Climate policy: How to deal with ambiguity?

Abstract

We study the impact of ambiguity and ambiguity attitudes on optimal adaptation and mitigation decisions when the future environmental quality is ambiguous and the decision maker’s (DM) preferences are represented by the $\alpha$-Maxmin Expected Utility model. We show that ambiguity aversion plays a significant role in designing an optimal climate policy that is different from risk aversion. We also focus on the induced effects of changes in ambiguity, captured by the arrival of additional information. We state that a change in the informational structure may trigger more efforts of both mitigation and adaptation depending on both the DM’s attitude toward ambiguity and her environmental preferences.

Appendices

Appendices

1.1 A Proof of Proposition 1

Impact of risk aversion on mitigation Let us define $z\equiv q+\varphi ((1-\rho )T)>0$.

• If we consider that v(x) is a CARA utility function given by $v(x)=-\frac{1}{\beta }\exp \left( -\beta x\right)$, where $\beta$ is the coefficient of absolute risk aversion, Eq. (2) writes:

  $$\begin{aligned} -(y^{\prime }(T)+\rho )u^{\prime }(c(T,\rho )) = \int \limits _{{\underline{Q}}}^{{\overline{Q}}}\exp (-\beta z)h(q,T)dq, \end{aligned}$$

  (A.1)

  where we define $h(q,T)\equiv (1-\rho )\varphi ^{\prime }(\cdot )\ell (q,T)- \left( \alpha \frac{\partial F_{\underline{\theta }}(q,T)}{\partial T}+(1-\alpha )\frac{\partial F_{\overline{\theta }}(q,T)}{\partial T}\right) >0$ and $\ell (q,T)\equiv \left( \alpha f_{{\underline{\theta }}}(q,T)+(1-\alpha )f_{\overline{\theta }}(q,T)\right) >0$. Using Eq. (A.1), we can calculate the following derivatives¹⁷:

  $$\begin{aligned}&V^{\prime \prime }_{TT}(T,\rho )=y^{\prime \prime }(T)+(1-\rho )^2\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\left[ \varphi ^{\prime \prime }(\cdot )-\beta \varphi ^{\prime 2}(\cdot )\right] \exp (-\beta z)\ell (q,T)dq\nonumber \\&\quad -\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\exp (-\beta z)L_{TT}(q,T)dq+2\varphi ^{\prime }(\cdot )(1-\rho )\beta \nonumber \\&\quad \int \limits _{{\underline{Q}}}^{{\overline{Q}}}\exp {-\beta z}L_T(q,T)dq<0 \end{aligned}$$

  (A.2)

  where we define $L_T(q,T)\equiv \alpha \frac{\partial F_{{\underline{\theta }}}(q,T)}{\partial T}+(1-\alpha )\frac{\partial F_{{\overline{\theta }}}(q,T)}{\partial T}<0$ and, under Assumption 1, $L_{TT}(q,T)=\alpha \frac{\partial ^2 F_{{\underline{\theta }}}(q,T)}{\partial T^2}+(1-\alpha )\frac{\partial ^2 F_{{\overline{\theta }}}(q,T)}{\partial T^2}>0$, and

  $$\begin{aligned} V^{\prime \prime }_{T\beta }(T,\rho )=-\int \limits _{{\underline{Q}}}^{{\overline{Q}}}z\exp (-\beta z)h(q,T)dq<0 \end{aligned}$$

  (A.3)

  Using the implicit function theorem, $\frac{dT^{*}}{d\beta }=-\frac{V^{\prime \prime }_{T\beta }}{V^{\prime \prime }_{TT}}$, we can easily state that $\frac{dT^{*}}{d\beta }<0$.

• If v is a CRRA utility function given by $v(x)=\frac{x^{1-\kappa }}{1-\kappa }$, where $\kappa$ is the coefficient of relative risk aversion, Eq. (2) becomes:

  $$\begin{aligned} -(y^{\prime }(T)+\rho )u^{\prime }(c(T;\rho )) = \int \limits _{{\underline{Q}}}^{{\overline{Q}}}z^{-\kappa }h(q,T)dq \end{aligned}$$

  (A.4)

  Using the implicit functions theorem, the sign of $\dfrac{dT^{*}}{d\kappa }$ is given by the sign of $-\frac{V^{\prime \prime }_{T\kappa }}{V^{\prime \prime }_{TT}}$. We can calculate the following derivative:

  $$\begin{aligned} V^{\prime \prime }_{T\kappa }(T,\rho )=-\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\ln (z)\times z^{-\kappa }h(q,T)dq<0 \end{aligned}$$

  (A.5)

  Thus, $T^{*}$ is a decreasing function of $\kappa$ when ${\overline{Q}}$ is sufficiently high.

Impact of risk aversion on adaptation

• If v(x) is a CARA utility function given by $v(x)=-\frac{1}{\beta }\exp \left( -\beta x\right)$, where $\beta$ is the coefficient of absolute risk aversion, Eq. (3) writes:

  $$\begin{aligned} T u^{\prime }(c(T,\rho ))= & {} T \varphi ^{\prime }(\cdot ) \int \limits _{{\underline{Q}}}^{ {\overline{Q}}}\exp \left( -\beta z\right) \ell (q,T) dq \end{aligned}$$

  (A.6)

  Then, $\frac{d \rho ^{*}}{d\beta }=-\frac{V^{\prime \prime }_{\rho \beta }}{V^{\prime \prime }_{\rho \rho }}$. Using Eq. (A.6), we can calculate the following derivatives:

  $$\begin{aligned} V^{\prime \prime }_{\rho \beta }(T,\rho )=\varphi ^{\prime }(\cdot ) \int \limits _{{\underline{Q}}}^{ {\overline{Q}}}z\exp \left( -\beta z\right) \ell (q,T) dq > 0 \end{aligned}$$

  (A.7)

  and

  $$\begin{aligned}&V^{\prime \prime }_{\rho \rho }(T,\rho )\nonumber \\&\quad =T^2\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\left[ \varphi ^{\prime \prime }(\cdot ) \exp (-\beta z)-(\varphi ^{\prime }(\cdot ))^{2}\beta \exp (-\beta x)\right] \ell (q,T) dq<0. \end{aligned}$$

  (A.8)

  To conclude, we can easily state that $\frac{d\rho ^{*}}{d\beta }>0$ and thus, $(1-\rho ^{*})$ is a decreasing function of $\beta$.

• If v is a CRRA utility function given by $v(x)=\frac{x^{1-\kappa }}{1-\kappa }$, where $\kappa$ the coefficient of relative risk aversion, Eq. (3) becomes:

  $$\begin{aligned} T u^{\prime }(c(T,\rho ))=T \varphi ^{\prime }(\cdot ) \int \limits _{{\underline{Q}}}^{ {\overline{Q}}}z^{-\kappa } \ell (q,T) dq \end{aligned}$$

  (A.9)

  The sign of $\dfrac{d\rho ^{*}}{d\kappa }$ is given by the sign of $-\frac{V^{\prime \prime }_{\rho \kappa }}{V^{\prime \prime }_{\rho \rho }}$. We can calculate the following derivative:

  $$\begin{aligned} V^{\prime \prime }_{\rho \kappa }(T,\rho )=\varphi ^{\prime }(\cdot ) \int \limits _{{\underline{Q}}}^{ {\overline{Q}}}ln(z)z^{-\kappa } \ell (q,T)dq > 0 \end{aligned}$$

  (A.10)

  and Thus, $(1-\rho ^{*})$ is a decreasing function of $\gamma$.

1.2 B Proof of Proposition 2

Impact of ambiguity aversion on mitigation To measure the impact of ambiguity aversion on the optimal policy and using the implicit function theorem, $\dfrac{dT^{*}}{d\alpha }=-\frac{V^{\prime \prime }_{T\alpha }(T,\rho )}{V^{\prime \prime }_{TT}(T,\rho )}$. Through Assumption 2, integration by parts yields:

$$\begin{aligned} V^{\prime \prime }_{T\alpha } (T,\rho )= & {} -(1-\rho )\varphi ^{\prime }(\cdot )\int \limits _{{\underline{Q}}}^{{\overline{Q}}}v^{\prime \prime }(z)\left[ F_{\underline{\theta }}(q,T)-F_{{\overline{\theta }}}(q,T)\right] dq\nonumber \\&-\int \limits _{{\underline{Q}}}^{{\overline{Q}}}v^{\prime }(z) \left[ \frac{\partial F_{{\underline{\theta }}}(q,T)}{\partial T}-\frac{\partial F_{{\overline{\theta }}}(q,T)}{\partial T}\right] dq >0 \end{aligned}$$

(B.11)

Since $V^{\prime \prime }_{TT}(T,\rho )<0$, we can conclude that $\frac{dT^{*}}{d\alpha }>0$.

Impact of ambiguity aversion on adaptation

$$\begin{aligned} V^{\prime \prime }_{\rho \alpha }(T, \rho ) = T \varphi ^{\prime }(\cdot ) \int \limits _{{\underline{Q}}}^{ {\overline{Q}}}v^{\prime \prime }(z)\left( F_{\underline{\theta }}(q,T)-F_{{\overline{\theta }}}(q,T)\right) dq <0 \end{aligned}$$

(B.12)

The sign of $\frac{d\rho ^{*}}{d\alpha }$ is given by the sign of $-\frac{V^{\prime \prime }_{\rho \alpha }(T,\rho )}{V^{\prime \prime }_{\rho \rho }(T,\rho )}$. Since $V^{\prime \prime }_{\rho \rho }(T,\rho )<0$, we can conclude that $\frac{d\rho ^{*}}{d\alpha }<0$ and consequently the adaptation level, $(1-\rho ^{*})$, is an increasing function of $\alpha$.

1.3 C Proof of Proposition 3

Impact of a new scenario on mitigation Let us consider a new scenario ${\hat{\theta }}$, associated with a new welfare function denoted ${\hat{V}}(T,\rho )$ and a subsequent “new” optimal tax ${\hat{T}}=\arg \max \limits _{T} {\hat{V}}(T,\rho )$. Two configurations might occur:

• When ${\hat{\theta }}$ is worst than all the initial scenarios, then $F_{{\underline{\theta }}}(q,T)\le F_{{\hat{\theta }}}(q,T)$. We can state that ${\hat{T}}>T^{*}\Leftrightarrow \frac{\partial V({\hat{T}},\rho )}{\partial T}\ < 0$ which is equivalent to ${\hat{T}}>T^{*}\Leftrightarrow \frac{\partial V({\hat{T}},\rho )}{\partial T}\ <\frac{\partial \hat{V}({\hat{T}},\rho )}{\partial T}$ and to ${\hat{T}}>T^{*}\Leftrightarrow \frac{\partial V({\hat{T}},\rho )}{\partial T}\ - \frac{\partial \hat{V}({\hat{T}},\rho )}{\partial T} < 0$. After some computations, this inequality is satisfied if

  $$\begin{aligned}&-\alpha (1-\rho )\varphi ^{\prime }(\cdot )\left[ \int \limits _{ {\underline{Q}}}^{{\overline{Q}}}v^{\prime \prime }(z) \left( F_{{\underline{\theta }}}(q,T)-F_{{\hat{\theta }}}(q,T)\right) dq\right] \\&\quad -\alpha \int \limits _{{\underline{Q}}}^{{\overline{Q}}}v^{\prime }(z) \left( \frac{\partial F_{{\underline{\theta }} }(q,T)}{\partial T}-\frac{\partial F_{{\hat{\theta }}}(q,T)}{\partial T}\right) dq<0 \end{aligned}$$

  Through Assumption 2, this inequality is always verified for $\alpha >0$, meaning that ${\hat{T}}>T^{*}$.

• When ${\hat{\theta }}$ is better than the existing ones, then $F_{{\hat{\theta }}}(q,T)\le F_{{\overline{\theta }}}(q,T)$. As previously, ${\hat{T}}>T^{*}\Leftrightarrow \frac{\partial V({\hat{T}},\rho )}{\partial T}\ <\frac{\partial \hat{V}({\hat{T}},\rho )}{\partial T}$. A similar reasoning allows us to state that such an inequality is never satisfied for $\alpha <1$, meaning that ${\hat{T}}<T^{*}$.

Impact of a new scenario on adaptation Let us consider a new scenario ${\hat{\theta }}$, associated with a new welfare function denoted ${\hat{V}}(T,\rho )$ and a subsequent “new” optimal tax ${\hat{\rho }}=\arg \max \limits _{\rho } {\hat{V}}(T,\rho )$. Two configurations might occur:

• 1. When ${\hat{\theta }}$ is worst than all the initial scenarios then $F_{{\underline{\theta }}}(q,T)\le F_{{\hat{\theta }}}(q,T)$. We can state that ${\hat{\rho }}< \rho ^{*}\Leftrightarrow \frac{\partial V(T,{\hat{\rho }})}{\partial \rho }\ > 0$ which is equivalent to $\frac{\partial V(T,{\hat{\rho }})}{\partial \rho }\ >\frac{\partial {\hat{V}}(T,{\hat{\rho }})}{\partial \rho }$. Consequently, to prove that ${\hat{\rho }}< \rho ^{*}$, it is enough to prove that $\frac{\partial V(T,\rho )}{\partial \rho }\ >\frac{\partial {\hat{V}}(T,\rho )}{\partial \rho }$. After some computations, this is satisfied if

  $$\begin{aligned}&\alpha T\varphi ^{\prime }(\cdot )\left[ \int \limits _{ {\underline{Q}}}^{{\overline{Q}}}v^{\prime \prime }(x) \left( F_{{\underline{\theta }}}(q,T)-F_{{\hat{\theta }}}(q,T)\right) dq\right] \ge 0\,\,\text {for}\,\, \alpha \ge 0 \end{aligned}$$

  (C.13)

  The above inequality is always verified, and thus adaptation strictly increases when bad news arrives, when the DM is not strictly optimistic. For an optimistic DM, a bad news does not influence adaptation.

• 2. When ${\hat{\theta }}$ is better than the existing ones, then $F_{{\hat{\theta }}}(q,T)\le F_{{\overline{\theta }}}(q,T)$. A similar reasoning allows us to show that ${\hat{\rho }}> \rho ^{*}$ for $\alpha <1$ and ${\hat{\rho }}= \rho ^{*}$ for $\alpha =1$.

1.4 D Proof of Proposition 4

Impact of two new scenarios on mitigation Let us consider two additional scenarios ${\tilde{\theta }}$, being worst than all the others and ${\widehat{\theta }}$, being better than all the existing ones. Let us now denote by $\breve{T}$, the optimal tax in this case, that is when the DM maximizes the following welfare function:

$$\begin{aligned} \breve{V}(T,\rho)&= u(c(T,\rho ))+\alpha _{DM} \int \limits _{{\underline{Q}}}^{{\overline{Q}}}v(z)f_{\widehat{\theta }}(q,T)dq\nonumber \\& \quad +(1-\alpha _{DM})\int \limits _{{\underline{Q}}}^{{\overline{Q}}}v(z) f_{{\tilde{\theta }}}(q,T)dq. \end{aligned}$$

(D.14)

Then, $\breve{T}>T^{*}\Leftrightarrow \frac{\partial V(T^{*},\rho )}{\partial T} <\frac{\partial \breve{V}(T^{*},\rho )}{\partial T}$. This inequality is satisfied iff

$$\begin{aligned}&-(1-\rho )\varphi ^{\prime }(\cdot )\left[ \int \limits _{ {\underline{Q}}}^{{\overline{Q}}}v^{\prime \prime }(z) \left[ \alpha _{DM} \left( F_{{\underline{\theta }}}(q,T^{*})-F_{{\tilde{\theta }}}(q,T^{*})\right) \right. \right. \\&\quad \left. \left. +(1-\alpha _{DM})\left( F_{{\overline{\theta }}}(q,T^{*})-F_{{\widehat{\theta }}}(q,T^{*})\right) \right] dq\right] \\&\quad -\int \limits _{{\underline{Q}}}^{{\overline{Q}}}v^{\prime }(z)\left[ \alpha _{DM} \left( \frac{\partial F_{{\underline{\theta }} }(q,T^{*})}{\partial T}-\frac{\partial F_{{\tilde{\theta }} }(q,T^{*})}{\partial T}\right) \right. \\&\quad \left. +(1-\alpha _{DM})\left( \frac{\partial F_{{\overline{\theta }} }(q,T^{*})}{\partial T}-\frac{\partial F_{{\widehat{\theta }}}(q,T^{*})}{\partial T}\right) \right] dq<0 \end{aligned}$$

which is equivalent to

$$\begin{aligned}&\alpha _{DM} > \alpha _T(T^{*}) \\&\quad \equiv \frac{-(1-\rho )\varphi ^{\prime }(\cdot )\int v^{\prime \prime }(z) \left( F_{{\overline{\theta }}}-F_{{\widehat{\theta }}}\right) dq+\int v^{\prime }(z)\left( \frac{\partial F_{{\overline{\theta }} }}{\partial T}-\frac{\partial F_{{\widehat{\theta }}}}{\partial T}\right) dq}{-(1-\rho )\varphi ^{\prime }(\cdot )\int v^{\prime \prime }(z) \left( F_{\overline{\theta }}-F_{{\widehat{\theta }}}+F_{{\tilde{\theta }}}-F_{\underline{\theta }}\right) dq+\int v^{\prime }(z)\left( \frac{\partial F_{{\widehat{\theta }}}}{\partial T}-\frac{\partial F_{{\overline{\theta }} }}{\partial T}+\frac{\partial F_{{\underline{\theta }} }}{\partial T}-\frac{\partial F_{{\tilde{\theta }} }}{\partial T}\right) dq}. \end{aligned}$$

Impact of two new scenario on adaptation We denote by $(1-\breve{\rho })T$, the optimal adaptation level in this case. Using Eq. (D.14), we can show that $\breve{\rho }<\rho ^{*}\Leftrightarrow \frac{\partial V(T,\rho )}{\partial \rho } > \frac{\partial \breve{V}(T,\rho )}{\partial \rho }$. This inequality is satisfied iff

$$\begin{aligned}&\alpha _{DM} \varphi ^{\prime }(\cdot )\left[ \int \limits _{ {\underline{Q}}}^{{\overline{Q}}}v^{\prime }(z) \left( f_{{\underline{\theta }}}(q,T)-f_{{\tilde{\theta }}}(q,T)\right) dq\right] \\&\quad +(1-\alpha _{DM}) \varphi ^{\prime }(\cdot )\left[ \int \limits _{ {\underline{Q}}}^{{\overline{Q}}}v^{\prime }(z) \left( f_{{\overline{\theta }}}(q,T)-f_{{\hat{\theta }}}(q,T)\right) dq\right] < 0 \end{aligned}$$

which is equivalent to

$$\begin{aligned} \alpha _{DM} > \alpha _\rho (\rho ^{*}) \equiv \dfrac{\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\left[ F_{{\tilde{\theta }}}(q,T)-F_{{\overline{\theta }}}(q,T) \right] v^{''}(z)dq }{\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\left[ F_{{\tilde{\theta }}}(q,T)-F_{{\overline{\theta }}}(q,T)+ F_{{\underline{\theta }}}(q,T)- F_{{\hat{\theta }}}(q,T)\right] v^{''}(z)dq } \end{aligned}$$

which proves the result.

1.5 E Proof of Lemma 1

To demonstrate that $T^{*}$ and $\rho ^{*}$ are complements at the optimum, let us differentiate equation (3) with respect to T:

$$\begin{aligned} V^{\prime \prime }_{\rho T}(T,\rho )= & {} 1-\varphi ^{\prime \prime }(\cdot )(1-\rho )T\int \limits _{{\underline{Q}}}^{{\overline{Q}}}v^{\prime }(z)\ell (q)dq\nonumber -\varphi ^{\prime }(\cdot )\int \limits _{{\underline{Q}}}^{{\overline{Q}}}v^{\prime } (z)\ell (q)dq\\&-(\varphi ^{\prime }(\cdot ))^2T (1-\rho )\int \limits _{{\underline{Q}}}^{{\overline{Q}}}v^{\prime \prime } (z)\ell (q)dq\nonumber \\&-\varphi ^{\prime }(\cdot )T\int \limits _{{\underline{Q}}}^{{\overline{Q}}}v^{\prime } (z)\left[ \alpha \frac{\partial f_{{\overline{\theta }}}(q,T)}{\partial T}+(1-\alpha )\frac{\partial f_{{\underline{\theta }}}(q,T)}{\partial T}\right] dq \end{aligned}$$

Using Eq. (3) and integrating by part the last member of the right-hand side of equation (E.15), we deduce that:

$$\begin{aligned} V^{\prime \prime }_{\rho T}(T,\rho )=-\frac{(1-\rho )T\varphi ^{\prime \prime }(\cdot )}{\varphi ^{\prime }(\cdot )}-\varphi ^{\prime }(\cdot )T\int \limits _{{\underline{Q}}}^{{\overline{Q}}}v^{\prime \prime } (z)h(q,T)dq>0. \end{aligned}$$

Thus, $T^{*}$ and $\rho ^{*}$ are complements at the optimum.

1.6 F Proof of Proposition 5

1.6.1 F.1 Effect of Risk aversion

CARA utility function

$$\begin{aligned}V^{\prime \prime }_{\rho T}(T,\rho )& = -T\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\varphi ^{\prime }(\cdot )\beta \exp (-\beta z)L_T(q,T)dq\nonumber \\&\quad -T(1-\rho )\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\left[ \varphi ^{\prime \prime }(\cdot ) -\varphi ^{\prime 2}(\cdot )\beta \right] \exp (-\beta z)\ell (q,T) dq>0. \end{aligned}$$

(F.15)

Since v(x) is a CARA utility function, where $\beta$ is the coefficient of absolute risk aversion, total effects of risk aversion on the optimal climate policy design are given by

$$\begin{aligned} \frac{dT^*}{d\beta }=\frac{V^{\prime \prime }_{\rho T}V^{\prime \prime }_{\rho \beta }-V^{\prime \prime }_{T\beta }V^{\prime \prime }_{\rho \rho }}{ \Delta } \text { and } \frac{d\rho ^*}{d\beta }=\frac{V^{\prime \prime }_{\rho T}V^{\prime \prime }_{T\beta }-V^{\prime \prime }_{TT}V^{\prime \prime }_{\rho \beta }}{\Delta } \end{aligned}$$

where $\Delta =V^{\prime \prime }_{TT} V^{\prime \prime }_{\rho \rho }-(V^{\prime \prime }_{\rho ,T})^2$ is assumed to be strictly positive. The sign of $\frac{dT^*}{d\beta }$ is given by the sign of $V^{\prime \prime }_{\rho T}V^{\prime \prime }_{\rho \beta }-V^{\prime \prime }_{T\beta }V^{\prime \prime }_{\rho \rho }$.

After some computations, we obtain:

$$\begin{aligned}&V^{\prime \prime }_{\rho T}V^{\prime \prime }_{\rho \beta }-V^{\prime \prime }_{T\beta }V^{\prime \prime }_{\rho \rho }=T^2\left[ -\beta \varphi ^{\prime 2}(\cdot )\int \limits _{{\underline{Q}}}^{{\overline{Q}}}z \exp (-\beta z)\ell (q,T) dq \int \limits _{{\underline{Q}}}^{{\overline{Q}}}\exp (-\beta z)L_T(q,T) dq\right. \\&\quad \left. -\int \limits _{{\underline{Q}}}^{{\overline{Q}}}z \exp (-\beta z)L_T(q,Y)dq\int \limits _{{\underline{Q}}}^{{\overline{Q}}} \left[ \varphi ^{\prime \prime }(\cdot ) \exp (-\beta z)-\varphi ^{\prime 2}(\cdot )\beta \exp (-\beta z)\right] \ell (q,T) \right] . \end{aligned}$$

At the optimum, $\frac{dT^*}{d\beta }<0$ if:

$$\begin{aligned}&H(\beta )\equiv \beta \varphi ^{\prime 2}(\cdot )\int \limits _{{\underline{Q}}}^{{\overline{Q}}}z \exp (-\beta z)\ell (q,T) dq \int \limits _{{\underline{Q}}}^{{\overline{Q}}}\exp (-\beta z)L_T(q,T) dq\\&\quad +\int \limits _{{\underline{Q}}}^{{\overline{Q}}}z \exp (-\beta z)L_T(q,T)dq\int \limits _{{\underline{Q}}}^{{\overline{Q}}} \left[ \varphi ^{\prime \prime }(\cdot ) \exp (-\beta z)-\varphi ^{\prime 2}(\cdot )\beta \exp (-\beta z)\right] \ell (q,T)>0\\&\quad \Leftrightarrow \beta \varphi ^{\prime 2}(\cdot )(y^{\prime }(T)+1)\int \limits _{{\underline{Q}}}^{{\overline{Q}}}z \exp (-\beta z)\ell (q,T) dq\\&\quad +\left( \frac{\varphi ^{\prime \prime }(\cdot )-\varphi ^{\prime 2}(\cdot )\beta }{\varphi ^{\prime }(\cdot )}\right) \int \limits _{{\underline{Q}}}^{{\overline{Q}}}z \exp (-\beta z)L_T(q,T)dq>0. \end{aligned}$$

We can show that $H(0) > 0$ and $\lim \limits _{\beta \rightarrow +\infty }H(\beta )=0$. Consequently, and because H is continuous, there exists at least one value for $\beta$, ${\overline{\beta }}>0$, sufficiently small, such that for all $\beta \in ]0,{\overline{\beta }}]$, $H(\beta )> 0$ and thus $\frac{dT}{d\beta }<0$.

Similarly, since $\Delta >0$, $\text {sign}\frac{d\rho }{d\beta }=\text {sign}\left( V^{\prime \prime }_{\rho T}V^{\prime \prime }_{T\beta }-V^{\prime \prime }_{TT}V^{\prime \prime }_{\rho \beta }\right)$. We can write:

$$\begin{aligned}&V^{\prime \prime }_{\rho T}V^{\prime \prime }_{T\beta }-V^{\prime \prime }_{TT}V^{\prime \prime }_{\rho \beta }\\&\quad =\beta T \varphi ^{\prime }(\cdot )\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\exp (-\beta z)L_T(q,T) dq\left[ \varphi ^{\prime }(\cdot ) (1-\rho )\int \limits _{{\underline{Q}}}^{{\overline{Q}}}z\exp (-\beta z)\ell (q,T) dq\right. \\&\qquad \left. -\int \limits _{{\underline{Q}}}^{{\overline{Q}}}z\exp (-\beta z)L_T(q,T) dq\right] -Ty^{\prime \prime }(T)\varphi ^{\prime }(\cdot )\int \limits _{{\underline{Q}}}^{{\overline{Q}}}z\exp (-\beta z)\ell (q,T) dq\\&\qquad + T \varphi ^{\prime }(\cdot )\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\exp (-\beta z)L_{TT}(q,T) dq\int \limits _{{\underline{Q}}}^{{\overline{Q}}}z\exp (-\beta z)\ell (q,T) dq\\&\qquad -T(1-\rho )\left( \varphi ^{\prime \prime }(\cdot )-\beta \varphi ^{\prime 2}(\cdot )\right) \int \limits _{{\underline{Q}}}^{{\overline{Q}}}\exp (-\beta z)\ell (q,T) dq\int \limits _{{\underline{Q}}}^{{\overline{Q}}}z\exp (-\beta z)L_T(q,T) dq. \end{aligned}$$

At the equilibrium, we can rewrite this last expression as the following:

$$\begin{aligned}&V^{\prime \prime }_{\rho T}V^{\prime \prime }_{T\beta }-V^{\prime \prime }_{TT}V^{\prime \prime }_{\rho \beta }=-\beta T \varphi ^{\prime 2}(\cdot )(1-\rho )\left[ y^\prime (T)+1\right] \int \limits _{{\underline{Q}}}^{{\overline{Q}}}z\exp (-\beta z)\ell (q,T) dq\\&\quad -\beta T \varphi ^{\prime }(\cdot )\left[ y^\prime (T)+1\right] \int \limits _{{\underline{Q}}}^{{\overline{Q}}}z\exp (-\beta z)L_T(q,T) dq-Ty^{\prime \prime }(T)\varphi ^{\prime }(\cdot )\int \limits _{{\underline{Q}}}^{{\overline{Q}}}z\exp (-\beta z)\ell (q,T) dq\\&\quad + T \varphi ^{\prime }(\cdot )\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\exp (-\beta z)L_{TT}(q,T) dq\int \limits _{{\underline{Q}}}^{{\overline{Q}}}z\exp (-\beta z)\ell (q,T) dq\\&\quad -T(1-\rho )\frac{\varphi ^{\prime \prime }(\cdot )-\beta \varphi ^{\prime 2}(\cdot )}{\varphi ^{\prime }(\cdot )}\int \limits _{{\underline{Q}}}^{{\overline{Q}}}z\exp (-\beta z)L_T(q,T) dq. \end{aligned}$$

We can express:

$$\begin{aligned} V^{\prime \prime }_{\rho T}V^{\prime \prime }_{T\beta }-V^{\prime \prime }_{TT}V^{\prime \prime }_{\rho \beta }=-T\left[ (1-\rho )H(\beta )+ \varphi ^{\prime }(\cdot )J(\beta )\right] \end{aligned}$$

(F.16)

with $J(\beta )\equiv \int \limits _{{\underline{Q}}}^{{\overline{Q}}}z\exp (-\beta z)\left[ \beta (y^\prime (T)+1)L_T(q,T)+\left( y^{\prime \prime }(T)- \int \limits _{{\underline{Q}}}^{{\overline{Q}}}\exp (-\beta z)L_{TT}(q,T) dq\right) \ell (q,T) \right] dq$.

We can show that $J(0) < 0$ and $\lim \limits _{\beta \rightarrow +\infty }J(\beta )=0$. Consequently, and because J is continuous, there exists at least one value for $\beta$, $\overline{{\overline{\beta }}}>0$, sufficiently small, such that for all $\beta \in ]0,\overline{{\overline{\beta }}}]$, $J(\beta )< 0$.

$(1-\rho )H(0)+ \varphi ^{\prime }(\cdot )J(0)$ is positive when

$$\begin{aligned}&(1-\rho )\left( -\frac{\varphi ^{\prime \prime }(\cdot )}{\varphi ^{\prime }(\cdot )}\right) \int \limits _{{\underline{Q}}}^{{\overline{Q}}}z(-L_T(q,T)) dq>\left[ -y^{\prime \prime }(T)+\int \limits _{{\underline{Q}}}^{{\overline{Q}}}L_{TT}(q,T) dq\right] \int \limits _{{\underline{Q}}}^{{\overline{Q}}}z\ell (q,T) dq \end{aligned}$$

which is satisfied when $-\frac{\varphi ^{\prime \prime }(\cdot )}{\varphi ^{\prime }(\cdot )}$ is sufficiently high.

Consequently, as $-T\left[ (1-\rho )H(0)+ \varphi ^{\prime }(\cdot )J(0)\right] >0$ and $\lim \limits _{\beta \rightarrow +\infty }\left[ (1-\rho )H(\beta )+ \varphi ^{\prime }(\cdot )J(\beta )\right] =0$, and since $-T\left[ (1-\rho )H(\beta )+ \varphi ^{\prime }(\cdot )J(\beta )\right]$ is continuous, there exists at least one value for $\beta$, $\breve{\beta }>0$, sufficiently small, such that for all $\beta \in ]0,\breve{\beta }]$, $-T\left[ (1-\rho )H(\beta )+ \varphi ^{\prime }(\cdot )J(\beta )\right] < 0$ that is $\frac{d\rho ^*}{d\beta }<0.$

CRRA utility function

$$\begin{aligned}&V^{\prime \prime }_{\rho T}(T,\rho )=-T\kappa \int \limits _{{\underline{Q}}}^{{\overline{Q}}}\varphi ^{\prime }(\cdot ) z^{-\kappa -1}L_T(q,T)dq\nonumber \\&\quad -T(1-\rho )\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\left[ \varphi ^{\prime \prime }(\cdot )z -\varphi ^{\prime 2}(\cdot )\kappa \right] z^{-\kappa -1}\ell (q,T) dq>0. \end{aligned}$$

(F.17)

Since v(x) is a CRRA utility function, where $\kappa$ is the coefficient of relative risk aversion, total effects of risk aversion on the optimal climate policy design are given by

$$\begin{aligned} \frac{dT^*}{d\kappa }=\frac{V^{\prime \prime }_{\rho T}V^{\prime \prime }_{\rho \kappa }-V^{\prime \prime }_{T\kappa }V^{\prime \prime }_{\rho \rho }}{ \Delta } \text { and } \frac{d\rho ^*}{d\kappa }=\frac{V^{\prime \prime }_{\rho T}V^{\prime \prime }_{T\kappa }-V^{\prime \prime }_{TT}V^{\prime \prime }_{\rho \kappa }}{\Delta }. \end{aligned}$$

The sign of $\frac{dT^*}{d\kappa }$ is given by the sign of $K(\kappa ) \equiv V^{\prime \prime }_{\rho T}V^{\prime \prime }_{\rho \kappa }-V^{\prime \prime }_{T\kappa }V^{\prime \prime }_{\rho \rho }$ which is equal after some computations to the following expression:

$$\begin{aligned} K(\kappa )= & {} T^2\left[ -\kappa \varphi ^{\prime 2}(\cdot )\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\ln {z}\times z^{-\kappa } \ell (q,T) dq \int \limits _{{\underline{Q}}}^{{\overline{Q}}}z^{-\kappa -1}L_T(q,T) dq\right. \\&\left. +\frac{\kappa }{1-\kappa }(y^{\prime }(T)+1)\varphi ^{\prime 2}\int \limits _{{\underline{Q}}}^{{\overline{Q}}}z^{-\kappa -1} \ell (q,T) dq \right. \\&\quad \left. -\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\ln {z}\times z^{-\kappa }L_T(q,T)dq\int \limits _{{\underline{Q}}}^{{\overline{Q}}} \left[ \varphi ^{\prime \prime }(\cdot )z-\varphi ^{\prime 2}(\cdot )\kappa \right] z^{-\kappa - 1}\ell (q,T) \right] . \end{aligned}$$

We can show that $K(0) < 0$ and $\lim \limits _{\kappa \rightarrow +\infty }K(\kappa )=0$. Consequently, and because K is continuous, there exists at least one value for $\kappa$, ${\overline{\kappa }}>0$, sufficiently small, such that for all $\kappa \in ]0,{\overline{\kappa }}]$, $K(\kappa )< 0$ and thus $\frac{dT}{d\kappa }<0$.

Similarly, $\text {sign}\frac{d\rho }{d\kappa }=\text {sign}\left( V^{\prime \prime }_{\rho T}V^{\prime \prime }_{T\kappa }-V^{\prime \prime }_{TT}V^{\prime \prime }_{\rho \kappa }\right)$. At the optimum, we can write:

$$\begin{aligned}&V^{\prime \prime }_{\rho T}V^{\prime \prime }_{T\kappa }-V^{\prime \prime }_{TT}V^{\prime \prime }_{\rho \kappa }=-T(1-\rho )\frac{\varphi ^{\prime \prime }(\cdot )}{\varphi ^{\prime }(\cdot )}\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\ln {z}\times z^{-\kappa }L_T(q,T) dq\\&\quad -\kappa T(1-\rho )\varphi ^{\prime 2}(\cdot )\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\ln {z}\times z^{-\kappa }\ell (q,T) dq\int \limits _{{\underline{Q}}}^{{\overline{Q}}}z^{-\kappa -1}L_T(q,T) dq\\&\quad +\kappa T \varphi ^{\prime }(\cdot )\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\ln {z}\times z^{-\kappa }L_T(q) dq\int \limits _{{\underline{Q}}}^{{\overline{Q}}} z^{-\kappa -1}\left( (1-\rho )\varphi ^{\prime }\ell (q,T)-L_T(q,T)\right) dq\\&\quad -Ty^{\prime \prime }(T)\varphi ^{\prime }(\cdot )\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\ln {z}\times z^{-\kappa }\ell (q,T) dq + T \varphi ^{\prime }(\cdot )\int \limits _{{\underline{Q}}}^{{\overline{Q}}}z^{-\kappa }L_{TT}(q) dq\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\ln {z}\times z^{-\kappa }\ell (q,T) dq\\&\quad +2\frac{\kappa }{1-\kappa }T\varphi ^{\prime }(\cdot )(y^{\prime }(T)+1)\int \limits _{{\underline{Q}}}^{{\overline{Q}}}z^{-\kappa -1}\left( (1-\rho )\varphi ^{\prime }(\cdot )\ell (q,T)-L_T(q,T)\right) dq. \end{aligned}$$

Let us denote by $M(\kappa )$ this last expression, we can show that M(0) is negative when

$$\begin{aligned}&-(1-\rho )\frac{\varphi ^{\prime \prime }(\cdot )}{\varphi ^{\prime }(\cdot )}\left( -2(y^{\prime }(T)+1)-\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\ln {z}L_T(q,T) dq\right) \\&\quad -\varphi ^{\prime }(\cdot )\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\ln {z}\ell (q,T) dq \left( y^{\prime \prime }(T)-\int \limits _{{\underline{Q}}}^{{\overline{Q}}}L_{TT}(q,T) dq\right) \end{aligned}$$

is negative. That is satisfied when $-\frac{\varphi ^{\prime \prime }(\cdot )}{\varphi ^{\prime }(\cdot )}$ is sufficiently high.

Consequently, as $\lim \limits _{\kappa \rightarrow +\infty }M(\kappa )=0$, and since $M(\kappa )$ is continuous, there exists at least one value for $\kappa$, $\breve{\kappa }>0$, sufficiently small, such that for all $\kappa \in ]0,\breve{\kappa }]$, $\frac{d\rho ^*}{d\kappa }<0.$

1.6.2 F.2 Effects of Ambiguity Aversion

The total effects of a change in ambiguity aversion on the optimal design of the climate policy are given by:

$$\begin{aligned} \frac{dT^*}{d\alpha }=\frac{V^{\prime \prime }_{\rho T}V^{\prime \prime }_{\rho \alpha }-V^{\prime \prime }_{T\alpha }V^{\prime \prime }_{\rho \rho }}{ \Delta } \text { and } \frac{d\rho ^*}{d\alpha }=\frac{V^{\prime \prime }_{\rho T}V^{\prime \prime }_{T\alpha }-V^{\prime \prime }_{TT}V^{\prime \prime }_{\rho \alpha }}{\Delta } \end{aligned}$$

Since $\Delta >0$, we have that $\text {sign}\frac{dT^*}{d\alpha }=\text {sign} V^{\prime \prime }_{\rho T}V^{\prime \prime }_{\rho \alpha }-V^{\prime \prime }_{T\alpha }V^{\prime \prime }_{\rho \rho }$ and $\text {sign}\frac{d\rho ^*}{d\alpha }=\text {sign} V^{\prime \prime }_{\rho T}V^{\prime \prime }_{T\alpha }-V^{\prime \prime }_{TT}V^{\prime \prime }_{\rho \alpha }$.

CARA utility function

If v(x) is a CARA function so that $v(x)=-\frac{1}{\beta }exp(-\beta x)$, at the optimum, we first deduce that:

$$\begin{aligned}&V^{\prime \prime }_{\rho T}V^{\prime \prime }_{\rho \alpha }-V^{\prime \prime }_{T\alpha }V^{\prime \prime }_{\rho \rho }\\&\quad =T^{2}\left\{ (\varphi ^{\prime }(\cdot ))^{2}\beta ^{2}(y^{\prime }+1)\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\exp (-\beta x)(F_{{\underline{\theta }}}(q,T)-F_{\overline{\theta }}(q,T))dq\right\} \\&\qquad +T^{2}\left\{ \left( \frac{\varphi ^{\prime \prime }(\cdot )-(\varphi ^{\prime }(\cdot ))^{2}\beta }{\varphi ^{\prime }(\cdot )}\right) \int \limits _{{\underline{Q}}}^{{\overline{Q}}} \exp (-\beta x)\left( \frac{\partial F_{\underline{\theta }}(q,T)}{\partial T}-\frac{\partial F_{\overline{\theta }}(q,T)}{\partial T}\right) dq\right\} \end{aligned}$$

At the optimum, $\frac{dT^*}{d\alpha }>0$ if $G(\beta )>0$ with:

$$\begin{aligned}&G(\beta )=(\varphi ^{\prime }(\cdot ))^{2}\beta ^{2}(y^{\prime }+1)\int \limits _{{\underline{Q}}}^{{\overline{Q}}}\exp (-\beta z)(F_{{\underline{\theta }}}(q,T)-F_{{\overline{\theta }}}(q,T))dq \nonumber \\&\quad +\left( \frac{\varphi ^{\prime \prime }(\cdot )-(\varphi ^{\prime }(\cdot ))^{2}\beta }{\varphi ^{\prime }(\cdot )}\right) \int \limits _{{\underline{Q}}}^{{\overline{Q}}} \exp (-\beta z)\left( \frac{\partial F_{\underline{\theta }}(q,T)}{\partial T}-\frac{\partial F_{\overline{\theta }}(q,T)}{\partial T}\right) dq \end{aligned}$$

(F.18)

We can easily see that $G(0)>0$ and $\lim \limits _{\beta \rightarrow +\infty }G(\beta )=0$. Consequently, and because G is continuous, there exists at least one value for $\beta$, ${\hat{\beta }}>0$, sufficiently small, such that for all $\beta \in ]0,{\hat{\beta }}]$, $G(\beta )> 0$ and thus $\frac{dT^*}{d\alpha }>0$.

Now, let us turn to $\frac{d\rho ^*}{d\alpha }$, we obtain that:

$$\begin{aligned}&V^{\prime \prime }_{\rho T}V^{\prime \prime }_{T\alpha }-V^{\prime \prime }_{TT}V^{\prime \prime }_{\rho \alpha }\nonumber \\&\quad =T(1-\rho )\left( \frac{\varphi ^{\prime \prime }(\cdot )-(\varphi ^{\prime }(\cdot ))^{2}\beta }{\varphi ^{\prime }(\cdot )}\right) \int \limits _{{\underline{Q}}}^{{\overline{Q}}} \exp (-\beta z)\left( \frac{\partial F_{\underline{\theta }}(q,T)}{\partial T}-\frac{\partial F_{\overline{\theta }}(q,T)}{\partial T}\right) dq\nonumber \\&\qquad +T\varphi ^{\prime }(\cdot )\beta (y^{\prime }+1)\int \limits _{{\underline{Q}}}^{{\overline{Q}}} \exp (-\beta z)\left( \frac{\partial F_{{\underline{\theta }}}(q,T)}{\partial T}-\frac{\partial F_{{\overline{\theta }}}(q,T)}{\partial T}\right) dq\nonumber \\&\qquad +\left[ T\varphi ^{\prime }(\cdot )\beta y^{\prime \prime }(T)+T(\varphi ^{\prime }(\cdot )^{2}\beta ^{2} (1-\rho ) (y^{\prime }+1)\right] \int \limits _{{\underline{Q}}}^{{\overline{Q}}} \exp (-\beta z)\left( F_{{\underline{\theta }}}(q,T)-F_{\overline{\theta }}(q,T)\right) dq \nonumber \\&\qquad -T\varphi ^{\prime }(\cdot )\beta \int \limits _{{\underline{Q}}}^{{\overline{Q}}} \exp (-\beta z)L_{TT}(q,T)dq\int \limits _{{\underline{Q}}}^{{\overline{Q}}} \exp (-\beta z)\left( F_{{\underline{\theta }}}(q,T)-F_{\overline{\theta }}(q,T)\right) dq \end{aligned}$$

(F.19)

We can state that $\frac{d\rho ^*}{d\alpha }>0$ if $J(\beta )>0$, with $J(\beta )\equiv V^{\prime \prime }_{\rho T}V^{\prime \prime }_{T\alpha }-V^{\prime \prime }_{TT}V^{\prime \prime }_{\rho \alpha }$. We can easily see that $J(0)>0$ and $\lim \limits _{\beta \rightarrow +\infty }J(\beta )=0$. Consequently, and because J is continuous, there exists at least one value for $\beta$, ${\tilde{\beta }}>0$, sufficiently small, such that for all $\beta \in ]0,{\tilde{\beta }}]$, $J(\beta )> 0$ and thus $\frac{d\rho ^*}{d\alpha }>0$.

CRRA utility function

If v(x) is a CRRA function so that $v(x)=\frac{x^{1-\kappa }}{1-\kappa }$, at the optimum, we first deduce that:

$$\begin{aligned} N(\kappa )\equiv & {} V^{\prime \prime }_{\rho T}V^{\prime \prime }_{\rho \alpha }-V^{\prime \prime }_{T\alpha }V^{\prime \prime }_{\rho \rho }=T^{2}(\varphi ^{\prime }(\cdot ))^{2}\kappa ^{2}\left\{ \int \limits _{{\underline{Q}}}^{{\overline{Q}}}z^{-\kappa -1}(F_{{\underline{\theta }}}(q,T)\right. \\&\quad \left. -F_{\overline{\theta }}(q,T))dq\int \limits _{{\underline{Q}}}^{{\overline{Q}}}z^{-\kappa -1}L_T(q,T))dq\right. \\&\left. \int \limits _{{\underline{Q}}}^{{\overline{Q}}}z^{-\kappa }(F_{{\underline{\theta }}}(q,T)-F_{\overline{\theta }}(q,T))dq\int \limits _{{\underline{Q}}}^{{\overline{Q}}}z^{-\kappa -1}\ell (q,T))dq\right\} \\&\quad +T^{2}\frac{\varphi ^{\prime \prime }(\cdot )}{\varphi ^{\prime }(\cdot )}\int \limits _{{\underline{Q}}}^{{\overline{Q}}} z^{-\kappa }\left( \frac{\partial F_{\underline{\theta }}(q,T)}{\partial T}-\frac{\partial F_{\overline{\theta }}(q,T)}{\partial T}\right) dq \end{aligned}$$

At the optimum, $\frac{dT^*}{d\alpha }>0$ if $N(\kappa )>0$. We can easily see that $N(0)>0$ and $\lim \limits _{\kappa \rightarrow +\infty }N(\kappa )=0$. Consequently, and because N is continuous, there exists at least one value for $\kappa$, ${\hat{\kappa }}>0$, sufficiently small, such that for all $\kappa \in ]0,{\hat{\kappa }}]$, $N(\kappa )> 0$ and thus $\frac{dT^*}{d\alpha }>0$.

Now, let us turn to $\frac{d\rho ^*}{d\alpha }$, we obtain that:

$$\begin{aligned}&P(\kappa )\equiv V^{\prime \prime }_{\rho T}V^{\prime \prime }_{T\alpha }-V^{\prime \prime }_{TT}V^{\prime \prime }_{\rho \alpha }=T(1-\rho )\frac{\varphi ^{\prime \prime }(\cdot )}{\varphi ^{\prime }(\cdot )}\int \limits _{{\underline{Q}}}^{{\overline{Q}}} z^{-\kappa }\left( \frac{\partial F_{\underline{\theta }}(q,T)}{\partial T}-\frac{\partial F_{\overline{\theta }}(q,T)}{\partial T}\right) dq\nonumber \\&\quad +\kappa T\varphi ^{\prime }(\cdot )\int \limits _{{\underline{Q}}}^{{\overline{Q}}} z^{-\kappa }\left( \frac{\partial F_{\underline{\theta }}(q,T)}{\partial T}-\frac{\partial F_{\overline{\theta }}(q,T)}{\partial T}\right) dq\int \limits _{{\underline{Q}}}^{{\overline{Q}}} z^{-\kappa -1}\left( L_T(q,T)\right. \nonumber \\&\quad \left. -(1-\rho )\varphi ^{\prime }(\cdot )\ell (q,T)\right) dq\nonumber \\&\quad +\kappa T\varphi ^{\prime }(\cdot )\int \limits _{{\underline{Q}}}^{{\overline{Q}}} z^{-\kappa -1}\left( \frac{\partial F_{\underline{\theta }}(q,T)}{\partial T}-\frac{\partial F_{\overline{\theta }}(q,T)}{\partial T}\right) dq\nonumber \\&\quad \left[ y^{\prime \prime }(T)-\int \limits _{{\underline{Q}}}^{{\overline{Q}}} z^{-\kappa }L_{TT}(q,T)dq\right] \end{aligned}$$

(F.20)

We can state that $\frac{d\rho ^*}{d\alpha }>0$ if $P(\kappa )>0$. We can easily see that $P(0)>0$ and $\lim \limits _{\kappa \rightarrow +\infty }P(\kappa )=0$. Consequently, and because P is continuous, there exists at least one value for $\kappa$, ${\tilde{\kappa }}>0$, sufficiently small, such that for all $\kappa \in ]0,{\tilde{\kappa }}]$, $P(\kappa )> 0$ and thus $\frac{d\rho ^*}{d\alpha }>0$.

1.7 G Proof of Lemma 2

Let us prove that if an equilibrium $(T^{*}_h,T^{*}_{-h})$ exists, it may stable. For each country, the FOC writes:

$$\begin{aligned}&V_{T_h}^h\equiv V_{T_h}(T_h,T_{-h},\rho _h,\alpha _h)=\left( y^{\prime }(T_h)+\rho _h \right) u^{\prime }(c(T_h,\rho _h))\nonumber \\&\quad +(1-\rho _h )\varphi _h ^{\prime }(\cdot )\left[ \int \limits _{{\underline{Q}}}^{ {\overline{Q}}}v^{\prime }( q+\varphi _h(\cdot )) \left( \alpha _h f_{{\underline{\theta }}}(q,T_h+T_{-h})+(1-\alpha _h )f_{{\overline{\theta }}}(q,T_h+T_{-h})\right) dq\right] \nonumber \\&\quad +\int \limits _{{\underline{Q}}}^{{\overline{Q}}}v(q+\varphi _h(\cdot )) \left( \alpha _h \frac{\partial f_{\underline{\theta }}(q,T_h+T_{-h})}{\partial T_h}+(1-\alpha _h )\frac{\partial f_{{\overline{\theta }}}(q,T_h+T_{-h})}{\partial T_h}\right) dq \end{aligned}$$

(G.21)

with $\varphi _h(\cdot )=\varphi ((1-\rho _h)T_h)$.

From the FOC (G.21), and using the implicit function theorem, we obtain two best-response functions that correspond to the strategy adopted by the DM: $T_h=R_{h}(T_{-h})$. Furthermore, we can deduce the slope of each function $R_{h}$:

$$\begin{aligned} R^{\prime }_{h}(T_{-h})=-\frac{V^{h}_{T_{h}T_{-h}}}{V^{h}_{T_{h}T_{h}}} \end{aligned}$$

(G.22)

We can calculate the two second-order derivatives:

$$\begin{aligned}&V_{T_hT_h}^h = y^{\prime \prime }(T_h) u^{\prime }(c(T_h,\rho _h))+\left( y^{\prime }(T_h)+\rho _h \right) ^2 u^{\prime \prime }(c(T_h,\rho _h))\nonumber \\&\quad -2(1-\rho _h )\varphi _h ^{\prime 2 }(\cdot )\nonumber \\&\quad \int \limits _{{\underline{Q}}}^{ {\overline{Q}}}v^{\prime \prime }( q+\varphi _h(\cdot )) L_T^h(q,T)dq+(1-\rho _h )^2\varphi _h ^{\prime \prime }(\cdot ) \int \limits _{{\underline{Q}}}^{ {\overline{Q}}}v^{\prime }( q+\varphi _h(\cdot )) \ell ^h(q,T)dq\nonumber \\&\quad -\varphi _h ^{\prime }(\cdot ) \int \limits _{{\underline{Q}}}^{ {\overline{Q}}}v^{\prime }( q+\varphi _h(\cdot )) L_{TT}^h(q,T)dq + 2(1-\rho _h )^2\varphi _h ^{\prime 2 }(\cdot )\nonumber \\&\quad \int \limits _{{\underline{Q}}}^{ {\overline{Q}}}v^{\prime \prime }( q+\varphi _h(\cdot )) \ell ^h(q,T)dq<0 \end{aligned}$$

(G.23)

$$\begin{aligned}&V_{T_hT_{-h}}^h =- \int \limits _{{\underline{Q}}}^{ {\overline{Q}}}v^{\prime }( q+\varphi _h(\cdot )) L_{TT}^h(q,T)dq -(1-\rho _h )\varphi _h ^{\prime }(\cdot )\nonumber \\&\quad \int \limits _{{\underline{Q}}}^{ {\overline{Q}}}v^{\prime \prime }( q+\varphi _h(\cdot )) L_T^h(q,T)dq<0 \end{aligned}$$

(G.24)

with $L^h_T(q,T)= \alpha _h \frac{\partial F_{\underline{\theta }}(q,T_h+T_{-h})}{\partial T_h}+(1-\alpha _h ) \frac{\partial F_{{\overline{\theta }}}(q,T_h+T_{-h})}{\partial T_h}$ and $L^h_{TT}(q,T)= \alpha _h \frac{\partial ^2 F_{\underline{\theta }}(q,T_h+T_{-h})}{\partial T_h^2}+(1-\alpha _h ) \frac{\partial ^2 F_{{\overline{\theta }}}(q,T_h+T_{-h})}{\partial T_h^2}$.

An equilibrium $T^{*}$ exists and is a pair $(T^{*}_{h},T^{*}_{-h})$ if there exists a value of $T^{*}_{h}$ such that $R_{h}^{-1}(T^{*}_{h})=R_{-h}(T^{*}_{h})$.

Let us now consider that such an equilibrium exists, then it is stable if the following condition is satisfied:

$$\begin{aligned} \left| R_{h}^{-1\prime }(T^{*}_{h})\right|>\left| R_{-h}^{\prime }(T^{*}_{h})\right| \,\, \Leftrightarrow \,\, V^{-h}_{T_{-h}T_{-h}}V^{h}_{T_{h}T_{h}}>V^{-h}_{T_{-h}T_{h}}V^{h}_{T_{h}T_{-h}} \end{aligned}$$

(G.25)

1.8 H Proof of Proposition 6

To evaluate the impact of ambiguity aversion of the DM on the level of the optimal carbon tax in country h, we differentiate equation (4) with respect to $\alpha _h$. Total effect of ambiguity aversion is given by:

$$\begin{aligned} \frac{dT_h}{d\alpha _h}=\frac{V^{h}_{T_{h}\alpha _{h}}V^{-h}_{T_{-h}T_{-h}}}{V^{h}_{T_{h}T_{-h}}V^{-h}_{T_{-h}T_{h}}-V^{h}_{T_{h}T_{h}}V^{-h}_{T_{-h}T_{-h}}} \end{aligned}$$

(H.26)

Using Eqs. (G.23) and (B.11), we know that the numerator is negative. Overall, we state that $\frac{dT_h}{d\alpha _h}>0$ if $V^{h}_{T_{h}T_{-h}}V^{-h}_{T_{-h}T_{h}}-V^{h}_{T_{h}T_{h}}V^{-h}_{T_{-h}T_{-h}}<0$ which is true under condition (G.25).

We can also assess the effect of ambiguity aversion, $\alpha _{-h}$, of the other DM on the optimal carbon tax in the country h. Total effects of ambiguity aversion are given by:

$$\begin{aligned} \frac{dT_h}{d\alpha _{-h}}=\frac{-V^{h}_{T_{h}T_{-h}}V^{-h}_{T_{-h}\alpha _{-h}}}{V^{-h}_{T_{h}T_{-h}}V^{h}_{T_{h}T_{-h}}-V^{-h}_{T_{-h}T_{-h}}V^{h}_{T_{h}T_{h}}} \end{aligned}$$

(H.27)

Using Eqs. (B.11) and (G.24), the numerator is positive. Since $V^{h}_{T_{h}T_{-h}}V^{-h}_{T_{-h}T_{h}}-V^{h}_{T_{h}T_{h}}V^{-h}_{T_{-h}T_{-h}}<0$ under condition (G.25), we obtain that $\frac{dT_h}{d\alpha _{-h}}<0$.

Let us now turn to the effect of ambiguity aversion of $\hbox {DM}_h$ on the global mitigation, $T_{h}+T_{-h}$. We have to study the sign of the following expression:

$$\begin{aligned} \frac{dT_h}{d\alpha _h}+\frac{dT_{-h}}{d\alpha _{h}}=\frac{V^{h}_{T_{h}\alpha _{h}}\left( V^{-h}_{T_{-h}T_{-h}}-V^{-h}_{T_{-h}T_{h}}\right) }{V^{h}_{T_{-h}T_{h}}V^{-h}_{T_{-h}T_{h}}-V^{h}_{T_{h}T_{h}}V^{-h}_{T_{-h}T_{-h}}} \end{aligned}$$

(H.28)

The denominator is negative under condition (G.25), and we know that $V^{h}_{T_{h}\alpha _{h}}>0$. Thus, $\frac{dT_h}{d\alpha _h}+\frac{dT_{-h}}{d\alpha _{h}}>0 \Leftrightarrow V^{\prime \prime }_{-h,T_{-h}T_{-h}}<V^{\prime \prime }_{-h,T_{-h}T_{h}}<0$ which is true if $\varphi ^\prime (\cdot )$ is sufficiently large.