Unilateral Emission Cuts and Carbon Leakages in a Dynamic North–South Trade Model

Abstract

The effects of a unilateral cut in emissions are analyzed in a dynamic two-country model. Two (intermediate) goods are produced, one of which uses the fossil fuel. These inputs are traded and combine to produce a final good. The effect of a cut in fossil fuel use by a bloc (e.g. Annexure 1 countries in the Kyoto Protocol) depends on whether the fuel is priced at marginal cost or above. In the latter case a “green paradox” may appear. The paper’s contribution lies in analyzing the dynamics and trade implications of the unilateral action.

Appendices

Appendix 1

Differentiating Eqs. (15a)–(15d), we have:

$$\begin{aligned}&\left[ \begin{array}{cccc} {{U}^{\prime \prime }(C_1 )}&{} {-\beta (1+r_1 ){U}^{\prime \prime }(C_2 )}&{} {-\beta {U}^{\prime }(C_2 )}&{} 0 \\ 1&{} 0&{} 0&{} 1 \\ 0&{} 1&{} 0&{} {-\Omega _K^2 /z^{2}} \\ 0&{} 0&{} 1&{} {-\Omega _{KK}^2 /z^{2}} \\ \end{array} \right] \left[ \begin{array}{c} {dC_1 } \\ {dC_2 } \\ {dr_1 } \\ {dI_1 } \\ \end{array} \right] \nonumber \\&\quad =\left[ \begin{array}{c} 0 \\ {\{\Omega _p^1 -(\Omega _1 z_p^1 /z^{1})\}/z^{1}} \\ 0 \\ 0 \\ \end{array} \right] dp_1 + \left[ \begin{array}{c} 0 \\ {\Omega _q^1 /z^{1}} \\ 0 \\ 0 \\ \end{array} \right] dq_1\nonumber \\&\quad +\left[ \begin{array}{c} 0 \\ 0 \\ {\{\Omega _p^2 -(\Omega _2 z_p^2 /z^{2})\}/z^{2}} \\ {\{\Omega _{pK}^2 -(\Omega _K^2 z_p^2 /z^{2})\}/z^{2}} \\ \end{array} \right] dp_2 +\left[ \begin{array}{c} 0 \\ 0 \\ {\Omega _q^2 /z^{2}} \\ {\Omega _{Kq}^2 /z^{2}} \\ \end{array} \right] dq_2 \end{aligned}$$

(23)

Notationally, a change in the price of the clean energy (m) is just like the one for q i.e.

$$\begin{aligned}&\left[ 0 \quad {\Omega _m^1 /z^{1}}\quad 0\quad 0 \right] ^{\prime } \hbox {for}\, \hbox {dm}_{1}\, \hbox {and}\, \left[ 0\quad 0\quad {\Omega _m^2 /z^{2}}\quad {\Omega _{Km}^2 /z^{2}} \right] ^{\prime } \,\hbox {for}\, \hbox {dm}_{2} \end{aligned}$$

If we take the home country (North) to be the importer of the X-good (that uses fossil fuel), then a rise in $\hbox {p}_{1}$ increases both $\hbox {C}_{1}$ and $\hbox {I}_{1}$. These are just the terms of trade effects on real income. A rise in $\hbox {p}_{1}$ would then reduce both $\hbox {C}_{1}^{*}$ and $\hbox {I}_{1}^{*}$ . An increase in $\hbox {q}_{1}$ decreases $\hbox {C}_{1},\hbox { I}_{1},\,C_{1}^{*}$ and $I_{1}^{*}$. The partial effects for the North used in the text are given below (for the South the expressions are identical except $\Omega ^{*2}$ replaces $\Omega ^{2}$ etc.):

$$\begin{aligned} \partial I_1/\partial p_2&= -\left[ \left\{ \Omega _{Kp}^2 {-}\left( z_p^2 \Omega _K^2 /z^{2}\right) \right\} \beta {U}^{\prime }(C_2){+}\left\{ \Omega _p^2 -z_p^2 C_2\right\} (\beta {U}^{\prime \prime }(C_2)(1+r_1 )\right] /(\Sigma z^{2})\\ \partial C_2 /\partial p_2&= -\left[ \left\{ \Omega _{Kp}^2 -\left( z_p^2 \Omega _K^{*2} /z^{2}\right) \right\} \beta {U}^{\prime }(C_2)\Omega _K^2 -\left\{ \Omega _p^2 -z_p^2 C_2\right\} \right. \\&\times \left. \left\{ \beta {U}^{\prime \prime }(C_2)\Omega _{KK}^2 +{U}^{\prime \prime }(C_1 )\right\} \right] /\left( \Sigma z^{2}\right) \\ \partial C_2 /\partial q_2&= \left[ \left\{ \beta {U}^{\prime \prime }(C_2)\Omega _{KK}^2 +{U}^{\prime \prime }(C_1 )\right\} \Omega _q^2 -\beta {U}^{\prime \prime }(C_2)\Omega _{Kq}^2 \right] /\left( \Sigma z^{2}\right) <0\\ \partial C_1 /\partial q_1&= -\Omega _q^1 \beta \left\{ {U}^{\prime }(C_2 )\Omega _{KK}^2 +(1+r_1 ){U}^{\prime \prime }(C_2 )\Omega _K^2 \right\} /\left( \Sigma z^{1}\right) <0\\ \partial I_1 /\partial q_1&= {U}^{\prime \prime }(C_1 )\Omega _q^1 /\left( \Sigma z^{1}\right) >0\\ \partial I_1 /\partial p_2&= \left[ {U}^{\prime \prime }(C_1 )+\beta {U}^{\prime }(C_2 )\left\{ \Omega _{pK}^2 -\left( \Omega _K^2 z_p^2 /z^{2}\right) \right\} +(1+r_1 )\beta {U}^{\prime \prime }(C_2 )\right. \\&\left. \times \left\{ \Omega _p^2 -\left( \Omega _2 z_p^2 /z^{2}\right) \right\} \right] /\left( \Sigma z^{2}\right) \\ \partial I_1 /\partial q_2&= \left[ {U}^{\prime \prime }(C_1 )+\beta \left\{ {U}^{\prime }(C_2 )\Omega _{Kq}^2 +(1+r_1 ){U}^{\prime \prime }(C_2 )\Omega _q^2 \right\} \right] /\left( \Sigma z^{2}\right) \\ \partial C_1 /\partial p_1&= \left\{ \Omega _p^1 -\left( \Omega _1 z_p^1 /z^{1}\right) \right\} \beta \left\{ {U}^{\prime }(C_2 )\Omega _{KK}^2 +(1+r_1 ){U}^{\prime \prime }(C_2 )\Omega _K^2 \right\} /\left( \Sigma z^{1}\right) >0\\ \partial I_1 /\partial p_1&= \left\{ \Omega _p^1 -\left( \Omega _1 z_p^1 /z^{1}\right) \right\} {U}^{\prime \prime }(C_1 )/\left( \Sigma z^{1}\right) >0 \end{aligned}$$

where $\Sigma \equiv \beta \left\{ {U}^{\prime }(C_2)\Omega _{KK}^2 +(1+r_1 ){U}^{\prime \prime }(C_2 )\Omega _K^2 \right\} +{U}^{\prime \prime }(C_1 )<0$

Appendix 2

Outline of proof of Proposition 1

We want to show that there is leakage but $<$100 % i.e.

$$\begin{aligned} 0\ge \frac{d{R}_2^{*}}{d\overline{R}_2^{*}}=\frac{dR_2^*}{dp_2 }\frac{dp_2 }{d\overline{R}_2 }\ge -1 \end{aligned}$$

(24)

The market-clearing condition in the intermediate goods market for period 2 is given by Eq. (16b):

$$\begin{aligned} \Omega _1 \left( 1,p_2 ,\overline{K}_1 +I_1 ,q_2 ;m \right) +\Omega _1^*\left( 1,p_2 ,\overline{K}_1^*+I_1^*,q_2 ;m^{*} \right) =\tilde{X}_2 +\tilde{X}_2^*\end{aligned}$$

Differentiating this and setting, $R_2 =\overline{R}_2$ we obtain the change in the price $\hbox {p}_{2}$:

To obtain the leakage, we note that

$$\begin{aligned} R_2^*=R^{*}(p_2 ,K_2^*,(p_2 )) \end{aligned}$$

(25)

The first term is the effect of p$_{2 }$on the product mix and the second is the capital accumulation effect of p$_{2}$. So we have:

$$\begin{aligned} \frac{dR_2^*}{dp_2 }=\frac{\partial R_2^*}{\partial p_2 }+\left( {\frac{\partial R_2^*}{\partial K_2^*}} \right) \left( \frac{\partial I_1^*}{\partial p_2 }\right) \end{aligned}$$

(26)

As noted above, $\partial I^{*}_{1} / \partial p_{2} >0$. We have:

$0\ge \frac{dR_2^*}{dp_2 }\ge \frac{\partial R_2^*}{\partial p_2 }$ because $\left( {\frac{\partial R_2^*}{\partial K_2^*}} \right) \left( \frac{\partial I_1^*}{\partial p_2 }\right) \ge 0$.

In the South due to consumption smoothing, investment declines; this implies that emissions in the South are lower than it would have been had we not taken into account the dynamic effects. Thus an upper bound to carbon leakage is given by the “static” leakage (i.e. holding $dI_1^*=0$).

$$\begin{aligned} \left| {\frac{dR_2^*}{dp_2 }} \right| =\left| {\frac{\partial R_2^*}{\partial p_2 }+\left( {\frac{\partial R_2^*}{\partial K_2^*}} \right) \left( \frac{\partial I_1^*}{\partial p_2 }\right) } \right| \le \left| {\frac{\partial R_2^*}{\partial p_2 }} \right| \end{aligned}$$

(27)

The first term in curly brackets on the right hand side of Eq. (27) is the static effect (negative) and the second one is the dynamic effect (positive).

Note that the change in fossil fuel use is obtained from the two equations, equating the marginal product of capital intersectorally and the marginal product of the fossil fuel to its given marginal cost. (we need to solve for S and substitute): That is, we use the three marginal productivity conditions:

$$\begin{aligned} F_K \left( {K_2^{X*} ,R_2^*,Z^{X*}} \right)&= p_2 G_K \left( {K_2^*-K_2^{X*} ,S_2^*,Z^{Y*}} \right) \end{aligned}$$

(28)

$$\begin{aligned} F_R \left( {K_2^{X*} ,R_2^*,Z^{X*}} \right)&= c.\end{aligned}$$

(29)

$$\begin{aligned} G_S \left( {K_2^*-K_2^{X*} ,S_2^*,Z^{Y*}} \right)&= m\Rightarrow dS_2^*=-\left( \frac{G_{KS}^*}{G_{SS}^*}\right) \left( {dK_2^*-dK_2^{X*} } \right) \end{aligned}$$

(30)

After substitution we have:

$$\begin{aligned} \left[ \begin{array}{cc} {[F_{KK}^*+p_1 (G_{SS}^*)^{-1}\{G_{KK}^*G_{SS}^*-(G_{KS}^*)^{2}\}]}&{} {F_{KR}^*} \\ {F_{KR}^*}&{} {F_{RR}^*} \\ \end{array} \right] \left[ \begin{array}{c} {dK_2^{X*} } \\ {dR_2^*} \\ \end{array} \right] = \left[ \begin{array}{c} {G_K^*dp_2 } \\ 0 \\ \end{array} \right] \end{aligned}$$

And so:

$$\begin{aligned}&\frac{\partial R_2^*}{\partial p_2 }=-G_K^*F_{KR}^*\Big /\left[ \left\{ F_{KK}^*F_{RR}^*-\left( F_{KR}^*\right) ^{2}\right\} +p_1 F_{RR}^*\left( G_{SS}^*\right) ^{-1}\right. \nonumber \\&\left. \quad \left\{ G_{KK}^*G_{SS}^*-\left( G_{KS}^*\right) ^{2}\right\} \right] <0 \end{aligned}$$

(31)

Now note that as the X-intermediate sector expands due to a fall in $\hbox {p}_{2}$, the marginal product of capital in the Y sector rises. And this causes both $K_2^{X*}$ and $R_2^*$ to rise as shown in Fig. 1 below. The line RR is the combination of $K_2^{X*}$ and $R_2^*$ that gives a marginal product of R. The line KK is the intersectoral equality of the marginal product of capital.

Fig. 1

Period one fossil fuel reduction with marginal cost pricing

The slopes are as shown because $[\{F_{KK}^*F_{RR}^*-(F_{KR}^*)^{2}\}+p_1 F_{RR}^*(G_{SS}^*)^{-1}\{G_{KK}^*G_{SS}^*-(G_{KS}^*)^{2}\}]>0$. The initial equilibrium is at A and the new one with $d\overline{R}_2 <0$ at B. The vertical distance of $d\overline{R}_2 <0$ is given by the point on $\hbox {K}'\hbox {K}'$ vertically below A. So from the diagram as the KK line shifts down with $d\overline{R}_2 <0$, there is an increase in $dR_2^*$but is less than $\left| {d\overline{{R}}_2}\right|$ (KK has a slope steeper than RR). The algebra is trivial.

Therefore we have: $1<\left( \left| {\frac{dR_2^*}{d\overline{R}_2 }} \right| \right) _{dI_1^*=0} <\left| {\frac{dR_2^*}{d\overline{R}_2 }} \right| <0 \blacksquare$

Proof of Proposition 2

It proceeds as above for period 2. For period 1, we have:

$$\begin{aligned} \frac{dp_1 }{d\overline{R}_1 }&= \frac{\left\{ {\left( {\frac{z_1^1 }{z^{1}}} \right) \Omega _{\overline{R}}^1 -\Omega _{\mathop {1R}\limits }^1 } \right\} }{\left\{ \Omega _{1p}^1 +\Omega _{1p}^{*1}\right\} }\\ R_1^*&= R^{*} (p_1 )\\ \frac{dR_1^*}{dp_1 }&= -G_K^*F_{KR}^*\Big /\left[ \left\{ F_{KK}^*F_{RR}^*-\left( F_{KR}^*\right) ^{2}\right\} \right. \nonumber \\&\left. +p_1 F_{RR}^*\left( G_{SS}^*\right) ^{-1}\left\{ G_{KK}^*G_{SS}^*-\left( G_{KS}^*\right) ^{2}\right\} \right] <0 \blacksquare \end{aligned}$$

Appendix 3

Equations for Sect. 4.1:

$$\begin{aligned} R_1^*&= -\Omega _q \left( 1,p_1 ,\overline{{K}}_1^*,q_1 ;m^{*}\right) \quad R_2^*=-\Omega _q \left( 1,p_2 ,K_2^*,q_2 ;m^{*}\right) \\ R_1&= -\Omega _q \left( 1,p_1 ,\overline{{K}}_1 ,q_1 ;m \right) \,\quad R_2 =\overline{{R}}_2\\ d\overline{{R}}_2&= d\Omega _q^{*1} +d\Omega _q^1 +d\Omega _q^{*2} \end{aligned}$$

We have from Eqs. (12), (16a) and (16b) to solve for $\hbox {p}_{2},\hbox { p}_{1}$ and $\hbox {q}_{1}$:

$$\begin{aligned}&\left\{ \Omega _{qp}^{*2} +\Omega _{qK}^*\left( \partial I_{*1} /\partial p_2 \right) +\Omega _{pq}^2 +\Omega _{qK} \left( \partial I_1 /\partial p_2 \right) \right\} dp_2 \nonumber \\&+\left\{ \Omega _{qK} \left( \partial I_1 /\partial p_1 \right) +\Omega _{qK}^*\left( \partial I_{*1} /\partial p_1 \right) \right\} dp_1\nonumber \\&\quad \left[ \left\{ \Omega _{qq}^{*2} +\Omega _{qK}^*\left( \partial I_{*1} /\partial q_2 \right) \right\} \rho +\left( \Omega _{qq}^{*1} +\Omega _{qq}^1 \right) \right] dq_1 =d\overline{{R}}_2 \end{aligned}$$

(32)

$$\begin{aligned}&\left( \Omega _{pp}^1 +\Omega _{pp}^{*1} \right) dp_1 +\left( \Omega _{pq}^1 +\Omega _{pq}^{*1} \right) dq_1 =0\end{aligned}$$

(33)

$$\begin{aligned}&\left\{ \Omega _{qp}^{*2} \!+\!\Omega _{qK}^*\left( \partial I_{*1} /\partial p_2 \right) \!+\!\Omega _{qK} \left( \partial I_1 /\partial p_2 \right) \right\} dp_2 {+}\Omega _{qK}^*\left( \partial I_{*1} /\partial p_1 \right) dp_1 {+}\left( \Omega _{qp}^1 {+}\Omega _{qp}^{*1} \right) dp_1 \nonumber \\&\quad +\left\{ \Omega _{qq}^{*2} \rho +\Omega _{qq}^1 +\Omega _{qq}^{*1} +\Omega _{qK}^*\rho \left( \partial I_{*1} /\partial q_1 \right) \right\} dq_1 =\left( 1-\Omega _{qK} \left( \partial I_1 /\partial p_2 \right) \right) d\overline{{R}}_2 \end{aligned}$$

(34)

Equations for Sect. 4.2:

In Sect. 4, any policy to restrict fossil fuel use in period 2 also impacts period 1 via a change in $\hbox {q}_{1}$ and hence $\hbox {p}_{1}$. As in Sect. 3.2, a cap on fossil fuel use at $\overline{{R}}_1$ in period 1, of course, will have an impact on $\hbox {p}_{2}$ via investment. But now there is the additional channel of a fall in $\hbox {q}_{2}$.

$$\begin{aligned} R_2&= -\Omega _q \left( 1,p_2 ,K_2 ,q_2 ;m \right) \quad \,\,\, R_1^*=-\Omega _q \left( 1,p_1 ,\overline{{K}}_1^*,q_1 ;m^{*}\right) \\ R_2^*&= -\Omega _q \left( 1,p_2 ,K_2^*,q_2 ;m^{*}\right) \quad R_1 =\overline{{R}}_1 \end{aligned}$$

Equation (12) implies:

$$\begin{aligned} d\overline{{R}}_1 =d\Omega _q^{*1} +d\Omega _q^2 +d\Omega _q^{*2} \end{aligned}$$

Equations (12), (16a) and (16b) give (again to solve for $\hbox {p}_{2},\hbox { p}_{1}$ and $\hbox {q}_{1}$):

$$\begin{aligned}&\left\{ \Omega _{qp}^{*2} +\Omega _{qK}^*\left( \partial I_{*1} /\partial p_2 \right) +\Omega _{pq}^2 +\Omega _{qK} \left( \partial I_1 /\partial p_2 \right) \right\} dp_2 \nonumber \\&\quad +\left[ \left( \Omega _{qq}^2 +\Omega _{qq}^{*2} +\Omega _{qK}^*\left( \partial I_{*1} /\partial q_2 \right) + \Omega _{qK} \left( \partial I_1 /\partial q_2 \right) \right) \rho +\Omega _{qq}^{*1} \right] dq_1\nonumber \\&\quad +\left\{ \Omega _{qK} \left( \partial I_1 /\partial p_1 \right) +\Omega _{qK}^*\left( \partial I_{*1} /\partial p_1 \right) +\left( \Omega _{pp}^1 +\Omega _{pp}^{*1} \right) \right\} dp_1 =d\overline{{R}}_1 \end{aligned}$$

(35)

$$\begin{aligned}&\left( \Omega _{pp}^1 +\Omega _{pp}^{*1} \right) dp_1 +\left( \Omega _{pq}^1 +\Omega _{pq}^{*1} \right) dq_1 =\left\{ \left( 1-\left( z_p^1 /z^{1}\right) \right) \Omega _{\overline{{R}}}^1 -\Omega _{p\overline{{R}}}^1 \right\} d\overline{{R}}_1\end{aligned}$$

(36)

$$\begin{aligned}&\left\{ \Omega _{pp}^{*2} +\Omega _{pK}^{*2} \left( \partial I_{*1} /\partial p_2 \right) +\Omega _{pp}^2 +\Omega _{pK}^2 \left( \partial I_1 /\partial p_2 \right) \right\} dp_2\nonumber \\&\quad +\left\{ \Omega _{pq}^{*2} \rho +\Omega _{pq}^2 \rho +\Omega _{qK}^{*2} \rho \left( \partial I_{*1} /\partial q_1 \right) +\Omega _{qK}^2 \rho \left( \partial I_1 /\partial q_1 \right) \right\} dq_1 +\left\{ \Omega _{pK}^{2*} \left( \partial I_{*1} /\partial p_1 \right) \right. \nonumber \\&\quad \left. +\,\Omega _{pK}^2 \left( \partial I_1 /\partial p_1 \right) \right\} dp_1 =0 \end{aligned}$$

(37)