In recent years, the global economy has witnessed rapid growth, leading to a surge in the development of the new energy vehicle industry. However, uncertainties such as geopolitical conflicts have caused fluctuations in energy supply, while demand continues to rise steadily. Concurrently, traditional industries and the automotive sector face increasing energy shortages and environmental issues like smog and exhaust emissions. Against this backdrop, electric cars have emerged as a critical solution in the transportation sector due to their eco-friendly and energy-efficient characteristics. Many automakers are intensifying efforts in research and mass production of electric car models to reduce reliance on fossil fuels and achieve energy conservation and emission reduction. As the largest domestic manufacturer of electric cars in China, BYD sold over 3 million units in 2023, representing a growth rate of more than 150%, making it a exemplary company in China’s export of “new three” industries. Additionally, tech giants like Huawei and Xiaomi have entered the electric car market, further driving innovation and competition.
Nevertheless, with the increasing adoption of electric cars, particularly in the China EV sector, the issue of recycling and re-processing end-of-life power batteries has become prominent. In 2020, approximately 200,000 tons of power batteries entered the recycling phase, and it is projected that by 2025, this figure will reach 780,000 tons. However, the average lifespan of a power battery is only 5–8 years, and once its capacity drops below 70%–80% of the original, it becomes prone to safety hazards such as insufficient braking or even combustion. The large volume of retired batteries, which are difficult to assess accurately, poses significant challenges for automakers. If not handled properly, these batteries could cause severe environmental pollution. Moreover, retired electric car power batteries still retain a certain amount of stored energy, and failure to recycle them would result in resource wastage. Therefore, the Chinese government has introduced the “New Energy Vehicle Industry Development Plan (2021–2035),” emphasizing the “strengthening of the traceability management platform for electric car power batteries” to improve the recycling and reutilization system. In practice, companies like GEM have developed lifecycle information systems for power batteries, enabling enterprises to upload data on the entire process of battery recycling, disassembly, storage, and echelon utilization, seamlessly connecting with the national traceability platform to ensure that battery sources are traceable, destinations are trackable, and nodes are controllable.
After recycling used batteries, companies must test the remaining capacity of electric car power batteries using instruments and manually disassemble battery packs, incurring high testing and disassembly costs. Currently, the recycling rate for electric car power batteries is only around 20%, facing numerous challenges in the recycling process. In light of this, government subsidies are urgently needed to enhance the recycling rate of power batteries. Similar to government subsidies in green supply chains, some local governments have already provided subsidies to the recycling industry. For instance, Shanghai offers “a subsidy of 1,000 yuan per set of recycled electric car power batteries to automakers,” representing a volume-based recycling subsidy policy. Hefei proposes “a recycling reward of no more than 20 yuan per kilowatt-hour based on the recycled battery capacity,” indicating a capacity-based recycling subsidy policy. Building on previous research on production subsidies during manufacturing, this paper examines the impact of three differentiated subsidy policies on the operational decisions, profits, and social welfare of enterprises in the electric car supply chain.
Literature Review
Recycling of Electric Car Power Batteries
Research on the recycling of electric car power batteries has focused on recycling frameworks and channel models. For example, Rao et al. (2023) demonstrated through case analysis that collaborative recycling models for electric car power batteries can effectively reduce the recycling distance and time costs for vehicle owners, enhance their recycling enthusiasm, and validate the effectiveness of multi-party collaborative recycling models. Fang, Li, & Govindan (2024) studied four joint recycling strategies from the perspective of electric car manufacturers, finding that deposit-refund systems can effectively improve recycling rates, and when recycling competition is intense, integrated cooperation models are optimal; otherwise, non-cooperative models are preferred. Zhang, Li, & Tian (2023) and Zhang, Tian, & Han (2022) both constructed four power battery recycling models to study the recycling decisions of enterprises in the electric car power battery supply chain. Xie et al. (2020) found that when the profit margin from recycling and remanufacturing meets certain conditions, battery producers tend to choose multi-channel recycling models. Zhao & Ma (2022) investigated the impact of external environments and coordination contracts on the decisions of new energy vehicles, revealing that battery manufacturers and recyclers are affected by low-capacity battery prices, while the production of automakers and the recycling volume of third-party recyclers are largely unaffected by the environment. Xing & Wang (2025) constructed a supply chain consisting of battery suppliers, third-party recyclers, and electric car manufacturers. Through Stackelberg game analysis, they examined the impact of blockchain adoption and government reward-punishment mechanisms on recycling channel selection, recycling rates, and social, economic, and environmental benefits.
Subsidy Policies in Supply Chains
Government subsidy policies play a key role in the electric car supply chain. For instance, Zhang & Chen (2021) studied the impact of government subsidies on the closed-loop supply chain for battery echelon utilization under four scenarios, finding that subsidies can reduce retail prices, increase recycling rates, and improve member profits, with recycling subsidies more significantly enhancing recycling rates and third-party recycler profits. Gu et al. (2021) established a two-period closed-loop supply chain model involving battery manufacturers, secondary users, and the government for battery secondary use, showing that the government would consider providing recycling subsidies only when the remaining capacity of recycled batteries is relatively high or the remanufacturing rate is relatively low. Tang et al. (2018) considered three single recycling channel models and three competitive dual recycling channel models, evaluating the impact of reward-punishment mechanisms and policies on power battery recycling. They found that strengthened reward-punishment mechanisms can be applied to models achieving higher recycling rates, while environmental awareness affects the social efficiency and benefits of power battery recycling.
Literature Comparison and Contributions
In the field of electric car research, previous literature has primarily focused on recycling subsidies, often neglecting production subsidies, and rarely integrated recycling subsidies with the characteristics of electric car power batteries (e.g., recycling volume and battery capacity) into the subsidy framework. A detailed comparison with relevant literature is provided in Table 1.
| Literature | Subsidy Type or Scenario | Considers Social Welfare | Main Research Content |
|---|---|---|---|
| Fang, Li & Govindan (2024) | One-time fixed recycling cost subsidy | Yes | Optimal reverse channel selection for new entrant automakers and impact of recycling subsidies on social welfare |
| Xie et al. (2020) | None | No | Contract coordination in battery producer-led closed-loop supply chains under different recycling models |
| Zhang & Chen (2021) | No subsidy, subsidy to retailers, manufacturers, or 3PR | No | Optimal decisions and coordination in power battery closed-loop supply chains under four scenarios |
| Gu et al. (2021) | Recycling subsidy | Yes | Conditions for battery secondary reuse in two-period supply chains and optimal government incentive policies |
| Zhang, Tian & Han (2022) | Recycling subsidy | No | Recycling mode selection and carbon reduction decisions under carbon quota trading policies |
| Lou et al. (2023) | Recycling subsidy | Yes | Operational decisions in electric car power battery supply chains under different recycling subsidy policies |
| Guo, Du & Zhao (2024) | Recycling subsidy | No | Impact of carbon quota and used battery recycling subsidy mechanisms in system dynamics models |
| This paper | Production/Recycling subsidy | Yes | Impact of government production/recycling subsidies on operational decisions and social welfare in electric car supply chains |
The innovations of this paper are reflected in the research perspective and scenarios considered. First, from the perspective of whether the government provides subsidies and the differentiation of subsidy targets. Previous literature often considers different subsidies given to the same entity or the same subsidy given to different entities, failing to fully reflect real-world scenarios. This paper thoroughly considers three differentiated subsidies, deeply studying the impact mechanism of government production/recycling subsidies on the operational decisions and social welfare of the electric car supply chain, revealing some counterintuitive or key conclusions: (1) Production subsidies are more conducive to improving battery capacity than recycling subsidies; (2) Production subsidies benefit demand and the profits of manufacturers and suppliers, while recycling subsidies benefit recycling volume and third-party recycler profits. The magnitude of recycling volume under the two recycling subsidy scenarios depends on the relationship between volume-based and capacity-based subsidies. Second, the scenarios considered optimize previous research. Previous literature primarily focuses on corporate profits, whereas this paper further explores social welfare and, in numerical analysis, evolves production subsidies into three scenarios. Specifically: (1) It considers subsidies and social welfare rather than solely measuring enterprise choices with profits; (2) In numerical analysis, it fully considers three scenarios of government production subsidies (i.e., taxation, no taxation/no subsidy, subsidy), compensating for the lack of mathematical analysis regarding no subsidy and taxation; (3) In extended models, it constructs two special cases where the manufacturer acts as the leader and the manufacturer merges with the supplier by paying a fee to form a centralized system, accommodating potential future scenarios.
Model Description and Basic Assumptions
System Description and Basic Assumptions
A supply chain for electric cars is constructed, consisting of a power battery supplier, an electric car manufacturer, a third-party power battery recycler (3PR), and consumers. The 3PR is responsible for recycling used power batteries, delivering them to the power battery supplier for producing new power batteries, which are then sold to the electric car manufacturer for producing complete vehicles and retailing to consumers. To explore the differential outcomes of previous production subsidies and the emerging recycling subsidy policies in Shanghai and Hefei, this paper constructs three scenarios: production subsidy based on demand volume (D), recycling subsidy based on recycling volume (Q), and recycling subsidy based on recycled battery capacity (B), studying the operational decisions, profits, and social welfare comparisons under differentiated subsidy methods and different subsidy targets. The economic structure is illustrated in Figure 1.
Assumption 1: The product demand function is $$D = \alpha – \beta p + k h$$, where $$\alpha$$ represents the potential market size, $$\beta$$ is the sensitivity coefficient of the retail price, and $$k$$ indicates consumers’ preference for battery capacity, with $$k > 0$$. Similar to the quadratic cost function depicting low-carbon technology innovation, the supplier’s power battery R&D cost is depicted as $$C_h = \frac{k_h h^2}{2}$$, where $$k_h$$ is the scale coefficient of R&D cost.
Assumption 2: The recycling quantity function for used power batteries is $$Q = Q_0 + \lambda p_r + \gamma e$$, where $$Q_0$$ represents the recycling quantity under无偿 conditions, $$\lambda$$ and $$\gamma$$ are influence coefficients on decision variables, with the recycling quantity constraint $$Q \leq D$$. The 3PR recycles used batteries, incurring a recycling effort cost of $$C_e = \frac{k_e e^2}{2}$$, where $$k_e$$ is the scale coefficient of recycling effort cost.
Assumption 3: The recycling fee $$b$$ for used power batteries by the power battery supplier is exogenous, and $$b > p_r$$, ensuring recycling motivation.
Assumption 4: The government either provides the power battery supplier with a production subsidy coefficient $$s_d$$ based on demand volume, or provides the 3PR with a recycling subsidy coefficient $$s_q$$ based on recycling volume or $$s_h$$ based on recycled battery capacity.
The variables and parameters used in this paper are listed in Table 2.
| Parameter | Definition |
|---|---|
| $$m$$ | Unit raw material cost of power battery |
| $$c_m$$ | Unit manufacturing cost of power battery |
| $$c_n$$ | Unit production cost of electric car |
| $$k_h$$ | Battery capacity R&D cost coefficient |
| $$b$$ | Buyback fee of power battery supplier, with $$b > p_r$$ |
| $$\alpha$$ | Potential market size |
| $$\beta$$ | Retail price sensitivity coefficient |
| $$k$$ | Consumer preference for battery capacity |
| $$Q_0$$ | Recycling quantity under无偿 conditions |
| $$\lambda$$ | Influence coefficient of recycling fee |
| $$\gamma$$ | Influence coefficient of recycling effort level |
| $$k_e$$ | Scale coefficient of recycling effort cost |
| $$s_d$$ | Unit production subsidy based on demand volume (in numerical analysis, considered positive as subsidy, negative as tax) |
| $$s_q$$ | Unit recycling subsidy based on recycling volume |
| $$s_h$$ | Unit recycling subsidy based on recycled battery capacity |
| Decision Variable | Definition |
| $$w$$ | Wholesale price of power battery, with $$w > m + c_m$$ |
| $$p$$ | Retail price of complete vehicle, with $$p > w + c_n$$ |
| $$h$$ | Battery capacity of power battery |
| $$e$$ | Recycling effort level |
| $$p_r$$ | Recycling fee for used power batteries by 3PR |
| Function | Definition |
| $$D$$ | Demand function |
| $$Q$$ | Recycling quantity |
| $$C_e$$ | Recycling effort cost |
| $$C_h$$ | Battery R&D cost |
| $$\pi_i^j$$ | Profit function of enterprise $$i$$ under scenario $$j$$, where $$i \in \{S, M, 3PR\}$$, $$j \in \{D, Q, B\}$$ |
Profit Functions and Game Sequence Under Three Subsidy Scenarios
Scenario 1: Production subsidy based on demand volume (superscript D)
The profit functions of the power battery supplier, electric car manufacturer, and 3PR are respectively:
$$\pi^D_S = (w – c_m – m + s_d) \cdot D – b \cdot Q – C_h$$
$$\pi^D_M = (p – w – c_n) \cdot D$$
$$\pi^D_{3PR} = (b – p_r) \cdot Q – C_e$$
where $$s_d \cdot D$$ is the production subsidy based on demand volume.
Scenario 2: Recycling subsidy based on recycling volume (superscript Q)
The profit functions are:
$$\pi^Q_S = (w – c_m – m) \cdot D – b \cdot Q – C_h$$
$$\pi^Q_M = (p – w – c_n) \cdot D$$
$$\pi^Q_{3PR} = (b – p_r + s_q) \cdot Q – C_e$$
where $$s_q \cdot Q$$ is the recycling subsidy based on recycling volume.
Scenario 3: Recycling subsidy based on recycled battery capacity (superscript B)
The profit functions are:
$$\pi^B_S = (w – c_m – m) \cdot D – b \cdot Q – C_h$$
$$\pi^B_M = (p – w – c_n) \cdot D$$
$$\pi^B_{3PR} = (b – p_r + s_h h) \cdot Q – C_e$$
where $$s_h h \cdot Q$$ is the recycling subsidy based on recycled battery capacity.
Game Sequence: First, the power battery supplier sets the wholesale price $$w$$ and battery capacity $$h$$; then, the electric car manufacturer sets the retail price $$p$$; finally, the 3PR sets the recycling effort level $$e$$ and the recycling fee for used power batteries $$p_r$$.
Social Welfare Functions Under Three Subsidy Scenarios
Similar to the construction of social welfare $$SW$$ by Xu et al. (2024), it consists of three parts: total supply chain profit $$\pi$$, consumer surplus $$CS$$, and government subsidy expenditure $$GS$$, where $$\pi = \pi^i_S + \pi^i_M + \pi^i_{3PR}$$ for $$i = \{D, Q, B\}$$. Let $$p_1$$ denote the price when the demand for electric cars is 0, i.e., $$p_1 = (\alpha + k h)/\beta$$, then consumer surplus is expressed as $$CS = \frac{(p_1 – p) D}{2}$$. Government subsidy expenditure $$GS$$ depends on the government’s subsidy method. Thus, the social welfare function is expressed as $$SW = \pi + CS – GS$$. The social welfare under each strategy is:
$$SW^D = \pi^D_S + \pi^D_M + \pi^D_{3PR} + \frac{k^2 h \beta [\alpha – (c_m + c_n + m – s_d) \beta]^2}{2 M^2} – s_d \cdot D^D$$
$$SW^Q = \pi^Q_S + \pi^Q_M + \pi^Q_{3PR} + \frac{k^2 h \beta [\alpha – (c_m + c_n + m) \beta]^2}{2 M^2} – s_q \cdot Q^Q$$
$$SW^B = \pi^B_S + \pi^B_M + \pi^B_{3PR} + \frac{k^2 h \beta [\alpha – (c_m + c_n + m) \beta]^2}{2 M^2} – s_h \cdot h^B \cdot Q^B$$
Equilibrium Decisions and Comparisons Under Three Subsidy Scenarios
Equilibrium Decisions Under Three Subsidy Scenarios
Using backward induction, the equilibrium decisions, demand, and profits under the three scenarios are obtained, as shown in Table 3 and Table 4.
| Decision Variable | Production Subsidy Based on Demand (D) | Recycling Subsidy Based on Volume (Q) | Recycling Subsidy Based on Capacity (B) |
|---|---|---|---|
| Wholesale Price $$w$$ | $$w^{D*} = \frac{2(A – s_d \beta) k_h – (c_m + m – s_d) k^2}{M}$$ | $$w^{Q*} = \frac{2 A k_h – (c_m + m) k^2}{M}$$ | $$w^{B*} = \frac{2 A k_h – (c_m + m) k^2}{M}$$ |
| Battery Capacity $$h$$ | $$h^{D*} = \frac{k (B + s_d \beta)}{M}$$ | $$h^{Q*} = \frac{k B}{M}$$ | $$h^{B*} = \frac{k B}{M}$$ |
| Retail Price $$p$$ | $$p^{D*} = \frac{\alpha + k h^{D*} + \beta (c_n + w^{D*})}{2 \beta}$$ | $$p^{Q*} = \frac{\alpha + k h^{Q*} + \beta (c_n + w^{Q*})}{2 \beta}$$ | $$p^{B*} = \frac{\alpha + k h^{B*} + \beta (c_n + w^{B*})}{2 \beta}$$ |
| Recycling Effort $$e$$ | $$e^{D*} = \frac{\gamma (Q_0 + b \lambda)}{N}$$ | $$e^{Q*} = \frac{\gamma [Q_0 + (b + s_q) \lambda]}{N}$$ | $$e^{B*} = \frac{\gamma [Q_0 + (b + s_h h^{B*}) \lambda]}{N}$$ |
| Recycling Fee $$p_r$$ | $$p_r^{D*} = b – \frac{(Q_0 + b \lambda) k_e}{N}$$ | $$p_r^{Q*} = b + s_q – \frac{[Q_0 + (b + s_q) \lambda] k_e}{N}$$ | $$p_r^{B*} = b + s_h h^{B*} – \frac{[Q_0 + (b + s_h h^{B*}) \lambda] k_e}{N}$$ |
Where the assumptions for negative definiteness of the Hessian matrix and uniqueness of equilibrium solutions are: $$\gamma < \lambda k_e$$ and $$k < 2 \sqrt{\beta k_h}$$, $$b < \frac{Q_0 k_e}{\lambda k_e – \gamma^2}$$, $$\alpha > \max\{\hat{\alpha}_1, \hat{\alpha}_2\}$$ with $$\hat{\alpha}_1 = (c_n – c_m – m) \beta$$, $$\hat{\alpha}_2 = (c_n + c_m + m) \beta$$, $$M = 4 \beta k_h – k^2$$, $$N = 2 \lambda k_e – \gamma^2$$, $$A = \alpha – (c_n – c_m – m) \beta$$, $$B = \alpha – (c_n + c_m + m) \beta$$.
| Variable | Production Subsidy Based on Demand (D) | Recycling Subsidy Based on Volume (Q) | Recycling Subsidy Based on Capacity (B) |
|---|---|---|---|
| Demand $$D$$ | $$D^{D*} = \frac{\beta k_h (B + s_d \beta)}{M}$$ | $$D^{Q*} = \frac{\beta k_h B}{M}$$ | $$D^{B*} = \frac{\beta k_h B}{M}$$ |
| Recycling Quantity $$Q$$ | $$Q^{D*} = \frac{\lambda k_e (Q_0 + b \lambda)}{N}$$ | $$Q^{Q*} = \frac{\lambda k_e [Q_0 + (b + s_q) \lambda]}{N}$$ | $$Q^{B*} = \frac{\lambda k_e (Q_0 + b \lambda)}{N} + \frac{\lambda k_e (k s_h B \lambda)}{M N}$$ |
| Supplier Profit $$\pi_S$$ | $$\pi_S^{D*} = (w^{D*} – c_m – m + s_d) \cdot D^{D*} – b \cdot Q^{D*} – \frac{k_h (h^{D*})^2}{2}$$ | $$\pi_S^{Q*} = (w^{Q*} – c_m – m) \cdot D^{Q*} – b \cdot Q^{Q*} – \frac{k_h (h^{Q*})^2}{2}$$ | $$\pi_S^{B*} = (w^{B*} – c_m – m) \cdot D^{B*} – b \cdot Q^{B*} – \frac{k_h (h^{B*})^2}{2}$$ |
| Manufacturer Profit $$\pi_M$$ | $$\pi_M^{D*} = \frac{\beta k_h^2 (B + s_d \beta)^2}{M^2}$$ | $$\pi_M^{Q*} = \frac{\beta k_h^2 B^2}{M^2}$$ | $$\pi_M^{B*} = \frac{\beta k_h^2 B^2}{M^2}$$ |
| 3PR Profit $$\pi_{3PR}$$ | $$\pi_{3PR}^{D*} = \frac{k_e (Q_0 + b \lambda)^2}{2 N}$$ | $$\pi_{3PR}^{Q*} = \frac{k_e [Q_0 + (b + s_q) \lambda]^2}{2 N}$$ | $$\pi_{3PR}^{B*} = \frac{k_e [k s_h B \lambda + M (Q_0 + b \lambda)]^2}{2 N M^2}$$ |
| Consumer Surplus $$CS$$ | $$\frac{k^2 h \beta (B + s_d \beta)^2}{2 M^2}$$ | $$\frac{k^2 h \beta B^2}{2 M^2}$$ | $$\frac{k^2 h \beta B^2}{2 M^2}$$ |
| Government Subsidy $$GS$$ | $$s_d \cdot D^{D*}$$ | $$s_q \cdot Q^{Q*}$$ | $$s_h \cdot h^{B*} \cdot Q^{B*}$$ |
| Social Welfare $$SW$$ | $$SW^{D*}$$ | $$SW^{Q*}$$ | $$SW^{B*}$$ |
From the sensitivity analysis of equilibrium solutions under the three subsidy scenarios, we obtain Corollary 1 and 2.
Corollary 1: Regardless of the subsidy scenario, the higher the battery capacity R&D cost coefficient $$k_h$$, the lower the consumers’ preference for battery capacity $$k$$, and consequently the lower the manufacturer’s profit and consumer demand (i.e., $$\frac{\partial D^{i*}}{\partial k_h} < 0$$, $$\frac{\partial D^{i*}}{\partial k} > 0$$, $$\frac{\partial \pi_M^{i*}}{\partial k_h} < 0$$, $$\frac{\partial \pi_M^{i*}}{\partial k} > 0$$ for $$i = \{D, Q, B\}$$).
Corollary 1 indicates that consumers’ preference for battery capacity $$k$$ incentivizes a positive effect (i.e., $$\frac{\partial D^{i*}}{\partial k} > 0$$ and $$\frac{\partial \pi_M^{i*}}{\partial k} > 0$$). However, high battery capacity implies high R&D costs, increasing the production cost per vehicle and compressing profit margins. Therefore, as the battery capacity R&D cost coefficient increases, the manufacturer’s profit and consumer demand decrease (i.e., $$\frac{\partial D^{i*}}{\partial k_h} < 0$$ and $$\frac{\partial \pi_M^{i*}}{\partial k_h} < 0$$). Thus, manufacturers need to balance the “negative effect of high R&D costs” and the “positive effect of large battery capacity and consumer preference promoting demand” to make overall optimal equilibrium decisions. Management implication: Manufacturers need to balance the trade-off between increasing battery capacity and R&D costs to maximize their own profits and meet consumer demand. Through technological upgrades or increased production scale, they can improve battery capacity while reducing R&D costs to maintain competitiveness. In the past, governments have stimulated consumer demand for high-capacity battery vehicles by providing corresponding subsidy policies, such as offering more subsidies or incentives to consumers purchasing high-capacity battery models, indicating the robustness of Corollary 1’s conclusions and verifying the practical value of subsidies.
Corollary 2: Regardless of the subsidy scenario, the recycling quantity of used products and the profit of the 3PR are proportional to the recycling quantity under无偿 conditions $$Q_0$$ and the influence coefficient of recycling effort level $$\gamma$$ (i.e., $$\frac{\partial Q^{i*}}{\partial Q_0} > 0$$, $$\frac{\partial \pi_{3PR}^{i*}}{\partial Q_0} > 0$$, $$\frac{\partial Q^{i*}}{\partial \gamma} > 0$$, $$\frac{\partial \pi_{3PR}^{i*}}{\partial \gamma} > 0$$ for $$i = \{D, Q, B\}$$); however, they are proportional to the influence coefficient of recycling fee $$\lambda$$ only when $$\lambda$$ is large, i.e., when $$\lambda > \max\{\hat{\lambda}_1, \hat{\lambda}_2\}$$, $$\frac{\partial Q^{i*}}{\partial \lambda} > 0$$, $$\frac{\partial \pi_{3PR}^{i*}}{\partial \lambda} > 0$$, where $$\hat{\lambda}_1 = \frac{b \gamma^2 + \sqrt{b \gamma^2 (2 k_e Q_0 + b \gamma^2)}}{2 b k_e}$$, $$\hat{\lambda}_2 = \frac{Q_0}{b} + \frac{\gamma^2}{k_e}$$.
Regardless of the subsidy scenario, the recycling quantity of used products and the profit of the 3PR are proportional to the recycling quantity under无偿 conditions $$Q_0$$ and the influence coefficient of recycling effort level $$\gamma$$ (i.e., $$\frac{\partial Q^{i*}}{\partial Q_0} > 0$$, $$\frac{\partial \pi_{3PR}^{i*}}{\partial Q_0} > 0$$, $$\frac{\partial Q^{i*}}{\partial \gamma} > 0$$, and $$\frac{\partial \pi_{3PR}^{i*}}{\partial \gamma} > 0$$). This is because, on one hand, the more used products obtained for free, the lower the operating costs for the 3PR; on the other hand, subsidies provide economic incentives, enabling the 3PR to directly or indirectly obtain additional income from recycling used products, increasing the motivation for the 3PR to engage in recycling activities. Management implication: In the short term, subsidies can directly or indirectly motivate the 3PR to engage in recycling activities. In the long run, subsidies encourage enterprises to expand recycling scale, helping them reduce costs in the recycling process, including collection, sorting, and processing. Therefore, for the government, on one hand, it should strengthen the promotion of environmental concepts, and on the other hand, formulate reasonable subsidy policies to promote the increase of recycling volume and enhance the recycling motivation of the 3PR.
Additionally, when the influence coefficient of recycling fee $$\lambda$$ is large, the recycling quantity of used products and the profit of the 3PR are proportional to it (when $$\lambda > \max\{\hat{\lambda}_1, \hat{\lambda}_2\}$$, $$\frac{\partial Q^{i*}}{\partial \lambda} > 0$$, $$\frac{\partial \pi_{3PR}^{i*}}{\partial \lambda} > 0$$). A larger influence coefficient of recycling fee means that the recycling variable cost (i.e., recycling fee) has a greater direct impact on recycling quantity, indirectly affecting the profit of the 3PR. In this case, the 3PR will pay more attention to controlling recycling costs, finding a balance between controlling recycling fees and increasing recycling volume. Management implication: The 3PR should take measures to reduce unit recycling costs, such as optimizing recycling processes, improving recycling efficiency, and strengthening the promotion of environmental concepts to reduce unit recycling fees.
Mathematical Comparison of Equilibrium Decisions and Profits Under Three Subsidy Scenarios
Proposition 1: Production subsidies are more conducive to improving battery capacity than recycling subsidies (i.e., $$h^{D*} > h^{Q*} = h^{B*}$$); recycling subsidies are conducive to promoting recycling effort and increasing recycling prices (i.e., $$e^{D*} < \min\{e^{Q*}, e^{B*}\}$$ and $$p_r^{D*} < \min\{p_r^{Q*}, p_r^{B*}\}$$). Furthermore, the magnitude of recycling decisions under the two recycling subsidy scenarios depends on the relationship between $$s_q$$ and $$s_h$$.
This is a counterintuitive conclusion. Generally, production subsidies directly increase the supplier’s profit, while recycling subsidies have a direct positive effect on recycling, indirectly promoting battery capacity R&D and improvement. However, Proposition 1 shows that production subsidies are more conducive to improving battery capacity than recycling subsidies (i.e., $$h^{D*} > h^{Q*} = h^{B*}$$). The potential reason may be that production subsidies reduce product pricing, enhancing consumers’ purchasing power, especially for those with poorer economic conditions, which helps expand the consumer base; simultaneously, production subsidies can increase consumers’ confidence in electric cars because they know the government is taking measures to support electric cars, thereby stimulating more consumption behavior. Under the premise of increased potential demand, suppliers and manufacturers convert sufficient profits into investment in battery R&D, thereby improving battery capacity. However, recycling subsidies mainly have a positive effect on the 3PR, with limited indirect improvement in the profits of suppliers and manufacturers. Management implication: Based on the positive effects of production subsidies, manufacturing enterprises should continuously innovate, provide new products or services that meet market demand, and invest in R&D and innovation to maintain their competitiveness. For the government, it should design reasonable subsidy mechanisms (such as determining subsidy amount, subsidy targets, subsidy period, etc.) to ensure that subsidies effectively stimulate consumer demand and indirectly increase the profits of upstream enterprises.
Additionally, recycling subsidies are conducive to promoting recycling effort and increasing recycling prices (i.e., $$e^{D*} < \min\{e^{Q*}, e^{B*}\}$$ and $$p_r^{D*} < \min\{p_r^{Q*}, p_r^{B*}\}$$). Under the two recycling subsidy scenarios, the larger $$s_q$$ or the smaller $$s_h$$, the higher the recycling price under the recycling subsidy based on volume scenario (i.e., $$p_r^{Q*} > p_r^{B*}$$). This means that under unit recycling subsidy based on volume, recyclers will try to increase recycling volume as much as possible to obtain more subsidies, while under unit recycling subsidy based on battery capacity, recyclers will pay more attention to the capacity of recycled batteries rather than the volume. Management implication: For the government, when formulating relevant policies, it should reasonably choose appropriate subsidy targets and amounts, and in the future, establish effective performance evaluation mechanisms to monitor and evaluate the implementation effects of recycling subsidy policies, adjust policies in a timely manner, and ensure their effectiveness and sustainability.
Proposition 2: Production subsidies are conducive to increasing demand and the profits of manufacturers and suppliers (i.e., $$D^{D*} > D^{Q*} = D^{B*}$$, $$\pi_M^{D*} > \pi_M^{Q*} = \pi_M^{B*}$$, and $$\pi_S^{D*} > \max\{\pi_S^{Q*}, \pi_S^{B*}\}$$); recycling subsidies are conducive to increasing recycling volume and the profit of the 3PR (i.e., $$Q^{D*} < \min\{Q^{Q*}, Q^{B*}\}$$ and $$\pi_{3PR}^{D*} < \min\{\pi_{3PR}^{Q*}, \pi_{3PR}^{B*}\}$$). Furthermore, the magnitude of recycling volume under the two recycling subsidy scenarios depends on the relationship between $$s_q$$ and $$s_h$$.
From Proposition 2, production subsidies benefit suppliers and manufacturers (i.e., $$\pi_M^{D*} > \pi_M^{Q*} = \pi_M^{B*}$$ and $$\pi_S^{D*} > \max\{\pi_S^{Q*}, \pi_S^{B*}\}$$). This is because production subsidies can reduce the production costs of suppliers and manufacturers, enhance their competitive advantage, and they will also expand production scale, forming economies of scale. Recycling subsidies benefit the 3PR (i.e., $$\pi_{3PR}^{D*} < \min\{\pi_{3PR}^{Q*}, \pi_{3PR}^{B*}\}$$). This is because recycling subsidies can serve as an incentive mechanism, making individuals and the 3PR more motivated to actively participate in recycling, thereby reducing resource waste and achieving the goals of reduction, reuse, and recycling. Management implication: Enterprises should closely monitor changes in government policies; enterprises should strive for production/recycling subsidies from the government to improve production or recycling enthusiasm, ensuring their own profit maximization. For the government, whether production or recycling subsidies, they help create more potential demand, thereby promoting economic growth and sustainable development. However, when formulating subsidy policies, the government should ensure that the formulation and distribution process of subsidy policies are transparent and fair, preventing unfair competition and resource waste.
Simulation Comparison of Social Welfare Under Three Subsidy Scenarios
Considering that the three subsidy forms have inconsistent impacts on the interests of multiple enterprises in the supply chain, it is necessary to further explore the impact mechanism of different subsidy forms on the overall profit of the supply chain and even social welfare. Therefore, the simulation comparison in this section can not only verify the conclusions of the above propositions but may also yield new research results. When selecting parameter default values or ranges, on one hand, it must ensure that the basic assumptions and the non-negativity of equilibrium solutions/equilibrium profits are satisfied; on the other hand, it should尽可能展现 the selection motives of different enterprises between strategies; in addition, some real-world scenarios need to be considered, such as considering $$s_d = [-0.4, 0, 0.4]$$, i.e., considering taxation, no government involvement, and subsidy scenarios. Under this rule, the default values of the parameters are: $$m = 0.38$$, $$c_m = 0.43$$, $$c_n = 0.86$$, $$b = 0.45$$, $$\alpha = 2.5$$, $$k = 0.52$$, $$k_e = 1.30$$, $$Q_0 = 0.14$$, $$\lambda = 0.40$$, $$k_h = 0.48$$, $$\gamma = 0.11$$, $$s_d = [-0.4, 0, 0.4]$$, $$s_q = 0.60$$, $$s_h = 0.45$$, $$\beta \in [0.4, 0.8]$$. Figure 2 shows the impact of price sensitivity coefficient $$\beta$$ and production subsidy $$s_d$$ on social welfare $$SW^*$$.
From Figure 2, it can be seen that as the price sensitivity coefficient $$\beta$$ increases, social welfare under all three scenarios gradually decreases (i.e., $$\frac{\partial SW^{i*}}{\partial \beta} < 0$$). This may be because a larger price sensitivity coefficient makes consumers more sensitive to retail prices and only accept relatively lower prices. On one hand, the market is prone to supply-demand imbalances, leading to market failures, such as market monopoly or excessive competition; on the other hand, it makes price fluctuations and market demand uncertainty greater, and this uncertainty makes it difficult for enterprises to make long-term plans, affecting production decisions, while consumers’ purchasing decisions also fluctuate, affecting social welfare under dual effects, leading to its decline. Additionally, when the production subsidy based on demand volume is negative, i.e., taxation is implemented (as in Figure 2(1) with $$s_d = -0.4$$), the government theoretically gains more revenue, i.e., taxes, but counterintuitively, social welfare significantly decreases (i.e., when $$s_d = -0.4$$, $$SW^{D*} < \min\{SW^{Q*}, SW^{B*}\}$$). This potentially means that corporate profits significantly decrease, i.e., taxation increases the operating costs of enterprises, causing them to shrink production scale or increase pricing to compensate for the operating costs brought by taxes, ultimately, social welfare反而 decreases. Conversely, with the cancellation of taxation (as in Figure 2(2) with $$s_d = 0$$) and the emergence of subsidies (as in Figure 2(3) with $$s_d = 0.4$$), social welfare under the production subsidy based on demand volume (D) scenario gradually increases and exceeds that under the recycling subsidy based on volume (Q) scenario and the recycling subsidy based on battery capacity (B) scenario (i.e., when $$s_d = 0.4$$, $$\max\{SW^{Q*}, SW^{B*}\} < SW^{D*}$$; $$SW^{D*}|_{s_d=-0.4} < SW^{D*}|_{s_d=0} < SW^{D*}|_{s_d=0.4}$$). This is because when the government provides subsidies, production subsidies can reduce product prices, thereby increasing consumers’ purchasing power and improving effective market demand. This helps improve resource utilization efficiency, producers’ income, and consumer purchase utility, thereby enhancing social welfare. For the electric car supply chain and the government, subsidies are currently a better equilibrium choice than taxation.
Extended Models
Consider two extended models: (1) changing the game sequence, with the manufacturer as the leader; (2) the manufacturer merges with the supplier by paying a fee $$F$$, finding the threshold of $$F$$ to form a centralized system.
(1) Manufacturer as the Leader (subscript m)
Assume no government subsidies (i.e., $$s_d = s_q = s_h = 0$$), the profit functions of the original power battery supplier, electric car manufacturer, and 3PR are respectively:
$$\pi^N_S = (w – c_m – m) \cdot D – b \cdot Q – C_h$$
$$\pi^N_M = (p – w – c_n) \cdot D$$
$$\pi^N_{3PR} = (b – p_r) \cdot Q – C_e$$
If the manufacturer acts as the leader, the game sequence is updated as: first, the electric car manufacturer sets the marginal profit $$m_p$$ for the electric car (then $$p = m_p + w$$); then, the power battery supplier sets the wholesale price $$w$$ and battery capacity $$h$$; finally, the 3PR sets the recycling effort level $$e$$ and recycling fee $$p_r$$.
The profit functions of the power battery supplier, electric car manufacturer, and 3PR are respectively:
$$\pi^N_{Sm} = (w – c_m – m) \cdot D – b \cdot Q – C_h$$
$$\pi^N_{Mm} = (m_p – c_n) \cdot D$$
$$\pi^N_{3PRm} = (b – p_r) \cdot Q – C_e$$
Using backward induction, the equilibrium decisions, equilibrium profits, and equilibrium social welfare under the manufacturer-as-leader scenario can be obtained, and compared with the original model to obtain Proposition 3.
Proposition 3: $$\pi^{N*}_{Mm} > \pi^{N*}_M$$. When $$\frac{4 k_h \beta}{3} < k < 2 \sqrt{k_h \beta}$$, $$\pi^{N*}_{Sm} > \pi^{N*}_S$$; otherwise, $$\pi^{N*}_{Sm} \leq \pi^{N*}_S$$. $$SW^{N*}_m > SW^{N*}$$.
Proposition 3 shows that compared to the basic model, the manufacturer as the leader is beneficial to itself (i.e., $$\pi^{N*}_{Mm} > \pi^{N*}_M$$), meaning that whoever掌握 leadership has an advantage. Additionally, if the manufacturer observes that in reality, consumers’ preference for battery capacity is high and profitable (i.e., when $$\frac{4 k_h \beta}{3} < k < 2 \sqrt{k_h \beta}$$), it can lead the entire supply chain to improve its own and the supplier’s profits and even social welfare. Management implication: Enterprises should focus on leadership and consumer preference, prioritizing the improvement of battery capacity, which is the key to achieving profit growth and social welfare improvement.
(2) Manufacturer Merges Supplier Scenario (subscript F)
Assume no government subsidies (i.e., $$s_d = s_q = s_h = 0$$), the profit function of the centralized supply chain is:
$$\pi^N_{MF} = (p – c_n – c_m – m) \cdot D – b \cdot Q – C_h$$
Using backward induction, the equilibrium decisions and equilibrium profits under the centralized system can be obtained, and compared with the basic model to obtain Proposition 4.
Proposition 4: When $$F < \hat{F} = \frac{2 k_h^3 \beta^2 B^2}{M^2 (2 \beta k_h – k^2)}$$, $$\pi^{N*}_{MF} > \pi^{N*}_M + \pi^{N*}_S$$; otherwise, $$\pi^{N*}_{MF} \leq \pi^{N*}_M + \pi^{N*}_S$$.
Proposition 4 shows that only when the merger cost $$F$$ is below a certain threshold can the manufacturer merging with the supplier to form a centralized system outperform the decentralized supply chain in the basic model. In particular, when the merger cost is the threshold $$\hat{F}$$, the manufacturer, regardless of choosing centralization or decentralization, has consistent overall supply chain profit. Management implication: Enterprises should evaluate merger costs to ensure they are below the benefit threshold, realizing the advantages of a centralized supply chain and avoiding excessive investment leading to profit decline.
Conclusion
This paper constructs a supply chain for electric cars consisting of a power battery supplier, an electric car manufacturer, a 3PR, and consumers. In the basic model, three government subsidy scenarios are considered: production subsidy based on demand volume, recycling subsidy based on recycling volume, and recycling subsidy based on recycled battery capacity, studying the impact of government production subsidies or recycling subsidies for used power batteries on the operational decisions, corporate profits, and social welfare of the electric car supply chain. The results show: (1) Regardless of the subsidy scenario, the manufacturer’s profit is proportional to consumers’ preference for battery capacity; the recycling quantity of used products and the profit of the 3PR are proportional to the recycling quantity under无偿 conditions and the influence coefficient of recycling effort level; (2) Production subsidies are more conducive to improving battery capacity than recycling subsidies and are conducive to increasing demand and the profits of manufacturers and suppliers; recycling subsidies are conducive to improving recycling effort and recycling prices and increasing recycling volume and the profit of the 3PR. Furthermore, the magnitude of recycling decisions and recycling volume under the two recycling subsidy scenarios depends on the relationship between volume-based and capacity-based subsidies; (3) As the price sensitivity coefficient increases, social welfare under all three subsidy scenarios gradually decreases; as the production subsidy gradually increases, social welfare under this scenario gradually increases and exceeds that under the two recycling subsidy scenarios. In the extended models, constructing scenarios where the manufacturer acts as the leader and the manufacturer merges with the supplier, the study finds: (1) If the manufacturer observes that consumers’ preference for battery capacity is high and profitable, it can lead the entire supply chain to improve its own and the supplier’s profits and even social welfare; (2) Only when the merger cost is below a certain threshold can the manufacturer merging with the supplier to form a centralized system outperform the decentralized supply chain.
Based on the above research conclusions, the following management implications and policy recommendations can be drawn:
(1) Manufacturers need to balance the trade-off between increasing battery capacity and R&D costs to maximize their own profits and meet consumer demand. Based on the positive effects of production subsidies, manufacturing enterprises should, through technological upgrades or increased production scale, improve battery capacity while reducing R&D costs to maintain competitiveness.
(2) In the short term, recycling subsidies can directly or indirectly motivate the 3PR to engage in recycling activities. In the long run, subsidies encourage enterprises to expand recycling scale, helping them reduce costs in the recycling process, including collection, sorting, and processing. Therefore, for the government, on one hand, it should strengthen the promotion of environmental concepts, and on the other hand, formulate reasonable subsidy policies to promote the increase of recycling volume and enhance the recycling motivation of the 3PR.
(3) For the government, it should design reasonable subsidy mechanisms (e.g., providing more subsidies or incentives to consumers purchasing high-capacity battery models, and determining subsidy amount, subsidy targets, subsidy period, etc.), establish effective performance evaluation mechanisms, further stimulate consumer demand for high-capacity battery models, thereby increasing manufacturers’ sales and profits, promoting industrial development, and meeting consumer demand.
This paper still has certain limitations in the research. Although government production or recycling subsidies have certain positive effects on promoting electric cars, they may induce fraud, requiring simultaneous adoption of multiple positive measures, such as increasing corporate taxes, providing subsidies to consumers, etc. Additionally, the electric car supply chain involves many participants, and electric car brands vary. The parameter assignments in the mathematical analysis may not be entirely accurate, warranting further data collection and analysis, and using tools like Python for crawling and business data analysis.