A Complementarity Approach for an Environmental–Economic Game with Coupling Emission Constraints

Abstract

We consider an environmental–economic game where players face not only Cournot competition but also coupling environmental and individual capacity constraints. Under the complementarity problem framework, we study the existence of the (normalized) Nash equilibrium, computability of the equilibria, and the closed form expressions of the optimal weights. We also report numerical results of two examples as well as the insights gained from them.

Appendix: Proofs

Proof of Theorem 1

We first look at the case when β > 0. It follows from Eq. (3) that given the rival firms’ strategies (production levels), each firm’s objective function is a strongly concave function since β > 0. Therefore, each firm’s optimization problem has a unique solution given that it is feasible. Define the feasible set of the game as

$$ \mathcal F \, \triangleq \, \{p \in \mathbb R^{\nu}_+ \,|\, p \leq \mbox{CAP}, e^Tp \leq A\}, $$

where $\mbox{CAP} \triangleq \left(\mbox{CAP}_f\right)_{f=1,\cdots,\nu}$ and $e \triangleq \left(e_f\right)_{f=1,\cdots,\nu}$. Given a vector $p\in \mathcal F$, and an index f ∈ {1, ⋯ ,ν}, define the set

$$\mathcal F_f (p_{-f}) \, \triangleq \, \{q \in \mathbb R_+ \, |\, q\leq \mbox{CAP}_f, e_fq+ e_{-f}^Tp_{-f} \leq A\},$$

where $e_{-f} \triangleq \left(e_g \right)_{g\neq f}$ is the vector of coefficients e _g of all rival firms. For any given $p\in \mathcal F$, it is clear that $\mathcal F_f(p_{-f})$ is nonempty for all f. Therefore, the individual firm’s optimization problem must have a unique solution. In this case, we use $\mbox{argmax}_{p_f\in \mathcal F_f(p_{-f})}\{h_f(p_f,p_{-f})\}$ to refer to the unique solution, instead of the solution set. For any $p \in \mathcal F$, define the map

$$ \Phi(p) \, \triangleq \,\displaystyle{ \left( \mbox{argmax}_{p_f\in \mathcal F_f(p_{-f})}\{h(p_f,p_{-f})\}\right)_{f=1,\cdots,\nu} }. $$

It can be easily verified that Φ(p) is a self map, that is, $\Phi(p) \in \mathcal F$ for all $p\in \mathcal F$. Applying classical optimization theory, we can show that Φ(p) is a continuous function of p. In fact, the KKT optimality condition for an individual firm’s optimization problem is as follows: if $p^*_f = \mbox{argmax}_{p_f\in \mathcal F_f(p_{-f})}\{h(p_f,p_{-f})\}$, then there exist multipliers λ _f , μ _f , and ξ _f such that

$$ \begin{array}{rcl} 2\beta p^*_f + \beta Q_{-f} + d_fe_f + c_f+\lambda_f e_f + \mu_f -\alpha- \xi_f & = & 0,\\[5pt] 0 \, \leq \, \lambda_f \, \perp \, A - e_fp^*_f-E_{-f} & \geq & 0,\\[5pt] 0 \, \leq \, \mu_f \, \perp \, \mbox{CAP}_f - p^*_f & \geq & 0,\\[5pt] 0 \, \leq \, \xi_f \, \perp \, p^*_f & \geq & 0. \end{array} $$

(19)

Solving ξ _f from the first equation and substituting into the last complementarity condition, we obtain the following complementarity problem

$$\begin{array}{rll} &&{\kern-1pt} \leq{\kern-1pt} p^*_f {\kern-1pt}\! \perp{\kern-1pt} 2\beta p^*_f \!+\! \beta Q_{-f} \!+\! d_fe_f {\kern-.5pt}+{\kern-.5pt} c_f{\kern-.5pt}+{\kern-.5pt}\lambda_f e_f {\kern-.5pt}+{\kern-.5pt} \mu_f {\kern-1pt} -{\kern-1pt} \alpha \! \geq\! 0, \\ &&\leq \lambda_f \perp A - e_fp^*_f-E_{-f} \, \geq \, 0,\\ &&\leq \mu_f \perp \mbox{CAP}_f - p^*_f \geq 0. \end{array} $$

(20)

Notice that in Eq. (20), d _f , e _f , c _f , and $\mbox{CAP}_f$ are given data, and Q _− f and E _− f depends on the rival’s strategy. Notice also that Q _− f and E _− f are both determined by p _f , therefore, we denote Eq. (20) by LCP _f (q _− f ), and the solution set by $\mbox{SOL}_f(q_f)$. We now consider the set-valued map $\Xi_f: q_f \rightrightarrows \mbox{SOL}_f(q_{-f})$. It is clear that the Ξ _f is a polyhedron multifunction, namely, the graph of Ξ _f can be represented by a union of finite number of polyhedra. Let matrix

$$B \, \triangleq \, \left[\begin{array}{ccc}1&0&0\\0&0&0\\0&0&0\end{array}\right],$$

define the map $\widehat{\Xi}_f : q_{-f} \rightrightarrows B\mbox{SOL}_f(q_{-f})$. By the uniqueness of the solution to the individual optimization problem, we know that $\widehat{\Xi}$ is a single-valued polyhedron multifunction. Thus, by a result in [3, Exercise 5.6.14], $\widehat{\Xi}_f : q_{-f} \rightrightarrows B\mbox{SOL}_f(q_{-f})$ is in fact a piecewise affine function and hence continuous. Therefore, the map from p _− f to $p^*_f = B\mbox{SOL}_f(q_{-f})$ is a continuous map and so is the map Φ(p). Notice that the set $\mathcal F$ is compact convex, and the function Φ(p) is a continuous self map from $\mathcal F$ to $\mathcal F$. By Brouwer’s fixed point theorem, Φ(p) must have a fixed point $p^* \in \mathcal F$, namely, Φ(p ^*) = p ^*. By the definition of the equilibrium state of the game, p ^* must be an equilibrium state.

When β = 0, the individual firm’s optimization problem is still convex but not strongly convex, and hence the map Φ(p) is a set-valued map. Therefore, one needs to apply the Kakutani fixed theorem to conclude the equilibrium existence. The details are omitted here. □

Proof Lemma 2

Given a vector $y= \left(\begin{array}{c}p\\ \kappa \\ \mu \end{array}\right) \gvertneqq 0$, we have

$$\mathbf H^N y\, = \, \left(\begin{array}{c}\beta(I_{\nu\times\nu}+\mathbf 1_{\nu\times\nu})p + \kappa We + \mu\\[5pt] -e^Tp\\-p \end{array}\right).$$

Suppose p _f  > 0 for some f ∈ {1, ⋯ ,ν}, then the fth component in

$$\beta(I_{\nu\times\nu}+\mathbf 1_{\nu\times\nu})p + \kappa We + \mu$$

is given by

$$\beta\left(p_f + \sum\limits_{g=1}^{\nu}p_f\right) + \kappa w_fe_f + \mu_f \, \geq \, 0.$$

Therefore, we may assume that p _f  = 0 for all f = 1, ⋯ ,ν. Now, if κ > 0, the corresponding component in H ^N is e ^T p = 0. If μ _f  > 0 for some f ∈ {1, ⋯ ,ν}, the corresponding component in H ^N is p _f  = 0. This concludes the proof.□

Proof of Theorem 2

We have shown that H ^N is semimonotone in Lemma 2. Now assume that there exists τ > 0 such that SOL$(h^N + \tau \mathbf 1_{2\nu+1}, \mathbf H^N)$ contains a sequence of solutions $(p^k,\kappa^k,\mu^k)_{k=1}^{\infty}$ such that

$$\begin{array}{rll} &&{\kern-6pt} 0 \, \leq \, p^k \perp \beta p^k + \beta \mathbf 1_{\nu\times\nu}p^k + d + c+ \kappa^k We \\ &&{\kern12pt} + \mu^k - \alpha \mathbf 1_{\nu} + \tau \mathbf 1_{\nu} \, \geq \, 0, \\ &&{\kern-6pt} 0 \, \leq \, \kappa^k \perp A - e^T p^k + \tau \, \geq \,0,\\ &&{\kern-6pt} 0 \, \leq \, \mu^k \perp \mbox{CAP}- p^k + \tau \mathbf 1_{\nu}\, \geq \, 0. \end{array}$$

(21)

It is clear that p ^k must be bounded since $p^k \leq \mbox{CAP} + \tau \mathbf 1_{\nu}$, that is, $p_f^k \leq \overline{\mbox{CAP}}_f + \tau$ for each f = 1, ⋯ ,ν. We claim that κ ^k must be bounded. Suppose not, then we can assume without loss of generality that κ ^k > 0 for all k sufficiently large. Therefore, by the second complementarity condition in Eq. (21), we conclude that

$$A - e^T p^k + \tau \, = \,0,$$

for all k sufficiently large. We can also see that $e_f \kappa^k$ is unbounded for all f = 1, ⋯ ,ν since e _f  > 0 for all f = 1, ⋯ ,ν. Therefore we have

$$\beta p_f^k + \beta \mathbf 1_{\nu}^Tp^k + d_f + c_f+ \kappa^k \frac{1}{w_f}e_f + \mu_f^k - \alpha + \tau$$

must be unbounded for all f = 1, ⋯ ,ν since α is a fixed number. Now, by the first complementarity condition in Eq. (21), we deduce that p ^k = 0 for all k sufficiently large. However,

$$ e^Tp^k =A + \tau \, > \,0, $$

for all k sufficiently large. This is a contradiction. Similarly, if μ ^k is unbounded, that is, $\mu_f^k$ is unbounded for some f ∈ {1, ⋯ ,ν}, then

$$ p^k_f \, = \, \mbox{CAP}_f+1 \, > \, 0 $$

for all k sufficiently large. Moreover,

$$ \beta p_f^k + \beta \mathbf 1_{\nu}^Tp^k + d_f + c_f+ \kappa^k \frac{1}{w_f}e_f + \mu_f^k - \alpha + \tau $$

must be unbounded. By the first complementarity condition in Eq. (21), we must have $p^k_f = 0$ for all k sufficiently large. This is again a contradiction. Thus, we can conclude that μ ^k must also be bounded. Therefore, for any given τ, SOL$(h^N + \tau \mathbf 1_{2\nu+1}, \mathbf H^N)$ is bounded. By Lemma 1, the theorem holds readily. □

Remark 3

As pointed out by one of the reviewer, this proof actually follows the proof of Theorem 1 in the study by Rosen [15], but has been presented in the particular context of this paper. We include it here for completeness.