Research on the Cooperation Model of New Energy Vehicle Supply Chain under the Background of Government Subsidies Declining

Abstract

This paper studies the impact of the decrease in government subsidies on the selection of the cooperation model of vehicle manufacturers’ in the new energy vehicle supply chain, and uses the mathematical modeling method to establish MA (cooperation between vehicle manufacturers and battery suppliers with better battery life), MB (cooperation between vehicle manufacturers and battery suppliers with similar battery life) MAB (vehicle manufacturer cooperates with both) and N (does not cooperate with both) four cooperation models. The results show that vehicle manufacturers and battery suppliers A and B have cooperation motives, but whether vehicle manufacturers prefer to cooperate with battery supplier A or battery supplier B is related to the decrease in subsidies. In addition, it studies the impact of the decrease in subsidies on the sales price, market demand and supply chain profit of new energy vehicles.

1. Introduction

In recent years, the Chinese new energy vehicle industry has developed rapidly under the strong promotion of the national subsidy policy [1,2]. However, the subsidy policy is only a vehicle market-oriented means. With the increase in the number of new energy vehicles and the continuous increase in the number of subsidies, government finance is under great pressure, and the decrease in subsidies is inevitable [3,4]. In April 2020, the Ministry of Finance, the Ministry of Industry and Information Technology, the Ministry of Science and Technology and the National Development and Reform Commission issued a notice on improving the financial subsidy policy for the promotion and application of new energy vehicles. It indicated that the implementation period of the subsidy policy for the promotion and application of new energy vehicles would be extended to 2022, but the subsidy standard would decrease by 10% and 20%, respectively, from the previous year. The decrease in subsidies is bound to affect the operation of new energy vehicle enterprises. Yuan [5] found that since the announcement of the “fall off” subsidy policies, the domestic new energy vehicle market environment has changed dramatically, the sales growth of new energy vehicles has slowed down significantly, and some new energy vehicles and parts enterprises have had operating difficulties or even closed down. If the subsidies are cancelled in 2023, the new energy vehicles will be completely handed over to the market for competition. This will change the competitive pattern of the new energy vehicle market and have a significant impact on the supply chain of new energy vehicles [6,7,8].

To slow down the negative impact of the “fall off” subsidy policies on new energy vehicle enterprises, it is necessary for them to find a new way to be competitive. As is known to all, the core technology of new energy vehicles is battery technology [9]. Furthermore, the amount of the government subsidy is related to the mileage of the new energy vehicle purchased by the consumer [10]. The greater the mileage, the greater the subsidy amount. The mileage depends precisely on the performance of the battery. But most vehicle manufacturers are unable to produce batteries and rely heavily on battery suppliers. As for batteries, there are two ways for new energy vehicle manufacturers to obtain them: buy batteries from battery suppliers and produce new energy vehicles independently, or cooperate with battery suppliers to jointly produce new energy vehicles based on a cost-sharing and revenue-sharing contract model [11]. Therefore, against the background of decreasing subsidies, the key to how new energy vehicle manufacturers can break through the bottleneck and gain a competitive advantage lies in batteries. The methods of cooperation between car suppliers and battery suppliers are also gradually becoming an issue worthy of study.

In order to break the dilemma of the “fall off” subsidy policies, this paper designs four cooperation models for a new energy vehicle supply chain with one car manufacturer and two battery suppliers with different product standards: the MA model, involving an agreement between the car manufacturer and a battery supplier who supplies quality batteries; the MB model, involving an agreement between the car manufacturer and a supplier who supplies average-quality batteries; the MAB model involving an agreement between the car manufacturer and two battery suppliers, and the non-cooperative model in the form of purchase. It explores the influence of the subsidy decrease on decision variables in different cooperation models, to improve corporate profits and achieve the Pareto optimum of supply chain profits. The paper also enriches and expands the research on new energy vehicles and provides a reliable basis for the scientific decision-making of new energy vehicle enterprises.

The remainder of this article is organized as follows. In Section 2, we present the literature review. Section 3 sets forth the research hypotheses and introduces the four models. In Section 4, we solve the models to obtain the optimal wholesale prices, selling prices, demands and profits separately. We perform a comparative analysis of these models in Section 5. Numerical simulations are in Section 6. Finally, Section 7 summarizes the findings of this paper and directions for further research.

2. Literature Review

Different government subsidy levels and strategies will affect the new energy vehicle supply chain to different degrees. Zan et al. [12] constructed an evolutionary game model of government, enterprises and consumers to reveal the evolutionary process of enterprise production and consumer purchase behavior in different subsidy modes. Zheng et al. [13] used the propensity score matching method to compare and analyze the incentive effect of government subsidies and tax incentives on the R&D investment of listed companies of new energy vehicles. AHN et al. [14] investigated this question by analyzing the data from 489 Korean innovative manufacturing firms using a propensity score matching analysis. The results show that R&D subsidies stimulate firms to choose partners more adventurously by going outside the traditional value chains and regional boundaries. Zhao et al. [15] studied the impact of government subsidies on the supply chain of new energy vehicles by using the Stackelberg game theory. Jiang et al. [16] studied the impact of government subsidies on the choice of technology introduction or independent research and development of new energy vehicle enterprises by building an evolutionary game model. Cheng et al. [17] established the decision-making model of new energy vehicle supply chain under subsidy by using game theory to analyze the subsidy mechanism and its impact. Zhao et al. [18] established closed-loop supply chain pricing models for different subsidy objects to study the impact of different subsidy objects on the pricing decisions and profit distribution of supply chain members. Trianni et al. [19] analyzed the effect of public R&D subsidies on private R&D expenditure in a sample of French firms during the period 1993–2009. They found evidence of either no additionality or substitution effects between public and private R&D expenditure.

Based on different subsidy policies, many scholars have conducted research on the production optimization decision of new energy vehicle supply chain. Xiong et al. [20] constructed the optimal pricing model of new energy vehicle manufacturers and solved the optimal pricing mode in different subsidy modes. The results show that the main body of government subsidies should gradually shift from consumers to manufacturers. Shen et al. [21] constructed a model of the new energy vehicle supply chain in two models of direct subsidy manufacturer and direct subsidy seller. The study compares the optimal decisions of the automotive supply chain parties and government subsidy policies under the different models. Oikawa et al. [22] believe that inclusive subsidies stimulate the research and development of new energy vehicles with greater ease than selective subsidies. However, with the implementation of the government subsidy withdrawal policy, the supply chain of new energy vehicles has been greatly affected. Many scholars have conducted further research in the context of subsidy decrease. Wu and others [23] used the method of evolutionary game to build a tripartite collaborative innovation game matrix of new energy vehicle enterprises, universities and the government, and studied the impact of relevant parameters on the strategy. The results show that the increased speeds in the aspiration for participating in collaborative innovation are different for the three parties, and the aspiration of the government drops as that of new energy automobile enterprises and universities increases. Yu et al. [24] explained the impact of subsidy decrease and double points on the optimal decision-making of automobile enterprises and distributors based on Stackelberg game under the decrease and double points policy. The research shows that when the Double Integral policy is not implemented, the demand for new energy vehicles and the profit of manufacturers will decrease with the decrease in subsidies. Zheng et al. [25] studied whether manufacturers’ technological innovation can effectively replace government subsidies under the background of decreasing subsidies. The results show that only with the low market acceptance of NEV does the manufacturer’s technical innovation play as significant a role as the government subsidy does, which means that the subsidy reduction can effectively motivate the technical innovation and upgrade the industry technology.

In terms of supply chain cooperation, some scholars have studied the cooperation strategies of two-stage supply chains. Fan et al. [26] considered the electric vehicle supply chain composed of a battery supplier, a well-known brand manufacturer (manufacturer a) and an ordinary brand manufacturer (manufacturer B), and studied the non-cooperation strategy. The cooperation strategy between battery supplier and manufacturer A and the co-operation strategy between the battery supplier and manufacturer B. Li et al. [27] constructed a supply chain composed of a single ordinary manufacturer, a single green manufacturer and a single retailer, considered the cooperation models such as non-cooperation, horizontal cooperation, vertical cooperation and all cooperation among members, and studied the impact of consumers’ green preference on product pricing, greenness and the total profit of the supply chain in different models. Hu et al. [28] constructed a supply chain composed of one retailer and two manufacturers. The retailer established a revenue sharing contract relationship with one manufacturer and a wholesale price discount contract relationship with the other manufacturer. Hafezalkotob [29] developed a price competition model of two green and regular supply chains under the influences of government financial intervention. Results reveal that the environmental protection and social responsibility tendencies of the government have measurable impacts on the government’s revenue as well as on the profits of supply chains and their members. Madani et al. [30] studied the competition between ordinary supply chains and green supply chains in the case of government intervention. The results show that the government can influence the decisions of supply chain members.

Another group of scholars has studied the cooperation strategies in the three-tier supply chain. Wang [31] studied the decision-making and profit distribution of supply chain members in the three situations of no cooperation in the supply chain, cooperation between manufacturers and disassemblers, and cooperation between manufacturers and retailers. Shi et al. [32] established four cooperation strategies: manufacturers do not cooperate, manufacturers only cooperate with retailers, manufacturers only cooperate with recyclers, and manufacturers cooperate with retailers and recyclers to study the impact of reward and punishment mechanism on manufacturers’ cooperation choice. Xiao et al. [33] considered the closed-loop supply chain composed of a single manufacturer, retailer and recycler, and studied the impact of the reward and punishment mechanism of recovery rate on the manufacturer’s cooperation strategy in the closed-loop supply chain with capacity constraints using the Stackelberg theory. Zhang et al. [34] established four cooperation models: manufacturers only cooperate with recyclers, manufacturers only cooperate with retailers, manufacturers cooperate with both, and manufacturers do not cooperate with both, and solved the optimal control strategy using different cooperation strategies.

The literature review above shows that although the existing literature has studied government subsidies and the policy of subsidy decrease, there are fewer studies on the quantitative analysis of subsidy decrease and the optimization of the supply chain in the context of subsidy decrease. In addition, most of the studies on supply chain cooperation models are focused on other industries. Fewer studies have been conducted on the supply chain cooperation model of the new energy vehicle industry in the context of subsidy decrease. Therefore, to address the shortcomings or limitations of existing studies, the paper investigates the problem of selecting cooperation models for vehicle manufacturers in the supply chain of new energy vehicles in the context of subsidy decrease. This paper designs four different cooperation models and compares them. The relationship between the subsidy retreat coefficients on each decision variable in different cooperation models is investigated.

3. Model Assumptions and Parameter Description

3.1. Model Assumptions

This paper considers building a supply chain model composed of a vehicle manufacturer, battery supplier A (producing battery A with a good battery life) and battery supplier B (producing battery B with average battery life). Figure 1 shows four cooperation models between vehicle manufacturers and battery suppliers. In model N, the vehicle manufacturer occupies a dominant position; in model MA, the alliance between vehicle manufacturer and battery supplier A occupies a dominant position; in the model MB, the alliance between vehicle manufacturer and battery supplier B dominates; in the model MAB, the vehicle manufacturer and battery suppliers A and B make a joint decision.

In the paper, the following assumptions and explanations are put forward:

(1)

There are differences in consumers’ perceived value of new energy vehicles. The perceived value of new energy vehicle $X$ is $v_X$, $v_X$ is evenly distributed in [$v_{X1}$,$v_{X2}$], and the perceived value of new energy vehicle $Y$ is $v_Y$, $v_Y$ is evenly distributed in [$v_{Y1}$,$v_{Y2}$]. Assuming that there is only a difference in endurance between new energy vehicles $X$ and $Y$, and the consumers are rational, that is, if the endurance of new energy vehicle $X$ is better than $Y$, there are $v_{X1}>v_{Y2}$.

(2)

Consumers’ demand for new energy vehicles is related to its utility function $u$. $u$ is related to consumers’ perceived value of new energy vehicles $v$, sales price $p$ and government subsidy $h$, that is, $u=v-\left(p-h\right)$. When consumer utility function $u>0$, consumers will buy new energy vehicles.

(3)

Assuming that battery A has a good life and battery $B$ has an average life, there are $c_A>c_B$, of which $c_A$ is the production cost of battery $A$ and $c_B$ is the production cost of battery $B$.

(4)

Assuming that the endurance of new energy vehicle $X$ is good and that of $Y$ is average, there is $h_X>h_Y$, where $h_X$ is the government subsidy obtained by consumers when purchasing new energy vehicle $X$ and $h_Y$ is the government subsidy obtained by consumers when purchasing new energy vehicle $Y$.

(5)

Assuming that new energy vehicles are sold through the direct sales channel of the vehicle manufacturer, that is, there is no retailer, and the sales price of new energy vehicles is determined by the vehicle manufacturer.

(6)

Assuming that the vehicle manufacturer occupies a dominant position in the supply chain, battery suppliers A and B have the same bargaining power.

(7)

Each new energy vehicle is equipped with one unit of battery, and the output of battery is the same as that of new energy vehicles; the output and sales of new energy vehicles are the same.

3.2. Parameter Description

Description of basic symbols is provided in

Table 1. Description of basic symbols.

Parameter Symbol	Definition
ci, i = A, B	Production cost of battery A/B
cM	Remaining production costs of new energy vehicles X and Y
wi	Wholesale price of battery A/B
pj, j = X/Y	Sales price of new energy vehicle X/Y
hj	Subsidy amount for consumers to purchase new energy vehicle X/Y
$\theta$	Subsidy reduction coefficient
vj	Consumers’ perceived value of new energy vehicle X/Y
uj	Utility function of consumers’ purchase of new energy vehicle X/Y
$\alpha$	Potential market scale of new energy vehicle X
$\beta$	Potential market scale of new energy vehicle Y
dj	Consumer demand for new energy vehicle X/Y
d	Total market demand of new energy vehicles
${\mathsf{\Pi }}_j^N$	Profit of battery supplier A/B under model N
${\mathsf{\Pi }}_M^N$	Profit of vehicle manufacturers under model N
${\mathsf{\Pi }}^N$	Total profit of supply chain under model N
${\mathsf{\Pi }}_{M+A}^{MA}$	Overall profit of vehicle manufacturer and battery supplier A under model MA
${\mathsf{\Pi }}_B^{MA}$	Profit of battery supplier B under model MA
${\mathsf{\Pi }}^{MA}$	Total profit of supply chain under model MA
${\mathsf{\Pi }}_{M+B}^{MB}$	Overall profit of vehicle manufacturer and battery supplier B under model MB
${\mathsf{\Pi }}_A^{MB}$	Profit of battery supplier A under model MB
${\mathsf{\Pi }}^{MB}$	Total profit of supply chain under model MB
${\mathsf{\Pi }}^{MAB}$	Total profit of supply chain under model MAB

.

4. Different New Energy Vehicle Supply Chain Cooperation Models

According to the assumption of Ke et al. [35] on consumer demand, considering the general linear demand form, it is assumed that consumer demand is only affected by the sales price of new energy vehicles and the amount of government subsidies. In this case, the utility function of consumers’ purchase of new energy vehicle X/Y is:

$$u_X=v_X-\left[p_X-\left(1-\theta \right)h_X\right]$$

(1)

$$u_Y=v_Y-\left[p_Y-\left(1-\theta \right)h_Y\right]$$

(2)

In the formulas, $v_j$ is the consumer perceived value, $p_j$ is the sales price of new energy vehicles, $\left(1-\theta \right)h_j$ is the amount of subsidy after the subsidy reduction.

When consumers buy new energy vehicle X/Y, they must meet condition $u_X>0$ or $u_Y>0$. When consumers’ perceived value satisfies $v_X>p_X-\left(1-\theta \right)h_X$ or $v_Y>p_Y-\left(1-\theta \right)h_Y$, they will buy the vehicle. Their purchase behavior is shown in Figure 2. Hence, according to the proportion of consumer valuation, the demand for new energy vehicle X/Y is:

$$d_X=a{\displaystyle {\int }_{P_X-(1-\theta )h_X}^{v_{X2}}1d{v}_X=}a[v_{X2}-p_X+(1-\theta )h_X]$$

(3)

$$d_Y=\beta {\displaystyle {\int }_{P_Y-(1-\theta )h_Y}^{v_{Y2}}1d{v}_Y=}\beta [v_{Y2}-p_Y+(1-\theta )h_Y]$$

(4)

In these formulas, α, β are the potential market scale of two type of new energy vehicles, and the integral parts are the purchase behavior of consumers.

Therefore, the total demand for new energy vehicles is:

$$d=d_X+d_Y=\alpha \left[v_{X2}-p_X+\left(1-\theta \right)h_X\right]+\beta \left[v_{Y2}-p_Y+\left(1-\theta \right)h_Y\right]$$

(5)

4.1. Model N

Under model N, there is no cooperative relationship between the vehicle manufacturer and battery suppliers A and B due to the oversupply of batteries, and the vehicle manufacturer occupies a dominant position in the supply chain. The vehicle manufacturer decides the sales price of new energy vehicles X and Y, estimates the market demand $d_X$ and $d_Y$, and then purchases batteries A and B at wholesale prices of $w_A$ and $w_B$ from battery suppliers A and B, respectively. At this time, the profit functions of vehicle manufacturer and battery supplier A and B are, respectively:

$${\Pi }_M^N=\left(p_X^N-w_A^N-c_M\right)d_X^N+\left(p_Y^N-w_B^N-c_M\right)d_Y^N$$

(6)

$${\Pi }_A^N=\left(w_A^N-c_A\right)d_X^N$$

(7)

$${\Pi }_B^N=\left(w_B^N-c_B\right)d_Y^N$$

(8)

In the Equation (6), the first part is the profit of the vehicle manufacturer from selling new energy vehicles A, and the second part is the profit of the vehicle manufacturer from selling new energy vehicles B. Equations (7) and (8), respectively, are the profits of battery supplier A and B.

Using the reverse induction method, the optimal sales price of battery A is:

$$w_A^{*N}=\frac{v_{X2}-c_M+c_A+\left(1-\theta \right)h_X}{2}$$

(9)

The optimal wholesale price of battery B is:

$$w_B^{*N}=\frac{v_{Y2}-c_M+c_B+\left(1-\theta \right)h_Y}{2}$$

(10)

The optimal selling price of new energy vehicle X is:

$$p_X^{{*}_N}=\frac{3v_{X2}+c_M+c_A+3\left(1-\theta \right)h_X}{4}$$

(11)

The optimal selling price of new energy vehicle Y is:

$$p_Y^{{*}_N}=\frac{3v_{Y2}+c_M+c_B+3\left(1-\theta \right)h_Y}{4}$$

(12)

The market demand for new energy vehicle X is:

$$d_X^{*N}=\frac{\alpha \left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}{4}$$

(13)

The market demand for new energy vehicle Y is:

$$d_Y^{*N}=\frac{\beta \left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}{4}$$

(14)

The total market demand for new energy vehicles is:

$$d^{*N}=d_X^{*N}+d_Y^{*N}=\frac{\alpha \left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]+\beta \left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}{4}$$

(15)

The maximum profit of the vehicle manufacturer is:

$${\Pi }_M^{{*}_N}=\frac{\alpha {\left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}^2+\beta {\left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}^2}{16}$$

(16)

The maximum profit of battery supplier A is:

$${\Pi }_A^{{*}_N}=\frac{\alpha {\left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}^2}{8}$$

(17)

The maximum profit of battery supplier B is:

$${\Pi }_B^{{*}_N}=\frac{\beta {\left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}^2}{8}$$

(18)

The total profit of the supply chain is:

$${\Pi }^{{*}_N}={\Pi }_M^{{*}_N}+{\Pi }_A^{{*}_N}+{\Pi }_B^{{*}_N}=\frac{3\alpha {\left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}^2+3\beta {\left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}^2}{16}$$

(19)

4.2. Model MA

In the model MA, the vehicle manufacturer and battery supplier A make a joint decision. They jointly determine the sales prices of new energy vehicles X and Y, estimate the market demand of new energy vehicle Y, and then purchase battery B at a wholesale price from battery supplier B. At this time, the profit functions of the alliance between the vehicle manufacturer and battery supplier A and battery supplier B, respectively, are:

$${\Pi }_{M+A}^{MA}=\left(p_X^{MA}-c_A-c_M\right)d_X^{MA}+\left(p_Y^{MA}-w_B^{MA}-c_M\right)d_Y^{MA}$$

(20)

$${\Pi }_B^{MA}=\left(w_B^{MA}-c_B\right)d_Y^{MA}$$

(21)

Using the reverse induction method (the same as model N), the optimal wholesale price of battery B is:

$$w_B^{*MA}=\frac{v_{Y2}-c_M+c_B+\left(1-\theta \right)h_Y}{2}$$

(22)

The optimal selling price of new energy vehicle X is:

$$p_X^{{*}_{MA}}=\frac{v_{X2}+c_M+c_A+\left(1-\theta \right)h_X}{2}$$

(23)

The optimal selling price of new energy vehicle Y is:

$$p_Y^{*MA}=\frac{3v_{Y2}+c_M+c_B+3\left(1-\theta \right)h_Y}{4}$$

(24)

The market demand for new energy vehicle X is:

$$d_X^{*MA}=\frac{\alpha \left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}{2}$$

(25)

The market demand for new energy vehicle Y is:

$$d_Y^{*MA}=\frac{\beta \left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}{4}$$

(26)

The total market demand for new energy vehicles is:

$$d^{*MA}=d_X^{*MA}+d_Y^{*MA}=\frac{2\alpha \left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\left]+\beta \right[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}{4}$$

(27)

The maximum profit of the vehicle manufacturer and battery supplier A alliance is:

$${\Pi }_{M+A}^{*MA}=\frac{4\alpha {\left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}^2+\beta {\left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}^2}{16}$$

(28)

The maximum profit of battery supplier B is:

$${\Pi }_B^{*MA}=\frac{\beta {\left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}^2}{8}$$

(29)

The total profit of the supply chain is:

$${\Pi }^{*MA}={\Pi }_{M+A}^{*MA}+{\Pi }_B^{*MA}=\frac{4\alpha {\left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}^2+3\beta {\left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}^2}{16}$$

(30)

4.3. Model MB

In model MB, the vehicle manufacturer and battery supplier B make a joint decision. They jointly determine the sales prices of new energy vehicles X and Y, estimate the market demand $d_X$ of new energy vehicle X, and then purchase battery A at a wholesale price of $w_A$ from battery supplier A. At this time, the profit functions of the alliance between vehicle manufacturer and battery supplier B and battery supplier A, respectively, are:

$${\Pi }_{M+B}^{MB}=\left(p_X^{MB}-w_A^{MB}-c_M\right)d_X^{MB}+\left(p_Y^{MB}-c_B-c_M\right)d_Y^{MB}$$

(31)

$${\Pi }_A^{MB}=\left(w_A^{MB}-c_A\right)d_X^{MB}$$

(32)

Using the reverse induction method (the same as model N), the optimal wholesale price of battery A is:

$$w_A^{*MB}=\frac{v_{X2}-c_M+c_A+\left(1-\theta \right)h_X}{2}$$

(33)

The optimal selling price of new energy vehicle X is:

$$p_X^{*MB}=\frac{3v_{X2}+c_M+c_A+3\left(1-\theta \right)h_X}{4}$$

(34)

The optimal selling price of new energy vehicle Y is:

$$p_Y^{*MB}=\frac{v_{Y2}+c_M+c_B+\left(1-\theta \right)h_Y}{2}$$

(35)

The market demand for new energy vehicle X is:

$$d_X^{*MB}=\frac{\alpha \left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}{4}$$

(36)

The market demand of new energy vehicle Y is:

$$d_Y^{*MB}=\frac{\beta \left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}{2}$$

(37)

The total market demand for new energy vehicles is:

$$d^{*MB}=d_X^{*MB}+d_Y^{*MB}=\frac{\alpha \left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\left]+2\beta \right[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}{4}$$

(38)

The maximum profit of the vehicle manufacturer and battery supplier B alliance is:

$${\Pi }_{M+B}^{*MB}=\frac{\alpha {\left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}^2+4\beta {\left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}^2}{16}$$

(39)

The maximum profit of battery supplier A is:

$${\Pi }_A^{*MB}=\frac{\beta {\left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}^2}{8}$$

(40)

The total profit of the supply chain is:

$${\Pi }^{*MB}={\Pi }_{M+B}^{*MB}+{\Pi }_A^{*MB}=\frac{3\alpha {\left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}^2+4\beta {\left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}^2}{16}$$

(41)

4.4. Model MAB

In the model MAB, the vehicle manufacturer and battery suppliers A and B make joint decisions. The three are regarded as a whole. They pursue the maximization of the total profit of the supply chain, and jointly determine the sales price of new energy vehicles X and Y. At this time, the profit function of the alliance between the vehicle manufacturer and battery suppliers A and B is:

$${\Pi }_{M+A+B}^{MAB}=\left(p_X^{MAB}-c_A-c_M\right)d_X^{MAB}+\left(p_Y^{MAB}-c_B-c_M\right)d_Y^{MAB}$$

(42)

Using the reverse induction method (the same as model N), the sales price of new energy vehicle X is:

$$p_X^{*MAB}=\frac{v_{X2}+c_M+c_A+\left(1-\theta \right)h_X}{2}$$

(43)

The sales price of new energy vehicle Y is:

$$p_Y^{{*}_{MAB}}=\frac{v_{Y2}+c_M+c_B+\left(1-\theta \right)h_Y}{2}$$

(44)

The market demand for new energy vehicle X is:

$$d_X^{*MAB}=\frac{\alpha \left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}{2}$$

(45)

The market demand for new energy vehicle Y is:

$$d_Y^{*MAB}=\frac{\beta \left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}{2}$$

(46)

The total market demand for new energy vehicles is:

$$d^{*MAB}=d_Y^{*MAB}+d_Y^{*MAB}=\frac{\alpha \left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\left]+\beta \right[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}{2}$$

(47)

The maximum profit of the alliance of vehicle manufacturer, battery supplier A and B is:

$${\Pi }_{M+A+B}^{*MAB}=\frac{\alpha {\left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}^2+\beta {\left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}^2}{4}$$

(48)

The total profit of the supply chain is:

$${\Pi }^{*MAB}={\Pi }_{M+A+B}^{*MAB}=\frac{\alpha {\left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}^2+\beta {\left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}^2}{4}$$

(49)

5. Model Comparison

Firstly, the paper compares the equilibrium results using different cooperation models, explores the size relationship of decision variables in different cooperation models, and explains the reasons; Then, it studies the cooperation motivation and cooperation tendency between vehicle manufacturers and battery suppliers A and B. Finally, the relationship between the total profit of supply chain using different cooperation models is studied.

Proposition 1. 

Relationship between sales prices of new energy vehicles.

(1)

The relationship between the sales price of new energy vehicle X is as follows:

$$p_X^{*N}=p_X^{*MB}>p_X^{*MA}=p_X^{*MAB}$$

(50)

(2)

The relationship between the sales price of new energy vehicle Y is as follows:

$$p_Y^{*N}=p_Y^{*MA}>p_Y^{*MB}=p_Y^{*MAB}$$

(51)

It is proved that the sales price of new energy vehicle X in model MA and model MB is poor, and the following is obtained:

$$p_X^{*MA}-p_X^{*MB}=-\frac{v_{X2}-c_M-c_A+\left(1-\theta \right)h_X}{4}$$

(52)

The difference between the sales price of new energy vehicle Y in model MA and model MB is as follows:

$$p_Y^{*MA}-p_Y^{*MB}=-\frac{v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y}{4}$$

(53)

It is easy to obtain $p_X^{*MA}<p_X^{*MB}$ and $p_Y^{*MA}>p_Y^{*MB}$.

Conclusion (1) of proposition 1 shows that the sales price of new energy vehicle X under model N and model MB is the same, the sales price of new energy vehicle X under model MA and model MAB is the same, and the sales price of new energy vehicle X under model N and model MB is greater than that of new energy vehicle X under model MA and model MAB. This is because compared with model N and model MB, the adoption of model MA and model MAB can reduce the battery cost required by the vehicle manufacturer to produce new energy vehicle X. It makes the price advantage in the sales of new energy vehicle X, and appropriately reduces the sales price of new energy vehicle X. Similarly, compared with model N and model MA, the adoption of model MB and model MAB can appropriately reduce the sales price of new energy vehicle Y, as shown in conclusion (2).

Proposition 2. 

Relationship between wholesale prices of batteries.

(1)

In model N and model MB, the relationship between the wholesale price of battery A is: $w_A^{*MB}=w_A^{*N}$

(2)

In model N and model MA, the relationship between the wholesale price of battery B is: $w_B^{*MA}=w_B^{*N}$

Proposition 2 shows that the wholesale price of battery A is the same under model N and model MB, and the wholesale price of battery B is the same under model N and model MA. In other words, whether the vehicle manufacturer cooperates with battery supplier A or B does not affect the battery wholesale price of non-cooperative battery suppliers. The reason is that the sales price of new energy vehicle X under model N and model MB is the same, and the sales price of new energy vehicle Y under model N and model MA is the same (see proposition 1). Therefore, the wholesale price of battery A under model MB remains the same as that of model N, and the wholesale price of battery B under model MA remains the same as that of model N. In the case of cooperation, the supply relationship between vehicle manufacturers and battery suppliers and the competitive environment of the supply chain remain unchanged, so the wholesale price of batteries remains unchanged.

Proposition 3. 

Relationship between the market demand of new energy vehicles.

(1)

The relationship between the market demand for new energy vehicles X is as follows:

$$d_X^{*N}=d_X^{*MB}<d_X^{*MA}=d_X^{*MAB}$$

(54)

(2)

The relationship between the market demand for new energy vehicles Y is as follows:

$$d_Y^{*N}=d_Y^{*MA}<d_Y^{*MB}=d_Y^{*MAB}$$

(55)

(3)

If ${\theta }_1=\frac{\alpha v_{X2}-\beta v_{Y2}+\beta c_M-\alpha c_M+\beta c_B-\alpha c_A+\alpha h_X-\beta h_Y}{\alpha h_X-\beta h_Y}$, the subsidy gradient coefficient is small, that is, when $\theta <{\theta }_1$, $d^{*N}<d^{*MB}<d^{*MA}<d^{*MAB}$. When the subsidy gradient coefficient is large, that is, when $\theta >{\theta }_1$, $d^{*N}<d^{*MA}<d^{*MB}<d^{*MAB}$.

It is proved that the market demand of new energy vehicle X in model MA and model MB is poor, and it is obtained that:

$$d_X^{*MA}-d_X^{*MB}=\frac{\alpha \left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}{4}$$

(56)

The difference between the market demand for new energy vehicle X in model MA and model MB is as follows:

$$d_Y^{*MA}-d_Y^{*MB}=-\frac{\beta \left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}{4}$$

(57)

It is easy to obtain $d_X^{*MA}>d_X^{*MB}$ and $d_Y^{*MA}<d_Y^{*MB}$.

The total market demand for new energy vehicles in the four models is poor, and it is easy to draw conclusion (3).

Conclusion (1) of proposition 3 shows that the market demand of new energy vehicle X using model N and model MB is the same, the market demand of new energy vehicle X using model MA and model MAB is the same, and the market demand of new energy vehicle X using model N and model MB is less than that of new energy vehicle X using model MB and model MAB. Similarly, it can be seen from conclusion (2) that the market demand of new energy vehicle Y using model N and model MA is less than that using model MA and model MAB. This is because the vehicle manufacturer can reduce the cost of its battery and the sales price of the corresponding new energy vehicle by cooperating with any battery supplier. In this way, the market demand for new energy vehicles can be expanded.

Conclusion (3) shows that the total market demand for new energy vehicles using model MAB is the largest, the total market demand for new energy vehicles using model N is the smallest, and the total market demand for new energy vehicles using model MA and model MB is between model N and model MAB. This is because when vehicle manufacturers cooperate with battery suppliers A and B, they can minimize the cost of their batteries and reduce the sales price of new energy vehicles X and Y to maximize the total market demand of new energy vehicles. The battery cost of new energy vehicles is the highest when the vehicle manufacturer does not cooperate with any battery supplier, so the sales price of the two types of new energy vehicles is the highest and the total market demand for new energy vehicles is the lowest. In addition, the value of subsidy reduction coefficient will affect the size relationship between the total market demand for new energy vehicles using model MA and model MB. Therefore, when the subsidy reduction coefficient is taken in different ranges, the size relationship between the total market demand for new energy vehicles under model MA and model MB is different.

Proposition 4. 

Motivation and tendency of cooperation between vehicle manufacturers and battery supplier A and B.

(1)

The vehicle manufacturer and battery supplier A have cooperation motivation, i.e., ${\Pi }_{M+A}^{*MA}>{\Pi }_M^{*N}+{\Pi }_A^{*N}$

(2)

The vehicle manufacturer and battery supplier B have cooperation motivation, i.e., ${\Pi }_{M+B}^{*MB}>{\Pi }_M^{*N}+{\Pi }_B^{*N}$

(3)

When ${\theta }_2=\frac{\sqrt{\alpha }{v}_{X2}-\sqrt{\beta }{v}_{Y2}-\sqrt{\alpha }{c}_M+\sqrt{\beta }{c}_M-\sqrt{\alpha }{c}_A+\sqrt{\beta }{c}_B+\sqrt{\alpha }{h}_X-\sqrt{\beta }{h}_Y}{\sqrt{\alpha }{h}_X-\sqrt{\beta }{h}_Y}$, the subsidy slope coefficient is small, that is, when $\theta <{\theta }_2$, the vehicle manufacturer is more inclined to cooperate with battery supplier A, that is, ${\Pi }_{M+A}^{*MA}-\left({\Pi }_M^{*N}+{\Pi }_A^{*N}\right)>{\Pi }_{M+B}^{*MB}-\left({\Pi }_M^{*N}+{\Pi }_B^{*N}\right)$; When the subsidy gradient coefficient is large, i.e., $\theta >{\theta }_2$, the vehicle manufacturer is more inclined to cooperate with battery supplier B, i.e., ${\Pi }_{M+A}^{*MA}-\left({\Pi }_M^{*N}+{\Pi }_A^{*N}\right)<{\Pi }_{M+B}^{*MB}-\left({\Pi }_M^{*N}+{\Pi }_B^{*N}\right)$.

This proves that the total profits of the vehicle manufacturer and battery supplier A in model N and model MA are different, and the following is obtained:

$${\Pi }_{M+A}^{*MA}-\left({\Pi }_M^{*N}+{\Pi }_A^{*N}\right)=\frac{\alpha {\left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}^2}{16}$$

(58)

The difference between the total profits of the vehicle manufacturer and battery supplier B in model N and model MB is as follows:

$${\Pi }_{M+B}^{*MB}-\left({\Pi }_M^{*N}+{\Pi }_B^{*N}\right)=\frac{\beta {\left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}^2}{16}$$

(59)

It is easy to obtain ${\Pi }_{M+A}^{*MA}>{\Pi }_M^{*N}+{\Pi }_A^{*N}$ and ${\Pi }_{M+B}^{*MB}>{\Pi }_M^{*N}+{\Pi }_B^{*N}$.

The difference between the total profits of the vehicle manufacturer and the battery supplier A in model N and model MA and the total profits of the vehicle manufacturer and the battery supplier B in model N and model MB is calculated as follows:

$$\left[{\Pi }_{M+A}^{*MA}-\left({\Pi }_M^{*N}+{\Pi }_A^{*N}\right)]-[{\Pi }_{M+B}^{*MB}-\left({\Pi }_M^{*N}+{\Pi }_B^{*N}\right)\right]=\frac{\alpha {\left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}^2-\beta {\left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}^2}{16}$$

(60)

Make $\frac{\alpha {\left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}^2-\beta {\left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}^2}{16}>0$ and $\frac{\alpha {\left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}^2-\beta {\left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}^2}{16}<0$, respectively, and it is easy to obtain the conclusion (3).

Conclusion (1) and (2) of proposition 4 show that the profit of the alliance of vehicle manufacturer and battery supplier A in model MA is greater than the total profit of vehicle manufacturer and battery supplier A in model N, and the profit of the alliance of vehicle manufacturer and battery supplier B in model MB is greater than the total profit of vehicle manufacturer and battery supplier B in model N. This is because the cooperation between vehicle manufacturers and battery suppliers can eliminate the double marginal effect in the state of non-cooperation. In this way, the total profit of both sides can be increased and a win-win result can be achieved. Conclusion (3) shows that when the subsidy reduction coefficient is taken in different ranges, compared with model N, the relationship between the profit increased by the cooperation between the vehicle manufacturer and battery supplier A in model MA and the profit increased by the cooperation between the vehicle manufacturer and battery supplier B in model MB is different. In other words, the value of subsidy reduction coefficient will affect the tendency of vehicle manufacturers to cooperate with different battery suppliers.

Proposition 5. 

Relationship between total profits of supply chain.

(1)

When the subsidy gradient coefficient is small, that is, when $\theta <{\theta }_2$, ${\Pi }^{*N}<{\Pi }^{*MB}<{\Pi }^{*MA}<{\Pi }^{*MAB}$;

(2)

When the subsidy gradient coefficient is large, that is, when $\theta >{\theta }_2$, ${\Pi }^{*N}<{\Pi }^{*MA}<{\Pi }^{*MB}<{\Pi }^{*MAB}$.

It is proved that the total profit of the supply chain in model MA and model N is poor, and the following is obtained:

$${\Pi }^{*MA}-{\Pi }^{*N}=\frac{\alpha {\left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}^2}{16}$$

(61)

The total profit of the supply chain in model MB minus the total profit of the supply chain in model N is:

$${\Pi }^{*MB}-{\Pi }^{*N}=\frac{\beta {\left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}^2}{16}$$

(62)

The total profit of the supply chain in the model MAB minus the total profit of the supply chain in model MA is:

$${\Pi }^{*MAB}-{\Pi }^{*MA}=\frac{\beta {\left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}^2}{16}$$

(63)

The total profit of the supply chain in the model MAB minus the total profit of the supply chain in model MB is:

$${\Pi }^{*MAB}-{\Pi }^{*MB}=\frac{\alpha {\left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}^2}{16}$$

(64)

It is easy to obtain ${\Pi }^{*MA}>{\Pi }^{*N}$, ${\Pi }^{*MB}>{\Pi }^{*N}$, ${\Pi }^{*MAB}>{\Pi }^{*MA}$ and ${\Pi }^{*MAB}>{\Pi }^{*MB}$.

The total profit of the supply chain in model MA minus the total profit of the supply chain in model MB is:

$${\Pi }^{*MA}-{\Pi }^{*MB}=\frac{\alpha {\left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}^2-\beta {\left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}^2}{16}$$

(65)

Let $\frac{\alpha {\left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}^2-\beta {\left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}^2}{16}>0$ and $\frac{\alpha {\left[v_{X2}-c_M-c_A+\left(1-\theta \right)h_X\right]}^2-\beta {\left[v_{Y2}-c_M-c_B+\left(1-\theta \right)h_Y\right]}^2}{16}<0$, respectively. Combined with the above proof, it is easy to obtain conclusions (1) and (2).

Proposition 5 shows that the total profit of supply chain using model MAB is the highest, the total profit of supply chain using model N is the lowest, and the total profit of supply chain using model MA and model MB is between model N and model MAB. This is because the cooperation between the vehicle manufacturer and battery suppliers A and B can maximize and eliminate the double marginal effect between the vehicle manufacturer and battery supplier A and B, so as to maximize the total profit of the supply chain. When the vehicle manufacturer cooperates with one of the battery suppliers, the double marginal effect between the vehicle manufacturer and the other battery supplier still exists, and the total profit of the supply chain is less than the total profit of the supply chain in the model MAB. When the vehicle manufacturer does not cooperate with any battery supplier, the double marginal effect between the vehicle manufacturer and the battery supplier is the largest and the total profit of the supply chain is the smallest. In addition, the value of subsidy setback coefficient will affect the relationship between the total profit of supply chain using model MA and model MB. Therefore, when the subsidy setback coefficient is taken in different ranges, the relationship between the total profit of supply chain using model MA and model MB is different.

6. Numerical Analysis

This paper will verify the conclusions of the above cooperation models and the comparison of cooperation models through a numerical example, mainly including the following two aspects: (1) compare the optimal pricing of batteries and new energy vehicles in different models, the market demand for new energy vehicles and the maximum profit of supply chain; (2) this paper studies the influence of the value of subsidy reduction coefficient on the pricing of new energy vehicles, the market demand for new energy vehicles and the total profit of supply chain using different models.

According to industry reports and previous studies [10,36], and in combination with the definition and relationship of model parameters in this paper, it is assumed that $\alpha =6$, $\beta =4$, $v_{X1}=4.1$, $v_{X2}=5$, $v_{Y1}=3.1$, $v_{Y2}=4$, $c_A=2$, $c_B=0.8$, $c_M=2.2$, $h_X=0.3$, $h_Y=0.2$, $\theta =0.3$. By substituting the specific values of various parameters, the pricing of batteries and new energy vehicles, the market demand for new energy vehicles and the total profit of the supply chain using different models are obtained, as shown in

Table 2. Comparison of supply chain models under four different cooperation models.

Variable Name	Model N	Model MA	Model MB	Model MAB	Comparison
$w_A$	2.51	NA	2.51	NA	$N=MB$
$w_B$	1.37	1.37	NA	NA	$N=MA$
$p_X$	4.96	4.71	4.96	4.71	$N=MB>MA=MAB$
$p_Y$	3.86	3.86	3.57	3.57	$N=MA>MB=MAB$
$d_X$	1.52	3.03	1.52	3.03	$N=MB<MA=MAB$
$d_Y$	1.14	1.14	2.28	2.28	$N=MA<MB=MAB$
$d$	2.66	4.17	3.8	5.31	$N<MB<MA<MAB$
$\Pi$	2.12	2.5	2.45	2.83	$N<MB<MA<MAB$

.

Next, combined with a numerical example, this paper studies the influence of subsidy reduction coefficient ($\theta$) on the pricing of new energy vehicles, the market demand for new energy vehicles and the total profit of supply chain under different cooperation models (model N, model MA, model MB, model MAB).

It can be seen from Figure 3, Figure 4, Figure 5 and Figure 6 that the sales price of new energy vehicles decreases with the increase in the subsidy decline coefficient, and the market demand for new energy vehicles and the total profit of the supply chain decrease with the increase in subsidy decline coefficient. In other words, the sales price, market demand and total profit of the supply chain of new energy vehicles are inversely proportional to the subsidy decline coefficient. This shows that after the decrease in government subsidies, market demand and supply chain profits will be reduced, and vehicle manufacturers also need to appropriately reduce the sales price of new energy vehicles. It can be seen from Figure 6 that when the subsidy gradient coefficient is within [0, 0.88], the total profit of the supply chain using model MA is greater than that using model MB. When the subsidy gradient coefficient is within [0.88, 1], the total profit of the supply chain using model MB is greater than that using model MA. Therefore, for vehicle manufacturers, the value range of the subsidy setback coefficient needs to be considered in the selection of model MA and model MB.

7. Conclusions

This paper studies the selection of a vehicle manufacturers’ cooperation model with battery suppliers in the supply chain of new energy vehicles in the light of decreasing subsidies, and formulates four cooperation models, comparing the pricing of batteries and new energy vehicles, the market demand for new energy vehicles and the total profit of supply chain under different cooperation models. It explores the influence of subsidy reduction coefficient on the pricing, the market demand and the total profit of the supply chain in different cooperation models. The results of the study are as follows.

(1)

Under the non-cooperation model (N), the sales price of new energy vehicles is the highest, while the market demand for the vehicles and the total profit of supply chain are the lowest. When vehicle manufacturers cooperate with two battery suppliers (MAB), the sales price is the lowest, whereas the market demand and the total profit of supply chain are the largest.

(2)

Under the decrease of government subsidies, regardless of the cooperative model, the sales price, the market demand and supply chain profit of new energy vehicles all decrease, and they are inversely proportional to the decline coefficient of subsidies.

(3)

Vehicle manufacturers and battery suppliers with good battery (MA) or with average battery (MB) are both motivated to cooperate. Moreover, the cooperation tendency of vehicle manufacturers is related to the level of decrease in subsidy. When the decrease in subsidies is less, vehicle manufacturers prefer to cooperate with battery suppliers with better batteries; When subsidies decrease more, the manufacturers prefer to cooperate with the suppliers with average batteries.

(4)

When vehicle manufacturers only cooperate with battery supplier with good batteries (MA) or with average batteries (MB), the total profit of the supply chain is also related to the decrease in subsidies. When the decrease in subsidies is less, the total profit of the supply chain in MA model is higher than that in MB model. However, when the “fall off” subsidy is greater, the total profit of the supply chain in MB model is higher than that in MA model.

To sum up, in the background of government subsidies decreasing, if vehicle manufacturers want to use batteries as a breakthrough to gain a competitive advantage, they need to fully consider various factors such as the battery life, the cost of battery, the degree of subsidies decreasing, and then choose the cooperation model with battery suppliers, thus maximizing the profit of the enterprise and the supply chain.