Supply Chain Coordination of New Energy Vehicles under a Novel Shareholding Strategy

Abstract

As important methods of ecofriendly transportation, the supply chain coordination of new energy vehicles (NEVs) is an important issue in the field of sustainability. This study constructs a Stackelberg game composed of a power battery supplier and an NEV manufacturer. To better describe the coordination relationship in the NEV supply chain, we introduce the Nash bargaining framework into the fairness concern preference utility function. Through a comprehensive discussion of shareholding ratios and external environment factors, we discover that the traditional shareholding strategy fails to coordinate the NEV supply chain effectively, as enterprises seek to avoid losing management control, which occurs when excessive shares are held by others. In this context, this study proposes a novel industry–university–research (IUR) shareholding strategy, which can more easily achieve supply chain coordination and improve social welfare. In particular, this study reveals the superiority of the novel strategy in eliminating the double-marginal effect caused by high fairness concern preference among NEV enterprises. Based on these facts, we provide enterprises with optimal strategies under different conditions and offer a government-optimal subsidy to maximize the social welfare function.

1. Introduction

As global warming has intensified in recent decades, new energy industries have been promoted worldwide to mitigate the environmental effects of conventional energy sources. The concept of low-carbon energy has been receiving more attention from enterprises and consumers; this also influences global automakers, shifting their main focus to new energy vehicles (NEVs) [1].

Surprisingly, the upstream and downstream of the NEV industry are in entirely different situations. Upstream enterprises face significant research and development (R&D) costs to stay in step with the swift progression of the global NEV industry. The high prices of power batteries are the principal hindrance to downstream NEV competitiveness and exacerbate the imbalance between upstream and downstream NEV enterprises. Thus, differently from the high profits attained by battery suppliers, downstream companies hardly benefit from selling NEVs. This huge contrast between the upstream and the downstream stakeholders reflects the fact that the current development of the NEV supply chain is not coordinated, and that the cooperation with R&D is not as close as it should be.

To overcome the above-mentioned dilemma of downstream companies, subsidies from the government are needed. Yalabik and Fairchild [2] support that subsidies are more effective than fines in dealing with pollution, especially among “dirty” industries. With these subsidies, these companies can deploy more resources in devising environmentally friendly technologies. Furthermore, government subsidies and price discounts also play an important role in promoting new technologies in the mainstream market [3]. As NEVs provide an alternative solution for reducing carbon emissions with electric power, they also require support from the government. Subsidies invested in NEV companies help to improve the penetration rate of the NEV market [4]. However, government subsidies for NEV companies also have several disadvantages. Firstly, the financial burden caused by large-scale subsidies cannot be ignored. Secondly, some enterprises will manufacture unqualified NEVs in an effort to take unfair advantage of the available subsidies. Therefore, government subsidies for NEV companies have gradually reduced.

As subsidies from the government have decreased, we need to find alternative strategies that can ensure that supply chain members receive greater benefits. Supply chain participants should aim for enhanced collaboration, a well-established practice that can be an effective means of attaining superior performance within a low-carbon supply chain [5]. Optimal cooperation entails achieving complete supply chain coordination. In a state of perfect coordination, supply chain performance matches that of a centralized supply chain, and the optimal action set for supply chain members represents the sole equilibrium solution [6].

To provide solutions to current NEV companies, researchers have come up with several methods of adjusting the current supply chain. However, their methods have several main drawbacks.

1. Some researchers consider only “absolute fairness” in the preference of fairness concerns, which is quite different from practical circumstances.

2. Many studies explore the coordination of supply chains that are still focused on traditional contracts.

3. Many previous studies have ignored the effect of the external environment on the NEV supply chain.

Considering these limitations, we propose a supply chain configuration comprising a power battery supplier and an NEV manufacturer. Our method asserts that the “relative fairness” of the NEV supply is more appropriate in describing the utility of members in the supply chain. Furthermore, we study and expand the shareholding strategy instead of traditional contracts to better relate to reality. Last but not least, the study of the external environment is helpful in providing more helpful suggestions for the government and enterprises. To be more specific, this study contributes to the field by answering the following questions:

(1) Can a shareholding strategy improve NEV supply chain performance? Under what conditions can NEV supply chain coordination be achieved?

(2) How can we extend the shareholding strategy to ensure that coordination conditions are easier to achieve?

(3) What is the impact of the external environment on supply chain performance when supply chain members adopt a shareholding strategy?

By answering the above questions, we try to realize the perfect coordination of the supply chain and demonstrate the superior performance of our novel coordination strategy in the NEV supply chain. With a comprehensive analysis from the perspective of enterprises and governments, the findings of the present study can be used as a reference by policymakers in relevant companies and governments. We can summarize the contributions of this paper in the following three points:

(1) This paper proposes a new cross-shareholding strategy considering IUR cooperation to eliminate the “double marginal effect”. Based on the results, we find that the cross-shareholding strategy can solve free-riding problems under a strict constraint. To make the conclusion more general, we introduce IUR, an advanced system, into the cooperation of the NEV supply chain.

(2) This study integrates the Nash bargaining fairness framework into the NEV supply chain, emphasizing “relative fairness” based on organizational behavior theory. It posits that enterprises prioritize relative fairness, which inherently combines the power and contribution of players, as opposed to solely emphasizing absolute fairness or profits.

(3) We make valuable management recommendations for governments and enterprises based on three external environmental factors: government subsidy, consumer’s low-carbon preferences, and R&D capabilities. With our proposals, enterprises can select the optimal strategy to maximize utility, while governments can determine the optimal subsidy to maximize social welfare.

The following paper is organized into four sections: a literature review is presented in Section 2; the model settings and equilibrium analysis are described in Section 3; numerical studies are laid out in Section 4; the conclusions of this paper are provided in Section 5.

2. Literature Review

There are three research directions that are closely related to this study: NEV supply chains, cooperative R&D in supply chains, and fairness concerns in supply chains.

2.1. NEV Supply Chains

Although current governments and consumers demonstrate a strong desire for low-carbon transportation, traditional automakers are not able to shift their sales totally to NEVs because of the upstream–downstream dilemma described in Section 1. Only 14% of all new vehicles sold were electric in 2022 [7]. Thus, the transition from fuel vehicles to NEVs is still in the primary stage and requires the efforts of all parties in society, especially the government.

The government should act as a beacon in NEV supply chains, subsidizing those environmentally friendly enterprises while penalizing manufacturers opposed to the low-carbon concept. However, the performance of the subsidies and penalties affects the enterprises to different extents. As depicted in work by Yalabik and Fairchild [2], subsidies seem to be more effective than directly penalizing the “dirty” industry. Meath, et al. [8] also point out the government investments attributed to reducing the environmental costs of enterprises and improving their competitiveness based on the careful study of 202 enterprises. Furthermore, subsidies enhance the penetration rate of the NEV market, promoting the development and innovation of those NEV enterprises. Therefore, unlike previous research focusing on government regulatory policies [9,10], this study is devoted to exploring the impacts of government incentives on the NEV supply chain. Government incentives can be broadly segmented into two main types: production-based subsidies on a per-unit basis and subsidies designed to promote innovation efforts. Contrary to common sense, Chen, et al. [11] point out that the two subsides used at the same time may not help as well as utilizing only one of them. Hence, in this paper, we only consider the innovation effort subsidy.

With the sponsoring of the government, the NEV supply chain has received more attention recently. Fan, et al. [12] investigate an NEV supply chain comprising battery suppliers, renowned brand manufacturers, and generic brand manufacturers—their research focuses on the impact of brand effects within the supply chain under three cooperation strategies. Yu, et al. [13] explore four R&D collaboration contracts. They find that members should adopt vertical R&D collaboration contracts when both R&D efficiency and sales efficiency are at a high level. Han, et al. [14] aim to explore the influence mechanism of policy transitions in the NEV cooperative innovation network, highlighting that the government should focus on enhancing innovation cooperation rather than hastily reducing subsidies. The innovation cooperation may occur among NEV supply chain members or between R&D groups and external institutions.

However, the above papers do not account for the influence of several important exogenous variables in the coordination strategies. Building upon the insights derived from the aforementioned studies, this study delves into examining how government subsidies, consumers’ low-carbon preferences, and R&D capabilities influence the coordinated NEV supply chain.

2.2. Cooperative R&D in Supply Chains

In the high-tech intensive industry, core technology always occupies a crucial position. Strong innovation capabilities help enterprises remain competitive in industries with high R&D intensity and rapid technological change. However, Bustinza, et al. [15] found that completely internal innovation is no longer sufficient for companies to maintain a rapid response to cutting-edge technology. Instead, the R&D of products has become prominent. Although product development requires a higher level of fixed investment, it also offers opportunities for developing economies on a larger scale and scope [15]. To alleviate the significant costs brought by R&D, more and more enterprises seek cooperative R&D in NEV, such as BWM, BYD, etc. Zhang and Liu [16] believe that the success of supply chain management lies in building proper synergies and incentives. Manufacturers should form good partnerships with outsourcing organizations. The partnership can help manufacturers manage and focus on core manufacturing activities to make their products more differentiated and innovative.

Some researchers focus on cooperative R&D partners in supply chains. Tether [17] uses the data of innovative enterprises in the UK and finds that there are four kinds of R&D cooperation modes: cooperating with competitors, suppliers, customers, and IUR. Belderbos, et al. [18] study the combination strategy of these four R&D partners rather than using a single strategy. Belderbos, et al. [19] find that R&D-intensive enterprises are willing to cooperate with universities in R&D. This is because companies prefer to choose partners with more R&D experiences and knowledge. In the NEV supply chain, the core technology of the power battery is still mainly based on patented technology. This means that the core technology of the power battery is exclusive. Therefore, vertical cooperation is more acceptable than horizontal cooperation for the suppliers. The supplier’s choice to partner with IUR is another good option. As for the NEV manufacturers, they are seeking to obtain power batteries from corporations because the price of power batteries is rising and the supply is unstable. Fan, et al. [12] mention that there are two main ways to obtain power batteries for manufacturers, including direct purchase and vertical cooperation. Hence, we examine the R&D of carbon emission reduction in vertical cooperation and introduce IUR into a new cooperation strategy.

There are other researchers who focus on cooperative R&D contracts in supply chains. Hong and Guo [5] conclude that a two-part tariff contract is better than a price-only, green-marketing, and cost-sharing contract. They find that partnerships can facilitate green supply chains. Li, et al. [20] examine three contract strategies for the green supply chain under two contracting formats. Hagedoorn [21] believes that contract R&D partnerships are suitable for short-term joint R&D between enterprises. In other words, he argues that collaborative R&D is unstable when relying solely on contract strategies. Therefore, considering practical realities, we explore fairness in R&D strategies. We introduce a shareholding strategy, a financial tool, into the NEV supply chain.

Some researchers have also noted the importance of shareholding strategies. Ren, et al. [22] propose three shareholding strategies within a supplier-led green supply chain to investigate the operational mechanisms underlying them. Zhang and Qin [23] find that the cross-shareholding strategy can effectively coordinate the transportation supply chain under the carbon tax policy. Sun, et al. [24] find that the cross-shareholding strategy can perfectly coordinate the upstream-led, low-carbon, closed-loop supply chain. Zhu, et al. [25] compare the effects of cross-shareholding and unidirectional shareholding on the NEV market scale, R&D investment, and social welfare. These studies have demonstrated the effectiveness of shareholding strategies in supply chain coordination. However, the above papers focus on the effectiveness of the shareholding strategy and do not explore the constraints associated with the strategy when it is effective. In addition, the impact of government subsidies on the NEV supply chain has not been addressed.

2.3. Fairness Concerns in Supply Chains

Contemporary researches within the domain of behavioral economics acknowledge the prevalence of decision makers who frequently demonstrate instances of bounded rationality and various social preferences, including fairness concerns. An enterprise’s fairness concern relates to the equitable allocation of profits among parties in the supply chain, constituting a fundamental issue in supply chain cooperation, as emphasized in the work of Liu, et al. [26]. This focus has become particularly pronounced after the COVID-19 pandemic, the global energy crisis, and the frequency of various extreme weather events. Supply chain members seem to be taking fairness more seriously [27]. Xue, et al. [28] find that the technology giant, Apple, appears to suffer from the “fairness” problem. When Qualcomm refused to supply its microchips to Apple, Apple initiated a billion-dollar legal battle to defend its fairness. All this can be attributed to fairness concerns among enterprises in the supply chain, especially when one of the enterprises feels that it is not obtaining a relatively fair profit.

In recent times, an increasing number of researchers have displayed a growing interest in the examination of social preferences within the realm of supply chain research. For instance, Jian, et al. [29] explored the dynamics of a green supply chain comprising a retailer and a manufacturer and analyzed the implications of both centralized and decentralized decision-making frameworks with the retailer’s sales effort. They notably formulated a Stackelberg game model that incorporates the manufacturer’s fairness concern as a pivotal factor in this context. Furthermore, the work by Liu, et al. [26] focuses on how retailers’ fairness concerns can influence cooperative relationships within a sustainable supply chain. This study also explores supply chain coordination, especially when considering fairness concerns as an interval variable. In a parallel vein, Zhang, et al. [30] contributed to this discourse by investigating how the fairness concerns of green retailers impact product eco-friendliness and profitability. Their study further explores strategies for the equitable distribution of surplus profits in the presence of such fairness concerns, employing cooperative game theory as a guiding framework. Additionally, Liu, et al. [31] engaged in an analysis of the repercussions of fairness concerns on equilibrium decisions and profitability within the supply chain context, contributing to the understanding of adaptable price transmission mechanisms. This research extends the discourse on fairness concerns and their implications, aligning with the adaptable price transmission mechanism concept delineated by Fan, et al. [12].

While the aforementioned studies aptly address the concept of fairness concerns, they face challenges in providing a precise depiction of how enterprises involved in the NEVs supply chain perceive and manifest fairness concerns. To more accurately capture and represent the fairness concerns specific to NEV companies, our research adopts the Nash bargaining fairness-concerns framework. This framework is chosen for its emphasis on relative fairness, which takes into account the power and contribution of each party within the supply chain through a mechanism of self-enforcement, as opposed to an absolute or universal fairness standard [32,33]. This means that supply chain members do not simply take each other’s profit as the standard of perceived fairness but take their own “fairness profit” as the standard.

In a similar vein, other scholars have explored the ramifications of fairness preference concerns by employing the Nash bargaining fairness reference framework. Notably, Li, et al. [34] incorporated the Nash bargaining fairness reference to investigate the impact of fairness preference concerns within the context of a dual-channel supply chain. Their study delves into the multifaceted interplay between fairness considerations, channel coordination, and contracting mechanisms. Additionally, they quantitatively assess the effects of fairness concerns and the retailer’s bargaining power on three critical dimensions, namely, coordination performance, wholesale price, and overall channel efficiency. Moreover, Guan, et al. [32] tackled a supply chain coordination problem characterized by a manufacturer and a retailer, both of whom harbor Nash bargaining fairness concerns. Their work further contributes to the understanding of how fairness considerations manifest within the supply chain domain. Under the proposed conditions, any channel dominated by members can achieve optimal decisions under the proposed conditions, improving product quality when all members are concerned about fairness.

Although several researchers have studied the Nash bargaining fairness-concerns framework, few studies introduced it into the supply chain of NEVs. Our paper adopts the Nash bargaining fairness-concerns framework because we believe that people often pay more attention to whether they obtain “what they deserve” instead of what others receive. This “fairness preference” is more consistent with the current situation of the NEV supply chain.

To elucidate the connections and distinctions between our research and prior literature,

Table 1. Summary and comparison of the prior works of literature.

Literature	NEV Supply Chain	External Environments	Nash Bargaining Fairness Concerns	Cooperation Strategy
Li, et al. [20]				RS, CS
Hong and Guo [5]		√		T, CS
Yu, et al. [13]	√			F, RS
Li, et al. [34]			√	RS
Ren, et al. [22]				SH
Zhao and Ma [35]	√			CS-RS
Zhang and Qin [23]		√		SH
Yu, et al. [36]	√	√		CS
Sun, et al. [24]		√		SH-T
Zhu, et al. [25]	√			SH
Our paper	√	√	√	SH, C-SH-I

Notions: CS—cost-sharing, RS—revenue sharing, T—two-tariff, F—fixed payment, CS-RS—a new contract combining CS and RS, SH—shareholding, C-SH-I—IUR shareholding strategy.

provides a comparative analysis of this study with existing works.

Based on these studies, the NEV supply chain is a highly technology-intensive industry, and this paper studies three kinds of R&D cooperation strategies. These include cost-sharing, shareholding in vertical cooperation, and a novel IUR shareholding strategy.

3. Model Settings and Equilibrium Analysis

In this section, we begin with the problem description and the model assumptions to investigate the optimal decisions of two benchmark models (integrated and decentralized supply chain) and two cross-shareholding strategies (traditional and novel).

3.1. Problem Description

This paper considers a two-level supply chain system consisting of a power battery supplier and an NEV manufacturer. The NEV supply chain is accountable for the production and R&D of power batteries, while the NEV manufacturer purchases power batteries from the power battery supplier and is responsible for the whole vehicle’s manufacturing. Both parties make decisions based on their own fairness utility maximization. According to the strong position of power battery suppliers in reality, this paper examines the NEV supply chain under the Stackelberg game led by suppliers. The decision-making sequence is that the suppliers first decide the wholesale price of power batteries and the effort of emission reduction innovation, after which the manufacturers follow by setting the retail price. The cost of emission-reduction efforts is borne by the supplier alone. In the supply chain, the government is accountable for subsidizing the R&D of power batteries to reduce emissions.

Initially, this paper analyzes the impact of shareholding strategy on profit distribution and emission reduction innovation in the NEV supply chain. Then, an extension model based on a cross-shareholding strategy is proposed. To enhance clarity, we provide a comprehensive listing of model notations in

Table 2. Notations.

Indexes
DECISION VARIABLE
p	Unit retail price of NEV
ω	Unit wholesale price of power battery
θ	Carbon emission reduction
PARAMETERS
Q	Demand for NEVs
a	The maximum demand in the NEV market
b	Price sensitivity coefficient of NEV demand
λ	Low-carbon preference coefficient of consumers
c	Supplier’s unit manufacturing cost for the power battery
V	Other fixed costs for NEVs
α	Fairness concerns preference coefficient of the power battery supplier
β	Fairness concerns preference coefficient of the NEV manufacturer
ϕ	Proportion of government subsidy
η	Coefficient of R&D investment for carbon emission reduction
k	Forward shareholding ratio
j	Backward shareholding ratio
g	Investment ratio of the power battery supplier for IUR
r	Environmental governance cost coefficient
OTHER SYMBOLS
Π	Profit
U	Fairness concerns utility
SW	Social welfare
SUBSCRIPT
()e	Represents the relevant parameters of the power battery supplier
()v	Represents the relevant parameters of the NEV manufacturer
()r	Represents the relevant parameters of the IUR institutions
SUPERSCRIPT
()c	Represents the relevant results in the centralized NEV supply chain
()d	Represents the relevant results in the decentralized NEV supply chain
()j,k	Represents the relevant results in the NEV supply chain under the traditional shareholding strategy
()g	Represents the relevant results in the NEV supply chain under the novel shareholding strategy

.

3.2. Model Assumption

In order to ensure the model’s alignment with realistic conditions, the subsequent section outlines the underlying assumptions:

Assumption 1.

There is only one supplier and one manufacturer in this supply chain, and the market is completely clear. The information between the supplier and the manufacturer is completely symmetric, and the supplier is in the decision-making leadership position and has the ability to make preferential decisions.

Assumption 2.

Similar to the research by Gurnani, et al. [37], the demand function of the NEV manufacturer is

$$Q=a-bp+\lambda \theta$$

(1)

where a is the market potential of NEVs; p is the selling price of NEVs; b is the price sensitivity of consumers $(0<b<1)$, where the smaller it is, the higher consumers’ expected prices of NEV products is; $\lambda$ is consumers’ low-carbon preferences $(0<\lambda <1)$, where the larger it is, the higher consumers’ demand for the low-carbon degree of NEV is; and $\theta$ represents the emission reduction of the NEV supply chain $(0<\theta )$.

Assumption 3.

Power battery suppliers achieve emission reduction of power batteries through technological innovation, so their cost is consistent with the classical technology innovation cost function [38]. The carbon emission reduction cost function is the convex function of $\theta$:

$$c\left(\theta \right)={\displaystyle \frac{1}{2}}\left(1-\varphi \right)\eta {\theta }^2$$

(2)

$\eta$ is the low-carbon R&D investment coefficient of the supplier’s emission-reduction effort. It reflects the R&D innovation ability of the supplier to reduce emissions. To prevent irrational and inconsequential scenarios, we assume $\eta >{\displaystyle \frac{-{\lambda }^2}{2b\left(-1+\varphi \right)}}$. The proportion of government subsidies for carbon emission reduction R&D is $\varphi$.

Assumption 4.

In the supply chain of NEVs, it should be ensured that both battery suppliers and car manufacturers have positive revenues. The following constraints are obtained.

$$\begin{array}{c}p-\omega -V>0\hfill \\ \omega -c>0\hfill \end{array}$$

Assumption 5.

In the NEV industry, different from the traditional automobile industry, manufacturers have weak bargaining power, and the supply chain is dominated by suppliers. In this context, this study uses the “Bounded Rationality”—that is, suppliers and manufacturers do not aim at profit maximization of their own enterprises only, but have fairness concerns [39]. Thus, supply chain members take fairness utility maximization as the decision-making goal. The Nash bargaining fairness-concerns framework is introduced to describe the fairness concern utility function [34], and the self-perceived fairness profit is taken as the reference point of the profit.

For simplicity and without loss of generality, we utilize a linear format to represent the utility of each participant in a two-member supply chain as follows [33,40]:

$$U_i={\Pi }_i+{\mu }_i\left({\Pi }_i-{\overline{\Pi }}_i\right)$$

(3)

where $U_i$ accounts for the profit and fairness concerns of the NEV supply chain member. Parameter ${\mu }_i$ measures the fairness concerned level of enterprises. When ${\mu }_i$ is high, the members are more concerned with fairness. The fairness concern preferences of the supplier and the manufacturer are assumed to be $\alpha$ and $\beta$, respectively ($\alpha ,\beta >0$). Thus, the fairness utility functions of the supplier and the manufacturer in this paper are as follows:

$$U_e={\Pi }_e+\alpha \left({\Pi }_e-{\displaystyle \frac{1+\alpha }{2+\alpha +\beta }}\Pi \right)$$

(4)

$$U_V={\Pi }_v+\beta \left({\Pi }_v-{\displaystyle \frac{1+\beta }{2+\alpha +\beta }}\Pi \right)$$

(5)

Among them, there is $\Pi ={\Pi }_e+{\Pi }_v=\overline{{\Pi }_e}+\overline{{\Pi }_v}$, and $\Pi$ represents the NEV supply chain profit. That is to say, the supplier believes that it should own ${\displaystyle \frac{n}{1+n}}$ of the total profit of the supply chain.

Assumption 6.

In this paper, it is assumed that the government only subsidizes the efforts of emission reduction R&D in the NEV supply chain. The emission reduction cost of NEVs is subsidized by $\varphi (0<\varphi <1)$, and the classical social welfare function [41] is used to describe the following:

$$SW=\Pi +C_S-R-L$$

(6)

where $C_S$ is consumer surplus, $R$ is the government’s subsidy for emission reduction R&D, and $L$ is the economic loss caused by NEVs to the environment, which is called environmental loss in this paper. The consumer surplus $C_S={\displaystyle \frac{1}{2b}}{Q}^2$ is a convex function of the demand. The government subsidy for emission reduction R&D is $R={\displaystyle \frac{1}{2}}\varphi \eta {\theta }^2$. The environmental loss is characterized by the environmental governance cost coefficient $r$ $(r>0)$, which is a convex function of carbon emissions of NEVs. Then the government’s social welfare function in this paper is

$$SW={\Pi }_e+{\Pi }_v+{\displaystyle \frac{1}{2b}}{Q}^2-{\displaystyle \frac{1}{2}}\varphi \eta {\theta }^2-r{\left(1-\theta \right)}^2$$

(7)

3.3. Benchmark Model

As price-only contracts are prevalent in practice, we begin by examining the decisions of supply chain members under this established benchmark. We discuss a centralized NEV supply chain and a decentralized NEV supply chain, respectively. We form a two-stage supply chain model, including a power battery supplier and an NEV manufacturer with the price-only contract.

3.3.1. The Centralized Supply Chain (Case C)

Within the centralized system (Case C), the supply chain functions as a unified entity, determining retail price $p$ and carbon emission reduction level $\theta$ with the objective of maximizing the collective profitability of the total supply chain. The model of Case C is shown in Figure 1.

Theorem 1.

In Case C, we can obtain the equilibrium carbon emission reduction level ${\theta }^c$ , the equilibrium retail price $p^c$ , and the equilibrium quantity $Q^c$. On this basis, we can obtain the equilibrium profit of the whole supply chain ${\Pi }^c$ and the corresponding social welfare ${SW}^c$.

The decision-making objectives of the supply chain can be formulated as

$$\underset{p,\theta }{\mathrm{m}\mathrm{a}\mathrm{x}}{\Pi }^c=\left(p-c-V\right)Q-{\displaystyle \frac{1}{2}}\left(1-\varphi \right)\eta {\theta }^2$$

(8)

Solving this function, we can prove Theorem 1 (the solutions are shown in

Table A1. The formulas corresponding to the symbols.

Formulas	Conditions
$T=a-b(c+V)$	$T>0$
$M^c=4b\eta (-1+\varphi )$	$M^c+2{\lambda }^2<0$
$M^d=b(4+\alpha )(2+\beta )\eta (-1+\varphi )$	$M^d+2{\lambda }^2<0$
$M^{k,j}=b(-4-\alpha +j(2+\alpha +\beta ))(-2-\beta +k(2+\alpha +\beta ))\eta (-1+\varphi )$	$M^{k,j}+2{\lambda }^2<0$ $\left\{\begin{array}{c}j+k\ne 1\\ 0<j<{\displaystyle \frac{2+\alpha }{2+\alpha +\beta }}or {\displaystyle \frac{2+\alpha }{2+\alpha +\beta }}<j<{\displaystyle \frac{2+\alpha +\left(3+\alpha \right)\beta }{\left(1+\beta \right)\left(2+\alpha +\beta \right)}}\\ 0<k<{\displaystyle \frac{2+\alpha +(3+\alpha )\beta -j(1+\beta )(2+\alpha +\beta )}{(2+\alpha +\beta )(3+\alpha -j(2+\alpha +\beta ))}}\end{array}\right.$
$M^g=-bg(-4-\alpha +j(2+\alpha +\beta ))(-2-\beta +k(2+\alpha +\beta ))$	$M^g+2{\lambda }^2<0$ $\left\{\begin{array}{c}j+k\ne 1\\ 0<j<{\displaystyle \frac{2+\alpha }{2+\alpha +\beta }}or {\displaystyle \frac{2+\alpha }{2+\alpha +\beta }}<j<{\displaystyle \frac{2+\alpha +\left(3+\alpha \right)\beta }{\left(1+\beta \right)\left(2+\alpha +\beta \right)}}\\ 0<k<{\displaystyle \frac{2+\alpha +\left(3+\alpha \right)\beta -j\left(1+\beta \right)\left(2+\alpha +\beta \right)}{\left(2+\alpha +\beta \right)\left(3+\alpha -j\left(2+\alpha +\beta \right)\right)}}\\ g>{\displaystyle \frac{2{\lambda }^2}{b(-4-\alpha +j(2+\alpha +\beta ))(-2-\beta +k(2+\alpha +\beta ))}}\end{array}\right.$

,

Table A2. Equilibrium solutions of decision variables.

Case	Equilibrium Solutions
Case C	$p^c=c+V+{\displaystyle \frac{M^{c}T}{2b(2{\lambda }^2+M^c)}}$ ${\theta }^c=-{\displaystyle \frac{2T\lambda }{2{\lambda }^2+M^c}}$
Case D	$p^d=c+V+{\displaystyle \frac{M^{d}T(3+\alpha )}{b(4+\alpha )(2{\lambda }^2+M^d)}}$ ${\omega }^d=c+{\displaystyle \frac{M^{d}T{(2+\alpha )}^2}{b(2+\alpha +\beta )(4+\alpha )(2{\lambda }^2+M^d)}}$ ${\theta }^d=-{\displaystyle \frac{2T\lambda }{2{\lambda }^2+M^d}}$
Case C-SH	$p^{k,j}=c+V+{\displaystyle \frac{M^{k,j}T(-3-\alpha +j(2+\alpha +\beta ))}{b(-4-\alpha +j(2+\alpha +\beta ))(2{\lambda }^2+M^{k,j})}}$ ${\omega }^{k,j}=c+{\displaystyle \frac{M^{k,j}T{(2+\alpha -j(2+\alpha +\beta ))}^2}{b(-1+j+k)(2+\alpha +\beta )(-4-\alpha +j(2+\alpha +\beta ))(2{\lambda }^2+M^{k,j})}}$ ${\theta }^{k,j}=-{\displaystyle \frac{2T\lambda }{2{\lambda }^2+M^{k,j}}}$
Case C-SH-I	$p^g=c+V+{\displaystyle \frac{M^{g}T(-3-\alpha +j(2+\alpha +\beta ))}{b(-4-\alpha +j(2+\alpha +\beta ))(2{\lambda }^2+M^g)}}$ ${\omega }^g=c+{\displaystyle \frac{M^{g}T{((-1+j)(2+\alpha )+j\beta )}^2}{b(-1+j+k)(2+\alpha +\beta )(-4-\alpha +j(2+\alpha +\beta ))(2{\lambda }^2+M^g)}}$ ${\theta }^g=-{\displaystyle \frac{2T\lambda }{2{\lambda }^2+M^g}}$

,

Table A3. Equilibrium solutions of other variables.

Case	Equilibrium Solutions
Case C	$Q^c={\displaystyle \frac{M^{c}T}{2(2{\lambda }^2+M^c)}}$ ${\Pi }^c={\displaystyle \frac{M^c{T}^2}{4b(2{\lambda }^2+M^c)}}$
Case D	$Q^d={\displaystyle \frac{M^{d}T}{(4+\alpha )(2{\lambda }^2+M^d)}}$ ${U_e}^d={\displaystyle \frac{M^d{T}^2(1+\alpha )}{b(4+\alpha )(2+\alpha +\beta )(2{\lambda }^2+M^d)}}$ ${U_v}^d=M^d{T}^2{\displaystyle \frac{(1+\beta )(-2\beta (4+\alpha ){\lambda }^2+M^d(2+\alpha )(2+\beta ))}{b{(4+\alpha )}^2(2+\beta )(2+\alpha +\beta ){(2{\lambda }^2+M^d)}^2}}$ $U^d={\displaystyle \frac{M^d{T}^2(M^d(2+\beta )(2(3+\beta )+\alpha (6+\alpha +\beta ))+2(4+\alpha )(2-{\beta }^2+\alpha (2+\beta )){\lambda }^2)}{b{(4+\alpha )}^2(2+\beta )(2+\alpha +\beta ){(2{\lambda }^2+M^d)}^2}}$
Case C-SH	$Q^{k,j}={\displaystyle \frac{M^{k,j}T}{(4+\alpha -j(2+\alpha +\beta ))(2{\lambda }^2+M^{k,j})}}$ ${U_e}^{k,j}={\displaystyle \frac{M^{k,j}{T}^2(1+\alpha )}{b(4+\alpha -j(2+\alpha +\beta ))(2+\alpha +\beta )(2{\lambda }^2+M^{k,j})}}$ ${U_v}^{k,j}={\displaystyle \frac{M^{k,j}{T}^2\left(1+\beta \right)(M^{k,j}((-1+j)(2+\alpha )+j\beta )(2+\beta -k(2+\alpha +\beta ))+2(-4-\alpha +j(2+\alpha +\beta ))(-\beta +k(2+\alpha +\beta )){\lambda }^2)}{b{\left(4+\alpha -j\left(2+\alpha +\beta \right)\right)}^2\left(2+\alpha +\beta \right)\left(-2-\beta +k\left(2+\alpha +\beta \right)\right){(2{\lambda }^2+M^{k,j})}^2}}$ $U^{k,j}={\displaystyle \frac{M^{k,j}{T}^2\left(1+\beta \right)(M^{k,j}((-1+j)(2+\alpha )+j\beta )(2+\beta -k(2+\alpha +\beta ))+2(-4-\alpha +j(2+\alpha +\beta ))(-\beta +k(2+\alpha +\beta )){\lambda }^2)}{b{\left(4+\alpha -j\left(2+\alpha +\beta \right)\right)}^2\left(2+\alpha +\beta \right)\left(-2-\beta +k\left(2+\alpha +\beta \right)\right){(2{\lambda }^2+M^{k,j})}^2}}+{\displaystyle \frac{M^{k,j}{T}^2(1+\alpha )}{b(4+\alpha -j(2+\alpha +\beta ))(2+\alpha +\beta )(2{\lambda }^2+M^{k,j})}}$
Case C-SH-I	$Q^g={\displaystyle \frac{M^{g}T}{(4+\alpha -j(2+\alpha +\beta ))(2{\lambda }^2+M^g)}}$ ${U_e}^g={\displaystyle \frac{M^g{T}^2(1+\alpha )}{b(4+\alpha -j(2+\alpha +\beta ))(2+\alpha +\beta )(2{\lambda }^2+M^g)}}$ ${U_v}^g={\displaystyle \frac{M^g{T}^2\left(1+\beta \right)\left(M^g\left(\left(-1+j\right)\left(2+\alpha \right)+j\beta \right)\left(2+\beta -k\left(2+\alpha +\beta \right)\right)+2\left(-4-\alpha +j\left(2+\alpha +\beta \right)\right)\left(-\beta +k\left(2+\alpha +\beta \right)\right){\lambda }^2\right)}{b{\left(4+\alpha -j\left(2+\alpha +\beta \right)\right)}^2\left(2+\alpha +\beta \right)\left(-2-\beta +k\left(2+\alpha +\beta \right)\right){\left(2{\lambda }^2+M^g\right)}^2}}$ $U^g={\displaystyle \frac{M^g{T}^2\left(1+\beta \right)\left(M^g\left(\left(-1+j\right)\left(2+\alpha \right)+j\beta \right)\left(2+\beta -k\left(2+\alpha +\beta \right)\right)+2\left(-4-\alpha +j\left(2+\alpha +\beta \right)\right)\left(-\beta +k\left(2+\alpha +\beta \right)\right){\lambda }^2\right)}{b{\left(4+\alpha -j\left(2+\alpha +\beta \right)\right)}^2\left(2+\alpha +\beta \right)\left(-2-\beta +k\left(2+\alpha +\beta \right)\right){\left(2{\lambda }^2+M^g\right)}^2}}$ $+{\displaystyle \frac{M^g{T}^2(1+\alpha )}{b(4+\alpha -j(2+\alpha +\beta ))(2+\alpha +\beta )(2{\lambda }^2+M^g)}}$ ${{\Pi }_r}^g={\displaystyle \frac{2T^2{\lambda }^2(g+{\eta }_r(-1+\varphi ))}{{(2{\lambda }^2+M^g)}^2}}$

and

Table A4. Social welfare.

Case	Social Welfare
Case C	${SW}^c={\displaystyle \frac{2T^2\eta (-{\lambda }^2+3b\eta {(-1+\varphi )}^2)}{{(2{\lambda }^2+M^c)}^2}}-r(1+{\displaystyle \frac{2T\lambda }{{2\lambda }^2+M^c}}{)}^2$
Case D	${SW}^d={\displaystyle \frac{T^2\eta (-4{\lambda }^2+b(7+2\alpha ){(2+\beta )}^2\eta {(-1+\varphi )}^2)}{2{(2{\lambda }^2+M^d)}^2}}-r(1+{\displaystyle \frac{2T\lambda }{2{\lambda }^2+M^d}}{)}^2$
Case C-SH	${SW}^{k,j}={\displaystyle \frac{T^2\eta (-4{\lambda }^2+b(7+2\alpha -2j(2+\alpha +\beta )){(2+\beta -k(2+\alpha +\beta ))}^2\eta {(-1+\varphi )}^2)}{2{(2{\lambda }^2+M^{k,j})}^2}}-r(1+{\displaystyle \frac{2T\lambda }{2{\lambda }^2+M^{k,j}}}{)}^2$
Case C-SH-I	${SW}^g={\displaystyle \frac{T^2(-4{\eta }_r{\lambda }^2+b{g}^2(7+2\alpha -2j(2+\alpha +\beta )){(2+\beta -k(2+\alpha +\beta ))}^2)}{2{(2{\lambda }^2+M^g)}^2}}-r(1+{\displaystyle \frac{2T\lambda }{2{\lambda }^2+M^g}}{)}^2$

in Appendix A).

Proof 

The proof is by the backward induction and can be found in many classical studies on the subject. □

Since ${\Pi }^c$ is concave in $\theta ,p$, we have $\eta >{\displaystyle \frac{{\lambda }^2}{2b(1-\varphi )}}$, which means that the R&D capacity of the supplier is always limited.

3.3.2. The Decentralized Supply Chain (Case D)

In the decentralized supply chain (Case D), the power battery supplier holds a leading position and prioritizes decisions on emission reduction levels and wholesale prices. Then, the NEV manufacturer lags behind in pricing the sales price, and finally, consumers purchase. Thus, a two-stage Stackelberg pricing game is established. We employ backward induction to ascertain the Stackelberg equilibrium. The model of Case D is shown in Figure 2.

Theorem 2.

In Case D, we can obtain the equilibrium carbon emission reduction level ${\theta }^d$, the equilibrium retail price $p^d$, the equilibrium wholesale price ${\omega }^d$, and the equilibrium quantity $Q^d$. On this basis, we can obtain the equilibrium utilities of the whole supply chain, power battery supplier, and NEV manufacturer $U^d$, ${U_e}^d$, ${U_v}^d$ and the corresponding social welfare ${SW}^d$.

In Case D, the profit ${\Pi }_e$ of power battery supplier and ${\Pi }_v$ of NEV manufacturer are

$${\Pi }_e=\left(\omega -c\right)Q-{\displaystyle \frac{1}{2}}\left(1-\varphi \right)\eta {\theta }^2$$

(9)

$${\Pi }_v=\left(p-\omega -V\right)Q$$

(10)

As explained in Assumption 5, this paper uses the fairness concerns utility function to characterize the utility of suppliers. The objective functions of suppliers and manufacturers are, respectively,

$$\left\{\begin{array}{c}\underset{\omega ,\theta }{\mathrm{m}\mathrm{a}\mathrm{x}}{U}_e\\ \underset{p}{\mathrm{m}\mathrm{a}\mathrm{x}}{U}_V\end{array}\right.$$

(11)

Solving this game, we can prove Theorem 2 (the solutions are shown in Table A1, Table A2, Table A3 and Table A4 in Appendix A).

Proof 

The proof is analogous to the one provided for Theorem 1. □

It is easy to verify that ${\theta }^c>{\theta }^d$ and ${{\Pi }^c>U}^d$—that is, both $\theta$ and $U$ are lower than them in Case C. The fairness concern preferences of enterprises make the double marginal effect more prominent. Therefore, we need a coordination strategy to coordinate the NEV supply chain. The following study discusses the supply chain coordination ability of shareholding strategy, a widely used financial tool for achieving supply chain coordination.

3.4. NEV Supply Chain Coordination Using Two Strategies

3.4.1. Shareholding Strategy (Case C-SH)

Rather than contracts, many companies may employ standardized financial instruments to align their interests. Supply chain participants can acquire the shares of another member to gain from their operations, and in turn, the target member can also be motivated to achieve mutual benefits.

In Case C-SH, the power battery supplier holds a leading position and prioritizes decisions on emission reduction levels and wholesale prices. Then, the NEV manufacturer makes decisions on sales prices, and finally, consumers purchase. Thus, a two-stage Stackelberg pricing game is formed. The model of Case C-SH is shown in Figure 3.

Theorem 3.

In Case C-SH, we can obtain the equilibrium carbon emission reduction level ${\theta }^{k,j}$, the equilibrium retail price $p^{k,j}$, the equilibrium wholesale price ${\omega }^{k,j}$, and the equilibrium quantity $Q^{k,j}$. On this basis, we can obtain the equilibrium utilities of the whole supply chain, power battery supplier, and NEV manufacturer $U^{k,j}$, ${U_e}^{k,j}$, ${U_v}^{k,j}$ and the corresponding social welfare ${SW}^{k,j}$.

In Case C-SH, the power battery supplier and the NEV manufacturer play the Stackelberg game dominated by the supplier. The supplier’s profit ${\Pi }_e$ and the manufacturer’s profit ${\Pi }_v$ are, respectively,

$${\Pi }_e=(1-k)((\omega -c)Q-{\displaystyle \frac{1}{2}}\eta {\theta }^2(1-\varphi ))+j(p-\omega -V)Q$$

(12)

$${\Pi }_v=(1-j)(p-\omega -V)Q+k((\omega -c)Q-{\displaystyle \frac{1}{2}}\eta {\theta }^2(1-\varphi ))$$

(13)

Similarly, we can solve this game to prove Theorem 3 (the solutions are shown in Table A1, Table A2, Table A3 and Table A4 in Appendix A, and the detailed proof is given in Appendix B).

Proposition 1.

Case C-SH can perfectly coordinate the NEV supply chain (i) when $\alpha =\beta$, $k+j=1$ (ii) when $\alpha \ne \beta$, $j={\displaystyle \frac{1+\alpha }{2+\alpha +\beta }}$, and $k={\displaystyle \frac{1+\beta }{2+\alpha +\beta }}$. The detailed proof is given in Appendix B.

Based on the foregoing findings, we find that the NEV supply chain cannot achieve perfect coordination under existing constraint $\omega >c$ in Case C-SH. Therefore, we can relax the constraint. When $\omega =c$, the supplier sells the power batteries to the manufacturer at cost price and gains profits by holding the manufacturer. Then, the NEV supply chain can achieve perfect coordination, and the total utility and decision variables are consistent with those in the Case C.

Therefore, Proposition 1 reveals that when both parties have the same fairness concerns preference, if the sum of each other’s shareholding ratios is 1, then the supply chain can achieve perfect coordination. Alternatively, if the fairness concern preferences are different, their shareholding ratios must satisfy a strict limitation connecting with the fairness concern preferences.

Corollary 1.

The supply chain members accept Case C-SH to achieve the perfect coordination. (i) When $\alpha =\beta$, the value of $j,k$ should satisfy $j_1<j<j_2$ and $k+j=1$, and (ii) when $\alpha \ne \beta$, the value of $j,k$ should satisfy $j={\displaystyle \frac{1+\alpha }{2+\alpha +\beta }}$ and $k={\displaystyle \frac{1+\beta }{2+\alpha +\beta }}$. $j_1,j_2$ are given in 

Table A5. Boundary values.

Boundary Values
$j_1={\displaystyle \frac{4(1+\alpha ){\lambda }^2+b{(2+\alpha )}^3\eta (-1+\varphi )}{2(1+\alpha )(2{\lambda }^2+b(2+\alpha )(4+\alpha )\eta (-1+\varphi ))}}$
$j_2={\displaystyle \frac{8(1+\alpha ){\lambda }^4+2b(24+\alpha (32+\alpha (10+\alpha )))\eta {\lambda }^2(-1+\varphi )+b^2{(2+\alpha )}^4(6+\alpha ){\eta }^2{(-1+\varphi )}^2}{2(1+\alpha ){(2{\lambda }^2+b(2+\alpha )(4+\alpha )\eta (-1+\varphi ))}^2}}$
$j_3={\displaystyle \frac{2+\beta +\alpha (4+\alpha +2\beta )}{{(2+\alpha +\beta )}^2}}$
$j_4={\displaystyle \frac{\left(\begin{array}{c}(2{\left(2{\lambda }^2+b\left(4+\alpha \right)\left(2+\beta \right)\eta \left(-1+\varphi \right)\right)}^2+\alpha (2+\alpha ){\left(2{\lambda }^2+b\left(4+\alpha \right)\left(2+\beta \right)\eta \left(-1+\varphi \right)\right)}^2\\ +\beta {\left(2{\lambda }^2+b\left(4+\alpha \right)\left(2+\beta \right)\eta \left(-1+\varphi \right)\right)}^2+2(1+\beta )({\lambda }^2+2b\eta (-1+\varphi ))\\ (2\beta {\lambda }^2-b(2+\alpha ){\left(2+\beta \right)}^2\eta (-1+\varphi )))(-2\beta {\lambda }^2+b(2+\alpha ){(2+\beta )}^2\eta (-1+\varphi ))\end{array}\right)}{-(1+\alpha )(2+\alpha +\beta ){(2{\lambda }^2+b(4+\alpha )(2+\beta )\eta (-1+\varphi ))}^2(2\beta {\lambda }^2-b(2+\alpha ){(2+\beta )}^2\eta (-1+\varphi ))}}$
${\eta }_1={\displaystyle \frac{({\alpha }^2(2+\beta )+{(2+\beta )}^2+\alpha (4+\beta -{\beta }^2)-j(2+\alpha +\beta )(2-{\beta }^2+\alpha (2+\beta ))){\lambda }^2}{2b(\alpha -\beta )(2+\beta )(-1-\alpha +j(2+\alpha +\beta ))(-1+\varphi )}}$
$k_g={\displaystyle \frac{{\lambda }^2(g(2-{\beta }^2+\alpha (2+\beta ))+(2+\alpha +\beta )\eta (-1+\varphi ))+2b(\alpha -\beta )\eta (g(2+\beta )+\eta (-1+\varphi ))(-1+\varphi )}{g(\alpha -\beta )(2+\alpha +\beta )({\lambda }^2+2b\eta (-1+\varphi ))}}$
$j_g={\displaystyle \frac{{\lambda }^2(2g(1+\beta )+(2+\alpha +{\alpha }^2+\beta -\alpha \beta )\eta (-1+\varphi ))+2b(1+\alpha )(\alpha -\beta ){\eta }^2{(-1+\varphi )}^2}{(\alpha -\beta )(2+\alpha +\beta )\eta ({\lambda }^2+2b\eta (-1+\varphi ))(-1+\varphi )}}$

 in Appendix A, and detailed proof is given in Appendix B.

According to Corollary 1, we find that when Case C-SH perfectly coordinate the NEV supply chain, the shareholding ratios are unique and sum to one, unless the enterprises have the same fairness concerns preference. Moreover, there is always an enterprise that is over 50% owned, which means that the enterprise risks losing management rights. Therefore, to solve these problems, based on IUR’s high R&D capability, we try to propose a new shareholding strategy in the next section.

3.4.2. Extended Model: A Novel IUR Shareholding Strategy (Case C-SH-I)

Due to the limitation of Case C-SH, we propose a cross-shareholding R&D cooperation strategy with IUR (Case C-SH-I) to coordinate the NEV supply chain. In areas where technology development is fast, the selection of R&D partners often favors those considered crucial sources of knowledge in the innovation process. It is generally believed that universities and research institutions have stronger R&D capabilities and can obtain better and more stable innovation degree with the same degree of investment $(0<{\eta }_r<\eta )$. Different from the basic model, in this case, the supplier entrusts all the emission-reduction R&D to a third party, IUR, and bears the associated costs.

In Case C-SH-I, the power battery supplier holds a leading position and gives priority to making decisions on the emission reduction level and wholesale price. Then, the NEV manufacturer makes decision on the sales price, and finally, consumers purchase. Thus, a two-stage Stackelberg pricing game is formed. The model of Case C-SH-I is shown in Figure 4.

In particular, IUR is responsible for low-carbon research and development according to the requirements of battery manufacturers. We assume that the R&D cost function paid by the supplier to IUR is still a convex function of the power battery emission reduction. The investment ratio of supplier for IUR is $g(0<g)$. Even if IUR is defined as a non-profit organization, it must ensure that its profits are non-negative to maintain effective R&D efforts.

Theorem 4.

In Case C-SH-I, we can obtain the equilibrium carbon emission reduction level ${\theta }^g$, the equilibrium retail price $p^g$, the equilibrium wholesale price ${\omega }^g$, and the equilibrium quantity $Q^g$. On this basis, we can obtain the equilibrium utilities of the whole supply chain, power battery supplier, and NEV manufacturer $U^g$, ${U_e}^g$, ${U_v}^g$ and the corresponding social welfare ${SW}^g$.

The profits of IUR ${\Pi }_r$, power battery suppliers ${\Pi }_e$, and NEV manufacturers ${\Pi }_v$ are

$${\Pi }_r={\displaystyle \frac{1}{2}}g{\theta }^2-{\displaystyle \frac{1}{2}}{\eta }_r{\theta }^2(1-\varphi )$$

(14)

$${\Pi }_e=(1-k)((\omega -c)Q-{\displaystyle \frac{1}{2}}g{\theta }^2)+j(p-\omega -V)Q$$

(15)

$${\Pi }_v=(1-j)(p-\omega -V)Q+k((\omega -c)Q-{\displaystyle \frac{1}{2}}g{\theta }^2)$$

(16)

Similarly, we can solve this game to prove Theorem 4 (the solutions are shown in Table A1, Table A2, Table A3 and Table A4 in Appendix A, and detailed proof is given in Appendix B).

Proposition 2.

Case C-SH-I can perfectly coordinate the NEV supply chain (i) when $\alpha =\beta$, $k+j=1 and \left(ii\right) when \alpha \ne \beta , j=j_g$ and $k=k_g.$ $j_g, k_g$ are given in Table A5 in Appendix A, and the detailed proof is given in Appendix B.

Based on the foregoing findings, we find that the NEV supply chain cannot achieve perfect coordination under existing constraint $\omega >c$ in Case C-SH-I. Therefore, we can relax the constraint. When $\omega =c$, the NEV supply chain can achieve perfect coordination, and the total utility and decision variables are consistent with those in Case C.

Different from Case C-SH, Proposition 2 reveals that, when the fairness concern preferences are different, their shareholding ratios can be adjusted by the investment ratio of the supplier. This means that the shareholding ratio of both parties can be maintained below 50%. That is, using Case C-SH-I to coordinate the supply chain will not compromise the management rights of the enterprise. The above findings indicate that the novel IUR shareholding strategy can solve the problems existing in the common cross-shareholding strategy.

Corollary 2.

The supply chain members accept Case C-SH-I to achieve perfect coordination. (i) When $\alpha =\beta$, the value of $j,k$ should satisfy $j_1<j<j_2$ and $k+j=1$; (ii) $\alpha >\beta$, the value of $j,k, \eta$ should satisfy ${\displaystyle \frac{1+\alpha }{2+\alpha +\beta }}<j<j_4$, $k=k_g$, $\eta >-{\displaystyle \frac{{\lambda }^2}{-2b+2b\varphi }}$; and (iii) $\alpha <\beta$, the value of $j,k, \eta$ should satisfy ${\displaystyle \frac{1+\alpha }{2+\alpha +\beta }}<j<j_4$, $k=k_g$, ${\eta }_1>\eta >-{\displaystyle \frac{{\lambda }^2}{-2b+2b\varphi }}$. $j_1, j_2, j_4, j_g, k_g, {\eta }_1$ are given in Table A5 in Appendix A.

Proof 

The proof is analogous to the one provided for Corollary 1. □

By comparing Corollary 1 and Corollary 2, we find that the novel shareholding strategy can effectively adjust the shareholding ratio via the investment ratio ($g)$. The advantage of Case C-SH-I is that the shareholding ratios are no longer unique, and no party needs to be owned more than 50%. However, we find that the external environmental factors affecting the utilities of the NEV supply chain are not robust. Different levels of investment in IUR and varying fairness concern preferences will exhibit different monotonicity. We will use numerical examples to verify the models in Section 4.

3.5. Sensitivity Analysis

To make more intuitive comparisons and analyses, the sensitivity analyses of the above models are concentrated in this section.

Corollary 3.

We observe that the coordination strategies of the NEV supply chain do not affect the sensitivity of the optimal results to the external environmental factors. Specific sensitivity conclusions can be found in 

Table 3. The sensitivity analysis conclusions of the external environmental factors.

	$\mathit{p}$	$\mathit{\omega }$	$\mathit{\theta }$	${\mathit{U}}_{\mathit{e}}$	${\mathit{U}}_{\mathit{v}}$	$\mathit{U}$
$\varphi$	+	+	+	+	+	+
$\lambda$	+	+	+	+	+	+
$\eta$	−	−	−	−	−	−

Notions: +—monotonically increasing, −—monotonically decreasing.

.

Table 3 shows that in all models, retail price $p$, wholesale price $\omega$, carbon emission reduction level $\theta$, the utility of supplier $U_e$, the utility of manufacturer $U_v$, and the total utility $U$ (i) increase in $\varphi , \lambda$ and (ii) decrease in $\eta$.

The results show that the government subsidies and consumers’ low carbon preferences support the NEV supply chain in producing more ecofriendly NEVs. Although ecofriendly NEVs come with higher prices, enterprises can achieve greater utility in the market where consumers have a higher level of low-carbon preferences. Regarding $\eta$, the high coefficient of R&D investment has been a pain point for manufacturers, leading them to forgo higher emission reduction levels. Hence, when the low carbon degree of NEVs is insufficient, the supply chain needs to leverage price advantages to retain customers.

Corollary 4.

We observe that when the NEV supply chain adopts a coordination strategy and achieves perfect coordination, (i) the enterprises’ fairness concern preferences will no longer affect the retail price, wholesale price, carbon emission reduction level, and total utility. (ii) The sensitivities of fairness concern preferences to the optimal utilities of the supplier and the manufacturer are affected by the shareholding ratios. Specific sensitivity conclusions can be found in 

Table 4. The sensitivity analysis conclusions of the fairness concern preferences.

				$U_e$	$U_v$
Case D		α β		* −	− *
Case C-SH	A	$\alpha /\beta$	$j_1<j<{\displaystyle \frac{1}{2}}$ ${\displaystyle \frac{1}{2}}<j<j_2$	− +	+ −
	B	$\alpha$	$j={\displaystyle \frac{1+\alpha }{2+\alpha +\beta }}$	+	−
		$\beta$		−	+
Case C-SH-I	A	$\alpha /\beta$	$j_1<j<{\displaystyle \frac{1}{2}}$ ${\displaystyle \frac{1}{2}}<j<j_2$	− +	+ −
	B/C	$\alpha$	${\displaystyle \frac{1+\alpha }{2+\alpha +\beta }}<j<j_3$ $j_3<j<j_4$	− +	+ −
		$\beta$	${\displaystyle \frac{1+\alpha }{2+\alpha +\beta }}<j<j_4$	+	−

Notions: +—monotonically increasing, −—monotonically decreasing, *—uncertain.

. (For the convenience of discussion, coordination strategies are classified and discussed. The specific constraints of different types are shown in 

Table A6. Types of coordination strategies.

Case	Types
$\mathrm{C}\mathrm{a}\mathrm{s}\mathrm{e} \mathrm{C}\text{-}\mathrm{S}\mathrm{H}$	$\begin{array}{c}\left(a\right) \left\{\begin{array}{c} j_1<j<j_2\\ k+j=1\end{array}\right.; \alpha =\beta \to \mathrm{Case} \mathrm{C}\text{-}\mathrm{S}\mathrm{H}\text{-}A\\ \left(b\right) \left\{\begin{array}{c}k={\displaystyle \frac{1+\beta }{2+\alpha +\beta }}\\ j={\displaystyle \frac{1+\alpha }{2+\alpha +\beta }}\end{array}\right.; \alpha \ne \beta \to \mathrm{Case} \mathrm{C}\text{-}\mathrm{SH}\text{-}\mathrm{B}\end{array}$
$\mathrm{C}\mathrm{a}\mathrm{s}\mathrm{e} \mathrm{C}\text{-}\mathrm{S}\mathrm{H}\text{-}\mathrm{I}$	$\left(a\right) \alpha =\beta \to \left\{\begin{array}{c} j_1<j<j_2\\ k+j=1\end{array}\right.\to \mathrm{C}\mathrm{a}\mathrm{s}\mathrm{e} \mathrm{C}\text{-}\mathrm{S}\mathrm{H}\text{-}\mathrm{I}\text{-}A$ $\left(b\right) \alpha <\beta \to \left\{\begin{array}{c} {\displaystyle \frac{1+\alpha }{2+\alpha +\beta }}<j<j_3\\ k=k_g\\ {\eta }_1>\eta >{\displaystyle \frac{{\lambda }^2}{2b-2b\varphi }}\end{array}\right.\to \mathrm{C}\mathrm{a}\mathrm{s}\mathrm{e} \mathrm{C}\text{-}\mathrm{S}\mathrm{H}\text{-}\mathrm{I}\text{-}B$ $\left(c\right) \alpha >\beta \to \left\{\begin{array}{c} {\displaystyle \frac{1+\alpha }{2+\alpha +\beta }}<j<j_3\\ k=k_g\\ \eta >{\displaystyle \frac{{\lambda }^2}{2b-2b\varphi }}\end{array}\right.\to \mathrm{C}\mathrm{a}\mathrm{s}\mathrm{e} \mathrm{C}\text{-}\mathrm{S}\mathrm{H}\text{-}\mathrm{I}\text{-}C$

 in Appendix A).

Table 4 shows that the coordination strategies can make the influence of fairness preferences on the optimal utilities become regular. We observe that in a decentralized supply chain (Case D), an increase in fairness preferences hurts the optimal utility of the other side but does not necessarily benefit itself. Then, in Case C-SH, when $\alpha =\beta$ and $j<0.5$, the increase in $\alpha (\beta )$ is beneficial to the manufacturer and harmful to the supplier. Otherwise $(j>0.5)$, we will obtain the opposite results. This suggests that the parties with the higher shareholding ratios always “benefit at the expense of others” by increasing their own fairness preference. However, when $\alpha \ne \beta$, no matter what the shareholding ratio is, the fairness preferences always benefit themselves and harm others.

In Case C-SH-I, when $\alpha =\beta$, the conclusions are consistent with those in Case C-SH. When $\alpha \ne \beta$ and $j_3<j$, the self-serving tendency of the supplier still exists. Interestingly, when $j_3>j$, the object of increasing utility due to the growth of the fairness preference changes. In this case, the increase in one’s own fairness preference will increase the utility of the other party and inhibit one’s own utility. This shows that the novel cross-shareholding strategy has advantages in restraining the excessive fairness preferences of NEV supply chain enterprises.

3.6. Optimal Government Subsidies for NEV Supply Chain

In this study, the government, as a supporter of the NEV supply chain, provides subsidies for the low-carbon R&D of power batteries. The government does not participate in the game, but it can still decide the optimal subsidy from the perspective of maximizing social welfare. Therefore, in this section, we explore the optimal government subsidy ratios and social welfare.

Theorem 5.

For the government, to maximize social welfare without directly intervening in the decisions of supply chain members, we can obtain the optimal subsidy ratios for different cases of ${\varphi }^c, {\varphi }^d, {\varphi }^{k,j}, {\varphi }^g$. The detailed proof is given in Appendix B.

According to Theorem 5, we have ${SW}^d<{SW}^c={SW}^{k,j}<{SW}^g$ and ${\varphi }^c={\varphi }^{k,j}<{\varphi }^g<{\varphi }^d$. In other words, when Case C-SH achieves perfect coordination in the NEV supply chain, it can align the social welfare function with Case C. This implies that if a cross-shareholding strategy is used to achieve perfect coordination in the NEV supply chain, the government can make subsidy decisions based on the centralized supply chain. This can enhance government decision making efficiency, enabling the attainment of optimal social welfare without the need to monitor changes in shareholdings among enterprises. In Case C-SH-I, the involvement of research institutions has led to an increase in social welfare by enhancing R&D capabilities. In other words, in Case C-SH-I, achieving equivalent social welfare requires fewer subsidies from the government. It significantly reduces the government’s financial burden and facilitates the implementation of subsidy reduction plans. However, the maximum social welfare valued by the government may fall within the negative range of $\varphi$. In such instances, theoretically, the government should withdraw subsidies to achieve maximum social welfare. Therefore, it is worth investigating how $\varphi$ values may influence enterprises’ utilities, as detailed in the numerical analysis within Section 4.

4. Numerical Studies

Numerical studies investigate the impact of three strategies on the utility of power battery suppliers and NEV manufacturers, while numerical analysis reflects the influence of external factors on coordinated supply chains. Industry reports and previous research suggest the following parameter values: $\alpha =1$, $\beta =2$, $b=0.7$, $a=10$, $c=3$, $V=5$, $\lambda =1$, $\varphi =0.3$, $\eta =3$, $g=1.5$, ${\eta }_r=2$, $r=1.5$. With these given data, we obtain the following observations.

4.1. The Impact of Strategies on NEV Supply Chain and the Optimal Strategies

This section first discusses the impact of changes in $j,k$ on the utility of the NEV supply chain under two cross-shareholding strategies. Next, we explore the optimal coordination (improvement) strategies for suppliers and manufacturers under different scenarios.

4.1.1. The Impact of Two Shareholding Strategies on NEV Supply Chain

This section explores the effects of using two shareholding strategies on NEV supply chain utility. We incorporate the hypothesis values into the optimal results shown in Table A3 in Appendix A.

In Figure 5, we observe the following conclusions. For the whole NEV supply chain, when one of the values of $j$ or $k$ is fixed, the total utility first increases and then decreases as the other ratio increases. When $j$ and $k$ are the same, the total utility in Case C-SH-I is always greater than in Case C-SH. When $j$ and $k$ are at a high level, the total utility is lower than that in Case D. In this setting, only the total utility in Case C-SH-I can equal that in Case C. For the power battery supplier, $j$ and $k$ consistently promote utility growth. We observe that ${U_e}^g>{U_e}^{j,k}>{U_e}^d$, with the exception of $k$ at a relatively high level. This means that as long as the manufacturer’s shareholding ratio is not too large, the supplier can always benefit from the shareholding strategies. For the NEV manufacturer, the impacts of $j$ and $k$ on the manufacturer’s utility are similar to their impacts on the total utility. Thus, only when $j$ and $k$ are within the appropriate ranges will the manufacturer consider the shareholding strategies.

Solving ${U_v}^{k,j}{{,U}_v}^g$ with the coordination conditions yields $\omega =c$, which violates the constraints (see the proofs of Propositions 1 and 2 in Appendix B; the proof of the same formula below is omitted). Therefore, Case C-SH and Case C-SH-I cannot perfectly coordinate the supply chain without relaxing the constraint in this numerical study. This implies that if the enterprises fail to reach an agreement, the strategies can be used to achieve Pareto improvement rather than perfect coordination in the NEV supply chain. Conversely, when $\omega =c$, the NEV supply chain can be perfectly coordinated by the strategies.

4.1.2. The Optimal Strategy for the NEV Supply Chain Members

This section investigates the optimal strategy for the NEV supply chain members in two situations: with or without constraints.

First, when the enterprises fail to reach an agreement (i.e., $\omega \ne c)$, they can still use the strategies to achieve Pareto improvement. Figure 6 illustrates the optimal strategy set for enterprises at various levels of shareholding ratios. For the power battery supplier, cross-shareholding strategies can effectively achieve Pareto improvement in most situations. Given the issue of corporate control ($0<j,k<0.5$), the optimal choice for suppliers is always Case C-SH-I. Similarly, for the NEV manufacturer, decision makers are inclined to choose Case C-SH-I.

Then, when suppliers and manufacturers reach an agreement on $\omega =c$, the coordination conditions can be obtained by solving ${U_v}^{k,j}$ and ${U_v}^g$. In Case C-SH, when $j=0.4$ and $k=0.6$, we have $U^{j,k}={\Pi }^c$, ${U_e}^{j,k}>{U_e}^d$, ${U_v}^{j,k}>{U_v}^d$, ${U_e}^{j,k}<{U_v}^{j,k}$. In Case C-SH-I, when $j=0.58$ and $k=0.27$, we have $U^g={\Pi }^c$, ${U_e}^g>{U_e}^d$, ${U_v}^g>{U_v}^d{{,U}_e}^g>{U_v}^g$. At these points, the NEV supply chain can be perfectly coordinated. Figure 7 illustrates the optimal strategy set for enterprises at various levels of shareholding ratios. Similarly, considering the issue of corporate control, ($0<j,k<0.5$). For the power battery supplier, the optimal strategy is Case C-SH-I. For the NEV manufacturer, the optimal strategy is Case C-SH. In contrast to the supplier, the manufacturer is more inclined to support low forward shareholding ratios. When $j$ is sufficiently low, the manufacturer can achieve a higher utility in the Case C-SH.

Hence, supply chain members will choose Case C-SH-I to coordinate the NEV supply chain within the acceptable range of $j,k$. Consequently, the NEV supply chain achieves Pareto improvement. However, compared to Figure 6, the Pareto interval for achieving utility improvement has become smaller. This indicates that the NEV supply chain does not need to relax the constraint if the goal is solely to achieve Pareto improvements.

Based on the above numerical study, if achieving Pareto improvement is the sole objective for the NEV supply chain, enterprises do not need to align with $\omega =c$. Within the range of $0<j,k<0.5$, utilizing Case C-SH-I is the optimal strategy for firms. However, if both parties aim for perfect coordination, enterprises need to align with $\omega =c$, where $j=0.58$ and $k=0.27$. It is apparent that the value of $j$ does not match the ideal scenario. Nevertheless, if the supplier adjusts $g$ to a value greater than 1.76, then $0<j,k<0.5$. In this scenario, the supply chain achieves perfect coordination, and the coordination strategy does not affect the management rights. These conclusions highlight the advancement and adaptability of the cross-shareholding strategy that integrates IUR in coordinating the NEV supply chain.

4.2. The Impact of the External Environments on the NEV Supply Chain

This section studies the impact of three external environmental factors, consumer low-carbon preference, government subsidy, and carbon emission reduction R&D capability, on the NEV supply chain. According to the conclusions in Section 4.1, considering the issue of ownership among supply chain enterprises ($0<j,k<0.5$), we set the investment ratio $g$ to 1.8. Hence, we have $k_g=0.46$, $j_g=0.49$, which are the shareholding ratios that can make the NEV supply chain achieve perfect coordination under the calculation example. This implies that Case C-SH-I can perfectly coordinate the NEV supply chain without affecting the management rights. At this point, NEV supply chain members have two possible options: (i) use the Case C-SH-I strategy for perfect coordination of the supply chain ($\omega =c$); (ii) use the Case C-SH-I strategy to achieve Pareto improvement in the supply chain ($\omega >c$). To control variables and simplify computations, we assume ${\theta }^c={\theta }^g={\theta }^{g*}$ and $j_{g^{*}}=0.4$, where the superscript “$g^{*}$” represents the results of Option (ii). Next, we will discuss the impact of one external factor on the NEV supply chain when other parameters are held constant.

4.2.1. The Impact of Consumer’s Low-Carbon Preference on NEV Supply Chain

Consumer preferences have always been a focal influencing factor for both enterprises and governments. Consumer demand for low-carbon products affects both the utility of the supply chain and social welfare. In this section, $\lambda$ is the independent variable, so we have $k_g={\displaystyle \frac{-1.67+0.77{\lambda }^2}{-2.94+{\lambda }^2}}, j_g={\displaystyle \frac{-1.18+0.23{\lambda }^2}{-2.94+{\lambda }^2}}, k_{g^{*}}=0.49, j_{g^{*}}=0.4.$

In Figure 8, we observe the following conclusions.

In Case D, the impact of the low-carbon preferences of consumers $\lambda$ on utilities and social welfare is minimal. This suggests that non-cooperative NEV supply chain members lack the incentive to increase consumers’ low-carbon preferences. When $0<j,k<0.5$, we have $0.857<\lambda <1$; this range represents the effective coordination (imp rovement) range (the dashed line interval in the figure; comments are omitted below) of the strategy on $\lambda$. Within this range, both options significantly enhance the utility of the supply chain and societal welfare. For the manufacturer, the total supply chain, and the government, Option (i) is consistently the optimal choice. However, for suppliers, Option (i) is the optimal choice only when $\lambda$ is sufficiently high. Therefore, in this numerical study, since the manufacturer’s utility from Option (i) far exceeds that of Option (ii), the manufacturer will vigorously promote an increase in low-carbon preferences to ensure the supplier also selects Option (i), achieving higher utility. Furthermore, Option (i) provides a more equitable utility distribution within the NEV supply chain while maintaining supplier utility.

This research indicates that using Case C-SH-I for perfect supply chain coordination can further incentivize both the government and enterprises to promote an increase in consumer low-carbon preferences. Simultaneously, higher consumer low-carbon preferences also benefit the goals of both the government and enterprises.

4.2.2. The Impact of Government Subsidy on NEV Supply Chain

A government subsidy, as an indispensable factor in the current stage of the NEV supply chain, has always been a significant means for the government to guide enterprises toward innovative R&D. However, the excessive burden on public finances caused by high subsidies has made the extent of government subsidies a critical topic in NEV supply chain coordination. Furthermore, we aim to ascertain whether high subsidies invariably result in enhanced utilities for enterprises. In this section, $\varphi$ is the independent variable, so we have $k_g=-0.01+{\displaystyle \frac{0.17}{0.76-\varphi }}+0.33\varphi , j_g={\displaystyle \frac{0.42+(-0.99+0.40\varphi )\varphi }{0.76+\varphi (-1.76+\varphi )}}, k_{g^{*}}=0.05(15.12+8.34(-1+\varphi )), j_{g^{*}}=0.4.$

In Figure 9, we observe the following conclusions.

In Case D, the impact of the government subsidy $\varphi$ on utilities and social welfare is weak. Only when subsidies increase to unreasonable levels does $\varphi$ significantly impact utility and social welfare. This indicates that government subsidies for R&D in a loose supply chain would not yield satisfactory results. When $0<j,k<0.5$, we have $0.278<\varphi <0.325$; this range represents the effective coordination (improvement) range of the strategy on $\varphi$. Within the effective range, both options significantly enhance the utility of the supply chain and societal welfare. For suppliers, manufacturers, the total supply chain, and the government, Option (i) is the optimal choice, yet their expectations for the value of $\varphi$ are different. For manufacturers and the total supply chain, the utilities increase in $\varphi$. Conversely, suppliers and the government prefer $\varphi$ at lower levels. In this numerical study, the effective range of coordination strategies exceeds the optimal subsidy of the government. Hence, the government would choose the minimum value within this range to subsidize R&D. Consequently, suppliers would always advocate for a “subsidy reduction” policy. Although the utility for the manufacturer decreases in $\varphi$, it still remains significantly higher than that in Option (ii).

Counterintuitively, the effect of government subsidies $\varphi$ on the NEV supply chain is not simply to promote them. This alerts the government and enterprises that they cannot solely rely on increasing subsidy levels to obtain greater utility. In some cases, higher subsidies may even have counterproductive effects. The study in this section also reveals that the “subsidy reduction” policy is feasible. However, due to its impact on coordinating the NEV supply chain, the government may not always achieve the optimal subsidy level and optimal social welfare.

4.2.3. The Impact of Carbon Emission Reduction R&D Capability on NEV Supply Chain

The carbon emission reduction R&D capability of the supplier has always been a significant factor contributing to the high prices of NEVs. To meet consumers’ low-carbon preferences and satisfy the government’s carbon emission requirements for NEVs, suppliers have been striving to enhance their carbon emission reduction R&D capabilities. However, enhancing R&D capabilities is a challenging task for suppliers. Therefore, this study incorporates IUR with high R&D capabilities into the cross-shareholding strategy to coordinate the NEV supply chain. In this section, $\eta$ is the independent variable, so we have $k_g=0.32-{\displaystyle \frac{0.74}{1.02-\eta }}-0.08\eta , j_g=0.4+{\displaystyle \frac{1.86}{1.02-\eta }}+{\displaystyle \frac{3.09}{\eta }}, k_{g^{*}}=0.05(15.12-1.96\eta ), j_{g^{*}}=0.4.$

In Figure 10, we observe the following conclusions.

In Case D, the impacts of $\eta$ on utilities and social welfare are weak and negative. The effect of $\eta$ on utility is evident only when $\eta$ is at a low level. This indicates that if the existing $\eta$ is at a high level, NEV supply chain members lack motivation to enhance their R&D capabilities. When $0<j,k<0.5$, we have $2.893<\eta <3.096$; this range represents the effective coordination (improvement) range of the strategy on $\eta$. Within the effective range, both options significantly enhance the utility of the supply chain and societal welfare. For suppliers, manufacturers, the total supply chain, and the government, Option (i) is consistently the optimal choice. In this numerical study, we find that in Case C-SH-I, since carbon emission reduction R&D is entrusted to IUR, the change in $\eta$ reflects the change in the NEV supply chain’s demand for $\theta$. Due to the coordination needs of the NEV supply chain, the expected $\theta$ of the supply chain equals ${\theta }^c$. In other words, if the R&D capability of the supplier is initially weak, the corresponding expected $\theta$ will not be too high. Therefore, when $\eta$ is sufficiently low, it is impractical for the supplier to outsource R&D. On one hand, this is because their R&D capabilities are already sufficiently high, and outsourcing R&D may actually increase costs. On the other hand, the supply chain has high expectations for $\theta$, and continuing government subsidies at the original level would increase financial pressure.

The above results indicate that enterprises should consider their R&D capabilities when selecting a case and adjust their investment in IUR appropriately to maximize utility. For the government, when enterprises or IUR already possess strong R&D capabilities, reducing subsidy levels is advisable to ensure the stability of social welfare.

By comparing Figure 8, Figure 9 and Figure 10, we can confirm the necessity of supply chain coordination. When $\lambda ,\varphi ,\eta$ are at the same level, the utilities are always higher in a coordinated supply chain compared to a decentralized one. In addition, the results reveal the NEV supply chain’s reliance on the government. The results show that when $\lambda$ is at a low level, its effect on utility is minimal. This suggests that governments need to help NEV companies promote low-carbon preferences among consumers at an early stage. When $\lambda$ is at a high level, NEV enterprises will naturally lead consumers to have low-carbon preferences. As for the government subsidies, the results show that in all cases, the government subsidies do not have a significant effect on utilities. This aligns with our speculation that government subsidies should be used to help enterprises overcome capital shortages but cannot significantly increase utility.

Therefore, when the NEV industry matures, the government’s subsidy should be gradually withdrawn. In the current NEV supply chain, the level of carbon emission reduction R&D of power batteries is low, and cooperation with IUR is a general trend. In Case C-SH-I with a fixed $g$, the manufacturer will experience a loss in utility due to the decrease in $\eta$. Therefore, only when the R&D capability of the supplier is low can the Case C-SH-I with a fixed $g$ achieve perfect coordination of the NEV supply chain. Otherwise, manufacturers will either exit the coordination strategy or aim only for Pareto improvement.

4.3. The Regulatory Role of Investment Ratio on the Coordinated Range

Through Section 4.2, we discover that to meet the constraint of $0<j,k<0.5$, the range of the external environmental factors is restricted. Once beyond this range, Case C-SH-I may lose effectiveness due to concerns about enterprise management rights. Interestingly, we find that the investment ratio $g$ can precisely address this issue. In previous studies, we set $g$ as a fixed value. However, in reality, as the external environmental factors change, the coordination conditions of the NEV supply chain will also change. Therefore, suppliers should adjust $g$ according to the changes in the external environmental factors to achieve perfect coordination of the NEV supply chain with manufacturers.

This section discusses the regulatory role of $g$ in achieving perfect coordination in Case C-SH-I of the NEV supply chain. Assuming all other variables remain constant, we study the impact of one single external environmental factor and $g$. The value of $g$ must satisfy two conditions: (1) ensuring non-negative profits for IUR; (2) reducing costs for suppliers in R&D. Therefore, we have ${\eta }_r\left(1-\varphi \right)<g<\eta (1-\varphi )$.

In Figure 11, we can observe that compared to Figure 8 in Section 4.2.1, the effective range of $\lambda$ expands from (0.57, 1) to (0, 1) due to the regulatory effect of $g$. Similarly, compared to Figure 9 in Section 4.2.2, the effective range of $\varphi$ extends from (0.278, 0.325) to (0, 0.440); compared to Figure 10 in Section 4.2.3, the effective range of $\eta$ extends from (2.893, 3.096) to (2.157, $+\infty$). Specifically, it should be noted that since $k$ is non-differentiable with respect to both $\varphi$ and $g$, the effective range cannot be directly observed through a top-down view of the three-dimensional graph. Therefore, to present the results more concisely and intuitively, we will separately use inequality equations for solving and plotting.

Furthermore, we can calculate that the optimal subsidy $\varphi =0.228$, and the optimal social welfare $SW=16.45$ can be realized by adjusting $g$ to (1.88, 2.28). This means that the regulating effect of $g$ solves the problem in Section 4.2.2, which is that the government cannot adopt the optimal subsidy.

The above conclusions demonstrate that in Case C-SH-I, with any low-carbon preference and carbon emission reduction R&D capability, suppliers can adjust their investment in IUR to achieve perfect coordination in the NEV supply chain. Regarding the range of $\varphi$, while excessively high subsidies still cannot achieve effective coordination, $\varphi$ being able to take on zero means that the government’s subsidy reduction can occur without affecting perfect coordination. These results highlight the superiority of Case C-SH-I in achieving perfect coordination in the NEV supply chain. Thanks to the regulatory effect of $g$, the coordination of the supply chain is more stable.

5. Conclusions, Shortcomings, and Prospects

5.1. Concluding Remarks and Major Findings

In this study, we have conducted an analytical study on the coordination of the NEV supply chain considering Nash bargaining fairness concerns preference. Some significant findings and management insights are summarized below.

A superior coordination strategy:

To address the existing imbalance in the distribution of interests in the new energy vehicle (NEV) supply chain and to more effectively tackle the issue of “free-riding”, we attempt to achieve perfect coordination in the NEV supply chain using a shareholding strategy. We explore the conditions for achieving perfect coordination in the NEV supply chain, as well as the conditions under which enterprises may accept coordination or improvement.

Contrary to previous understanding, we find that under both cross-shareholding strategies, perfect coordination in the NEV supply chain is achieved only when the supplier and manufacturer engage in cost-based transactions for power batteries. To investigate the impact of coordination strategies on the NEV supply chain, we have conducted several numerical studies. The results indicate that when both parties fail to reach an agreement on $\omega =c$, both shareholding strategies can achieve Pareto improvement in the NEV supply chain, and the novel strategy is the optimal choice for both parties. Conversely, when both parties reach an agreement on $\omega =c$, the novel strategy can achieve perfect coordination in the NEV supply chain, with utility benefits surpassing those of Pareto improvement for both parties.

We can also explain the feasibility of cost-based transactions for power batteries from a real-world perspective. Due to the formation of a stable mutual shareholding “alliance” between the two parties, although the supplier cannot directly profit from sales contracts, they can receive dividends from the shares held by the manufacturer, and the utility derived from these dividends exceeds the original sales utility. The advantage of this model is that the manufacturer can significantly reduce financial pressure and allocate more resources to other core competencies. The supplier can also achieve higher utility and alleviate pressure on their own low-carbon emission reduction research and development for power batteries. With the continuous growth in the sales of NEVs, the formation of such alliances between equally strong enterprises supports both parties in maximizing benefits amid increasingly fierce market competition.

Based on the above conclusions, this study proposes a cross-shareholding strategy based on cost-based transactions for power batteries.

However, we identified another issue: when using a traditional cross-shareholding strategy to achieve perfect coordination, the shareholding ratios among NEV supply chain members are fixed and sum to 1. In other words, at least one member’s shareholding ratio will exceed 0.5. Generally, shareholding affects an enterprise’s management authority, and holding more than half of the shares often implies actual control over the enterprise’s management. To ensure their own management authority, the shareholding ratios in the cross-shareholding strategy should all be below 0.5. Obviously, the traditional cross-shareholding strategy has limitations.

Therefore, we propose a novel IUR shareholding strategy, which can more easily accomplish the perfect coordination of the NEV supply chain. When the new strategy perfectly coordinates the NEV supply chain, the R&D of carbon emission reduction in power batteries is entrusted to IUR, and manufacturers buy power batteries at their original cost. This study finds that this new shareholding strategy can solve the problems that traditional shareholding strategies cannot solve. We find that the supplier can adjust its investment intensity in IUR to achieve perfect coordination in the NEV supply chain, where shareholding ratios are below 0.5.

Through sensitivity analysis, we find that the novel strategy has an advantage in suppressing the double marginal effect caused by excessive fairness concern preference among supply chain members. When the strategy is used to coordinate the NEV supply chain, NEV enterprises can no longer increase utility by increasing their own fairness concern preferences.

Last but not least, through numerical analysis, we explore the effective range of external environmental factors in the coordinated NEV supply chain. We find that the effective range can be expanded by adjusting the supplier’s investment factor $g$. This demonstrates the flexibility and stability of the novel strategy in achieving perfect coordination of the supply chain.

Managerial implications:

In addition, the results of this study also offer value management recommendations for the government and the NEV enterprises. We explore the impacts of three external environmental factors on the supply chain utility: government subsidies, consumer low-carbon preferences, and R&D capabilities. Building upon this, we compare the utility and social welfare between two scenarios: achieving Pareto improvement in the NEV supply chain and achieving perfect coordination in the NEV supply chain using the novel IUR shareholding strategy.

For NEV enterprises, we find that outsourcing carbon emission reduction R&D to IUR is always a wise decision. Under the same supplier R&D capability $\eta$, the NEV supply chain coordinated with the novel strategy always achieves optimal utilities. We find that $\eta$ no longer affects supply chain directly but influences utility by impacting supply chain coordination objectives. The higher the R&D ability of the supplier, the greater the goal of perfect supply chain coordination. When $\eta$ is sufficiently small, it is obviously unwise for the supply chain to outsource R&D. However, in the current NEV supply chain, when the R&D capability of suppliers is at a low level, improving the R&D capability can have a significant impact.

For the government, our numerical findings suggest that it should not over-subsidize a coordinated (improved) NEV supply chain. Counterintuitively, high government subsidies may harm the utility and social welfare of the NEV supply chain. We find that when the novel strategy is used to achieve coordination, the government can always achieve higher social welfare with lower subsidies. This means that implementing the novel strategy can reduce the financial burden on the government.

As for consumers, enterprises and the government should work together to promote consumers’ low-carbon preferences $\lambda$. We suggest that governments should help NEV enterprises promote $\lambda$ at an early stage since the utility of enterprises’ promotion of $\lambda$ is low at this time. However, when $\lambda$ is at a high level, NEV enterprises will spontaneously lead consumers to adopt low-carbon preferences. When supply chains are coordinated, the benefits of increasing consumers’ low-carbon preferences are amplified. In other words, when the novel strategy is used to achieve coordination, enterprises and the government will be more motivated to promote consumers’ low-carbon preferences.

5.2. Shortcomings and Prospects

Although the validity of the model has been verified from the numerical analysis, there is still a lack of industrial verification. This is because the current NEV supply chain data on power battery R&D remain confidential. Moreover, most of the shareholding strategies between NEV enterprises are still in their early stage, and the coordination effect is still difficult to observe. We hope that industry data can be obtained in future studies to further enhance the model.

For future research, we can study coordination strategies for different contracts involving IUR. Specifically, the various ways suppliers invest in IUR may affect the NEV supply chain’s coordination as well. Furthermore, if IUR becomes a for-profit organization, the NEV supply chain will transform into a three-stage game, and determining how to achieve supply chain coordination will present an interesting challenge. Additionally, research on NEV supply chain risk should also be considered. It is worth exploring the impact of supply chain coordination in scenarios where there is a lack of personnel or supplies in the NEV supply chain.