Appendix 1: Derivations

Region Welfare Maximization

By differentiating the regional welfare (5) with respect to consumptions tax, we get

$$\begin{aligned} \frac{{\partial W^{1} }}{{\partial v^{1} }} & = u_{x}^{1} \frac{{\partial \bar{x}^{1} }}{{\partial v^{1} }} + u_{y}^{1} \frac{{\partial \bar{y}}}{{\partial v^{1} }} + u_{z}^{1} \frac{{\partial \bar{z}^{1} }}{{\partial v^{1} }} - c_{x}^{{x1}} \frac{{\partial x^{1} }}{{\partial v^{1} }} - c_{y}^{{y1}} \frac{{\partial y^{1} }}{{\partial v^{1} }} - c_{z}^{{z1}} \frac{{\partial z^{1} }}{{\partial v^{1} }} - c_{e}^{{y1}} \frac{{\partial e^{{y1}} }}{{\partial v^{1} }} \\ & \quad - \,c_{e}^{{z1}} \frac{{\partial e^{{z1}} }}{{\partial v^{1} }} - \tau \left[ {\frac{{\partial e^{{y1}} }}{{\partial v^{1} }} + \frac{{\partial e^{{y2}} }}{{\partial v^{1} }} + \frac{{\partial e^{{y3}} }}{{\partial v^{1} }} + \frac{{\partial e^{{z1}} }}{{\partial v^{1} }} + \frac{{\partial e^{{z2}} }}{{\partial v^{1} }} + \frac{{\partial e^{{z3}} }}{{\partial v^{1} }}} \right] \\ \end{aligned}$$

Recall the conditions and assumptions from (2) and (3), and we then get

$$= p^{x} \frac{{\partial \bar{x}^{1} }}{{\partial v^{1} }} + \left( {p^{y} + v^{1} } \right)\frac{{\partial \bar{y}^{1} }}{{\partial v^{1} }} + p^{z1} \frac{{\partial \bar{z}^{1} }}{{\partial v^{1} }} - p^{x} \frac{{\partial x^{1} }}{{\partial v^{1} }} - \left( {p^{y} + s^{1} } \right)\frac{{\partial y^{1} }}{{\partial v^{1} }} - p^{z1} \frac{{\partial z^{1} }}{{\partial v^{1} }} + t^{1} \frac{{\partial e^{y1} }}{{\partial v^{1} }} + t^{1} \frac{{\partial e^{z1} }}{{\partial v^{1} }} - \tau \left[ {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{y2} }}{{\partial v^{1} }} + \frac{{\partial e^{y3} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }} + \frac{{\partial e^{z2} }}{{\partial v^{1} }} + \frac{{\partial e^{z3} }}{{\partial v^{1} }}} \right]$$

We further simplify the equation

$$= p^{x} \frac{{\partial \bar{x}^{1} }}{{\partial v^{1} }} - p^{x} \frac{{\partial x^{1} }}{{\partial v^{1} }} + \left( {p^{y} + v^{1} } \right)\frac{{\partial \bar{y}^{1} }}{{\partial v^{1} }} + p^{z1} \frac{{\partial \bar{z}^{1} }}{{\partial v^{1} }} - p^{z1} \frac{{\partial z^{1} }}{{\partial v^{1} }} - \left( {p^{y} + s^{1} } \right)\frac{{\partial y^{1} }}{{\partial v^{1} }} + t^{1} \left( {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }}} \right) - \tau \left[ {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{y2} }}{{\partial v^{1} }} + \frac{{\partial e^{y3} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }} + \frac{{\partial e^{z2} }}{{\partial v^{1} }} + \frac{{\partial e^{z3} }}{{\partial v^{1} }}} \right]$$

Since there is no trade of the good z, i.e. \(\left( {\frac{{\partial \bar{z}^{1} }}{{\partial v^{1} }} = \frac{{\partial z^{1} }}{{\partial v^{1} }}} \right)\):

$$= p^{x} \left( {\frac{{\partial \bar{x}^{1} }}{{\partial v^{1} }} - \frac{{\partial x^{1} }}{{\partial v^{1} }}} \right) + \left( {p^{y} + v^{1} } \right)\frac{{\partial \bar{y}^{1} }}{{\partial v^{1} }} - \left( {p^{y} + s^{1} } \right)\frac{{\partial y^{1} }}{{\partial v^{1} }} + t^{1} \left( {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }}} \right) - \tau \left[ {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{y2} }}{{\partial v^{1} }} + \frac{{\partial e^{y3} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }} + \frac{{\partial e^{z2} }}{{\partial v^{1} }} + \frac{{\partial e^{z3} }}{{\partial v^{1} }}} \right]$$

Recall (4), further we differentiate (4) w.r.t. consumption tax, remembering the product rule:

$$\frac{{\partial p^{y} }}{{\partial v^{1} }}\left( {y^{1} - \bar{y}^{1} } \right) + p^{y} \left( {\frac{{\partial y^{1} }}{{\partial v^{1} }} - \frac{{\partial \bar{y}^{1} }}{{\partial v^{1} }}} \right) + \frac{{\partial p^{x} }}{{\partial v^{1} }}\left( {x^{1} - \bar{x}^{1} } \right) + p^{x} \left( {\frac{{\partial x^{1} }}{{\partial v^{1} }} - \frac{{\partial \bar{x}^{1} }}{{\partial v^{1} }}} \right) = 0$$

solving this for px

$$p^{x} = \frac{{\left( {p^{y} \left( {\frac{{\partial y^{1} }}{{\partial v^{1} }} - \frac{{\partial \bar{y}^{1} }}{{\partial v^{1} }}} \right) + \frac{{\partial p^{y} }}{{\partial v^{1} }}\left( {y^{1} - \bar{y}^{1} } \right) + \frac{{\partial p^{x} }}{{\partial v^{1} }}\left( {x^{1} - \bar{x}^{1} } \right)} \right)}}{{ - \left( {\frac{{\partial x^{1} }}{{\partial v^{1} }} - \frac{{\partial \bar{x}^{1} }}{{\partial v^{1} }}} \right)}}$$

we insert this into our equation for px

$$\begin{aligned} \frac{{\partial W^{1} }}{{\partial v^{1} }} & = \left[ {\frac{{\left( {p^{y} \left( {\frac{{\partial y^{1} }}{{\partial v^{1} }} - \frac{{\partial \bar{y}^{1} }}{{\partial v^{1} }}} \right) + \frac{{\partial p^{y} }}{{\partial v^{1} }}\left( {y^{1} - \bar{y}^{1} } \right) + \frac{{\partial p^{x} }}{{\partial v^{1} }}\left( {x^{1} - \bar{x}^{1} } \right)} \right)}}{{ - \left( {\frac{{\partial x^{1} }}{{\partial v^{1} }} - \frac{{\partial \bar{x}^{1} }}{{\partial v^{1} }}} \right)}}} \right]\left( {\frac{{\partial \bar{x}^{1} }}{{\partial v^{1} }} - \frac{{\partial x^{1} }}{{\partial v^{1} }}} \right) \\ & \quad + \left( {p^{y} + v^{1} } \right)\frac{{\partial \bar{y}^{1} }}{{\partial v^{1} }} - \left( {p^{y} + s^{1} } \right)\frac{{\partial y^{1} }}{{\partial v^{1} }} + t^{1} \left( {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }}} \right) - \tau \left[ {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{y2} }}{{\partial v^{1} }} + \frac{{\partial e^{y3} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }} + \frac{{\partial e^{z2} }}{{\partial v^{1} }} + \frac{{\partial e^{z3} }}{{\partial v^{1} }}} \right] \\ \end{aligned}$$

and since

$$- \frac{{\left( {\frac{{\partial \bar{x}^{1} }}{{\partial v^{1} }} - \frac{{\partial x^{1} }}{{\partial v^{1} }}} \right)}}{{\left( {\frac{{\partial x^{1} }}{{\partial v^{1} }} - \frac{{\partial \bar{x}^{1} }}{{\partial v^{1} }}} \right)}} = \frac{{\left( {\frac{{\partial \bar{x}^{1} }}{{\partial v^{1} }} - \frac{{\partial x^{1} }}{{\partial v^{1} }}} \right)}}{{\left( {\frac{{\partial \bar{x}^{1} }}{{\partial v^{1} }} - \frac{{\partial x^{1} }}{{\partial v^{1} }}} \right)}} = 1$$

We can further simplify:

$$\begin{aligned} & = p^{y} \left( {\frac{{\partial y^{1} }}{{\partial v^{1} }} - \frac{{\partial \bar{y}^{1} }}{{\partial v^{1} }}} \right) + \frac{{\partial p^{y} }}{{\partial v^{1} }}\left( {y^{1} - \bar{y}^{1} } \right) + \frac{{\partial p^{x} }}{{\partial v^{1} }}\left( {x^{1} - \bar{x}^{1} } \right) + \left( {p^{y} + v^{1} } \right)\frac{{\partial \bar{y}^{1} }}{{\partial v^{1} }} - \left( {p^{y} + s^{1} } \right)\frac{{\partial y^{1} }}{{\partial v^{1} }} \\ & \quad + \,t^{1} \left( {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }}} \right) - \tau \left[ {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{y2} }}{{\partial v^{1} }} + \frac{{\partial e^{y3} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }} + \frac{{\partial e^{z2} }}{{\partial v^{1} }} + \frac{{\partial e^{z3} }}{{\partial v^{1} }}} \right] \\ \end{aligned}$$

$$\begin{aligned} & = p^{y} \left( {\frac{{\partial y^{1} }}{{\partial v^{1} }} - \frac{{\partial \bar{y}^{1} }}{{\partial v^{1} }} + \frac{{\partial \bar{y}^{1} }}{{\partial v^{1} }} - \frac{{\partial y^{1} }}{{\partial v^{1} }}} \right) + \frac{{\partial p^{y} }}{{\partial v^{1} }}\left( {y^{1} - \bar{y}^{1} } \right) + \frac{{\partial p^{x} }}{{\partial v^{1} }}\left( {x^{1} - \bar{x}^{1} } \right) + v^{1} \frac{{\partial \bar{y}^{1} }}{{\partial v^{1} }} - s^{1} \frac{{\partial y^{1} }}{{\partial v^{1} }} \\ & \quad + t^{1} \left( {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }}} \right) - \tau \left[ {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }} + \frac{{\partial e^{y2} }}{{\partial v^{1} }} + \frac{{\partial e^{z2} }}{{\partial v^{1} }} + \frac{{\partial e^{y3} }}{{\partial v^{1} }} + \frac{{\partial e^{z3} }}{{\partial v^{1} }}} \right] \\ \end{aligned}$$

Recall the constraint on emission in region 1 and 2, \(\bar{E} = e^{y1} + e^{y2} + e^{z1} + e^{z2}\). By differentiating this w.r.t the consumption tax, we have that:

$$\frac{{\partial \bar{E}}}{{\partial v^{1} }} = \frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{y2} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }} + \frac{{\partial e^{z2} }}{{\partial v^{1} }} = 0$$

By this assumption, our equation can now be expressed as:

$$= p^{y} \left( {\frac{{\partial y^{1} }}{{\partial v^{1} }} - \frac{{\partial \bar{y}^{1} }}{{\partial v^{1} }} + \frac{{\partial \bar{y}^{1} }}{{\partial v^{1} }} - \frac{{\partial y^{1} }}{{\partial v^{1} }}} \right) + \frac{{\partial p^{y} }}{{\partial v^{1} }}\left( {y^{1} - \bar{y}^{1} } \right) + \frac{{\partial p^{x} }}{{\partial v^{1} }}\left( {x^{1} - \bar{x}^{1} } \right) + v^{1} \frac{{\partial \bar{y}^{1} }}{{\partial v^{1} }} - s^{1} \frac{{\partial y^{1} }}{{\partial v^{1} }} + t^{1} \left( {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }}} \right) - \tau \left[ {\frac{{\partial e^{y3} }}{{\partial v^{1} }} + \frac{{\partial e^{z3} }}{{\partial v^{1} }}} \right]$$

and simplified to

$$= v^{1} \frac{{\partial \bar{y}^{1} }}{{\partial v^{1} }} - s^{1} \frac{{\partial y^{1} }}{{\partial v^{1} }} + \frac{{\partial p^{y} }}{{\partial v^{1} }}\left( {y^{1} - \bar{y}} \right) + \frac{{\partial p^{x} }}{{\partial v^{1} }}\left( {x^{1} - \bar{x}^{1} } \right) + t^{1} \left( {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }}} \right) - \tau \left[ {\frac{{\partial e^{y3} }}{{\partial v^{1} }} + \frac{{\partial e^{z3} }}{{\partial v^{1} }}} \right]$$

And we finally arrive at (11), by moving v1 on the other side of the equal sign

$$v^{1*} = \left( {\frac{{\partial \bar{y}^{1} }}{{\partial v^{1} }}} \right)^{ - 1} \left[ {s^{1} \frac{{\partial y^{1} }}{{\partial v^{1} }} - \frac{{\partial p^{y} }}{{\partial v^{1} }}\left( {y^{1} - \bar{y}^{1} } \right) - \frac{{\partial p^{x} }}{{\partial v^{1} }}\left( {x^{1} - \bar{x}^{1} } \right) + \left( { - t^{1} } \right)\left( {\frac{{\partial e^{y1} }}{{\partial y^{1} }}\frac{{\partial y^{1} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial z^{1} }}\frac{{\partial z^{1} }}{{\partial v^{1} }}} \right) + \tau \left( {\frac{{\partial e^{y3} }}{{\partial y^{3} }}\frac{{\partial y^{3} }}{{\partial v^{1} }} + \frac{{\partial e^{z3} }}{{\partial z^{3} }}\frac{{\partial z^{3} }}{{\partial v^{1} }}} \right)} \right].$$ (11)

Global Welfare Maximization

By differentiating the global welfare w.r.t consumption tax in region 1, we get

$$\frac{{\partial W^{G} }}{{\partial v^{1} }} = \mathop \sum \limits_{j = 1,2,3} \left[ {u_{x}^{j} \frac{{\partial \bar{x}^{j} }}{{\partial v^{1} }} + u_{y}^{j} \frac{{\partial \bar{y}^{j} }}{{\partial v^{1} }} + u_{z}^{j} \frac{{\partial \bar{z}^{j} }}{{\partial v^{1} }} - c_{x}^{xj} \frac{{\partial x^{j} }}{{\partial v^{1} }} - c_{y}^{yj} \frac{{\partial y^{j} }}{{\partial v^{1} }} - c_{z}^{zj} \frac{{\partial z^{j} }}{{\partial v^{1} }} - \left( {\tau + c_{e}^{yj} } \right)\frac{{\partial e^{yj} }}{{\partial v^{1} }} - \left( {\tau + c_{e}^{zj} } \right)\frac{{\partial e^{zj} }}{{\partial v^{1} }}} \right]$$

From our assumption in (2), (3), (5) and (6) we get

$$\begin{aligned} \frac{{\partial W^{G} }}{{\partial v^{1} }} & = \mathop \sum \limits_{j = 1,2,3} \left[ {p^{x} \frac{{\partial \bar{x}^{j} }}{{\partial v^{1} }} + \left( {p^{y} + v^{j} } \right)\frac{{\partial \bar{y}^{j} }}{{\partial v^{1} }} + p^{zj} \frac{{\partial \bar{z}^{j} }}{{\partial v^{1} }} - p^{x} \frac{{\partial x^{j} }}{{\partial v^{1} }} - \left( {p^{y} + s^{j} } \right)\frac{{\partial y^{j} }}{{\partial v^{1} }} - p^{zj} \frac{{\partial z^{j} }}{{\partial v^{1} }}} \right] \\ & \quad - \left( {\tau + c_{e}^{y1} } \right)\frac{{\partial e^{y1} }}{{\partial v^{1} }} - \left( {\tau + c_{e}^{z1} } \right)\frac{{\partial e^{z1} }}{{\partial v^{1} }} - \left( {\tau + c_{e}^{y2} } \right)\frac{{\partial e^{y2} }}{{\partial v^{1} }} - \left( {\tau + c_{e}^{z2} } \right)\frac{{\partial e^{z2} }}{{\partial v^{1} }} \\ & \quad - \left( {\tau + c_{e}^{y3} } \right)\frac{{\partial e^{y3} }}{{\partial v^{1} }} - \left( {\tau + c_{e}^{z3} } \right)\frac{{\partial e^{z3} }}{{\partial v^{1} }} \\ \end{aligned}$$

$$\begin{aligned} & = \mathop \sum \limits_{j = 1,2,3} \left[ {p^{x} \frac{{\partial \bar{x}^{j} }}{{\partial v^{1} }} - p^{x} \frac{{\partial x^{j} }}{{\partial v^{1} }} + \left( {p^{y} + v^{j} } \right)\frac{{\partial \bar{y}^{j} }}{{\partial v^{1} }} - \left( {p^{y} + s^{j} } \right)\frac{{\partial y^{j} }}{{\partial v^{1} }} + p^{zj} \frac{{\partial \bar{z}^{j} }}{{\partial v^{1} }} - p^{zj} \frac{{\partial z^{j} }}{{\partial v^{1} }}} \right] - \left( {\tau - t^{1} } \right)\frac{{\partial e^{y1} }}{{\partial v^{1} }} \\ & \quad - \left( {\tau - t^{1} } \right)\frac{{\partial e^{z1} }}{{\partial v^{1} }} - \left( {\tau - t^{2} } \right)\frac{{\partial e^{y2} }}{{\partial v^{1} }} - \left( {\tau - t^{2} } \right)\frac{{\partial e^{z2} }}{{\partial v^{1} }} - \left( {\tau + c_{e}^{y3} } \right)\frac{{\partial e^{y3} }}{{\partial v^{1} }} - \left( {\tau + c_{e}^{z3} } \right)\frac{{\partial e^{z3} }}{{\partial v^{1} }} \\ \end{aligned}$$

Since good z is non-tradable, the production in region j is equal to consumption in the same region. Also recall that \(c_{e}^{y3} = c_{e}^{z3} = 0\) and \(t^{1} = t^{2}\)

$$\begin{aligned} & = \mathop \sum \limits_{j = 1,2,3} \left[ {p^{x} \left( {\frac{{\partial \bar{x}^{j} }}{{\partial v^{1} }} - \frac{{\partial x^{j} }}{{\partial v^{1} }}} \right) + \left( {p^{y} + v^{j} } \right)\frac{{\partial \bar{y}^{j} }}{{\partial v^{1} }} + p^{zj} \left( {\frac{{\partial \bar{z}^{j} }}{{\partial v^{1} }} - \frac{{\partial z^{j} }}{{\partial v^{1} }}} \right) - \left( {p^{y} + s^{j} } \right)\frac{{\partial y^{j} }}{{\partial v^{1} }}} \right] \\ & \quad + \left( {t^{1} - \tau } \right)\left( {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }}} \right) + \left( {t^{1} - \tau } \right)\left( {\frac{{\partial e^{y2} }}{{\partial v^{1} }} + \frac{{\partial e^{z2} }}{{\partial v^{1} }}} \right) - \tau \left( {\frac{{\partial e^{y3} }}{{\partial v^{1} }} + \frac{{\partial e^{z3} }}{{\partial v^{1} }}} \right) \\ \end{aligned}$$

Again, we use our assumptions from (4), differentiate w.r.t consumption tax and solve it for px (remembering the product rule):

$$\begin{aligned} & \frac{{\partial p^{y} }}{{\partial v^{1} }}\left( {y^{j} - \bar{y}^{j} } \right) + p^{y} \left( {\frac{{\partial y^{j} }}{{\partial v^{1} }} - \frac{{\partial \bar{y}^{j} }}{{\partial v^{1} }}} \right) + \frac{{\partial p^{x} }}{{\partial v^{1} }}\left( {x^{j} - \bar{x}^{j} } \right) + p^{x} \left( {\frac{{\partial x^{j} }}{{\partial v^{1} }} - \frac{{\partial \bar{x}^{j} }}{{\partial v^{1} }}} \right) = 0 \\ & p^{x} = \frac{{\left( {p^{y} \left( {\frac{{\partial y^{j} }}{{\partial v^{1} }} - \frac{{\partial \bar{y}^{j} }}{{\partial v^{1} }}} \right) + \frac{{\partial p^{y} }}{{\partial v^{1} }}\left( {y^{j} - \bar{y}^{j} } \right) + \frac{{\partial p^{x} }}{{\partial v^{1} }}\left( {x^{j} - \bar{x}^{j} } \right)} \right)}}{{ - \left( {\frac{{\partial x^{j} }}{{\partial v^{1} }} - \frac{{\partial \bar{x}^{j} }}{{\partial v^{1} }}} \right)}} \\ \end{aligned}$$

Insert this for px into our equation:

$$\begin{aligned} & \mathop \sum \limits_{j = 1,2,3} \left[ {\frac{{\left( {p^{y} \left( {\frac{{\partial y^{j} }}{{\partial v^{1} }} - \frac{{\partial \bar{y}^{j} }}{{\partial v^{1} }}} \right) + \frac{{\partial p^{y} }}{{\partial v^{1} }}\left( {y^{j} - \bar{y}^{j} } \right) + \frac{{\partial p^{x} }}{{\partial v^{1} }}\left( {x^{j} - \bar{x}^{j} } \right)} \right)}}{{ - \left( {\frac{{\partial x^{j} }}{{\partial v^{1} }} - \frac{{\partial \bar{x}^{j} }}{{\partial v^{1} }}} \right)}}\left( {\frac{{\partial \bar{x}^{j} }}{{\partial v^{1} }} - \frac{{\partial x^{j} }}{{\partial v^{1} }}} \right) + \left( {p^{y} + v^{j} } \right)\frac{{\partial \bar{y}^{j} }}{{\partial v^{1} }} + p^{zj} \left( {\frac{{\partial \bar{z}^{j} }}{{\partial v^{1} }} - \frac{{\partial z^{j} }}{{\partial v^{1} }}} \right) - \left( {p^{y} + s^{j} } \right)\frac{{\partial y^{j} }}{{\partial v^{1} }}} \right] \\ & \quad + \left( {t^{1} - \tau } \right)\left( {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }}} \right) + \left( {t^{1} - \tau } \right)\left( {\frac{{\partial e^{y2} }}{{\partial v^{1} }} + \frac{{\partial e^{z2} }}{{\partial v^{1} }}} \right) - \tau \left( {\frac{{\partial e^{y3} }}{{\partial v^{1} }} + \frac{{\partial e^{z3} }}{{\partial v^{1} }}} \right) \\ \end{aligned}$$

Since

$$\frac{{\left( {\frac{{\partial \bar{x}^{j} }}{{\partial v^{1} }} - \frac{{\partial x^{j} }}{{\partial v^{1} }}} \right)}}{{\left( {\frac{{\partial \bar{x}^{j} }}{{\partial v^{1} }} - \frac{{\partial x^{j} }}{{\partial v^{1} }}} \right)}} = 1$$

The equation can be simplified to

$$\begin{aligned} & = \mathop \sum \limits_{j = 1,2,3} \left[ {p^{y} \left( {\frac{{\partial y^{j} }}{{\partial v^{1} }} - \frac{{\partial \bar{y}^{j} }}{{\partial v^{1} }}} \right) + \frac{{\partial p^{y} }}{{\partial v^{1} }}\left( {y^{j} - \bar{y}^{j} } \right) + \frac{{\partial p^{x} }}{{\partial v^{1} }}\left( {x^{j} - \bar{x}^{j} } \right) + \left( {p^{y} + v^{j} } \right)\frac{{\partial \bar{y}^{j} }}{{\partial v^{1} }} - \left( {p^{y} + s^{j} } \right)\frac{{\partial y^{j} }}{{\partial v^{1} }}} \right] \\ & \quad + \left( {t^{1} - \tau } \right)\left( {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }}} \right) + \left( {t^{1} - \tau } \right)\left( {\frac{{\partial e^{y2} }}{{\partial v^{1} }} + \frac{{\partial e^{z2} }}{{\partial v^{1} }}} \right) - \tau \left( {\frac{{\partial e^{y3} }}{{\partial v^{1} }} + \frac{{\partial e^{z3} }}{{\partial v^{1} }}} \right) \\ \end{aligned}$$

$$\begin{aligned} & = \mathop \sum \limits_{j = 1,2,3} \left[ {p^{y} \left( {\frac{{\partial y^{j} }}{{\partial v^{1} }} - \frac{{\partial \bar{y}^{j} }}{{\partial v^{1} }} + \frac{{\partial \bar{y}^{j} }}{{\partial v^{1} }} - \frac{{\partial y^{j} }}{{\partial v^{1} }}} \right) + \frac{{\partial p^{y} }}{{\partial v^{1} }}\left( {y^{j} - \bar{y}^{j} } \right) + \frac{{\partial p^{x} }}{{\partial v^{1} }}\left( {x^{j} - \bar{x}^{j} } \right) + v^{j} \frac{{\partial \bar{y}^{j} }}{{\partial v^{1} }} - s^{j} \frac{{\partial y^{j} }}{{\partial v^{1} }}} \right] \\ & \quad + \left( {t^{1} - \tau } \right)\left( {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }}} \right) + \left( {t^{1} - \tau } \right)\left( {\frac{{\partial e^{y2} }}{{\partial v^{1} }} + \frac{{\partial e^{z2} }}{{\partial v^{1} }}} \right) - \tau \left( {\frac{{\partial e^{y3} }}{{\partial v^{1} }} + \frac{{\partial e^{z3} }}{{\partial v^{1} }}} \right) \\ \end{aligned}$$

$$\begin{aligned} & = \mathop \sum \limits_{j = 1,2,3} \left[ {v^{j} \frac{{\partial \bar{y}^{j} }}{{\partial v^{1} }} - s^{j} \frac{{\partial y^{j} }}{{\partial v^{1} }} + \frac{{\partial p^{y} }}{{\partial v^{1} }}\left( {y^{j} - \bar{y}^{j} } \right) + \frac{{\partial p^{x} }}{{\partial v^{1} }}\left( {x^{j} - \bar{x}^{j} } \right)} \right] \\ & \quad + \left( {t^{1} - \tau } \right)\left( {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }}} \right) + \left( {t^{1} - \tau } \right)\left( {\frac{{\partial e^{y2} }}{{\partial v^{1} }} + \frac{{\partial e^{z2} }}{{\partial v^{1} }}} \right) - \tau \left( {\frac{{\partial e^{y3} }}{{\partial v^{1} }} + \frac{{\partial e^{z3} }}{{\partial v^{1} }}} \right) \\ \end{aligned}$$

Recall our assumption from (1):

$$\begin{aligned} & \bar{x}^{1} + \bar{x}^{2} + \bar{x}^{3} = x^{1} + x^{2} + x^{3} \\ & \bar{y}^{1} + \bar{y}^{2} + \bar{y}^{3} = y^{1} + y^{2} + y^{3} \\ \end{aligned}$$

And we can rewrite our equation to

$$= \mathop \sum \limits_{j = 1,2,3} \left[ {v^{j} \frac{{\partial \bar{y}^{j} }}{{\partial v^{1} }} - s^{j} \frac{{\partial y^{j} }}{{\partial v^{1} }}} \right] + \left( {t^{1} - \tau } \right)\left( {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }}} \right) + \left( {t^{2} - \tau } \right)\left( {\frac{{\partial e^{y2} }}{{\partial v^{1} }} + \frac{{\partial e^{z2} }}{{\partial v^{1} }}} \right) - \tau \left( {\frac{{\partial e^{y3} }}{{\partial v^{1} }} + \frac{{\partial e^{z3} }}{{\partial v^{1} }}} \right)$$

Since the consumption tax is only introduced in region 1, and OBA in region 1 and 2, we can re-write to:

$$= \left( {v^{1} \frac{{\partial \bar{y}^{1} }}{{\partial v^{1} }} - s^{1} \frac{{\partial y^{1} }}{{\partial v^{1} }} - s^{2} \frac{{\partial y^{2} }}{{\partial v^{1} }}} \right) + \left( {t^{1} - \tau } \right)\left( {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }}} \right) + \left( {t^{1} - \tau } \right)\left( {\frac{{\partial e^{y2} }}{{\partial v^{1} }} + \frac{{\partial e^{z2} }}{{\partial v^{1} }}} \right) - \tau \left( {\frac{{\partial e^{y3} }}{{\partial v^{1} }} + \frac{{\partial e^{z3} }}{{\partial v^{1} }}} \right)$$

From (2) \(s^{1} = s^{2}\) and \(t^{1} = t^{2}\)

$$v^{1G*} = \left( {\frac{{\partial \bar{y}^{1} }}{{\partial v^{1} }}} \right)^{ - 1} \left[ {s^{1} \left( {\frac{{\partial y^{1} }}{{\partial v^{1} }} + \frac{{\partial y^{2} }}{{\partial v^{1} }}} \right) + \left( {\tau - t^{1} } \right)\left( {\frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }} + \frac{{\partial e^{y2} }}{{\partial v^{1} }} + \frac{{\partial e^{z2} }}{{\partial v^{1} }}} \right) + \tau \left( {\frac{{\partial e^{y3} }}{{\partial v^{1} }} + \frac{{\partial e^{z3} }}{{\partial v^{1} }}} \right)} \right]$$

Remembering our emission constraint \(\frac{{\partial \bar{E}}}{{\partial v^{1} }} = \frac{{\partial e^{y1} }}{{\partial v^{1} }} + \frac{{\partial e^{y2} }}{{\partial v^{1} }} + \frac{{\partial e^{z1} }}{{\partial v^{1} }} + \frac{{\partial e^{z2} }}{{\partial v^{1} }} = 0\), and we finally arrive at (9)

$$v^{1G*} = \left( {\frac{{\partial \bar{y}^{1} }}{{\partial v^{1} }}} \right)^{ - 1} \left[ {s^{1} \left( {\frac{{\partial y^{1} }}{{\partial v^{1} }} + \frac{{\partial y^{2} }}{{\partial v^{1} }}} \right) + \tau \left( {\frac{{\partial e^{y3} }}{{\partial y^{3} }}\frac{{\partial y^{3} }}{{\partial v^{1} }} + \frac{{\partial e^{z3} }}{{\partial z^{3} }}\frac{{\partial z^{3} }}{{\partial v^{1} }}} \right)} \right].$$ (12)

Appendix 2: Summary of the Numerical CGE Model

Set of regions R: NOREU, ROW Set of goods g: x, y, z r (alias j): Index for regions

\(S^{gr}\) : Production of good g in r \(S_{FE}^{r}\) : Production of FE in r \(D^{gr}\) : Aggregated consumer demand of good g in r \(KL^{gr}\) : Value-added composite for g in r \(KLF^{r}\) : Value-added composite for FE in r \(A^{gr}\) : Armington aggregate of g in r \(IM^{gr}\) : Import aggregate of g in r \(W^{r}\) : Consumption composite in r \(p^{g,r}\) : Price of g in r \(p_{FE}^{r}\) : Price of Primary fossil FE in r \(p_{KL}^{gr}\) : Price of value added for g in r \(p_{KLF}^{r}\) : Price of value added for FE in r \(p_{L}^{r}\) : Price of labor (wage rate) in r \(p_{K}^{r}\) : Price of capital (rental rate) in r \(p_{Q}^{r}\) : Rent for primary energy resource in r \(p_{A}^{gr}\) : Price of Armington aggregate of g in r \(p_{IM}^{gr}\) : Price of aggregate imports of g in r \(p_{CO2}^{r}\) : Price of CO2 emission in r \(p_{W}^{r}\) : Price of consumption composite in r \(o^{gr}\) : Output-based allocation on g in r \(v^{gr}\) : Consumption tax on g in r

\(\sigma_{KLE}^{r}\) : Substitution between value-added and energy g in r \(\sigma_{KL}^{r}\) : Substitution between value-added g in r \(\sigma_{Q}^{r}\) : Substitution between value-added and natural resource in FE in r \(\sigma_{LN}^{r}\) : Substitution between value-added in FE in r \(\sigma_{A}^{gr}\) : Substitution between import and domestic g in r \(\sigma_{IM}^{gr}\) : Substitution between imports from different g in r \(\sigma_{W}^{r}\) : Substitution between goods to consumption \(\theta_{FE}^{gr}\) : Cost Share of FE in production of g in r \(\theta_{KL}^{gr}\) : Cost Share of labor in production of g in r \(\theta_{Q}^{r}\) : Cost Share of natural resource in production of FE in r \(\theta_{LN}^{r}\) : Cost Share of labor in production of FE in r \(\theta_{A}^{gr}\) : Cost Share of domestic goods g in consumption in r \(\theta_{IM}^{gr}\) : Cost Share of different imports goods g in consumption in r \(L_{0}^{gr}\) : Labor endowment in sector g in region r \(L_{0,FE}^{r}\) : Labor endowment in FE in region r \(K_{0}^{gr}\) : Capital endowment in sector g in region r \(K_{0,FE}^{r}\) : Capital endowment in FE in region r \(Q_{0}^{r}\) : Resource endowment of primary fossil energy in region r \(CO2_{MAX}^{r}\) : CO 2 emission allowance in region r \(\kappa_{CO2}^{r}\) : Coefficient for primary fossil energy of CO 2 emission in region r

Zero Profit Conditions

Production of goods except for fossil primary energy:

$$\pi_{S}^{gr} = \left( {\theta_{FE}^{gr} \left( {p_{FE}^{r} + \kappa_{{{\text{CO}}_{2} }}^{r} p_{{{\text{CO}}_{2} }}^{gr} } \right)^{{\left( {1 - \sigma_{KLE}^{r} } \right)}} + \left( {1 - \theta_{FE}^{gr} } \right)p_{KL}^{{gr(1 - \sigma_{KLE}^{r} )}} } \right)^{{\left( {\frac{1}{{1 - \sigma_{KLE}^{r} }}} \right)}} \ge p^{gr} + o^{gr} \quad \bot S^{gr}$$

Sector specific value-added aggregate for x, y and z:

$$\pi_{KL}^{gr} = \left( {\theta_{KL}^{gr} p_{L}^{{r(1 - \sigma_{KL}^{gr} )}} + \left( {1 - \theta_{KL}^{gr} } \right)p_{K}^{{r(1 - \sigma_{KL}^{gr} )}} } \right) ^{{\left( {\frac{1}{{1 - \sigma_{KL}^{gr} }}} \right)}} \ge p_{KL}^{gr} \quad \bot KL^{gr}$$

Production of fossil primary energy:

$$\pi_{FE}^{r} = \left( {\theta_{Q}^{r} p_{Q}^{{r(1 - \sigma_{Q}^{r} )}} + \left( {1 - \theta_{Q}^{r} } \right)p_{KLF}^{{r(1 - \sigma_{Q}^{r} )}} } \right)^{{\left( {\frac{1}{{1 - \sigma_{Q}^{r} }}} \right)}} \ge p_{FE}^{r} \quad \bot S_{FE}^{r}$$

Sector specific value-added aggregate for FE:

$$\pi_{KLF}^{r} = \left( {\theta_{LN}^{r} p_{L}^{{r(1 - \sigma_{LN}^{r} )}} + \left( {1 - \theta_{LN}^{r} } \right)p_{K}^{{r(1 - \sigma_{LN}^{r} )}} } \right) ^{{\left( {\frac{1}{{1 - \sigma_{LN}^{r} }}} \right)}} \ge p_{KLF}^{r} \quad \bot KLF^{r}$$

Armington aggregate except for FE:

$$\pi_{A}^{gr} = \left( {\theta_{A}^{gr} \left( {p^{gr} + v^{gr} } \right)^{{\left( {1 - \sigma_{A}^{gr} } \right)}} + \left( {1 - \theta_{A}^{gr} } \right)p_{IM}^{{gr(1 - \sigma_{A}^{gr} )}} } \right)^{{\left( {\frac{1}{{1 - \sigma_{A}^{gr} }}} \right)}} \ge p_{A}^{gr} \quad \bot A^{gr}$$

Import composite except for FE:

$$\pi_{IM}^{gr} = \left( {\mathop \sum \limits_{j

e r} \theta_{IM}^{gjr} \left( {p^{gj} + v^{gr} } \right)^{{\left( {1 - \sigma_{IM}^{gr} } \right)}} } \right)^{{\left( {\frac{1}{{1 - \sigma_{IM}^{gr} }}} \right)}} \ge p_{IM}^{gr} \quad \bot IM^{gr}$$

Consumption composite:

$$\pi_{W}^{r} = \left( {\theta_{W}^{xr} p_{A}^{{xr(1 - \sigma_{W}^{r} )}} + \theta_{W}^{yr} p_{A}^{{yr(1 - \sigma_{W}^{r} )}} + \theta_{W}^{zr} p_{A}^{{zr(1 - \sigma_{W}^{r} )}} } \right)^{{\left( {\frac{1}{{1 - \sigma_{w}^{r} }}} \right)}} \ge p_{W}^{r} \quad \bot W^{r} .$$

Market Clearing Conditions

Labor:

$$\mathop \sum \limits_{g} L_{0}^{gr} + L_{0,FE}^{r} \ge \mathop \sum \limits_{g} KL^{gr} \frac{{\partial \pi_{KL}^{gr} }}{{\partial p_{L}^{r} }} + KLF^{r} \frac{{\partial \pi_{KLF}^{r} }}{{\partial p_{L}^{r} }}\quad \bot p_{L}^{r}$$

Capital:

$$\mathop \sum \limits_{g} K_{0}^{gr} + K_{0,FE}^{r} \ge \mathop \sum \limits_{g} KL^{gr} \frac{{\partial \pi_{KL}^{gr} }}{{\partial p_{K}^{r} }} + KLF^{r} \frac{{\partial \pi_{KLF}^{r} }}{{\partial p_{K}^{r} }}\quad \bot p_{K}^{r}$$

Primary fossil energy resource:

$$Q_{0}^{r} \ge S_{FE}^{r} \frac{{\partial \pi_{FE}^{r} }}{{\partial p_{Q}^{r} }} \quad \bot p_{Q}^{r}$$

Value-added except FE:

$$KL^{gr} \ge S^{gr} \frac{{\partial \pi_{S}^{gr} }}{{\partial p_{KL}^{gr} }} \quad \bot p_{KL}^{gr}$$

Value-added FE:

$$KLF^{r} \ge S_{FE}^{r} \frac{{\partial \pi_{FE}^{r} }}{{\partial p_{KLF}^{r} }}\quad \bot p_{KLF}^{r}$$

Armington aggregate:

$$A^{gr} \ge W^{r} \frac{{\partial \pi_{W}^{r} }}{{\partial p_{A}^{gr} }}\quad \bot p_{A}^{gr}$$

Import aggregate:

$$IM^{gr} \ge A^{gr} \frac{{\partial \pi_{A}^{gr} }}{{\partial p_{IM}^{gr} }}\quad \bot p_{IM}^{gr}$$

Supply–demand balance of goods, except FE:

$$S^{gr} \ge A^{gr} \frac{{\partial \pi_{A}^{gr} }}{{\partial p^{gr} }} + \mathop \sum \limits_{j

e r} IM^{gj} \frac{{\partial \pi_{IM}^{gj} }}{{\partial p^{gj} }}\quad \bot p^{gr}$$

Supply–demand balance of FE:

$$S_{FE}^{r} \ge \mathop \sum \limits_{g} S^{gr} \frac{{\partial \pi_{S}^{gr} }}{{\partial \left( {p_{FE}^{r} + \kappa_{CO2}^{r} p_{CO2}^{gr} } \right)}}\quad \bot p_{FE}^{r}$$

Demand of goods:

$$D^{gr} \ge A^{gr} \frac{{\partial \pi_{A}^{gr} }}{{\partial p^{gr} }} + IM^{gr} \frac{{\partial \pi_{IM}^{gr} }}{{\partial p^{gr} }}\quad \bot D^{gr}$$

CO 2 Emission in region:

$$CO2_{MAX}^{r} \ge \kappa_{CO2}^{r} S_{FE}^{r} \quad \bot p_{CO2}^{r}$$

Consumption by consumers

$$p_{W}^{r} W^{r} \ge p_{L}^{r} \left( {\mathop \sum \limits_{g} L_{0}^{gr} + L_{0,FE}^{r} } \right) + p_{K}^{r} \left( {\mathop \sum \limits_{g} K_{0}^{gr} + K_{0,FE}^{r} } \right) + p_{Q}^{r} Q_{0}^{r} + p_{CO2}^{r} CO2_{MAX}^{r} - S^{gr} o^{gr} + D^{gr} v^{gr} \quad \bot p_{W}^{r}$$

See Figs. 8 and 9.

Fig. 8 Nesting in production, except for fossil fuel energy Full size image

Fig. 9 Nesting in production of fossil fuel energy Full size image

Appendix 3: Mapping of WIOD Sectors

Table 2 shows the mapping of the 56 WIOD sectors to three composite sectors in our model.