Investment and Adaptation as Commitment Devices in Climate Politics

Abstract

The strategic roles of adaptation and technological investment in international climate politics have been analyzed within various approaches. What makes this paper unique is that we investigate the combined impact of adaptation and investment on global mitigation and we compare the subgame-perfect equilibria for different sequences of decisions. Considering a purely non-cooperative, game-theoretic framework, we find that by investment countries commit to lower national contributions to the global public good of mitigation. Moreover, the order of adaptation before mitigation might reinforce this strategic effect of technological investments. As a consequence, the subgame-perfect equilibrium for symmetric countries yields a globally lower level of mitigation, and higher global costs of climate change when countries engage in advanced adaptation. Besides this theoretical contribution, the paper proposes some strategies to combat the unfortunate ‘rush to adaptation’ which can be currently observed in climate politics.

Appendices

Appendix 1: Intermediate and Late Adaptation (Asymmetric Case)

1.1 Mitigation and Intermediate Adaptation

The impact intermediate adaptation has on mitigation in the third stage can be determined by differentiating the first-order conditions of mitigation in home and foreign (5) with respect to adaptation¹⁷

$$\begin{aligned} \left( \begin{array}{c} dM\\ dm \end{array} \right) =-\frac{1}{\det _{1}}\left( \begin{array}{cc} \left[ c_{11}+d_{11}\right] D_{12} &{} -D_{11}d_{12}\\ -d_{11}D_{12} &{} \left[ C_{11}+D_{11}\right] d_{12} \end{array}\right) \left( \begin{array}{c}dA\\ da\end{array} \right) . \end{aligned}$$

(14)

From (14) it can be seen that domestic adaptation has a negative (positive) impact on domestic (foreign) mitigation, $\frac{\partial M}{\partial A}=-\left[ c_{11}+d_{11}\right] \cdot D_{12}/\det _{1}<0$ and $\frac{\partial m}{\partial A}=d_{11}\cdot D_{12}/\det _{1}>0$. However, as $\frac{\partial \left[ M+m\right] }{\partial A}=-c_{11}\cdot D_{12}/\det _{1}<0$ the overall effect of adaptation on global mitigation is negative. The impact of foreign adaptation is analogous.

1.2 Impact of a Reverse Sequencing

In order to determine the impact of a reverse sequencing of adaptation and mitigation has on the subgame-perfect equilibria in stage 2 (including stage 3), we have to analyze the comparative statics of the choices on $(M,m,A,a)$. The first-order conditions with respect to mitigation, (4) and (5), are identical in either case of sequencing, but the optimal choices on adaptation, (3) and (6) differ with respect to the strategic term. However, we can integrate both in a single approach such that the first-order conditions for home and foreign, respectively, are given by

$$\begin{aligned} G_{1}+D_{2}+\delta D_{1}\frac{\partial m}{\partial A}&=0\end{aligned}$$

(15)

$$\begin{aligned} g_{1}+d_{2}+\delta d_{1}\frac{\partial M}{\partial a}&=0. \end{aligned}$$

(16)

Here, the parameter $\delta$ serves to distinguish the different cases, $\delta =0$ represents the standard case late adaptation, and $\delta =1$ stands for intermediate adaptation. For convenience we assume that $\frac{\partial m}{\partial A}$ and $\frac{\partial M}{\partial a}$ are approximately constant and, thus, independent of mitigation and adaptation itself.

Proof for part i ) of Proposition 1: In order to analyze the impact of sequencing, we totally differentiate the first-order conditions (15) and (16) of the decisions on adaptation regarding $\delta$.

Inserting (14) and rearranging terms yields

$$\begin{aligned} \begin{pmatrix} dA\\ da \end{pmatrix}&=\frac{-\begin{pmatrix} \frac{\omega \cdot c_{11}\left[ 1+\frac{D_{11}}{C_{11}}\right] +d_{11}\left[ g_{11}+d_{22}\right] -\left[ d_{12}\right] ^{2}}{c_{11}\left[ 1+\frac{D_{11}}{C_{11}}\right] +d_{11}} &{} \frac{\left[ D_{12}+\delta D_{11}\frac{\partial m}{\partial A}\right] d_{12}}{c_{11}\left[ 1+\frac{D_{11}}{C_{11}}\right] +d_{11}}\\ &{} \\ \frac{\left[ d_{12}+\delta d_{11}\frac{\partial M}{\partial a}\right] D_{12}}{C_{11}\left[ 1+\frac{d_{11}}{c_{11}}\right] +D_{11}} &{} \frac{\Omega \cdot C_{11}\left[ 1+\frac{d_{11}}{c_{11}}\right] +D_{11}\left[ G_{11}+D_{22}\right] -\left[ D_{12}\right] ^{2}}{C_{11}\left[ 1+\frac{d_{11}}{c_{11}}\right] +D_{11}} \end{pmatrix}}{\det _{2}}\nonumber \\&\quad \times \begin{pmatrix} D_{1}\frac{\partial m}{\partial A}\\ d_{1}\frac{\partial M}{\partial a} \end{pmatrix} d\delta , \end{aligned}$$

(17)

with $\omega = g_{11}+d_{22}+\delta d_{12}\frac{\partial M}{\partial a}$ and $\Omega = G_{11}+D_{22}+\delta D_{12}\frac{\partial m}{\partial A}$.

All elements of the 2 $\times$ 2-matrix above are positive and both elements of the vector are negative. Furthermore, it can be shown (after some tedious math) that the appropriate determinant $det_{2}$ is positive as well. Hence, the levels of adaptation in home and foreign increase in $\delta$, i.e. $\frac{\partial A}{\partial \delta }>0$ and $\frac{\partial a}{\partial \delta }>0$. Therefore, the equilibrium levels of adaptation are higher in each country in case of advanced adaptation, i.e. $A^{\blacktriangle }>A {{}^\bullet }_{{}}.$

Proof for part ii ) of Proposition 1: The first-order conditions with respect to mitigation, (4) and (5), are identical and do not directly depend on $\delta$. Thus, mitigation in home and foreign is just indirectly affected by the sequencing of adaptation and mitigation, which can be represented by $\frac{\partial M}{\partial \delta }=\frac{\partial M}{\partial A}\frac{\partial A}{\partial \delta }+\frac{\partial M}{\partial a} \frac{\partial a}{\partial \delta }$ and, accordingly, $\frac{\partial m}{\partial \delta }=\frac{\partial m}{\partial A}\frac{\partial A}{\partial \delta }+\frac{\partial m}{\partial a}\frac{\partial a}{\partial \delta }.$ Due to opposing effects of increasing adaptation in home and foreign on mitigation (see “Mitigation and Intermediate Adaptation” in “Appendix 1”), the signs of $\frac{\partial M}{\partial \delta }$ and $\frac{\partial m}{\partial \delta }$ cannot be determined unambiguously for asymmetric countries. However, the overall impact of intermediate adaptation yields a globally lower level of mitigation since $\frac{\partial \left[ M+m\right] }{\partial \delta }=\frac{\partial \left[ M+m\right] }{\partial A}\frac{\partial A}{\partial \delta }+\frac{\partial \left[ M+m\right] }{\partial a}\frac{\partial a}{\partial \delta }<0$ due to $\frac{\partial \left[ M+m\right] }{\partial A},\frac{\partial \left[ M+m\right] }{\partial a}<0$ (cf. “Mitigation and Intermediate Adaption” in “Appendix 1”). Thus, the total level of mitigation in equilibrium decreases with $\delta \in [0;1]$ such that $\left[ M^{\blacktriangle }+m^{\blacktriangle }\right] <\left[ M{{}^\bullet } +m {{}^\bullet } \right]$. Therefore, at least in one of the two countries the level of mitigation is lower for intermediate adaptation. Moreover, for symmetric countries it can be shown that $\frac{\partial M}{\partial \delta } =\frac{\partial m}{\partial \delta }=-\frac{C_{11}D_{12}}{\det _{1}} \frac{\partial A}{\partial \delta }<0$, i.e. mitigation in both home and foreign is smaller when adaptation is advanced. As all best-response functions are continuous, the same result holds true even for slightly asymmetric countries.

1.3 Impact of Enhanced Investment

In the following, we determine the effects of investment on mitigation and adaptation, respectively. Starting point for the comparative statics are (15) and (16) for the choice on adaptation and (4) or (5) for mitigation. Again, the parameter $\delta$ helps to distinguish between late ($\delta =0$) and intermediate adaptation ($\delta =1$). As before, we assume that $\frac{\partial m}{\partial A}$ and $\frac{\partial M}{\partial a}$ are approximately constant and, thus, independent of mitigation and adaptation itself.

First, we totally differentiate the first-order conditions of the decisions on mitigation of home and foreign (4) and (5), respectively, which are identical for either sequencing. This yields

$$\begin{aligned}&\left( \begin{array}{ccc} C_{11}+D_{11} &{} &{} D_{11}\\ d_{11} &{} &{} c_{11}+d_{11} \end{array} \right) \left( \begin{array}{c} dM\\ dm \end{array} \right) +\left( \begin{array}{cc} D_{12} &{} 0\\ 0 &{} d_{12} \end{array}\right) \left( \begin{array}{c} dA\\ da \end{array}\right) \nonumber \\&\quad =-\left( \begin{array}{cc} C_{12} &{} 0\\ 0 &{} c_{12} \end{array} \right) \left( \begin{array}{c} dI\\ di \end{array} \right) . \end{aligned}$$

(18)

Second, we totally differentiate the first-order conditions of the decisions on adaptation, (15) and (16), with respect to the endogenous variables

$$\begin{aligned} \left( \begin{array}{c} dA\\ da \end{array} \right) =-\left( \begin{array}{ll} \frac{ D_{12}+\delta D_{11}\frac{\partial m}{\partial A} }{ G_{11} +D_{22}+\delta D_{12}\frac{\partial m}{\partial A} } &{} \frac{D_{12}+\delta D_{11}\frac{\partial m}{\partial A} }{ G_{11}+D_{22}+\delta D_{12} \frac{\partial m}{\partial A} }\\ &{} \\ \frac{ d_{12}+\delta d_{11}\frac{\partial M}{\partial a} }{ g_{11} +d_{22}+\delta d_{12}\frac{\partial M}{\partial a} } &{} \frac{ d_{12}+\delta d_{11}\frac{\partial M}{\partial a} }{ g_{11}+d_{22}+\delta d_{12} \frac{\partial M}{\partial a}} \end{array} \right) \left( \begin{array}{c} dM\\ dm \end{array} \right) , \end{aligned}$$

(19)

which shows that adaptation is a substitute to domestic and foreign mitigation.

In order to determine the effect of investment on mitigation, we substitute $(dA; da)$ from (19) into (18). Rearranging terms and solving the equation system for the change in mitigation yields

$$\begin{aligned} \left( \begin{array}{c} dM\\ dm \end{array} \right) =\frac{-\left( \begin{array}{ll} C_{12}\left[ c_{11}+\frac{d_{11}\left[ g_{11}+d_{22}\right] -[d_{12}]^{2} }{g_{11}+d_{22}+\delta d_{12}\frac{\partial M}{\partial a}}\right] &{} -\frac{c_{12}\left[ D_{11}\left[ G_{11}+D_{22}\right] -\left[ D_{12}\right] ^{2}\right] }{G_{11}+D_{22}+\delta D_{12}\frac{\partial m}{\partial A}}\\ &{} \\ -\frac{C_{12}\left[ d_{11}\left[ g_{11}+d_{22}\right] -\left[ d_{12}\right] ^{2}\right] }{g_{11}+d_{22}+\delta d_{12}\frac{\partial M}{\partial a}} &{} c_{12}\left[ C_{11}+\frac{D_{11}\left[ G_{11}+D_{22}\right] -\left[ D_{12}\right] ^{2}}{G_{11}+D_{22}+\delta D_{12}\frac{\partial m}{\partial A} }\right] \end{array} \right) \left( \begin{array}{c} dI\\ di \end{array}\right) }{\det _{3}}, \end{aligned}$$

(20)

where determinant $\det _{3}=\left[ C_{11}+\frac{D_{11}\left[ G_{11} +D_{22}\right] -\left[ D_{12}\right] ^{2}}{G_{11}+D_{22}+\delta D_{12} \frac{\partial m}{\partial A}}\right] \left[ c_{11}+\frac{d_{11}\left[ g_{11}+d_{22}\right] -[d_{12}]^{2}}{g_{11}+d_{22}+\delta d_{12}\frac{\partial M}{\partial a}}\right] -\left[ \frac{ D_{11}\left[ G_{11} +D_{22}\right] -\left[ D_{12}\right] ^{2} }{G_{11}+D_{22}+\delta D_{12} \frac{\partial m}{\partial A}} \right] \left[ \frac{ d_{11}\left[ g_{11}+d_{22}\right] -\left[ d_{12}\right] ^{2} }{g_{11}+d_{22}+\delta d_{12}\frac{\partial M}{\partial a}} \right]$ is always positive such that the Nash equilibrium is again stable and unique (Tirole 1988). Comparative statics show that domestic investment is a strategic complement (substitute) to domestic (foreign) mitigation, i.e. $\frac{\partial M}{\partial I} =-\frac{C_{12}}{\det _{3}}\left[ c_{11}+\frac{d_{11}\left[ g_{11} +d_{22}\right] -[d_{12}]^{2}}{g_{11}+d_{22}+\delta d_{12}\frac{\partial M}{\partial a}}\right] >0$ and $\frac{\partial m}{\partial I}=\frac{C_{12} }{\det _{3}}\left[ \frac{d_{11}\left[ g_{11}+d_{22}\right] -\left[ d_{12}\right] ^{2}}{g_{11}+d_{22}+\delta d_{12}\frac{\partial M}{\partial a} }\right] <0$. Moreover, investment encourages mitigation efforts globally $\frac{\partial (M+m)}{\partial I}=-C_{12}\cdot c_{11}/\det _{3}>0.$ The first two relations directly follow from the convexity of the damage functions, i.e. $d_{11}d_{22}-\left[ d_{12}\right] ^{2}>0$. Furthermore, the denominators are positive irrespective of the sequential choice of adaptation and mitigation since $\frac{\partial M}{\partial a},\frac{\partial m}{\partial A}>0$ [cf. (14)].

Next, we determine the strategic effect of investment on adaptation. Although technological investment at stage one has no direct impact on adaptation [cf. (19)], there are crabwise implications through the mitigation choices. By inserting (20) into (19), we substitute $(dM; dm)$. Rearranging terms and solving the equation system for the changes in adaptation yields

$$\begin{aligned} \left( \begin{array}{c} dA\\ da \end{array} \right) =\frac{\left( \begin{array}{ll} C_{12}c_{11}\frac{ D_{12}+\delta D_{11}\frac{\partial m}{\partial A} }{ G_{11}+D_{22}+\delta D_{12}\frac{\partial m}{\partial A} } &{} c_{12} C_{11} \frac{ D_{12}+\delta D_{11}\frac{\partial m}{\partial A} }{ G_{11} +D_{22}+\delta D_{12}\frac{\partial m}{\partial A} }\\ &{} \\ C_{12}c_{11} \frac{ d_{12}+\delta d_{11}\frac{\partial M}{\partial a} }{ g_{11}+d_{22}+\delta d_{12}\frac{\partial M}{\partial a} } &{} c_{12} C_{11} \frac{ d_{12}+\delta d_{11}\frac{\partial M}{\partial a} }{ g_{11} +d_{22}+\delta d_{12}\frac{\partial M}{\partial a} } \end{array} \right) \left( \begin{array}{c} dI\\ di \end{array} \right) }{\det _{3}}. \end{aligned}$$

(21)

Comparative statics show that adaptation is a substitute to each domestic and foreign investment, i.e. $\frac{\partial A}{\partial I}<0$ and $\frac{\partial A}{\partial i}<0.$

Appendix 2: Symmetric Subgame Perfect Equilibria

1.1 Late Versus Intermediate Adaptation

We investigate the effect of sequencing on the subgame-perfect equilibria when investment is decided in the first stage. Inverting the 3 $\times$ 3 matrix in (11) we determine the signs of the comparative statics¹⁸

$$\begin{aligned} \text {sign}\left( \begin{array}{c} \frac{dA}{d\delta }\\ \\ \frac{dM}{d\delta }\\ \\ \frac{dI}{d\delta } \end{array} \right) =-\text {sign}\left( \begin{array}{c} \underset{>0}{\underbrace{A_{11} \cdot \frac{\partial m}{\partial a} \frac{\partial a}{\partial I} }}+ \underset{<0}{\underbrace{A_{31} \cdot {\frac{\partial m}{\partial A}}}}\\ \underset{<0}{\underbrace{A_{12} \cdot \frac{\partial m}{\partial a} \frac{\partial a}{\partial I}}} + \underset{>0}{\underbrace{A_{32} \cdot \frac{\partial m}{\partial A}}}\\ \underset{<0}{\underbrace{A_{13} \cdot \frac{\partial m}{\partial a} \frac{\partial a}{\partial I}}} + \underset{>0}{\underbrace{A_{33} \cdot \frac{\partial m}{\partial A}}} \end{array}\right) _{,} \end{aligned}$$

(22)

where $A_{ij}$ is an abbreviation for the cofactors of the 3 $\times$ 3-matrix of (11), i.e.

$$\begin{aligned} A_{11}&= -2C_{12}\left[ D_{12}+\delta D_{11} \frac{\partial m}{\partial A}\right] >0 \\ A_{12}&= C_{12}\left[ G_{11}+D_{22}+\delta D_{12}\frac{\partial m}{\partial A} \right] <0 \\ A_{13}&= 2\left[ D_{12}^{2}- D_{11}D_{22} \right] -C_{11}\left[ G_{11}+D_{22}+\delta D_{12} \frac{\partial m}{\partial A}\right] -2D_{11}G_{11}<0 \\ A_{31}&= C_{12}^{2}-C_{11}C_{22}-2D_{11}\left[ C_{22}-C_{12}\frac{dm}{dI} \right] -F_{11} \left[ C_{11} + 2D_{11}\right] <0 \\ A_{32}&= D_{12}\left[ F_{11}+C_{22}-C_{12} \frac{dm}{dI} \right] >0\\ A_{33}&= D_{12} \left[ C_{11}\frac{dm}{dI}-C_{12} \right] >0. \end{aligned}$$

The signs of the cofactors can be determined (after some tedious math) by substituting $\frac{d m}{d I}$ from (20), inserting the definition of $\det _{3},$ and rearranging terms such that $\left[ C_{12}-C_{11}\frac{d m}{d I}\right] <0$ and $\left[ C_{22}-C_{12}\frac{d m}{d I}\right] >0.$

The two strategic terms that appear when adaptation is advanced have countervailing effects on the equilibrium levels of investment, mitigation and adaptation. However, it can be shown that the direct strategic effects of adaptation on mitigation $A_{3j} \cdot {\frac{\partial m}{\partial A}}$ outweigh the indirect ones of investment on mitigation $A_{1j} \cdot \frac{\partial m}{\partial a}\frac{\partial a}{\partial I}$ in each case if the cross-derivative of the mitigation cost function $C_{12}$ is sufficiently small. Having a closer look at the strategic effects, we observe that the indirect strategic term crucially depends on the cross-derivative $C_{12}$, while the direct one does not [see (14) and (21)]. For $C_{12} \rightarrow 0$ the indirect effects disappear since $\frac{\partial m}{\partial a}\frac{\partial a}{\partial I} \rightarrow 0$ as well as $A_{11}, A_{12} \rightarrow 0$, i.e. the sizes of the indirect effects vanish when $C_{12}$ is sufficiently small. However, the sizes of the direct effects of adaptation on mitigation, $\frac{\partial m}{\partial A}$ and the cofactors $A_{3j}$ do not vanish when $C_{12}$ approaches zero. Thus, in case of a sufficiently small level of $C_{12}$, the direct effects dominate which yields $dA/d\delta >0,$ $dM/d\delta <0$ and $dI/d\delta <0.$ Continuity implies that the results hold true even for slightly asymmetric countries.

1.2 Late Versus Early Adaptation

Comparative statics of the third stage yield $\frac{\partial m}{\partial I}=\frac{d_{11}C_{12}}{det_{1}}<0$ and $\frac{\partial m}{\partial A}=\frac{d_{11}D_{12}}{det_{1}}>0$, i.e. domestic investment has a negative impact on foreign mitigation and domestic adaptation induces an increase in foreign mitigation. Again we assume that the direct effect dominates the indirect one $\frac{d m}{d A} \equiv \frac{\partial m}{\partial A} + \frac{\partial m}{\partial i} \frac{\partial i}{\partial A }>0$.

Analogous to the comparison of late and intermediate adaptation, totally differentiating (12) with respect to $\beta$ and inverting the 3 $\times$ 3-matrix, the signs of the effects of early adaptation on the equilibrium allocation can be derived

$$\begin{aligned} \text {sign}\left( \begin{array}{c} \frac{dA}{d\beta }\\ \\ \frac{dM}{d\beta }\\ \\ \frac{dI}{d\beta } \end{array} \right) \!\!\!=\!-\text {sign}\left( \! \begin{array}{c} \underset{<0}{\underbrace{C_{12}^{2}-C_{11}C_{22} +2D_{11}\left[ C_{12} \frac{\partial m}{\partial I}-C_{11}C_{22}\right] - F_{11} \left[ C_{11} + 2 D_{11}\right] }}\\ \underset{>0}{\underbrace{D_{12} \left[ C_{22}-C_{12}\frac{\partial m}{\partial I}+F_{11}\right] }}\\ \underset{>0}{\underbrace{D_{12}\left[ C_{11} \frac{\partial m}{\partial I}-C_{12}\right] }}\\ \end{array} \!\right) \!. \end{aligned}$$

(23)

The signs are unambiguous: $dA/d\beta >0,$ $dM/d\beta <0$ and $dI/d\beta <0$. Consequently, early adaptation is globally inferior in comparison to late adaptation at least for symmetric countries.