The U.S.-China Supply Competition for Rare Earth Elements: a Dynamic Game View

Abstract

Rare earth elements govern today’s high-tech world and are deemed to be essential for the attainment of sustainable development goals. Since the 1990s, these elements have been predominantly supplied by one single actor, China. However, due to the increasing relevance of their availability, the United States, who imports 80% of its rare earths from China, recently announced its plan to (re-)enter the rare earths supply market. This paper analyzes the strategic interactions among these two countries in open-loop and Markovian strategy spaces. Particular interest is devoted to the impact of heterogeneous supply concepts on (1) the theoretical optimal timing for the United States (U.S.) to enter the non-renewable resource market; (2) China’s optimal supply reaction to the U.S.’ entry announcement; (3) the central planner outcome; and (4) the profitability of the suppliers’ extraction behavior. By setting up a continuous-time differential game model, we show that in the absence of arbitrage opportunity, (1) the U.S. should always postpone the production launch until its rare earths reserves coincide with those of China; (2) China’s monopolistic supply is not shaped by the selected strategy; (3) while the duopolistic Markovian behavior is initially more lucrative than open-loop commitment, the opposite situation emerges as the competition proceeds; and (4) on balance, both countries are financially better off when committing to an open-loop supply path.

Appendices

Appendix 1

1.1 Proof Proposition 1

The second-period optimal control problem for both countries is

$$\begin{aligned} \pi _i^{II}(R_C^{*},T^*)=\max _{q_i^{II}(t)} \int _{T^*}^{+\infty }\mathrm {e}^{-rt}P^{II}(Q^{II}(t))\,q_i^{II}(t)\,\mathrm {d} t, \end{aligned}$$

subject to

$$\begin{aligned} \int _{T^*}^{+\infty }\!\!\!\!q_i^{II}(t)\,\mathrm {d} t \le R_i(T^*)= \left\{ \begin{array}{l} R_C^*,\;\mathrm {for}\;i=C\\ R_A(0)\;\mathrm {given},\;\mathrm {for}\;i=A \end{array} \right. , \;\; q_i^{II}(t)\ge 0, \end{aligned}$$

and

$$\begin{aligned} \dot{R}_i(t)=-q_i^{II}(t),\;\;t\in \;[T^*,+\infty ). \end{aligned}$$

Since the first constraint of the optimization problem is equivalent to

$$\begin{aligned} \int _{T^*}^t \! q_i^{II}(\tau )\,\mathrm {d} \tau \le R_i(T^*)\,,\;t\in [T^*,+\infty ), \end{aligned}$$

the Lagrangian is set up as follows:

$$\begin{aligned} \mathcal {L}_i^{II}\Big (q_i^{II}(t), \lambda _i^{II}(t), \alpha _i^{II}\Big ) = & \ P^{II}(Q^{II}(t))q_i^{II}(t)-\lambda _i^{II}(t)q_i^{II}(t)\\&-\alpha _i^{II}\left( R_i(T^*)-\int _{T^*}^{t}\!q_i^{II}(\tau )\,\mathrm {d} \tau \right) , \end{aligned}$$

where $\lambda _i^{II}(t)$ is the shadow price and $\alpha _i^{II}$ is the static Lagrange multiplier. The standard first-order conditions (FOCs) are

$$\begin{aligned} \left\{ \begin{array}{l} \dot{\lambda }_i^{II}(t)= r\,\lambda _i^{II}(t),\\ {\partial \over \partial q_i^{II}(t)}\mathcal {L}_i^{II}\big (q_i^{II}(t), \lambda _i^{II}(t), \alpha _i^{II}\big ) = {\partial \over \partial q_i^{II}(t)}RV_i^{II}(t) -\lambda _i^{II}(t)\,+\,\alpha _i^{II}{\partial \over \partial q_i^{II}(t)}\int _{T^*}^t \! q_i^{II}(\tau )\,\mathrm {d} \tau =0,\\ \alpha _i^{II}\ge 0,\;\;R_i(T^*)-\int _{T^*}^{t}\!q_i^{II}(\tau )\,\mathrm {d} \tau \ge 0, \;\; \alpha _i^{II}\left( R_i(T^*)-\int _{T^*}^{t}\!q_i^{II}(\tau )\,\mathrm {d} \tau \right) =0, \end{array} \right. \end{aligned}$$

where the revenue of country $i=\{A,C\}$ is $RV_i^{II}(t)=P^{II}(t)q_i^{II}(t)$. Based on the remarks in Subsection 3, it is not optimal for i to exhaust its resources in finite time; thus, $R_i(T^*)>\int _{T^*}^{t}\!q_i^{II}(\tau )\,\mathrm {d} \tau$, and hence $\alpha _i^{II}=0$.

The first FOC yields that i’s shadow price $\lambda _i^{II}(t)$ of its remaining reserve $R_i(t)$ grows at interest rate r:

$$\begin{aligned} \lambda _i^{II}(t)= \lambda _i^{II}(T^*)\mathrm {e}^{r(t-T^*)}. \end{aligned}$$

(40)

The second FOC and the fact that $\alpha _i^{II}=0$ show that the shadow price does actually correspond to:

$$\begin{aligned} {\partial RV_i^{II}(t) \over \partial q_i^{II}(t)}= \lambda _i^{II}(t). \end{aligned}$$

(41)

Furthermore, since the revenue of country i is

$$\begin{aligned} RV_i^{II}(t)=P^{II}(t)q_i^{II}(t)=a\left( Q^{II}(t)\right) ^{\alpha -1}q_i^{II}(t)=a\left( q_A^{II}(t)+q_C^{II}(t)\right) ^{\alpha -1}q_i^{II}(t), \end{aligned}$$

the partial derivative of its revenue function with respect to its supply path is

$$\begin{aligned} {\partial RV_i^{II}(t) \over \partial q_i^{II}(t)}=a\left( Q^{II}(t)\right) ^{\alpha -1} \left( 1-{(1-\alpha )q_i^{II}(t) \over q_A^{II}(t)+q_C^{II}(t)}\right) = \lambda _i^{II}(t), \end{aligned}$$

(42)

which yields

$$\begin{aligned} {1-{(1-\alpha )q_i^{II}(t) \over q_A^{II}(t)+q_C^{II}(t)} \over 1-{(1-\alpha )q_j^{II}(t) \over q_A^{II}(t)+q_C^{II}(t)}}={\lambda _i^{II}(T^*) \over \lambda _j^{II}(T^*)}={\lambda _i^{II}(t) \over \lambda _j^{II}(t)}, \end{aligned}$$

(43)

where $i,j=\{A,C\}$ and $i \ne j$. After rearranging Eq. (43), we find

$$\begin{aligned} q_A^{II}(t) = {\lambda _C^{II}(T^*)-\alpha \lambda _A^{II}(T^*) \over \lambda _A^{II}(T^*)-\alpha \lambda _C^{II}(T^*)}\,q_C^{II}(t). \end{aligned}$$

(44)

When integrating Eq. (44) over $[T^*,+\infty )$ and by assuming that over $[T^*,+\infty )$ the total REE reserve is exhausted, as from an economical viewpoint it is not optimal to leave some elements in the deposit (no market value), we get

$$\begin{aligned} \frac{R_A(0)}{R_C(T^*)}= {\lambda _C^{II}(T^*)-\alpha \lambda _A^{II}(T^*) \over \lambda _A^{II}(T^*)-\alpha \lambda _C^{II}(T^*)}, \end{aligned}$$

(45)

that is,

$$\begin{aligned} \lambda _A^{II}(T^*)=\frac{R_C^*+\alpha R_A(0)}{R_A(0)+\alpha R_C^*}\; \lambda _C^{II}(T^*). \end{aligned}$$

Combining (44) and (45) yields

$$\begin{aligned} {q_A^{II}(t) \over q_C^{II}(t)}={R_A(0) \over R_C^*},\; \forall t\ge T^*. \end{aligned}$$

(46)

In view of Eq. (46), the price function in (1) is

$$\begin{aligned} P^{II}(t)=a\left( q_A^{II}(t)+q_C^{II}(t)\right) ^{\alpha -1}=a \left( {R_A(0)\over R_C^*}+1\right) ^{\alpha -1} \left( q_C^{II}(t)\right) ^{\alpha -1} \end{aligned}$$

(47)

$$\begin{aligned} \left[ P^{II}(t)=a \left( 1+{R_C^*\over R_A(0)}\right) ^{\alpha -1} \left( q_A^{II}(t)\right) ^{\alpha -1}\right] , \end{aligned}$$

and the revenue of country C [A] is

$$\begin{aligned} RV_C^{II}(t)= P^{II}(t)q_C^{II}(t)=a \left( {R_A(0)\over R_C^*}+1\right) ^{\alpha -1}(q_C^{II}(t))^{\alpha } \end{aligned}$$

(48)

$$\begin{aligned} \left[ RV_A^{II}(t)= P^{II}(t)q_A^{II}(t)=a \left( 1+{R_C^* \over R_A(0)}\right) ^{\alpha -1}(q_A^{II}(t))^{\alpha }\right] , \end{aligned}$$

so that on the one hand the country i’s marginal revenue corresponds to:

$$\begin{aligned} {\mathrm {d} RV_i^{II}(t) \over \mathrm {d} q_i^{II}(t)}=\alpha P^{II}(t). \end{aligned}$$

(49)

On the other hand, the findings of (42) and (46) imply that the derivative of country C’s revenue function with respect to its supply path leads to:

$$\begin{aligned} {\mathrm {d} RV_C^{II}(t) \over \mathrm {d} q_C^{II}(t)}=\lambda _C^{II}(t)+q_C^{II}(t) \, a (\alpha -1) \left( q_A^{II}+q_C^{II}(t)\right) ^{\alpha -2}. \end{aligned}$$

(50)

Combining (49) and (50) yields

$$\begin{aligned} \lambda _C^{II}(t)={q_C^{II}(t)+ \alpha q_A^{II}(t) \over q_A^{II}(t)+q_C^{II}(t)} \, P^{II}(t), \end{aligned}$$

and based on Eq. (46), this is

$$\begin{aligned} \lambda _C^{II}(t)={R_C^* + \alpha R_A(0) \over R_A(0)+R_C^*} \, P^{II}(t) \end{aligned}$$

(51)

$$\begin{aligned} \left[ \lambda _A^{II}(t)={R_A(0)+\alpha R_C^* \over R_A(0)+R_C^*} \, P^{II}(t)\right] . \end{aligned}$$

When substituting (47) into (51), we obtain

$$\begin{aligned} q_C^{II}(t) = {R_C^*\over R_A(0)+ R_C^*} \left( R_A(0)+R_C^* \over R_A(0) + \alpha R_C^*\right) ^{1\over \alpha -1} \left( {\lambda _C^{II}(T^*)\over a }\right) ^{{1\over \alpha -1}} \mathrm {e}^{{r(t-T^*)\over \alpha -1}}. \end{aligned}$$

(52)

Integrating Eq. (52) over $[T^*,+\infty )$ yields

$$\begin{aligned} \lambda _C^{II}(T^*)= a \left( R_C^* +\alpha R_A(0) \over R_A(0)+R_C^*\right) \left( {r\over 1-\alpha }\left( R_A(0)+R_C^*\right) \right) ^{\alpha -1}. \end{aligned}$$

(53)

After substituting (53) into (52), we find the extraction rate of country C:

$$\begin{aligned} q_C^{II}(t)= {r\over 1-\alpha }\;R_C^*\;\mathrm {e}^{{r(t-T^*)\over \alpha -1}}, \end{aligned}$$

(54)

which, combined with Eq. (47), yields the duopoly market price of the REE:

$$\begin{aligned} P^{II}(t)=a \left( {r\over 1-\alpha }\left( R_A(0)+R_C^*\right) \right) ^{\alpha -1}\mathrm {e}^{r(t-T^*)}. \end{aligned}$$

(55)

Then, by substituting (54) into (46), we get the extraction rate of country A:

$$\begin{aligned} q_A^{II}(t)={R_A(0)\over R_C^*}\;q_C^{II}(t) = {r\over 1-\alpha }\;R_A(0)\; \mathrm {e}^{{r(t-T^*)\over \alpha -1}}. \end{aligned}$$

(56)

Thus, the aggregate revenue for player i is

$$\begin{aligned} \Pi _i=\int _{T^*}^\infty e^{-rt}P^{II}q_i^{II}dt=a\left( {r\over 1-\alpha }(R_A(0)+R_C^*)\right) ^{\alpha -1}R_i(T^*), \end{aligned}$$

with $i=A$, $R_i(T^*)=R_A(0)$ and $i=C$, $R_i(T^*)=R_C^*$. Obviously, the second period revenue difference between the two players only depends on the difference initial reserve at time $t=T^*$. Nonetheless, given $0<\alpha <1$, the revenue, via price, decreases with the initial aggregate reserve $R_A(0)+R_C^*$. Thus, from player A’s point of view, it is more beneficial to wait to enter the supply market as late as possible, such that player C’s reserve decreases sufficiently low to yield higher market price. Then with the same initial reserve, player A can generate higher revenue. In other words, player A would not enter the market when $R_C(t)>R_A(0)$. Obviously, player C sees the same opportunity and waits for player A to supply the market as early as possible. If at some time T, player C’s reserve, $R_C(T)<R_A(0)$, player C should stop supply to the market. Otherwise, player C would lose revenue while player A gains, which is an arbitrage opportunity. So $R_C(T)<R_A(0)$ can not happen either.

As a conclusion, player A enter the market if and only if $R_C(T)= R_A(0)$.

This finishes the proof. The results are presented in Proposition 1 and its corollary of Subsection 4.1.

Appendix 2

1.1 Proof that $T^*=0$ is Not Optimal for Player A

In this subsection, we show that $T^*=0$ can not be optimal. To do so, we first suppose that from the beginning of the game, that is, $t=0$, player A and C jointly supply to the market. Then, we show that under this assumption, player A’s revenue, denoted as $\Pi _A^{b}$ from her initial reserve, is lower than the revenue from Proposition 1.

The objective of player i is

$$\begin{aligned} \Pi _i^{b}&=\max _{q_i^b}\int _0^{+\infty } e^{-rt}P^b(t) q_i^b(t) dt\\&=\max _{q_i^b}\int _0^{+\infty } e^{-rt}a(q_A^{b}+q_C^{b})^{\alpha -1} q_i^b(t) dt, \end{aligned}$$

subject to

$$\begin{aligned} \dot{R}_i=-q_i^b,\;\;\mathrm{and }\;\;\; \int _0^{+\infty }q_i^b(t) dt\le R_i(0), \;\; q_i(t)\ge 0. \end{aligned}$$

The same calculation under open-loop strategy as the proof of Proposition 1, it follows that the optimal supply of player i is

$$\begin{aligned} q_i^b(t)= {r\over 1-\alpha }R_i(0) e^{rt\over \alpha -1},\;\; i=A, C, \end{aligned}$$

the market price is

$$\begin{aligned} P^b(t)= a\left( {r\over 1-\alpha } (R_A(0)+R_C(0)) \right) ^{\alpha -1} e^{rt}, \end{aligned}$$

and optimal revenue for player i under the current situation is

$$\begin{aligned} \Pi _i^b=a\left( {r\over 1-\alpha } (R_A(0)+R_C(0)) \right) ^{\alpha -1} R_i(0),\;\; i=A, C. \end{aligned}$$

Recall, the optimal revenue for player A when enters at $t=T^*>0$ is

$$\begin{aligned} \Pi _A=\int _{T^*}^\infty e^{-rt}P^{II}q_A^{II}dt=a\left( {r\over 1-\alpha }(R_A(0)+R_C^*)\right) ^{\alpha -1}R_A(0). \end{aligned}$$

Thus, given the same initial reserve, $R_A(0), R_C(0)$, and $0<\alpha <1$, player A faces higher price than earlier entering, that is,

$$\begin{aligned} (R_A(0)+R_C^*)^{\alpha -1}<(R_A(0)+R_C(0))^{\alpha -1}. \end{aligned}$$

Therefore,

$$\begin{aligned} \Pi _A^b < \Pi _A. \end{aligned}$$

A similar proof can be done for Markovian strategy. As a conclusion, $T^*=0$ can not happen.

Appendix 3

1.1 Proof of Lemma 1

Recall that the optimal control problem faced by the competitors in the second time period looks as follows:

$$\begin{aligned} \pi _i^{II}(R_C^{*},T^*)=\max _{q_i^{II}\!\left( R_i,t\right) \ge 0} \int _{T^*}^{+\infty }\!\!\!\!e^{-r(t-T^*)}P^{II}\!\!\left( Q\!\left( R_i,t\right) \right) \,q_i^{II}\!\!\left( R_i,t\right) \,\mathrm {d} t, \end{aligned}$$

subject to

$$\begin{aligned} \int _{T^*}^{+\infty }\!\!\!\!q_i^{II}\!\!\left( R_i,t\right) \, \mathrm {d} t \le R_i(T^*)= \left\{ \begin{array}{l} R_A(0)\;\mathrm {given},\;\mathrm {if}\;i=A,\\ R_C^*,\;\mathrm {if}\;i=C, \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \dot{R}_i(t)=-q_i^{II}\!\!\left( R_i,t\right) ,\; t \in [T^*,+\infty ). \end{aligned}$$

The stationary Hamilton-Jacobi-Bellman (HJB) equation of the above problem is thus given by

$$\begin{aligned} r\, W_i(R_i)=\max _{q_i^{II}}\left\{ P^{II}\!(Q)\,q_i^{II}-W_i'(R_i)\,q_i^{II}\right\} , \end{aligned}$$

which yields the corresponding first-order condition (FOC)

$$\begin{aligned} W_i'(R_i)=a\,\left( q_i^{II}+q_j^{II}\right) ^{\alpha -2}\,\left( \alpha q_i^{II}+q_j^{II}\right) . \end{aligned}$$

It follows that

$$\begin{aligned} {W_i'(R_i)\over W_j'(R_j)} = {{\alpha q_i^{II}+q_j^{II}} \over {\alpha q_j^{II}+q_i^{II}}}, \end{aligned}$$

where $i,j = \{A,C\}$ and $i \ne j$.

Moreover, given that country i’s Markovian supply strategy $q_i^{II}\!\!\left( R_i\right)$ depends only on its own reserves $R_i$ and not on the ones of country j, we can apply the envelope theorem to differentiating both sides of the HJB equation with respect to $R_i$. This gives

$$\begin{aligned} r\, W_i'(R_i) = W_i''(R_i)\, \left( -q_i^{II}\right) , \end{aligned}$$

which, based on the dynamic equation of the above problem, can be rewritten as

$$\begin{aligned} r= {W_i''(R_i) \over W_i'(R_i)}\, \dot{R}_i. \end{aligned}$$

(57)

From Eq. (57) it then follows that

$$\begin{aligned} {\mathrm {d} \over \mathrm {d} t}\left( {1 \over W_i'(R_i)}\right) = -{{W_i''(R_i)}\over \left( W_i'(R_i)\right) ^{2}}\,\dot{R}_i = -{1\over W_i'(R_i)}\,r. \end{aligned}$$

(58)

Finally, solving the differential equation of (58) leads to:

$$\begin{aligned} W_i'(R_i(t)) = W_i'(R_i(T^*))\, \mathrm {e}^{r(t-T^*)}, \end{aligned}$$

(59)

where $W_i'(R_i(T^*))={\mathrm {d} W_i\left( R_i(T^*)\right) \over \mathrm {d} R_i}$, and so

$$\begin{aligned} {W_i'(R_i(t)) \over W_j'(R_j(t))} = { W_i'(R_i(T^*)) \over W_j'(R_j(T^*))}. \end{aligned}$$

(60)

At this point, we can combine the first order condition from the last page and Eq. (60) to find that

$$\begin{aligned} {{\alpha q_i^{II}+q_j^{II}} \over {\alpha q_j^{II}+q_i^{II}}}={ W_i'(R_i(T^*)) \over W_j'(R_i(T^*))}, \end{aligned}$$

which, after rearranging, corresponds to:

$$\begin{aligned} q_i^{II}\!(R_i(t))=\frac{\alpha W_i'(R_i(T^*)) - W_j'(R_j(T^*))}{\alpha W_j'(R_j(T^*)) - W_i'(R_i(T^*))}\;q_j^{II}\!(R_j(t)). \end{aligned}$$

(61)

Under the assumption that both competitors fully exhaust their initial reserves $R_i(T^*)$ over the second-period planning horizon because, from an economical viewpoint, it cannot be optimal to leave some REEs in the deposit (no market value), integrating Eq. (61) over $[T^*, +\infty )$ yields

$$\begin{aligned} \frac{R_i(T^*)}{R_j(T^*)}=\frac{\alpha W_i'(R_i(T^*))-W_j'(R_j(T^*))}{\alpha W_j'(R_j(T^*)) -W_i'(R_i(T^*))}, \end{aligned}$$

(62)

which can also be expressed as

$$\begin{aligned} W_i'(R_i(T^*))={\alpha R_i(T^*) + R_j(T^*) \over \alpha R_j(T^*) + R_i(T^*)}\, W_j'(R_j(T^*)). \end{aligned}$$

(63)

Then, after combining Eqs. (61) and (62), we have

$$\begin{aligned} {q_A^{II}\!(R_A(t)) \over q_C^{II}\!(R_C(t))}={R_A(0) \over R_C^*},\; \forall t\ge T^*. \end{aligned}$$

The conjecture that there is no arbitrage opportunity for any of the countries at the entry time $T^*$ means that

$$\begin{aligned} W_i'(R_i(T^*))=W_j'(R_j(T^*)). \end{aligned}$$

From Eqs. (59) and (63), we obtain that the above no arbitrage condition is satisfied if and only if

$$\begin{aligned} R_C^*=R_A(0). \end{aligned}$$

This finishes the proof of Lemma 1 in Sect. 5.1.