1. Introduction
Rare earth elements (REEs) are now vital for a vast array of modern technologies related to the transition to a low carbon economy, such as energy generation and storage, energy efficient lights, electric cars and catalytic converters, as well as military and aerospace applications (Golev et al., Reference Golev, Scott, Erskine, Ali and Ballantyne2014).Footnote 1 According to the US Geological Service Mineral Commodity Summaries 2016 report (USGS, 2016), REEs reserves worldwide amount to 130 million tons. China and Brazil hold the largest shares of such reserves with 16.9 per cent and 42.3 per cent respectively, followed by Australia (2.5 per cent), India (2.4 per cent) and the United States (2 per cent). Regarding mine extraction, out of the 124,000 metric tons estimated to have been produced in 2015, China contributed 87.5 per cent, followed by Australia with 8.3 per cent and the United States with 3.4 per cent (Fernandez, Reference Fernandez2017). Although the USA long dominated the rare earth industry, from the mid-1960s to the mid-1980s, China has become the main producer and now holds a quasi-monopoly. This leading status is mainly attributed to lower labor costs and lower environmental standards (Campbell, Reference Campbell2014; Muller et al., Reference Muller, Schweizer and Seiler2016).
REEs processing is water and energy-intensive and requires chemicals use (US EPA, 2012). The mining and processing of REEs usually result in significant environmental impacts despite increasing efforts towards more efficient waste management (see e.g. An et al., Reference An, Zhang and Adom2019, for the case of industrial waste management utilities in China). Many deposits are actually characterized by high concentrations of radioactive elements such as uranium and thorium (Massari and Ruberti, Reference Massari and Ruberti2013) and acidic substances (Elshkaki and Graedel, Reference Elshkaki and Graedel2014) that are released into the environment without being treated (Folger, Reference Folger2011). The Asian Rare Earth company which was located in Malaysia between 1982 and 1992 has often been reported as an example of radioactive pollution associated with the processing of monazite ores (Ichihara and Harding, Reference Ichihara and Harding1995). In China, mining of abundant ionic clay resources induces significant damages due to severe erosion, air, water and soil pollution, as well as biodiversity loss (Packey and Kingsnorth, Reference Packey and Kingsnorth2016). Human health issues are also reported. The case of Baotou in Inner Mongolia is an emblematic location in which inhabitants are affected by cancers, respiratory diseases and dental losses (Schüler et al., Reference Schüler, Buchert, Liu, Dittrich and Merz2011), whereas the radioactive sludge lake makes the land around this city unsuitable for agriculture.
Despite their name, REEs are not all rare (Falconnet, Reference Falconnet1985; Wübbeke, Reference Wübbeke2013). Rare earth ores contain one or several of the 17 elements, which makes several elements relatively abundant compared to others. The balance problem arises when the market demand for several REEs is not balanced with their natural abundance in REE ores (Elshkaki and Graedel, Reference Elshkaki and Graedel2014). It is a major concern for extractors in that they bear storage costs for abundant REEs. For them, the balance problem is a more important issue than the availability of REEs. For instance, some REEs such as neodymium, dysprosium, terbium and lanthanum are not abundant and are high in demand (Binnemans, Reference Binnemans2014; Binnemans and Jones, Reference Binnemans and Jones2015), whereas other elements such as cerium are abundant and low in demand (Golev et al., Reference Golev, Scott, Erskine, Ali and Ballantyne2014).Footnote 2
Several options have been proposed so far to mitigate the balance problem (Binnemans et al., Reference Binnemans, Jones, Acker, Blanpain, Mishra and Apelian2013a; Binnemans, Reference Binnemans2014; Binnemans and Jones, Reference Binnemans and Jones2015). In this paper, we focus on the recycling of REEs that could postpone the extraction (Ba and Mahenc, Reference Ba and Mahenc2018) and contribute to enhancing environmental quality (Duraiappah et al., Reference Duraiappah, Xin and Van Beukering2002). For example, the supply of neodymium and dysprosium from their recycling is expected to cover about 5 per cent of the demand by 2050 (Elshkaki and Graedel, Reference Elshkaki and Graedel2013). Several countries and corporations have already started to recycle REEs. China recovers REEs up to a maximum level of 95 per cent (Yang et al., Reference Yang, Zhang and Fang2014) while Japan recycles a third of REEs used in the production of magnets (Hetzel and Bataille, Reference Hetzel and Bataille2014). The Solvay Group has recently developed the process for recovering REEs from lamp phosphors, batteries, magnets and tailings in France and in Belgium (Binnemans et al., Reference Binnemans, Jones, Blanpain, Gerven, Yang, Walton and Buchert2013b). Hitachi Ltd has devised technologies to recycle rare earth magnets from hard disk drives (Hitachi, 2010, cited in Binnemans et al., Reference Binnemans, Jones, Blanpain, Gerven, Yang, Walton and Buchert2013b). Osram is able to recover REEs from used phosphors (Binnemans and Jones, Reference Binnemans and Jones2014). Other processes now allow the recovery of REEs from the scrap generated in the various end-uses sectors (Schüler et al., Reference Schüler, Buchert, Liu, Dittrich and Merz2011). It is worth stressing that mining companies such as Molycorp have also implemented recycling schemes for magnets in order to reduce the overproduction of some abundant REEs (Binnemans et al., Reference Binnemans, Jones, Blanpain, Gerven, Yang, Walton and Buchert2013b). These efforts made by several companies are still insufficient (OECD, 2015). One can, therefore, wonder whether environmental policies can have an effect on the intensity of recycling.
Recycling has been the subject of several theoretical investigations. First, some papers focus on the relationship between recycling and natural resource exhaustion. Andre and Cerdà (Reference Andre and Cerdà2006), Weikard and Seyhan (Reference Weikard and Seyhan2009) and Seyhan et al. (Reference Seyhan, Weikard and Ierland2012) show that recycling delays the depletion of these resources. Ba and Mahenc (Reference Ba and Mahenc2018) analyze the extent to which taking into account recycling challenges the Hotelling rule. Several other papers explore the impact of recycling on market power. The results are somewhat contradictory. One the one hand, recycling does not substantially affect the extractor's long-run market power (see Gaskins, Reference Gaskins1974; Swan, Reference Swan1980; Martin, Reference Martin1982; Suslow, Reference Suslow1986; Hollander and Lasserre, Reference Hollander and Lasserre1988; Grant, Reference Grant1999). On the other hand, recycling increases the extractor's market power (see Gaudet and Long, Reference Gaudet and Long2003; Baksi and Long, Reference Baksi and Long2009). Finally some papers analyze instruments that favor recycling (see, for example, Yokoo and Kinnaman, Reference Yokoo and Kinnaman2013; Gupt, Reference Gupt2015).
The main purpose of this paper is to investigate the extent to which recycling and environmental taxes can alter both the balance problem and the pollution generated by REEs extraction. To the best of our knowledge, our contribution is the first one that takes into account the balance problem in an economic framework. It is also the first one that designs environmental taxes in the presence of recycling.
There are two ways of modeling the production process of REEs. In the first, the extraction is a joint production process. The valuation of the co-products ensures the profitability of extraction. In the second, the extractor only cares about specific elements while others are byproducts. The latter only provides extra value to the mining project and does not influence the optimal extraction. Fizaine (Reference Fizaine2013), for instance, analyzes the link between mining byproducts and the primary products. This paper formalizes the equilibrium between the supply and the demand for the primary ore only. The price elasticity of the byproduct supply is null because the extractor overlooks it. Yet, an equilibrium exists on the byproduct market between the inelastic supply and the demand. Fizaine (Reference Fizaine2013) analyzes the market of minor metals but does not address the balance problem, which is precisely what we intend to do in this paper.
We rely on a two-period framework where a monopolist extracts two types of REEs, namely abundant and non-abundant REEs. In the second period of the game, the monopolist engages in competition with one firm that recycles part of the non-abundant REEs consumed in the first period. This brings our model close to Ba and Mahenc's (Reference Ba and Mahenc2018) model. It is however different, in that the pollution and the balance problem are both taken into account.
Our results are the following. We show that recycling always reduces extracted quantities thereby mitigating the balance problem and environmental damages. Other results depend on the amount of scrap that can be recycled. If the recycling activity is not limited by this quantity, it does not change extraction in the first period. Otherwise, the extractor adopts a foreclosure strategy in period one and reduces extraction of REE. The existence of both pollution and market power in each period does not allow the optimum to be reached. We, therefore, propose to implement environmental taxes on extracted quantities. If the environmental tax reduces extraction and favors recycling when recycling is not limited by the available scrap, it can decrease recycling in the opposite case. This result suggests that the regulator has to be very cautious if he wants the use of environmental taxes to indirectly favor recycling. The second-best levels of environmental taxes depend on the marginal damage, on the market power as well as on the recycling. It is also worth noting that the environmental taxes are never equal to the marginal damage.
The remainder of the paper is structured as follows. Section 2 describes the assumptions of the model and the first-best outcome. We consider the decentralized economy with recycling in section 3. We introduce an exogenous environmental regulation in section 4, and second-best environmental taxation in section 5. Section 6 concludes the paper. Technical proofs have been relegated to the appendix.
2. The model
In this section we present the assumptions of the model and, as a benchmark, the first-best outcome.
2.1. Assumptions
We consider a two-period model where one firm extracts REEs from one mine. The extracted ore contains abundant and non-abundant REEs. Let 
$x_{t}$ denote the supply of non-abundant REEs and 
$\bar {x}_{t}$ the supply of abundant REEs, where t = 1,2 is a time index. Since both types of REEs are extracted from the same ore, the extraction of one type mechanically induces the extraction of the other type such that 
$\bar {x}_{t}=\alpha x_{t}$, where α is a positive parameter. The extraction cost is denoted by 
$ C_{t}(x_{t})$ that has the usual properties (
$C^{\prime }>0$ and 
$C^{\prime \prime }>0)$. For the sake of simplification, we assume that the discount factor is normalized to one.
We assume that the market of non-abundant REEs is cleared while the supply of abundant REEs exceeds the demand such as 
$\bar {x}_{t}>x_{t}{}^{d}\ \forall \ \bar {P}_{t}\geqslant 0$, where 
$x_{t}{}^{d}$ is the demand of abundant REEs and 
$\bar {P}_{t}$ their unit price.Footnote 3 The balance problem for abundant REEs incurs a storage cost borne by the extractor: 
$c_{s}\sum _{t=1}^{2}(\bar {x}_{t}-x_{t}^{d})$, where 
$c_{s}>0$ is the marginal cost of storage.
In the first period, the extractor is a monopolist whereas in the second period it faces one recycler who recycles a quantity 
$r\leq kx_{1}$. The parameter k denotes the recycling technology efficiency with 
$k\in [0;1] $. If the strict inequality 
$r<kx_{1}$ holds, it means that depreciation occurs during recycling. Note that the non-abundant REEs are recycled at a cost 
$C_{r}(r)$ that is an increasing and convex function.
Extracted and recycled quantities of non-abundant REEs are perfectly substitutable. 
$P_{t}$ is their prevailing market price. The inverse demand function is 
$P_{t}=P(Q_{t})$ with 
$P^{\prime }<0$ and 
$P^{\prime \prime }\leqslant 0$. We denote 
$Q_{t}$ the total quantity of non-abundant REEs supplied such that 
$Q_{1}=x_{1}$ and 
$Q_{2}=x_{2}+r$. We assume stationary demand functions.
We assume that the extraction of REEs causes pollution. The damage induced by REEs therefore depends on the quantities extracted and not on a pollution stock. Hence the damage function is written as 
$D(x_{t}),$ which is an increasing and convex function.
In the rest of the paper, our results depend crucially on whether 
$r\leq kx_{1}$. In order to ensure concavity in the extractor's profit and to have tractable results, we restrict our analysis to 
$P^{\prime \prime }=0$ and 
$C^{\prime \prime \prime }=0$ when 
$r=kx_{1}$.
2.2. The first-best outcome
We define the first-best outcome as a situation where there is no market power and strategic interactions and that takes into account environmental damages. We consider a benevolent regulator acting under perfect information, who maximizes social welfare under the constraint on available scrap that can be used by the recycler. The program of the regulator is:
\begin{align} & {\rm Max}\ W(x_{1},x_{2},r,\lambda)=\int_{0}^{x_{1}}P(u)\, d u+ Sc(x_{1}^{d})+x_{1}^{d}\bar{P}_{1}-C_{1}(x_{1})\nonumber\\ &\hspace*{76pt}\quad -c_{s}[\alpha x_{1}-x_{1}^{d}]-D(x_{1})+\int_{0}^{x_{2}+r}P(z)\, d z+Sc(x_{2}^{d}) \nonumber\\ &\hspace*{76pt}\quad +x_{2}^{d}\bar{P}_{2}-C_{2}(x_{2})-c_{s}[\alpha x_{1}+\alpha x_{2}-x_{1}^{d}-x_{2}^{d}] \nonumber\\ &\hspace*{76pt}\quad -C_{r}(r)-D(x_{2}), \nonumber\\ & {\rm s.t.}\ r\leq kx_{1},\end{align}
where λ is a Kuhn-Tucker multiplier and 
$Sc(x_{1}^{d})$ and 
$ Sc(x_{2}^{d})$ the consumer surplus from abundant REEs. The first-order conditions (FOCs) are:
\begin{align} & P_{1}(x_{1})-C_{1}^{\prime }(x_{1})-2\alpha c_{s}-D^{\prime}(x_{1})+\lambda k=0,\end{align}
\begin{align} & P_{2}(x_{2}+r)-C_{2}^{\prime }(x_{2})-\alpha c_{s}-D^{\prime }(x_{2})=0,\end{align}
\begin{align} & P_{2}(x_{2}+r)-C_{r}^{\prime }(r)-\lambda =0,\end{align}
\begin{align} & \lambda \lbrack kx_{1}-r]=0.\end{align}Let us explore the first-best outcome by distinguishing the following two cases:
The non-binding case: when the recycling constraint is not binding – i.e., 
$r<kx_{1}$ and 
$\lambda =0$ – we get, after several rearrangements of (2), (3) and (4):
\begin{align} P_{1}(x_{1}^{\ast nc}) &= C_{1}^{\prime }(x_{1}^{\ast nc})+2\alpha c_{s}+D^{\prime }(x_{1}^{\ast nc}),\end{align}
\begin{align} P_{2}(x_{2}^{\ast nc}+r^{\ast nc}) &= C_{2}^{\prime }(x_{2}^{\ast nc})+\alpha c_{s}+D^{\prime }(x_{2}^{\ast nc}),\end{align}
\begin{align} P_{2}(x_{2}^{\ast nc}+r^{\ast nc}) &= C_{r}^{\prime }(r^{\ast nc}),\end{align}
where the superscript 
$\ast nc$ means the non-constrained first-best. In this case, the REEs value set in each period is equal to the private marginal costs augmented by the marginal environmental damage induced by extraction. Recycling and extracted quantities in period 2 are such that social marginal costs of production are identical. We find: 
$\frac {\partial x_{1}^{\ast nc}}{\partial c_{s}}<0$, 
$\frac {\partial x_{2}^{\ast nc}}{\partial c_{s}}<0$, 
$\frac {\partial r^{\ast nc}}{\partial c_{s}}>0$ (see appendix A.1). The balance problem leads to a decrease in extracted quantities in both periods and favors recycling.
The binding case: when the recycling constraint is binding, we have 
$r=kx_{1}$ and 
$\lambda >0$. Rearranging equations (2), (3) and (4) gives the following optimal conditions:
\begin{align} P_{1}(x_{1}^{\ast c})-C_{1}^{\prime }(x_{1}^{\ast c})-2\alpha c_{s}-D^{\prime }(x_{1}^{\ast c}) +k[P_{2}(x_{2}^{\ast c} +kx_{1}^{\ast c})-C_{r}^{\prime }(kx_{1}^{\ast c})]&=0,\end{align}
\begin{align} P_{2}(x_{2}^{\ast c}+kx_{1}^{\ast c})-C_{2}^{\prime }(x_{2}^{\ast c})-\alpha c_{s}-D^{\prime }(x_{2}^{\ast c})&=0,\end{align}
where the superscript 
$\ast c$ refers to the constrained first-best. The regulator defines the level of the quantity extracted in period 1 taking into account the impact of this extraction during period 2. We find: 
$\frac {\partial x_{1}^{\ast c}}{\partial c_{s}}<0$, 
$\frac {\partial x_{2}^{\ast c}}{\partial c_{s}}<0$ and 
$\frac {\partial r^{\ast c}}{\partial c_{s}}<0$ (see appendix B.1). If the balance problem reduces extraction in both periods, it also reduces the level of recycled quantities.
From equations (9) and (6), the extracted quantities in period 1 are higher in the binding case than in the non-binding case: at the first-best – due to the convex damage function – the regulator allows an increase in the damage in the first period in order to favor recycling in period 2.Footnote 4
3. Recycling
In this section, we analyze the effect of recycling in the decentralized economy. We first investigate the economy without recycling and then with recycling.
3.1. Equilibrium without recycling
Without recycling, the extractor acts as a monopolist in both periods. The profit of the extractor is the sum of revenues earned in both periods from selling both types of REEs minus extraction and storage costs:
\begin{align*} \pi ^{e}(x_{1},x_{2}) &= P_{1}(x_{1})x_{1}-C_{1}(x_{1})-c_{s} [\bar{x}_{1}-x_{1}^{d}]+x_{1}^{d}\bar{P}_{1}+P_{2}(x_{2})x_{2}\\ &\quad -C_{2}(x_{2})+x_{2}^{d} \bar{P}_{2}-c_{s}[\bar{x}_{1}-x_{1}^{d}+\bar{x}_{2}-x_{2}^{d}]. \end{align*}FOCs take into account the relationship between both types of REEs:
\begin{align} & P_{1}(x_{1}^{wr})+P_{1}^{\prime }(x_{1}^{wr})x_{1}^{wr}-C_{1}^{\prime} (x_{1}^{wr})-2\alpha c_{s}=0,\end{align}
\begin{align} & P_{2}(x_{2}^{wr})+P_{2}^{\prime }(x_{2}^{wr})x_{2}^{wr}-C_{2}^{\prime} (x_{2}^{wr})-\alpha c_{s}=0,\end{align}where the superscript wr means without recycling. Equations (11) and (12) indicate that in each period the price of the non-abundant REEs is equal to the sum of the marginal costs of extraction and storage, adjusted for the monopoly market power. The balance problem, by introducing storage costs, leads to reduced extraction in each period. The reduction is more pronounced in period 1 because the storage cost is reduced in period 2. We have:
\[ x_{1}^{wr}<x_{2}^{wr}. \]Proposition 1 due to the balance problem, extracted quantities without recycling in period 2 are higher than extracted quantities in period 1.
It is fair to state that this result is (likely) not robust to the extension to the infinite horizon. If we relax the assumption of stationary demand functions, Proposition 1 does not hold anymore. For example, if the demand shrinks sufficiently from period 1 to period 2, first period production could be larger than second period production. In the same vein, if we introduce a discount factor, the extractor undervalues the storage cost borne in period 2. Hence it increases extraction in period 1 compared to the case where the discount factor is equal to one.
3.2. Equilibrium with recycling
Recycling occurs only in the second period. Using backward induction, we first find the equilibrium quantities in period 2 and we then solve for the quantity produced by the extractor in period 1.
3.2.1. The second stage: the equilibrium quantities in period 2
Let us define the subgame-perfect Nash equilibrium in period 2. The extractor's profit function in the second period is the following:
\[ \pi ^{e}(x_{2},r)=P_{2}(x_{2}+r)x_{2}-C_{2}(x_{2})+x_{2}^{d}\bar{P}_{2}-c_{s} [\alpha x_{1}-x_{1}^{d}+\alpha x_{2}-x_{2}^{d}]. \]The FOC gives:
\[ P_{2}(x_{2}+r)+P_{2}^{\prime }(x_{2}+r)x_{2}-C_{2}^{\prime }(x_{2})-\alpha c_{s}=0. \]The recycler maximizes its profit, subject to the constraint on the available resource:
\begin{align*} \pi ^{r}(r,x_{2}) &= P_{2}(x_{2}+r)r-C_{r}(r) \\ r &\leq kx_{1}. \end{align*}We find:
\begin{equation} \left\{\begin{array}{@{}ll@{}} P_{2}(x_{2}+r)+P_{2}^{\prime }(x_{2}+r)r-C_{r}^{\prime }(r)=0 \quad {\rm if } r<kx_{1} \\ r=kx_{1} \quad {\rm otherwise,} \end{array}\right.\end{equation}which represents the recycler's best response function. The equilibrium depends on whether the recycling constraint is binding (denoted by the superscript c) or not (denoted by the superscript nc).
• If the available quantity of scrap is higher than the unconstrained profit-maximizing quantity, the extractor and the recycler produce the quantities that satisfy the following FOCs:
(14)The Implicit Function Theorem on FOCs given by (14) shows that reaction functions are decreasing. The recycled scrap and the extracted output are strategic substitutes. Recycling reduces the quantity of non-abundant REEs which is extracted by the monopolist in the second period. This behavior was coined the ‘business-stealing’ effect after Mankiw and Whinston (Reference Mankiw and Whinston1986). The strategic response of existing firms to new entry results in reducing their production when a new entrant ‘steals business’ from incumbent firms. Solving the system given by (14) gives the non-constrained subgame-perfect Nash equilibrium
\begin{equation}\left\{\begin{array}{@{}l@{}} P_{2}(x_{2}^{{\rm nc}}+r^{{\rm nc}})+P_{2}^{\prime}(x_{2}^{{\rm nc}}+r^{{\rm nc}})x_{2}^{{\rm nc}}-C_{2}^{\prime } (x_{2}^{{\rm nc}})-\alpha c_{s}=0 \\ P_{2}(x_{2}^{{\rm nc}}+r^{{\rm nc}})+P_{2}^{\prime}(x_{2}^{{\rm nc}}+r^{{\rm nc}})r^{{\rm nc}}-C_{r}^{{\prime }}(r^{{\rm nc}})=0. \end{array}\right.\end{equation}$(x_{2}^{{\rm nc}},r^{{\rm nc}})$. The extracted quantities and the level of recycled scrap in period 2 do not depend on the quantity extracted in period 1.Footnote 5• If the available quantity of scrap is lower than the unconstrained profit-maximizing quantity, the equilibrium in period 2 is given by:
(15)Solving this system gives the constrained subgame-perfect Nash equilibrium\begin{equation}\left\{\begin{array}{@{}l@{}} P_{2}(x_{2}^{c}+r^{c})+P_{2}^{\prime}(x_{2}^{c}+r^{c}) x_{2}^{c}-C_{2}^{{\prime }}(x_{2}^{c})-\alpha c_{s}=0 \\ r=kx_{1}^{c}. \end{array}\right.\end{equation}$ (x_{2}^{c},r^{c})$. We find $x_{2}^{c}=f(x_{1})$, with $\frac {dx_{2}^{c}}{ dx_{1}}<0$. Hence, the extraction level in period 1 will affect the level of recycling as well as the extracted quantity in period 2.
Proposition 2 whatever the level of recycled scrap, the recycling activity reduces extraction in the second period.
3.2.2. The first stage: the equilibrium quantities in period 1
In order to obtain the equilibrium quantity in period 1, we replace equilibrium quantities in period 2 in the extractor's profit function. The quantity in the first stage depends on the subgame-perfect Nash equilibrium obtained in the second stage.
The non-binding case: if the available quantity of scrap does not constrain the recycler from producing, we find:
\begin{align*} \pi ^{e}(x_{1},x_{2}^{{\rm nc}},r^{{\rm nc}}) &= P_{1}(x_{1})x_{1}-C_{1}(x_{1})+x_{1}^{d} \bar{P}_{1}-c_{s}[\alpha x_{1}-x_{1}^{d}] \\ &\quad +P_{2}(x_{2}^{{\rm nc}}+r^{{\rm nc}})x_{2}^{{\rm nc}}-C_{2}(x_{2}^{{\rm nc}})+x_{2}^{d} \bar{P}_{2}\\ &\quad -c_{s}[\alpha x_{1}-x_{1}^{d}+\alpha x_{2}^{{\rm nc}}-x_{2}^{d}]. \end{align*}The FOC is:
\begin{equation} P_{1}(x_{1}^{{\rm nc}})+P_{1}^{\prime }(x_{1}^{{\rm nc}})x_{1}^{{\rm nc}}-C_{1}^{\prime} (x_{1}^{{\rm nc}})-2\alpha c_{s}=0.\end{equation}Recycling does not affect the quantity of REEs extracted by the monopolist in the first period (equation (16) is similar to equation (11)). As expressed above, recycling slows down extraction in the second period. Thus, recycling helps to mitigate the balance problem by reducing the stock of abundant REEs and also contributes to reducing pollution in the second period. As far as total quantities traded in period 2, we find
\[ Q_{2}^{{\rm nc}}\gtrless Q_{1}^{{\rm nc}}=Q_{1}^{wr}. \]On the one hand, without recycling, the storage cost induces more extraction in the second period than in the first. On the other hand, recycling reduces extraction in the second period. Thus, depending on the magnitude of the storage cost, the total quantities can either increase or decrease between periods 1 and 2.
Moreover we find that (see appendix A.2):
\[ \frac{\partial x_{1}^{{\rm nc}}}{\partial c_{s}}<0;\quad \frac{\partial x_{2}^{{\rm nc}}}{\partial c_{s}}<0;\quad \frac{\partial r^{{\rm nc}}}{\partial c_{s}}>0. \]If the storage costs reduce extracted quantities in each period, they boost the recycled quantity in period 2. The effects of the balance problem are similar to the ones under ‘first-best’.
Proposition 3 in the non-binding case, recycling does not change extraction in period 1. The balance problem favors recycling in period 2 and mitigates environmental damages in both periods.
The binding case: if the collected scrap constrains the production level of the recycler, the profit of the extractor reads as follows:
\begin{align*} \pi ^{e}(x_{1}) &= P_{1}(x_{1})x_{1}-C_{1}(x_{1})+x_{1}^{d} \bar{P}_{1}-c_{s}[\alpha x_{1}-x_{1}^{d}]+P_{2}(x_{2}^{c}(x_{1})+kx_{1})) x_{2}^{c}(x_{1})\\ &\quad -C_{2}(x_{2}^{c}(x_{1})) +x_{2}^{d}\bar{P}_{2}-c_{s}[\alpha x_{1}-x_{1}^{d}+\alpha x_{2}^{c}(x_{1})-x_{2}^{d}]. \end{align*}The FOC is the following:
\begin{align} & P_{1}(x_{1}^{c})+P_{1}^{\prime }(x_{1}^{c})x_{1}-C_{1}^{\prime} (x_{1}^{c})-2\alpha c_{s}+\frac{dx_{2}^{c}(x_{1}^{c})}{dx_{1}^{c}} [P_{2}(x_{1}^{c}) \nonumber\\ &\quad +P_{2}^{\prime}(x_{1}^{c})x_{2}^{c}(x_{1}^{c})-C_{2}^{\prime } (x_{1}^{c})-\alpha c_{s}]+kP_{2}^{\prime }(x_{1}^{c})x_{2}^{c}(x_{1}^{c})=0.\end{align}
Comparing equations (16) and (17) gives 
$ x_{1}^{c}<x_{1}^{{\rm nc}}$. Contrary to the non-binding case, recycling reduces the first period extracted quantity of REEs. By reducing extraction, the extractor, acting as a leader, curtails recycling in the second period. That enables future competition to be reduced. The extractor adopts a foreclosure strategy in order to keep strong market power in period 2. Hence, recycling strengthens the market power in period 1. Finally, we obtain the following result on global quantities:
\[ Q_{2}^{{\rm nc}}\gtrless Q_{1}^{c}. \]We also find (see appendix B.2):
\[ \frac{\partial x_{1}^{c}}{\partial c_{s}}<0;\quad \frac{\partial x_{2}^{c}}{\partial c_{s}}<0;\quad \frac{\partial r^{c}}{\partial c_{s}}<0. \]
If a high storage cost favors recycling in the non-binding case, it reduces recycling if 
$r=kx_{1}$. In this case, the balance problem indirectly limits recycling and, hence, competition in the second period.
Proposition 4 in the binding case, recycling leads to foreclosure behavior in the first period.Footnote 6 The balance problem limits recycling and, hence, triggers environmental damages in the second period.
Comparing equations (14) and (16) with equations (6) to (8), and equations (15) and (17) with (9) and (10), shows that the market equilibrium never reaches the first-best. In each case, the storage cost of abundant REEs induces the extractor to take the balance problem into account. Both market power and pollution, however, prevent the first-best outcome from being reached. As is widely acknowledged in the literature, one way to restore the social optimum is to tax negative externalities. Below we will analyze what will happen with the implementation of a tax scheme by the benevolent regulator.
4. Exogenous environmental regulation
In order to internalize the negative externality, i.e., pollution induced by extraction, the regulator sets environmental taxes 
$\tau _{t}$, which it levies on each extracted unity in each period. As in section 3, we solve the game by backward induction. We first define equilibrium quantities in period 2, then in period 1.
4.1. The second stage: the equilibrium quantities in period 2
If an environmental tax is levied on the extraction, the extractor's profit maximization program in period 2 becomes:
\[ \pi ^{e}(x_{1},x_{2}{\bf, }r)=P_{2}(x_{2}+r)x_{2}-C_{2}(x_{2})+y_{2}^{d} \bar{P}_{2}-c_{s}[\alpha x_{1}-y_{1}^{d}+\alpha x_{2}-y_{2}^{d}]-\tau_{2}x_{2}. \]The profit function of the recycler is not modified by the environmental taxation. Extracted quantities in period 2 depend on the constraint on recycling:
• If the available quantity is higher than the unconstrained profit-maximizing quantity, the extractor and the recycler produce the quantities that satisfy the following FOCs:
(18)Solving this system gives the unconstrained subgame-perfect Nash equilibrium\begin{equation} \left\{\begin{array}{@{}l@{}} P_{2}(x_{2}^{{\rm nct}}+r^{{\rm nct}})+P_{2}^{\prime }(x_{2}^{{\rm nct}}+ r^{{\rm nct}})x_{2}^{{\rm nct}}-C_{2}^{\prime }(x_{2}^{{\rm nct}})-\alpha c_{s}-\tau _{2}=0 \\ P_{2}(x_{2}^{{\rm nct}}+r^{{\rm nct}})+P_{2}^{\prime}(x_{2}^{{\rm nct}}+r^{{\rm nct}})r^{{\rm nct}}-C_{r}^{\prime }(r^{{\rm nct}})=0. \end{array}\right.\end{equation}$(x_{2}^{{\rm nct}}(\tau _{2}),r^{{\rm nct}}(\tau _{2}))$, with $\frac {\partial x_{2}^{{\rm nct}}}{\partial \tau _{2}}<0$ and $\frac {\partial r^{{\rm nct}}}{\partial \tau _{2}}>0$ (see appendix A.3). Equilibrium quantities in period 2 do not depend on the extracted quantities in period 1.• If the recycler is limited by the available quantity of scrap, the constrained subgame-perfect Nash equilibrium is given by:
(19)Solving this system gives\begin{equation} \left\{\begin{array}{@{}l@{}} P_{2}(x_{2}^{{\rm ct}}+r^{{\rm ct}})+P_{2}^{\prime}(x_{2}^{{\rm ct}}+r^{{\rm ct}})x_{2}^{{\rm ct}}-C_{2}^{\prime } (x_{2}^{{\rm ct}})-\alpha c_{s}-\tau_{2}=0 \\ r^{{\rm ct}}=kx_{1}. \end{array}\right.\end{equation}$x_{2}^{{\rm ct}}(x_{1},\tau _{2})$, with $\frac {\partial x_{2}^{{\rm ct}}}{\partial x_{1}}<0$.
4.2. The first stage: the equilibrium quantities in period 1
Quantities in period 1 depend on the outcome of the subgame-perfect Nash equilibrium.
The non-binding case. If 
$r<kx_{1}$, quantities in period 2 do not depend on 
$x_{1}$. The extractor maximizes its profit in the first period. We have:
\begin{align*} \pi ^{e}(x_{1}) &= P_{1}(x_{1})x_{1}-C_{1}(x_{1})+x_{1}^{d} \bar{P}_{1}-c_{s}[\alpha x_{1}-y_{1}^{d}]-\tau_{1}x_{1}\\ &\quad +P_{2}(x_{2}^{{\rm nct}}+r^{{\rm nct}})x_{2}^{{\rm nct}}-C_{2}(x_{2}^{{\rm nct}}) \\ &\quad +y_{2}^{d}\bar{P}_{2}-c_{s}[\alpha x_{1}-y_{1}^{d}+ \alpha x_{2}^{{\rm nct}}-y_{2}^{d}]-\tau _{2}x_{2}^{{\rm nct}} \end{align*}
\begin{equation} P_{1}(x_{1}^{{\rm nct}})+P_{1}^{\prime }(x_{1}^{{\rm nct}})x_{1}-C_{1}^{\prime} (x_{1}^{{\rm nct}})-2\alpha c_{s}-\tau _{1}=0.\end{equation}
If we compare equation (20) to equation (16), we show that the extractor reduces the extracted quantity in period 1 under environmental taxation. Solving equation (20) enables us to obtain 
$x_{1}^{{\rm nct}}$ as a function of 
$\tau _{1}$, with 
$\frac {\partial x_{1}^{{\rm nct}}}{\partial \tau _{1} }<0$ (appendix A.3). Each per-period extracted quantity decreases with the per-period tax rate. Hence taxes increase the recycled output, since recycling and the second period extracted output are strategic substitutes.
Proposition 5 when the recycling activity is not bounded by the available scrap, environmental taxation favors recycling.
The binding case. We replace 
$x_{2}^{{\rm ct}}(x_{1},\tau _{2})$ and 
$ r^{{\rm ct}}=kx_{1}$ in the profit of the extractor. We obtain:
\begin{align*} \pi ^{e}(x_{1}) &= P_{1}(x_{1})x_{1}-C_{1}(x_{1})+x_{1}^{d} \bar{P}_{1}-c_{s}[\alpha x_{1}-y_{1}^{d}]-\tau_{1}x_{1}\\ &\quad +P_{2} (x_{2}^{{\rm ct}}(x_{1},\tau _{2})+kx_{1})x_{2}^{{\rm ct}}(x_{1},\tau_{2}) \\ &\quad -C_{2}(x_{2}^{{\rm ct}}(x_{1},\tau _{2}))+y_{2}^{d}\bar{P}_{2}-c_{s} [\alpha x_{1}-y_{1}^{d}+\alpha x_{2}^{{\rm ct}}(x_{1},\tau _{2})-y_{2}^{d}]- \tau_{2}x_{2}^{{\rm ct}}(x_{1},\tau _{2}). \end{align*}The FOC reads as follows:
\begin{equation} P_{1}+P_{1}^{\prime }x_{1}^{{\rm ct}}-C_{1}^{{\prime }}-2\alpha c_{s}-\tau _{1}+ \frac{\partial x_{2}^{{\rm ct}}}{\partial x_{1}^{{\rm ct}}}[P_{2}^{\prime}x_{2}^{{\rm ct}}+ P_{2}-C_{2}^{\prime }-\alpha c_{s}-\tau _{2}]+P_{2}^{\prime}kx_{2}^{{\rm ct}}=0.\end{equation}
Solving equation (21) yields 
$x_{1}^{{\rm ct}}(\tau _{1},\tau _{2})$ and hence 
$x_{2}^{{\rm ct}}(\tau _{1},\tau _{2})$ with 
$\frac {dx_{1}^{{\rm ct}}}{d\tau _{1}}= \frac {dx_{2}^{{\rm ct}}}{d\tau _{2}}<0$ and 
$\frac {dx_{1}^{{\rm ct}}}{d\tau _{2}}=\frac {dx_{2}^{{\rm ct}}}{d\tau _{1}}>0$ (see appendix B.3). Depending on 
$\tau _{1}$ and 
$ \tau _{2}$, the extracted quantities in the first period (second period) can increase if 
$\tau _{2}$ (
$\tau _{1}$) is high enough. Hence the recycling activity increases with the tax in the second period – as in the binding case – but decreases with the tax in the first period. Thus environmental taxes can disadvantage recycling activity.
Proposition 6 environmental taxation can reduce recycling when the recycling activity is limited by available scrap.
5. The second-best environmental regulation
The regulator determines the second-best environmental taxesFootnote 7 maximizing the welfare function (given by equation (1)), replacing quantities depending on the tax levels found in the preceding section.Footnote 8 The solution depends on the subgame-perfect Nash equilibrium, i.e., whether the recycler is limited or not by the collected scrap quantity.
5.1. The non-binding case
We replace 
$x_{1}^{{\rm nct}}(\tau _{1})$, 
$x_{2}^{{\rm nct}}(\tau _{2})$ and 
$ r^{{\rm nct}}(\tau _{2})$ in the welfare function that we maximize with respect to 
$\tau _{1}$ and 
$\tau _{2}$. The FOCs are the following:
\begin{equation} \begin{array}{@{}l@{}} \dfrac{dx_{1}}{d\tau _{1}}[P_{1}(x_{1})-C_{1}^{\prime }(x_{1})-2\alpha c_{s}-D^{\prime }(x_{1})]=0, \\ \dfrac{dx_{2}}{d\tau _{2}}[P(x_{2}+r)-C_{2}^{\prime }(x_{2})-\alpha c_{s}-D^{\prime }(x_{2})]\\ \quad +\dfrac{dr}{d\tau _{2}}[P_{2}(x_{2}+r)-C_{r}^{\prime}(r)]=0. \end{array}\end{equation}Substituting (18) and (20) into (22) yields the following pair of tax rates:
\begin{equation} \begin{array}{@{}l@{}} \tau _{1}^{{\rm nct}}=\underset{{\rm {\bf Usual\ result}}}{\underbrace{D^{\prime } (x_{1})+P^{\prime }(x_{1})}}x_{1} \\ \tau _{2}^{{\rm nct}}=\underset{{\rm {\bf Usual\ result}}}{\underbrace{D^{\prime } (x_{2})+P^{\prime }(x_{2}+r)x_{2}}}+\underset{{\rm {\bf Recycling\ effect}}}{\underbrace{\frac{\frac{dr(\tau _{2})}{d\tau _{2}}}{ \frac{dx_{2}(\tau _{2})}{d\tau _{2}}}[P_{2}^{\prime }(x_{2}+r)r]}} \end{array}.\end{equation}
The tax rate in period 1 depends only on the active distortions over this period. One is the distortion from the negative externality generated by the pollution, the other is the distortion from the extractor's market power in the market of non-abundant REEs. Since 
$P_{1}^{\prime }(x_{1})<0$, the tax rate is lower than the marginal damage (see Barnett, Reference Barnett1980).Footnote 9 The benevolent regulator sets the tax at this level in order to reduce the tendency of the monopolist to underproduce. In this case, the tax could be either positive or negative. Its sign depends on the prevailing distortion. If the distortion from the environmental damage is larger than the distortion from the extractor's market power, i.e., 
$D^{\prime }(x_{1})>-P_{1}^{\prime }(x_{1})x_{1}$, then 
$\tau _{1}^{{\rm nct}}>0$ and turns out to be a tax. Otherwise, 
$\tau _{1}^{{\rm nct}}<0$ and 
$\tau _{1}^{{\rm nct}}$ is a subsidy. It is worth noting that recycling does not influence the first period tax rate because it does not affect the extraction in that period.
According to equation (23), the second-period tax depends only on distortions in this period. It is also composed of both usual distortions but is adjusted by an additional term emanating from the recycling activity. As 
$\frac {dr(\tau _{2})}{d\tau _{2}}/\frac {dx_{2}(\tau _{2})}{d\tau _{2}}<0$ and 
$[P_{2}^{\prime }(x_{2}+r)r]<0$, the recycling effect is positive. The regulator increases further the second period tax rate in order to foster recycling. If this recycling effect is strong enough, the second period tax rate will be higher than the marginal damage. Note that the recycling effect catches the capacity of 
$\tau _{2}$ to modify the price in the second period. Thus the regulator increases the tax in the second period in order to both favor competition and reduce environmental damage. If we replace 
$ \tau _{1}^{{\rm nct}}$ and 
$\tau _{2}^{{\rm nct}}$ in equations (18) and (20), we find:
\begin{align} & P(x_{1})-C_{1}^{{\prime }}(x_{1})-2\alpha c_{s}-D^{\prime }(x_{1})=0\end{align}
\begin{align} & P(x_{2}+r)-C_{2}^{{\prime }}(x_{2})-\alpha c_{s}-D^{\prime }(x_{2})- \frac{\frac{dr(\tau _{2})}{d\tau _{2}}}{\frac{dx_{2}(\tau _{2})}{d\tau _{2}}} P^{\prime }(x_{2}+r)r=0.\end{align}As equation (24) is similar to equation (6), second-best environmental taxation in the first period enables the first-best outcome to be reached. The tax internalizes both market failures induced by market power and pollution. As equation (25) is different from equation (7), a tax in period 2 cannot simultaneously cope with distortions enacted by market power and the environmental damage while taking into account the recycled output. Second-best taxation in the second period therefore enables a second-best outcome to be reached.
The taxation effect on the balance problem depends on the sign of 
$\tau _{1}^{{\rm nct}}$ and 
$\tau _{2}^{{\rm nct}}$. The balance problem is enhanced if tax rates are negative. On the contrary, positive tax rates lead to reduced extracted quantities, thereby alleviating the balance problem.
Proposition 7 due to market power, the second-best tax in period 1 is lower than the marginal damage which leads to achieving the first-best quantity. In period 2, recycling increases the tax whereas the market power reduces it. Finally the second-best tax level in period 2 can be higher than the marginal damage.
5.2. The binding case
In this case, we replace 
$x_{1}^{{\rm ct}}(\tau _{1},\tau _{2})$, 
$x_{2}^{{\rm ct}}(\tau _{1,}\tau _{2})$ and 
$r^{{\rm ct}}(\tau _{1},\tau _{2})$ in the welfare function and maximize with respect to 
$\tau _{1}$ and 
$\tau _{2}$. After rearranging the first-order conditions, we find:
\begin{align}\begin{split} &\frac{dx_{1}}{d\tau _{1}}[P_{1}(x_{1})-C_{1}^{\prime }(x_{1})-2\alpha c_{s}-D^{\prime }(x_{1})+P_{2}(x_{2}+kx_{1})k-kC_{r}^{{\prime }}(kx_{1})]\\ &\quad +\frac{dx_{2}}{d\tau _{1}}[P_{2}(x_{2}+kx_{1})-C_{2}^{\prime }(x_{2})-\alpha c_{s}-D^{\prime }(x_{2})]=0, \\ &\frac{dx_{1}}{d\tau _{2}}[P_{1}(x_{1})-C_{1}^{\prime }(x_{1})-2\alpha c_{s}-D^{\prime }(x_{1})+P_{2}(x_{2}+kx_{1})k-kC_{r}^{\prime }(kx_{1})] \\ &\quad +\frac{dx_{2}}{d\tau _{2}}[P_{2}(x_{2}+kx_{1})-C_{2}^{\prime }(x_{2})-\alpha c_{s}-D^{\prime }(x_{2})]=0. \end{split}\end{align}Substituting (19) and (21) in (26), we find the following taxes:
\begin{equation}\begin{array}{@{}l@{}} \tau _{1}^{{\rm ct}}=\underset{{\rm {\bf Usual\ result}}}{\underbrace{P^{\prime} (x_{1})x_{1}+D^{\prime }(x_{1})}}\underset{{\rm {\bf Recycling\ effect}}} {\underbrace{-k[P_{2}(x_{2}+kx_{1})-C_{r}^{\prime }(kx_{1})-P_{2}^{\prime} (x_{2}+kx_{1})x_{2}]}} \\ \tau _{2}^{{\rm ct}}=\underset{{\rm {\bf Usual\ result}}}{\underbrace{P_{2}^{\prime } (x_{2}+kx_{1})x_{2}+D^{\prime }(x_{2})}}\,\,.\end{array}\end{equation}
As the quantity extracted in period 1 has an effect on period 2, the design of 
$\tau _{1}^{{\rm ct}}$ has to consider effects in both periods. The first two terms catch usual effects in period 1 and other terms take into account effects in period 2. 
$\tau _{1}^{{\rm ct}}$ diminishes with the marginal profit of the recycler and with the price variation induced by recycling. Finally 
$ \tau _{1}^{{\rm ct}}$ is always inferior to the marginal damage and so is 
$\tau _{2}^{{\rm ct}}$. Replacing both taxes in (19) and (21) yields the equations (9) and (10). The regulator is able to implement the first-best outcome provided the recycling constraint is binding.
Proposition 8 the second-best taxation scheme enables the first-best quantities to be reached in each period. Tax in period 1 
$($period 
$2)$ takes into account market power and recycling 
$($market power
$)$. Both taxes are lower than the marginal damage.
6. Conclusion
The extraction of REEs raises serious pollution problems and leads to the balance problem. This paper theoretically analyzed the effect of recycling and environmental tax regulation on both the balance problem and pollution. It contributes to the theoretical analysis of green policies aiming at downscaling resource use while promoting recycling activities.
We set up a Cournot model in which one firm involved in the extraction sector simultaneously supplies two types of REEs – abundant and non-abundant – over two consecutive periods. In the second period, it competes with a recycler of non-abundant REEs that are used in the first period. We first showed that recycling always reduces extracted quantities, thus mitigating both the balance problem and the environmental damages. Our results crucially depend on whether the recycler can recycle the whole quantity of scrap it wants. If its activity is not limited, the first period extracted quantities are unchanged. Otherwise, the extractor adopts a foreclosure strategy that consists of reducing its extraction in the first period. Because there are distortions in the economy (pollution and market power), both equilibria are not optimal. Owing to those distortions, second-best environmental taxation is introduced in each period. We showed that environmental taxation always favors recycling when the constraint on the scrap availability is not binding, whereas recycled quantities can decrease if the constraint is binding. From this point of view, the regulator should pay careful attention when shaping environmental taxation in order to indirectly favor recycling. In addition, we established that the second-best levels of environmental taxes depend on the marginal damage, on the market power as well as on the recycling.
This article is, according to our knowledge, the first to theoretically investigate the balance problem and to design environmental taxes in the presence of recycling. It can be considered as providing several theoretical insights that could be valuable for the proper functioning of the circular economy. It is important to note that this normative analysis is conducted on a global scale since the extractor and the recycler belong to the same economy. It, therefore, overlooks the fact that rare earth ores are mainly extracted in China and that recycling activities are disseminated in several countries. Taking into account this geopolitical specificity would warrant another game theoretical set-up, i.e., a game between a monopoly and a competitive fringe under the assumption that REEs are traded without transportation costs. The environmental taxes would also be implemented in the extracting country while not taking into account the global consumer surplus, as it is done in this article – but only the consumer surplus from the quantity consumed in this country. These new assumptions would not fundamentally change the extractor strategy but, rather, the environmental tax levels. According to our results, if China implements environmental taxes on extraction of REEs, recycling would not necessarily be boosted. China could instead strategically implement environmental taxes in order to reduce recycling if the available scrap is low.
This paper can be extended in several directions. We do not consider that the stock of unsold abundant REEs can be polluting. Likewise, we do not take into account that REEs recycling activity can also be polluting. This would lead the regulator to potentially consider other damages and, consequently, to introduce other environmental taxes. That would change the level of recycling and therefore extraction. We also neglect the strategic aspects for a country related to the holding of REEs. In this case, the regulator may wish to develop recycling to ensure the security of rare earth supply (Golev et al., Reference Golev, Scott, Erskine, Ali and Ballantyne2014). An extension of our paper would be to not consider the balance problem but the environmental tax implementation when there is an equilibrium on the byproduct market. Lastly, it would be interesting to extend this model over infinite time in order to assess whether the asymmetry between periods would still hold. This analysis would take into account the REE stock. Further research is needed to investigate these different questions.
Appendix A: The non-binding case
A.1. The first-best
Concavity of the program
From (2), (3) and (4) we find: 
$H(W(x_{1},x_{2},r))=\left [\arraycolsep3pt\begin {smallmatrix} P_{1}^{\prime }-C_{1}^{\prime \prime }-D_{1}^{\prime \prime } & 0 & 0 \\ 0 & P_{2}^{\prime }-C_{2}^{\prime \prime }-D_{2}^{\prime \prime } & P_{2}^{\prime } \\ 0 & P_{2}^{\prime } & P_{2}^{\prime }-C_{r}^{\prime \prime } \end {smallmatrix}\right ]$, with 
$M_{1}=P_{1}^{\prime }-C_{1}^{\prime \prime }-D_{1}^{\prime \prime }<0$; 
$M_{2}=[P_{1}^{\prime }-C_{1}^{\prime \prime }-D_{1}^{\prime \prime }][P_{2}^{\prime }-C_{2}^{\prime \prime }-D_{2}^{\prime \prime }]>0$; 
$M_{3}=[P_{1}^{\prime }-C_{1}^{\prime \prime }-D_{1}^{\prime \prime }]\{[P_{2}^{\prime }-C_{2}^{\prime \prime }-D_{2}^{\prime \prime }][P_{2}^{\prime }-C_{r}^{\prime \prime }]-P_{2}^{\prime 2}\}=[P_{1}^{\prime }-C_{1}^{\prime \prime }-D_{1}^{\prime \prime }]\underset {\nabla }{ \underbrace {\{[-C_{2}^{\prime \prime }-D_{2}^{\prime \prime }][P_{2}^{\prime }-C_{r}^{\prime \prime }]+P_{2}^{\prime }[-C_{r}^{\prime \prime }]\}}}<0$.
Effect of a change in $c_{s}$
$x_{1}^{\ast nc}(c_{s})$ solves 
$P_{1}(x_{1}^{\ast nc})-C_{1}^{\prime }(x_{1}^{\ast nc})-2\alpha c_{s}-D^{\prime }(x_{1}^{\ast nc})=0.$ We set 
$F(x_{1},c_{s})=P_{1}(x_{1})-C_{1}^{\prime }(x_{1})-2\alpha c_{s}-D^{\prime }(x_{1}).$ We apply the Implicit Function Theorem, and we find:
\begin{align*}& \frac{\partial x_{1}^{\ast nc}}{\partial c_{s}}=-\frac{\partial F(x_{1},c_{s})/\partial c_{s}}{\partial F(x_{1},c_{s})/\partial x_{1}}= -\frac{-2\alpha }{M_{1}}<0\end{align*}
$x_{2}^{\ast nc}(c_{s})\ {\rm and}\ r^{\ast nc}(c_{s})$ solve:
\begin{align*}\left\{\begin{array}{@{}l@{}} P_{2}(x_{2}^{\ast nc}(c_{s})+r^{\ast nc}(c_{s}))-C_{2}^{\prime }(x_{2}^{\ast nc} (c_{s}))-\alpha c_{s}-D_{2}^{\prime }(x_{2}^{\ast nc}(c_{s}))=0 \\ P_{2}(x_{2}^{\ast nc}(c_{s})+r^{\ast nc}(c_{s}))-C_{r}^{\prime}(r^{\ast nc}(c_{s}))=0. \end{array}\right. \end{align*}
If we differentiate this system with respect to 
$c_{s}$, we obtain after simplification:
\begin{align*} & \left\{\begin{array}{@{}l@{}} P_{2}^{\prime }\dfrac{dx_{2}^{\ast nc}}{dc_{s}}+P_{2}^{\prime } \dfrac{dr^{\ast nc}}{dc_{s}}-C_{2}^{\prime \prime } \dfrac{dx_{2}^{\ast nc}}{dc_{s}}-D_{2}^{\prime \prime } \dfrac{dx_{2}^{\ast nc}}{dc_{s}}=\alpha \\ P_{2}^{\prime }\dfrac{dx_{2}^{\ast nc}}{dc_{s}}+P_{2}^{\prime } \dfrac{dr^{\ast nc}}{dc_{s}}-C_{r}^{\prime \prime } \dfrac{dr^{\ast nc}}{dc_{s}}=0 \end{array}\right.\\ & \left[\begin{array}{@{}cc@{}} P_{2}^{\prime }-C_{2}^{\prime \prime }-D_{2}^{\prime \prime } & P_{2}^{\prime } \\ P_{2}^{\prime } & P_{2}^{\prime }-C_{r}^{\prime \prime } \end{array}\right] \left[\begin{array}{@{}c@{}} \dfrac{dx_{2}^{\ast nc}}{dc_{s}} \\ \dfrac{dr^{\ast nc}}{dc_{s}} \end{array}\right] =\left[\begin{array}{@{}c@{}} \alpha \\ 0 \end{array}\right]\\ & \left[\begin{array}{@{}c@{}} \dfrac{dx_{2}^{\ast nc}}{dc_{s}} \\ \dfrac{dr^{\ast nc}}{dc_{s}} \end{array}\right] =\frac{1}{\nabla }\left[\begin{array}{@{}c@{}} \alpha (P_{2}^{\prime }-C_{r}^{\prime \prime }) \\ -\alpha P_{2}^{\prime } \end{array}\right]\quad {\rm with\ the\ property\ that}\ \left[\begin{array}{@{}c@{}} (<0) \\ (>0) \end{array}\right],\quad {\rm with}\ \nabla >0. \end{align*}A.2. The recycling activity
Stability of the equilibrium.
As we have 
$\pi _{x_{2}x_{2}}^{e}=2P_{2}^{\prime }+P_{2}^{\prime \prime }x_{2}-C_{2}^{\prime \prime }<0$; 
$\pi _{x_{2}r}^{e}=P_{2}^{\prime }+P_{2}^{\prime \prime }x_{2}<0$; 
$\pi _{rr}^{r}=2P_{2}^{\prime }+P_{2}^{\prime \prime }r-C_{r}^{\prime \prime }<0$; 
$\pi _{rx_{2}}^{r}=P_{2}^{\prime }+P_{2}^{\prime \prime }r<0$. We find: 
$\pi _{x_{2}x_{2}}^{e}<\pi _{x_{2}r}^{e}<0$ and 
$\pi _{rr}^{r}<\pi _{x_{2}r}^{r}<0$, so 
$\Delta =\pi _{x_{2}x_{2}}^{e}\pi _{rr}^{r}-\pi _{x_{2}r}^{r}\pi _{rx_{2}}^{e}>0$. As 
$\pi _{x_{2}x_{2}}^{e}<0,\pi _{rr}^{r}<0$ and 
$\Delta >0,$ the Gale-Nikaido condition is satisfied, meaning global uniqueness of the Cournot equilibrium. 
$\Delta >0$ is also the Routh-Hurwitz condition for reaction function stability. As 
$\pi _{x_{2}r}^{e}<0$ and 
$\pi _{rx_{2}}^{r}<0$, the quantity 
$x_{2}$ and r are strategic substitutes.
Effect of a change in $c_{s}$
From (16), 
$x_{1}^{{\rm nc}}(c_{s})$ solves 
$P_{1}(x_{1}^{{\rm nc}})+P_{1}^{\prime }(x_{1}^{{\rm nc}})x_{1}^{{\rm nc}}-C_{1}^{\prime }(x_{1}^{{\rm nc}})-2\alpha c_{s}=0$. We set: 
$F(x_{1},c_{s})=\pi _{x_{1}}=P_{1}(x_{1})+P_{1}^{\prime }(x_{1})x_{1}-C_{1}^{\prime }(x_{1})-2\alpha c_{s}$. We apply the Implicit Function Theorem, and we find:
\[ \frac{\partial x_{1}^{{\rm nc}}}{\partial c_{s}}=-\frac{\partial F(x_{1},c_{s})/\partial c_{s}}{\partial F(x_{1},c_{s})/\partial x_{1}}=- \frac{-2\alpha }{2P_{1}^{\prime }+P_{1}^{\prime \prime}x_{1}-C_{1}^{\prime \prime }}= -\frac{-2\alpha }{\pi _{x_{1}x_{1}}}<0. \]
$x_{2}^{{\rm nc}}(c_{s})$ and 
$r^{{\rm nc}}(c_{s})$ solve:
\[ \left\{\begin{array}{@{}l@{}} P_{2}(x_{2}^{{\rm nc}}(c_{s})+r^{{\rm nc}}(c_{s}))+P_{2}^{\prime} (x_{2}^{{\rm nc}}(c_{s})+r^{{\rm nc}}(c_{s}))x_{2}^{{\rm nc}}(c_{s})-C_{2}^{\prime} (x_{2}^{{\rm nc}}(c_{s}))=\alpha c_{s} \\ P_{2}(x_{2}^{{\rm nc}}(c_{s})+r^{{\rm nc}}(c_{s}))+P_{2}^{\prime} (x_{2}^{{\rm nc}}(c_{s})+r^{{\rm nc}}(c_{s}))r^{{\rm nc}}(c_{s})-C_{r}^{\prime} (r^{{\rm nc}}(c_{s}))=0. \end{array}\right. \]
If we differentiate this system with respect to 
$c_{s}$, we obtain after simplification:
\begin{align*} & \left\{\begin{array}{@{}l@{}} \dfrac{dx_{2}^{{\rm nc}}}{dc_{s}}[2P_{2}^{\prime }+P_{2}^{\prime \prime} x_{2}^{{\rm nc}}-C_{2}^{\prime \prime }]+\dfrac{dr^{{\rm nc}}}{dc_{s}}[P_{2}^{\prime}+ P_{2}^{\prime \prime }x_{2}^{{\rm nc}}]=\alpha \\ \frac{dx_{2}^{{\rm nc}}}{dc_{s}}[P_{2}^{\prime }+P_{2}^{\prime \prime }r^{{\rm nc}}]+ \dfrac{dr^{{\rm nc}}}{dc_{s}}[2P_{2}^{\prime }+P_{2}^{\prime \prime} r^{{\rm nc}}-C_{r}^{\prime \prime }]=0 \end{array}\right. \\ & \left\{\begin{array}{@{}l@{}} \dfrac{dx_{2}^{{\rm nc}}}{dc_{s}}[\pi _{x_{2}x_{2}}^{e}]+\dfrac{dr^{{\rm nc}}}{dc_{s}} [\pi_{x_{2}r}^{e}]=\alpha \\ \dfrac{dx_{2}^{{\rm nc}}}{dc_{s}}[\pi _{rx_{2}}^{r}]+\dfrac{dr^{{\rm nc}}}{dc_{s}}[\pi_{rr}^{r}]=0 \end{array}\right. \\ & \left[\begin{array}{@{}c@{}} \dfrac{dx_{2}^{{\rm nc}}}{dc_{s}} \\ \dfrac{dr^{{\rm nc}}}{dc_{s}} \end{array}\right] =\frac{1}{\Delta }\left[\begin{array}{@{}cc@{}} \pi _{rr}^{r} & -\pi _{rx_{2}}^{e} \\ -\pi _{x_{2}r}^{r} & \pi _{x_{2}x_{2}}^{e} \end{array}\right] \left[\begin{array}{@{}c@{}} \alpha \\ 0 \end{array}\right]\\ & \left[\begin{array}{@{}c@{}} \dfrac{dx_{2}^{{\rm nc}}}{dc_{s}} \\ \dfrac{dr^{{\rm nc}}}{dc_{s}} \end{array}\right] =\frac{1}{\Delta }\left[\begin{array}{@{}c@{}} \alpha \pi _{rr}^{r} \\ -\alpha \pi _{x_{2}r}^{r} \end{array}\right]\quad {\rm with\ the\ property\ that}\\ &\left[\begin{array}{@{}c@{}} (<0) \\ (>0) \end{array}\right]\quad {\rm with}\ \Delta =\pi _{x_{2}x_{2}}^{e}\pi _{rr}^{r}- \pi_{x_{2}r}^{r}\pi _{rx_{2}}^{e}>0. \end{align*}A.3. Exogenous environmental regulation
Effect of a change in $\tau _{1}$
$x_{1}^{{\rm nct}}(c_{s},\tau _{1})$ solves: 
$P_{1}^{\prime }(x_{1}^{{\rm nct}}(c_{s},\tau _{1}))x_{1}^{{\rm nct}}(c_{s},\tau _{1})+P_{1}(x_{1}^{{\rm nct}}(c_{s},\tau _{1}))-C_{1}^{\prime }(x_{1}^{{\rm nct}}(c_{s},\tau _{1}))-2\alpha c_{s}-\tau _{1}=0.$ We set: 
$ F(x_{1},c_{s},\tau _{1})=P_{1}^{\prime }(x_{1})x_{1}+P_{1}(x_{1})-C_{1}^{\prime }(x_{1})-2\alpha c_{s}-\tau _{1}$. We apply the Implicit Function Theorem, and we find: 
$\frac {\partial x_{1}^{{\rm nct}}}{\partial \tau _{1}}=-\frac {\partial F(x_{1},c_{s},\tau _{1})/\partial \tau _{1}}{\partial F(x_{1},c_{s},\tau _{1})/\partial x_{1}}=- \frac {-1}{P_{1}^{\prime \prime }x_{1}+2P_{1}^{\prime }-C_{1}^{\prime \prime }(x_{1})}<0$.
Effect of a change in $\tau _{2}$
$x_{2}^{{\rm nct}}(c_{s},\tau _{2})$ and 
$r^{{\rm nct}}(c_{s},\tau _{2})$ solve:
\[ \left\{\begin{array}{@{}l@{}} P_{2}^{\prime }(x_{2}^{{\rm nct}}(c_{s},\tau _{2})+r^{{\rm nct}}(c_{s},\tau_{2})) x_{2}^{{\rm nct}}(c_{s},\tau _{2})+P_{2}(x_{2}^{{\rm nct}}(c_{s},\tau_{2})+ r^{{\rm nct}}(c_{s},\tau _{2}))-C_{2}^{\prime }(x_{2}^{{\rm nct}}(c_{s},\tau_{2}))\\ \quad -\alpha c_{s}-\tau _{2}=0 \\ P_{2}(x_{2}^{{\rm nct}}(c_{s},\tau _{2})+r^{{\rm nct}}(c_{s},\tau _{2}))+P_{2}^{\prime} (x_{2}^{{\rm nct}}(c_{s},\tau _{2})+r^{{\rm nct}}(c_{s},\tau _{2})r^{{\rm nct}}-C_{r}^{\prime} (r^{^{{\rm nct}}}(c_{s},\tau _{2}))=0. \end{array}\right. \]
If we differentiate this system with respect to 
$\tau _{2}$, we obtain after simplification:
\begin{align*} & \left\{\begin{array}{@{}l@{}} \dfrac{dx_{2}^{{\rm nct}}}{d\tau _{2}}[2P_{2}^{\prime }+P_{2}^{\prime \prime }x_{2}^{{\rm nct}}-C_{2}^{\prime \prime }]+\dfrac{dr^{{\rm nct}}}{d\tau _{2}} [P_{2}^{\prime }+P_{2}^{\prime \prime }x_{2}^{{\rm nct}}]=1 \\ \dfrac{dx_{2}^{{\rm nct}}}{d\tau _{2}}[P_{2}^{\prime }+P_{2}^{\prime \prime }r^{{\rm nct}}]+\dfrac{dr^{{\rm nct}}}{d\tau _{2}}[2P_{2}^{\prime }+P_{2}^{\prime \prime }r^{{\rm nct}}-C_{r}^{\prime \prime }]=0 \end{array}\right.\\ & \left\{\begin{array}{@{}l@{}} \dfrac{dx_{2}^{{\rm nct}}}{d\tau _{2}}[\pi _{x_{2}x_{2}}^{e}]+ \dfrac{dr^{{\rm nct}}}{d\tau_{2}}[\pi _{x_{2}r}^{e}]=1 \\ \dfrac{dx_{2}^{{\rm nct}}}{d\tau _{2}}[\pi _{rx_{2}}^{r}]+ \dfrac{dr^{{\rm nct}}}{d\tau _{2}}[\pi _{rr}^{r}]=0 \end{array}\right.\\ & \left[\begin{array}{@{}c@{}} \dfrac{dx_{2}^{{\rm nct}}}{d\tau _{2}} \\ \dfrac{dr^{{\rm nct}}}{d\tau _{2}} \end{array}\right] =\frac{1}{\Delta }\left[\begin{array}{@{}cc@{}} \pi _{rr}^{r} & -\pi _{rx_{2}}^{e} \\ -\pi _{x_{2}r}^{r} & \pi _{x_{2}x_{2}}^{e} \end{array}\right] \left[\begin{array}{@{}c@{}} 1 \\ 0 \end{array}\right]\\ & \left[\begin{array}{@{}c@{}} \dfrac{dx_{2}^{{\rm nct}}}{d\tau _{2}} \\ \dfrac{dr^{{\rm nct}}}{d\tau _{2}} \end{array}\right] =\frac{1}{\Delta }\left[\begin{array}{@{}c@{}} \pi _{rr}^{r} \\ -\pi _{x_{2}r}^{r} \end{array}\right]\quad {\rm with\ the\ property\ that}\ \left[\begin{array}{@{}c@{}} (<0) \\ (>0) \end{array}\right]\quad {\rm with}\ \Delta >0. \end{align*}Appendix B: The binding case
We assume 
$P^{\prime \prime }=0$ and 
$C^{\prime \prime \prime }=0$.
B.1. The first-best
Concavity of the program.
\[ H(W(x_{1},x_{2},r))=\left[\begin{array}{@{}cc@{}} P_{1}^{\prime }-C_{1}^{\prime \prime }-D_{1}^{\prime \prime}+ k^{2}P_{2}^{\prime }-k^{2}C_{r}^{\prime \prime } & kP_{2}^{\prime } \\ kP_{2}^{\prime } & P_{2}^{\prime }-C_{2}^{\prime \prime }-D_{2}^{\prime\prime} \end{array}\right] \]
$W_{x_{1}x_{1}}=[P_{1}^{\prime }-C_{1}^{\prime \prime }-D_{1}^{\prime \prime }+k^{2}P_{2}^{\prime }-k^{2}C_{r}^{\prime \prime }]<0$; 
$W_{x_{1}x_{2}}=kP_{2}^{\prime }=W_{x_{2}x_{1}}<0$; 
$W_{x_{2}x_{2}}=P_{2}^{\prime }-C_{2}^{\prime \prime }-D_{2}^{\prime \prime }<0$.
${\rm Det}\,H=\Psi =[P_{1}^{\prime }-C_{1}^{\prime \prime }-D_{1}^{\prime \prime }+k^{2}P_{2}^{\prime }-k^{2}C_{r}^{\prime \prime }][P_{2}^{\prime }-C_{2}^{\prime \prime }-D_{2}^{\prime \prime }]-[kP_{2}^{\prime }]^{2}=[P_{1}^{\prime }-C_{1}^{\prime \prime }-D_{1}^{\prime \prime }-k^{2}C_{r}^{\prime \prime }][P_{2}^{\prime }-C_{2}^{\prime \prime }-D_{2}^{\prime \prime }]+k^{2}P_{2}^{\prime }[-C_{2}^{\prime \prime }-D_{2}^{\prime \prime }]>0$.
Effect of a change in $\mathbf {c}_{s}$
$x_{1}^{\ast c}(c_{s})$ and 
$x_{2}^{\ast c}(c_{s})$ solve:
\[ \left\{\begin{array}{@{}l@{}} P_{1}(x_{1}^{\ast c}(c_{s}))-C_{1}^{\prime }(x_{1}^{\ast c}(c_{s}))-2\alpha c_{s}-D_{1}^{\prime }(x_{1}^{\ast c}(c_{s}))+k[P_{2}(x_{2}^{\ast c}(c_{s})+kx_{1}^{\ast c}(c_{s}))\\ \quad -C_{r}^{\prime }(kx_{1}^{\ast c}(c_{s}))]=0\\ P_{2}(x_{2}^{\ast c}(c_{s})+kx_{1}^{\ast c}(c_{s}))-C_{2}^{\prime} (x_{2}^{\ast c}(c_{s}))-\alpha c_{s}-D_{2}^{\prime }(x_{2}^{\ast c}(c_{s}))=0. \end{array}\right. \]
If we differentiate this system with respect to 
$C_{s}$, we obtain after simplification:
\begin{align*} & \left\{\begin{array}{@{}l@{}} P_{1}^{\prime }\dfrac{dx_{1}^{\ast c}}{dc_{s}}-C_{1}^{\prime \prime } \dfrac{dx_{1}^{\ast c}}{dc_{s}}-D_{1}^{\prime \prime }\dfrac{dx_{1}^{\ast c}}{dc_{s}} +k[P_{2}^{\prime }\left[\dfrac{dx_{2}^{\ast c}}{dc_{s}}+k \dfrac{dx_{1}^{\ast c}}{dc_{s}}\right]-k^{2}C_{r}^{\prime \prime } \dfrac{dx_{1}^{\ast c}}{dc_{s}}]=2\alpha\\ P_{2}^{\prime }\left[\dfrac{dx_{2}^{\ast c}}{dc_{s}}+k \dfrac{dx_{1}^{\ast c}}{dc_{s}}\right]-C_{2}^{\prime \prime } \dfrac{dx_{2}^{\ast c}}{dc_{s}}-D_{2}^{\prime \prime}\dfrac{dx_{2}^{\ast c}}{dc_{s}}=\alpha \end{array}\right.\\ & \left\{\begin{array}{@{}l@{}} \dfrac{dx_{1}^{\ast c}}{dc_{s}}[P_{1}^{\prime }-C_{1}^{\prime \prime}- D_{1}^{\prime \prime }+k^{2}P_{2}^{\prime }-k^{2}C_{r}^{\prime \prime }]+ \dfrac{dx_{2}^{\ast c}}{dc_{s}}kP_{2}^{\prime }=2\alpha \\ \dfrac{dx_{2}^{\ast c}}{dc_{s}}[P_{2}^{\prime }-C_{2}^{\prime \prime}- D_{2}^{\prime \prime }]+P_{2}^{\prime }k\dfrac{dx_{1}^{\ast c}}{dc_{s}}]=\alpha \end{array}\right.\\ & \left[\begin{array}{@{}c@{}} \dfrac{dx_{1}^{\ast c}}{dc_{s}} \\ \dfrac{dx_{2}^{\ast c}}{dc_{s}} \end{array}\right] =\frac{1}{\Psi }\left[\begin{array}{@{}cc@{}} P_{2}^{\prime }-C_{2}^{\prime \prime }-D_{2}^{\prime \prime } & -kP_{2}^{\prime } \\ -kP_{2}^{\prime } & P_{1}^{\prime }-C_{1}^{\prime \prime }-D_{1}^{\prime \prime }+k^{2}P_{2}^{\prime }-k^{2}C_{r}^{\prime \prime } \end{array}\right] \left[\begin{array}{@{}c@{}} 2\alpha \\ \alpha \end{array}\right]\\ & {\rm with}\ \Psi >0,\ {\rm we\ obtain}: \left\{\begin{array}{@{}l@{}} \dfrac{dx_{1}^{\ast c}}{dc_{s}}=\dfrac{\alpha }{\Psi }\{P_{2}^{\prime} (2-k)-2C_{2}^{\prime \prime }-2D^{\prime \prime }]\}<0 \\ \dfrac{dx_{2}^{\ast c}}{dc_{s}}=\dfrac{\alpha }{\Psi } \{P^{\prime}[k^{2}-2k+1]-C_{1}^{\prime \prime }-D_{1}^{\prime \prime}-k^{2}C_{r}^{\prime \prime }\}<0 \end{array}\right. \end{align*}B.2. The recycling activity
Variation of $x_{2}^{c}$ with respect to $x_{1:}^{c}$
From (15), we note 
$ F(x_{1},x_{2})=P_{2}(x_{2}+kx_{1})+P_{2}^{\prime }(x_{2}+kx_{1})x_{2}-C_{2}^{{\prime }}(x_{2})-\alpha c_{s}=0$. So 
$\frac {\partial x_{2}^{c}}{\partial x_{1}^{c}}=-\frac {\partial F(x_{2},x_{1})/\partial x_{1}}{\partial F(x_{2},x_{1})/\partial x_{2}}=- \frac {kP_{2}^{\prime }}{2P_{2}^{\prime }-C_{2}^{\prime \prime }}<0$.
Concavity of the profit in the first stage
From (17), 
$\Pi _{x_{1}}^{e}=P_{1}(x_{1}^{c})+P_{1}^{\prime }(x_{1}^{c})x_{1}^{c}-C_{1}^{{\prime }} (x_{1}^{c})-2\alpha c_{s}+\frac {dx_{2}^{c}}{dx_{1}^{c}}[P_{2}(x_{2}^{c}(x_{1}^{c})+kx_{1}^{c})+P_{2}^{\prime }(x_{2}^{c}\break (x_{1}^{c})+kx_{1}^{c})x_{2}^{c}(x_{1}^{c})-C_{2}^{\prime }(x_{2}^{c}(x_{1}^{c}))-\alpha c_{s}]+kP_{2}^{\prime }(x_{2}^{c}(x_{1}^{c})+kx_{1}^{c})x_{2}^{c}(x_{1}^{c})$
\begin{align*} \Pi _{x_{1}x_{1}}^{e} &= 2P_{1}^{\prime }-C_{1}^{\prime \prime }+ \frac{dx_{2}^{c}}{dx_{1}^{c}}\{ \frac{dx_{2}^{c}}{dx_{1}^{c}} [2P_{2}^{\prime }-C_{2}^{\prime \prime }]+2kP_{2}^{\prime }\} \\ &= \frac{1}{[2P_{2}^{\prime }-C_{2}^{\prime \prime }]}\{[2P_{1}^{\prime }-C_{1}^{\prime \prime }][2P_{2}^{\prime }-C_{2}^{\prime \prime}]-[kP_{2}^{\prime }]^{2}\}\\ &= \frac{1}{[2P_{2}^{\prime}-C_{2}^{\prime \prime }]}\{ \underset{ \Theta >0}{\underbrace{[P^{\prime }]^{2}(4-k)-C_{1}^{\prime \prime }[2P_{2}^{\prime }-C_{2}^{\prime \prime }]-2P_{1}^{\prime }C_{2}^{\prime \prime }}}\}<0. \end{align*} Effect of a change in $c_{s}$
At the equilibrium, 
$x_{1}^{c}(c_{s})$ and 
$x_{2}^{c}(c_{s})$ solve:
\[ \left\{\begin{array}{@{}l@{}} P_{1}(x_{1}^{c})+P_{1}^{\prime }(x_{1}^{c})x_{1}^{c}-C_{1}^{{\prime }}(x_{1}^{c})-2\alpha c_{s}+ \dfrac{dx_{2}^{c}}{dx_{1}^{c}}[P_{2}(x_{2}^{c}+kx_{1}^{c})\\\quad +P_{2}^{\prime}(x_{2}^{c}+kx_{1}^{c})x_{2}^{c}-C_{2}^{\prime }(x_{2}^{c})-\alpha c_{s}]+ kP_{2}^{\prime }(x_{2}^{c}+kx_{1}^{c})x_{2}^{c}=0 \\ P_{2}(x_{2}^{c}+kx_{1}^{c})+P_{2}^{\prime} (x_{2}^{c}+kx_{1}^{c})x_{2}^{c}-C_{2}^{\prime }(x_{2}^{c})-\alpha c_{s}=0. \end{array}\right. \]
If we differentiate this system with respect to 
$c_{s}$, we obtain:
\begin{align*} & \left\{\begin{array}{@{}l@{}} \dfrac{dx_{1}^{c}}{dc_{s}}[2P_{1}^{\prime }+P_{1}^{\prime \prime}x_{1}^{c}-C_{1}^{\prime \prime } +\dfrac{dx_{2}^{c}(x_{1}^{c})}{dx_{1}^{c}}[P_{2}^{\prime }k+kP_{2}^{\prime \prime }x_{2}^{c}+ k^{2}P_{2}^{\prime \prime}x_{2}^{c}]\\ \quad +\dfrac{dx_{2}^{c}}{dc_{s}} \left[kP_{2}^{\prime \prime}x_{2}^{c}+kP_{2}^{\prime }+\dfrac{dx_{2}^{c}(x_{1}^{c})}{dx_{1}^{c}} (2P_{2}^{\prime }-C_{2}^{\prime \prime }+P_{2}^{\prime \prime }x_{2})\right] =2\alpha +\dfrac{dx_{2}^{c}(x_{1}^{c})}{dx_{1}^{c}}\alpha\\ \dfrac{dx_{1}^{c}}{dc_{s}}[kP_{2}^{\prime }+kP_{2}^{\prime \prime }x_{2}^{c}]+ \dfrac{dx_{2}^{c}}{dc_{s}}[2P_{2}^{\prime }+P_{2}^{\prime \prime}x_{2}^{c}-C_{2}^{\prime \prime }]=\alpha \end{array}\right.\\ & \left[\begin{array}{@{}c@{}} \dfrac{dx_{1}^{c}}{dc_{s}} \\ \dfrac{dx_{2}^{c}}{dc_{s}} \end{array}\right] =\frac{1}{\Theta}\left[\begin{array}{@{}cc@{}} 2P_{2}^{\prime }-C_{2}^{\prime \prime } & -\left[kP_{2}^{\prime }+ \dfrac{dx_{2}^{c}(x_{1}^{c})}{dx_{1}^{c}}(2P_{2}^{\prime }-C_{2}^{\prime \prime })\right]\\ -[kP_{2}^{\prime }] & 2P_{1}^{\prime }-C_{1}^{\prime \prime }+ \dfrac{dx_{2}^{c}(x_{1}^{c})}{dx_{1}^{c}}[P_{2}^{\prime }k] \end{array}\right] \left[\begin{array}{@{}c@{}} \alpha \left[2+\dfrac{dx_{2}^{c}(x_{1}^{c})}{dx_{1}^{c}}\right] \\ \alpha \end{array}\right]\\ & {\rm With}\ \Theta =[2P_{1}^{\prime }-C_{1}^{\prime \prime}] [2P_{2}^{\prime }-C_{2}^{\prime }]-[kP_{2}^{\prime }]^{2}>0,\\ &{\rm we\ find:}\ \left\{\begin{array}{@{}l@{}} \dfrac{dx_{1}^{c}}{dc_{s}}=\dfrac{1}{\Theta }[\alpha P_{2}^{\prime} [4-k]-2\alpha C_{2}^{\prime \prime }]<0 \\ \dfrac{dx_{2}^{c}}{dc_{s}}=\frac{\alpha }{\Theta }[2P^{\prime}(1-k)-C_{1}^{\prime \prime }]<0. \end{array}\right. \end{align*}B.3. Exogenous environmental regulation
Effect of a change in $\tau _{1}$ and $\tau _{2}$
$x_{1}^{{\rm ct}}(c_{s},\tau _{1},\tau _{2})$ and 
$x_{2}^{{\rm ct}}(c_{s},\tau _{1},\tau _{2})$ solve:
\[ \left\{\begin{array}{@{}l@{}} P_{1}(x_{1}^{{\rm ct}}(c_{s},\tau _{1},\tau _{2}))+P_{1}^{\prime} (x_{1}^{{\rm ct}}(c_{s},\tau _{1},\tau _{2}))x_{1}^{{\rm ct}}(c_{s},\tau _{1},\tau_{2})-C_{1}^{\prime } (x_{1}^{{\rm ct}}(c_{s},\tau _{1},\tau _{2}))-2\alpha c_{s}-\tau _{1} \\ \quad +\,\dfrac{dx_{2}^{c}}{dx_{1}^{c}}[P_{2}(x_{2}^{{\rm ct}}(c_{s},\tau _{1}, \tau_{2})+kx_{1}^{{\rm ct}}(c_{s},\tau _{1},\tau _{2}))+P_{2}^{\prime} (x_{2}^{{\rm ct}}(c_{s},\tau _{1},\tau _{2})\\ \qquad +\,kx_{1}^{{\rm ct}}(c_{s},\tau _{1},\tau_{2})) x_{2}^{{\rm ct}}(c_{s},\tau _{1},\tau _{2}) -C_{2}^{\prime }(x_{2}^{{\rm ct}}(c_{s},\tau _{1},\tau _{2}))\\\noalign{\vskip5pt} \quad-\,\alpha c_{s}-\tau_{2}]+kP_{2}^{\prime } (x_{2}^{{\rm ct}}(c_{s},\tau _{1},\tau_{2})+kx_{1}^{{\rm ct}}(c_{s},\tau _{1},\tau _{2}))x_{2}^{{\rm ct}} (c_{s},\tau _{1},\tau_{2})=0 \\ P_{2}(x_{2}^{{\rm ct}}(c_{s},\tau _{1},\tau _{2})+kx_{1}^{{\rm ct}}(c_{s},\tau _{1},\tau_{2}))+ P_{2}^{\prime }(x_{2}^{{\rm ct}}(c_{s},\tau _{1},\tau_{2})+kx_{1}^{{\rm ct}}(c_{s},\tau _{1},\tau _{2}))x_{2} \\ \quad -\,C_{2}^{{\prime }}(x_{2}^{{\rm ct}}(c_{s},\tau _{1},\tau_{2}))-\alpha C_{s}-\tau _{2}=0. \end{array}\right. \]
If we differentiate this system with respect to 
$\tau _{1}$ and 
$\tau _{2}$, we obtain after simplification:
\begin{align*} & \left\{\begin{array}{@{}l@{}} 2P_{1}^{\prime }\dfrac{dx_{1}^{{\rm ct}}}{d\tau _{1}}-C_{1}^{\prime \prime } \dfrac{dx_{1}^{{\rm ct}}}{d\tau _{1}}+\dfrac{dx_{2}^{{\rm ct}}}{dx_{1}^{{\rm ct}}} \left\{P_{2}^{\prime }\left[\dfrac{dx_{2}^{{\rm ct}}}{d\tau _{1}}+ k\dfrac{dx_{1}^{{\rm ct}}}{d\tau _{1}}\right]+P_{2}^{\prime } \dfrac{dx_{2}^{{\rm ct}}}{d\tau _{1}}-C_{2}^{\prime \prime } \frac{dx_{2}^{{\rm ct}}}{d\tau _{1}}\right\} +kP_{2}^{\prime }\dfrac{dx_{2}^{{\rm ct}}}{d\tau_{1}}=1 \\ 2P_{1}^{\prime }\dfrac{dx_{1}^{{\rm ct}}}{d\tau _{2}}-C_{1}^{\prime \prime } \dfrac{dx_{1}^{{\rm ct}}}{d\tau _{2}}+\dfrac{dx_{2}^{{\rm ct}}}{dx_{1}^{{\rm ct}}} \left\{P_{2}^{\prime }\left[\dfrac{dx_{2}^{{\rm ct}}}{d\tau _{2}}+ k\dfrac{dx_{1}^{{\rm ct}}}{d\tau _{2}}\right]+P_{2}^{\prime } \dfrac{dx_{2}^{{\rm ct}}}{d\tau _{2}}-C_{2}^{\prime \prime } \dfrac{dx_{2}^{{\rm ct}}}{d\tau _{2}}\right\} +kP_{2}^{\prime } \dfrac{dx_{2}^{{\rm ct}}}{d\tau_{2}}=\dfrac{dx_{2}^{{\rm ct}}}{dx_{1}^{{\rm ct}}} \\ P_{2}^{\prime }\left[\dfrac{dx_{2}^{{\rm ct}}}{d\tau _{1}}+k\dfrac{dx_{1}^{{\rm ct}}}{d\tau _{1}}\right]+ P_{2}^{\prime }\dfrac{dx_{2}^{{\rm ct}}}{d\tau _{1}}-C_{2}^{\prime \prime } \dfrac{dx_{2}^{{\rm ct}}}{d\tau _{1}}=0 \\ P_{2}^{\prime }\left[\dfrac{dx_{2}^{{\rm ct}}}{d\tau _{2}}+k\dfrac{dx_{1}^{{\rm ct}}}{d\tau _{2}}\right]+ P_{2}^{\prime }\dfrac{dx_{2}^{{\rm ct}}}{d\tau _{2}}-C_{2}^{\prime \prime } \dfrac{dx_{2}^{{\rm ct}}}{d\tau _{2}}=1 \end{array}\right.\\ & \left[\begin{array}{@{}c@{}} \dfrac{dx_{1}^{{\rm ct}}}{d\tau _{1}} \\ \dfrac{dx_{2}^{{\rm ct}}}{d\tau _{1}} \\ \dfrac{dx_{1}^{{\rm ct}}}{d\tau _{2}} \\ \dfrac{dx_{2}^{{\rm ct}}}{d\tau _{2}} \end{array}\right] =\left[\begin{array}{@{}cccc@{}} A & 0 & 0 & 0 \\ 0 & 0 & A & 0 \\ C & D & 0 & 0 \\ 0 & 0 & C & D \end{array}\right]^{-1}\left[\begin{array}{@{}c@{}} 1 \\ \dfrac{dx_{2}^{{\rm ct}}}{dx_{1}^{{\rm ct}}} \\ 0 \\ 1 \end{array}\right], \end{align*}
with: 
$A=2P_{1}^{\prime }-C_{1}^{\prime \prime }-\frac {(kP_{2}^{\prime })^{2}}{2P_{2}^{\prime }-C_{2}^{\prime \prime }}<0$; 
$B=- \frac {kP_{2}^{\prime }}{2P_{2}^{\prime }-C_{2}^{\prime \prime }} [2P_{2}^{\prime }-C_{2}^{\prime \prime }]+kP_{2}^{\prime }=0$; 
$C=P_{2}^{\prime }k<0$; 
$D=2P_{2}^{\prime }-C_{2}^{\prime \prime }<0$.
\begin{align*} \left[\begin{array}{@{}c@{}} \dfrac{dx_{1}^{{\rm ct}}}{d\tau _{1}} \\ \dfrac{dx_{2}^{{\rm ct}}}{d\tau _{1}} \\ \dfrac{dx_{1}^{{\rm ct}}}{d\tau _{2}} \\ \dfrac{dx_{2}^{{\rm ct}}}{d\tau _{2}} \end{array}\right] &= \left[\begin{array}{@{}cccc@{}} \dfrac{1}{A} & 0 & 0 & 0 \\ -\dfrac{C}{AD} & 0 & \dfrac{1}{D} & 0 \\ 0 & \dfrac{1}{A} & 0 & 0 \\ 0 & -\dfrac{C}{AD} & 0 & \dfrac{1}{D} \end{array}\right] \left[\begin{array}{@{}c@{}} 1 \\ \dfrac{dx_{2}^{{\rm ct}}}{dx_{1}^{{\rm ct}}} \\ 0 \\ 1 \end{array}\right]\\ &\Rightarrow \left\{\begin{array}{@{}l@{}} \dfrac{dx_{1}^{{\rm ct}}}{d\tau _{1}}=\dfrac{1}{A}=\dfrac{1}{2P_{1}^{\prime }-C_{1}^{\prime \prime }-\dfrac{(kP_{2}^{\prime })^{2}}{2P_{2}^{\prime }-C_{2}^{\prime \prime }}}=\dfrac{dx_{2}^{{\rm ct}}}{d\tau _{2}}<0 \\ \dfrac{dx_{2}^{{\rm ct}}}{d\tau _{1}}=-\dfrac{C}{AD}=-\dfrac{P_{2}^{\prime }k}{ 2P_{1}^{\prime }-C_{1}^{\prime \prime }-\dfrac{(kP_{2}^{\prime })^{2}}{ 2P_{2}^{\prime }-C_{2}^{\prime \prime }}}\dfrac{1}{[2P_{2}^{\prime }-C_{2}^{\prime \prime }]}=\dfrac{dx_{1}^{{\rm ct}}}{d\tau _{2}}>0 \\ \dfrac{dx_{1}^{{\rm ct}}}{d\tau _{2}}=\dfrac{1}{A}\dfrac{dx_{2}^{c}}{dx_{1}^{c}}= \dfrac{1}{2P_{1}^{\prime }-C_{1}^{\prime \prime }-\dfrac{(kP_{2}^{\prime })^{2} }{2P_{2}^{\prime }-C_{2}^{\prime \prime }}}\left(-\dfrac{kP_{2}^{\prime }}{ 2P_{2}^{\prime }-C_{2}^{\prime \prime }}\right)=\dfrac{dx_{2}^{{\rm ct}}}{d\tau _{1}}>0\\ \dfrac{dx_{2}^{{\rm ct}}}{d\tau _{2}}=-\dfrac{C}{AD}\dfrac{dx_{2}^{c}}{dx_{1}^{c}}+ \dfrac{1}{D}=\dfrac{1}{2P_{1}^{\prime }-C_{1}^{\prime \prime }- \dfrac{(kP_{2}^{\prime })^{2}}{2P_{2}^{\prime }-C_{2}^{\prime \prime }}}= \dfrac{dx_{1}^{{\rm ct}}}{d\tau _{1}}<0. \end{array}\right. \end{align*}Total variation of extracted quantities
$dx_{i}^{{\rm ct}}=\frac {\partial x_{i}^{{\rm ct}}}{\partial \tau _{1}}d\tau _{1}+\frac {\partial x_{i}^{{\rm ct}}}{\partial \tau _{2}}d\tau _{2}\lessgtr 0$ depending on 
$d\tau _{1}$ and 
$d\tau _{2}$. If 
$d\tau _{1}=d\tau _{2}=d\tau >0$, 
$dx_{i}=d\tau \left[\frac {\partial x_{i}^{{\rm ct}}}{\partial \tau _{1}}+ \frac {\partial x_{i}^{{\rm ct}}}{\partial \tau _{2}}\right]=d\tau \left[\frac {1}{ 2P_{1}^{\prime }-C_{1}^{\prime \prime }-\frac {(kP_{2}^{\prime })^{2}}{ 2P_{2}^{\prime }-C_{2}^{\prime \prime }}}\right]\left[1-\frac {kP_{2}^{\prime }}{ 2P_{2}^{\prime }-C_{2}^{\prime \prime }}\right]<0$.
