin{array}{c}\text{(A.2)}& p_1^o=\frac{1+w\delta }{2\delta },e_1^o=-\frac{{-1+w\delta }^2\lambda }{4\delta -1+{\varphi }_1^o}.\end{array}$$

Substituting $p_i$ and $e_i$ into ${\pi }_M$ and solving the first-order condition of ${\pi }_M$ with respect to $A$ yield, we get$$\begin{array}{}\text{(A.3)}& A^o=\frac{1}{2}w\beta 1-w\delta .\end{array}$$

Using the above results and solving the first-order condition of ${\pi }_M$ with respect to ${\varphi }_1$ and ${\varphi }_2$ yield, we have$$\begin{array}{}\text{(A.4)}& {\varphi }_i^o=\frac{5w\delta -1}{1+3w\delta }.\end{array}$$

Substituting (A.4) into $e_i^o$, ${\pi }_M^o$, and ${\pi }_R^o$, we get the optimal solutions shown in Proposition 10.

Proposition 11.

It can be obtained by solving the first-order condition of $A^o$ and $e_i^o$ as follows:$$\begin{array}{c}\text{(A.5)}& \frac{\partial A^o}{\partial w}=-\frac{1}{2}\beta -1+2w\delta =0,\\ \text{(A.6)}& \frac{\partial e_i^o}{\partial w}=-\frac{1}{4}-1+3w\delta \lambda =0.\end{array}$$

Solving equations (A.5) and (A.6), we get the result as shown in Proposition 11.

Proposition 12.

Using backward induction and solving the first-order condition of ${\pi }_R$, we get$$\begin{array}{c}\text{(A.7)}& p_2^t=\frac{1+w\delta }{2\delta },e_2^t=-\frac{{-1+w\delta }^2\theta \lambda }{4\delta -1+{\varphi }_2^o}.\end{array}$$

Substituting $p_2$ and ${\text{e}}_2$ into ${\pi }_r$ and solving the first-order condition of ${\pi }_R$, we have$$\begin{array}{c}\text{(A.8)}& p_1^t=\frac{1+w\delta }{2\delta },e_1^t=-\frac{{-1+w\delta }^2\lambda }{4\delta -1+{\varphi }_1^o}.\end{array}$$

Substituting $p_i$ and $e_i$ into ${\pi }_M$ and solving the first-order condition of ${\pi }_M$ with respect to $A$ yield, we get$$\begin{array}{}\text{(A.9)}& A^t=\frac{w\beta -1+w\delta }{2-1+\psi }.\end{array}$$

Using the above results and solving the first-order condition of ${\pi }_M$ with respect to ${\varphi }_1$ and ${\varphi }_2$ yield, we have$$\begin{array}{}\text{(A.10)}& {\varphi }_i^t=\frac{5w\delta -1}{1+3w\delta }.\end{array}$$

Using the above results and solving the first-order condition of ${\pi }_R$ with respect to $\psi$, we get$$\begin{array}{}\text{(A.11)}& \psi =1-2w\delta .\end{array}$$

Substituting equations (A.10) and (A.11) into $e_i^o$, $A^t$, ${\pi }_M^o$, and ${\pi }_R^o$, we get the optimal solutions shown in Proposition 11.

Proposition 13.

The proof is similar to Proposition 10 and thus is omitted.

Proposition 14.

To prove this, we need to compare the subsidy rate for the platform advertising under different strategies:$$\begin{array}{c}\text{(A.12)}& {\varphi }_i^o={\varphi }_i^t=\frac{5w\delta -1}{1+3w\delta },\\ {\varphi }_i^r-{\varphi }_i^t=\frac{4-1+\eta -w\delta +w^2{\delta }^2-\eta -w\delta \eta }{1+3w\delta 2w\delta +2\eta +w\delta \eta -{\eta }^2}>0.\end{array}$$

Proposition 15.

To prove this we need to compare the national advertising level under different strategies:$$\begin{array}{c}\text{(A.13)}& A^o=\frac{1}{2}w\beta 1-w\delta ,A^t=\frac{w\beta -1+w\delta }{2-1+\psi },\\ A^r=-\frac{\beta w\delta -\eta w\delta +\eta +w\delta \eta -{\eta }^2}{4\delta {\eta }^2},\\ A^r-A^o=\frac{\beta -1+\eta w^2{\delta }^2+2w^2{\delta }^2\eta -{\eta }^2}{4\delta {\eta }^2}>0,\\ A^t-A^o=\frac{\beta -1+w\delta -1+2w\delta }{4\delta }>0.\end{array}$$

When $A^r=A^t$ satisfies, we get $w={\eta }^{3/2}3\sqrt{\eta }-\sqrt{-4+5\eta }/21+\eta \delta$ or $w={\eta }^{3/2}3\sqrt{\eta }+\sqrt{-4+5\eta }/21+\eta \delta$. Because we have the condition as $0<w<1/2\delta$, we can obtain the result in Proposition 15.

Proposition 16.

To prove this, we need to compare the platform advertising level under different strategies:$$\begin{array}{c}\text{(A.14)}& e_i^o=-\frac{-1+w\delta 1+3w\delta \lambda }{8\delta },e_i^t=-\frac{-1+w\delta 1+3w\delta \lambda }{8\delta },\\ e_i^r=-\frac{w\delta -\eta 2w\delta +2\eta +w\delta \eta -{\eta }^2\lambda }{8\delta {\eta }^2}.\end{array}$$

We can obtain that $e_i^o=e_i^t$. When $e_i^r=e_i^t$ satisfies, we get $w=-\sqrt{1-\eta }/\delta \sqrt{2+\eta -3{\eta }^2/{\eta }^2}$ or $w=\sqrt{1-\eta }/\delta \sqrt{2+\eta -3{\eta }^2/{\eta }^2}$. Because we have the condition as $w<1/2\delta$, we can obtain the result in Proposition 16.

Proposition 17.

To prove this, we need to compare the retail price under different strategies:$$\begin{array}{c}\text{(A.15)}& p_i^o=\frac{1+w\delta }{2\delta },p_i^t=\frac{1+w\delta }{2\delta },\\ p_i^r=\frac{w\delta +\eta }{2\delta \eta }.\end{array}$$

We can obtain that $p_i^o=p_i^t$ and $p_i^r-p_i^t=-w-1+\eta /2\eta >0$. We can obtain the result in Proposition 17.

Proposition 18.

It can be obtained by solving the first-order condition of $\partial {\varphi }_i^r/\partial \eta$ and $\partial e_i^r/\partial \eta$ as follows:$$\begin{array}{c}\text{(A.16)}& \frac{\partial {\varphi }_i^r}{\partial \eta }=\frac{4w^2{\delta }^2-2w\delta \eta -{\eta }^2}{{2w\delta +2\eta +w\delta \eta -{\eta }^2}^2},\frac{\partial e_i^r}{\partial \eta }=\frac{4w^2{\delta }^2+w^2{\delta }^2\eta -{\eta }^3\lambda }{8\delta {\eta }^3}.\end{array}$$

We can obtain that if $w^2{\delta }^2-2w\delta \eta -{\eta }^2>0$, then $\partial {\varphi }_i^r/\partial \eta >0$, and if $4w^2{\delta }^2+w^2{\delta }^2\eta -{\eta }^3>0$, then $\partial e_i^r/\partial \eta >0$. To compare $w^2{\delta }^2-2w\delta \eta -{\eta }^2$, $4w^2{\delta }^2+w^2{\delta }^2\eta -{\eta }^3$, and 0, we get Figure 7.

Through Figure 7, we can obtain the result in Proposition 18.

[figure omitted; refer to PDF]

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