Unilateral Climate Policy: Harmful or Even Disastrous?

Abstract

This paper deals with possible foreign reactions to unilateral carbon demand reducing policies. It differentiates between demand side and supply side reactions as well as between intra- and inter-temporal shifts in greenhouse gas emissions. In our model, we integrate a stock-dependent marginal physical cost of extracting fossil fuels into Eichner and Pethig’s (Int Econ Rev 52(3):767–805, 2011) general equilibrium carbon leakage model. The results are as follows: Under similar but somewhat tighter conditions than those derived by Eichner and Pethig (Int Econ Rev 52(3):767–805, 2011), a weak green paradox arises. Furthermore, a strong green paradox can arise in our model under supplementary constraints. That means a “green” policy measure might not only lead to a harmful acceleration of fossil fuel extraction but to an increase in the cumulative climate damages at the same time. In some of these cases there is even a cumulative extraction expansion, which we consider disastrous.

Appendix

1.1 The Fossil Fuel Market

Throughout the appendix the commodity in period one is chosen as numeraire. Rearranging of (7)–(9), (20), and (21) yields:

$$\begin{aligned} p_{e1}-X^{E1}_{e_{F1}}-p_{x2}X^{E2}_{e_{F1}}&= 0, \end{aligned}$$

(39)

$$\begin{aligned} p_{e2}-p_{x2}X^{E2}_{e_{F2}}&= 0, \end{aligned}$$

(40)

$$\begin{aligned} e_{Ft}-\overline{e}_{At}-e_{Nt}&= 0,\quad t=1,2, \end{aligned}$$

(41)

$$\begin{aligned} X^{A1}_{\overline{e}_{A1}}-p_{e1}-\pi _1&= 0, \end{aligned}$$

(42)

$$\begin{aligned} p_{x2}X^{A2}_{\overline{e}_{A2}}-p_{e2}-\pi _2&= 0, \end{aligned}$$

(43)

$$\begin{aligned} X^{N1}_{e_{N1}}-p_{e1}&= 0, \end{aligned}$$

(44)

$$\begin{aligned} p_{x2}X^{N2}_{e_{N2}}-p_{e2}&= 0. \end{aligned}$$

(45)

Total differentiation of (39)–(45) yields:

$$\begin{aligned}&\displaystyle \mathrm{d}p_{e1}-X^{E1}_{e_{F1}e_{F1}}\mathrm{d}e_{F1}-X^{E2}_{e_{F1}}\mathrm{d}p_{x2} -p_{x2}\Bigg [X^{E2}_{e_{F1}e_{F1}}\mathrm{d}e_{F1}+X^{E2}_{e_{F1}e_{F2}}\mathrm{d}e_{F2}\Bigg ]=0,\end{aligned}$$

(46)

$$\begin{aligned}&\displaystyle \mathrm{d}p_{e2}-X^{E2}_{e_{F2}}\mathrm{d}p_{x2}-p_{x2}\Bigg [X^{E2}_{e_{F2}e_{F1}} \mathrm{d}e_{F1}+X^{E2}_{e_{F2}e_{F2}}\mathrm{d}e_{F2}\Bigg ]=0,\end{aligned}$$

(47)

$$\begin{aligned}&\displaystyle \mathrm{d}e_{Ft}-\mathrm{d}\overline{e}_{At}-\mathrm{d}e_{Nt}=0,\quad t=1,2,\end{aligned}$$

(48)

$$\begin{aligned}&\displaystyle X^{A1}_{\overline{e}_{A1}\overline{e}_{A1}}\mathrm{d}\overline{e}_{A1} -\mathrm{d}p_{e1}-\mathrm{d}\pi _1=0,\end{aligned}$$

(49)

$$\begin{aligned}&\displaystyle X^{A2}_{\overline{e}_{A2}}\mathrm{d}p_{x2}+p_{x2}X^{A2}_{\overline{e}_{A2} \overline{e}_{A2}}\mathrm{d}\overline{e}_{A2}-\mathrm{d}p_{e2}-\mathrm{d}\pi _2=0, \end{aligned}$$

(50)

$$\begin{aligned}&\displaystyle \frac{\widehat{e}_{N1}}{\widehat{p}_{e1}}-\eta _{N1}=0,\end{aligned}$$

(51)

$$\begin{aligned}&\displaystyle \frac{\widehat{e}_{N2}}{\widehat{p}_{e2}-\widehat{p}_{x2}}-\eta _{N2}=0, \end{aligned}$$

(52)

where $\eta _{Nt}:=\frac{X^{Nt}_{e_{Nt}}}{e_{Nt}X^{Nt}_{e_{Nt} e_{Nt}}}<0$ for $t=1,2$.

Inserting (51) and (52) in (48) and afterwards inserting in (46)–(47) yields:

$$\begin{aligned}&\mathrm{d}p_{e1}-X^{E1}_{e_{F1}e_{F1}}\Bigg [\mathrm{d}\overline{e}_{A1}+e_{N1} \eta _{N1}\widehat{p}_{e1}\Bigg ]-X^{E2}_{e_{F1}}\mathrm{d}p_{x2}-p_{x2} \Bigg [X^{E2}_{e_{F1}e_{F1}}\Bigg [\mathrm{d}\overline{e}_{A1}\nonumber \\&\left. \quad +\,e_{N1}\eta _{N1}\widehat{p}_{e1}\Bigg ]+X^{E2}_{e_{F1}e_{F2}}\Bigg [\mathrm{d}\overline{e}_{A2}+e_{N2}\eta _{N2}\left[ \widehat{p}_{e2}- \widehat{p}_{x2}\right] \Bigg ]\right] =0,\end{aligned}$$

(53)

$$\begin{aligned}&\mathrm{d}p_{e2}-X^{E2}_{e_{F2}}\mathrm{d}p_{x2}-p_{x2}\Bigg [X^{E2}_{e_{F2} e_{F1}}\Bigg [\mathrm{d}\overline{e}_{A1}+e_{N1}\eta _{N1}\widehat{p}_{e1}\Bigg ] \nonumber \\&\quad +\,X^{E2}_{e_{F2}e_{F2}}\Bigg [\mathrm{d}\overline{e}_{A2}+e_{N2}\eta _{N2}\left[ \widehat{p}_{e2}-\widehat{p}_{x2} \right] \Bigg ]\Bigg ]=0. \end{aligned}$$

(54)

Inserting (53) in (54) yields:

$$\begin{aligned} \mathrm{d}p_{e1}&= -\frac{\frac{p_{e1}}{e_{N1}\eta _{N1}}[\Gamma _0-p_{e1} [p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]]}{\Gamma _0}\mathrm{d}\overline{e}_{A1}\nonumber \\&+\,\,\frac{p_{e1}p_{e2}p_{x2}X^{E2}_{e_{F1}e_{F2}}}{\Gamma _0}\mathrm{d}\overline{e}_{A2}+\frac{p_{e1}X^{E2}_{e_{F1}}[p_{e2}-p_{x2}X^{E2}_{ e_{F2}e_{F2}}e_{N2}\eta _{N2}]}{\Gamma _0}\mathrm{d}p_{x2},\end{aligned}$$

(55)

$$\begin{aligned} \mathrm{d}p_{e2}&= \frac{p_{e1}p_{e2}p_{x2}X^{E2}_{e_{F2}e_{F1}}}{\Gamma _0} \mathrm{d}\overline{e}_{A1}\nonumber \\&-\,\,\frac{\frac{p_{e2}}{e_{N2}\eta _{N2}}[\Gamma _0-p_{e2}[p_{e1} -[X^{E1}_{e_{F1}e_{F1}}+p_{x2}X^{E2}_{e_{F1}e_{F1}}]e_{N1} \eta _{N1}]]}{\Gamma _0}\mathrm{d}\overline{e}_{A2}\nonumber \\&+\,\,\frac{X^{E2}_{e_{F2}} \Gamma _3}{\Gamma _0}\mathrm{d}p_{x2}, \end{aligned}$$

(56)

where $\Gamma _0=[p_{e1}-[X^{E1}_{e_{F1}e_{F1}}+p_{x2} X^{E2}_{e_{F1}e_{F1}}]e_{N1}\eta _{N1}][p_{e2}-p_{x2} X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]-p_{x2}X^{E2}_{e_{F1} e_{F2}}e_{N1}\eta _{N1}\cdot p_{x2}X^{E2}_{e_{F2}e_{F1}} e_{N2}\eta _{N2}$ and $\Gamma _3=\Gamma _0+\frac{p_{x2} X^{E2}_{e_{F1}}}{p_{e2}}\cdot p_{x2}X^{E2}_{e_{F2} e_{F1}}e_{N1}\eta _{N1}\cdot p_{e2}$.

Inserting (51) and (52) in (48) and afterwards inserting (55) and (56) yields:

$$\begin{aligned} \mathrm{d}e_{F1}&= \frac{p_{e1}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2} \eta _{N2}]}{\Gamma _0}\mathrm{d}\overline{e}_{A1}+\frac{p_{e2}p_{x2} X^{E2}_{e_{F1}e_{F2}}e_{N1}\eta _{N1}}{\Gamma _0}\mathrm{d}\overline{e}_{A2} \nonumber \\&+\,\,\frac{X^{E2}_{e_{F1}}e_{N1}\eta _{N1}[p_{e2}-p_{x2}X^{E2}_{e_{F2} e_{F2}}e_{N2}\eta _{N2}]}{\Gamma _0}\mathrm{d}p_{x2},\end{aligned}$$

(57)

$$\begin{aligned} \mathrm{d}e_{F2}&= \frac{p_{e1}p_{x2}X^{E2}_{e_{F2}e_{F1}}e_{N2}\eta _{N2}}{\Gamma _0}\mathrm{d}\overline{e}_{A1}+\frac{p_{e2}[p_{e1}-[X^{E1}_{e_{F1} e_{F1}}+p_{x2}X^{E2}_{e_{F1}e_{F1}}]e_{N1}\eta _{N1}]}{\Gamma _0}\mathrm{d}\overline{e}_{A2}\nonumber \\&+\,\,\frac{X^{E2}_{e_{F1}}e_{N1}\eta _{N1}[p_{x2}X^{E2}_{e_{F2}e_{F1}} e_{N2}\eta _{N2}]}{\Gamma _0}\mathrm{d}p_{x2}. \end{aligned}$$

(58)

Adding (57)–(58) yields:

$$\begin{aligned} \mathrm{d}e_{F1}+\mathrm{d}e_{F2}=\mathrm{d}e_{F\varSigma }=\frac{p_{e1}\Gamma _1}{\Gamma _0}\mathrm{d}\overline{e}_{A1}+\frac{p_{e2}\Gamma _2}{\Gamma _0}\mathrm{d}\overline{e}_{A2}+\frac{X^{E2}_{e_{F1}}e_{N1}\eta _{N1}\Gamma _1}{\Gamma _0}\mathrm{d}p_{x2}, \end{aligned}$$

(59)

where $\Gamma _1=p_{e2}+[p_{x2}X^{E2}_{e_{F2}e_{F1}}-p_{x2} X^{E2}_{e_{F2}e_{F2}}]e_{N2}\eta _{N2}$ and $\Gamma _2=p_{e1} -[X^{E1}_{e_{F1}e_{F1}}+p_{x2}X^{E2}_{e_{F1}e_{F1}}-p_{x2} X^{E2}_{e_{F1}e_{F2}}]e_{N1}\eta _{N1}$.

1.2 The Commodity Market

The relative commodity demand of $A,N,F \text{ and } E$ is equal to:

$$\begin{aligned} q^d=\frac{\sum x_{i1}}{\sum x_{i2}}=\frac{x_{A1}+x_{N1}+x_{F1} +X^{E1}}{x_{A2}+x_{N2}+x_{F2}+X^{E2}},\quad i=A,N,F,E. \end{aligned}$$

(60)

Inserting (22) and (27) in (60) yields:

$$\begin{aligned} q^d=\left( \frac{\alpha _1p_{x2}}{\alpha _2}\right) ^\sigma -\left( \frac{\alpha _1p_{x2}}{\alpha _2}\right) ^\sigma \frac{X^{E2}}{X^{A2}+X^{N2}}+\frac{X^{E1}}{X^{A2}+X^{N2}}. \end{aligned}$$

(61)

Total differentiation of (61) and afterwards inserting (39)–(45) and (27) yields:

$$\begin{aligned} \mathrm{d}{q}^d&= \left( \frac{\alpha _1p_{x2}}{\alpha _2}\right) ^\sigma \sigma \widehat{p}_{x2}-\left( \frac{\alpha _1p_{x2}}{\alpha _2}\right) ^\sigma \sigma \widehat{p}_{x2}\frac{X^{E2}}{X^{A2}+X^{N2}}\nonumber \\&-\left( \frac{\alpha _1p_{x2}}{\alpha _2}\right) ^\sigma \frac{\mathrm{d}X^{E2}(X^{A2}+X^{N2})-X^{E2}(\mathrm{d}X^{A2}+\mathrm{d}X^{N2})}{(X^{A2} +X^{N2})^2}\nonumber \\&+\,\,\frac{\mathrm{d}X^{E1}(X^{A2}+X^{N2})-X^{E1}(\mathrm{d}X^{A2}+\mathrm{d}X^{N2})}{(X^{A2}+X^{N2})^2}\nonumber \\&= \frac{x^s_{A1}+x^s_{N1}-x_{E1}}{x^s_{A2}+x^s_{N2}}\sigma \widehat{p}_{x2}-\frac{\frac{\pi _2}{p_{x2}}\left( \frac{x^s_{A1} +x^s_{N1}}{x^s_{A2}+x^s_{N2}}-\frac{x^s_{A1}+x^s_{N1}-x_{E1}}{x^s_{A2} +x^s_{N2}-x_{E2}}\right) }{x^s_{A2}+x^s_{N2}}\mathrm{d}{\overline{e}}_{A2} \nonumber \\&+\,\,\frac{X^{E1}_{e_{F1}}-X^{E2}_{e_{F1}}\cdot \frac{x^s_{A1}+x^s_{N1} -x_{E1}}{x^s_{A2}+x^s_{N2}-x_{E2}}}{x^s_{A2}+x^s_{N2}}\mathrm{d}{e}_{F1} -\frac{X^{E2}_{e_{F2}}\cdot \frac{x^s_{A1}+x^s_{N1}}{x^s_{A2} +x^s_{N2}}}{x^s_{A2}+x^s_{N2}}\mathrm{d}{e}_{F2}. \end{aligned}$$

(62)

The relative commodity supply of $A \text{ and } N$ is equal to:

$$\begin{aligned} q^s=\frac{\sum x^s_{j1}}{\sum x^s_{j2}}=\frac{X^{A1}+X^{N1}}{X^{A2}+X^{N2}},\quad j=A,N. \end{aligned}$$

(63)

Total differentiation of (63) and afterwards inserting (39)–(45) yields:

$$\begin{aligned}&\!\!\!\mathrm{d}{q}^s=\frac{(X^{A1}_{\overline{e}_{A1}}\mathrm{d}{\overline{e}}_{A1} +X^{N1}_{e_{N1}}\mathrm{d}{e}_{N1})(X^{A2}+X^{N2})-(X^{A1}+X^{N1}) (X^{A2}_{\overline{e}_{A2}}\mathrm{d}{\overline{e}}_{A2}+X^{N2}_{e_{N2}} \mathrm{d}{e}_{N2})}{(X^{A2}+X^{N2})^2}\nonumber \\&\,=\frac{\pi _1}{x^s_{A2}+x^s_{N2}}\mathrm{d}{\overline{e}}_{A1}{-} \frac{\frac{\pi _2}{p_{x2}}\cdot \frac{x^s_{A1}{+}x^s_{N1}}{x^s_{A2} +x^s_{N2}}}{x^s_{A2}+x^s_{N2}}\mathrm{d}{\overline{e}}_{A2}+\,\frac{X^{E1}_{e_{F1}}+p_{x2}X^{E2}_{e_{F1}}}{x^s_{A2}+x^s_{N2}} \mathrm{d}{e}_{F1}{-}\frac{X^{E2}_{e_{F2}}\cdot \frac{x^s_{A1}{+}x^s_{N1}}{x^s_{A2}+x^s_{N2}}}{x^s_{A2}+x^s_{N2}}\mathrm{d}{e}_{F2}.\nonumber \\ \end{aligned}$$

(64)

Equating (62) and (64) yields:

$$\begin{aligned} \mathrm{d}{p}_{x2}=\frac{p_{x2}}{\sigma }\left( \frac{\pi _1}{x^s_{A1} +x^s_{N1}-x_{E1}}\mathrm{d}{\overline{e}}_{A1}-\frac{\pi _2}{p_{x2} (x^s_{A2}+x^s_{N2}-x_{E2})}\mathrm{d}{\overline{e}}_{A2}+\Theta \mathrm{d}{e}_{F1}\right) , \end{aligned}$$

(65)

where $\Theta =\frac{p_{x2}X^{E2}_{e_{F1}}}{x^s_{A1}+x^s_{N1}-x_{E1}} +\frac{p_{x2}X^{E2}_{e_{F1}}}{p_{x2}(x^s_{A2}+x^s_{N2}-x_{E2})}$.

1.3 The Combined Market

1.3.1 The Quantities on the Combined Market

Inserting (65) in (57) for $\mathrm{d}{\overline{e}}_{A2}=0$ yields:

$$\begin{aligned} \mathrm{d}{e}_{F1}&= \frac{p_{e1}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2} \eta _{N2}]}{\Gamma _0}\mathrm{d}\overline{e}_{A1}+\frac{X^{E2}_{e_{F1}}e_{N1} \eta _{N1}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}{\Gamma _0}\nonumber \\&\cdot \frac{p_{x2}}{\sigma }\left( \frac{\pi _1}{x^s_{A1}+x^s_{N1} -x_{E1}}\mathrm{d}\overline{e}_{A1}+\Theta \mathrm{d}{e}_{F1}\right) \nonumber \\&= \frac{[p_{e1}\sigma +p_{x2}X^{E2}_{e_{F1}}\frac{\pi _1e_{N1}\eta _{N1}}{x^s_{A1}+x^s_{N1}-x_{E1}}][p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2} \eta _{N2}]}{\sigma \Gamma _0-p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1} \eta _{N1}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]} \mathrm{d}\overline{e}_{A1}. \end{aligned}$$

(66)

Inserting (65) and (66) in (58) for $\mathrm{d}{\overline{e}}_{A2}=0$ yields:

$$\begin{aligned} \mathrm{d}{e}_{F2}&= \frac{p_{e1}p_{x2}X^{E2}_{e_{F2}e_{F1}}e_{N2} \eta _{N2}}{\Gamma _0}\mathrm{d}\overline{e}_{A1}+\frac{X^{E2}_{e_{F1}} e_{N1}\eta _{N1}[p_{x2}X^{E2}_{e_{F2}e_{F1}}e_{N2}\eta _{N2}]}{\Gamma _0} \nonumber \\&\cdot \frac{p_{x2}}{\sigma }\left( \frac{\pi _1}{x^s_{A1}+x^s_{N1} -x_{E1}}\mathrm{d}\overline{e}_{A1}\right. \nonumber \\&\left. \quad \quad \quad +\,\,\Theta \frac{[p_{e1}\sigma +p_{x2}X^{E2}_{e_{F1}} \frac{\pi _1e_{N1}\eta _{N1}}{x^s_{A1}+x^s_{N1}-x_{E1}}] [p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}{\sigma \Gamma _0-p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1}\eta _{N1}[p_{e2} -p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}\mathrm{d}\overline{e}_{A1} \right) \nonumber \\&= \frac{[p_{e1}\sigma +p_{x2}X^{E2}_{e_{F1}}\frac{\pi _1e_{N1} \eta _{N1}}{x^s_{A1}+x^s_{N1}-x_{E1}}]p_{x2}X^{E2}_{e_{F2}e_{F1}} e_{N2}\eta _{N2}}{\sigma \Gamma _0-p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1} \eta _{N1}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]} \mathrm{d}\overline{e}_{A1}. \end{aligned}$$

(67)

Adding (66) and (67) yields:

$$\begin{aligned} \mathrm{d}{e}_{F1}+\mathrm{d}{e}_{F2}=\mathrm{d}{e}_{F\varSigma }=\frac{[p_{e1}\sigma \!\!+\!\!p_{x2}X^{E2}_{e_{F1}}\frac{\pi _1e_{N1}\eta _{N1}}{x^s_{A1}\!+\!x^s_{N1} \!-\!x_{E1}}]\Gamma _1}{\sigma \Gamma _0-p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1}\eta _{N1}[p_{e2}\!-\!p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]} \mathrm{d}\overline{e}_{A1}.\qquad \end{aligned}$$

(68)

Inserting (66) and (68) in (29) yields:

$$\begin{aligned}&\mathrm{d}D(e_{F1},e_{F\varSigma })\gtreqless 0\nonumber \\&\Leftrightarrow \quad \Bigg [p_{e1}\sigma +p_{x2}X^{E2}_{e_{F1}} \frac{\pi _1e_{N1}\eta _{N1}}{x^s_{A1}\!+\!x^s_{N1}-x_{E1}}\Bigg ]\Bigg [[p_{e2} -p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]+\lambda \Gamma _1\Bigg ] \mathrm{d}\overline{e}_{A1}\gtreqless 0.\nonumber \\ \end{aligned}$$

(69)

Inserting (65) in (57) for $\mathrm{d}{\overline{e}}_{A1}=0$ yields:

$$\begin{aligned} \mathrm{d}{e}_{F1}&= \frac{p_{e2}p_{x2}X^{E2}_{e_{F1}e_{F2}}e_{N1} \eta _{N1}}{\Gamma _0}\mathrm{d}\overline{e}_{A2}+\frac{X^{E2}_{e_{F1}} e_{N1}\eta _{N1}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}{\Gamma _0}\nonumber \\&\cdot \frac{p_{x2}}{\sigma }\left( -\frac{\pi _2}{p_{x2}(x^s_{A2} +x^s_{N2}-x_{E2})}\mathrm{d}\overline{e}_{A2}+\Theta \mathrm{d}e_{F1}\right) \nonumber \\&= \frac{p_{e2}\sigma p_{x2}X^{E2}_{e_{F1}e_{F2}}e_{N1}\eta _{N1}}{\sigma \Gamma _0-p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1}\eta _{N1}[p_{e2} -p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}\mathrm{d}\overline{e}_{A2} \nonumber \\&-\frac{p_{x2}X^{E2}_{e_{F1}}\frac{\pi _2e_{N1}\eta _{N1}}{p_{x2} (x^s_{A2}+x^s_{N2}-x_{E2})}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}} e_{N2}\eta _{N2}]}{\sigma \Gamma _0-p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1} \eta _{N1}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}\mathrm{d}\overline{e}_{A2}. \end{aligned}$$

(70)

Inserting (65) and (70) in (58) for $\mathrm{d}{\overline{e}}_{A1}=0$ yields:

$$\begin{aligned} \mathrm{d}{e}_{F2}&= \frac{p_{e2}[p_{e1}-[X^{E1}_{e_{F1}e_{F1}}+p_{x2} X^{E2}_{e_{F1}e_{F1}}]e_{N1}\eta _{N1}]}{\Gamma _0}\mathrm{d}\overline{e}_{A2} \nonumber \\&+\,\,\frac{X^{E2}_{e_{F1}}e_{N1}\eta _{N1}[p_{x2}X^{E2}_{e_{F2}e_{F1}} e_{N2}\eta _{N2}]}{\Gamma _0}\cdot \frac{p_{x2}}{\sigma }\left( -\frac{\pi _2}{p_{x2}(x^s_{A2}+x^s_{N2}-x_{E2})}\mathrm{d}\overline{e}_{A2} \right. \nonumber \\&+\,\,\Theta \frac{p_{e2}\sigma p_{x2}X^{E2}_{e_{F1}e_{F2}}e_{N1}\eta _{N1}}{\sigma \Gamma _0-p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1}\eta _{N1}[p_{e2} -p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}\mathrm{d}\overline{e}_{A2} \nonumber \\&\left. -\,\,\Theta \frac{p_{x2}X^{E2}_{e_{F1}}\frac{\pi _2e_{N1}\eta _{N1}}{p_{x2}(x^s_{A2}+x^s_{N2}-x_{E2})}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}} e_{N2}\eta _{N2}]}{\sigma \Gamma _0-p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1} \eta _{N1}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}\mathrm{d}\overline{e}_{A2}\right) \nonumber \\&= \frac{p_{e2}\sigma [p_{e1}-[X^{E1}_{e_{F1}e_{F1}}+p_{x2}X^{E2}_{e_{F1} e_{F1}}]e_{N1}\eta _{N1}]}{\sigma \Gamma _0-p_{x2}X^{E2}_{e_{F1}} \Theta e_{N1}\eta _{N1}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2} \eta _{N2}]}\mathrm{d}\overline{e}_{A2}\nonumber \\&-\frac{p_{x2}X^{E2}_{e_{F1}}\frac{\pi _2e_{N1}\eta _{N1}}{p_{x2} (x^s_{A2}+x^s_{N2}-x_{E2})}p_{x2}X^{E2}_{e_{F2}e_{F1}}e_{N2}\eta _{N2} +p_{e2}p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1}\eta _{N1}}{\sigma \Gamma _0 -p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1}\eta _{N1}[p_{e2}-p_{x2} X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}\mathrm{d}\overline{e}_{A2}.\nonumber \\ \end{aligned}$$

(71)

Adding (70) and (71) yields:

$$\begin{aligned} \mathrm{d}{e}_{F1}\!{+}\!\mathrm{d}{e}_{F2}\!=\!\mathrm{d}{e}_{F\varSigma }\!=\!\frac{p_{e2}\sigma \Gamma _2{-}p_{x2}X^{E2}_{e_{F1}}\frac{\pi _2e_{N1}\eta _{N1}}{p_{x2} (x^s_{A2}\!+\!x^s_{N2}\!-\!x_{E2})}\Gamma _1\!-\!p_{e2}p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1}\eta _{N1}}{\sigma \Gamma _0\!-\!p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1}\eta _{N1}[p_{e2}\!-\!p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2} \eta _{N2}]}\mathrm{d}\overline{e}_{A2}.\nonumber \\ \end{aligned}$$

(72)

Inserting (70) and (72) in (29) yields:

$$\begin{aligned}&\mathrm{d}D(e_{F1},e_{F\varSigma })\gtreqless 0\nonumber \\&\quad \Leftrightarrow \quad \Bigg [p_{e2}\sigma [p_{x2}X^{E2}_{e_{F1}e_{F2}} e_{N1}\eta _{N1}+\lambda \Gamma _2]-p_{x2}X^{E2}_{e_{F1}}\frac{\pi _2 e_{N1}\eta _{N1}}{p_{x2}(x^s_{A2}+x^s_{N2}-x_{E2})}\nonumber \\&\qquad \qquad \cdot [[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}] +\lambda \Gamma _1]-\lambda p_{e2}p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1} \eta _{N1}\Bigg ]\mathrm{d}\overline{e}_{A2}\gtreqless 0.\nonumber \\ \end{aligned}$$

(73)

1.3.2 The Prices on the Combined Market

Inserting (66) in (65) for $\mathrm{d}{\overline{e}}_{A2}=0$ yields:

$$\begin{aligned} \frac{\mathrm{d}{p}_{x2}}{\mathrm{d}{\overline{e}}_{A1}}&= \frac{p_{x2}}{\sigma } \left( \frac{\pi _1}{x^s_{A1}+x^s_{N1}-x_{E1}}\right. \nonumber \\&\left. +\,\,\Theta \frac{[p_{e1}\sigma +p_{x2}X^{E2}_{e_{F1}} \frac{\pi _1e_{N1}\eta _{N1}}{x^s_{A1}+x^s_{N1}-x_{E1}}][p_{e2} -p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}{\sigma \Gamma _0 -p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1}\eta _{N1}[p_{e2}-p_{x2} X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}\right) \nonumber \\&= \frac{p_{x2}\frac{\pi _1}{x^s_{A1}+x^s_{N1}-x_{E1}}\Gamma _0 +p_{e1}p_{x2}\Theta [p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2} \eta _{N2}]}{\sigma \Gamma _0-p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1} \eta _{N1}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]} \nonumber \\&> 0. \end{aligned}$$

(74)

Inserting (70) in (65) for $\mathrm{d}{\overline{e}}_{A1}=0$ yields:

$$\begin{aligned} \frac{\mathrm{d}{p}_{x2}}{\mathrm{d}{\overline{e}}_{A2}}&= \frac{p_{x2}}{\sigma } \left( -\frac{\pi _2}{p_{x2}(x^s_{A2}+x^s_{N2}-x_{E2})}\right. \nonumber \\&\left. +\,\,\Theta \frac{p_{e2}\sigma p_{x2}X^{E2}_{e_{F1}e_{F2}}e_{N1} \eta _{N1}-p_{x2}X^{E2}_{e_{F1}}\frac{\pi _2e_{N1}\eta _{N1}}{p_{x2} (x^s_{A2}+x^s_{N2}-x_{E2})}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2} \eta _{N2}]}{\sigma \Gamma _0-p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1}\eta _{N1} [p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}\right) \nonumber \\&= -\frac{p_{x2}\frac{\pi _2}{p_{x2}(x^s_{A2}+x^s_{N2}-x_{E2})} \Gamma _0-p_{e2}p_{x2}\Theta p_{x2}X^{E2}_{e_{F1}e_{F2}}e_{N1} \eta _{N1}}{\sigma \Gamma _0-p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1} \eta _{N1}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}\nonumber \\&< 0. \end{aligned}$$

(75)

Inserting (74) in (55) for $\mathrm{d}{\overline{e}}_{A2}=0$ yields:

$$\begin{aligned} \frac{\mathrm{d}{p}_{e1}}{\mathrm{d}{\overline{e}}_{A1}}&= -\frac{\frac{p_{e1}}{e_{N1}\eta _{N1}}[\Gamma _0-p_{e1}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}} e_{N2}\eta _{N2}]]}{\Gamma _0}\nonumber \\&\quad +\,\,\frac{p_{e1}X^{E2}_{e_{F1}}[p_{e2}-p_{x2} X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}{\Gamma _0}\nonumber \\&\cdot \frac{p_{x2}\frac{\pi _1}{x^s_{A1}+x^s_{N1}-x_{E1}}\Gamma _0 +p_{e1}p_{x2}\Theta [p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2} \eta _{N2}]}{\sigma \Gamma _0-p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1} \eta _{N1}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}\nonumber \\&= -\,\,\frac{\frac{p_{e1}}{e_{N1}\eta _{N1}}\sigma [\Gamma _0-p_{e1} [p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]]}{\sigma \Gamma _0-p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1}\eta _{N1}[p_{e2} -p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}\nonumber \\&+\,\,\frac{p_{e1}p_{x2}X^{E2}_{e_{F1}}[\frac{\pi _1}{x^s_{A1}+x^s_{N1} -x_{E1}}+\Theta ][p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}{\sigma \Gamma _0-p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1}\eta _{N1}[p_{e2} -p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}\nonumber \\&> 0. \end{aligned}$$

(76)

Inserting (74) in (56) for $\mathrm{d}{\overline{e}}_{A2}=0$ yields:

$$\begin{aligned} \frac{\mathrm{d}{p}_{e2}}{\mathrm{d}{\overline{e}}_{A1}}&= \frac{p_{e1}p_{e2} p_{x2}X^{E2}_{e_{F2}e_{F1}}}{\Gamma _0}+\frac{X^{E2}_{e_{F2}}\Gamma _3}{\Gamma _0}\nonumber \\&\cdot \frac{p_{x2}\frac{\pi _1}{x^s_{A1}+x^s_{N1}-x_{E1}}\Gamma _0 +p_{e1}p_{x2}\Theta [p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2} \eta _{N2}]}{\sigma \Gamma _0-p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1} \eta _{N1}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}\nonumber \\&= \frac{p_{e1}p_{e2}\sigma p_{x2}X^{E2}_{e_{F2}e_{F1}}+p_{x2} X^{E2}_{e_{F2}}\frac{\pi _1}{x^s_{A1}+x^s_{N1}-x_{E1}}\Gamma _3 +p_{e1}p_{x2}X^{E2}_{e_{F2}}\Theta [p_{e2}-p_{x2}X^{E2}_{e_{F2} e_{F2}}e_{N2}\eta _{N2}]}{\sigma \Gamma _0-p_{x2}X^{E2}_{e_{F1}} \Theta e_{N1}\eta _{N1}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2} \eta _{N2}]}\nonumber \\&> 0\Leftarrow \Gamma _3>0. \end{aligned}$$

(77)

Inserting (75) in (55) for $\mathrm{d}{\overline{e}}_{A1}=0$ yields:

$$\begin{aligned} \frac{\mathrm{d}{p}_{e1}}{\mathrm{d}{\overline{e}}_{A2}}&= \frac{p_{e1}p_{e2} p_{x2}X^{E2}_{e_{F1}e_{F2}}}{\Gamma _0}+\frac{p_{e1}X^{E2}_{e_{F1}} [p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}{\Gamma _0}\nonumber \\&\cdot -\,\frac{p_{x2}\frac{\pi _2}{p_{x2}(x^s_{A2}+x^s_{N2}-x_{E2})} \Gamma _0-p_{e2}p_{x2}\Theta p_{x2}X^{E2}_{e_{F1}e_{F2}}e_{N1} \eta _{N1}}{\sigma \Gamma _0-p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1} \eta _{N1}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}\nonumber \\&= \frac{p_{e1}p_{e2}\sigma p_{x2}X^{E2}_{e_{F1}e_{F2}}-p_{e1}p_{x2} X^{E2}_{e_{F1}}\frac{\pi _2}{p_{x2}(x^s_{A2}+x^s_{N2}-x_{E2})}[p_{e2} -p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]}{\sigma \Gamma _0-p_{x2} X^{E2}_{e_{F1}}\Theta e_{N1}\eta _{N1}[p_{e2}-p_{x2}X^{E2}_{e_{F2} e_{F2}}e_{N2}\eta _{N2}]}\nonumber \\&= \frac{p_{e1}}{e_{N1}\eta _{N1}}\cdot \frac{\mathrm{d}{e}_{F1}}{\mathrm{d}{\overline{e}}_{A2}}\gtreqless 0\Leftrightarrow \frac{\mathrm{d}{e}_{F1}}{\mathrm{d}{\overline{e}}_{A2}}\lesseqgtr 0. \end{aligned}$$

(78)

Inserting (75) in (56) for $\mathrm{d}{\overline{e}}_{A1}=0$ yields:

$$\begin{aligned} \frac{\mathrm{d}{p}_{e2}}{\mathrm{d}{\overline{e}}_{A2}}&= -\,\,\frac{\frac{p_{e2}}{e_{N2}\eta _{N2}}[\Gamma _0-p_{e2}[p_{e1}-[X^{E1}_{e_{F1}e_{F1}} +p_{x2}X^{E2}_{e_{F1}e_{F1}}]e_{N1}\eta _{N1}]]}{\Gamma _0} +\frac{X^{E2}_{e_{F2}}\Gamma _3}{\Gamma _0}\nonumber \\&\cdot -\frac{p_{x2}\frac{\pi _2}{p_{x2}(x^s_{A2}+x^s_{N2}-x_{E2})} \Gamma _0-p_{e2}p_{x2}\Theta p_{x2}X^{E2}_{e_{F1}e_{F2}}e_{N1} \eta _{N1}}{\sigma \Gamma _0-p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1} \eta _{N1}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]} \nonumber \\&= -\frac{\frac{p_{e2}}{e_{N2}\eta _{N2}}\sigma [\Gamma _0-p_{e2} [p_{e1}-[X^{E1}_{e_{F1}e_{F1}}+p_{x2}X^{E2}_{e_{F1}e_{F1}}]e_{N1} \eta _{N1}]]}{\sigma \Gamma _0-p_{x2}X^{E2}_{e_{F1}}\Theta e_{N1} \eta _{N1}[p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}]} \nonumber \\&\!-\!\frac{p_{x2}X^{E2}_{e_{F2}}\frac{\pi _2}{p_{x2}(x^s_{A2}+x^s_{N2} \!-\!x_{E2})}\Gamma _3{-}p_{e2}p_{x2}\Theta p_{x2}e_{N1}\eta _{N1} [X^{E2}_{e_{F2}}X^{E2}_{e_{F1}e_{F2}}\!-\!X^{E2}_{e_{F1}} X^{E2}_{e_{F2}e_{F2}}]}{\sigma \Gamma _0\!{-}\!p_{x2}X^{E2}_{e_{F1}} \Theta e_{N1}\eta _{N1}[p_{e2}\!-\!p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2} \eta _{N2}]}.\nonumber \\ \end{aligned}$$

(79)

1.4 The Gammas

$$\begin{aligned} \Gamma _0&= \Bigg [p_{e1}-\Bigg [X^{E1}_{e_{F1}e_{F1}}+p_{x2}X^{E2}_{e_{F1}e_{F1}}\Bigg ] e_{N1}\eta _{N1}\Bigg ]\Bigg [p_{e2}-p_{x2}X^{E2}_{e_{F2}e_{F2}}e_{N2}\eta _{N2}\Bigg ] \nonumber \\&-p_{x2}X^{E2}_{e_{F1}e_{F2}}e_{N1}\eta _{N1}\cdot p_{x2}X^{E2}_{e_{F2} e_{F1}}e_{N2}\eta _{N2}\nonumber \\&= \frac{p_{e2}e_{N2}\left| \eta _{N2}\right| p_{e1}e_{N1}\left| \eta _{N1}\right| }{e_{F1}\eta _{F1,2}e_{F2}\eta _{F2,1}}\cdot \left[ \left( \frac{e_{F1} \eta _{F1,2}}{e_{N2}\left| \eta _{N2}\right| }+\frac{e_{F1}\eta _{F1,2}}{e_{F2} \eta _{F2,2}}\right) \right. \\&\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left. \cdot \left( \frac{e_{F2}\eta _{F2,1}}{e_{N1} \left| \eta _{N1}\right| }+\frac{e_{F2}\eta _{F2,1}}{e_{F1}\eta _{F1,1}} \right) -1\right] ,\end{aligned}$$

(80)

$$\begin{aligned} \Gamma _1&= p_{e2}+\Bigg [p_{x2}X^{E2}_{e_{F2}e_{F1}}-p_{x2}X^{E2}_{e_{F2} e_{F2}}\Bigg ]e_{N2}\eta _{N2}\nonumber \\&= \frac{p_{e2}e_{N2}\left| \eta _{N2}\right| }{e_{F1}\eta _{F1,2}}\cdot \left( \frac{e_{F1}\eta _{F1,2}}{e_{N2}\left| \eta _{N2}\right| }+\frac{e_{F1} \eta _{F1,2}}{e_{F2}\eta _{F2,2}}-1\right) ,\nonumber \\&\gtreqless 0\quad \Leftrightarrow \quad \frac{e_{F1}\eta _{F1,2}}{e_{N2} \left| \eta _{N2}\right| }+\frac{e_{F1}\eta _{F1,2}}{e_{F2}\eta _{F2,2}} \gtreqless 1,\end{aligned}$$

(81)

$$\begin{aligned} \Gamma _2&= p_{e1}-\Bigg [X^{E1}_{e_{F1}e_{F1}}+p_{x2}X^{E2}_{e_{F1}e_{F1}} -p_{x2}X^{E2}_{e_{F1}e_{F2}}\Bigg ]e_{N1}\eta _{N1}\nonumber \\&= \frac{p_{e1}e_{N1}\left| \eta _{N1}\right| }{e_{F2}\eta _{F2,1}}\cdot \left( \frac{e_{F2}\eta _{F2,1}}{e_{N1}\left| \eta _{N1}\right| }+\frac{e_{F2} \eta _{F2,1}}{e_{F1}\eta _{F1,1}}-1\right) ,\\&\gtreqless 0\quad \Leftrightarrow \quad \frac{e_{F2}\eta _{F2,1}}{e_{N1} \left| \eta _{N1}\right| }+\frac{e_{F2}\eta _{F2,1}}{e_{F1}\eta _{F1,1}} \gtreqless 1,\end{aligned}$$

(82)

$$\begin{aligned} \Gamma _3&= \Gamma _0+\frac{p_{x2}X^{E2}_{e_{F1}}}{p_{e2}}\cdot p_{x2}X^{E2}_{e_{F2}e_{F1}}e_{N1}\eta _{N1}\cdot p_{e2} \nonumber \\&= \frac{p_{e2}e_{N2}\left| \eta _{N2}\right| p_{e1}e_{N1}\left| \eta _{N1}\right| }{e_{F1}\eta _{F1,2}e_{F2}\eta _{F2,1}}\cdot \left[ \left( \frac{e_{F1} \eta _{F1,2}}{e_{N2}\left| \eta _{N2}\right| }+\frac{e_{F1}\eta _{F1,2}}{e_{F2} \eta _{F2,2}}\right) \right. \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \left. \cdot \left( \frac{e_{F2}\eta _{F2,1}}{e_{N1} \left| \eta _{N1}\right| }+\frac{e_{F2}\eta _{F2,1}}{e_{F1}\eta _{F1,1}} \!\right) -1\!-\!\frac{p_{x2}X^{E2}_{e_{F1}}}{p_{e1}}\cdot \frac{e_{F2} \eta _{F2,1}}{e_{N2}\left| \eta _{N2}\right| }\!\right] \!,\nonumber \\&> \underline{\Gamma _3}=p_{x2}X^{E2}_{e_{F1}}[p_{x2}X^{E2}_{e_{F2}} +p_{x2}X^{E2}_{e_{F2}e_{F1}}e_{N1}\eta _{N1}]\nonumber \\&= p_{x2}X^{E2}_{e_{F1}}\cdot p_{x2}X^{E2}_{e_{F2}e_{F1}}e_{N1} \left| \eta _{N1}\right| \left( \frac{e_{F1}\left| \eta _{F1,2}\right| }{e_{N1} \left| \eta _{N1}\right| }-1\right) ,\nonumber \\&\gtreqless 0\quad \Leftrightarrow \quad \frac{e_{F1}\eta _{F1,2}}{e_{N1} \left| \eta _{N1}\right| }\gtreqless 1, \end{aligned}$$

(83)

where ${\underline{\Gamma }_3}$ is a lower limit for $\Gamma _3$.