Adaptation, mitigation and risk-taking in climate policy

Abstract

The future consequences of climate change are highly uncertain and estimates of economic damages differ widely. Governments try to cope with these risks by investing in mitigation and adaptation measures. In contrast to most of the existing literature, we explicitly model the decision of risk averse governments on mitigation and adaptation policies. We also consider the interaction of the two strategies in presence of uncertainty. Mitigation efforts of a single country trigger crowding out as other countries will reduce their mitigation efforts. This may even lead to lower mitigation on the global scale. In contrast, a unilateral commitment to large adaptation efforts benefits the single country and can reduce the global risk from climate change at the expense of other countries.

Appendix

1.1 Reaction curve

The derivatives of Eq. (3) amount to

$$\begin{aligned} f_{m_1 m_1}= & {} U_{\mu \mu } \cdot (-c_{m_1} - \alpha _m \cdot \bar{\mu }_1)^2 + U_{\mu \sigma } \cdot (-c_{m_1} - \alpha _m \cdot \bar{\mu }_1) \cdot \alpha _m \cdot \bar{\sigma }_1 \nonumber \\&+\; U_{\mu } \cdot (- c_{m_1 m_1} - \alpha _{mm} \cdot \bar{\mu }_1) + U_{\sigma \mu } \cdot (-c_{m_1} - \alpha _m \cdot \bar{\mu }_1) \cdot \alpha _m \cdot \bar{\sigma }_1 \nonumber \\&+\; U_{\sigma \sigma } \cdot (\alpha _m \cdot \bar{\sigma }_1)^2 + U_{\sigma } \cdot \alpha _{mm} \cdot \bar{\sigma }_1 \end{aligned}$$

(A.21)

and

$$\begin{aligned} f_{m_1 m_2}= & {} U_{\mu \mu } \cdot (-c_{m_1} - \alpha _m \cdot \bar{\mu }_1) \cdot (- \alpha _m \cdot \bar{\mu }_1) + U_{\mu \sigma } \cdot (-c_{m_1} - \alpha _m \cdot \bar{\mu }_1) \cdot \alpha _m \cdot \bar{\sigma }_1 \nonumber \\&+ \;U_{\mu } \cdot (- \alpha _{mm} \cdot \bar{\mu }_1) + U_{\sigma \mu } \cdot (- \alpha _m \cdot \bar{\mu }_1) \cdot \alpha _m \cdot \bar{\sigma }_1 \nonumber \\&+\; U_{\sigma \sigma } \cdot (\alpha _m \cdot \bar{\sigma }_1)^2 + U_{\sigma } \cdot \alpha _{mm} \cdot \bar{\sigma }_1. \end{aligned}$$

(A.22)

Note that Eqs. (A.21) and (A.22) have similar structures. Hence, we can write

$$\begin{aligned} f_{m_1 m_2} = f_{m_1 m_1} + U_{\mu \mu } \cdot (-c_{m_1} - \alpha _m \cdot \bar{\mu }_1) \cdot c_{m_1} + U_{\mu \sigma } \cdot \alpha _m \cdot \bar{\sigma }_1 \cdot c_{m_1} + U_{\mu } \cdot c_{m_1 m_1}. \end{aligned}$$

(A.23)

Substituting in (6) leads to Eq. (7).

1.2 Absolute risk aversion

Let S be the slope of the indifference curve: [$S\equiv \frac{d \mu _i}{d\sigma _i}|_{\bar{U}}=-\frac{U_{\sigma }}{U_{\mu }}$ with $U_\mu >0$, $U_\sigma <0$ and $i=1,2$]. Absolute risk aversion is decreasing (increasing) if the slope of the indifference curve decreases (increases) with $\mu _i$ (see, for instance, Sinn 1989). Taking the derivative of S with respect to $\mu _i$ yields

$$\begin{aligned} \frac{\partial S}{\partial \mu _i} = - \frac{U_{\mu \sigma } U_{\mu }- U_{\mu \mu } U_{\sigma }}{{U_{\mu }}^2}. \end{aligned}$$

(A.24)

Hence, absolute risk aversion is

$$\begin{aligned} \left\{ \begin{array}{c} \text {decreasing} \\ \text {constant} \\ \text {increasing} \\ \end{array} \right\} \quad \text {for} \quad U_{\mu \sigma } U_{\mu }- U_{\mu \mu } U_{\sigma } \left\{ \begin{array}{c} > \\ = \\ < \\ \end{array} \right\} 0. \end{aligned}$$

(A.25)

1.3 Second-order condition

The second-order conditions are

$$\begin{aligned} f_{a_1 a_1}<0 \qquad f_{m_1 m_1}<0 \qquad |D|= f_{a_1 a_1} f_{m_1 m_1} - (f_{a_1 m_1})^2 >0. \end{aligned}$$

(A.26)

We can rewrite the second-order condition $|D|>0$ as

$$\begin{aligned} f_{a_1 a_1} f_{m_1 m_1} - (f_{a_1 m_1})^2= & {} [A+U_{\mu }\mu _{{a_1} {a_1}}+U_{\sigma }\sigma _{{a_1} {a_1}}][A+U_{\mu }\mu _{m m}+U_{\sigma }\sigma _{m m}]\\&-\,[A+ U_{\mu }\mu _{{a_1} m}+U_{\sigma }\sigma _{{a_1} m}]^2\\= & {} A[U_{\mu }\mu _{m m}+ U_{\sigma }\sigma _{m m}]\;+\; A[U_{\mu }\mu _{{a_1} {a_1}}+U_{\sigma }\sigma _{{a_1} {a_1}}]\\&+\,[U_{\mu }\mu _{{a_1} {a_1}} + U_{\sigma }\sigma _{{a_1} {a_1}}][U_{\mu }\mu _{m m}+U_{\sigma }\sigma _{m m}]\\&-\,2A[U_{\mu }\mu _{{a_1} m}+U_{\sigma }\sigma _{{a_1} m}]-[U_{\mu }\mu _{{a_1} m}+ U_{\sigma }\sigma _{{a_1} m}]^2\\= & {} A[-U_{\mu }\bar{\mu }_1+ U_{\sigma }\bar{\sigma }_1] [\alpha _{m m}+\alpha _{{a_1} {a_1}}- 2 \alpha _{{a_1} m}] \\&+\,[-U_{\mu }\bar{\mu }_1+U_{\sigma }\bar{\sigma }_1]^{2} [\alpha _{{a_1} {a_1}}\alpha _{m m}-\alpha _{{a_1} m}^2] \end{aligned}$$

with

$$\begin{aligned} A \equiv U_{\mu \mu }\mu _{i}\mu _{j}+U_{\mu \sigma }\mu _{i}\sigma _{j}+U_{\sigma \mu }\mu _{j}\sigma _{i}+ U_{\sigma \sigma }\sigma _{i}\sigma _{j} \end{aligned}$$

(A.27)

and $i,j=\{a,m\}$. The second-order condition is always fulfilled if we have

• (A1) $A<0$

• (A2) $\alpha _{ii}-\alpha _{ij}>0$,

i.e. the cross-derivative $U_{\mu \sigma }$ should not be too strongly positive and an increase in climate measure i reduces the marginal productivity $\alpha _{i}$ more than an increase in measure j. The latter implies some degree of complementarity between adaptation and mitigation. In the case with perfect substitutability, we have $\alpha _{ij}=0$.

1.4 Crowding out

The change of mitigation of country i as reaction on an increase of mitigation by country j is given by:

$$\begin{aligned} \frac{d m_i}{d m_j}= -\frac{f_{m_i m_j} f_{a_i a_i} - \{f_{a_i m_i}\}^2}{f_{m_i m_i} f_{a_i a_i} - \{f_{a_i m_i}\}^2}. \end{aligned}$$

(A.28)

Taking the derivative of (12) with respect to $m_{i}$ yields.

$$\begin{aligned} f_{m_i m_i}= & {} U_{\mu \mu } \cdot (-c_{m_i} - \alpha _m \cdot \bar{\mu }_i)^2 + U_{\mu \sigma } \cdot (-c_{m_i} - \alpha _m \cdot \bar{\mu }_i) \cdot \alpha _m \cdot \bar{\sigma }_i \nonumber \\&+\; U_{\mu } \cdot (- c_{m_i m_i} - \alpha _{mm} \cdot \bar{\mu }_i) + U_{\sigma \mu } \cdot (-c_{m_i} - \alpha _m \cdot \bar{\mu }_i) \cdot \alpha _m \cdot \bar{\sigma }_i \nonumber \\&+\; U_{\sigma \sigma } \cdot (\alpha _m \cdot \bar{\sigma }_i)^2 + U_{\sigma } \cdot \alpha _{mm} \cdot \bar{\sigma }_i. \end{aligned}$$

(A.29)

The corresponding derivative with respect to $m_{j}$ is

$$\begin{aligned} f_{m_i m_j}= & {} U_{\mu \mu } \cdot (-c_{m_i} - \alpha _m \cdot \bar{\mu }_i) \cdot (- \alpha _m \cdot \bar{\mu }_i) + U_{\mu \sigma } \cdot (-c_{m_i} - \alpha _m \cdot \bar{\mu }_i) \cdot \alpha _m \cdot \bar{\sigma }_i \nonumber \\&+ \;U_{\mu } \cdot (- \alpha _{mm} \cdot \bar{\mu }_i) + U_{\sigma \mu } \cdot (- \alpha _m \cdot \bar{\mu }_i) \cdot \alpha _m \cdot \bar{\sigma }_i\nonumber \\&+\; U_{\sigma \sigma } \cdot (\alpha _m \cdot \bar{\sigma }_i)^2 + U_{\sigma } \cdot \alpha _{mm} \cdot \bar{\sigma }_i. \end{aligned}$$

(A.30)

As Eqs. (A.29) and (A.30) have similar structures, we can also write

$$\begin{aligned} f_{m_i m_j} = f_{m_i m_i} + U_{\mu \mu } \cdot (-c_{m_i} - \alpha _m \cdot \bar{\mu }_i) \cdot c_{m_i} + U_{\mu \sigma } \cdot \alpha _m \cdot \bar{\sigma }_i \cdot c_{m_i} + U_{\mu } \cdot c_{m_i m_i}. \end{aligned}$$

(A.31)

Finally, using $f_{a_i m_i}=f_{a_i m_j}=f_{m_i a_i}$, Eq. (A.28) can be rewritten as

$$\begin{aligned} \frac{d m_i}{d m_j}= - \frac{f_{a_i a_i}\left\{ f_{m_i m_i} + [U_{\mu \sigma }U_{\mu }-U_{\mu \mu }U_{\sigma }] \cdot \frac{\sigma _{i_{m}}}{U_{\mu }} c_{m_i} + c_{m_i m_i} U_{\mu }\right\} -\left\{ f_{a_i m_i}\right\} ^2}{f_{a_i a_i} f_{m_i m_i}- \left\{ f_{a_i m_i}\right\} ^2}. \end{aligned}$$

(A.32)

A comparison with Eq. (7) immediately shows that the condition for over-crowding out is the same in the extended model as in our baseline model of Sect. 2.