2. Some stylized facts

This section presents key findings on REEs based on data from the International Energy Agency (IEA) and the US Geological Survey.Footnote ⁶ According to the US Geological Survey (2023), REEs are abundant in the Earth’s crust, but minable concentrations can be challenging to locate compared to other mineral commodities. Additionally, REE processing operations can result in adverse environmental impacts, and China’s dominance across various parts of the REE supply chain, including mining, separation, and refining, poses challenges (IEA, 2021). This dominance extends to light rare-earth elements, such as neodymium, praseodymium, lanthanum, and cerium, as well as to heavy rare-earth elements, including dysprosium, terbium, ytterbium, and yttrium.

After the mid-1990s, China rapidly established itself as the dominant global supplier of REEs, a group of 17 chemically similar elements crucial to modern technologies, surpassing the United States’ previous dominance. By the late 1990s, China controlled around 95% of the global supply of REEs. This dominance was achieved through a combination of low production costs, vast resource deposits, and substantial government investment in REE extraction and processing capabilities. China’s stronghold in this sector has had a profound impact on global industries, particularly those related to clean energy technologies and advanced electronics.

China’s advantage in the REE market extends beyond its production capacity. It also holds significant REE reserves, giving it a long-term strategic position. According to data from the US Geological Survey (2023), China leads the world in REE reserves, followed by Vietnam, Russia, and Brazil. These countries collectively possess substantial untapped REE resources, although they have not yet reached the same production scale as China. This reserve distribution highlights the concentration of REE resources in a few key regions globally, further intensifying China’s dominance in the market.

Despite China’s dominance, its share of global REE mining has declined gradually in recent years. Nevertheless, it continues to exert considerable control over downstream REE operations, particularly at the refining and processing stages. As of 2019, China held 90% of the global market share in REE processing operations, according to the IEA (IEA, 2021). This is a critical advantage, as raw REEs extracted from the earth must undergo complex chemical processing before they can be used in advanced manufacturing processes.

Although other countries have processing facilities, they are limited to handling light REEs. For instance, Malaysia has four processing plants, while France, India, and Estonia also contribute to light REE processing. However, China remains the dominant player in processing heavy REEs, which are rarer and more difficult to refine, but essential for several high-tech applications. This gives China a substantial edge in the global REE market, particularly in industries that rely on these elements for advanced technologies such as wind turbines, solar technology, battery production, and electric vehicles (EVs).

China’s dominant position has not gone unchallenged. In 2010, China significantly reduced its exports of REEs, leading to a surge in prices and raising concerns about global supply chain vulnerabilities. In response, various countries and companies have initiated efforts to diversify REE production and reduce reliance on China. According to recent reports, approximately 20 new REE mining and processing projects are underway in Australia, Canada, and the United States. These projects aim to increase the global supply of REEs and provide alternative sources for industries that rely heavily on these critical elements. Some of these projects are expected to commence operations in the mid-to-late 2020s, contributing to the diversification of the REE supply chain.

Rare earth elements are indispensable for producing clean energy technologies, particularly in the manufacturing of wind turbines, solar panels, batteries, and EVs. These technologies rely heavily on REEs to produce permanent magnets, a key component in the motors of wind turbines and EVs. Among the various REEs, neodymium, dysprosium, praseodymium, and terbium are considered the most critical for clean energy applications. These elements are used in the production of permanent-magnet synchronous motors, which are widely used in electric vehicles and play a crucial role in improving their energy efficiency. However, several challenges have arisen due to the increasing demand for REEs in the clean energy sector. One significant issue is the lack of established recycling pathways for REEs, which makes it difficult to reuse these elements after they are used in products such as EVs and wind turbines. This has led to a growing dependence on primary sources of REEs. Furthermore, the high demand for REEs has driven prices sharply higher. For example, the price of neodymium, a key element for permanent magnets, surged from approximately USD 60 per kilogram in June 2020 to over USD 120 per kilogram by February 2021. This dramatic price increase has led some manufacturers to explore alternatives.

The price trends of key REEs such as neodymium, dysprosium, and terbium from 2018 to 2022 reflect the growing pressures on the REE market. These elements have seen significant price increases due to their critical role in emerging clean energy technologies. According to data from the US Geological Survey (2023), the price of terbium has risen sharply in recent years, reaching over USD 2,000 per kilogram by 2022. Meanwhile, the prices of neodymium and dysprosium have also increased, albeit at a more gradual pace.

Looking ahead, the demand for REEs is expected to grow significantly, driven by the global shift toward clean energy and the increasing adoption of electric vehicles. According to the IEA (2021), the demand for REEs is projected to triple by 2040 under the Stated Policies Scenario (STEPS) and could increase sevenfold under the sustainable development scenario (SDS). This anticipated growth underscores the importance of securing stable and diversified supply chains for REEs to support the global energy transition and meet the needs of the rapidly expanding clean energy sector.

3. The moving-horizon decision framework

This section introduces our solution concepts based on moving-horizon strategies implemented via MPC techniques. The MPC technique is a control strategy that aims to determine an optimal decision policy by solving an optimization problem at each time step. This optimization problem predicts the system’s future behavior over a finite time horizon and finds the sequence of control actions that minimize a predefined cost function while satisfying constraints on the system’s states and control inputs.Footnote ⁷

We begin by addressing the problem of a single agent, following the framework outlined by Grüne and Pannek (Reference Grüne and Pannek2013). Subsequently, we extend this to cover strategic interactions among multiple players, introducing the concept of Nash MPC equilibrium as discussed by Saltari et al. (Reference Saltari, Semmler and Di Bartolomeo2022). Usually, the MPC procedure is written in continuous time, but it can be converted to discrete time, often using the Euler discretization method (see Grüne et al. Reference Grüne, Semmler and Stieler2015).

Consider the following state-space model:

(1) \begin{equation}x(t+1)=f(x(t),u(t)),\mathrm{with}\, {x}(0)=x_{0},\end{equation}

where $x(t)\in X\subseteq \mathbb{R}^{n}$ is the state of the system at time $t$ , $u(t)\in U\subseteq \mathbb{R}^{m}$ is the control input at time $t$ , and $f\colon \mathbb{R}^{n}\times \mathbb{R}^{m}\rightarrow \mathbb{R}^{n}$ is the system dynamics function that defines how the state evolves based on the current state and control input. In the model, $X$ is the set of allowable states, and $U$ is the set of admissible control inputs.

At each time step $t$ , the objective of MPC is to minimize a cost function ( $J$ ) that represents the system’s performance over a finite time horizon ( $T$ ): $J=\sum _{k=0}^{T-1}L(x(k),u(k))+V(x(T))$ . The cost function is typically composed of a period cost, $L(x(t),u(t))$ , which penalizes deviations from a desired state or control effort at each time step, and a terminal cost, $V(x(T))$ , which penalizes the state at the end of the prediction horizon).

Formally, for each $t\in \{1,N-1\}$ , the MPC optimization problem can be written as follows:

(2) \begin{equation} \mathop{\min}\limits_{u(t),u(t+1),\ldots ,u(t+T-1)}J=\sum _{k=0}^{T-1}L(x(t+k),u(t+k))+V(x(t+T))\end{equation}

Subject to: $x(t+1)=f(x(t),u(t))$ and $x(t)\in X,u(t)\in U,x(t-1)\in x_{C}$ , where $x_{C}$ is the current state of the system at $t$ .

Defining by $u^{\ast}(i\vee j)$ the optimal control that solves the above problem for the control at time $i\in [1,N-1]$ when the MPC is solved at time $j\in [1,N-1]$ , the optimal MPC control is just the sequence $\{u^{*}(i|i)\}_{i=0}^{N-1}$ , i.e., $u^{*}(1|1),u^{*}(2|2),\ldots ,u^{*}(N-1|N-1)$ .

In a nutshell, at each time step $t$ , MPC finds the optimal control sequence that minimizes $J$ subject to the state-space model between $t$ and $T$ , given the current state of the system as the initial condition. The MPC controller computes an optimal control sequence for the horizon, ( $u^{*}(t)$ , $u^{*}(t+1)$ , …, $u^{*}(t+T-1)$ ). However, only the first control action $u^{*}(t)$ is applied to the system at time $t$ . At the next step, $t+1$ , the system’s state is re-measured. The optimization problem is solved again for the updated state $x(t+1)$ , generating a new optimal control sequence for the next horizon $T$ . This process is repeated at each step, creating a receding horizon approach, where the optimization problem is continuously updated based on the latest state information.

Now, our objective is to illustrate how to extend the MPC solution to incorporate strategic interactions among multiple policymakers with $N$ periods of a decision horizon and $T$ periods of a planning horizon (Saltari et al. Reference Saltari, Semmler and Di Bartolomeo2022). For brevity, we will consider a scenario involving two players. In this context, the evolution over the planning horizon is influenced by the decisions made by two policymakers:

(3) \begin{equation}x(t+1)=f(x(t),u^{1}(t),u^{2}(t)),\mathrm{with}\ {x}(0)=x_{0},\end{equation}

Here, player 1 sets $u^{1}(t)\in U_{1}\subseteq \mathbb{R}^{{m_{1}}}$ , player 2 sets $u^{2}(t)\in U_{2}\subseteq \mathbb{R}^{{m_{2}}}$ , with $m_{1}+m_{2}=m$ , and $x(t)\in X\subseteq \mathbb{R}^{n}$ .

Each player $k\in \{1,2\}$ seeks to optimize their objective function over a finite time horizon ( $T$ ). The function is denoted by $J^{k}$ and depends on their state and control variables. Formally, $J^{k}=\sum _{k=0}^{T-1}L^{i}(x(k),u^{1}(t),u^{2}(t))+V^{i}(x(T))$ . Once again, we assume that players solve their problems by iteratively solving an optimal control problem at each sampling instant, following a receding decision horizon approach. The MPC two players’ optimization problem, for each $t\in \{0,N\}$ , can be written as follows:

(4) \begin{equation}\begin{cases} \mathop{\min}\limits_{u^{1}\left(t\right),u^{1}\left(t+1\right),\ldots ,u^{1}\left(t+T-1\right)}J^{1}=\sum _{k=0}^{T-1}L\left(x\left(t+k\right),u^{1}\left(t+k\right),u^{2}\left(t+k\right)\right)+V\left(x\left(t+T\right)\right)\\[10pt]\mathop{\min}\limits_{u^{2}\left(t\right),u^{2}\left(t+1\right),\ldots ,u^{2}\left(t+T-1\right)}J^{2}=\sum _{k=0}^{T-1}L\left(x\left(t+k\right),u^{1}\left(t+k\right),u^{2}\left(t+k\right)\right)+V\left(x\left(t+T\right)\right) \end{cases}\end{equation}

Subject to: $x(t+1)=f(x(t),u^{1}(t),u^{2}(t))$ and $x(t)\in X,u^{1}(t)\in U_{1},u^{2}(t)\in U_{2}$ , $x(t-1)\in x_{C}$ , where $x_{C}$ is the current state of the system at $t$ .Footnote ⁸

The Nash MPC solution is then just the sequence, $\{u^{1\ast}(i|i),u^{2\ast}(i|i)\}^{N-1}_{i=0}$ i.e., for each $i\in [1,N-1]$ , the pair of the optimal controls that solve the above problem for the control at time $i$ when the MPC is solved at time $i$ . In other words, for each player, the chosen control or strategy maximizes the objective function, given the other player’s strategy and the initial state. The control is, however, applied over the decision horizon by executing only the first control actions at the current time step, then solving the optimization problem again at the next step with an updated state measurement, repeating this process in a receding-horizon fashion until time $N-1$ .

The Nash MPC equilibrium defines a strategy profile in which each player optimizes its objective function, taking into account the expected behavior of the other players and the game’s dynamics. A plausible definition of a Nash equilibrium would then be that at each instant of time, t, the respective control decisions of the players, defined by the controls u(t) would converge toward some constants. This concept can be generalized to include more than two players. However, it is essential to recognize that this definition resembles a traditional Nash solution concept, albeit with certain restrictions on the strategy space. Additionally, regarding existence and stability, it should be noted that the regularity conditions identified by Grüne and Pannek (Reference Grüne and Pannek2013) for individual agents may not necessarily apply in situations involving distinct payoff functions and strategic interactions among players (Grüne, et al., Reference Grüne, Semmler and Stieler2015, sect. 3; Saltari et al. Reference Saltari, Semmler and Di Bartolomeo2022).Footnote ⁹

4. The economic model

The rest of the paper focuses on the dynamics of China’s and ROW’s market shares. This is analyzed using an extended version of the theoretical model of natural resource extraction by Groot et al. (Reference Groot, Withagen and De Zeeuw2003), adapted to a duopoly framework. In this model, China maintains a leading position due to its vast resource stocks and lower marginal extraction costs than other countries.

Our analysis considers the rarity of REEs, related to the difficulty of discovery and refinement, rather than their scarcity in the Earth’s crust. This limited access to resources is captured in our model through the dynamics of the stock of available (discovered) resources. Unlike Hotelling’s models for non-renewable resources,Footnote ¹⁰ which focus on exhaustibility, we emphasize the concept of (relative) scarcity. In this respect, we assume that when REEs become scarce, it is more likely that new deposits that were previously economically unviable will be discovered and that production processes will improve.Footnote ¹¹

Formally, the evolution of the discovered reserves of REEs, $s_{t},$ is influenced by the detection of new resources at a rate $f\geq 0$ and the world’s exploitation of those resources. As noted earlier and as drawn from Greiner et al. (Reference Greiner, Semmler and Mette2012), we assume that the discovery rate depends on resources yet to be discovered. Our optimization problem considers $e_{t}^{C}$ and $e_{t}^{R}$ as control variables, representing extraction activities driven by China and those of the ROW, while $s_{t}$ is the state variable representing the stock of discovered resources. Hence, the discovered reserves of REEs evolve as follows:

(5) \begin{equation}\dot{s_{t}}=f(s_{t})-e_{t}^{W}\ s_{0} = \overline{s}\ \ \text{for}\ \ t\in R_{0}^{+} \end{equation}

where $f'(s_{t})\lt 0$ , i.e., the effort in search of REEs and improvement of their utilization and the entire supply chain depends inversely on their availability. The second term, $e_{t}^{W}$ , is extraction due to the activities of the dominant Chinese position and those of the rest of the world, i.e., $e_{t}^{W}=e_{t}^{C}+e_{t}^{R}$ . The constant $\overline{s}$ represents the initial level of known resources. We assume that increases in REEs are linear, i.e., $f(s_{t},s^{0})=\theta (s^{z}-s_{t})$ , where $s^{z}$ as the parameter of resources and $\theta$ are positive parameters. Equation (5) modifies Greiner et al. (Reference Greiner, Semmler and Mette2012) to account for relative scarcity.

China extracts $e_{t}^{C}$ at time $t$ with constant marginal extraction costs $m^{C}$ , while ROW pursues its extraction $e_{t}^{R}$ with marginal extraction costs $m^{R}$ . As in Groot et al. (Reference Groot, Withagen and De Zeeuw2003), we assume a linear demand function of the total supply ( $e_{t}^{w}$ ):

(6) \begin{equation} p_{t}=\overline{p}-e_{t}^{w} \end{equation}

where $\overline{p}$ represents the choke price, i.e., a level above which demand for the resource ( $e_{t}^{W}$ ) will be zero.Footnote ¹² To ensure that China and the ROW have sufficient incentives to extract, we assume their marginal extraction costs are lower than the choke price.

Given the strategies of the other group, both players aim to maximize their discounted profits at each point in time $t=R_{0}^{+}$ , using the Nash MPC equilibrium.

The market leader (China) maximizes the following optimization problem in each instant of time $t$ :

(7) \begin{eqnarray}&& \mathop{\max}\limits_{e_{t}^{C}} \int_{t}^{t+T} e^{-rt} \left(\overline{p}-e_{t}^{w}-m^{C}\right) e_{t}^{C}dt\nonumber\\[10pt]&&\text{s.t.}\ \ \dot{s_{t}}=\theta (s^{z}-s_{t})-e_{t}^{w}, \quad e_{t}^{w}=e_{t}^{C}+e_{t}^{R} \end{eqnarray}

Here, $r$ is the discount rate. The length $T$ is the agent’s planning horizon.

Similarly, in each instant of time $t$ , the follower seeks to maximize its profits; its optimization problem is formulated as follows:

(8) \begin{eqnarray}&& \mathop{\max}\limits_{e_{t}^{R}}\int _{t}^{t+T}e^{-rt}\left(\overline{p}\left(1+z_{R}\right)-e_{t}^{w}-m^{R}\right)e_{t}^{R}dt\nonumber\\[10pt]&& \text{s.t.}\ \ \dot{s_{t}}=\theta (s^{z}-s_{t})-e_{t}^{w}, \quad e_{t}^{w}=e_{t}^{C}+e_{t}^{R}\end{eqnarray}

where $z_{R}$ is a subsidy to support production in the rest of the world. Given China’s longstanding dominance in the market, with access to greater resources and infrastructure, we expect it to maintain its leadership role in the duopoly. However, China’s dominant position would lead the ROW to adopt policies to encourage new entrants into the duopoly market.Footnote ¹³ We eventually capture this scenario by assuming $z_{R}\gt 0$ .

The iterative solutions of problems (7) and (8) search for a fixed point at each time $t$ . The optimal strategies are mutually consistent in the game equilibrium, resulting in a sequence of solutions. In this model, at each moment in time $t$ , the solutions to the market dynamics solve China’s extraction path, $e_{t}^{C}$ , and theROW’s extraction, $e_{t}^{R}$ , based on their optimal strategies.Footnote ¹⁴

5. The policy experiments

The dynamics we assess involve the gradual reduction of Chinese monopoly power in REE markets, within a duopoly in which the simultaneous interaction of leader and follower firms is crucial. We consider three scenarios or phases. In the first scenario, China maintains a highly dominant position, and a weak duopoly of the ROW can only be achieved with public support for other countries’ economies. Next, we examine the effects of technological advancements by other countries, which significantly reduce the gap while still maintaining a cost differential with China. Finally, we assume a REE market characterized by a duopoly, with no public support. In this scenario, however, we continue to assume a dominant position for China, albeit reduced, as the resource factor becomes less significant compared to the factors that characterize monopolistic power in this market.

To be concrete, our scenarios can be formally described as follows.

1. 1. Non-competitive duopoly market. China is assumed to have a dominant position with low marginal costs compared to the ROW ( $m^{C}\lt m^{R}$ ). Despite China allowing new suppliers, its strong market position makes it challenging for the ROW to gain higher market share. The ROW implements a support policy in the form of a production subsidy to enter the market ( $z_{R}\gt 0$ ).

2. 2. Duopoly market with support. The marginal extraction cost of the ROW decreases due to technological advancements, thereby making the ROW more competitive. Compared to the previous scenario, we assume that $m^{R}-m^{C}$ falls, but remains positive. We maintain the assumption that the ROW implements a support policy in the form of a subsidy ( $z_{R}\gt 0$ ).

3. 3. Duopoly market without support. This scenario examines the impact of the absence of supporting policies on the REE market structure ( $z_{R}=0$ ), keeping China’s dominant position ( $m^{R}-m^{C}\gt 0$ ).

Overall, these scenarios provide insights into the dynamics of the REE market, the role of supportive policies, and the implications of policymakers’ myopia on resource extraction.

The solution necessitates numerical simulations conducted using the MPC routines described in Saltari et al. (Reference Saltari, Semmler and Di Bartolomeo2022). The calibration of the scenarios is as follows. For all scenarios, we normalize $m^{C}$ = 1.8, $s^{z}$ =31, $\overline{p}$ = 7, and $r$ = 0.03 as plausible values.Footnote ¹⁵ Conversely, $m^{R}$ and $z_{R}$ vary depending on the scenario.Footnote ¹⁶ Expressly, in the primary scenario, we assume a high $m^{R}$ ,i.e., around 3 times higher than the cost faced by the Chinese industry and a supporting subsidy of 30% of the choke price. In the scenario with lower marginal costs in the ROW, we set $m^{R}$ , 1.5 times higher than China’s, and the same as the previous scenario $z_{R}$ = 0.3. Finally, in the scenario without a supporting policy, we keep $m^{R}$ the same as the second scenario, but we assume $z_{R}$ = 0. For a complete list of the parameters and values used in the numerical simulations, see the Appendix.

In all scenarios, we consider that policymakers operate under limited information or constrained information-processing capacity, leading to imprecise responses.Footnote ¹⁷ This limitation is addressed by adopting the MPC approach, acknowledging that policymakers’ decision-making processes, the decision horizon, do not involve maximizing over the complete planning horizon. Instead, we incorporate varying forecasting horizons, with each value representing a different degree of policymakers’ myopia. By adopting this approach, we effectively capture and analyze the impact of myopic decision-making on policy outcomes.Footnote ¹⁸ Regarding the decision horizon $T,$ which measures varying levels of myopia, we explore three decision horizons ( $T$ ): 3, 5, and 7, corresponding to high (3), medium (5), and low (7) levels of policymakers’ myopia, respectively. It should be noted that while these scenarios assume fixed policies, in practice, governments may adjust them dynamically in response to evolving market conditions. Future research could explore adaptive policy changes that better reflect real-world conditions.