Unleashing the Power of Closed-Loop Supply Chains: A Stackelberg Game Analysis of Rare Earth Resources Recycling (2024)

1. Introduction

Rare earth elements exhibit exceptional optical, electrical, and magnetic properties, rendering them indispensable in the fabrication of various advanced materials such as permanent magnets, luminescent compounds, hydrogen storage materials, and high-temperature superconductors [1,2]. Owing to their paramount significance in the realm of high technology, rare earths have garnered the status of a strategic essential mineral in numerous nations worldwide. The mounting urgency of global warming has spurred a growing consensus among countries to align with the imperatives of the Paris Agreement actively, propelling the development of clean energy technologies and initiatives for carbon reduction. In consonance with this trajectory, China formally unveiled the strategic objectives of achieving “carbon peaking” and “carbon neutrality” in February 2021 [3]. As part of the effort to reduce carbon emissions, the burgeoning expansion of sectors such as new energy vehicles (NEVs), wind power generation, and high-efficiency industrial motors has led to an accelerated increase in the demand for and production of rare earth materials, including rare earth permanent magnets (REPMs). In particular, the rapid progress of NEVs has led to a steady escalation in the depletion of non-renewable rare earth resources over time. Projections indicate that the global NEV sector will consume approximately 100,000 tons of REPMs by 2032, accounting for approximately one-third of total REPM consumption [4]. Despite holding only 37% of the world’s reserves, since the 1990s, China has emerged as a major supplier of rare earth raw materials, contributing more than 80% of global supply, with a staggering share of over 90% for heavy rare earths [5]. Despite this, three decades of unregulated and excessive mining has led to the depletion of China’s rare earth reserves. This over-exploitation has resulted in the depletion of key mining regions, accompanied by environmental impacts such as soil erosion, land subsidence, and secondary geological incidents [6,7]. China has embarked on a multifaceted approach in response to the critical challenges posed by the depletion of reserves and environmental degradation caused by excessive rare earth mining. This strategy encompasses limiting mining volumes, curtailing mining licenses for companies, and imposing stringent regulations on rare earth mining areas [8,9]. Through years of concerted remediation efforts, China has achieved notable progress in the regulation of rare earth mining and smelting. However, this has also led to a substantial reduction in the production of rare earths, to the extent that it has become inadequate to fulfill domestic demand. A case in point is the year 2018, wherein China’s domestic demand for rare earth oxides (REO) reached 180,000 tons, while the allocated mining quota stood at merely 120,000 tons, resulting in a glaring supply–demand gap of 60,000 tons [10]. Concurrently, other rare earth-producing nations worldwide have encountered challenges in expanding their production capacities due to a combination of environmental concerns and market dynamics. For instance, the United States, in a bid to prevent radioactive waste contamination, has shuttered the Mountain Pass smelting, separation, and mining facilities. In contrast to other nations, China possesses a notably comprehensive rare earth industry chain, particularly excelling in the domains of mining, beneficiation, and smelting separation processes. These sectors have historically maintained a dominant position within the global market. Consequently, set against the backdrop of an upsurge in global demand for rare earths, the stringent oversight exercised by major rare earth-producing countries, notably China, over their mining and smelting operations is poised to exert an immense influence on the stability of the worldwide rare earth supply chain. Projections indicate that by the year 2030, China is anticipated to continue furnishing more than fifty percent of the globe’s praseodymium and neodymium elements. However, despite this anticipated contribution, the international community is still poised to grapple with a substantial demand shortfall of approximately 47,000 tons [11].

Given the challenges associated with expanding rare earth production, the recycling of rare earth resources emerges as a viable and effective strategy to mitigate the environmental pressures of mining and address the supply–demand imbalance within the rare earth market. When compared to conventional supply methods, rare earth recycling presents several distinct advantages. Primarily, establishing a rare earth mine can entail a protracted timeline of 8 to 15 years, with an additional 20 years required for full production [12], alongside a substantial investment totaling around USD 1 billion [13]. In contrast, the financial outlay necessary for establishing a production line dedicated to recycled rare earths is significantly lower, with a much shorter construction period. Secondly, the recycling of rare earths is inherently more environmentally friendly and emits fewer carbon emissions [14]. This attribute assumes pronounced importance within the global drive for carbon reduction. Lastly, rare earth ores often encompass a diverse array of rare earth elements (REEs), implying that attempts to enhance the supply of a specific REE through intensified mining could inadvertently lead to an oversupply of other REEs [15]. By contrast, rare earth recycling typically targets specific REEs, thereby circumventing the risk of oversupply while effectively addressing shortages. Various nations worldwide have recognized the crucial significance of rare earth recycling. The European Union, in particular, invested EUR 140 million in the SUSMAGPRO initiative to investigate sustainable recycling and reuse of rare earth permanent magnets (REPMs) [16]. As the preeminent global exporter of rare earths, China has similarly introduced a series of incentivizing policies to foster the growth of the rare earth recycling sector. A case in point is the “Catalogue of Preferential Value Added Tax for Products and Services for Comprehensive Utilization of Resources” published by the Ministry of Finance, which outlines a 30% instant refund on value-added tax (VAT) for comprehensive resource utilization encompassing rare earth products, processing waste, end-of-life rare earth products, and dismantled materials [17].

Rare earth recycling, encompassing various tiers of the supply chain and involving diverse stakeholders, manifests as a closed-loop supply process. Within this context, the Stackelberg game theory emerges as a highly pertinent approach for investigating the interplay among different stakeholders within a closed-loop supply chain. Given China’s comprehensive rare earth industry chain, this study adopts China as an illustrative example, constructing a closed-loop supply chain framework for the reverse supply of rare earths while accounting for the prevailing vertical integration present within China’s rare earth industry. Leveraging Stackelberg game theory, the research delves into the intricate dynamics characterizing the relationships among different entities in the supply chain. The primary objective of this study is to address the following inquiries: (1) How does the reverse supply of rare earths impact the downstream segment of the rare earth industry? (2) What is the influence of distinct recycling sources on rare earth groups and recycling entities individually? (3) In what manner do strategies adopted by governments and enterprises shape the closed-loop supply chain, and how can these strategies facilitate the achievement of “Pareto optimality” within the closed-loop supply chain framework?

2. Literature Review

2.1. Recycling of Rare Earths

Rare earth recycling draws from a diverse array of sources, with the most valuable source being rare earth permanent magnets (REPMs), owing to their high rare earth content and burgeoning utilization. Following closely are rare earth fluorescent materials. Extensive research has underscored the technical feasibility of recycling rare earths from waste materials. Notably, Sun et al. (2022) devised a process utilizing fluorinated wastewater to separate rare earths from ultrafine NdFeB waste, yielding nanoscale neodymium oxyfluoride [18]. Similarly, Chen et al. (2015), Maat et al. (2016), and Venkatesan et al. (2018) proposed an electrochemically assisted method for extracting REEs from waste NdFeB magnets, requiring only minimal quantities of sodium chloride and oxalic acid to yield high-purity REEs [19,20,21]. The economic value of rare earth recycling has also been a focal point of research. Ippolito et al. (2022) conducted a comparative assessment of two hydrometallurgical processes for recycling rare earths from fluorescent waste lamps, revealing positive economic potential for both methods [22]. Sprecher et al. (2014) evaluated the viability of recycling neodymium from discarded computer hard drives, highlighting the substantial potential for closed-loop REPM supply [23]. Jyothi et al. (2020) conducted a comprehensive review of various processes for recycling rare earth elements (REEs) from diverse sources alongside a life cycle assessment and policy considerations [24]. Their collective findings indicate that recycling REEs presents a significant avenue for generating value from waste. Schulze and Buchert (2016) utilized material flow analysis to demonstrate that end-of-life REPMs could potentially contribute up to 23% of the market supply [25]. Meanwhile, Machacek et al. (2015) emphasized that recycling REEs from REE phosphors and subsequent market supply could enhance the security of REE supply, serving as an autonomous source beyond China [26].

Nonetheless, an analysis encompassing the comprehensive economic implications of rare earth recycling through the lens of supply and demand, or market dynamics, has been relatively scarce in the literature. Jin et al. (2018) can be credited with pioneering this domain by employing operations research to delve into the profitability of recycling rare earths extracted from discarded computer hard drives [27]. Their study notably introduced consideration of uncertainties inherent in both supply and demand aspects, alongside proposing strategies for enhancing the overall profitability of reverse rare earth supply. Jin’s work represented a seminal contribution in terms of focusing on the closed-loop supply management of rare earths. However, their investigation omitted an examination of the market competition between reverse supply and forward supply, as well as the potential influence of government interventions and firm decisions on the dynamics of the supply chain. This research gap underscores the need for further studies that embrace a more holistic approach, taking into account these additional factors to provide a comprehensive understanding of the economic implications of rare earth recycling.

2.2. Research on Closed-Loop Supply Chain (CLSC)

The research landscape on closed-loop supply chains (CLSCs) has garnered substantial interest, with a major focus on areas such as recycling model selection, pricing decisions, and contract design. Notably, investigations into recycling models have proven pivotal. Savaskan et al. (2004) utilized game theory to compare CLSC models under distinct recycling approaches, demonstrating the potential for optimal profits and recycling rates when retailers engage in waste product recycling [28]. Building on this, Savaskan and Wassenhove (2006) examined the choice of recycling channels in the presence of competitive retailers [29]. Wang and Da (2010) delved into CLSC models involving retailer and third-party recycler recycling, analyzing how recycling efforts impact product pricing [30]. In terms of pricing decisions, Lu and Li (2016) crafted a CLSC model within a competitive environment for retailers, juxtaposing optimal prices and profits under two recycling paradigms [31]. Lu et al. (2018) extended this by establishing a CLSC model comprising a manufacturer, a retailer, and a third party, unraveling the implications of third-party economies of scale on supply chain channel structure [32]. Li et al. (2016) formulated a closed-loop supply chain model wherein manufacturers and third parties collaborated for optimal pricing strategies [33]. In the domain of contract design, Alamdar et al. (2018) devised a three-tiered CLSC incorporating a manufacturer, retailer, and recycler, proffering a coordination scheme to align decentralized decision-making with centralized decision-making outcomes [34]. In contrast, Xing et al. (2020) considered the scenario of two competing recyclers and proposed a coordination mechanism aligning decentralized decisions with centralized outcomes within a competitive context [35]. Additionally, Genc and De Giovanni (2018) contemplated consumer consumption and product returns, exploring the optimal rebate mechanism under both variable and fixed rebate models [36]. These various lines of inquiry illustrate the multifaceted nature of CLSC research, encompassing models, pricing dynamics, and contract arrangements to provide comprehensive insights into the complexities of managing closed-loop supply chains.

While there exists a substantial body of research on closed-loop supply chains (CLSCs), the current literature predominantly focuses on consumer-driven recycling scenarios. Furthermore, these studies commonly assume that the materials recovered and reverse-supplied in manufacturing are inherently different from the raw materials supplied in the forward direction. The context of rare earths introduces unique considerations that differentiate it from the conventional paradigm. Firstly, rare earth recycling is not solely reliant on end-of-life products sourced from consumers; it also encompasses oil sludge scrap generated during manufacturing. Secondly, both forward- and reverse-supplied substances are REOs; they are essentially identical. Consequently, this study is set apart from conventional research by constructing a CLSC framework that is tailored to the distinctive dynamics of rare earths.

In this distinct paradigm, the interplay between manufacturing-generated scrap and consumer-end products plays an important role. Additionally, the fact that recycled and raw materials are indistinguishable introduces intriguing considerations for logistics, pricing, and the overall supply chain structure. By addressing these unique attributes, this research endeavors to provide a comprehensive understanding of the intricacies inherent in the rare earth CLSC, thus contributing to a more nuanced comprehension of CLSCs.

3. Materials and Methods

The comprehensive rare earth industry chain and supply chain encompasses a range of activities, commencing with upstream processes like beneficiation, mining, and refining. Midstream activities involve smelting and separation to yield rare earth oxides (REOs), along with the production of diverse rare earth functional materials. The downstream phase is characterized by the creation of a variety of rare earth products, including rare earth motors and rare earth polishing powders. These products find application across diverse industries and sectors, such as NEVs, wind energy power generation, and optical glass [11]. Within this intricate web, processing residues laden with rare earth elements materialize in various forms, manifesting across upstream, midstream, and downstream segments of the rare earth supply chain. To illustrate, consider the example of NdFeB magnets; the production process of these magnets yields oil sludge scrap, constituting about 30% of the production volume. Remarkably, these wastes contain an equivalent amount of rare earth content as the final product itself [19]. This distinguishing feature sets rare earth recycling apart from conventional reverse supply investigations, which predominantly center on end-of-life products. Instead, rare earth recycling constitutes a dynamic process that spans both upstream and downstream realms of the rare earth industry. Therefore, the sources of recycling extend beyond end-of-life products to encompass the substantial rare earth waste generated at various stages of the rare earth supply chain.

After undergoing years of development, China has successfully established a more comprehensive and integrated rare earth industry chain and supply chain in comparison to other countries. This progress sets the stage for the current study, which aims to delve into the ramifications of government intervention and technological advancements within enterprises on the overarching advantages of rare earth recycling within the supply chain. In light of this objective, employing China—with its mature industry chain—as the contextual backdrop does not diminish the general applicability and significance of the study’s findings. On the contrary, China’s advanced rare earth industry framework serves as an ideal platform for investigating the intricate interplay between government policies, industrial innovation, and the broader benefits stemming from the recycling of rare earths.

Driven by imperatives surrounding resource conservation and the need for streamlined market oversight, China has consistently championed the integration of its rare earth industry. This strategic direction has been underpinned by the publication of a comprehensive array of policies aimed at facilitating such integration [8]. Presently, China has made substantial headway in the consolidation of rare earth mining, smelting, and material manufacturing enterprises. This concerted effort has culminated in the establishment of a well-structured strategic landscape comprising six prominent groups, among which feature entities such as China Rare Earth Group and China Northern Rare Earth Group. It is noteworthy that China’s commitment to industry-wide integration remains resolute, with a strategic vision to further advance integration within the rare earth sector. The overarching aspiration involves constructing a fully interconnected industrial chain for rare earths, with large-scale corporate groups serving as the central anchors [37]. In the context of rare earth industry integration, the processing of rare earths from mining to manufacturing rare earth materials can be viewed as the behavior of a single enterprise.

In summary, this study considers a CLSC consisting of a REG that integrates upstream and downstream businesses in the rare earth industry, as well as a rare earth recycler. The ensuing analysis is contingent upon the following underlying assumptions.

3.1. Underlying Assumptions

Assumption1.

In the market context, a REG is established to encompass the entirety of operations spanning from rare earth mining to the provisioning of rare earth materials. Concurrently, a dedicated recycler is present, tasked with the collection of oil sludge scrap sourced from the REG, as well as end-of-life products from consumers. This recycler undertakes the re-extraction of REOs from these materials, subsequently reselling them to the REG. Within this study, three distinct recycling scenarios are formulated based on varying recycling options pursued by the recycler: S1 (the recycler only recycles oil sludge scrap), S2 (the recycler only recycles end-of-life products), and S3 (the recycler recycles both end-of-life products and oil sludge scrap). The supply situation is shown in Figure 1.

Assumption2.

The market’s demand for rare earth materials is denoted as $D$ . This demand, quantified by the equation $D = N - ε P_{m}$ , is determined by three key factors: the selling price of rare earth materials ( $P_{m}$ ), the market volume ( $N$ ), and the price sensitivity coefficient ( $ε$ ).

Assumption3.

The recycler is equipped to engage in the recycling of both oil sludge scrap sourced from the REG and end-of-life products collected from consumers. Let $Q_{l}$ symbolize the quantity of recycled oil sludge scrap, influenced by the price denoted as $P_{l}$ for recycling such scrap, as well as the quantity A provided without charge. This relationship can be expressed as $Q_{l} = A + P_{l}$ . Similarly, the volume of end-of-life products reclaimed by the recycler is designated as $Q_{s},$ given by the equation $Q_{s} = A + P_{s}$ , where $P_{s}$ represents the price attributed to the recovery of end-of-life products. In the interest of simplification, we assume that the quantity obtained without charge for end-of-life products aligns with that for oil sludge scrap. Consequently, in cases where the recycler concurrently recycles both oil sludge scrap and end-of-life products, the aggregate quantity of recycling becomes $Q_{l + s} = Q_{l} + Q_{s} = 2 A + P_{l} + P_{s}$ . The preliminary processing costs incurred for waste stemming from different sources are not uniform. Let $C_{l}$ denote the preliminary processing cost for recycling oil sludge scrap, while $C_{s}$ signifies the processing cost for end-of-life products. Given the inherently greater complexity and additional steps involved in processing end-of-life products, we assume that $C_{l} < C_{s}$ , reflecting a disparity in these processing costs.

Assumption4.

The recycler plays a pivotal role in supplying REO to the REG. In this capacity, the recycler is tasked with extracting rare earth elements from the collected oil sludge scrap or end-of-life products. The rare earth recycling rate is denoted as $θ$ , reflecting the proportion of rare earth elements successfully reclaimed through recycling efforts. Notably, the recycler demonstrates a commitment to research and development (R&D) endeavors aimed at enhancing the rare earth recycling rate. The investment cost associated with these R&D initiatives adheres to a quadratic relationship expressed as $\frac{1}{2} m θ^{2}$ , where $m$ signifies the R&D cost coefficient ( $m > 0$ ).

Assumption5.

The production of a unit of rare earth material necessitates a consumption of α REOs. Consequently, the total REO consumption amounts to $α D$ . Within the REG, the mining and smelting division generates an appropriate quantity of REOs based on orders from the material manufacturing division. The REO production process encompasses various stages ranging from mining beneficiation to the ultimate separation and smelting of REOs. The cumulative cost of these processes, required to yield a unit of REO, is denoted as $C_{r s}$ . Given the prioritization of REO procurement from recyclers, the mining and smelting division within the REG is compelled to produce a quantity of REOs, represented as $Q_{r s} = α D - Q$ . In S1, the calculation is $Q_{r s} = α D - θ (A + P_{l})$ . In S2, it becomes $Q_{r s} = α D - θ (A + P_{s})$ . Lastly, for S3, the formulation is $Q_{r s} = α D - θ (2 A + P_{l} + P_{s})$ .

Assumption6.

Referring to the research conducted by Sun et al. (2022), the carbon emissions associated with producing a unit of REO by the Rare Earth Group are denoted as $R_{n}$ [38]. Conversely, the carbon emissions attributed to producing a unit of REO by a recycler are labeled as $R_{0}$ , where $R_{n} > R_{0}$ . Under the carbon trading framework, the REG and the recycler are allocated a predetermined count of free carbon credits, with $L_{1}$ designated for the REG and $L_{2}$ for the recycler, where $L_{1} > L_{2}$ . When these credits are depleted, the REG and recycler are obligated to acquire additional carbon credits from the carbon trading market at a price, $W$ .

Assumption7.

To foster increased rare earth recycling and production, the government initiates a subsidy mechanism. Specifically, for every unit of REO sold by the recycler, a subsidy amount denoted as $S$ is offered. This incentive not only curbs any potential hoarding of rare earth wastes to exploit the subsidy but also serves as an effective means to bolster the augmentation of the rare earth recycling rate.

All the parameters mentioned above can be found in Table 1.

3.2. Model Construction

3.2.1. Recycler Only Recycles Oil Sludge Scrap (S1)

In S1, the REG maximizes its own profit by determining the selling price $P_{m}$ for rare earth materials and the purchase price $P_{r}$ for rare earths from recyclers. The recycler maximizes its own profit by determining the recycling price $P_{l}$ for the oil sludge scrap. The revenue functions of the rare earth group and the recycler are:

$\prod_{G}^{1} (P_{m}, P_{r}) = (P_{m} - C_{m}) D + P_{l} Q_{l} + W (L_{1} - R_{n} Q_{r s}) - C_{r s} Q_{r s} - P_{r} θ Q_{l}$

(1)

$\prod_{R}^{1} (P_{l}) = P_{r} θ Q_{l} - P_{l} Q_{l} + W (L_{2} - R_{0} Q_{r s}) + S θ Q_{l} - C_{l} Q_{l} - \frac{1}{2} m θ^{2}$

(2)

Since the REG holds a dominant position in the market while the recycler is in a follower role, determining the equilibrium solution involves employing the inverse induction approach. The process commences with taking the derivative of the recycler’s profit function (Equation (2)) and setting it equal to 0, yielding the following expression:

$P_{l} = \frac{θ P_{r} - C_{l} + θ S - θ W R_{0} - A}{2}$

(3)

Taking Equation (3) into Equation (1), we get:

$\begin{array}{l} \prod_{G}^{1} (P_{m}, P_{r}) = (P_{m} - C_{m}) \times (N - ε P_{m}) + \frac{θ P_{r} - C_{l} + θ S - θ W R_{0} - A}{2} \times \frac{θ P_{r} - C_{l} + θ S - θ W R_{0} + A}{2} + W (L_{1} - R_{n} \times (α (N - ε P_{m}) - \\ θ \times \frac{θ P_{r} - C_{l} + θ S - θ W R_{0} + A}{2})) - C_{r s} \times (α (N - ε P_{m}) - θ \times \frac{θ P_{r} - C_{l} + θ S - θ W R_{0} + A}{2}) \end{array}$

(4)

The Hessian matrix of Equation (4) with respect to $P_{m}$ and $P_{r}$ is:

$[\begin{matrix} - 2 ε & 0 \\ 0 & - \frac{θ^{2}}{2} \end{matrix}]$

(5)

The eigenvalues of this matrix are $- 2 ε (< 0)$ and $- \frac{θ^{2}}{2} (< 0)$ . As both eigenvalues are negative, the Hessian matrix qualifies as a negative definite matrix. Consequently, Equation (4) is established as a concave function concerning variables $P_{m}$ and $P_{r}$ . This property ensures the presence of a maximum value within this function.

By solving for the partial derivatives of Equation (4) with respect to $P_{m}$ and $P_{r}$ and setting these derivatives equal to 0, the maximum values for $P_{m}$ and $P_{r}$ can be determined:

$P_{m}^{1} = \frac{N + ε {α C}_{r s} + ε C_{m} + ε α W R_{n}}{2 ε}$

(6)

$P_{r}^{1} = \frac{θ C_{r s} - A + θ W R_{n}}{θ}$

(7)

Taking Equation (7) into Equation (3), the maximum value for $P_{l}$ can be determined:

$P_{l}^{1} = \frac{θ C_{r s} - C_{l} + θ S + θ W (R_{n} - R_{0}) - 2 A}{2}$

(8)

3.2.2. Recycler Only Recycles End-of-Life Products (S2)

In S2, the REG maximizes its own profit by determining the selling price $P_{m}$ for rare earth materials and the purchase price $P_{r}$ for rare earths from recyclers. The recycler maximizes its own profit by determining the recycling price $P_{s}$ for end-of-life products. The revenue functions of the REG and the recycler are (the calculation process is detailed in Appendix A):

$\prod_{G}^{2} (P_{m}, P_{r}) = (P_{m} - C_{m}) D + W (L_{1} - R_{n} Q_{r s}) - C_{r s} Q_{r s} - P_{r} θ Q_{s}$

(9)

$\prod_{R}^{2} (P_{s}) = P_{r} θ Q_{s} - P_{s} Q_{s} + W (L_{2} - R_{0} Q_{r s}) + S θ Q_{s} - C_{s} Q_{s} - \frac{1}{2} m θ^{2}$

(10)

Using the same approach, $P_{m}^{2}$ , $P_{r}^{2}$ and $P_{s}^{2}$ can be obtained as:

$P_{m}^{2} = \frac{N + ε {α C}_{s} + ε C_{m} + ε α W R_{n}}{2 ε}$

(11)

$P_{r}^{2} = \frac{C_{s} + θ C_{r s} - A + θ W (R_{n} + R_{0}) - θ S}{2 θ}$

(12)

$P_{s}^{2} = \frac{θ C_{r s} - C_{s} - 3 A + θ S + θ W (R_{n} - R_{0})}{4}$

(13)

3.2.3. Recycler Recycles Both End-of-Life Products and Oil Sludge Scrap (S3)

In S3, the REG maximizes its profit by determining the selling price $P_{m}$ for rare earth materials and the purchase price $P_{r}$ for rare earths from recyclers. The recycler maximizes its profit by determining the recycling price $P_{l}$ for the oil sludge scrap and the recycling price $P_{s}$ for end-of-life products. The revenue functions of the REG and the recycler are (the calculation process is detailed in Appendix A):

$\prod_{G}^{3} (P_{m}, P_{r}) = (P_{m} - C_{m}) D + Q_{l} P_{l} + W (L_{1} - R_{n} Q_{r s}) - C_{r s} Q_{r s} - P_{r} θ Q_{l + s}$

(14)

$\prod_{R}^{3} (P_{l}, P_{s}) = P_{r} θ Q_{l + s} - P_{l} Q_{l} - P_{s} Q_{s} + W (L - R_{0} Q_{r s}) + S θ Q - C_{l} Q_{l} - C_{s} Q_{s} - \frac{1}{2} m θ^{2}$

(15)

Using the same approach, $P_{m}^{3}$ , $P_{r}^{3}$ and $P_{s}^{3}$ can be obtained as:

$P_{m}^{3} = \frac{N + ε {α C}_{s} + ε C_{m} + ε α w R_{n}}{2 ε}$

(16)

$P_{r}^{3} = \frac{C_{s} - 2 A + 2 θ C_{r s} - θ S + θ W {(R}_{0} + 2 R_{n})}{3 θ}$

(17)

$P_{l}^{3} = \frac{C_{s} - 3 C_{l} - 5 A + 2 θ (C_{r s} + S + W (R_{n} - R_{0}))}{6}$

(18)

$P_{s}^{3} = \frac{θ C_{r s} - C_{s} + θ (S + W (R_{n} - R_{0}))}{3} - \frac{5 A}{6}$

(19)

4. Results

4.1. Theoretical Analysis

Proposition1.

$P_{m}^{1} = P_{m}^{2} = P_{m}^{3}$ , $D^{1} = D^{2} = D^{3}$ .

Proposition 1 shows that the optimal selling prices of rare earth materials are equal across the three recycling scenarios, and the market demand for rare earth materials remains consistent. This demonstrates that the recycling of rare earths does not impact the positive supply market for rare earth materials.

Proposition2.

$P_{l}^{1} > P_{l}^{3}$ , $P_{s}^{2} < P_{s}^{3}$ .

Proof of Proposition2.

$P_{l}^{1} - P_{l}^{3} = \frac{{θ C}_{r s} - C_{s} - A + θ S + θ W (R_{n} - R_{0})}{6} > 0$ , $P_{s}^{2} - P_{s}^{3} = - \frac{θ C_{r s} - C_{s} - A + θ S + θ W (R_{n} - R_{0})}{12} < 0$ . □

Under S3, the recycler reduces the recycling price for oil sludge scrap while increasing the recycling price for end-of-life products. The proposition suggests that the recycler does not merely engage in a straightforward addition of recycling quantities from both sources; instead, it actively adjusts the recycling prices and quantities to maintain a balanced overall recycling strategy.

Proposition3.

$\frac{\partial P_{l}^{1}}{\partial w} = \frac{\partial Q_{l}^{1}}{\partial w} = \frac{θ (R_{n} - R_{0})}{2} > \frac{\partial P_{s}^{2}}{\partial w} = \frac{\partial Q_{s}^{2}}{\partial w} = \frac{θ (R_{n} - R_{0})}{4} > 0$ .

An increase in carbon trading price ( $W$ ) positively affects rare earth recycling activities. The increase in recycling price for oil sludge scrap due to $W$ is larger than the increase for end-of-life products. This suggests that the recycling of oil sludge scrap is more sensitive to changes in carbon trading prices.

Proposition4.

$\frac{\partial P_{l}^{3}}{\partial C_{s}} = \frac{1}{6} > 0$ , $\frac{\partial P_{s}^{3}}{\partial C_{s}} = - \frac{1}{3} < 0$ , $\frac{\partial P_{l}^{3}}{\partial C_{l}} = - \frac{1}{2} < 0$ , $\frac{\partial P_{s}^{3}}{\partial C_{l}} = 0$ , $\frac{\partial (P_{l}^{3} + P_{s}^{3})}{\partial C_{l}} < \frac{\partial (P_{l}^{3} + P_{s}^{3})}{\partial C_{s}} < 0$ .

Under S3, a distinct response is observed in the recycler’s behavior. Specifically, an increase in the processing cost of end-of-life products ( $C_{s}$ ) leads to a reduction in the recycling price ( $P_{s}^{3}$ ) for end-of-life products, which in turn results in an increased quantity of oil sludge scrap being recycled. This can be interpreted as the recycler strategically adjusting its recycling approach to balance the impact of higher processing costs. However, when the processing cost of oil sludge scrap $(C_{l}$ ) increases, the recycler reacts by decreasing the recycling quantity of oil sludge scrap without a corresponding increase in the recycling price of end-of-life products. This suggests that when $C_{l}$ increases, the recycler is more inclined to prioritize reducing the recycling quantity rather than significantly altering the pricing of end-of-life products. It is important to note that while an increase in either $C_{l}$ or $C_{s}$ might initially lead to adjustments in the recycling quantities and prices, these changes eventually contribute to a decrease in the total quantity recycled by the recycler.

Proposition5.

$\frac{\partial P_{r}^{1}}{\partial C_{r s}} = 1 > \frac{\partial P_{r}^{3}}{\partial C_{r s}} = \frac{2}{3} > \frac{\partial P_{r}^{2}}{\partial C_{r s}} = \frac{1}{2} > 0$ .

An increase in the cost of rare earth mining raises the purchase price of recycled rare earths in all three scenarios. The greatest increase in purchase price occurs in S1, followed by S3 and then S2. This indicates that higher mining costs prompt REG to seek more affordable recycled rare earths.

4.2. Numerical Simulation Analysis

The term “rare earth” collectively encompasses a series of elements with diverse applications across various sectors. These elements not only exhibit different market demands but also vary significantly in market prices. Accordingly, this study concentrates primarily on REPM (rare earth permanent magnet) market dynamics, incorporating supplementary criteria to establish precise parameter values.

According to empirical market data, the current market price for REPMs stands at approximately CNY 400,000 per ton. As of 2023, the global demand for REPMs has surged to nearly 350,000 tons, as reported by China Securities in 2022. Consequently, with $P_{m}$ fixed at 400,000 and $D$ at 350,000, the resultant calculations yield $N$ as 800,421 and $ε$ as 1.12.

It is noteworthy that the composition of REOs within REPMs constitutes approximately 30% to 40% of the aggregate content. Thus, the conversion ratio between REPMs and REOs is designated as $α = 0.4$ , signifying that 0.4 tons of REOs are consumed per ton of REPMs produced. According to data sourced from China Nonferrous Metals Network, the mining expenses for ionic rare earths in southern China are estimated at CNY 600,000 per ton [38]. Meanwhile, the mining costs for light rare earths in northern regions are anticipated to be comparatively lower, thereby establishing an average mining cost for rare earths at CNY 500,000 per ton within this study. Factoring in the conversion relationship between rare earth ore and REOs, the production cost of REOs computes to CNY 1,000,000 per ton. Notably, it is pivotal to allocate costs when performing calculations since this study employs the REPM market demand and pricing as proxies for the broader rare earth material market. Specifically, the requisite REEs for REPMs encompass praseodymium and neodymium, constituting 20% of the REO composition. Accordingly, the positive supply cost of REO ( $C_{r s}$ ), is determined to be CNY 200,000 per ton.

Turning to the context of carbon emissions, this study adopts the insights from Sun et al. (2022) [39]. Herein, the carbon emission price ( $W$ ) is set at CNY 646 per ton. Concurrently, the carbon emissions associated with traditional REO production methods are represented by the unit carbon emission $R_{n}$ , which stands at 3. Furthermore, accounting for carbon mitigation strategies, a carbon credit volume of $L_{1}$ is established at 250,000 tons, remaining beneath the requisite carbon emissions of the designated REG. This strategic configuration aligns with China’s policy to curtail environmentally intensive enterprises and foster emission reduction through carbon trading mechanisms. Finally, it is posited that the recycler’s carbon emission intensity $R_{0}$ is pegged at 1.5, equating to half of $R_{n}$ , accompanied by a carbon credit allocation of $L_{2}$ , set at 100,000 tons, constituting 0.4 times the magnitude of $L_{1}$ .

The specific parameter settings are shown in Table 2.

4.2.1. Analysis of the Effects of Recycler R&D ( $θ$ ) on the Supply Chain

This section delves into a comprehensive analysis of the influence of the rare earth recycling rate, denoted as $θ,$ on various critical factors, such as the purchase price of recycled rare earths ( $P_{r}$ ), recycling prices for oil sludge scrap ( $P_{l}$ ), and end-of-life products ( $P_{s}$ ), as well as the quantity of REOs produced by REG ( $Q_{r s}$ ). Given the diverse composition of rare earth materials, the scope of the recycling rate $θ$ is confined to the realistic range of $0.1 \leq θ \leq 0.5$ in order to ensure the fidelity of the simulations. The ensuing outcomes are depicted in Figure 2, Figure 3 and Figure 4.

As illustrated in Figure 2, when operating under S1, the $P_{r}$ displays an ascending trend in correspondence with the escalation of the rare earth recycling rate ( $θ$ ). Conversely, under S2 and S3, $P_{r}$ registers a diminishing pattern as $θ$ augments. The dynamic observed in S1 is indicative of a symbiotic relationship between the REG and the recycler, yielding an evident cooperative synergy. This indicates that in S1, an augmentation in $θ$ induces a swift elevation in profitability for both REG and recycler. In contrast, S2 and S3 are characterized by the emergence of a greater influx of recycled rare earths due to the higher $θ$ , causing $P_{r}$ to follow the established principles of market dynamics and, consequently, exhibit a decline.

Figure 3 provides insights into the behavior of recycling prices ( $P_{l}$ and $P_{s}$ ) across all three scenarios. The $P_{l}^{1}$ outpaces the $P_{s}^{2}$ , underscoring the substantial discrepancy between the two sources. Notably, the surge in $θ$ precipitates an overall rise in both $P_{l}$ and $P_{s}$ across all scenarios. Moreover, the trajectory of $P_{l}^{1}$ exhibits a steeper incline compared to $P_{s}^{2}$ . In S3, an intriguing phenomenon emerges, wherein $P_{l}^{3}$ trails behind $P_{l}^{1}$ while $P_{s}^{3}$ surpasses $P_{s}^{2}$ . This indicates that the recycler, when engaged in a combined recycling approach, allocates focus towards amplifying the recycling of end-of-life products while reducing the recycling of oil sludge scrap. The holistic understanding drawn from Figure 4 unveils that S3 witnesses the lowest quantity of REOs in the forward supply ( $Q_{r s}$ ), signifying the recycler’s heightened engagement in rare earth recycling. Following suit, S2 claims the second lowest position, and S3 claims the lowest position. Despite S3’s curtailed emphasis on oil sludge scrap recycling, the augmented recycling of end-of-life products bolsters the aggregate recycling output, rendering it superior to both S1 and S2.

4.2.2. Comparative Analysis of the Effects of Government Intervention ( $S$ and $W$ ) on the Supply Chain

In the context of a market economy, governmental influence over the rare earth recycling sector can manifest through market mechanisms or direct intervention. This article explores the efficacy of employing a carbon price mechanism and direct government subsidies as regulatory tools within the rare earth recycling market. Analytical findings are presented in Figure 5, Figure 6 and Figure 7.

Figure 5 illustrates the contrasting impacts of government subsidies and carbon trading prices on the purchase price ( $P_{r}$ ) of recycled rare earths. Notably, an increase in government subsidy ( $S$ ) leads to a decrease in $P_{r}$ within scenarios S2 and S3. Concurrently, the data presented in Figure 5a reveals that an augmented government subsidy results in elevated recycling prices for recyclers across all three scenarios. This, in turn, leads to increased quantities of rare earths being recycled. This adherence to conventional market dynamics, wherein higher recycling quantities due to government subsidies typically depress the marketing price of recycled rare earths, is not observed in S1. The rationale behind this phenomenon could be analogous to what was depicted in Figure 2 of Section 4.2.1. In S1, the purchase price ( $P_{r}^{1}$ ) rises with the quantity of oil sludge scrap recycled. However, the increase in $P_{r}^{1}$ in this section is counteracted by the decrease resulting from the influence of variable $S$ .

Figure 6 underscores the relationship between both government subsidies and carbon trading prices with recycling prices in all three scenarios. Specifically, Figure 6a reveals that government subsidies exert the most pronounced influence on $P_{l}^{1}$ when recyclers exclusively handle oil sludge scrap. The effectiveness of government subsidies diminishes when the recycler is engaged in a combined recycling approach (S3), with the lowest impact observed in scenarios involving only end-of-life product recycling (S2). Figure 6b highlights that while an increase in carbon trading prices leads to higher recycling prices across all scenarios, the magnitude of this effect remains relatively moderate.

In the context of conventional rare earth production, as depicted in Figure 7, heightened government subsidies ( $S$ ) and carbon trading prices ( $W$ ) result in reduced conventional rare earth production across all scenarios. Notably, the influence of government subsidies on conventional rare earth production is markedly more pronounced than that of carbon trading prices. Reduced conventional rare earth production signifies an increased reliance on recycled rare earths by rare earth groups and incentivizes recyclers to engage in greater recycling activities. However, it is crucial to note that the limited impact of carbon trading prices on the purchase price does not significantly stimulate recyclers to increase their recycling efforts.

4.2.3. Comparative Analysis of Supply Chain Profits

As rational economic entities operating within the marketplace, enterprises conduct their transactions with the overarching aim of optimizing their profits. In this section, we meticulously dissect and compare the profit dynamics of both the REG and the recycler across the three distinct scenarios. The results are shown in Figure 8, Figure 9 and Figure 10.

Figure 8 illustrates the profit trends of both the REG and the recycler as a function of the rare earth recycling rate ( $θ$ ) across all three scenarios. Notably, profits for both entities increase as $θ$ rises. In the case of the REG, S3 yields the highest profit due to its combination of robust recycled rare earth production and a moderate purchase price. Conversely, for the recycler, S1 emerges as the most profitable. This intriguing divergence suggests that while S3 results in the highest sales volume of recycled rare earths for the recycler, it does not translate into higher profits, primarily due to the lower purchase price ( $P_{r}$ ) in S3. Notably, the difference in profitability between Scenarios 1 and 3 is relatively modest, especially when $θ < 0.25$ , where the profits in both scenarios converge. Further analysis from Section 4.2.1 underscores that $P_{r}^{1}$ increases with $θ$ , while $P_{r}^{2}$ decreases with $θ$ . Consequently, an increase in $θ$ leads to a faster rise in the recycler’s profit in S1 compared to S2.

Figure 9 showcases the profit trends of both the REG and the recycler in response to variations in government subsidy levels across all three scenarios. Despite the government exclusively subsidizing the recycler, the ripple effect along the supply chain results in increased profits for the REG, as well. Notably, the REG registers its highest profit in S3, experiencing the most pronounced impact from government subsidies. On the other hand, the recycler attains its highest profit in S1 and witnesses the strongest influence of government subsidies. Moreover, the disparity between the recycler’s profits in S1 and S3 is relatively modest.

Figure 10 sheds light on the contrasting effects of increasing carbon trading prices on the REG and the recycler within all three scenarios. Across the board, a rise in carbon trading prices results in decreased profits for the REG. Conversely, this same increase in carbon trading prices leads to higher profits for the recycler across all scenarios. This underscores the recycler’s status as the beneficiary of elevated carbon trading prices.

5. Discussion

The reverse supply of rare earth resources represents a pivotal avenue for diversifying the sources of rare earth supply, thereby bolstering supply stability and sufficiency. This study constructs a second-level CLSC model from a supply chain management perspective, taking into account the intricate dynamics of the rare earth industry. We investigate the impact of enterprises’ R&D efforts and government intervention on the reverse supply of rare earths.

Our findings reveal notable trends in profitability for both the REG and the recycler across different scenarios. Specifically, for the REG, the highest profit consistently occurs under S3, while for the recycler, S1 consistently yields the highest profit. Conversely, S2 consistently generates the least profit for both the REG and the recycler. Interestingly, despite the larger recycling quantity in S3, it does not translate into the highest profit for the recycler. This discrepancy arises because as recycling quantities expand, the recycler faces higher recycling costs and must sell recycled rare earths at lower prices. Consequently, blindly expanding production capacity can erode the recycler’s profitability.

To facilitate a direct comparison between S1 and S2, we simplified the model by ignoring other factors influencing recycling quantities, focusing solely on recycling prices and processing costs. Compared with the actual situation, we reduce the difficulty of recycling end-of-life products. However, S2 still does not exhibit advantages over S1, both in terms of profitability and recycling volume. In S1, where the recycler focuses on sludge scrap recycling, the REG benefits not only from cost savings through the substitution of conventional rare earths with recycled ones but also from the sale of the sludge scrap. Consequently, the REG in S1 is inclined to offer a higher purchase price ( $P_{r}^{1}$ ) to the recycler for recycled rare earths, even when higher recycling rates ( $θ$ ) and government subsidies (S) lead to decreased $P_{r}^{2}$ and $P_{r}^{3}$ . This collaborative recycling dynamic in S1 fosters a win-win partnership between the REG and the recycler.

Regarding the influence of government subsidies, our analysis demonstrates that government subsidies not only boost the quantity of rare earth recycling and enhance recycling profitability but also positively impact REG profits. Notably, the essence of the REG’s profit increase lies in the reduction of $P_{r}$ due to government subsidies. In essence, the REG, with pricing power, can “plunder” a portion of the recycler’s profit gain from subsidies by lowering the purchase price when it observes government support for the recycler. This phenomenon is akin to previous research findings, such as Zhao et al. (2020), which highlighted that government subsidies to third-party recyclers often result in profit transfers to manufacturers [40]. This implies that government subsidies may have a more limited impact than anticipated in incentivizing recyclers to increase their recycling activities.

Carbon trading, as a tool for government market guidance, also stimulates rare earth recycling. However, carbon trading operates through distinct mechanisms compared to government subsidies. Firstly, carbon trading price increases lead to rising purchase prices ( $P_{r}$ ) in all three scenarios, indicating that the carbon trading mechanism encourages REGs to purchase more recycled rare earths to reduce carbon emissions costs. Secondly, unlike REGs, the recycler possesses surplus carbon credits and can benefit from carbon trading in the carbon market. Consequently, green enterprises like recyclers are the beneficiaries of carbon trading, as supported by research by Xia et al. (2020), highlighting the positive correlation between carbon trading prices and the profits of low-carbon manufacturers [41]. Carbon trading can effectively incentivize enterprises to engage in green innovation activities [42], contributing to the development of the rare earth recycling market. However, it is important to note that carbon trading prices are influenced by market dynamics and are not solely government-controlled, limiting their influence on rare earth recycling quantities. Thus, carbon trading may primarily serve as a means for recyclers to ensure basic profitability and survival.

In conclusion, while government subsidies and carbon trading represent valuable tools for government intervention in rare earth recycling, their impact on increasing reverse supply is not as significant as expected. An increase in the rare earth recycling rate ( $θ$ ) stands out as the most potent driver, significantly augmenting reverse rare earth supply and profitability for both the REG and the recycler. Notably, for the recycler, the effect of increasing $θ$ far surpasses the cumulative impact of government subsidies and carbon trading. These findings align with the study of Rademaker et al. (2013), who emphasized the critical importance of developing recycling technology and infrastructure for future rare earth recycling efforts [43]. Furthermore, Jiang and Xu (2023) underscore the importance of technological innovation over government subsidies in enhancing enterprise efficiency [44].

6. Conclusions and Recommendations

6.1. Conclusions

This study has established a two-level closed-loop supply chain comprising the REG and the recycler, employing the Stackelberg game theory to determine equilibrium solutions for both supply sides. It has comprehensively analyzed the impact of enterprise R&D and government intervention on the rare earth closed-loop supply chain, leading to the following conclusions.

(1): Recycling scenarios and reverse REO supply do not exert an influence on downstream rare earth material prices or market demand. The downstream market remains relatively stable in this context.
(2): In S1, there is a mutually beneficial synergy between the REG and the recycler, with the recycler realizing the highest profit. Although S3 has the highest production of recycled rare earths, the recycler’s profit is dwarfed by that of S1. This underscores that maximizing production capacity does not necessarily result in the highest profit, emphasizing the need for the recycler to carefully evaluate production capacity in line with market dynamics. The REG achieves the highest profit in S3, followed by S1, and the lowest in S2. This result is due to the lower cost of recycling rare earths compared to conventional rare earths, resulting in higher profits for the REG as the reverse supply of rare earths increases.
(3): Government subsidy and carbon trading, as two forms of government intervention, both yield positive effects on rare earth recycling. However, they operate through distinct mechanisms and have different impacts. Government subsidies directly increase the profits of recyclers, but some of the increased profits will be “plundered” by REGs. In contrast, carbon trading aims to mitigate profit losses for REGs by incentivizing them to voluntarily purchase more recycled rare earths to reduce carbon emissions costs. This, in turn, increases the profitability of recyclers. Nonetheless, it is important to note that government subsidies and carbon trading have limited capacity to drive rare earth recycling and primarily serve as a means for recyclers to maintain basic profitability and survival.
(4): An increase in the rare earth recycling rate significantly boosts the production of recycled rare earths, with significant positive effects on the profits of both REGs and recyclers. Consequently, compared to government policy tools, R&D and technological innovation initiatives undertaken by enterprises emerge as the primary drivers for augmenting the quantity of recycled rare earths and enhancing the overall profitability of the rare earth closed-loop supply chain. This highlights the pivotal role of technological advancements in driving sustainability and efficiency within rare earth recycling.

This study makes contributions to both academia and management practice. Academically, it fully considers the technical characteristics of rare earth recycling and proposes a CLSC model tailored for the rare earth industry. Additionally, based on the characteristics of rare earth recycling sources, the proposed model considers the recovery of rare earths from both end-of-life products and oil sludge scrap. Previous research has often focused solely on the recovery from end-of-life products, neglecting oil sludge scrap (or industrial waste) as a source of recovery. From a managerial perspective, the conclusions drawn from this study assist recyclers in adjusting their production and operational strategies, as well as help the government refine their market intervention strategies. Based on market forecasts, recyclers can better select recovery channels and determine R&D investments, while the government can adjust subsidy and carbon trading policies more effectively to promote the rare earth recovery market efficiently.

6.2. Recommendations

Based on the above conclusions, we propose the following recommendations.

(1): Improve the multiple market guidance mechanism. While carbon trading and government subsidies have positively impacted the rare earth recycling market, our findings suggest that current models fall short of fully driving the sector. To address this, the government should refine these mechanisms. For carbon trading, policy reforms tailored to the unique dynamics of rare earth recycling are essential. This includes revising existing models and ensuring transparency and fairness in the carbon trading market to better stimulate rare earth recycling. In the case of government subsidies, it is crucial to define clear objectives and adopt more precise models specific to rare earth recycling. This will prevent undue profit transfers and optimize subsidy efficiency.
(2): While the recycling of oil sludge scraps currently dominates the recycling market, the future holds immense potential for the end-of-life product recycling market. As the number of discarded appliances and new energy vehicles continues to rise, this segment presents significant opportunities. This study indicates that once a certain threshold of total recycling quantities is reached, recyclers tend to shift their focus towards increasing the recovery of end-of-life products rather than oil sludge scraps. To further promote this transition and optimize the recycling market, the integration of rare earth enterprises becomes essential. Integrating recyclers into the REGs can facilitate greater emphasis on recycling oil sludge scraps, helping to reach the critical threshold in total recycling quantities. Subsequently, this will stimulate the recycling market for end-of-life products, fostering a more comprehensive and sustainable rare earth recycling ecosystem.
(3): Enhancing the rare earth recycling rate emerges as a highly effective strategy for augmenting recyclers’ profitability. Instead of passively awaiting policy and market changes, recyclers should take a proactive stance by intensifying their R&D efforts. Actively pursuing technological innovation is crucial for elevating the rare earth recycling rate, reducing recycling costs, and ultimately enhancing their competitiveness in the market.

The actual market operation will be influenced by more complex factors. The three recovery scenarios considered in this paper may not fully reflect the intricate realities of the rare earth recovery market. For example, competition among recyclers is not accounted for in this study. Additionally, the policy tools available to governments in practice are more varied. Therefore, considering more complex market scenarios and a broader range of government intervention policies is a direction that future research needs to explore further.

Author Contributions

Conceptualization, methodology, and review and editing, X.W.; Experiment construction, method implementation, software, and writing—original draft, C.L.; result calibration, Y.Z.; Investigation and data curation, H.L. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Humanities and Social Sciences Foundation of Jiangxi province (NO. JC21123), and the Open Fund of Key Laboratory of Ionic Rare Earth Resources and Environment, Ministry of Natural Resources (NO. 2023IRERE402). The authors would like to thank the Editor and all anonymous reviewers for their insightful comments, which greatly helped to improve the presentation of the whole paper.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

All data in this article can be obtained by reasonably contacting the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

NEVs	new energy vehicles
REPMs	rare earth permanent magnets
REO	rare earth oxides
REE	rare earth element
CLSC	closed-loop supply chain

Appendix A

Appendix A.1. Recycler Only Recycles End-of-Life Products—S2

$\prod_{G}^{2} (P_{m}, P_{r}) = (P_{m} - C_{m}) D + W (L_{1} - R_{n} Q_{r s}) - C_{r s} Q_{r s} - P_{r} θ Q_{s}$

(A1)

$\prod_{R}^{2} (P_{s}) = P_{r} θ Q_{s} - P_{s} Q_{s} + w (L_{2} - R_{0} Q_{r s}) + S θ Q_{s} - C_{s} Q_{s} - \frac{1}{2} m θ^{2}$

(A2)

Use the inverse induction method by first taking the derivative of the recycler’s profit (Equation (A2)) and letting the derivative equal to 0. We can get:

$P_{s} = \frac{θ P_{r} - C_{s} + θ S - θ W R_{0} - A}{2}$

(A3)

Taking Equation (A3) into Equation (A1) we get:

$\prod_{G}^{2} (P_{m}, P_{r}) = (P_{m} - C_{m}) \times (N - ε P_{m}) + W (L_{1} - R_{n} (α (N - ε P_{m}) - θ \times \frac{θ P_{r} - C_{s} + θ S - θ W R_{0} + A}{2})) - C_{r s} \times (α (N - ε P_{m}) - θ \times \frac{θ P_{r} - C_{s} + θ S - θ W R_{0} + A}{2})$

(A4)

The Hessian matrix of Equation (A4) with respect to $P_{m}$ and $P_{r}$ is:

$[\begin{matrix} - 2 ε & 0 \\ 0 & - \frac{θ^{2}}{2} \end{matrix}]$

(A5)

Since the two eigenvalues of the matrix are $- 2 ε < 0$ and $- \frac{θ^{2}}{2} < 0$ , this Hessian matrix is a negative definite matrix. Therefore, Equation (A4) is a concave function with respect to $P_{m}$ and $P_{r}$ and there exists a maximum value.

Taking the partial derivatives of Equation (A4) for $P_{m}$ and $P_{r}$ , we get:

$\frac{\partial \prod_{G}^{2} (P_{m}, P_{r})}{\partial P_{m}} = N + ε α C_{r s} + ε C_{m} - 2 ε P_{m} + ε α W R_{n}$

(A6)

$\frac{\partial \prod_{G}^{2} (P_{m}, P_{r})}{\partial P_{r}} = \frac{θ (C_{s} - A + {θ C}_{r s} - 2 θ P_{r} - θ S + θ W (R_{n} + R_{0}))}{2}$

(A7)

letting the partial derivatives equal 0, we get:

$P_{m}^{2} = \frac{N + ε {α C}_{r s} + ε C_{m} + ε α W R_{n}}{2 ε}$

(A8)

$P_{r}^{2} = \frac{C_{s} + θ C_{r s} - A + θ W (R_{n} + R_{0}) - θ S}{2 θ}$

(A9)

Taking Equation (A9) into Equation (A3) we get:

$P_{s}^{2} = \frac{θ C_{r s} - C_{s} - 3 A + θ S + θ W (R_{n} - R_{0})}{4}$

(A10)

Appendix A.2. Recycler Recycles Both End-of-Life Products and Oil Sludge Scrap—S3

$\prod_{G}^{3} (P_{m}, P_{r}) = (P_{m} - C_{m}) D + Q_{l} P_{l} + W (L_{1} - R_{n} Q_{r s}) - C_{r s} Q_{r s} - P_{r} θ Q_{l + s}$

(A11)

$\prod_{R}^{3} (P_{l}, P_{s}) = P_{r} θ Q_{l + s} - P_{l} Q_{l} - P_{s} Q_{s} + W (L - R_{0} Q_{r s}) + S θ Q - C_{l} Q_{l} - C_{s} Q_{s} - \frac{1}{2} m θ^{2}$

(A12)

The Hessian matrix of Equation (A12) with respect to $P_{l}$ and $P_{s}$ is:

$[\begin{matrix} - 2 & 0 \\ 0 & - 2 \end{matrix}]$

(A13)

Since the two eigenvalues of the matrix are $- 2 < 0$ and $- 2 < 0$ , this Hessian matrix is a negative definite matrix. Therefore, Equation (A12) is a concave function with respect to $P_{l}$ and $P_{s}$ and there exists a maximum value.

Taking the partial derivatives of Equation (A12) for $P_{l}$ and $P_{s}$ , we get:

$\frac{\partial \prod_{R}^{3} (P_{l}, P_{s})}{\partial P_{l}} = θ P_{r} - C_{l} - 2 P_{l} - A + θ S - θ W R_{0}$

(A14)

$\frac{\partial \prod_{R}^{3} (P_{l}, P_{s})}{\partial P_{s}} = θ P_{r} - C_{s} - 2 P_{s} - A + θ S - θ W R_{0}$

(A15)

letting the partial derivatives equal 0, we get:

$P_{l}^{3} = \frac{θ P_{r} - C_{l} - A + θ S - θ W R_{0}}{2}$

(A16)

$P_{s}^{3} = \frac{θ P_{r} - C_{s} - A + θ S - θ W R_{0}}{2}$

(A17)

Taking Equation (A16) and Equation (A17) into Equation (A11) we get:

$\begin{array}{l} \prod_{G}^{3} (P_{m}, P_{r}) = (P_{m} - C_{m}) \times (N - ε P_{m}) + \frac{θ P_{r} - C_{l} + θ S - θ W R_{0} - A}{2} \times \frac{θ P_{r} - C_{l} + θ S - θ W R_{0} + A}{2} + W (L_{1} - R_{n} (α (N - ε P_{m}) - \\ θ \times \frac{2 θ P_{r} - C_{l} - C_{s} + 2 θ S - 2 θ W R_{0} + 2 A}{2})) - C_{r s} \times (α (N - ε P_{m}) - θ \times \frac{2 θ P_{r} - C_{l} - C_{s} + 2 θ S - 2 θ W R_{0} + 2 A}{2}) \end{array}$

(A18)

The Hessian matrix of Equation (A18) with respect to $P_{m}$ and $P_{r}$ is:

$[\begin{matrix} - 2 ε & 0 \\ 0 & - \frac{{3 θ}^{2}}{2} \end{matrix}]$

(A19)

Since the two eigenvalues of the matrix are $- 2 ε < 0$ and $- \frac{{3 θ}^{2}}{2} < 0$ , this Hessian matrix is a negative definite matrix. Therefore, Equation (A18) is a concave function with respect to $P_{m}$ and $P_{r}$ and there exists a maximum value.

Taking the partial derivatives of Equation (A18) for $P_{m}$ and $P_{r}$ , we get:

$\frac{\partial \prod_{G}^{3} (P_{m}, P_{r})}{\partial P_{m}} = N + ε α C_{r s} + ε C_{m} - 2 ε P_{m} + ε α W R_{n}$

(A20)

$\frac{\partial \prod_{G}^{3} (P_{m}, P_{r})}{\partial P_{r}} = \frac{θ (C_{s} - 2 A + 2 θ C_{r s} - 3 θ P_{r} - θ S + θ W R_{0} + 2 θ W R_{n})}{2}$

(A21)

letting the partial derivatives equal 0, we get:

$P_{m}^{3} = \frac{N + ε {α C}_{r s} + ε C_{m} + ε α W R_{n}}{2 ε}$

(A22)

$P_{r}^{3} = \frac{C_{s} - 2 A + 2 θ C_{r s} - θ S + θ W {(R}_{0} + 2 R_{n})}{3 θ}$

(A23)

Taking Equation (A23) into Equation (A16) and Equation (A17) we get:

$P_{l}^{3} = \frac{C_{s} - 3 C_{l} - 5 A + 2 θ (C_{r s} + S + W (R_{n} - R_{0}))}{6}$

(A24)

$P_{s}^{3} = \frac{θ C_{r s} - C_{s} + θ (S + W (R_{n} - R_{0}))}{3} - \frac{5 A}{6}$

(A25)

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Figure 1.The framework of the recycling process.

Figure 2.The effect of $θ$ on $P_{r}$ .

Figure 3.Effects of $θ$ on recycling price.

Figure 4.The effect of $θ$ on $Q_{r s}$ .

Figure 5.Effects of $S$ and $W$ on $P_{r}$ .

Figure 6.Effects of $S$ and $W$ on recycling prices.

Figure 7.Effects of $S$ and $W$ on $Q_{r s}$ .

Figure 8.Effects of $θ$ on profits.

Figure 9.Effects of $S$ on profits.

Figure 10.Effects of $W$ on profit.

Table 1.Parameters and meanings of variables.

Parameters	Meanings
$N$	Market Volume
$ε$	The price sensitivity coefficient
$P_{m}$	The price of rare earth materials
$P_{r}$	Purchase price of recycled rare earths
$P_{l}$	Recycling price of oil sludge scrap
$P_{s}$	Recycling price of end-of-life products
$C_{r s}$	The cost of REO production by REG
$C_{m}$	Manufacturing cost of rare earth materials
$C_{l}$	The processing cost of oil sludge scrap
$C_{s}$	The processing cost of end-of-life products
$Q_{l}$	Recycling quantity of oil sludge scrap
$Q_{s}$	Recycling quantity of end-of-life products
$Q_{l + s}$	The total quantity of recycling in the S3
$Q_{r s}$	Quantity of REOs produced by REG
$θ$	The rate of rare earth recycling
$m$	R&D cost coefficient
$α$	The REO consumption rate of rare earth materials
$W$	The price of carbon trading
$R_{n}$	Carbon emissions from REO production in the conventional way
$R_{0}$	Carbon emissions from the production of REOs by recycling
$L_{1}$	Carbon credits for REG
$L_{2}$	Carbon credits for recyclers
$S$	$The government subsidy$

Table 2.Value of parameters.

Parameter	Value	Parameter	Value
$N$	800,421	$m$	1,000,000
$ε$	1.12	$L_{1}$	250,000
$α$	0.4	$L_{2}$	100,000
$C_{r s}$	200,000	$W$	646
$C_{m}$	9180	$R_{n}$	3
$A$	5000	$R_{0}$	1.5

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Unleashing the Power of Closed-Loop Supply Chains: A Stackelberg Game Analysis of Rare Earth Resources Recycling (2024)

1. Introduction

2. Literature Review

2.1. Recycling of Rare Earths

2.2. Research on Closed-Loop Supply Chain (CLSC)

3. Materials and Methods

3.1. Underlying Assumptions

3.2. Model Construction

3.2.1. Recycler Only Recycles Oil Sludge Scrap (S1)

3.2.2. Recycler Only Recycles End-of-Life Products (S2)

3.2.3. Recycler Recycles Both End-of-Life Products and Oil Sludge Scrap (S3)

4. Results

4.1. Theoretical Analysis

4.2. Numerical Simulation Analysis

4.2.1. Analysis of the Effects of Recycler R&D (θ) on the Supply Chain

4.2.2. Comparative Analysis of the Effects of Government Intervention (S and W) on the Supply Chain

4.2.3. Comparative Analysis of Supply Chain Profits

5. Discussion

6. Conclusions and Recommendations

6.1. Conclusions

6.2. Recommendations

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

Abbreviations

Appendix A

Appendix A.1. Recycler Only Recycles End-of-Life Products—S2

Appendix A.2. Recycler Recycles Both End-of-Life Products and Oil Sludge Scrap—S3

References

References

4.2.1. Analysis of the Effects of Recycler R&D ( $θ$ ) on the Supply Chain

4.2.2. Comparative Analysis of the Effects of Government Intervention ( $S$ and $W$ ) on the Supply Chain