Appendix A
A.1 Equilibrium Analysis (Second Stage)
A.1.1 Bertrand Competition
First, the presented model will be solved assuming price competition. We maximize profits with respect to domestic and foreign prices. For simplicity, we introduce \(\Gamma =\frac{1}{4\alpha ^{2}-\beta ^{2}}\) and solve for equilibrium prices
$$\begin{aligned} \left( \begin{array}{cc} p_{A}&{} q_{A} \\ p_{B} &{} q_{B}\end{array}\right) = \Gamma \left( \begin{array}{cc} 2\alpha &{} \beta \\ \beta &{} 2\alpha \end{array}\right) \left( \begin{array}{cc} 1+\alpha c_{A}&{} 1+\alpha \varsigma _{A} \\ 1+\alpha c_{B} &{} 1+\alpha \varsigma _{B}\end{array}\right) \end{aligned}$$
(A.1)
and equilibrium quantities
$$\begin{aligned} \left( \begin{array}{cc} x_{A}&{} y_{A} \\ x_{B} &{} y_{B}\end{array}\right) = \left( \begin{array}{cc}1 &{} 1 \\ 1 &{} 1\end{array}\right) -\Gamma \left( \begin{array}{cc}2\alpha ^{2}-\beta ^{2} &{} -\alpha \beta \\ alpha\beta &{} 2\alpha ^{2}-\beta ^{2}\end{array}\right) \left( \begin{array}{cc} 1+\alpha c_{A}&{} 1+\alpha \varsigma _{A} \\ 1+\alpha c_{B} &{} 1+\alpha \varsigma _{B}\end{array}\right) . \end{aligned}$$
(A.2)
A.1.2 Cournot Competition
To solve the third stage under Cournot competition, we need the inverse demand functions within the home and the foreign country
$$\begin{aligned} \left( \begin{array}{cc}p_{A}&{} q_{A}\\ p_{B} &{} q_{B}\end{array}\right) = \frac{\left( \begin{array}{cc} \alpha &{} \beta \\ \beta &{} \alpha \end{array} \right) \left( \begin{array}{cc}1-x_{A}&{} 1-y_{A}\\ 1-x_{B} &{} 1-y_{B}\end{array}\right) }{\alpha ^{2}-\beta ^{2}}. \end{aligned}$$
(A.3)
Maximizing profits, equilibrium quantities can be derived for the four firms that serve the domestic and foreign market
$$\begin{aligned} \left( \begin{array}{cc}x_{A} &{} y_{A} \\ x_{B} &{} y_{B}\end{array}\right) =\Gamma (\alpha +\beta ) \left( \begin{array}{cc} 2\alpha &{} -\beta \\ -\beta &{} 2\alpha \end{array}\right) \left( \begin{array}{cc}1-c_{A}(\alpha -\beta ) &{} 1-\varsigma _{A}(\alpha -\beta ) \\ 1-c_{B}(\alpha -\beta ) &{} 1-\varsigma _{B}(\alpha -\beta )\end{array}\right) . \end{aligned}$$
(A.4)
Using equilibrium quantities, we solve for equilibrium prices
$$\begin{aligned} \left( \begin{array}{c}p_{A} \\ p_{B} \\ q_{A} \\ q_{B}\end{array}\right) =\frac{\Gamma }{\alpha ^{2}-\beta ^{2}} \left( \begin{array}{cc}1 &{} c_{A}(2\alpha ^{2}-\beta ^{2})+\alpha \beta c_{B} \\ 1 &{} c_{B}(2\alpha ^{2}-\beta ^{2})+\alpha \beta c_{A} \\ 1 &{} \varsigma _{A}(2\alpha ^{2}-\beta ^{2})+\alpha \beta \varsigma _{B} \\ 1 &{} \varsigma _{B}(2\alpha ^{2}-\beta ^{2})+\alpha \beta \varsigma _{A}\end{array}\right) \left( \begin{array}{c}(\alpha +\beta )(2\alpha ^{2}-\beta ^{2}) \\ \alpha ^{2}-\beta ^{2}\end{array}\right) . \end{aligned}$$
(A.5)
A.2 Welfare
A.2.1 General Prerequisites
First, we wish to state a number of general prerequisites that hold for both types of competition. We start with producer surplus, which consists of profits from both firms located in Country A, i.e. \(\pi _{AA}\) and \(\pi _{AB}\), and add revenues from carbon pricing
$$\begin{aligned} PS_{A}+T_{A} ={\mathop {\underset{\text {profit from domestic sales}}{\underbrace{ (p_{A}-c-t_{A})x_{A}}}+\underset{\text {profit from exports}}{\underbrace{(q_{A}-c-(t_{A}-\delta _{2}))y_{A}}}+}\limits _{ \underset{\begin{array}{c} \text {carbon pricing revenues} \\ \text {from domestic production} \end{array}}{\underbrace{t_{A}{x}_{A}+ (t_{A}-\delta _{2} ){y}_{A}}}+\underset{\begin{array}{c} \text {revenues from}\\ \text {import BA} \end{array}}{\underbrace{\delta _{1} x_{B}}}.}} \end{aligned}$$
(A.6)
As revenues from carbon pricing levied on domestic production are costs incurred by Country A’s producers, they cancel out to zero, and (A.6) can be simplified to
$$\begin{aligned} PS_{A}+T_{A}=(p_{A}-c)x_{A}+(q_{A}-c)y_{A}+\delta _{1} x_{B}. \end{aligned}$$
(A.7)
For domestic consumer surplus, we take into account domestic demand for both products (\(x_{A}\) and \(x_{B})\) and obtain
$$\begin{aligned} CS_{A}=\frac{1}{2}(p_{A}^{0}-p_{A})x_{A}+\frac{1}{2} (p_{B}^{0}-p_{B})x_{B}, \end{aligned}$$
(A.8)
where \(p_{i}^{0}\) is the axis intercept of the inverse demand function \((i=A,B)\). Consumer surplus can be simplified to
$$\begin{aligned} CS_{A}=\frac{x_{A}^{2}+x_{B}^{2}}{2\alpha }. \end{aligned}$$
(A.9)
A.2.2 Import BA
To determine the impacts of import BA on producer surplus and public revenues, we differentiate (A.7) with respect to \(\tau \). With equilibrium prices (A.1) and quantities (A.2), we can determine the derivatives for Bertrand competition
$$\begin{aligned} \frac{\partial (PS_{A}^{B}+T_{A}^{B}-\varphi _{A}E)}{\partial \tau }&= \Gamma \alpha \left[ {2\alpha \beta }(p_{A}-c-t_{A}) + \frac{4\alpha ^{2}-\beta ^{2}}{\alpha }x_{B}\right] \nonumber \\&\quad +\frac{\alpha (\alpha -\beta )}{2\alpha -\beta }\left[ \varphi _{A}-\frac{\tau (2\alpha ^{2}-\beta ^{2})-\alpha \beta t_{A}}{2\alpha ^{2}-\beta ^{2}-\alpha \beta }\right] >0. \end{aligned}$$
(A.10)
With equilibrium quantities (A.4) and prices (A.5), we can determine the derivatives for Cournot competition
$$\begin{aligned} \frac{\partial (PS_{A}^{C}+T_{A}^{C}-\varphi _{A}E)}{\partial \tau }&=\Gamma \left[ 2\alpha x_{A}+(4\alpha ^{2}-\beta ^{2})x_{B}\right] \nonumber \\&\quad +\frac{\alpha ^{2}-\beta ^{2}}{2\alpha +\beta }\left[ \varphi _{A}-\frac{2\alpha \tau -\beta t_{A}}{2\alpha -\beta } \right] >0. \end{aligned}$$
(A.11)
In order to compute the effect on consumer surplus, we use equation (A.9) with equilibrium quantities (A.2) to obtain the impact for Bertrand competition
$$\begin{aligned} \frac{\partial CS_{A}^{B}}{\partial \tau }= \Gamma \left[ \alpha \beta x_{A}-(2\alpha ^{2}-\beta ^{2})x_{B}\right] <0, \end{aligned}$$
(A.12)
which decreases with \(\tau \) as \(x_{A}-x_{B}=-\Gamma \alpha (\Delta +\delta _{1}) (2\alpha ^{2}-\beta ^{2}-\alpha \beta )\le 0\) and \(\alpha \beta <2\alpha ^{2}-\beta ^{2}\). In Cournot competition, we obtain
$$\begin{aligned} \frac{\partial CS_{A}^{C}}{\partial \tau }=\frac{\Gamma }{\alpha }(\alpha ^{2}-\beta ^{2})(x_{A}\beta -2\alpha x_{B})<0, \end{aligned}$$
(A.13)
which decreases with \(\tau \) as \(x_{A}-x_{B}=\Gamma (\alpha ^{2}-\beta ^{2})(2\alpha +\beta )(\delta _{1}-\Delta )\le 0\) and \(\beta <2\alpha \).
A.2.3 Symmetric BA
For SBA, we differentiate (A.7) with respect to \(\theta \) to obtain the impact on domestic producer surplus and public revenues. With equilibrium prices (A.1) and quantities (A.2), we obtain the impact for Bertrand competition
$$\begin{aligned} \frac{\partial (PS_{A}^{B}+T_{A}^{B})}{\partial \theta }=\Gamma \left[ \begin{array}{c} {\alpha ^{2}\beta [2(p_{A}-c)-t_{A}]}\\ +\alpha [ 2\alpha ^{2}(t_{A}-\theta )-(q_{A}-c)\beta ^{2}]-\theta (2\alpha ^{2}-\beta ^{2})\alpha \end{array}\right] +x_{B}. \end{aligned}$$
(A.14)
With equilibrium quantities (A.4) and prices (A.5), we obtain the impact for Cournot competition
$$\begin{aligned} \frac{\partial (PS_{A}^{C}+T_{A}^{C})}{\partial \theta } =\Gamma \left[ \begin{array}{c} {\beta t_{A}(\alpha ^{2}-\beta ^{2})+2\alpha \beta x_{A}+2\alpha (\alpha ^{2}-\beta ^{2})(t_{A}-\theta )}\\ +\beta ^{2}y_{A}-\theta (\alpha ^{2}-\beta ^{2})2\alpha \end{array}\right] +x_{B}. \end{aligned}$$
(A.15)
The effect of SBA on consumer surplus is equal to the effect of IBA since including an additional export BA only influences the foreign market. Hence, the only relevant effect is that of the import BA which is captured in \(\frac{\partial CS_{A}}{\partial \tau }=\frac{\partial CS_{A}}{\partial \theta }\).