A Decision-Making Model for Echelon Utilization of Retired Batteries in Competitive Duopoly: the Role of Government Subsidy

: As the wave of retirements in the new energy vehicle sector approaches, China's 37 emphasis on echelon utilization has grown to optimize battery reuse and recycling. However, 38 the burgeoning industry will inevitably face the challenges of competition, impacting 39 stakeholders across the supply chain. In this study, we employ game theoretic models to 40 investigate the interplay between competitive dynamics and government subsidy policies 41 within this industry. Through the development of a competitive duopoly echelon utilization 42 supply chain model, our analysis offers valuable insights and recommendations for the 43 advancement of the industry. We find that increases in echelon utilization costs prompt 44 adjustments in retail prices by echelon utilization enterprises, third-party recyclers, and new 45 energy vehicle manufacturers. As competition intensifies, overall supply chain profitability 46 diminishes, resulting in a lose-lose situation, favoring only third-party recyclers, who raise 47 wholesale prices as new energy vehicle manufacturers reduce transfer payments. Conversely, 48 government subsidies stimulate higher total demand, benefiting subsidized echelon utilization 49 enterprises, which enhance consumer and social value. Non-subsidized enterprises are 50 compelled to raise retail prices. Given the positive impact of subsidies, governments should 51 prioritize support for enterprises demonstrating superior echelon utilization practices. 52


Introduction
In recent years, the global new energy vehicle market, encompassing regions like China and Europe, has seen remarkable growth driven by environmental concerns and industrial green transformation efforts (Li et al. 2023).With the rapid growth of the new energy vehicle industry, the first wave of retirement for core components, namely new energy batteries, is approaching (Chen et al. 2022;Gong et al. 2022).These batteries primarily consist of ternary lithium and lithium iron phosphate batteries (Wang et al. 2022).Inadequate processing of retired batteries poses a significant risk, as heavy metal elements found in both battery types can lead to water, soil, and air pollution, severely harming the ecological environment and hindering the achievement of environmental goals associated with promoting new energy vehicles (Feng et al. 2023).Failing to attain the desired environmental benefits may result in resource wastage and heightened environmental pollution.
Echelon utilization offers an effective means to manage the retirement of used batteries from new energy vehicles, mitigating environmental pollution concerns and fostering a positive corporate image (Zhang et al. 2020a).Echelon utilization involves classifying and reusing used batteries based on their remaining volume (Lai et al. 2021b).High-volume used batteries can be directly sold to consumers after minimal treatment, while low-volume ones can have their heavy metal elements extracted for remanufacturing (Zhao and Ma 2022;Chen et al. 2022).
Simultaneously, it presents eco-friendly solutions to the used battery dilemma, alleviating anxieties surrounding the new energy automobile industry's development and further propelling its growth.
In practice, the Chinese regulation named 'New Energy Vehicle Industry Development Plan (2021)(2022)(2023)(2024)(2025)(2026)(2027)(2028)(2029)(2030)(2031)(2032)(2033)(2034)(2035)' highlights the accelerated development of Chinese new energy vehicles 1 (Zhang et al. 2020a).In anticipation of an impending wave of retirements in this sector, the Plan emphasizes the need for an efficient battery recycling system, endorsing echelon utilization as a solution.Another regulation named 'Administrative Measures for the Echelon Utilization of New Energy Vehicle Power Batteries' further encourages the collaboration between echelon utilization enterprises, new energy vehicle manufacturers, battery manufacturers, and recyclers 2 .The US unveiled the National Lithium Development Blueprint 2021-2030, outlining objectives to optimize the utilization of retired lithium batteries, recycle crucial raw materials, and establish a robust and competitive lithium battery recycling value chain.
Simultaneously, the EU introduced the Strategic Research Agenda for batteries and the Battery Innovation Roadmap 2030, devising a comprehensive plan for medium and long-term carbon reduction in the battery industry.Additionally, it designed a carbon footprint management strategy for the entire battery chain (Chen et al.2022).Currently, 52 firms have met the Chinese standards outlined in the 'Industry Standard Conditions for Comprehensive Utilization of Waste Power Batteries for New Energy Vehicles,' indicating a growing number of enterprises in this field.It is anticipated that due to the ongoing efforts of the government, an increasing number of echelon utilization enterprises will meet the stipulated criteria.As the adoption of electric vehicles and renewable energy continues to rise, the importance of these enterprises in managing the life cycle of power batteries will only grow, making the industry increasingly competitive.The competition among power battery echelon utilization enterprises is driven by factors like investment in cost-efficient battery recycling systems (Zhang et al. 2023c), technological innovation (Lai et al. 2021a), environmental responsibility (Zhang et al. 2023a), market demand, andsubsidy (Bai et al. 2019;Bai et al. 2021).The continuous endeavors of echelon utilization enterprises are poised to generate a noteworthy challenge in the form of heightened competition within the industry.For example, in China, GEM 3 and ZCycle 4 are two prominent companies that engage in competition within the specialized field of power battery echelon utilization.GEM claims that it has built a new energy full life cycle value chain of "waste battery recycling -raw material remanufacturing -material remanufacturing -battery pack remanufacturing -reuse -echelon utilization", and its recycled used power batteries, excluding lead-acid batteries, constitute over 10% of China's overall scrap.ZCycle boasts advanced wet extraction technology and is committed to environmentally friendly waste power battery treatment with a low-carbon footprint.According to different types of retired batteries, ZCycle customizes safe application scenarios, in which retired ternary lithium batteries are used for manufacturing energy-saving street lights and retired iron phosphate batteries are used for forklifts and UPS power supply systems.Competition significantly shapes the decisionmaking process of both GEM and ZCycle, and the introduction of government subsidies and incentive policies further complicates their decision-making behavior.
In examining the real practices, questions surrounding decision-making under competition, the impact of the competitive environment on decisions, and the highly efficient strategic use of government subsidies for promoting echelon utilization have emerged.To mirror real-world dynamics, this paper aims to investigate the firms' echelon utilization strategies in a competitive duopoly environment.Furthermore, to investigate the government's role, we initially analyze firms' echelon utilization strategies without government subsidies, establishing a benchmark.Subsequently, we delve into their echelon utilization strategies in the presence of government subsidies.To be specific, this paper primarily focuses on echelon utilization enterprises, constructing a competitive duopoly supply chain to address three key issues: (1) Examining equilibrium scenarios in the echelon utilization competitive game, both with and without government subsidies.
(2) Investigating how changes in the echelon utilization enterprise environment influence decision-making and outcomes.
(3) Assessing the effects of government subsidies on echelon utilization enterprises and providing recommendations for effective subsidy strategies.
To meet the research objectives, we employ game theory methodology to analyze realworld dynamics and provide managerial recommendations.These models involve a multi-party game supply chain that includes two battery echelon utilization enterprises, one new energy vehicle manufacturer, one third-party recycler, and government entities.In this competitive environment, two echelon utilization companies engage in a game to define their echelon utilization strategies.By analyzing the model, this study conducts a comparative analysis of the game equilibrium outcomes in the absence of government subsidies and with the inclusion of government subsidies.This study makes significant contributions to understanding the practical impact of government subsidy policies.Examining the alterations and interconnections within different facets of the supply chain provides a more profound understanding of heightened competition and the ramifications of subsidy policies across the industry.The research discerns that government subsidy initiatives moderately bolster market demand, particularly for enterprises benefiting from subsidies.Nevertheless, this dynamic also precipitates a profitability downturn throughout the supply chain, engendering a mutually detrimental scenario.Additionally, the investigation illuminates the advantages realized by third-party recyclers operating under government subsidy policies.These entities achieve augmented profits by elevating wholesale prices, underscoring the constructive influence of subsidy policies on the evolution of the recycling industry.
The remainder of this paper is organized as follows.Section 2 briefly summarizes and reviews the previous research related to this research.Section 3 describes the research problem of this paper in detail and establishes the competitive duopoly supply chain models in two cases according to the relevant assumptions.Section 4 calculates the equilibrium situations in two cases and preliminarily analyzes the influence of the internal and external environment of echelon utilization enterprises and the subsidies.Section 5 further analyzes the impact of the environment and subsidies through numerical experiments and gives some relevant management suggestions.Section 6 summarizes the main conclusions of this paper and proposes the shortcomings of the research.We also give an outlook on possible future research directions.

Literature Review
This section reviews previous research in three primary categories: competition, government subsidies in closed-loop supply chain, and the management of new energy battery recycling and echelon utilization.While existing research has explored competition in various aspects of supply chains, including manufacturer, retailer, recycler, and supply chain competition, there has been limited investigation into the introduction of competition in the field of echelon utilization.

Research related to competition in or between closed-loop supply chain
Furthermore, research on competition among echelon utilization enterprises is also scarce.Our paper innovatively introduces the concept of competition into echelon utilization and uniquely centers on echelon utilization enterprises as the primary agents of competition.

Research related to government subsidizing the supply chain
Considerable research has examined the effects of various government subsidies on supply chains.Mitra et al. (2008) examined the impact of government subsidies on promoting remanufacturing activities, suggesting that partial subsidies to manufacturers could be effective.Mo et al. (2009) proposed the use of recycling tax incentives as a means for government intervention, based on their field investigations.Aksen et al. (2009), meanwhile, established supportive and legislative models, discovering that the supportive model necessitates more subsidies when recycling rates and profitability are the same.In the recent study, Zhang et al. (2020b) suggested strategic options that account for the effects of tax policy, subsidy policy, and tax-subsidy policy on the supply chain.
In the field of green products and green innovation, Li et al. (2018)  Despite the abundance of research into the mechanism of government subsidies in various industries, there has been little exploration of the role of these subsidies in the field of battery echelon utilization as well as into how the government should best disburse the subsidies.A major innovation of our research is in its study of how government subsidies in the field of echelon utilization affect the decision-making of echelon utilization enterprises.

Research related to the management of echelon utilization
The last decade has seen the emergence of a robust literature related to used battery recycling and echelon utilization.
Most recently, Lai et al. (2021b) investigated echelon utilization within the realm of battery recycling, conducting a systematic review on the echelon utilization and recycling processes of retired lithium-ion batteries (LIBs).Their research introduced two sorting methods aimed at enhancing the speed and accuracy of LIB sorting procedures.On a separate note, Turner et al. (2016) conducted an analysis of EPR policies, assessing their efficacy in managing the environmental costs and benefits of end-of-life (EOL) management across the European Union, Canada, and the United States.Their argument emphasized the necessity for these policies to extend their scope to encompass waste collection practices, the holistic life cycle impacts of EOL management, and the quality of recovered materials for greater effectiveness.

Gu et al. (2017) delved into the decision-making process of vehicle manufacturers concerning
battery recycling rates within the framework of government subsidies.Their findings revealed that either subsidy schemes or battery recycling initiatives could counteract the adverse effects of loss aversion on optimal production quantities and expected utility.Subsequently, Gu et al.
(2018) investigated the comparative aspects of new battery manufacturing versus battery recycling and reusing practices.They concluded that while battery recycling and reusing have the potential to curtail raw material consumption and thereby diminish the environmental impact, these practices might not necessarily yield substantial financial benefits.Zhang et al.
(2020a) examined China's echelon utilization policy, assessing its efficacy through the lenses of fundamental policy tools and the industrial chain process.They advocated for increased utilization of interactive impact instruments by the government and emphasized the need to refine the classification of these instruments, highlighting the importance of alignment between basic policy tools and the recycling industry chain for optimal effectiveness.
The above summary highlights that the present literature lacks a focus on the companies involved in echelon utilization practices.As a response, our paper focuses on the decisionmaking process of echelon utilization and how it can be influenced by the business environments and by government subsidies.

Model description
This research focuses on evaluating the influence of the internal and external business environment, along with government subsidies, on echelon utilization enterprises in a competitive setting.To achieve this, a competitive duopoly supply chain for echelon utilization is established, which comprises a new energy vehicle manufacturer, a third-party recycler, and two competing echelon utilization enterprises.The manufacturer sells new energy vehicles to consumers, and the echelon utilization enterprises entrust the third-party recycler to handle used battery recycling.The third-party recycler sells used batteries to the echelon utilization firms at a uniform price, facilitating the echelon utilization process through collaborative efforts.
In view of the nascent stage of echelon utilization for used new energy batteries in China, it is assumed that the number of used batteries reclaimed by recyclers exceeds the demand from the two echelon utilization enterprises.Consequently, the excess used batteries must be managed by the third-party recycler to avert potential ecological pollution.The used batteries are sold by the two echelon utilization enterprises to specialized consumers (e.g., power plants) for the initial stage of the echelon utilization process, encompassing dismantling, testing, and sorting.Subsequently, the processed used batteries are recycled and supplied to the new energy vehicle manufacturer for remanufacturing.
In this process, the new energy vehicle manufacturer holds a dominant position due to its influence on the entire process.The manufacturer determines the retail price, denoted as  for new energy vehicles.For model simplification, we assume that the actual recycling ratio and the unit recycling price of recyclers are exogenous.Consequently, the third-party recycler solely determines the wholesale price of used batteries (), while the two echelon utilization enterprises independently establish retail prices for their respective used batteries ( 1 , 2 ).The specific flow chart of the model is shown in Figure 1.
<Insert Figure 1> In Figure 1, the solid line illustrates the journey of new energy vehicles, spanning from manufacturing to recycling and, finally, reaching echelon utilization consumers.In contrast, the dotted line delineates the recycling of used batteries after the initial echelon utilization stage, followed by their transfer to the new energy vehicle manufacturer for remanufacturing.
To simplify the model,  1 and  2 in Figure 1 denote the unit costs associated with echelon utilization.These costs encompass activities such as dismantling, detection, classification, and recovery, and they are contingent solely upon the echelon utilization level of each enterprise.

Model Construction
Based on the descriptions in Section 3.1, we establish the decision-making sequence for all parties within the supply chain.Initially, the new energy vehicle manufacturer determines the sales price for its vehicles.Subsequently, the third-party recycler sets the wholesale price for used batteries.Finally, the two echelon utilization firms independently establish the retail prices for their respective used batteries.The model's relevant symbols and the decision variables for each party are detailed in Table 1.Assumption 3. Since echelon utilization is still in its initial phase in China, we assume that the demand for it is less than the actual number of batteries in need of recycling, with third-party recyclers needing to handle the resulting excess (Zhao and Ma 2022).Therefore, the relationship between the actual amount of recycling and the demand for echelon utilization services is as follows.

𝐺 > 𝑇 1 + 𝑇 2
Assumption 4. To make the model's conclusions more reasonable and in line with reality, there is a range of competition intensity between two echelon utilization enterprises.

𝐺 = 𝜃 * 𝑄
This study follows the aforementioned assumptions and employs a literature review approach, drawing insights from previous research and model construction.We utilize the backward induction method within a dynamic game with complete information to derive the decision-making process and relevant parameters at equilibrium.Subsequently, we apply sensitivity analysis and numerical experiments to explore the influence of the supply chain's internal dynamics.Lastly, we analyze the effects of government subsidies and offer managerial recommendations.

Model formulation without government subsidies
The profit functions of each party without government subsidies are as follows.
Profit function of the new energy vehicle manufacturer: The above profit formula 1 consists of two parts.The first half is the profit obtained from selling new energy vehicles in the forward supply chain, while the second represents the additional income through remanufacturing.
Profit function of the third-party recycler: The above profit formula 2 consists of two parts.The first half is the profit made by selling used batteries, with the second half the additional cost incurred by having to dispose of excess recycled batteries.
Profit function of echelon utilization enterprises: Echelon utilization enterprises generate profits from sales revenue in the marketplace and from transfer payments obtained from the new energy vehicle manufacturer.

Model formulation with government subsidies
Based on our research into the relevant policy documents, we consider the case in which the government subsidizes one echelon utilization firm based on the quantity of the echelon utilization product.To maintain symmetry, we study only the case where the government subsidizes echelon utilization enterprise 1, with the profit function as follows.
The profit function expressions of other parties are similar to those in Section 3.2.1.

Model Results and Analysis
In this section, we use the backward induction method in game theory to obtain the equilibrium outcomes for the two modes.Our analysis focuses on the decision-making of echelon utilization enterprises and assesses the impact of the business environment and subsidy levels.

The equilibrium situation without government subsidies
The equilibrium situation without government subsidies is shown in Proposition 1.
Proposition 1.The equilibrium situations of the game without government subsidies are: Proof.See Appendix.
By substituting the optimal decision variables  * ,  * ,  * ,  1 * , and  2 * into the defined functions of each variable, we get the actual market demand for new energy vehicles  * = 1 2 ( −   ), the actual recycling amount of used batteries  * = 1 2 ( −   ).
The market demand of those two echelon utilization enterprise  1 * and  2 * are as follows.
The profits of the vehicle manufacturer, the third-party recycler, and two competitive echelon utilization enterprises are listed below.
From Proposition 1, two corollaries can be derived.Corollary 1 evaluates how unit echelon utilization costs affect the optimal decision variables related to echelon utilization products.This analysis offers insights into whether echelon utilization enterprises should adjust their retail prices, whether the third-party recycler should modify wholesale prices for used batteries, and whether the new energy vehicle manufacturer should change transfer payments, in response to fluctuations in unit echelon utilization costs.
Corollary 1.The influence of the total unit cost of echelon utilization on the optimal decision variables related to echelon utilization products is shown in Table 2. Proof.See Appendix.
Table 2 reveals several significant findings.First, it demonstrates that when the costs of one echelon utilization enterprises increase, the competitor tends to lower their retail prices to capture a larger market share.Conversely, when the costs of one echelon utilization enterprises decrease, the competitor tends to raise retail prices to safeguard their profits.This result can be explained as follows: An increase in costs signifies a weaker competitive position, allowing competitors to raise their retail prices of echelon utilization products while still maintaining a competitive edge.
Second, as the unit cost of echelon utilization increases, the wholesale price of used batteries for the third-party recycler will decrease.This shows that as its unit cost increases, the third-party recycler tends to reduce the price, benefitting the echelon utilization industry by ensuring market demand.When the unit cost of echelon utilization services decreases, the third-party recycler tends to protect their interests by raising their retail prices.
Third, the effect of echelon utilization costs on transfer payments for remanufactured materials can be derived and represented in Figure 2, where we let  = 3000,  = 1.6,   = 800 ,   = 600 ,  0 = 3 ,  = 0.8,  = 100 ,  = 2, ℎ = 100 , and  = 0.8.According to Figure 2, with the increase in the cost of echelon utilization, the new energy vehicle manufacturer will increase the transfer payments to ease the pressure on echelon utilization enterprises.The new energy vehicle manufacturer hopes to maintain demand in the echelon utilization market, thereby stabilizing the resource of remanufactured materials.It should be noted here that the setting of this and subsequent parameters are obtained by partly referring to the research of Tang et al. ( 2018) and satisfying the basic constraints of the model.
<Insert Figure 2> Corollary 2 evaluates the impact of unit echelon utilization costs on the optimal profits of two echelon utilization enterprises.
Corollary 2. The profits earned by echelon utilization firms vary according to their operating costs, expressed as follows. (1) Corollary 2 explores how changes in the unit cost of echelon utilization impact the profits of echelon utilization enterprises.It shows that with the increase of echelon utilization costs, profits earned by the echelon utilization firms always decrease first and then increase later.This is because the change in cost affects both the retail price and the number of echelon utilization products.In the actual situation, considering the non-negativity of demand, price, and cost, these profits tend to change monotonically with the changes of  1 and  2 .That is, monotonically increasing or decreasing.

The equilibrium situation with government subsidies
According to the relevant proof method in Proposition 1, we can obtain the equilibrium situations with government subsidies as shown in Proposition 2.
Proof.Similar to the proof of Proposition 1.
By substituting the optimal decision variables   * ,   * ,   * ,  1  * , and  2  * into the defined functions of each variable, we get the actual market demand for new energy vehicles The market demand of those two echelon utilization enterprise  1  * and  2  * are as follows.
Under subsidies, the profits of the vehicle manufacturer, the third-party recycler, and two competitive echelon utilization enterprises are listed below.
From Proposition 2, some corollaries can be derived.Corollary 3 assesses the impact of government subsidy on the optimal decision variables related to echelon utilization products.This analysis offers insights into whether echelon utilization enterprises should adjust their retail prices, whether the third-party recycler should modify wholesale prices for used batteries, and whether the new energy vehicle manufacturer should change transfer payments, in response to changes in government subsidy.
Corollary 3. The influence of the government subsidy () on the optimal decision variables related to echelon utilization products is presented in Table 3. Proof.See Appendix.
Table 3 indicates several important findings.First, with the increase in government subsidies, the subsidized echelon utilization firm 1 will reduce the retail price, while unsubsidized firm 2 will increase it.It shows that the more government subsidies, the more the subsidized enterprises will reduce their retail price to stimulate demand in the echelon utilization market.In contrast, the unsubsidized firms will raise their retail price in order to protect their profit.
Second, the wholesale price of used batteries from third-party recyclers increases with government subsidies.It shows that the third-party recycler will raise the wholesale price because of the increase in government subsidies, in the hopes of earning some dividends from them.
Third, similar to the third-party recycler, the new energy vehicle manufacturer also hopes to earn some dividends from the subsidies.The impact of government subsidies on transfer payments is illustrated in Figure 3.We set  = 3000,  = 1.6,   = 800,   = 600,  0 = 3,  = 0.8,  = 100,  = 2, ℎ = 100,  = 0.8,  1 = 9,  2 = 5 in Figure 3.With the gradual increase in government subsidies, the new energy vehicle manufacturer will pay lower transfer payments.
<Insert Figure 3> The following Corollary 4 can be obtained by a sensitivity analysis of the equilibrium situation obtained in Proposition 2.
Corollary 4. The profits earned by the echelon utilization enterprises change with government subsidies as follows.
Proof.See Appendix.
Corollary 4 evaluates the impact of government subsidy on the optimal profits of two echelon utilization enterprises.It shows that with the increase in government subsidies, industry profits always decrease first and then increase later because the subsidies not only affect the price of echelon utilization products but also the demand for them.In the actual situation, considering the non-negativity of demand, price, cost, and government subsidies, echelon utilization company profits tend to change monotonically with the changes of , that is, monotonically increasing or decreasing.

Numerical Experiments
In this section, numerical analysis is employed to delve deeper into the equilibrium outcomes and derive pertinent managerial insights.

Influence of competition intensity on echelon utilization
In this section, our paper considers the effect of competition intensity on echelon utilization enterprises and third-party recyclers in the absence of subsidies.By referring to relevant literature and meeting the constraints of the model, we set  = 3000,  = 1.6,   = 800,   = 600,  0 = 3,  1 = 9,  2 = 5,  = 0.8,  = 100,  = 2, ℎ = 100, and substitute them in Proposition 1 to get Figures 4-6.Observation 1 can be drawn from the three Figures.
<Insert Figure 4 > <Insert Figure 5> <Insert Figure 6> Observation 1.With the increase in competition intensity(), echelon utilization firms will raise retail prices( 1 ,  2 ), leading to a decline in demand for their products and then in company profits.At the same time, third-party recycler profits will rise in proportion to the competition intensity-the higher it is, the faster profits will rise.
Figures 4-6, and Observation 1 show that: (1) it is intuitive that the echelon utilization enterprise with higher costs will earn fewer profits; (2) the increase in competition intensity will hurt those enterprises and cause them to excessively increase retail prices, leading to a lose-lose situation; and (3) with the increase in competition intensity, the third-party recycler will seize this opportunity to increase their wholesale prices to expand its revenue.

Influence of echelon utilization costs on the supply chain
This section investigates the influence of echelon utilization costs on the entire supply chain in the absence of government subsidies.We set  * =   * +   * + 1 * +  2 * to represent the total profit of the supply chain, with the numerical settings in this section consistent with those in Section 5.1.It should be noted that we set that  = 0.8, with the result of the numerical experiment shown in Figure 7, from which Observation 2 can be drawn.
<Insert Figure 7> Observation 2. The total profit of the supply chain ( * )will diminish as the echelon utilization costs (  1 ,  2 ) of both echelon utilization firms increase.In general, excessive echelon utilization costs will hurt the overall supply chain.
Figure 7 and Observation 2 demonstrate that, from the standpoint of the overall supply chain, it is logical that an increase in the echelon utilization cost of any echelon utilization enterprise under competition will result in a decrease in the total profit of the supply chain.
Hence, even within a competitive environment, echelon utilization enterprises with superior echelon utilization capabilities, meaning lower echelon utilization costs, should be preferred for cooperation.

Influence of government subsidies on echelon utilization enterprises
This section discusses the influence of government subsidies on market demand, retail price, and echelon utilization enterprise profits, with numerical settings consistent with those in Section 5.1.For the convenience of expression, we set the additional demand after government subsidies as ∆ 1 =  <Insert Figure 8> Observation 3. Government subsidies will increase demand for the subsidized echelon utilization firm and reduce demand for the unsubsidized firm, but the overall demand will increase.In other words, ∆ 1 increases as government subsidies increase, and ∆ 2 decreases as subsidies increase.
Observation 3 shows that government subsidies do affect competition such that the subsidized firm will increase market demand and gain market share while the unsubsidized firm will lose it.Yet, from the perspective of the entire market, government subsidies are beneficial overall because they can increase total demand and promote the development of the echelon utilization industry.Even subsidizing the firm with the higher costs will also increase total demand.When taken together with Observation 2, it becomes clear that the government should prioritize offering subsidies to the firm with the lower echelon utilization costs, so that the echelon utilization market can be further regulated and accelerated in this way.
<Insert Figure 9> Observation 4. Government subsidies will cause the recipient enterprise to lower their retail price, but will also cause their unsubsidized competitor to raise their retail price, with the resulting price decrease greater than the increase.Or, written another way, ∆ 1 decreases as government subsidies increase, and ∆ 2 increases as subsidies increase.
Observation 4 shows that government subsidies can significantly affect the retail prices of echelon utilization competitors, with the subsidized echelon enterprise reducing its price to benefit consumers, while the unsubsidized one increases its own in order to maintain its profits.
For the unsubsidized firm, government subsidies relax its willingness to compete and put it at a competitive disadvantage.From the perspective of the subsidized firm, government subsidies increase its willingness to compete but can also ultimately benefit consumers.This once again shows that when the government selects subjects for its subsidy policies, it should select the enterprises with the strongest capabilities, as determined by a thorough investigation.Doing so can optimize consumer benefit, thereby promoting the development of the industry efficiently.
<Insert Figure 10> Observation 5. Government subsidies will lead to higher profits for the recipient echelon utilization firm but reduced profits for the unsubsidized one, causing total market revenue to rise.Or, ∆ 1 increases as the government subsidies increase, and ∆ 2 decreases as subsidies increase.
Observation 5 shows that government subsidies can significantly affect the potential of echelon utilization competitors to earn additional profits.The subsidy recipient will obtain obvious additional profits, and the greater the subsidies, the faster the growth rate.At the same time, the unsubsidized firm will suffer additional losses, but the growth rate will gradually slow down as subsidies increase.When looking at the whole market, it becomes clear that subsidies can increase the total profit and play an important role in promoting development.Like suggestions mentioned in Observations 4 and 5, Figure 10 further illustrates that the government should be cautious from the perspective of industry profits when selecting subject firms to receive subsidies.(1) Without government subsidies, the echelon utilization enterprise will reduce retail prices to gain market share when costs rise for its competitors.When its costs increase, the firm tends to increase retail prices to ensure its profit.

Conclusions and Prospects for
In theory, with the increase in costs, the profits of the firms always decrease first and then increase, but after considering the actual situation, they often only change monotonically.
For the third-party recycler, when the costs of echelon utilization increase, it tends to reduce the wholesale price to ensure demand.Conversely, the recycler tends to increase the wholesale price to protect profit.When the costs of echelon utilization increase, the new energy vehicle manufacturer tends to increase the transfer payment to stabilize the source of remanufactured materials.
(2) In the case of government subsidies, the recipient echelon utilization enterprise is more likely to reduce the retail price to increase demand as subsidies increase, the unsubsidized competitor is willing to increase the retail price in consideration of its profits.This conclusion is similar to the findings of Gu et al. ( 2017), whose research also indicated that government subsidies increase the optimal output of enterprises.
As government subsidies increase, the echelon utilization firms' profits will theoretically decrease first and then increase.This viewpoint differs from existing research (Li et al, 2018;Mitra et al, 2008), which suggests that government subsidies are always beneficial for manufacturers.and considering the actual situation, they often only change monotonically.
The third-party recycler will raise wholesale prices as government subsidies increase as it hopes to enjoy its share of dividends from the subsidies.The new energy vehicle manufacturer, for its part, will reduce the transfer payment to enjoy its own share of dividends.
(3) With an increase in competition intensity, competing echelon utilization enterprises will raise retail prices in order to ensure their profits, which in turn may reduce market demand and eventually create a lose-lose situation.At the same time, the third-party recycler will gain an advantage over its competition and generate additional revenue from the opportunity.
(4) As far as the supply chain is concerned, the rise in echelon utilization costs will lead to a decline in total profits.For this reason, only those enterprises with lower costs should be selected for cooperation with the subsidy program.
(5) Government subsidies will increase demand for the subsidized firms, while possibly reducing market demand for echelon utilization products from the unsubsidized ones.This is still beneficial from the perspective of the market, however, as subsidies will lead to a rise in total demand.
In addition to demand, government subsidies will also significantly impact retail prices.
The subsidized echelon utilization competitor will cut its retail prices, to the benefit of consumers, while the unsubsidized one will raise retail prices to maximize revenue, benefiting itself.For the unsubsidized competing firm, although subsidies could allow it to ease its competitive will and avoid vicious competition, they would also weaken that firm's position.
For the subsidized competitor, although subsidies increase its willingness to compete, they will also benefit echelon utilization consumers in the end.
In addition, government subsidies can also affect the potential for echelon utilization enterprises to add revenue, with the subsidized competitor profiting more.This aligns with the findings of prior research (Zhang et al. 2023;Zhang et al. 2022); however, it overlooks the competition between subsidized and unsubsidized enterprises.In fact, we find that the greater the amount of government subsidies, the faster the growth of the subsidized firm.In contrast, the unsubsidized enterprise will suffer additional losses.That is, the more government subsidies received by its competitor, the lower the unsubsidized firm's growth rate.Therefore, from the perspective of the whole industry, subsidies can increase profit and play an important role in its development.We find, in fact, that government subsidies will always have an impact regardless of whether the recipient enterprise has higher or lower costs.We also know that the higher the echelon utilization costs, the greater the negative effect on the total profit of the supply chain.As such, for the benefit of the whole industry, we think the government should optimally choose to subsidize those echelon utilization enterprises that operate with lower costs.
Since our research focuses on echelon utilization enterprises, this paper may have the following deficiencies and parts that could be expanded in the future.( 1) To simplify the model and highlight the key points, we set the recycling ratio and the unit recycling price as constants.
In the future, we can treat them as decision variables.
(2) Our research focuses on the initial stage of echelon utilization.In the future, there may be further discussions where the supply and the demand are in balance or where the supply is less than the demand.(3) In our research, the sources of remanufactured materials all come from echelon utilization enterprises.It is possible to study the impact on echelon utilization enterprises when the third-party recycler or both the third-party recycler and echelon utilization enterprises provide remanufactured raw materials.According to monotonicity, (1) can be proved.(1) can thus be proved according to the principles of monotonicity.

Proof of
Over the past decade, numerous scholars have delved into research concerning competition within and between closed-loop supply chains.Additionally, studies have examined competition among manufacturers and retailers.For instance, Jena et al. (2014) investigated the effects of various cooperation models with retailers in the context of manufacturer competition.Likewise, Wang et al. (2017) explored the interplay between channel structure and price-and quality-based competition among two manufacturers characterized by customer loyalty asymmetry.In the realm of retailer competition research, Savaskan et al. (2016) pioneered the examination of competition intensity's influence on decision-making within the context of retailer competition in closed-loop supply chains.In a similar vein, Guo et al. (2020) investigated the impact of retail competition on the development of green products, revealing that heightened market competition intensity leads to a lower optimal level of greenness.Zhang et al. (2023b) constructed a two-period pricing game that involves two competing platforms to investigate the influence of network effects on optimal pricing strategies and profits.Numerous scholars have conducted research on competition within the recycling sector.. Liu et al. (2017) explored reverse channel choice decisions involving collection competition under three recycling options.Their findings suggest that manufacturers consistently benefit from the joint recycling mode with recyclers.In the context of echelon utilization, Tang et al. (2018) investigated mechanisms and policies under various recycling modes, including both single and competitive dual recycling channel modes.Wang et al. (2019) focused on remanufacturing space, exploring recycling competition between remanufacturers and recyclers, as well as between recyclers and retailers.Giri et al. (2019) addressed decisionmaking issues involving manufacturer-built recycling channels and traditional competition among recyclers, while Wei et al. (2019) analyzed the remanufacturer's decision-making in the context of dual recycling channels.Zhou et al. (2023) explored channel leadership and performance within a closed-loop supply chain, focusing on the interplay between an electric vehicle manufacturer, an electric vehicle recycler, and a third-party recycler in a competitive landscape.Conversely, Suvadarshini et al. (2023) examined a closed-loop supply chain involving an OEM, a retailer, and a third-party vendor.The study delves into how the OEM devises an optimized multi-channel recollection framework amid competitive efforts by recollection agents in retrieving used products.While competition has been explored extensively from various angles, such as customer service perspective by Boyaci et al. (2004), Cournot competition among supply chains considering leader position and cost structure as investigated by Majumder et al. (2008), and the influence of competition on product pricing strategy by Wang et al. (2017), relatively limited attention has been given to competition among echelon utilization enterprises.Our research breaks new ground by introducing competitive conditions into the echelon utilization domain, shedding light on its impact on decision-making within the supply chain.
find that consumption subsidies can improve social welfare and replacement subsidies can promote environmental protection.Yi et al. (2021) also introduced a novel extended producer responsibility (EPR) system emphasizing resource conservation.They argue that a combined tax subsidy approach can optimize social welfare and foster ecological innovation.Bai et al. (2021) introduce a threestage Stackelberg game model to find the optimal allocation of the subsidy budget among multiple products covered by the trade-in program.Bai et al. (2019) asserted that government R&D subsidies play a significant role in promoting green innovation among energy-intensive enterprises, although Yu et al. (2016) reached the opposite conclusion.Chang et al. (2019) highlighted the role of the joint taxsubsidy mechanism in incentivizing ecological innovation by manufacturers.Dusanee et al. (2013) examine the effects of technology subsidies and output subsidies on social welfare.Wang et al. (2020) investigated the impact of government subsidies and altruistic preferences on decision-making in a low-carbon e-commerce closed-loop supply chain.Zhang et al. (2022b) considered mode selection and coordination in a low-carbon closed-loop supply chain under compound government subsidies from a long-term dynamic perspective.
The price of the th echelon utilization enterprise's echelon utilization product, =1, 2The wholesale price of used batteries from the third-party recyclerTo align the model with real-world dynamics and facilitate the generation of meaningful results, this paper incorporates the following assumptions in the model construction, drawing inspiration from the work ofTang et al. (2018).Assumption 1.In the model constructed in this paper, the new energy vehicle manufacturer is in the leading position, with the third-party recycler and echelon utilization enterprises being the followers.And the third-party recycler makes decisions before the echelon utilization enterprises(Zhang et al. 2022a;Zhao and Ma 2022).Assumption 2. All parties in the dynamic game are profit-maximizing and exhibit complete rationality while possessing full information(Wang et al. 2020;Zhang et al. 2022a).

Assumption 5 .
For simplicity of representation, our model uses a linear function to express market demand for new energy vehicles and echelon utilization(Wei and Zhao 2015;Feng et al. 2023; Li et al. 2023;Zhang et al. 2022a).=  −  *   1 =  −  *  1 +  *  2  2 =  −  *  2 +  *  1Assumption 6.To facilitate calculation and highlight the focus of our research, we assume that the third-party recycler chooses the reasonable recycling ratio and the unit recycling price according to their actual situation and experience.In other words,  and ℎ are exogenous.The third-party recycler also provides one uniformly wholesale price of used batteries for two echelon utilization enterprises.Simultaneously, both echelon utilization enterprises are privy to this pricing information.Consequently, the actual amount of recycling can thus be expressed by the equation(Zhang et al. 2022a;Zhao and Ma 2022).
2 − 2ℎ +  + 2ℎ −  + (− + ) 1 + (− + ) 2 + 2  − 2  − 2  + 2  ) 2 + 4( − ) 0 (2 − 2ℎ 2 +  2 + 2ℎ −  − 8 + 4 + (− + ) 1 + (− + ) 2 + 2 2   − 2  + 8  − 4  − 2 2   + 2  )) Future ResearchDue to the rapid development of new energy vehicles, there is great potential for the echelon utilization of waste batteries in China, with various policies being introduced to support this.Although many scholars have studied the competition among different members of the supply chain, few have focused on the competition related to the echelon utilization of waste batteries.Based on this observation, this paper establishes a closed-loop competitive duopoly supply chain consisting of one new energy vehicle manufacturer, one third-party recycler, and two echelon utilization enterprises.This paper makes a contribution to the literature on echelon utilization of retired batteries by drawing attention to closed-loop supply chains with competitive duopoly and taking the influence of the government subsidy into consideration.By comparing and analyzing the decisions and equilibrium outcomes of each member in the supply chain, we find that:

Table 2 . The Influence of Unit Cost on the Optimal Decision Variables
* changes with  1 and  2 ↘  * changes with  1 and  2 ↗ (Notes:↗ indicates the same direction change, ↘ indicates the reverse direction change)

Table 1 .
Symbols and Decision Variables

Table 2 .
The Influence of Unit Cost on the Optimal Decision Variables

Table 3 .
The Influence of Government Subsidy on the Optimal Decision Variables

Appendix. Proofs Proof of Proposition 1:
We first calculate the second-order partial derivatives of  1 and  2 to  1 and  2 respectively.Since are thus both concave functions of  1 and  2 .Let the first-order partial derivatives of  1 and  2 to  1 and  2 be zero, so we can then get  1 * and  2 * and substitute them into   .Since   is a concave function of .Let the first-order partial derivative of   to  be zero, and we can get  * , then substitute them into   .If we let the second-order partial derivatives of   to  and  respectively, then we can get the Hessian Corollary 3: We can easily get the first-order partial derivative of  1 .According to Assumption 4, it is easy to know that 4 2 −  2 > 0 and 3 − 10 < 0 .From this, we can prove that We can easily get the first-order partial derivative of   * to The two conclusions in Corollary 4 are proved similarly, so we prove just (1) here.Because