Optimizing the Growing Dual Credit Requirements for Automobile Manufacturers in China’s Dual Credit Policy

: Dual credit policy (DCP) is a market-based mechanism introduced by the Chinese government to promote the new energy vehicle (NEV) industry and improve energy savings in China. To offer sufﬁcient impetus for the NEV industry while providing sufﬁcient transitional buffer time for automobile manufacturers (AMs), the government needs to scientiﬁcally design how to gradually increase its dual credit requirement for AMs year by year. To assist the multi-year DCP design, this paper proposes a generalized Nash equilibrium model to predict AMs’ short-term decisions (i.e., vehicle production and credit trading) and long-term decisions (i.e., investment in production capacity expansion and research and development) under any DCP, considering the interactions among AMs’ decisions, vehicle prices, and credit price. Based on the equilibrium model, we then develop a bi-level programming problem to optimize the multi-year DCP. With numerical experiments, we show that implementing the optimal DCP can effectively enhance the market share of NEVs. In the context of the optimal multi-year DCP, the credit requirements set by the government should maintain a relatively low threshold during the initial years, but rise rapidly after that. Such optimal DCP offers AMs sufﬁcient transition time while compelling a quick shift in their developmental strategies.


Introduction
The significant surge in carbon dioxide emissions has resulted in severe climate change, posing a critical threat to our ecosystems.To combat carbon emissions, 136 countries committed to achieving "carbon neutrality" targets by the end of 2021 [1].Encouraging the production and adoption of new energy vehicles (NEVs) has emerged as one of the most effective strategies to reduce carbon emissions [2,3].In line with the "NEV Industry Development Plan (2021-2035)" and the "Medium and Long-term Development Plan for the Automobile Industry" issued by the Chinese government, NEVs are projected to account for approximately 20% of total annual passenger car sales, with a targeted average fuel consumption rate of 4.0 L/100 KM by 2025 [4,5].Since 2010, the Chinese government has introduced several incentive policies, such as subsidies, priority licensing, and tax reductions, to promote NEV manufacturing and purchases.While these policies have driven NEV sales growth, they have also led to challenges, including fraudulent subsidy claims by automobile manufacturers and fiscal burdens for the government [6].
In September 2017, the Chinese government introduced the dual credit policy (DCP) as an innovative approach.The DCP comprises two types of credits: corporate average fuel consumption (CAFC) credits and new energy vehicle (NEV) credits [7].Each automobile manufacturer (AM) earns CAFC credits based on the difference between their actual vehicle fuel consumption and the government-set standard, and receives NEV credits based on their actual NEV production and government-set NEV production requirements.To avoid severe penalties (e.g., suspending the production of high-fuel-consumption vehicles), AMs must ensure compliance with dual credit targets annually.If AMs cannot meet the targets through production alone, they can achieve compliance by purchasing credits from other AMs with surplus credits through a government platform, with credit prices determined by the market.This policy creates a situation where traditional AMs, mainly producing gasoline vehicles (GVs), incur costs to meet the standards, while NEV producers can generate revenue through credit sales, effectively receiving subsidies.As a result, the government can promote NEV development without incurring fiscal expenditure.
To thrive under the DCP, AMs must strategically plan their production and credit trading to maximize profits while ensuring dual credit compliance.The credit price directly influences AMs' credit trading and production decisions for different vehicle types, and in turn, each AM's decisions impact the supply and demand of credits in the market, thus governing the credit price.Therefore, a primary challenge for AMs under the DCP is predicting credit prices, considering the interactions between the credit market and AMs' decisions.On the other hand, an AM's short-term production decisions are inherently constrained by its production capacity and technical level for different vehicle types.Hence, to achieve sustainable development under the DCP, AMs must strategically plan their longterm investment in capacity expansion and research and development (R&D) for different vehicle types.Previous studies have explored the impact of the DCP on AMs' short-term production decisions [8][9][10] and long-term R&D strategies [11][12][13][14][15][16][17].However, these studies often focus on the decisions of a single AM and assume exogenous credit prices without considering the interactions between the credit trading market and AMs' decisions.To the best of our knowledge, only Li et al. [18][19][20] and He et al. [21] have considered the interaction between credit prices and AMs' decision-making.Li et al. [18,19] proposed a market equilibrium model describing how credit prices interact with AMs' decisions within a year.Building upon this, Li et al. [20] expanded the single-year framework to a multi-year one and established a multi-year credit market dynamic equilibrium model.They investigated how different growth rates of the NEV discount coefficient affected production decisions.Nevertheless, the existing research primarily emphasizes AMs' shortterm production or R&D decisions under the DCP and overlooks the mutual constraints between short-term and long-term decisions.Therefore, it is necessary to consider AMs' long-term R&D and capacity decisions for different vehicle types and their impacts on short-term production and credit trading decisions.
To aid AMs in effectively planning their long-term and short-term investments in NEVs, the government releases dual credit requirements for multiple future years at once.Recognizing that technological advancements and production expansion for NEVs require time, the government gradually increases dual credit requirements for AMs instead of imposing immediate changes.Two coefficients, namely the annual corporate average fuel consumption (CAFC) coefficient and the new energy vehicle credit (NEVC) discount coefficient, were introduced by the government to adjust CAFC and NEV compliance values within the DCP.The annual CAFC coefficient indicates the ratio of the actual fuel consumption rate to the targeted fuel consumption rate.The coefficients from 2021 to 2024 are set to be 123%, 120%, 115%, and 108%, respectively [22].The NEVC discount coefficient for GVs denotes the number of NEV credits required for each GV produced by an AM.From 2021 to 2023, the coefficients are set at 0.14, 0.16, and 0.18 units, respectively [23].A lower annual CAFC coefficient (or higher NEVC discount coefficient) makes CAFC (or NEV) credit compliance more challenging for AMs.If the government decreases the annual CAFC coefficients or increases the NEVC discount coefficients too rapidly, traditional AMs reliant on GVs may face difficulties, potentially leading to collapse.Due to the significant importance of the automobile industry to the national economy and job market, the government is keen to avoid the collapse of many traditional AMs.Conversely, if changes occur too slowly, AMs may lack the impetus to transition from GVs to NEVs, resulting in an unachieved target NEV market share.Therefore, to promote NEV development in China, the government must carefully design the growth path of dual credit requirements for AMs.The existing research on DCP design primarily employs empirical analysis, system dynamics, and simulation to provide qualitative recommendations [6,24,25].However, there remains a lack of quantitative models to guide the government on gradually increas-ing dual credit requirements for AMs over multiple years to boost the NEV industry's development.
Considering the aforementioned research gap, this paper proposes a bi-level optimization model for designing a multi-year DCP.In the upper-level, the government optimizes the annual CAFC coefficients and NEVC discount coefficients for multiple years to maximize social welfare.In the lower-level, multiple AMs individually optimize their long-term (e.g., capacity expansion and R&D investment) and short-term (e.g., production) decisions for different types of GVs and NEVs each year, in response to the multi-year DCP.The prices of credits and vehicles each year are endogenously determined.Through numerical examples, we demonstrate how the government can gradually adjust the annual CAFC coefficient and NEVC discount coefficient to achieve maximal social welfare, and how AMs can make optimal production, capacity, and R&D investment decisions based on long-term profit maximization.Moreover, sensitivity analyses are conducted to assess the impact of consumers' willingness to pay for NEVs and the production and NEVs' R&D costs on the optimal decisions made by both the government and AMs.The main contributions of our paper are as follows: Firstly, we construct a generalized Nash equilibrium model considering the interplay among AMs' decisions, the credits market, and the demand for vehicles of different types under any multi-year DCP.We show that the generalized Nash equilibrium model can be equivalently transformed into a convex optimization problem, providing a unique equilibrium solution.By endogenously determining the prices of credits and vehicles, our approach offers a more realistic representation of AMs' decision-making under the DCP, enabling more precise decisions.
Secondly, unlike previous studies that narrowly focused on the short-term production decisions or R&D decisions of AMs under the DCP, our model considers the interconnected constraints between AM's short-term decisions (i.e., production, credit trading) and long-term decisions (i.e., investments in production capacity expansion and R&D).This innovative perspective allows us to uncover the impact of long-term DCP on short-term decisions and vice versa, provides new insights into the dynamics under the multi-year DCP, and facilitates better decision-making for AMs in the evolving automotive landscape.
Finally, compared to the previous studies for DCP design, which only provided qualitative suggestions without directly considering the response strategies of the AMs, we propose a bi-level model that describes the game relationship between the government's DCP design and the AMs' decision-making.This model enables a quantitative optimization of the DCP while comprehensively considering the response strategies of the AMs.In this model, the government can optimize the CAFC annual coefficients and NEVC discount coefficients each year to maximize social welfare regarding economic, social, and environmental benefits.We find that the government should implement a progressive DCP and adjust the credit requirements in response to the changes in the NEV market and cost factors.An optimized DCP can stimulate the R&D and production of NEVs for new AMs and provide consumers with a more economical way to purchase NEVs.This study provides valuable insights for the government to design a more scientific DCP and contributes to its energy-saving and NEV development goals.
The rest of this paper is organized as follows.Section 2 briefly introduces the dual credit calculation rules in DCP.Section 3 models AMs' decisions at generalized Nash equilibrium under any multi-year DCP, considering the interactions between AMs' decisions and the credit and vehicle markets.Section 4 establishes a bi-level optimization problem to quantitatively optimize the multi-year DCP considering the reactions of AMs.Numerical examples are presented in Section 5 to illustrate the properties of the optimal multi-year DCP.And finally, Section 6 concludes the paper.

Dual Credit Calculation Rules
Before introducing the model, we first briefly describe the calculation rules for CAFC and NEV credits under the DCP.
(1) For each AM, its CAFC credit is calculated as the product of the total vehicle production and the difference between the CAFC compliance value and the actual CAFC value.If the actual CAFC value is lower (higher) than the CAFC compliance value, the AM earns positive (negative) CAFC credits.
For each AM, its CAFC compliance value is determined by the production volumes and the target fuel consumption of different vehicle types.Let X and Y, respectively, be the sets of GV and NEV types, let g x t and g y t be the targets of the fuel consumption rate (related to the vehicle's overall mass and the number of seats) for GVs in type x ∈ X and NEVs in type y ∈ Y in year t, respectively, and let r t be the annual CAFC coefficient.Then, given the production of the GV type x ∈ X and the NEV type y ∈ Y for AM m in year t, i.e., f The where w y t w y t ≥ 1 represents the production discount multiplier for NEVs, as provided by the government in year t.Since the policy specifies the fuel consumption of NEVs to be zero for 2025, and the discount multiplier for NEVs in 2025 is set at 1 [22],we assume N y m,t = 0 and w y t = 1.From Equations (1) and (2), the CAFC credit for AM m in year t, denoted by CAFCC m,t (2) For each AM, the NEV credit (NEVC) is calculated as the difference between the actual NEVC value and the NEVC compliance value.If the actual NEVC value is higher (lower) than the NEVC compliance value, the AM obtains positive (negative) NEV credits.
The NEVC compliance value of an AM is calculated by multiplying the total production of GVs and the NEVC discount coefficient, which is set by the government specifically for GVs.Given the NEVC discount coefficient k t and the production f x m,t of the GVs for an AM m in year t, we calculate the NEVC compliance value NEVC req m,t f x m,t of the AM in year t as Obviously, the larger the value of k t , the greater the pressure on AMs to meet the NEV credit compliance caused by producing GVs. The Based on Equations ( 4) and ( 5), we can calculate the NEV credits for each AM in year t as (3) Credit offset and carryover.As in Li et al. [20], we assume that both types of credits cannot be carried forward.And to simplify analysis, we assume CAFC credits and NEV credits can offset each other at a ratio of 1:1.

Modeling AMs' Decisions at Generalized Nash Equilibrium under Any Multi-Year DCP
In this study, we considered an automobile market with multiple AMs with different initial production capacities and technical levels for different types of vehicles.Let M be the set of AMs.In the initial year t = 0, the government publishes the dual credit requirements for the following T years.Provided the coefficients (r,k) = (r t , k t , t = 1, 2, . . ., T) of the DCP, the AMs determine their production volumes, investment on capacity expansion, and R&D for every year t = {1, 2, . .., T} to maximize their profit while coping with the dual credit requirements every year.In the rest of this section, we first model AMs' optimal decisions as price takers in Section 3.1 and then establish the market equilibrium conditions that the credit and vehicle prices must meet in Sections 3.2 and 3.3.A generalized Nash equilibrium model is established in Section 3.4 considering the interplay among AMs' decisions, vehicle prices, and credit prices.Figure 1 depicts the following research framework describing the interactions among the government's coefficient setting, AMs' decision making, and market prices.Table 1 summarizes the input parameters, functions, and decision variables.optimal decisions as price takers in Section 3.1 and then establish the market equilibrium conditions that the credit and vehicle prices must meet in Sections 3.2 and 3.3.A generalized Nash equilibrium model is established in Section 3.4 considering the interplay among AMs' decisions, vehicle prices, and credit prices.Figure 1 depicts the following research framework describing the interactions among the government's coefficient setting, AMs' decision making, and market prices.Table 1 summarizes the input parameters, functions, and decision variables.Table 1.List of input parameters, functions, and decision variables.

Sets Description M
The set of AMs X The set of GV types Table 1.List of input parameters, functions, and decision variables.

M
The set of AMs X The set of GV types Y The set of NEV types T The total year in consideration AMs' initial state variables The production capacity of AM m ∈ M for GVs in type x ∈ X and NEVs in type y ∈ Y, respectively, in the initial year (t = 0) The technical level of AM m ∈ M for GVs in type x ∈ X and NEVs in type y ∈ Y, respectively, in the initial year (t = 0) The vehicle production cost functions for GVs in type x ∈ X and NEVs in type The capacity expansion cost functions for GVs in type x ∈ X and NEVs in type y ∈ Y by AM m ∈ M, respectively, in year t The R&D cost functions for GVs in type x ∈ X and NEVs in type y ∈ Y by AM m ∈ M, respectively, in year t The inverse demand functions for GVs in type x ∈ X and NEVs in type y ∈ Y, respectively, in year t

Decision variables of AMs
The production volumes for GVs in type x ∈ X and NEVs in type y ∈ Y, respectively, by AM m ∈ M in year t The technological improvements for GVs in type x ∈ X and NEVs in NEV type The production capacity expansion for GVs in type x ∈ X and NEVs in type y ∈ Y, respectively, by AM m ∈ M in year t The technical levels of AM m ∈ M for GVs in type x ∈ X and NEVs in type y ∈ Y, respectively, in year t Price and demand variables The demand for GVs in type x ∈ X and NEVs in type y ∈ Y, respectively, in year t P x t , P y t The price for GVs in type x ∈ X and NEVs in type y ∈ Y, respectively, in year t P t Credit price in year t

Decision variables of the government k t
The NEVC discount coefficient in year t (k t ≥ 0) r t The annual CAFC coefficient in year t (r t ≥ 1)

AM's Decision Model
Assuming that all AMs act as price takers in the credit and vehicle markets, the optimal production, capacity expansion, and R&D investment decisions for each AM under any given (P x t , x ∈ X, t = 1, . .., T), (P y t , y ∈ Y, t = 1, . .., T) and (P t , t = 1, . .., T) for AM m ∈ M can be obtained by solving the following optimization problem: where the objective function (7) represents the total profit of AM m ∈ M over T years.The first two terms within the brace represent the profit earned by the AM from the sales of GVs and NEVs (to calculate this value, we subtracted the production cost, R&D cost, and capacity expansion costs from the revenue obtained from vehicle sales).The third term refers to the AM's credit selling income.In the objective function, K m,t denotes the surplus of credits for AM m ∈ M in year t, whose value is calculated by Equation (10).When K m,t < 0, P t K m,t represents the AM's credit purchasing cost.Equations ( 8) and ( 9) imply that the production f x m,t and f y m,t of GVs in type x ∈ X and NEVs in type y ∈ Y each year must not exceed their respective maximum capacities of y m,j .And Equations ( 11) and ( 12) define the relationship between the AM's technical level for each type of vehicle in year t and the AM's R&D investment in previous years.

Proposition 1. Under the assumption that the cost functionsV
T are all continuously differentiable, strictly increasing, and convex functions, the above optimization problem (7)-( 13) is a convex optimization problem, whose solution is identical to the one that satisfies the following Karush-Kuhn-Tucker (KKT) conditions (the operator 'a⊥b' indicates the complementarity condition a where γ x m,t and γ y m,t correspond to the Lagrange multipliers of Equations ( 8) and ( 9 11) and (12), respectively.
Proof.Please refer to Appendix A.

Credit Price at Credit Market Equilibrium
The previous subsection discusses AM's production, capacity expansion, and R&D investment decision with exogenous credit and vehicle prices.Nevertheless, the credit price is endogenously determined by the decisions of AMs each year.When the equilibrium state of the credits market is reached, the credits price P t satisfies the following market clearing conditions. where and the vehicle price P x t for GVs in type x ∈ X in year t must satisfy the following market clearing conditions: Equation (24) states that when the price of GVs is positive, the market supply of GVs is equal to the demand, and the equilibrium price of GVs is zero when the supply exceeds the demand.
Similarly, the NEV market in year t must satisfy the following market equilibrium conditions.7)-( 13) illustrate how the prices of the credits and different vehicle types influence each AM's decisions; Equations ( 22)-( 26) describe how each AM's decisions affect supply and demand in the credit and vehicle markets, thereby determining the prices for credits and vehicles.When an equilibrium is reached, Equations ( 7)-( 13) and ( 22)-( 26) must be satisfied simultaneously.In the following proposition, we show that the generalized Nash equilibrium state can be obtained by solving an equivalent convex optimization problem.Proposition 2. The AM's optimal decisions at Nash equilibrium and the credit and vehicle prices at market equilibrium can be obtained by solving the following convex optimization problem, with the Lagrange multipliers of Equations ( 30)- (32) Proof.See Appendix B.

A Bi-Level Model for Optimal Multi-Year DCP
Based on the Nash equilibrium model in Section 3, we further propose a bi-level model to optimize the multi-year coefficients (r t , k t , t = 1, . . ., T) of the DCP from the government's perspective.The government's goal in implementing the DCP is to maximize social welfare.In this paper, we define social welfare as the sum of consumer surplus, AMs' profits, and environmental benefits.
The consumer surplus in the GV and NEV markets in year t, indicated by CSF t and CSE t respectively, can be calculated by: The total profit of all AMs in the year t, indicated by Π t , can be calculated by: In addition, the reduction in carbon emissions realized by implementing the DCP is an essential component of social welfare.In this study, we assumed the carbon emissions for each NEV to be zero, and the carbon emissions for each GV were calculated considering fuel consumption, average annual mileage, and lifespan.Let L (100 KM) denote the average annual mileage, β (KG/L) denote the carbon emissions resulting from unit fuel consumption, α (RMB/KG) denote the environmental cost incurred by AMs due to carbon emissions (equivalently, the carbon trading price), µ (year) denote the vehicle's lifespan, and φ denote the significant factor of carbon emission reduction benefits to social welfare.Given the production volume f x m,t = f x m,t , m ∈ M, x ∈ X and fuel consumption Z x m,t = Z x m,t , m ∈ M, x ∈ X of GVs produced by all AMs in year t, the total carbon cost function for all GVs in the market in year t can be expressed as follows: where the cost functions CE f x m,t , Z x m,t of carbon emission are continuously differentiable and strictly increasing convex functions with the production and fuel consumption of GVs, respectively.
Based on the above analysis, the government's DCP design problem can be expressed as the following bi-level problem P1:  27)- (33).where the objective function (38) is the sum of consumer surplus, AMs' profit, and environmental benefits, as detailed in Equations ( 34)-(37).Equations ( 27)-( 33) describe the optimal decisions of the AMs and the prices of the credits and vehicles at market equilibrium under any multi-year DCP.In our numerical experiments in the following, we transform the welfare.As shown in Figure 2, the optimal CAFC coefficient drops to the lower bound 1 in the second year and remains steady afterward.In contrast, the NEVC discount coefficient undergoes a progressive tightening over time.It stays at very low levels for the initial two years but then rises rapidly at an annual rate of approximately 0.4.This finding suggests that the government should set relatively lower dual credit requirements in the initial years to allow AMs a transition buffer.However, as the NEV capacity and technology improve, the compliance requirements for the DCP should escalate rapidly.
will decrease at a rate of 0.2% (decrease at a rate of 5%), i.e., ( )

Optimal Multi-Year DCP for Maximal Social Welfare
Under the above setup, we solved the bi-level problem in Section 4 to obtain the optimal annual CAFC coefficients and NEVC discount coefficients for maximal social welfare.As shown in Figure 2, the optimal CAFC coefficient drops to the lower bound 1 in the second year and remains steady afterward.In contrast, the NEVC discount coefficient undergoes a progressive tightening over time.It stays at very low levels for the initial two years but then rises rapidly at an annual rate of approximately 0.4.This finding suggests that the government should set relatively lower dual credit requirements in the initial years to allow AMs a transition buffer.However, as the NEV capacity and technology improve, the compliance requirements for the DCP should escalate rapidly.In Figure 3a, we observe an annual increase in social welfare with the optimal CAFC coefficients and NEVC discount coefficients.This increase primarily results from reduced In Figure 3a, we observe an annual increase in social welfare with the optimal CAFC coefficients and NEVC discount coefficients.This increase primarily results from reduced carbon emission costs and the growth of consumer surplus.Although the implementation of the DCP initially leads to losses for traditional AMs in the first eight years, they eventually begin to profit from the ninth year onward.Moreover, Figure 3b illustrates a steady rise in the market share of NEVs, increasing from 16% in the first year to 45% in the tenth year.As depicted in Figure 4a, in this example, sales of GVs continue to be the primary revenue source for traditional AMs over the decade, as consumers exhibit a sustained preference for purchasing them.The traditional AMs' credit expenditure is always positive and increasing during the ten years (Figure 4c), but the increasing speed is moderate under prudent adjustments to the production capacity and R&D investment in GVs and NEVs (Figure 4b).Traditional AMs generate positive CAFC credits by reducing the fuel consumption of GVs and narrow the NEV credit gap by increasing their capacity and production of NEVs.In contrast, for new AMs, as shown in Figure 5, revenue generated from credit sales emerge as their primary source of income.Under the DCP, NEVs are consistently priced below production costs to gain credits, leading to negative revenue from NEV sales for traditional and new AMs.As depicted in Figure 4a, in this example, sales of GVs continue to be the primary revenue source for traditional AMs over the decade, as consumers exhibit a sustained preference for purchasing them.The traditional AMs' credit expenditure is always positive and increasing during the ten years (Figure 4c), but the increasing speed is moderate under prudent adjustments to the production capacity and R&D investment in GVs and NEVs (Figure 4b).Traditional AMs generate positive CAFC credits by reducing the fuel consumption of GVs and narrow the NEV credit gap by increasing their capacity and production of NEVs.In contrast, for new AMs, as shown in Figure 5, revenue generated from credit sales emerge as their primary source of income.Under the DCP, NEVs are consistently priced below production costs to gain credits, leading to negative revenue from NEV sales for traditional and new AMs.
under prudent adjustments to the production capacity and R&D investment in GVs an NEVs (Figure 4b).Traditional AMs generate positive CAFC credits by reducing the fu consumption of GVs and narrow the NEV credit gap by increasing their capacity and pr duction of NEVs.In contrast, for new AMs, as shown in Figure 5, revenue generated fro credit sales emerge as their primary source of income.Under the DCP, NEVs are consis ently priced below production costs to gain credits, leading to negative revenue from NE sales for traditional and new AMs.

Sensitivity Analysis
Based on the optimal decisions mentioned above, we further examined the impact the NEV market factors and AMs' cost factors on the optimal decisions made by the go ernment and AMs.

Impacts of Customers' Willingness to Pay
In Section 5.1, we assume that the consumers' willingness to pay for NEVs

Sensitivity Analysis
Based on the optimal decisions mentioned above, we further examined the impact of the NEV market factors and AMs' cost factors on the optimal decisions made by the government and AMs.As depicted in Figure 6a, the optimal CAFC coefficients drop to the lower bound of 100% quickly under different values of A and B. Nonetheless, the optimal NEVC discount coefficients under different A and B demonstrate a significant disparity.Figure 6b indicates that as the consumers' willingness to pay for NEVs escalates (i.e., the larger the value of A) or the slope of the willingness-to-pay curve for NEVs lessens (i.e., the smaller the B), the NEVC discount coefficient increases, thereby heightening the compliance requirement for NEV credit.Therefore, the government's dual credit requirements will increase as the consumers' willingness to pay for NEVs increases annually.Yet, it is worth noting that even with accelerated growth in consumers' willingness to pay for NEVs, the optimal NEVC discount coefficient remained relatively small in the first two years.The above analysis suggests that the government should always offer AMs adequate buffer time during the initial phase of the DCP.As depicted in Figure 6a, the optimal CAFC coefficients drop to the lower bound of 100% quickly under different values of A and B. Nonetheless, the optimal NEVC discount coefficients under different A and B demonstrate a significant disparity.Figure 6b indicates that as the consumers' willingness to pay for NEVs escalates (i.e., the larger the value of A) or the slope of the willingness-to-pay curve for NEVs lessens (i.e., the smaller the B), the NEVC discount coefficient increases, thereby heightening the compliance requirement for NEV credit.Therefore, the government's dual credit requirements will increase as the consumers' willingness to pay for NEVs increases annually.Yet, it is worth noting that even with accelerated growth in consumers' willingness to pay for NEVs, the optimal NEVC discount coefficient remained relatively small in the first two years.The above analysis suggests that the government should always offer AMs adequate buffer time during the initial phase of the DCP.
As shown in Figure 7b,c, when the consumers' willingness to pay for NEVs is higher (i.e., the value of A is larger or the value of B is smaller), the price of NEVs in the same year is not necessarily higher.For the first four years, the prices of BEVs and PHEVs are lower under a larger value of A. This result is because the optimal DCP under a larger value of A imposes a higher NEV credit compliance requirement and induces a higher credit price (Figure 7d), encouraging AMs to produce more NEVs to increase the supply of NEV credits.After the fourth year, the larger the value of A, the lower the credit price and, consequently, the higher the price of NEVs.On the other hand, as observed in Figure 7a, when the willingness to pay for NEVs is higher (i.e., the larger the A or the smaller the B), the price of GVs is correspondingly higher in the same year.This is because the larger the A is, the larger the NEVC discount coefficient is in the same year, thereby escalating the cost and difficulty of credit compliance for GVs, and leading AMs to reduce the production of GVs.The impact brought about by a decrease in B is similar, but when B varies between 0% and −4%, the increase in the NEVC discount coefficient is relatively smaller.Therefore, as shown in Figure 7, the impacts of B on the prices of credit and different vehicle types are also minor.
As depicted in Figure 8, the higher the consumers' willingness to pay for NEVs in the same year (i.e., the larger the A or the smaller the B), the higher both the production and R&D costs of NEVs become, leading to an increased production share of NEVs.In the scenario analyzed, the DCP can guide the production share of NEVs to approximately 45% by the end of the tenth year if the consumers' willingness to pay for NEVs does not see an annual increase (A = 0%).In contrast, the production share of NEVs could surpass 55% by the end of the tenth year if there is a 4% annual growth in consumers' willingness to pay for NEVs.From the analysis above, it becomes clear that boosting consumers' willingness to pay for NEVs, by bolstering NEV promotion and publicity efforts, is a critical strategy for achieving carbon reduction.

Sensitivity Analysis of NEV Cost Factors
In Section 5.1, we set the constant coefficient c y of the production cost function and the constant coefficient λ y of the R&D cost function for NEVs over years.In this subsection, we explore how coefficients c y and λ y with different annual change rates impact the optimal DCP and AMs' decisions.We assumed that c y (t) = (1 − 100C%)c y (t − 1) and λ y (t) = (1 + 100D%)λ y (t − 1) with c y (0)= 18 and λ y (0)= 160, respectively.In the follow- ing example, we set C at 0%, +1%, and +2%, with D = 0%, and set D at 0%, +10%, and +20%, with C = 0%, and the results are presented in Figures 9-11  As shown in Figure 7b,c, when the consumers' willingness to pay for NEVs is higher (i.e., the value of A is larger or the value of B is smaller), the price of NEVs in the same year is not necessarily higher.For the first four years, the prices of BEVs and PHEVs are lower under a larger value of A. This result is because the optimal DCP under a larger value of A imposes a higher NEV credit compliance requirement and induces a higher credit price (Figure 7d), encouraging AMs to produce more NEVs to increase the supply of NEV credits.After the fourth year, the larger the value of A, the lower the credit price and, consequently, the higher the price of NEVs.On the other hand, as observed in Figure 7a, when the willingness to pay for NEVs is higher (i.e., the larger the A or the smaller the B), the price of GVs is correspondingly higher in the same year.This is because the larger the A is, the larger the NEVC discount coefficient is in the same year, thereby escalating the cost and difficulty of credit compliance for GVs, and leading AMs to reduce the production of GVs.The impact brought about by a decrease in B is similar, but when B varies between 0% and −4%, the increase in the NEVC discount coefficient is relatively smaller.Therefore, as shown in Figure 7, the impacts of B on the prices of credit and different vehicle types are also minor.
As depicted in Figure 8, the higher the consumers' willingness to pay for NEVs in the same year (i.e., the larger the A or the smaller the B), the higher both the production and R&D costs of NEVs become, leading to an increased production share of NEVs.In the scenario analyzed, the DCP can guide the production share of NEVs to approximately 45% by the end of the tenth year if the consumers' willingness to pay for NEVs does not see an annual increase (A = 0%).In contrast, the production share of NEVs could surpass 55% by the end of the tenth year if there is a 4% annual growth in consumers' willingness to pay for NEVs.From the analysis above, it becomes clear that boosting consumers' willingness to pay for NEVs, by bolstering NEV promotion and publicity efforts, is a critical strategy for achieving carbon reduction.As shown in Figure 9a, the optimal CAFC coefficient exhibits slight variation under different C and D values, rapidly converging to its lower bound of 100% in the subsequent year.However, as illustrated in Figure 9b, when the annual increase in R&D cost for NEVs is faster (i.e., the value of D is larger), the annual fluctuation in the NEVC discount coefficient remains minimal.In contrast, when the annual decrease in NEV production cost is more rapid (i.e., the value of C is larger), the growth of the NEVC discount coefficient becomes more rapid, leading to higher compliance requirements for NEV credits within the same year.
As shown in Figure 10 a-c, the lower the production cost of NEVs (i.e., the higher the C), the lower the prices of BEVs and PHEVs.Conversely, in the same year, the prices of GVs increase.This trend can be attributed to the fact that a smaller value of C results in a higher compliance requirement for NEV credits, thereby encouraging AMs to enhance NEV production while reducing GV production to meet this compliance (as shown in Figure 11a,b).Interestingly, a higher compliance requirement does not mean a higher credit price.As demonstrated in Figure 10d, an increase in the value of C initially leads to a higher credit price during the first four years.However, this trend reverses in the subsequent year with a decline in credit price.This outcome is due to the larger the value of C, the greater the initial R&D investment by AMs in the early stages (Figure 11d), leading to faster technological growth in NEVs (Figure 11f).Therefore, even though the NEV compliance requirements increase in the following years, the cost of compliance remains relatively low.The impact brought about by an increase in D differs.Under varying D values, the optimal DCP set by the government remains relatively unchanged.Therefore, the decisions made by AMs and credit prices demonstrate only minor variations.In this scenario, the price change curve overlaps under D values of 0%, +10%, and +20%.However, As shown in Figure 9a, the optimal CAFC coefficient exhibits slight variation under different C and D values, rapidly converging to its lower bound of 100% in the subsequent year.However, as illustrated in Figure 9b, when the annual increase in R&D cost for NEVs is faster (i.e., the value of D is larger), the annual fluctuation in the NEVC discount coefficient remains minimal.In contrast, when the annual decrease in NEV production cost is more rapid (i.e., the value of C is larger), the growth of the NEVC discount coefficient becomes more rapid, leading to higher compliance requirements for NEV credits within the same year.
As shown in Figure 10 a-c, the lower the production cost of NEVs (i.e., the higher the C), the lower the prices of BEVs and PHEVs.Conversely, in the same year, the prices of GVs increase.This trend can be attributed to the fact that a smaller value of C results in a higher compliance requirement for NEV credits, thereby encouraging AMs to enhance NEV production while reducing GV production to meet this compliance (as shown in Figure 11a,b).Interestingly, a higher compliance requirement does not mean a higher credit price.As demonstrated in Figure 10d, an increase in the value of C initially leads to a higher credit price during the first four years.However, this trend reverses in the subsequent year with a decline in credit price.This outcome is due to the larger the value of C, the greater the initial R&D investment by AMs in the early stages (Figure 11d), leading to faster technological growth in NEVs (Figure 11f).Therefore, even though the NEV compliance requirements increase in the following years, the cost of compliance remains relatively low.The impact brought about by an increase in D differs.Under varying D values, the optimal DCP set by the government remains relatively unchanged.Therefore, the decisions made by AMs and credit prices demonstrate only minor variations.In this scenario, the price change curve overlaps under D values of 0%, +10%, and +20%.However, if the production cost of NEVs decreases more rapidly (i.e., the smaller the C), both production and R&D costs for GVs decrease, whereas those for NEVs increase, consequently boosting the production share of NEVs.In the scenario analyzed, the DCP could guide the production share of NEVs to approximately 45% by the end of the tenth year if the production cost of NEVs remains constant annually (C = 0%).The share of NEVs could reach 55% at the end of the tenth year if there is a 2% annual decrease in the production cost of NEVs.From the analysis above, it is evident that minimizing the production cost of NEVs can effectively stimulate investment from AMs into the production and R&D of NEVs.This strategic shift could hasten the mass production of NEVs and the R&D strides in low-carbon technologies, which are critical to achieving our carbon emission reduction goals.

Conclusions
In this paper, we present a comprehensive study on the design and optimization of the DCP over multiple years.We propose a bi-level optimization model that captures the interplay between the government's DCP design and the decision-making of AMs.Our model considers the mutual interactions among AMs' short-term production decisions, credit trading, long-term capacity expansion, and R&D strategies.Through numerical experiments, we found that: (1) Considering the NEV technology and production capacity of AMs, a lower dual credit requirement should be introduced in the initial years, providing AMs with a transition period to adapt to the new policy.As the market matures and the adaptability of AMs enhances, the dual credit requirement should be gradually increased, thus promoting NEV development and increasing market share.
(2) An optimized DCP has the potential to turn credit sales into the primary revenue source for new AMs, thereby stimulating their production and development of NEVs.This policy also provides consumers with the advantage of purchasing NEVs at prices lower than their actual cost.In shaping and implementing the DCP, the government must balance the interests of AMs and consumers, promoting mutual economic benefits and ensuring policy fairness.
(3) The consumers' willingness to pay for NEVs and the NEV production cost are the two critical factors that affect the optimal DCP.When consumers' willingness to pay for NEVs increases or NEV production cost decreases faster yearly, the government should increase NEV discount coefficients, and AMs should reduce the production and R&D of GVs while enhancing that of NEVs.To prove that the problem ( 7)-( 13) is a convex optimization problem, it is sufficient to show that both f m (•) and g m (•) are convex.
To prove that f m (•) is convex, it suffices to show that the Hessian matrix of f m (•) is positive definite.For any specific t ∈ T and x ∈ X, the second-order partial derivatives are as follows: Since the function f m (•) maintains the same derivative structure for all instances of t ∈ T and x ∈ X, the Hessian matrix H m of f m (•) can be constructed by aligning the Hessian matrix H x m,t of each t ∈ T and x ∈ X along the diagonal:

Figure 1 .
Figure 1.The interactions among the government's coefficient setting, AMs' decision making, and market prices.

Figure 1 .
Figure 1.The interactions among the government's coefficient setting, AMs' decision making, and market prices.

Figure 2 .
Figure 2. The changes in the government's optimal CAFC coefficient and NEVC discount coefficient.

Figure 2 .
Figure 2. The changes in the government's optimal CAFC coefficient and NEVC discount coefficient.

Figure 3 .
Figure 3.The changes in the social welfare and the NEV share.(a) Social welfare and its various components; (b) NEV production share.

Figure 3 .
Figure 3.The changes in the social welfare and the NEV share.(a) Social welfare and its various components; (b) NEV production share.

Figure 4 .Figure 4 .Figure 5 .
Figure 4.The changes in the production, R&D decisions, credit trading volume, and benefits of traditional AMs.(a) Production decisions; (b) R&D decisions; (c) credit trading volume; (d) profit and its various components.
curve undergoes a yearly decrease of 0.05 (i.e.,

Figure 5 .
Figure 5.The changes in the production, R&D decisions, credit trading volume, and benefits of new AMs.(a) Production decisions; (b) R&D decisions; (c) credit trading volume; (d) profit and its various components.

Figure 8 .
Figure 8. Impacts of the changing rates A and B of the market factors y t a and y t b on the AMs' production and R&D decisions.(a) GV production; (b) NEV production; (c) R&D investment cost of GVs; (d) R&D investment cost of NEVs; (e) NEV production share.

Figure 8 .
Figure 8. Impacts of the changing rates A and B of the market factors a y t and b y t on the AMs' production and R&D decisions.(a) GV production; (b) NEV production; (c) R&D investment cost of GVs; (d) R&D investment cost of NEVs; (e) NEV production share. .

5. 3 . 2 .
Sensitivity Analysis of NEV Cost Factors In Section 5.1, we set the constant coefficient y c of the production cost function and the constant coefficient y λ of the R&D cost function for NEVs over years.In this subsec- tion, we explore how coefficients y c and y λ with different annual change rates impact the optimal DCP and AMs' decisions.We assumed that In the following example, we set C at 0%, +1%, and +2%, with D = 0%, and set D at 0%, +10%, and +20%, with C = 0%, and the results are presented in Figures 9-11 .

Figure 9 .
Figure 9. Impacts of the changing rates C and D of cost factors y c and y λ on the government's annual CAFC and NEVC discount coefficients.(a) Annual CAFC coefficient; (b) NEVC discount coefficient.

Figure 9 .
Figure 9. Impacts of the changing rates C and D of cost factors c y and λ y on the government's annual CAFC and NEVC discount coefficients.(a) Annual CAFC coefficient; (b) NEVC discount coefficient.

Figure 10 .Figure 10 .Figure 10 .Figure 11 .Figure 11 .
Figure 10.Impacts of the changing rates C and D of cost factors y c and y λ on the price of vehicles and credits.(a) GV market price; (b) BEV market price; (c) PHEV market price; (d) credit trading price.

Figure 11 .
Figure 11.Impacts of the changing rates C and D of cost factors c y and λ y on the AMs' production and R&D decisions.(a) GV production; (b) NEV production; (c) R&D investment cost of GVs; (d) R&D investment cost of NEVs; (e) fuel consumption for each GV; (f) credits for each NEV; (g) NEV production share.
actual NEVC value of an AM is determined by the production volumes and the technical level of different NEV types.Let Z ∈ Y for an AM m in year t, the actual NEVC value NEV act m,t f y m,t , Z y m,t of the AM in year t is as follows: , t = 1, . . ., T , as defined in Equations ( t +∑ y∈Y r t g signifies the total credit supply in the market.At market equilibrium, if P t > 0, the credit supply equals the demand in year t, if the credit supply is greater than the demand, then P t = 0. Hence, at market equilibrium, the demand d x t for GVs in type x ∈ X in year t must meet the following conditions: (3) x m,t ;Then, all the leading principal minors of H x m,t can be expressed as(3)represent the first-order, second-order, and third-order principal minors of the matrix H x m,t , respectively.Under the assumption that the cost functionsC x m,t f x m,t , ϕ x m,t θ x m,t , V x m,t R x m,t ,m ∈ M, x ∈ X,t ∈ T of GVs are all continuously differentiable, strictly increasing, and convex functions, all the leading principal minors D x m,t (1) ,D x m,t (2) , and D x m,t (3) are greater than zero.Therefore, the matrix H x m,t is positive definite.