Catastrophic Damages and the Optimal Carbon Tax Under Loss Aversion

Abstract

Recently, economists have begun to incorporate tipping points and catastrophic events into economy-climate models. It has been shown that the inclusion of tipping points amplifies the economic impacts of climate change and leads to much higher estimates of the social cost of carbon compared to the model that includes only non-catastrophic damages. All the estimates under catastrophic damages come from studies that assume full rationality. However, there is ample evidence that consumers exhibit loss aversion, meaning that they feel losses more strongly than equivalent gains. In this paper, we derive the optimal carbon tax in the Ramsey model under loss aversion and tipping points. We calibrate the model to generate a similar rate of return on capital, and thus pathways of capital and consumption, as a model with rational consumers in the business-as-usual scenario. We find that such a calibrated model generates an optimal carbon tax that is about three times higher than in the model with rational consumers in the optimal (OPT) scenario. A catastrophic event, which reduces the productivity of capital, results in a greater utility loss of loss-averse consumers compared to rational consumers. The optimal tax makes loss-averse consumers increase their precautionary savings before the shock, smoothing their consumption, which reduces welfare loss after the catastrophic event.

Appendices

Appendix 1: Baseline Model After the Climate Regime Shift

After the climate regime shift the social planner’s optimization problem is described by the Hamilton–Jacobi–Bellman (HJB) equation in the value function ${V}^{A}(K,d)$

$${\rho V}^{A}(K,P,d)=\underset{C}{\mathrm{max}}U\left({C}^{A}\right)+{V}_{K}^{A}(K,d)({Y}^{A}\left({K}^{A},d\right)-{C}^{A}).$$

(35)

Differentiating the value function with respect to consumption ${C}^{A}$ and capital ${K}^{A}$ gives the optimality conditions for consumption and capital after the climate regime shift

$$U{^{\prime}}\left({C}^{A}\right)={V}_{K}^{A}(K,d),$$

(36)

$${Y}_{t}^{A}\left({K}_{t}^{A},d\right)-{C}^{A}=-\frac{{V}_{K}^{A}(K,d)}{{V}_{KK}^{A}(K,d)}({Y}_{t}^{A}\left({K}_{t}^{A},d\right)-\theta ).$$

(37)

Combining the optimality condition (36) with optimality condition (37) and using that $\dot{{K}^{A}}=\dot{{C}^{A}}$, yields a differential equation for consumption after the regime shift ${C}^{A}$ as a function of capital ${K}^{A}$

$$\dot{{C}^{A}}=-\frac{U{^{\prime}}\left({C}^{A}\right)}{U{^{\prime}}{^{\prime}}\left({C}^{A}\right)}({Y}_{K}^{A}\left({K}^{A},d\right)-\theta ).$$

(12)

Using the standard CRRA utility function (7) and substituting for $U{^{\prime}}\left({C}^{A}\right)$ and $U{^{\prime}}{^{\prime}}\left({C}^{A}\right)$ the Keynes-Ramsey rule can be expressed as:

$$\dot{{C}^{A}}=\frac{{C}^{A}}{\sigma }({Y}_{K}^{A}\left({K}^{A},d\right)-\theta ).$$

(38)

Using the Cob-Douglas production function (30) the net output function can be expressed as:

$${Y}_{t}^{A}\left({K}_{t}^{A},d\right)=\underset{C,F}{\mathrm{max}}(1-d)A{{K}_{t}^{A}}^{\alpha }{({{F}_{t}^{A}}^{\kappa }{{R}_{t}^{A}}^{1-\kappa })}^{\beta }{L}^{1-\alpha -\beta }-(g{+{\tau }_{A})F}_{t}^{A} -b{R}_{t}^{A}- \delta {K}_{t}^{A}.$$

(39)

Differentiating the net output function with respect to fossil fuel use ${F}_{t}^{A}$ and rearranging gives the optimality condition for fossil fuel use after the climate regime shift:

$${F}^{A}={(g+{\tau }_{A})}^{\frac{1}{\kappa \beta -1}}{\left((1-d)A{{K}^{A}}^{\alpha }{{R}^{A}}^{(1-\kappa )\beta }{L}^{1-\alpha -\beta }\kappa \beta \right)}^{\frac{1}{1-\kappa \beta }}.$$

(40)

Substituting for ${F}^{A}$ from Eq. (40) into Eq. (39) and rearranging gives the net output function:

$${Y}^{A}\left(.\right)=\left(1-\beta \right){\left(\left(1-d\right)Z\left({P}^{A}\right)A{{K}^{A}}^{\alpha }{L}^{1-\alpha -\beta }{\beta }^{\beta }{\left(\frac{\kappa }{g+{\tau }_{A}}\right)}^{\beta \kappa }{\left(\frac{1-\kappa }{b}\right)}^{\beta \left(1-\kappa \right)}\right)}^{\frac{1}{1-\beta }}-\delta {K}^{A}.$$

(41)

Using that in the steady state $\dot{{C}^{A}}=0$ from Eq. (38) we have that

$${Y}_{K}^{A}\left({K}^{A},d\right)-\theta =0.$$

(42)

Next, using that ${C}^{A}=Y({K}^{A},d)$ (re-arranged from Eq. 12), the formula for net output function from Eqs. (37) and (42) we derive the steady state level of capital and consumption after the climate regime shift as:

$$\begin{aligned} \overline{{K^{A} }} & = \left( {\left( {\left( {1 - d} \right)Z\left( {\overline{{P^{A} }} } \right)AL^{1 - \alpha - \beta } \beta^{\beta } \left( {\frac{\kappa }{{g + \overline{{\tau_{A} }} }}} \right)^{\beta \kappa } \left( {\frac{1 - \kappa }{b}} \right)^{{\beta \left( {1 - \kappa } \right)}} } \right)^{{\frac{1}{1 - \beta }}} \left( {\frac{\alpha }{\theta + \delta }} \right)} \right)^{{\frac{1 - \beta }{{1 - \alpha - \beta }}}} , \\ \overline{{C^{A} }} & = \left( {1 - \beta } \right)\left( {\left( {1 - d} \right)Z\left( {\overline{{P^{A} }} } \right)AL^{1 - \alpha - \beta } \beta^{\beta } \left( {\frac{\kappa }{{g + \overline{{\tau_{A} }} }}} \right)^{\beta \kappa } \left( {\frac{1 - \kappa }{b}} \right)^{{\beta \left( {1 - \kappa } \right)}} } \right)^{{\frac{1}{1 - \beta }}} \overline{{K^{A} }}^{{\frac{\alpha }{1 - \beta }}} - \delta \overline{{K^{A} }} . \\ \end{aligned}$$

(43)

Appendix 2: Hamilton–Jacobi–Bellman Equation with Endogenous Hazard Rate

Starting at time t we can approximate the probability of a regime shift in a small time period $\Delta t$ by $H({P}_{t})\Delta t$. The value function $W(K,t)$ is the maximal expected value of the objective function at time $t$ for capital stock $K$ and can therefore be written as:

$$W(K,t)=\underset{C}{\mathrm{ max}}E\left({\int }_{0}^{t+\Delta t}U\left(C\right){e}^{-\theta t}dt+ (1-H(P)\Delta t )\mathrm{W}(\mathrm{K}+\Delta \mathrm{K},\mathrm{ t}+\Delta \mathrm{t})+H(P)\Delta t {V}^{A}(\mathrm{K}+\Delta \mathrm{K},d) {e}^{-\theta (t+\Delta t)}\right).$$

(44)

After approximating the integral ${\int }_{0}^{t+\Delta t}U\left({C}_{t}\right){e}^{-\theta t}dt$ by $U\left({C}_{t}\right){e}^{-\theta t}\Delta t$ and moving the left-hand side of the Eq. (44) to the right-hand side we get

$$0=\underset{C}{\mathrm{ max}}E\left(U\left(C\right){e}^{-\theta t}\Delta t+ (1-H(P)\Delta t )\mathrm{W}(\mathrm{K}+\Delta \mathrm{K},\mathrm{ t}+\Delta \mathrm{t})+H(P)\Delta t {V}^{A}(\mathrm{K}+\Delta \mathrm{K},d) {e}^{-\theta (t+\Delta t)}-W(K,t)\right),$$

(45)

which after simplifying and dividing by $\Delta t$ gives

$$0=\underset{C}{\mathrm{ max}}E\left(U\left(C\right){e}^{-\theta t}-H(P)\mathrm{ W}(\mathrm{K}+\Delta \mathrm{K},\mathrm{ t}+\Delta \mathrm{t})+H(P){V}^{A}(\mathrm{K}+\Delta \mathrm{K},d) {e}^{-\theta (t+\Delta t)} +\frac{\mathrm{W}(\mathrm{K}+\Delta \mathrm{K},\mathrm{ t}+\Delta \mathrm{t})-W(K,t)}{\Delta \mathrm{t}}\right).$$

(46)

Taking the limit of Eq. (45) for $\Delta \mathrm{t}\to 0$ and using that $\dot{{C}_{t}}={Y}_{t}\left({K}_{t}\right)-{C}_{t}-\delta {K}_{t}$ yields

$$0=\underset{C}{\mathrm{ max}}E\left(U\left(C\right){e}^{-\theta t} -H(P)\mathrm{W}(\mathrm{K},\mathrm{ t})+H(P) {V}^{A}(\mathrm{K},d) {e}^{-\theta t} +{W}_{K}(K,t)(Y\left(K\right)-C-\delta K\right)+{W}_{t}(K,t).$$

(47)

By setting ${V}^{B}(K)=W(K,t){e}^{\theta t}$ we have

$$0=\underset{C}{\mathrm{ max}}E\left(U\left(C\right){e}^{-\theta t} -H(P){V}^{B}(K){e}^{-\theta t}+H(P) {V}^{A}(\mathrm{K},d) {e}^{-\theta t} +{V}_{K}^{B}(K){e}^{-\theta t}(Y\left(K\right)-C-\delta K\right)+{V}^{B}(K){e}^{-\theta t}(-\theta ),$$

(48)

which can be simplified and rearranged into

$${\theta V}^{B}(K)=\underset{C}{\mathrm{ max}}E\left(U\left(C\right) -H(P)({V}^{B}(K)-{V}^{A}(\mathrm{K},d)) + {V}_{K}^{B}(K)(Y\left(K\right)-C-\delta K\right).$$

(49)

Therefore, assuming the Cobb–Douglas production function of the form $Y(K,L,F,R)$ and using the formula derived in Eq. (49) the deterministic Hamilton–Jacobi–Bellman equation in the value function ${V}^{B}(K,P)$ becomes

$${\theta V}^{B}(K,P)=\underset{C}{\mathrm{max}}U\left({C}^{B}\right)-H({P}^{B})({V}^{B}(K,P)-{V}^{A}(K,P, d))+{V}_{K}^{B}(K,P)(A{Y}^{B}\left({K}^{B},{L}^{B},{F}^{B},{R}^{B}\right)-(g{+\tau )F}_{t}^{B}- {bR}^{B}-{C}^{B}-\delta {K}^{B})+{V}_{P}^{B}(K,P)(\psi {F}^{B}-\gamma {P}^{B}).$$

(50)

Appendix 3: Baseline Model Before the Climate Regime Shift

Before the climate regime shift the social planner’s optimization problem is described by the Hamilton–Jacobi–Bellman equation in the value function ${V}^{B}(K,P)$

$${\theta V}^{B}(K,P)=\underset{C}{\mathrm{max}}U\left({C}^{B}\right)-H({P}^{B})({V}^{B}(K,P)-{V}^{A}(K,d))+{V}_{K}^{B}(K,P)({Y}^{B}\left({K}^{B},{F}^{B}\right)-(g{+\tau )F}_{t}^{B}-{C}^{B}-\delta {K}^{B})+{V}_{P}^{B}(K,P)(\psi {F}^{B}-\gamma {P}^{B}).$$

(51)

Differentiating the value function with respect to consumption ${C}^{B}$ and capital ${K}^{B}$ gives the optimality conditions for consumption and capital before the climate regime shift

$$U{^{\prime}}\left({C}^{B}\right)={V}_{K}^{B}(K,P),$$

(52)

$${Y}_{t}^{A}\left({K}_{t}^{B},\tau \right)-{C}^{B}=-\frac{{V}_{K}^{B}(K,P)}{{V}_{KK}^{B}(K,P)}\left({Y}_{t}^{B}\left({K}_{t}^{B},\tau \right)-\theta +H({P}^{B})\left(\frac{{V}_{K}^{A}(K,d)}{{V}_{K}^{B}(K,P)}-1\right)\right).$$

(53)

Combining the optimality condition (52) with optimality condition (53) and using that $\dot{{K}^{B}}=\dot{{C}^{B}}$, yields a differential equation for consumption before the regime shift ${C}^{B}$ as a function of capital ${K}^{B}$ and carbon emissions via the hazard function $H(P)$

$$\dot{{C}^{B}}=-\frac{U{^{\prime}}\left({C}^{B}\right)}{U{^{\prime}}{^{\prime}}\left({C}^{B}\right)}\left({Y}_{t}^{B}\left({K}_{t}^{B},\tau \right)-\theta +H({P}^{B})\left(\frac{U{^{\prime}}\left({C}^{A}\right)}{U{^{\prime}}\left({C}^{B}\right)}-1\right)\right),$$

(54)

which corresponds to the following Keynes-Ramsey rule

$$\dot{{C}^{B}}=-\frac{U{^{\prime}}\left({C}^{B}\right)}{U{^{\prime}}{^{\prime}}\left({C}^{B}\right)}({Y}_{K}^{B}\left({K}^{B},\tau \right)-\theta +\vartheta ),$$

(18)

where $\vartheta$ is the precautionary return on capital accumulation defined as

$$\vartheta =H({P}^{B}) \left(\frac{U{^{\prime}}\left({C}^{A}\right)}{U{^{\prime}}\left({C}^{B}\right)}-1\right).$$

(55)

Using the standard CRRA utility function (7) we have

$$\dot{{C}^{B}}=\frac{{C}^{B}}{\sigma }({Y}_{K}^{B}\left({K}^{B},\tau \right)-\theta +\vartheta ),$$

(56)

and the precautionary return on capital $\vartheta$ becomes

$$\theta =H(P) \left({\left(\frac{{C}^{B}}{{C}^{A}}\right)}^{\sigma }-1\right).$$

(21)

Using the Cob-Douglas production function (30) the net output function can be expressed as:

$${Y}_{t}^{B}\left({K}_{t}^{B},\tau \right)=\underset{C,F}{\mathrm{max}}Z({P}_{t}^{B})A{{K}_{t}^{A}}^{\alpha }{({{F}_{t}^{A}}^{\kappa }{{R}_{t}^{A}}^{1-\kappa })}^{\beta }{L}^{1-\alpha -\beta }-(g{+\tau )F}_{t}^{B} -b{R}_{t}^{B}- \delta {K}_{t}^{B}.$$

(15)

Differentiating the net output function with respect to fossil fuel use ${F}_{t}^{B}$ and rearranging gives the optimality condition for fossil fuel use before the climate regime shift

$${F}^{B}={(g+\tau )}^{\frac{1}{\kappa \beta -1}}{\left((1-d)A{{K}^{B}}^{\alpha }{{R}^{B}}^{(1-\kappa )\beta }{L}^{1-\alpha -\beta }\kappa \beta \right)}^{\frac{1}{1-\kappa \beta }}.$$

(57)

Substituting for ${F}^{B}$ from Eq. (53) into Eq. (50) and rearranging gives the net output function

$${Y}^{B}\left(.\right)=\left(1-\beta \right){\left(Z\left({P}^{B}\right)A{{K}^{B}}^{\alpha }{L}^{1-\alpha -\beta }{\beta }^{\beta }{\left(\frac{\kappa }{g+\tau }\right)}^{\beta \kappa }{\left(\frac{1-\kappa }{b}\right)}^{\beta \left(1-\kappa \right)}\right)}^{\frac{1}{1-\beta }}-\delta {K}^{B}.$$

(58)

Using that in the steady state $\dot{{C}^{B}}=0$ from Eq. (56) we have that

$${Y}_{K}^{B}\left({K}^{B},\tau \right)-\theta +\vartheta =0.$$

(59)

Next, using that ${C}^{B}=Y({K}^{B},\tau )+{\tau F}^{B}$ (re-arranged from Eq. 14a), the formula for net output function from Eqs. (58) and (59) we derive the steady state level of capital and consumption before the climate regime shift. Finally, using Eq. (56) we derive the precautionary return on capital in the steady state

$$\begin{aligned} \overline{{K^{B} }} & = \left( {\left( {Z\left( {\overline{{P^{B} }} } \right)AL^{1 - \alpha - \beta } \beta^{\beta } \left( {\frac{\kappa }{{g + \overline{\tau }}}} \right)^{\beta \kappa } \left( {\frac{1 - \kappa }{b}} \right)^{{\beta \left( {1 - \kappa } \right)}} } \right)^{{\frac{1}{1 - \beta }}} \left( {\frac{\alpha }{{\theta + \delta - \overline{\vartheta }}}} \right)} \right)^{{\frac{1 - \beta }{{1 - \alpha - \beta }}}} , \\ \overline{{C^{B} }} & = \left( {1 - \beta } \right)\left( {Z\left( {\overline{{P^{B} }} } \right)AL^{1 - \alpha - \beta } \beta^{\beta } \left( {\frac{\kappa }{{g + \overline{\tau }}}} \right)^{\beta \kappa } \left( {\frac{1 - \kappa }{b}} \right)^{{\beta \left( {1 - \kappa } \right)}} } \right)^{{\frac{1}{1 - \beta }}} \overline{{K^{B} }}^{{\frac{\alpha }{1 - \beta }}} - \delta \overline{{K^{B} }} , \\ \overline{\vartheta } & = H\left( {\overline{{P^{B} }} } \right) \left( {\left( {\frac{{\overline{{C^{B} }} }}{{\overline{{C^{A} }} }}} \right)^{\sigma } - 1} \right). \\ \end{aligned}$$

(60)

Appendix 4: Carbon Tax with Non-catastrophic and Catastrophic Damages in the Baseline Model

First, we differentiate the Hamilton–Jacobi–Bellman equation in the value function ${V}^{A}(K,P,d)$

$${\theta V}^{A}(K,P,d)=\underset{C,F}{\mathrm{max}}\left(U\left({C}^{A}\right)+{V}_{K}^{A}(K,P,d)({(1-d)Z({P}^{A})AY}^{A}\left({K}^{A},{F}^{A}\right)-{C}^{A}-(g+{\tau }_{A}){F}^{A}-\delta {K}^{A})+{V}_{P}^{A}(K,P)(\psi {F}^{A}-\gamma {P}^{A})\right)$$

(61)

with respect to fossil fuel use ${F}^{A}$, which gives:

$$0={V}_{K}^{B}(K,P)((1-d)Z({P}^{A}){Y}_{F}^{A}\left({K}^{A},{F}^{A}\right)-(g+{\tau }_{A})+{V}_{P}^{B}(K,P)\psi ,$$

(62)

and derive the optimality condition for fossil fuel use after the climate regime shift ${F}^{A}$ from the net output function ${Y}_{t}^{A}\left({K}_{t}^{A},{P}_{t}^{A},d\right)$ (39):

$$(1-d)Z({P}^{A})A{Y}_{F}^{A}\left({K}^{A},{F}^{A}\right) =(g+{\tau }_{A}).$$

(63)

By combining Eqs. (62) and (63) we derive the formula for the carbon tax after the catastrophe:

$${\tau }_{A}=-\psi \frac{{V}_{P}^{A}(K,P,d)}{{V}_{K}^{A}(K,P)}.$$

(10)

Then, we differentiate the modified Hamilton–Jacobi–Bellman equation in the value function ${V}^{B}(K,P)$

$${\rho V}^{B}\left(K,P\right)=\underset{C}{\mathrm{max}}U\left({C}^{B}\right)-H({P}^{B})({V}^{B}(K,P)-{V}^{A}(K,P,d))+{V}_{K}^{B}(K,P)(Z({P}^{B})A{Y}^{B}\left({K}^{B},{F}^{B}\right)-{C}^{B}-(g{+\tau )F}_{t}^{B}-\delta {K}^{B})+{V}_{P}^{B}(K,P)(\psi {F}^{B}-\gamma {P}^{B})$$

(64)

with respect to carbon emissions ${P}^{B}$ and capital ${K}^{B}$, which yields:

$${\dot{V}}_{P}^{B}(K,P)=(\theta +H({P}^{B})+\gamma )({V}_{P}^{B}(K,P)+H{^{\prime}}({P}^{B})({V}_{P}^{B}(K,P)-{V}^{A}(K,P,d))-H({P}^{B}){V}_{P}^{A}(K,P,d)-{V}_{K}^{B}(K,P)(Z{^{\prime}}({P}^{B}){Y}^{B}\left({K}^{B},\tau \right)),$$

(65)

$${\dot{V}}_{K}^{B}(K,P)=\left({Y}^{B}\left({K}^{B},\tau \right)-\theta +H({P}^{B})\right){V}_{K}^{B}(K,P)+H({P}^{B}){V}^{A}(K,P,d).$$

(66)

Using Eqs. (65) and (66) and the fact that

$$\frac{\dot{\tau }}{\tau }=\frac{{\dot{V}}_{P}^{B}(K,P)}{{V}_{P}^{B}(K,P)}-\frac{{\dot{V}}_{K}^{B}(K,P)}{{V}_{K}^{B}(K,P)}$$

(67)

we can derive the differential equation for the carbon tax:

$$\dot{\tau }=({Y}_{K}^{B}(K,P)+\gamma +H({P}^{B})+\theta )\tau -\psi H{^{\prime}}({P}^{B})\frac{{V}^{B}(K,P)-{V}^{A}(K,P,d)}{{V}_{K}^{B}(K,P)}-\psi \frac{H({P}^{B}){V}_{P}^{A}(K,P,d){+V}_{K}^{B}(K,P)Z{^{\prime}}({P}^{B}){Y}^{B}({K}^{B},\tau )}{{V}_{K}^{B}(K,P)}.$$

(68)

Using the optimality condition $U{^{\prime}}\left({C}^{B}\right)={V}_{K}^{B}(K,P)$ yields a differential equation for the carbon tax as a function of ${K}^{B}$ and ${C}^{B}$:

$$\dot{\tau }=({Y}_{K}^{B}(K,P)+\gamma +H({P}^{B})+\vartheta )\tau -\psi H{^{\prime}}({P}^{B})\frac{{V}^{B}(K,P)-{V}^{A}(K,P,d)}{U{^{\prime}}({C}^{B})}-\psi \frac{H({P}^{B}){V}_{P}^{A}(K,P,d){+V}_{K}^{B}(K,P)Z{^{\prime}}({P}^{B}){Y}^{B}({K}^{B},\tau )}{U{^{\prime}}({C}^{B})}.$$

(69)

Since in the steady state $\dot{\tau }=0$ using Eq. (69) we can derive the steady state carbon tax before the regime shift as:

$$\overline{\tau }=\psi H{^{\prime}}(\overline{{P }^{B}})\frac{{V}^{B}(K,P)-{V}^{A}(K,P,d)}{U{^{\prime}}(\overline{{C }^{B}}) ({Y}_{K}^{B}(K,P)+H(\overline{{P }^{B}})+\gamma +\theta )}-\frac{\psi H(\overline{{P }^{B}}){V}_{P}^{A}(K,P,d)}{U{^{\prime}}(\overline{{C }^{B}}) ({Y}_{K}^{B}(K,P)+H(\overline{{P }^{B}})+\gamma +\theta )}-\psi \frac{Z{^{\prime}}(\overline{{P }^{B}}){Y}^{B}(\overline{{K }^{B}},\overline{\tau })}{ {Y}_{K}^{B}(K,P)+H(\overline{{P }^{B}})+\gamma +\theta },$$

(24)

which using that ${Y}_{K}^{B}\left({K}^{B},\tau \right)+\vartheta =\theta$ (re-arranged from Eq. 63) can be written as:

$$\overline{\tau }=\psi H{^{\prime}}(\overline{{P }^{B}})\frac{{V}^{B}(K,P)-{V}^{A}(K,P,d)}{U{^{\prime}}(\overline{{C }^{B}}) (H(\overline{{P }^{B}})+\gamma +\theta )}-\frac{\psi H(\overline{{P }^{B}}){V}_{P}^{A}(K,P,d)}{U{^{\prime}}(\overline{{C }^{B}}) (H(\overline{{P }^{B}})+\gamma +\theta )}-\psi \frac{Z{^{\prime}}(\overline{{P }^{B}}){Y}^{B}(\overline{{K }^{B}},\overline{\tau })}{ H(\overline{{P }^{B}})+\gamma +\theta },$$

(70)

and using (10) translates into:

$$\overline{\tau }=\psi H{^{\prime}}(\overline{{P }^{B}})\frac{{V}^{B}(K,P)-{V}^{A}(K,P,d)}{U{^{\prime}}(\overline{{C }^{B}}) (H(\overline{{P }^{B}})+\gamma +\theta )}+\frac{H(\overline{{P }^{B}}){\tau }_{A}{V}_{K}^{A}(K,P,d)}{U{^{\prime}}(\overline{{C }^{B}}) (H(\overline{{P }^{B}})+\gamma +\theta )}-\psi \frac{Z{^{\prime}}(\overline{{P }^{B}}){Y}^{B}(\overline{{K }^{B}},\overline{\tau })}{ H(\overline{{P }^{B}})+\gamma +\theta }.$$

(25)

Substituting for ${V}^{B}(K,P)$ from (64), ${V}^{A}(K,P,d)$ and ${V}_{K}^{A}(K,P,d)$ from (35) and using the standard CRRA utility function (7) we derive the formula for the carbon tax in the steady state in the baseline model with catastrophic damages as:

$$\overline{\tau }=\psi H{^{\prime}}(\overline{{P }^{B}})\frac{{\overline{{C }^{B}}}^{\sigma }{\stackrel{-}{({C}^{B}}}^{1-\sigma }-{\overline{{C }^{A}}}^{1-\sigma })}{(1-\sigma )(H(\overline{{P }^{B}})+\gamma +\theta )(\theta +H(\overline{{P }^{B}}))}+\frac{H(\overline{{P }^{B}}){\tau }_{A}({\overline{{C }^{A}}}^{1-\sigma }-1){\overline{{C }^{B}}}^{\sigma }}{(1-\sigma )\theta (H(\overline{{P }^{B}})+\gamma +\theta )}-\psi \frac{Z{^{\prime}}(\overline{{P }^{B}}){Y}^{B}(\overline{{K }^{B}},\overline{\tau })}{ H(\overline{{P }^{B}})+\gamma +\theta }.$$

(27)

Appendix 5: Carbon Tax with Non-catastrophic and Catastrophic Damages in the Model with Loss Aversion

After the climate regime shift the social planner’s optimization problem in the model with loss aversion is described by the Hamilton–Jacobi–Bellman (HJB) equation in the value function ${V}^{A}(K,d)$ from Eq. (9), and differentiating the value function with respect to consumption ${C}^{A}$ gives a modified optimality conditions for consumption:

$${U}_{C}\left({C}^{A},{X}^{A}\right)={V}_{K}^{A}(K,d)+\rho {V}_{X}^{A}(K,P),$$

(71)

while the optimality condition for capital from Eq. (37) still holds.

Combining the optimality condition (69) with optimality condition (37), using a condition that $\dot{{U}_{C}\left(C,X\right)}=\dot{C}{U}_{CC}\left(C,X\right)+\dot{X}{U}_{CX}\left(C,X\right)$ and using that $\dot{{K}^{A}}=\dot{{C}^{A}}$, yields a differential equation for consumption after the regime shift ${C}^{A}$ as a function of capital ${K}^{A}$:

$$\dot{{C}^{A}}=-\frac{\theta \left(\left({C}^{A}-{X}^{A}\right){U}_{CX}\left({C}^{A},{X}^{A}\right)-\left(\rho +\theta \right){V}_{X}^{A}\left(K,P\right)+ {U}_{X}\left({C}^{A},{X}^{A}\right)\right)}{{U}_{CC}\left({C}^{A},{X}^{A}\right)}-\frac{{(U}_{C}\left({C}^{A},{X}^{A}\right)+{V}_{X}^{A}(K,P)\rho )\left({Y}_{K}^{A}\left({K}^{A},d\right)-\theta \right)}{{U}_{CC}\left({C}^{A},{X}^{A}\right)}.$$

(11)

Using utility function from Eq. (4), the formula describing accumulation of reference stock $\dot{X}$ from Eq. (8) and substituting for ${U}_{CX}\left({C}^{A},{X}^{A}\right)$ and ${U}_{CC}\left({C}^{A},{X}^{A}\right)$ the Keynes-Ramsey rule after the climate shock in the model with loss aversion can be expressed as:

$$\dot{{C}^{A}}=((\xi {{C}^{A}}^{-\sigma }-\nu (-1+\xi )\chi {(m-{C}^{A}+{X}^{A})}^{-1+\chi }+\rho {V}_{X}^{A}(K,P))(-{Y}_{K}^{A}\left({K}^{A},d\right)+\theta +(\nu (-1+\xi )\rho (-1+\chi )\chi ({C}^{A}-{X}^{A}){(m-{C}^{A}+{X}^{A})}^{-2+\chi })/(\xi {{C}^{A}}^{-\sigma }-\nu (-1+\xi )\chi {(m-{C}^{A}+{X}^{A})}^{-1+\chi }+\rho {V}_{X}^{A}(K,P))+\frac{\nu (-1+\xi )\rho \chi {(m-{C}^{A}+{X}^{A})}^{-1+\chi }}{\xi {{C}^{A}}^{-\sigma }-\nu (-1+\xi )\chi {(m-{C}^{A}+{X}^{A})}^{-1+\chi }+\rho {V}_{X}^{A}(K,P)}-\frac{\rho (\theta +\rho ){V}_{X}^{A}(K,P)}{\xi {{C}^{A}}^{-\sigma }-\nu (-1+\xi )\chi {(m-{C}^{A}+{X}^{A})}^{-1+\chi }+\rho {V}_{X}^{A}(K,P)}))/(-\xi \sigma {{C}^{A}}^{-1-\sigma }+\nu (-1+\xi )(-1+\chi )\chi {(m-{C}^{A}+{X}^{A})}^{-2+\chi }).$$

(72)

Before the climate regime shift the social planner’s optimization problem in the model with loss aversion is described by the Hamilton–Jacobi–Bellman equation in the value function ${V}^{B}(K,P)$ from Eq. (15), and differentiating the value function with respect to consumption ${C}^{B}$ gives a modified optimality conditions for consumption:

$${U}_{C}\left({C}^{B},{X}^{B}\right)={V}_{K}^{B}(K,P)+\rho {V}_{X}^{B}(K,P),$$

(73)

while the optimality condition for capital from Eq. (53) still holds.

Combining the optimality condition (71) with optimality condition (53), using a condition that $\dot{{U}_{C}\left(C,X\right)}=\dot{C}{U}_{CC}\left(C,X\right)+\dot{X}{U}_{CX}\left(C,X\right)$ and using that $\dot{{K}^{B}}=\dot{{C}^{B}}$, yields a differential equation for consumption before the regime shift ${C}^{B}$ as a function of capital ${K}^{B}$ and carbon emissions via the hazard function $H(P)$:

$$\dot{{C}^{B}}=-\frac{\theta (\left({C}^{B}-{X}^{B}\right){U}_{CX}\left({C}^{B},{X}^{B}\right)-\left(\rho +\theta \right){V}_{X}^{B}(K,P)+ {U}_{X}\left({C}^{A},{X}^{A}\right)) }{{U}_{CC}\left({C}^{B},{X}^{B}\right)}-\frac{{(U}_{C}\left({C}^{B},{X}^{B}\right)+{V}_{X}^{B}(K,P)\rho )\left({Y}_{K}^{B}\left({K}^{B},\tau \right)-\theta +\vartheta \right)}{{U}_{CC}\left({C}^{B},{X}^{B}\right)}.$$

(74)

Using utility function from Eq. (4), the formula describing accumulation of reference stock $\dot{X}$ from Eq. (8) and substituting for ${U}_{CX}\left({C}^{A},{X}^{A}\right)$ and ${U}_{CC}\left({C}^{A},{X}^{A}\right)$ the Keynes-Ramsey rule before the climate shock in the model with loss aversion can be expressed as:

$$\dot{{C}^{B}}=-(((\xi {{C}^{B}}^{-\sigma }-(-1+\xi )\chi {(m+{C}^{B}-{X}^{B})}^{-1+\chi }+\rho {V}_{X}^{B}(K,P))(-{Y}_{K}^{B}\left({K}^{B},\tau \right)+\theta -\vartheta -\frac{(-1+\xi )\rho (-1+\chi )\chi ({C}^{B}-{X}^{B}){(m+{C}^{B}-{X}^{B})}^{-2+\chi }}{\xi {{C}^{B}}^{-\sigma }-(-1+\xi )\chi {(m+{C}^{B}-{X}^{B})}^{-1+\chi }+\rho {V}_{X}^{A}(K,P)}+\frac{(-1+\xi )\rho \chi {(m+{C}^{B}-{X}^{B})}^{-1+\chi }}{\xi {{C}^{B}}^{-\sigma }-(-1+\xi )\chi {(m+{C}^{B}-{X}^{B})}^{-1+\chi }+\rho {V}_{X}^{A}(K,P)}-\frac{\rho (\theta -\vartheta +\rho ){V}_{X}^{A}(K,P)}{\xi {{C}^{B}}^{-\sigma }-(-1+\xi )\chi {(m+{C}^{B}-{X}^{B})}^{-1+\chi }+\rho {V}_{X}^{A}(K,P)}))/(\xi \sigma {{C}^{B}}^{-1-\sigma }+(-1+\xi )(-1+\chi )\chi {(m+{C}^{B}-{X}^{B})}^{-2+\chi })),$$

(75)

where

$$\vartheta =H({P}^{B}) \left(\frac{\xi {{C}^{A}}^{-\sigma }+\lambda (1-\xi )\chi {(m-{C}^{A}+{X}^{A})}^{-1+\chi }}{\xi {{C}^{B}}^{-\sigma }+(1-\xi )\chi {(m+{C}^{B}-{X}^{B})}^{-1+\chi }}-1\right).$$

(20)

Derivation of the carbon tax with non-catastrophic and catastrophic damages in the model with loss aversion follows Eqs. (61)–(68), however using the optimality condition ${U}_{C}\left({C}^{B},{X}^{B}\right)={V}_{K}^{B}(K,P)+\rho {V}_{X}^{B}(K,P)$ from Eq. (69) yields a modified differential equation for the carbon tax as a function of ${K}^{B}$ and ${C}^{B}$:

$$\dot{\tau }=({Y}_{K}^{B}(K,P)+\gamma +H({P}^{B})+\theta )\tau -\psi H{^{\prime}}({P}^{B})\frac{{V}^{B}(K,P)-{V}^{A}(K,P,d)}{{U}_{C}\left({C}^{B},{X}^{B}\right)}-\psi \frac{H({P}^{B}){V}_{P}^{A}(K,P,d)+({V}_{K}^{B}(K,P)+\rho {V}_{X}^{B}(K,P) )Z{^{\prime}}({P}^{B}){Y}^{B}({K}^{B},\tau )}{{U}_{C}\left({C}^{B},{X}^{B}\right)},$$

(23)

which in the steady state translates into:

$$\overline{\tau }=\psi H{^{\prime}}(\overline{{P }^{B}})\frac{{V}^{B}(K,P)-{V}^{A}(K,P,d)}{{U}_{C}(\overline{{C }^{B},{X}^{B}}) (H(\overline{{P }^{B}})+\gamma +\rho )}+\frac{H(\overline{{P }^{B}}){\tau }_{A}{V}_{K}^{A}(K,P,d)}{{U}_{C}(\overline{{C }^{B},{X}^{B}}) (H(\overline{{P }^{B}})+\gamma +\theta )}-\psi \frac{Z{^{\prime}}(\overline{{P }^{B}}){Y}^{B}(\overline{{K }^{B}},\overline{\tau })}{ H(\overline{{P }^{B}})+\gamma +\theta }.$$

(24)

Substituting for ${V}^{A}(K,d)$ from Eq. (9), ${V}^{B}(K,P)$ from Eq. (16) and using the utility function from Eq. (4) we derive the formula for the carbon tax in the steady state in the model with loss aversion and catastrophic and non-catastrophic damages as:

$$\overline{\tau }=\psi H{^{\prime}}(\overline{{P }^{B}})\frac{\xi ({\overline{{C }^{B}}}^{1-\sigma }-{\overline{{C }^{A}}}^{1-\sigma })}{(1-\sigma )(H(\overline{{P }^{B}})+\gamma +\theta )(\theta +H(\overline{{P }^{B}}))}+\psi H{^{\prime}}(\overline{{P }^{B}})\frac{((\xi -1)({m}^{\chi }(1+\nu )-\nu {(m-\overline{{C }^{A}}+\overline{{X }^{A}})}^{\chi }-{(m+\overline{{C }^{B}}-\overline{{X }^{B}})}^{\chi }))}{(H(\overline{{P }^{B}})+\gamma +\theta )(\theta +H(\overline{{P }^{B}}))}+\frac{H(\overline{{P }^{B}}){\tau }_{A}(\xi {\overline{{C }^{A}}}^{-\sigma }+(1-\xi )\chi {(m+\overline{{C }^{A}}-\overline{{X }^{A}})}^{-1+\chi }+\rho {V}_{X}^{A}(K,P,d))}{(\xi {\overline{{C }^{B}}}^{-\sigma }+(1-\xi )\chi {(m+\overline{{C }^{B}}-\overline{{X }^{B}})}^{-1+\chi })(H(\overline{{P }^{B}})+\gamma +\theta )}-\psi \frac{Z{^{\prime}}(\overline{{P }^{B}}){Y}^{B}(\overline{{K }^{B}},\overline{\tau })}{ H(\overline{{P }^{B}})+\gamma +\theta }.$$

(76)

Appendix 6: Derivation of the Rate of Return on Capital in the Model with Loss Aversion

In the model with loss aversion, a consumer takes into account the effects of her current consumption on her future reference stock. As a result, the dynamics of the reference stock enters the current value Hamiltonian:

$$H\equiv U\left(\frac{{C}_{t}}{{L}_{t}},\frac{{X}_{t}}{{L}_{t}}\right){L}_{t}+{\lambda }_{t}\left(Z\left({P}_{t}\right){Y}_{t}\left(.\right)-g{F}_{t}-b{R}_{t}-{C}_{t}-\delta {K}_{t}\right)+{\mu }_{t}\rho ({C}_{t}-{X}_{t})-{\pi }_{t}(\psi {F}_{t}-\gamma {P}_{t}),$$

(77)

where ${\mu }_{t}$ is the shadow value of the reference stock of consumption.

We derive the optimality conditions for capital, consumption and evolution of the shadow value of the reference stock:

$$\dot{{\lambda }_{t}}={\lambda }_{t}(\delta +\theta -Z\left({P}_{t}\right){Y}_{{K}_{t}}\left(.\right)),$$

(78)

$${U}_{{C}_{t}}\left({C}_{t},{X}_{t}\right)={\lambda }_{t}-{\mu }_{t}\rho ,$$

(79)

$$\dot{{\mu }_{t}}=\left(\rho +\theta \right){\mu }_{t}-{U}_{{X}_{t}}\left({C}_{t},{X}_{t}\right).$$

(80)

To derive the Euler equation in the model with loss aversion, we first differentiate Eq. (79) with respect to time, which gives:

$${{\dot{{\lambda }_{t}}=\dot{U}_{{C}_{t}}\left({C}_{t},\,{X}_{t}\right)}+\dot{{\mu }_{t}}\rho}.$$

(81)

Substituting Eq. (80) into (81) gives:

$$\dot{{\lambda }_{t}}=\dot{{U}_{{C}_{t}}\left({C}_{t},{X}_{t}\right)}+\left(\left(\rho +\theta \right){\mu }_{t}- {U}_{{X}_{t}}\left({C}_{t},{X}_{t}\right)\right)\rho .$$

(82)

After combining Eq. (82) with Eq. (78) and substituting ${\lambda }_{t}={{U}_{{C}_{t}}\left({C}_{t},{H}_{t}\right)-\mu }_{t}\rho$ (re-arranged from Eq. 79) to Eq. (82), we obtain:

$$\dot{{U}_{{C}_{t}}\left({C}_{t},{X}_{t}\right)}+\left(\left(\rho +\theta \right){\mu }_{t}- {U}_{{X}_{t}}\left({C}_{t},{X}_{t}\right)\right)\rho =\left({U}_{{C}_{t}}\left({C}_{t},{X}_{t}\right)+{\mu }_{t}\rho \right)\left(\delta +\theta -Z\left({P}_{t}\right){Y}_{{K}_{t}}\left(.\right)\right).$$

(83)

Using that $\dot{{U}_{{C}_{t}}\left({C}_{t},{X}_{t}\right)}=\dot{{C}_{t}}{U}_{{C}_{t}{C}_{t}}\left({C}_{t},{X}_{t}\right)+\dot{{X}_{t}}{U}_{{C}_{t}{X}_{t}}\left({C}_{t},{X}_{t}\right)$, substituting $\dot{{X}_{t}}=\rho \left({C}_{t}-{H}_{t}\right)$ (from Eq. 8) and rearranging, we receive the Euler equation:

$${\dot{C}}_{t}=\frac{({U}_{{C}_{t}}\left({C}_{t},{X}_{t}\right)+\rho {\mu }_{t})(\theta -{r}_{t})}{{U}_{{C}_{t}{C}_{t}}\left({C}_{t},{X}_{t}\right)}-\frac{\rho \left({C}_{t}-{X}_{t}\right){U}_{{C}_{t}{X}_{t}}\left({C}_{t},{X}_{t}\right)}{{U}_{{C}_{t}{C}_{t}}\left({C}_{t},{X}_{t}\right)}-\frac{\rho \left({\mu }_{t}\left(\rho +\theta \right)-{U}_{{X}_{t}}\left({C}_{t},{X}_{t}\right)\right)}{{U}_{{C}_{t}{C}_{t}}\left({C}_{t},{X}_{t}\right)},$$

(84)

where ${r}_{t}\equiv Z\left({P}_{t}\right){Y}_{{K}_{t}}\left(.\right)-\delta$.

After substituting formulas for ${U}_{{C}_{t}}\left({C}_{t},{X}_{t}\right)$, ${U}_{{X}_{t}}\left({C}_{t},{X}_{t}\right)$, ${U}_{{C}_{t}{X}_{t}}\left({C}_{t},{X}_{t}\right)$ and ${U}_{{C}_{t}{C}_{t}}\left({C}_{t},{X}_{t}\right)$ into Eq. (84), using the utility function from Eq. (4) and simplifying we receive:

$$\dot{C}_{t} = - \left(\left(\left( {\xi C_{t}^{ - \sigma } - \left( { - 1 + \xi } \right)\chi \left( {m + C_{t} - X_{t} } \right)^{ - 1 + \chi } + \rho \mu_{t} } \right)( - r + \theta - (\left( { - 1 + \xi } \right)\rho \left( { - 1 + \chi } \right) \chi \left( {C_{t} - X_{t} } \right)\left( {m + C_{t} - X_{t} } \right)^{ - 2 + \chi } )/\left( {\xi C_{t}^{ - \sigma } - \left( { - 1 + \xi } \right)\chi \left( {m + C_{t} - X_{t} } \right)^{ - 1 + \chi } + \rho \mu_{t} } \right) + \frac{{\left( { - 1 + \xi } \right)\rho \chi \left( {m + C_{t} - X_{t} } \right)^{ - 1 + \chi } }}{{\xi C_{t}^{ - \sigma } - \left( { - 1 + \xi } \right)\chi \left( {m + C_{t} - X_{t} } \right)^{ - 1 + \chi } + \rho \mu_{t} }} - \frac{{\rho \left( {\theta + \rho } \right)\mu_{t} }}{{\xi C_{t}^{ - \sigma } - \left( { - 1 + \xi } \right)\chi \left( {m + C_{t} - X_{t} } \right)^{ - 1 + \chi } + \rho \mu_{t} }})\right) /\left( {\xi \sigma C_{t}^{ - 1 - \sigma } + \left( { - 1 + \xi } \right)\left( { - 1 + \chi } \right)\chi \left( {m + C_{t} - X_{t} } \right)^{ - 2 + \chi } } \right)\right).$$

(85)

Using Eq. (84) we derive the rate of return on capital in the model with loss aversion:

$${r}_{t}=\theta -\frac{{\dot{C}}_{t}{U}_{{C}_{t}{C}_{t}}\left({C}_{t},{X}_{t}\right)}{\left({U}_{{C}_{t}}\left({C}_{t},{X}_{t}\right)+\rho {\mu }_{t}\right)}-\frac{\rho \left({C}_{t}-{X}_{t}\right){U}_{{C}_{t}{X}_{t}}\left({C}_{t},{X}_{t}\right)}{\left({U}_{{C}_{t}}\left({C}_{t},{X}_{t}\right)+\rho {\mu }_{t}\right)} -\frac{\rho \left({\mu }_{t}\left(\rho +\theta \right)-{U}_{{X}_{t}}\left({C}_{t},{X}_{t}\right)\right)}{\left({U}_{{C}_{t}}\left({C}_{t},{X}_{t}\right)+\rho {\mu }_{t}\right)}.$$

(86)

After substituting formulas for ${U}_{{C}_{t}}\left({C}_{t},{X}_{t}\right)$, ${U}_{{X}_{t}}\left({C}_{t},{X}_{t}\right)$, ${U}_{{C}_{t}{X}_{t}}\left({C}_{t},{X}_{t}\right)$ and ${U}_{{C}_{t}{C}_{t}}\left({C}_{t},{X}_{t}\right)$ into Eq. (86), using the utility function from Eq. (4) and simplifying we receive:

$${r}_{t}=\theta -\frac{{\dot{C}}_{t}(-\xi \sigma {{C}_{t}}^{-1-\sigma }+(1-\xi )(\chi -1)\chi {\left(m+{C}_{t}-{X}_{t}\right)}^{-2+\chi })}{\xi {{C}_{t}}^{-\sigma }-\left(\xi -1\right)\chi {\left(m+{C}_{t}-{X}_{t}\right)}^{-1+\chi }+\rho {\mu }_{t}}-\frac{\rho ({C}_{t}-{X}_{t})(\xi -1)(\chi -1)\chi {\left(m+{C}_{t}-{X}_{t}\right)}^{-2+\chi }}{\xi {{C}_{t}}^{-\sigma }-\left(\xi -1\right)\chi {\left(m+{C}_{t}-{X}_{t}\right)}^{-1+\chi }+\rho {\mu }_{t}}-\frac{\rho ({\mu }_{t}\left(\rho +\theta \right)-(1-\xi )\chi {\left(m+{C}_{t}-{X}_{t}\right)}^{-1+\chi })}{\xi {{C}_{t}}^{-\sigma }-\left(\xi -1\right)\chi {\left(m+{C}_{t}-{X}_{t}\right)}^{-1+\chi }+\rho {\mu }_{t}}.$$

(87)

Appendix 7: Carbon Tax with Gradual and Catastrophic Damages in the Model with Duffie-Epstein Preferences

In this section, we modify the model by van der Ploeg and de Zeeuw (2018) to compare the model with loss aversion to the E–Z preferences. Formally, the authors use the stochastic differential utility framework of Duffie and Epstein (1992) to approximate the solutions under the Epstein and Zin (1989) preferences. This is motivated by the fact that the E–Z preferences have been proposed for the discrete time, whereas van der Ploeg and de Zeeuw (2018) use a continuous-time Ramsey growth model with climate tipping.

In general, in the model with Duffie-Epstein preferences the social planner maximizes the modified expected value of the social welfare function:

$$\underset{C,F}{\mathrm{max}}E\left({\int }_{0}^{\infty }\Phi \left({C}_{t},{\widetilde{V}}_{t}\right){e}^{-\theta t}dt\right),$$

(88)

where value function ${\widetilde{V}}_{t}=\widetilde{V}({K}_{t},{P}_{t})$ is expressed by the following formula:

$$\Phi \left({C}_{t},{\widetilde{V}}_{t}\right)=\frac{\widetilde{\theta }}{1-{\sigma }_{I}}\frac{{{C}_{t}}^{1-{\sigma }_{I}}-{((1-{\sigma }_{R}){\widetilde{V}}_{t})}^{\frac{1-{\sigma }_{I}}{1-{\sigma }_{R}}}}{{((1-{\sigma }_{R}){\widetilde{V}}_{t})}^{\frac{{\sigma }_{R}-{\sigma }_{I}}{1-{\sigma }_{R}}}}$$

(89)

and $\Phi \left({C}_{t},{\widetilde{V}}_{t}\right)$ is the aggregator function for Duffie-Epstein preferences. Parameter ${\sigma }_{I}>1$ is $\frac{1}{EIS}$ (the intertemporal aspect) and ${\sigma }_{R}>0$ is RRA. The modified pure time preference parameter is $\widetilde{\theta }$ where $\widetilde{\theta }=\theta ({\sigma }_{I}-1){g}_{t}$.

The optimization problem after the climate regime shift gives the following Hamilton–Jacobi–Bellman (HJB) equation:

$$\widetilde{\theta }\left(\frac{1-{\sigma }_{R}}{1-{\sigma }_{I}}\right)\widetilde{{V}^{A}}\left(K,P,d\right)\underset{C,F}{=max}\left(\frac{\widetilde{\theta }}{1-{\sigma }_{I}}\right){{C}^{A}}^{1-{\sigma }_{I}}{\left(\left(1-{\sigma }_{R}\right)\widetilde{{V}^{A}}\left(K,P,d\right)\right)}^{\frac{{\sigma }_{I}-{\sigma }_{R}}{1-{\sigma }_{R}}}+\widetilde{{V}_{K}^{A}}\left(K,P\right)\left({\left(1-d\right)Z\left({P}^{A}\right)AY}^{A}\left({K}^{A},{F}^{A}\right)-{C}^{A}-\left(g+{\tau }_{A}\right){F}^{A}-{bR}^{A}-\delta {K}^{A}\right) +\widetilde{{V}_{P}^{A}}\left(K,P,d\right)\left(\psi {F}^{A}-\gamma {P}^{A}\right).$$

(90)

This yields the optimality condition for consumption before the climate regime shift:

$$\widetilde{\theta }{{C}^{A}}^{1-{\sigma }_{I}}{\left(\left(1-{\sigma }_{R}\right)\widetilde{{V}^{A}}(K,P,d)\right)}^{\frac{{\sigma }_{I}-{\sigma }_{R}}{1-{\sigma }_{R}}}=\widetilde{{V}_{K}^{A}}\left(K,P\right),$$

(91)

and the formula for the carbon tax after the catastrophe:

$${\tau }_{A}=-\psi \frac{\widetilde{{V}_{P}^{A}}(K,P,d)}{\widetilde{{V}_{K}^{A}}(K,P)}.$$

(92)

Together with Eqs. (2a) and (2b) this allows us to solve for the value function before the climate regime shift $\widetilde{{V}^{A}}\left(K,P,d\right):$

$$\widetilde{{V}^{A}}\left(K,P,d\right)=\frac{1}{1-{\sigma }_{R}}{\left(\left(1-{\sigma }_{I}\right)\widetilde{\theta }\widetilde{{V}^{A}}(K,P,d)\right)}^{\frac{1-{\sigma }_{R}}{1-{\sigma }_{I}}}.$$

(93)

The optimality condition for consumption after the climate regime shift thus boils down to the usual ${{C}^{A}}^{1-{\sigma }_{I}}=\widetilde{{V}_{K}^{A}}(K,P)$, which is independent of RRA as uncertainty has by this time been resolved.

The optimization problem before the climate regime shift gives the following Hamilton–Jacobi–Bellman (HJB) equation:

$$0=\underset{C,F}{max}\left(\Phi \left({{C}^{B}}_{t},{\widetilde{{V}^{B}}}_{t}\right)-H\left({P}^{B}\right)\left(\widetilde{{V}^{B}}\left(K,P\right)-\widetilde{{V}^{A}}\left(K,P,d\right)\right)+\widetilde{{V}_{P}^{B}}\left(K,P\right)\left(\psi {F}^{B}-\gamma {P}^{B}\right)\widetilde{{V}_{K}^{B}}\left(K,P\right)\left({Z\left({P}^{B}\right)AY}^{B}\left({K}^{B},{F}^{B}\right)-{C}^{B}-\left(g+\tau \right){F}^{B}-{bR}^{B}-\delta {K}^{B}\right)\right).$$

(94)

The optimality condition for consumption before the climate regime shift is:

$${{C}^{B}}^{-{\sigma }_{I}}={\left(\left(1-{\sigma }_{R}\right)\widetilde{{V}^{B}}(K,P,d)\right)}^{\frac{{\sigma }_{R}-{\sigma }_{I}}{1-{\sigma }_{R}}}\frac{\widetilde{{V}_{K}^{B}}(K,P)}{\widetilde{\theta }},$$

(95)

while the differential equation for consumption before the regime shift ${C}^{B}$ as a function of capital ${K}^{B}$ and carbon emissions via the hazard function $H(P)$ is:

$$\dot{{C}^{B}}=-\frac{{C}^{B}}{{\sigma }_{I}}\left({Y}_{t}^{B}\left({K}_{t}^{B},\tau \right)-\frac{{\sigma }_{R}-1}{1-{\sigma }_{I}}\widetilde{\theta }+\vartheta -\frac{\widetilde{\theta }{{C}^{B}}^{1-{\sigma }_{I}}}{1-{\sigma }_{I}}{({\sigma }_{I}-{\sigma }_{R})\left(\left(1-{\sigma }_{R}\right)\widetilde{{V}^{B}}(K,P,d)\right)}^{\frac{{\sigma }_{I}-1}{1-{\sigma }_{R}}}+\left(\frac{{\sigma }_{R}-{\sigma }_{I}}{1-{\sigma }_{R}}\right)\frac{\dot{\widetilde{{V}^{B}}}}{\widetilde{{V}^{B}}}\right),$$

(96)

where $\vartheta$ is the precautionary return on capital accumulation defined as:

$$\vartheta =H\left({P}^{B}\right)\left(\frac{\widetilde{\theta }\widetilde{{V}_{K}^{A}}\left(K,P,d\right)-\widetilde{{V}_{K}^{B}}\left(K,P\right)}{\widetilde{{V}_{K}^{B}}\left(K,P\right)}-1\right),$$

(97)

where:

$${{\widetilde{{V}_{K}^{B}}\left(K,P\right)=\widetilde{\theta }C}^{B}}^{1-{\sigma }_{I}}{\left(\left(1-{\sigma }_{R}\right)\widetilde{{V}^{B}}(K,P,d)\right)}^{\frac{{\sigma }_{R}-{\sigma }_{I}}{{\sigma }_{R}-1}}.$$

(98)

We can derive the carbon tax as:

$$\dot{\tau }=({Y}_{K}^{B}(K,P)+\gamma +H({P}^{B})+\widetilde{\theta })\tau -\psi H{^{\prime}}({P}^{B})\frac{\widetilde{{V}^{B}}(K,P)-\widetilde{{V}^{A}}(K,P,d)}{\widetilde{{V}_{K}^{B}}(K,P)}-\psi \frac{H\left({P}^{B}\right)\widetilde{{V}_{P}^{A}}\left(K,P,d\right)+\widetilde{{V}_{K}^{B}}(K,P)Z{^{\prime}}({P}^{B}){Y}^{B}({K}^{B},\tau )}{\widetilde{{V}_{K}^{B}}(K,P)}.$$

(99)

where the formula for $\widetilde{{V}_{K}^{B}}\left(K,P\right)$ is given by Eq. (98).