Transboundary Pollution Control and Competitiveness Concerns in a Two-Country Differential Game

Abstract

We analyze a transboundary pollution control problem in a heterogeneous two-country differential game setting in which regulators care for the implications of environmental policies on the competitiveness. We characterize the noncooperative and the cooperative solutions, showing that under both scenarios, in the presence of competitiveness considerations, heterogeneous countries will generally set different carbon taxes. This suggests, while implementing a mitigation policy is necessary to combat climate change, a universally homogeneous policy may not be optimal. Moreover, when countries are symmetric, except for their degree of competitiveness concerns, under noncooperation introduction of such concerns lowers the abatement policies in both countries, however, the self-effect is stronger than the cross-effect. Nevertheless, under cooperation, an increase in country j’s competitiveness concerns leads to more stringent policies in country i, while, the self-effect could be either positive or negative. The latter result emphasizes the importance of cooperation to tackle pollution in the presence of competitiveness concerns.

Appendices

Technical Appendix

Noncooperative Solution

In the noncooperative case, the solution to the problem Eqs. (3) and (4) should satisfy the following Hamilton–Jacobi–Bellman (HJB) equation, where $J_{i}^{n}\left( P\right)$ represents the country $i'$s regulator value function and $J_{i,P}^{n}=\frac{\partial J_{i}^{n}}{\partial P}$:

$$\begin{aligned} \theta J_{i}^{n}\left( P\right) = &\max _{0\le \tau _{i}<1}\left\{ \gamma _{i}P_{t}+\frac{1}{2}\alpha _{i}\tau _{i}^{2}+\beta _{i}\tau _{i}\left( \tau _{i}-\tau _{j}\right) \right. \\ & \left. +J_{i,P}^{n}\left[ \nu _{i}\left( 1-\tau _{i}\right) +\nu _{j}\left( 1-\tau _{j}\right) -\delta P\right] \right\} . \end{aligned}$$

(17)

The first-order condition yields:

$$\begin{aligned} \alpha _{i}\tau _{i}+2\beta _{i}\tau _{i}-\beta _{i}\tau _{j}-J_{i,P}^{n}\nu _{i}=0. \end{aligned}$$

(18)

We conjecture that the value function $J_{i}^{n}\left( P\right)$ has the following form:

$$\begin{aligned} J_{i}^{n}\left( P\right) =A_{i}^{n}+B_{i}^{n}P, \end{aligned}$$

(19)

where $A_{i}$ and $B_{i}$ are some constants to be determined. Plugging the first-order conditions for the regulators of the two countries and the conjectured value function into Eq. (17) and solving for $A_{i}$ and $B_{i}$ yields $B_{i}=\frac{\gamma _{i}}{\theta +\delta }$.² Using these results to substitute back into the first-order condition leads to noncooperative carbon tax rate given in Eq. (5). Substituting this into Eq. (4) and solving for P yields the noncooperative pollution trajectory given by: $P^n=\frac{\nu _{i}(1-\tau _{i}^n)+\nu _{j}(1-\tau _{j}^n)}{\delta } + \left[ P_0 - \frac{\nu _{i}(1-\tau _{i}^n)+\nu _{j}(1-\tau _{j}^n)}{\delta }\right] e^{-\delta t}$. Since $\tau _{i}^n$ and $\tau _{j}^n$ are constant and $\delta$ is strictly positive, it is straightforward to verify that the transversality condition $\lim _{t\rightarrow \infty }e^{-\theta t}J_i^n(P^n)=0$ is automatically satisfied. Since both the objective function and the state equation are convex in the control and state variables, it follows that the first-order conditions are both necessary and sufficient.

The derivatives of the carbon tax rate in Eq. (5) undoubtedly yield: $\frac{\partial \tau _{i}^{n}}{\partial \gamma _{i}}>0$, $\frac{\partial \tau _{i}^{n}}{\partial \gamma _{j}}>0$, $\frac{\partial \tau _{i}^{n}}{\partial \alpha _{i}}<0$, $\frac{\partial \tau _{i}^{n}}{\partial \alpha _{j}}<0$, $\frac{\partial \tau _{i}^{n}}{\partial \nu _{i}}>0$, $\frac{\partial \tau _{i}^{n}}{\partial \nu _{j}}>0$. The effect of the degree of the competitiveness concern is instead ambiguous, since the following results apply:

$$\begin{aligned} \frac{\partial \tau _{i}^{n}}{\partial \beta _{i}}=\frac{\left( \alpha _{j}+2\beta _{j}\right) \left( \alpha _{i}\nu _{j}\gamma _{j}-\left( 2\alpha _{j}+3\beta _{j}\right) \nu _{i}\gamma _{i}\right) }{\left( \delta +\theta \right) \left[ \alpha _{i}\left( \alpha _{j}+2\beta _{j}\right) +\beta _{i}\left( 2\alpha _{j}+3\beta _{j}\right) \right] ^{2}},\end{aligned}$$

(20)

$$\begin{aligned} \frac{\partial \tau _{j}^{n}}{\partial \beta _{i}}=\frac{\beta _{j}\left( \alpha _{i}\nu _{j}\gamma _{j}-\left( 2\alpha _{j}+3\beta _{j}\right) \nu _{i}\gamma _{i}\right) }{\left( \delta +\theta \right) \left[ \alpha _{i}\left( \alpha _{j}+2\beta _{j}\right) +\beta _{i}\left( 2\alpha _{j}+3\beta _{j}\right) \right] ^{2}}, \end{aligned}$$

(21)

i.e., $\frac{\partial \tau _{i}^{n}}{\partial \beta _{i}}\le (\ge )0$ if $\left( 2\alpha _{j}+3\beta _{j}\right) \nu _{i}\gamma _{i}\ge (\le )\alpha _{i}\nu _{j}\gamma _{j}$, while $\frac{\partial \tau _{i}^{n}}{\partial \beta _{j}}\le (\ge )0$ if $\left( 2\alpha _{i}+3\beta _{i}\right) \nu _{j}\gamma _{j}\ge (\le )\alpha _{j}\nu _{i}\gamma _{i}$.

The derivatives of the difference between the carbon tax rates in Eq. (6) are given by the following expressions which are again ambiguous:

$$\begin{aligned}& \frac{\partial (\tau _{i}^{N}-\tau _{j}^{N})}{\partial \beta _{i}}=\frac{\left( \alpha _{j}+\beta _{j}\right) \left( \alpha _{i}\gamma _{j}\nu _{j}-\gamma _{i}\nu _{i}\left( 2\alpha _{j}+3\beta _{j}\right) \right) }{\left( \delta +\theta \right) \left[ \alpha _{i}\left( \alpha _{j}+2\beta _{j}\right) +\beta _{i}\left( 2\alpha _{j}+3\beta _{j}\right) \right] ^{2}},\\& \frac{\partial (\tau _{i}^{N}-\tau _{j}^{N})}{\partial \beta _{j}}=-\frac{\left( \alpha _{i}+\beta _{i}\right) \left( \alpha _{j}\gamma _{i}\nu _{i}-\gamma _{j}\nu _{j}\left( 2\alpha _{i}+3\beta _{i}\right) \right) }{\left( \delta +\theta \right) \left[ \alpha _{i}\left( \alpha _{j}+2\beta _{j}\right) +\beta _{i}\left( 2\alpha _{j}+3\beta _{j}\right) \right] ^{2}}. \end{aligned}$$

However, $\frac{\partial \tau _{i}^{n}}{\partial \beta _{i}}$ and $\frac{\partial (\tau _{i}^{N}-\tau _{j}^{N})}{\partial \beta _{i}}$ has same sign, while signs of $\frac{\partial \tau _{i}^{n}}{\partial \beta _{j}}$ and $\frac{\partial (\tau _{i}^{N}-\tau _{j}^{N})}{\partial \beta _{j}}$ are opposite.

Cooperative Solution

In the cooperative case, the solution to the problem Eqs. (7) and (8) should satisfy the HJB equation, where now $J_{i}^{*}\left( P\right)$ represents the social (joint) value function and $J_{P}^{*}=\frac{\partial J^{*}}{\partial P}$:

$$\begin{aligned} \theta J^{*}\left( P\right) =\max _{0\le \tau _{i},\tau _{j}<1}&\left\{ \left( \gamma _{i}+\gamma _{j}\right) P_{t}+\frac{1}{2}\left( \alpha _{i}\tau _{i}^{2}+\alpha _{j}\tau _{j}^{2}\right)\right. \\ & \left. +\left( \beta _{i}\tau _{i}-\beta _{j}\tau _{j}\right) \left( \tau _{i}-\tau _{j}\right) +\right. \nonumber \\ & +\big .J_{P}^{*}\left[ \nu _{i}\left( 1-\tau _{i}\right) +\nu _{j}\left( 1-\tau _{j}\right) -\delta P\right] \big \}. \end{aligned}$$

(22)

The first-order condition yields:

$$\begin{aligned} \left( \alpha _{i}+2\beta _{i}\right) \tau _{i}=\left( \beta _{i}+\beta _{j}\right) \tau _{j}+\nu _{i}J_{p}^{*}. \end{aligned}$$

(23)

Our informed guess for the form of the value function is $J^{*}\left( P\right) =A^{*}+B^{*}P$, where $A^{*}$ and $B^{*}$ are constant to be determined. Replacing this conjectured value function and the first-order condition into Eq. (22) leads to $B^{*}=\frac{\gamma _{i}+\gamma _{j}}{\theta +\delta }$. Using the result to substitute back into the first-order condition leads to cooperative carbon tax rate given in Eq. (10). Substituting this into Eq. (8) and solving for P yields the cooperative pollution trajectory given by: $P^*=\frac{\nu _{i}(1-\tau _{i}^*)+\nu _{j}(1-\tau _{j}^*)}{\delta } + \left[ P_0 - \frac{\nu _{i}(1-\tau _{i}^*)+\nu _{j}(1-\tau _{j}^*)}{\delta }\right] e^{-\delta t}$. Also in this case, since $\tau _{i}^*$ and $\tau _{j}^*$ are constant and $\delta$ is strictly positive, the transversality condition $\lim _{t\rightarrow \infty }e^{-\theta t}J_i^*(P^*)=0$ turns out to be automatically satisfied. Since both the objective function and the state equation are convex in the control and state variables, it follows that the first-order conditions are both necessary and sufficient.

The derivatives of the carbon tax rate in Eq. (10) undoubtedly yield: $\frac{\partial \tau _{i}^{*}}{\partial \gamma _{i}}>0$, $\frac{\partial \tau _{i}^{*}}{\partial \gamma _{j}}>0$, $\frac{\partial \tau _{i}^{*}}{\partial \alpha _{i}}<0$, $\frac{\partial \tau _{i}^{*}}{\partial \alpha _{j}}<0$, $\frac{\partial \tau _{i}^{*}}{\partial \nu _{i}}>0$, $\frac{\partial \tau _{i}^{*}}{\partial \nu _{j}}>0$. The derivatives with respect to the degree of competitiveness concerns are instead ambiguous:

$$\begin{aligned} \frac{\partial \tau _{i}^{*}}{\partial \beta _{i}}=-\frac{\left( \gamma _{i}+\gamma _{j}\right) \left( 2\alpha _{j}^{2}\nu _{i}-\beta _{i}^{2}\nu _{j}-2\beta _{i}\beta _{j}\left( 2\nu _{i}+\nu _{j}\right) +\alpha _{j}\left( 6\beta _{j}\nu _{i}-2\beta _{i}\nu _{i}-\alpha _{i}\nu _{j}+2\beta _{j}\nu _{j}\right) +\beta _{j}\left( 4\beta _{j}\nu _{i}-2\alpha _{i}\nu _{j}+3\beta _{j}\nu _{j}\right) \right) }{\left( \delta +\theta \right) \left( 2\alpha _{j}\beta _{i}-\left( \beta _{i}-\beta _{j}\right) ^{2}+\alpha _{i}\left( \alpha _{j}+2\beta _{j}\right) \right) ^{2}}, \\ \frac{\partial \tau _{i}^{*}}{\partial \beta _{j}}=\frac{\left( \gamma _{i}+\gamma _{j}\right) \left( 2\beta _{i}\nu _{j}\left( \beta _{j}-\alpha _{i}\right) +\beta _{j}^{2}\left( 2\nu _{i}+\nu _{j}\right) -\beta _{i}^{2}\left( 2\nu _{i}+3\nu _{j}\right) +\alpha _{j}\left( 2\beta _{j}\nu _{i}+\alpha _{i}\nu _{j}+2\beta _{i}\left( \nu _{i}+\nu _{j}\right) \right) \right) }{\left( \delta +\theta \right) \left( 2\alpha _{j}\beta _{i}-\left( \beta _{i}-\beta _{j}\right) ^{2}+\alpha _{i}\left( \alpha _{j}+2\beta _{j}\right) \right) ^{2}}. \end{aligned}$$

The derivatives of the difference between the carbon tax rates in Eq. (11) are given by the following expressions which are again ambiguous:

Cooperation vs Noncooperation

The derivatives of the size of the distortion obtained by subtracting Eqs. (5) from (10) yield:

$$\begin{aligned}&\frac{\partial \left( \tau _{i}^{*}-\tau _{i}^{n}\right) }{\partial \beta _{i}}=\frac{\gamma _{i}+\gamma _{j}}{\left( \delta +\theta \right) }\left\{ \frac{\nu _{j}\left[ 2\alpha _{j}\beta _{i}-\left( \beta _{i}-\beta _{j}\right) ^{2}+\alpha _{i}\left( \alpha _{j}+2\beta _{j}\right) \right] -2\left( \alpha _{j}-\beta _{i}+\beta _{j}\right) \left[ \alpha _{j}\nu _{i}+\beta _{i}\nu _{j}+\beta _{j}\left( 2\nu _{i}+\nu _{j}\right) \right] }{\left[ 2\alpha _{j}\beta _{i}-\left( \beta _{i}-\beta _{j}\right) ^{2}+\alpha _{i}\left( \alpha _{j}+2\beta _{j}\right) \right] ^{2}}\right\} + \\ & \qquad \qquad \qquad \; \frac{1}{\left( \delta +\theta \right) }\left\{ \frac{\left( \alpha _{j}+2\beta _{j}\right) \left[ \nu _{i}\gamma _{i}\left( 2\alpha _{j}+3\beta _{j}\right) -\alpha _{i}\nu _{j}\gamma _{j}\right] }{\left[ \alpha _{i}\left( \alpha _{j}+2\beta _{j}\right) +\beta _{i}\left( 2\alpha _{j}+3\beta _{j}\right) \right] ^{2}}\right\} , \\& {1}\frac{\partial \left( \tau _{i}^{*}-\tau _{i}^{n}\right) }{\partial \beta _{j}}=\frac{\gamma _{i}+\gamma _{j}}{\left( \delta +\theta \right) }\left\{ \frac{\alpha _{j}\left[ 2\nu _{i}\left( \beta _{i}+\beta _{j}\right) +\nu _{j}\left( \alpha _{i}+2\beta _{i}\right) \right] +2\beta _{i}\nu _{j}\left( \beta _{j}-\alpha _{i}\right) +\beta _{j}^{2}\left( 2\nu _{i}+\nu _{j}\right) -\beta _{i}^{2}\left( 2\nu _{i}+3\nu _{j}\right) }{\left[ 2\alpha _{j}\beta _{i}-\left( \beta _{i}-\beta _{j}\right) ^{2}+\alpha _{i}\left( \alpha _{j}+2\beta _{j}\right) \right] ^{2}}\right\} + \\ & \qquad \qquad \qquad \; \; \frac{1}{\left( \delta +\theta \right) }\left\{ \frac{\beta _{i}\left[ \nu _{j}\gamma _{j}\left( 2\alpha _{i}+3\beta _{i}\right) -\alpha _{j}\nu _{i}\gamma _{i}\right] }{\left[ \alpha _{i}\left( \alpha _{j}+2\beta _{j}\right) +\beta _{i}\left( 2\alpha _{j}+3\beta _{j}\right) \right] ^{2}}\right\} . \end{aligned}$$

From the above expressions, it is clear that the sign of these derivatives cannot be determined unambiguously.