Strategic Investment in Climate Friendly Technologies: The Impact of Global Emissions Trading

Abstract

Our point of departure is that a group of industrialized countries invest in research and development (R&D) of greenhouse gas (GHG) abatement technologies. R&D investments influence the future GHG abatement choices of both industrialized and developing countries. We distinguish between investments that reduce industrialized countries’ abatement costs and investments that reduce developing countries’ abatement costs. Unlike earlier contributions, we include global trading in emission permits. This changes the nature of the game. With global permit trading, industrialized countries should in many cases invest strategically in technologies that only reduce abatement costs at home. This comes in addition to investments abroad. Second, we show that R&D investments always decrease total emissions. Finally, we find that the developing region receiving investments always benefits.

Appendices

Appendix 1: The Uniqueness of the Nash-Equilibrium in the Permit Trading Case

We have assumed that the Nash equilibrium following from (18) is unique. In this Appendix, we show that this is satisfied for $\frac{\partial ^{3}c_{i}}{(\partial e_{i})^{3}}=0$. For simplicity, we write the second order derivatives of the welfare function in the permit trading case as follows: $\frac{\partial ^{2}\bar{\omega }^{i}}{\partial \bar{e}_{i}\partial \bar{e}_{i}}=\bar{\omega }_{ii}^{i},\, \frac{\partial ^{2}\bar{\omega }^{i}}{ \partial \bar{e}_{i}\partial \bar{e}_{j}}=\bar{\omega }_{ij}^{i},\, i,\, j=h,a,\, i\ne j.$ The derivatives are given by:

$$\begin{aligned} \bar{\omega }_{ii}^{i}=b_{i}^{\prime \prime }-2\frac{\partial t}{\partial \bar{e}_{i}}+\frac{\partial t}{\partial \bar{e}_{i}}\frac{\partial e_{i}}{ \partial \bar{e}_{i}}-\frac{\partial ^{2}t}{(\partial \bar{e}_{i})^{2}}\left[ \bar{e}_{i}-e_{i}^{*}\right] ,\end{aligned}$$

(24)

$$\begin{aligned} \bar{\omega }_{ij}^{i}=b_{i}^{\prime \prime }-\frac{\partial t}{\partial \bar{e}_{j}}+\frac{\partial t}{\partial \bar{e}_{i}}\frac{\partial e_{i}}{\partial \bar{e}_{j}}-\frac{\partial ^{2}t}{\partial \bar{e}_{i}\partial \bar{e}_{j}}\left[ \bar{e}_{i}-e_{i}^{*}\right] . \end{aligned}$$

(25)

The second-order conditions of the regions’ optimization problems demand that $\bar{\omega }_{ii}^{i}<0.$ A sufficient condition for uniqueness is that the reaction functions are downward sloping with an absolute value of the slope less than one (Tirole 1988, chapter 5.4). The slope of the reaction function is given by $-\frac{\bar{\omega }_{ij}^{i}}{\bar{\omega } _{ii}^{i}}$.The sufficient condition for uniqueness is thus satisfied for $\bar{\omega }_{ij}^{i}<0$ and $\left| \bar{\omega }_{ii}^{i}\right| >\left| \bar{\omega }_{ij}^{i}\right| .$

Note that the derivative $\frac{\partial t}{\partial \bar{e}_{i}}$equals $\frac{\partial t}{\partial \bar{e}_{j}}$, $\frac{\partial ^{2}t}{(\partial \bar{e}_{i})^{2}}$ equals $\frac{\partial ^{2}t}{\partial \bar{e} _{i}\partial \bar{e}_{j}},$ and $\frac{\partial e_{i}}{\partial \bar{e}_{i}}$ equals $\frac{\partial e_{i}}{\partial \bar{e}_{j}}.$ Hence, for $\bar{\omega }_{ii}^{i}\,, \bar{\omega }_{ij}^{i}<0,$ we must have $\left| \bar{\omega }_{ii}^{i}\right| >\left| \bar{\omega }_{ij}^{i}\right| .\frac{ \partial ^{2}t}{(\partial \bar{e}_{i})^{2}}$ is equal to zero when $\frac{ \partial ^{3}c_{i}}{(\partial e_{i})^{3}}=0$. Then, since $b_{i}^{\prime \prime }<0\,, \frac{\partial t}{\partial \bar{e}_{i}}=\frac{\partial t}{ \partial \bar{e}_{j}}>0$ and $\frac{\partial e_{i}}{\partial \bar{e}_{i}}= \frac{\partial e_{i}}{\partial \bar{e}_{j}}\in \left\langle 0,1\right\rangle$, we have $\bar{\omega }_{ii}^{i}$, $\bar{\omega } _{ij}^{i}<0$ for $\frac{\partial ^{3}c_{i}}{(\partial e_{i})^{3}}=0$, and equilibrium is unique.

Appendix 2: The Numerical Example

1.1 Benefits and Costs

The benefits of emission reduction targets are given by:

$$\begin{aligned} b_{h}(\cdot )=\eta _{0}(\bar{e}_{h}+\bar{e}_{a})-\frac{\eta _{1}}{2}(\bar{e}_{h}+\bar{e}_{a})^{2},\\ b_{a}(\cdot )=\alpha _{0}(\bar{e}_{h}+\bar{e}_{a})-\frac{\alpha _{1}}{2}(\bar{e}_{h}+\bar{e}_{a})^{2}. \end{aligned}$$

While the costs of emission reductions are given by:

$$\begin{aligned} c_{h}(\cdot )=\frac{c_{h}}{2}(e_{h})^{2}-\sqrt{k_{h}}e_{h},\\ c_{a}(\cdot )=\frac{c_{a}}{2}(e_{a})^{2}-\sqrt{k_{a}}e_{a}. \end{aligned}$$

The level of investment $k_{i}$ is bounded such that $c_{i}e_{i}>2\sqrt{k_{i} }$. The derivative of the cost function with respect to $k_{i}$ can then be written:

$$\begin{aligned} \frac{\partial c_{i}(\cdot )}{\partial k_{i}}=\frac{-e_{i}}{2\sqrt{k_{i}}} \end{aligned}$$

For investments at home we have overinvestment if $\frac{e_{h}}{2\sqrt{k_{h}} }<p_{h}$, while for investments abroad we have overinvestment if $k_{a}>0$.

1.2 Parameter Values

The parameters in the benefit functions originate partly from Carbone et al. (2009). The initial emissions reduction for the home region is valued as $219$ $/ton $\hbox {CO}_{2}$, which is a GDP weighted average of the numbers given in Carbone et al. (2009) for the three countries: the US, EU and Japan. Moreover, we assume that the marginal benefit of emission reductions is halved if world emissions are halved. Hence, we have $\eta _{0}=219$ and $\eta _{1}=0.0049$. Following the same procedure for China, we obtain $\alpha _{0}=50$ and $\alpha _{1}=0.0011$.

The parameters $c_{i}$ are set by assuming that it costs 5 % of GDP to reduce region specific business as usual emissions by 50 %. This yields $c_{_{h}}=0.1$ and $c_{a}=0.014$. For the GDP figures we have used IMF World Economic Outlook year 2016 predictions (http://www.imf.org/external/pubs/ft/weo/2011/02/weodata/index.aspx). For the $CO_{2}$ emissions we have used CDIAC 2008 figures (http://cdiac.ornl.gov/). These are projected to 2016 by assuming the same emissions/GDP ratio in 2016 as in 2008.

Finally, for the prices of R&D we have $p_{h}\in \left[ 50,102\right]$ and $p_{h}=p_{a}/7$. This yields a cost reduction in the range 30–15 % for both regions in the sub-game perfect equilibrium with investments never amounting to more than 1.3 % of GDP in total ($k_{h}+k_{a}$), which to us seems reasonable. Note that with $p_{a}=p_{h}$, we would see cost reductions in the developing country far in excess of 30 % in the sub-game perfect equilibrium. This would not change the results qualitatively, but in the example we prefer to have similar cost reduction opportunities.

1.3 Solution Without Permit Trading

Welfare in Stage 2 can be expressed as follows:

$$\begin{aligned} W_{h}=\eta _{0}(\bar{e}_{h}+\bar{e}_{a})-\frac{\eta _{1}}{2}(\bar{e}_{h}+ \bar{e}_{a})^{2}-\frac{c_{h}}{2}(e_{h})^{2}+\sqrt{k_{h}}e_{h},\\ W_{a}=\alpha _{0}(\bar{e}_{h}+\bar{e}_{a})-\frac{\alpha _{1}}{2}(\bar{e}_{h}+ \bar{e}_{a})^{2}-\frac{c_{a}}{2}(e_{a})^{2}+\sqrt{k_{a}}e_{a}, \end{aligned}$$

where $\bar{e}_{h}=e_{h}$ and $\bar{e}_{a}=e_{a}$.

From the first-order conditions for a welfare maximum we obtain the following best response curves:

$$\begin{aligned} \eta _{0}-\eta _{1}(\bar{e}_{h}+\bar{e}_{a})-c_{h}\bar{e}_{h}+\sqrt{k_{h}}&= 0 \\ \alpha _{0}-\alpha _{1}(\bar{e}_{h}+\bar{e}_{a})-c_{a}\bar{e}_{a}+\sqrt{k_{a}}&= 0 \end{aligned}$$

$$\begin{aligned} \bar{e}_{h}=\frac{\eta _{0}+\sqrt{k_{h}}-\eta _{1}\bar{e}_{a}}{\eta _{1}+c_{h}},\quad \bar{e}_{a}=\frac{\alpha _{0}+\sqrt{k_{a}}-\alpha _{1} \bar{e}_{h}}{\alpha _{1}+c_{a}}. \end{aligned}$$

These are then plotted for values of $k_{i}$ such that $c_{i}=$ and $c_{i}=\, ,i=h,a$. We can also find the equilibrium values of $\bar{e}_{h}$ and $\bar{e}_{a}$:

$$\begin{aligned} \bar{e}_{h}^{N}=\frac{\alpha _{1}\eta _{0}-\alpha _{0}\eta _{1}+\eta _{0}c_{a}+(\alpha _{1}+c_{a})\sqrt{k_{h}}-\eta _{1}\sqrt{k_{a}}}{\eta _{1}c_{a}+\alpha _{1}c_{h}+c_{a}c_{h}} \end{aligned}$$

$$\begin{aligned} \bar{e}_{a}^{N}=\frac{\alpha _{0}\eta _{1}-\alpha _{1}\eta _{0}+\alpha _{0}c_{h}+(\eta _{1}+c_{h})\sqrt{k_{a}}-\alpha _{1}\sqrt{k_{h}}}{\eta _{1}c_{a}+\alpha _{1}c_{h}+c_{a}c_{h}} \end{aligned}$$

In order to find the equilibrium values of $k_{h}$ and $k_{a}$ we solve the following optimizing problem numerically:

$$\begin{aligned} \max _{k_{h},k_{a}}\left\{ \eta _{0}(\bar{e}_{h}^{N}+\bar{e}_{a}^{N})-\frac{\eta _{1}}{2}(\bar{e}_{h}^{N}+\bar{e}_{a}^{N})^{2}-\frac{c_{h}}{2}(\bar{e}_{h}^{N})^{2}+\sqrt{k_{h}}\bar{e}_{h}^{N}-p_{h}k_{h}-p_{a}k_{a}\right\} \end{aligned}$$

The results are reported in the text. For the derivative $\frac{\partial W_{a}}{\partial k_{a}}$ we have:

$$\begin{aligned} \frac{\partial W_{a}}{\partial k_{a}}=-\frac{\alpha _{0}-\alpha _{1}(\bar{e}_{h}+\bar{e}_{a})}{2\sqrt{k_{a}}}\frac{\eta _{1}}{\eta _{1}c_{a}+\alpha _{1}c_{h}+c_{a}c_{h}}+\frac{\bar{e}_{a}}{2\sqrt{k_{a}}} \end{aligned}$$

which can be plotted.

1.4 Solution with Global Permit Trading

Equilibrium in the permit market allows us to write $e_{h}\,, e_{a}$ and $t$ as functions of $\bar{e}_{h},\bar{e}_{a},k_{h}$ and $k_{a}$ as follows:

$$\begin{aligned} e_{h}=\frac{(\bar{e}_{a}+\bar{e}_{h})c_{a}-\sqrt{k_{a}}+\sqrt{k_{h}}}{ c_{a}+c_{h}}, \quad e_{a}=\frac{(\bar{e}_{a}+\bar{e}_{h})c_{h}+\sqrt{k_{a}}- \sqrt{k_{h}}}{c_{a}+c_{h}}, \end{aligned}$$

$$\begin{aligned} t=\frac{(\bar{e}_{a}+\bar{e}_{h})c_{h}c_{a}-c_{a}\sqrt{k_{h}}-c_{h}\sqrt{ k_{a}}}{c_{a}+c_{h}}. \end{aligned}$$

Welfare in Stage 2 can be expressed as follows:

$$\begin{aligned} W_{h}=\eta _{0}(\bar{e}_{h}+\bar{e}_{a})-\frac{\eta _{1}}{2}(\bar{e}_{h}+ \bar{e}_{a})^{2}-t\left[ \bar{e}_{h}-e_{h}\right] -\frac{c_{h}}{2} (e_{h})^{2}+\sqrt{k_{h}}e_{h}, \end{aligned}$$

$$\begin{aligned} W_{a}=\alpha _{0}(\bar{e}_{h}+\bar{e}_{a})-\frac{\alpha _{1}}{2}(\bar{e}_{h}+ \bar{e}_{a})^{2}-t\left[ \bar{e}_{a}-e_{a}\right] -\frac{c_{a}}{2} (e_{a})^{2}+\sqrt{k_{a}}e_{a}. \end{aligned}$$

Differentiating with respect to $\bar{e}_{h}$ and $\bar{e}_{a}$ yields the following first order conditions:

$$\begin{aligned} \eta _{0}-\eta _{1}(\bar{e}_{h}+\bar{e}_{a})-\frac{\partial t}{\partial \bar{e}_{h}}\left[ \bar{e}_{h}-e_{h}\right] -t=0,\\ \alpha _{0}-\alpha _{1}(\bar{e}_{h}+\bar{e}_{a})-\frac{\partial t}{\partial \bar{e}_{a}}\left[ \bar{e}_{a}-e_{a}\right] -t=0, \end{aligned}$$

which by inserting for $e_{i}$, $t$ and $\frac{\partial t}{\partial \bar{e} _{i}}$ yields the reaction functions. We can then find the Nash-equilibrium values of $\bar{e}_{h}$ and $\bar{e}_{a}$:

$$\begin{aligned} \bar{e}_{h}^{N}=\frac{\theta _{h}+c_{a}\left( 2c_{a}c_{h}+2\alpha _{1}c_{h}+\alpha _{1}c_{a}-\eta _{1}c_{a}\right) \sqrt{k_{h}}+c_{h}\left( \alpha _{1}c_{h}-\eta _{1}c_{h}-2\eta _{1}c_{a}\right) \sqrt{k_{a}}}{\varphi }\\ \bar{e}_{a}^{N}=\frac{\theta _{a}+c_{a}\left( \eta _{1}c_{a}-\alpha _{1}c_{a}-2\alpha _{1}c_{h}\right) \sqrt{k_{h}}+c_{h}\left( 2c_{a}c_{h}+2\eta _{1}c_{a}+\eta _{1}c_{h}-\alpha _{1}c_{h}\right) \sqrt{ k_{a}}}{\varphi } \end{aligned}$$

where $\theta _{h}=\left( 2\eta _{0}c_{a}+\eta _{0}c_{h}-\alpha _{0}c_{h}\right) c_{a}c_{h}-(\alpha _{0}\eta _{1}-\alpha _{1}\eta _{0})\left[ (c_{a})^{2}+(c_{h})^{2}+2c_{a}c_{h}\right]$ and $\theta _{a}=\left( 2\alpha _{0}c_{h}+\alpha _{0}c_{a}-\eta _{0}c_{a}\right) c_{a}c_{h}+\left( \alpha _{0}\eta _{1}-\alpha _{1}\eta _{0}\right) \left[ (c_{a})^{2}+(c_{h})^{2}+2c_{a}c_{h}\right]$ and $\varphi =\left[ 2c_{a}c_{h}+(\alpha _{1}+\eta _{1})(c_{h}+c_{a})\right] c_{a}c_{h}$.

In order to show that the effects of investments $k_{h}$ and $k_{a}$ are ambiguous, we use $c_{h}=c_{a}=c$. We then have:

$$\begin{aligned} sign\left[ \frac{d\bar{e}_{h}^{N}}{dk_{a}}\right]&= sign\left[ \alpha _{1}-3\eta _{1}\right] \\ sign\left[ \frac{d\bar{e}_{h}^{N}}{dk_{h}}\right]&= sign\left[ 2c+3\alpha _{1}-\eta _{1}\right] \\ sign\left[ \frac{d\bar{e}_{a}^{N}}{dk_{a}}\right]&= sign\left[ 2c+3\eta _{1}-\alpha _{1}\right] \\ sign\left[ \frac{d\bar{e}_{a}^{N}}{dk_{h}}\right]&= sign\left[ \eta _{1}-3\alpha _{1}\right] \end{aligned}$$

Note that all signs depend on the parameters of the benefit functions $\alpha _{1}$ and $\eta _{1}$.

In order to find the equilibrium values of $k_{h}$ and $k_{a}$ and the degree of overinvestment we solve the following optimization problem numerically:

$$\begin{aligned} \max _{k_{h},k_{a}}\left\{ \eta _{0}(\bar{e}_{h}^{N}+\bar{e}_{a}^{N})-\frac{ \eta _{1}}{2}(\bar{e}_{h}^{N}+\bar{e}_{a}^{N})^{2}+t(e_{h}-\bar{e}_{h}^{N})- \frac{c_{h}}{2}(e_{h})^{2}+\sqrt{k_{h}}e_{h}-p_{h}k_{h}-p_{a}k_{a}\right\} . \end{aligned}$$

The results are reported in the text. For the derivative $\frac{\partial W_{a}}{\partial k_{a}}$ in the with global permit trading case we have:

$$\begin{aligned} \frac{\partial W_{a}}{\partial k_{a}}=\frac{1}{2\sqrt{k_{a}}}\left[ \frac{ c_{h}\left( \alpha _{1}c_{h}-\eta _{1}c_{h}-2\eta _{1}c_{a}\right) \left[ \alpha _{0}-\alpha _{1}(\bar{e}_{h}+\bar{e}_{a})\right] }{\left[ 2c_{a}c_{h}+(\alpha _{1}+\eta _{1})(c_{h}+c_{a})\right] c_{a}c_{h}}+\frac{( \bar{e}_{a}-e_{a})c_{h}}{c_{a}+c_{h}}+e_{a}\right] \end{aligned}$$

which can be plotted.

Appendix 3: Proof of Proposition 3

In this Appendix we prove that $\frac{d\bar{e}_{h}^{N}}{dk_{i}}+\frac{d\bar{e }_{a}^{N}}{dk_{i}}>0.$ By totally differentiating the two equations (18), we obtain the effects of $k_{h}$ and $k_{a}$ as follows:

$$\begin{aligned} \frac{\partial \bar{e}_{h}^{N}}{\partial k_{h}}=\frac{\left[ \bar{\omega } _{ha}^{h}\bar{\omega }_{ak_{h}}^{a}-\bar{\omega }_{aa}^{a}\bar{\omega } _{hk_{h}}^{h}\right] }{D},\end{aligned}$$

(26)

$$\begin{aligned} \frac{\partial \bar{e}_{a}^{N}}{\partial k_{h}}=\frac{\left[ \bar{\omega } _{ah}^{a}\bar{\omega }_{hk_{h}}^{h}-\bar{\omega }_{hh}^{h}\bar{\omega } _{ak_{h}}^{a}\right] }{D},\end{aligned}$$

(27)

$$\begin{aligned} \frac{\partial \bar{e}_{h}^{N}}{\partial k_{a}}=\frac{\left[ \bar{\omega } _{ha}^{h}\bar{\omega }_{ak_{a}}^{a}-\bar{\omega }_{aa}^{a}\bar{\omega } _{hk_{a}}^{h}\right] }{D},\end{aligned}$$

(28)

$$\begin{aligned} \frac{\partial \bar{e}_{a}^{N}}{\partial k_{a}}=\frac{\left[ \bar{\omega } _{ah}^{a}\bar{\omega }_{hk_{a}}^{h}-\bar{\omega }_{hh}^{h}\bar{\omega } _{ak_{a}}^{a}\right] }{D}, \end{aligned}$$

(29)

where $\omega _{ii}^{i}$ and $\omega _{ij}^{i}$ are given by (24) and (25), respectively, and

$$\begin{aligned} \bar{\omega }_{ik_{i}}^{i}=-\frac{\partial t}{\partial k_{i}}+\frac{\partial t }{\partial \bar{e}_{i}}\frac{\partial e_{i}}{\partial k_{i}}-\frac{\partial ^{2}t}{\partial k_{i}\partial \bar{e}_{i}}\left[ \bar{e}_{i}-e_{i}\right] ,\end{aligned}$$

(30)

$$\begin{aligned} \bar{\omega }_{ik_{j}}^{i}=-\frac{\partial t}{\partial k_{j}}+\frac{\partial t }{\partial \bar{e}_{i}}\frac{\partial e_{i}}{\partial k_{j}}-\frac{\partial ^{2}t}{\partial k_{j}\partial \bar{e}_{i}}\left[ \bar{e}_{i}-e_{i}^{*} \right] , \end{aligned}$$

(31)

$$\begin{aligned} D=\bar{\omega }_{ii}^{i}\bar{\omega }_{jj}^{j}-\bar{\omega }_{ij}^{i}\bar{\omega }_{ji}^{j}>0 \end{aligned}$$

(see section “The uniqueness of the Nash-equilibrium in thepermit trading case” of Appendix)

By using (26)–(29) above we have:

$$\begin{aligned} \frac{\partial \bar{e}_{h}^{N}}{\partial k_{i}}+\frac{\partial \bar{e} _{a}^{N}}{\partial k_{i}}=\frac{\left[ \bar{\omega }_{ah}^{a}-\bar{\omega } _{aa}^{a}\right] \bar{\omega }_{hk_{i}}^{h}+\left[ \bar{\omega }_{ha}^{h}-\bar{ \omega }_{hh}^{h}\right] \bar{\omega }_{ak_{i}}^{a}}{D} \end{aligned}$$

The terms in the brackets are positive as $\bar{\omega }_{ii}^{i}\,, \bar{ \omega }_{ij}^{i}<0$ and $\left| \bar{\omega }_{ii}^{i}\right| >\left| \bar{\omega }_{ij}^{i}\right|$. Moreover, we also see from (24) and (25) that $\left[ \bar{\omega }_{ah}^{a}-\bar{\omega } _{aa}^{a}\right] =\left[ \bar{\omega }_{ha}^{h}-\bar{\omega }_{hh}^{h}\right] .$ Furthermore, the two first terms on the right-hand side of (30 ) are positive, and it can be shown that the sum of the two first terms on the right-hand side of (31) is positive.¹⁷ Since $\left[ \bar{e} _{h}-e_{h}^{*}\right] =-\left[ \bar{e}_{a}-e_{a}^{*}\right]$, then $\bar{\omega }_{hk_{i}}^{h}+\bar{\omega }_{ak_{i}}^{a}>0$, which implies that $\frac{\partial \bar{e}_{h}^{N}}{\partial k_{a}}+\frac{\partial \bar{e} _{a}^{N}}{\partial k_{a}}>0$, and $\frac{\partial \bar{e}_{h}^{N}}{\partial k_{h}}+\frac{\partial \bar{e}_{a}^{N}}{\partial k_{h}}>0$.

Appendix 4: Proof of Proposition 6

The derivative $\frac{\partial \bar{\omega }^{a}}{\partial k_{a}}$, given by ( 23), can be rearranged by using (21) and that $- \left[ \bar{e}_{a}^{N}-e_{a}\right] =\left[ \bar{e}_{h}^{N}-e_{h}\right]$:

$$\begin{aligned} \frac{\partial \bar{\omega }^{a}}{\partial k_{a}}=t\left[ \frac{\partial \bar{ e}_{h}^{N}}{\partial k_{a}}+\frac{\partial \bar{e}_{a}^{N}}{\partial k_{a}} \right] +\left| c_{ak}^{\prime }\right| -p_{a} \end{aligned}$$

We know from section “Proof of proposition 3” of Appendix that the sum of the terms in the bracket is positive. Hence, the condition $\left| c_{ak}^{\prime }\right| \ge p_{a}$ is sufficient for $\frac{\partial \bar{\omega }^{a}}{\partial k_{a}}\ge 0$, but not necessary. Since some R&D investment is profitable from a cost minimizing point of view, the equation $\left| c_{ak}^{\prime }\right| =p_{a}$ yields a positive $k_{a}$. Thus, $k_{a}^{*}$ must be positive.