Unilateral Climate Policies: Incentives and Effects

Abstract

We analyze the effect of climate policies using a two-region partial equilibrium model of resource extraction. The regions are heterogeneous in various aspects, such as in their climate policies and resource extraction costs. We obtain analytical and numerical conditions for a Green Paradox to occur as a consequence of a unilateral increase in carbon taxation and backstop subsidy. In order to assess the welfare and climate consequences of unilateral policy changes, we calibrate the model to real world parameter values. We find that forming a ‘climate coalition’ and introducing carbon taxation even in the absence of real climate concerns is the best course of action for the largest fossil fuel-using regions.

Appendices

Appendix 1: Analysis

1.1 Carbon Taxation

In order to obtain analytical results, we make some simplifying assumptions. In particular, we work with a demand function $D(p_i)$, satisfying $U^\prime (D(p_i))=p_i$, $i=y,x$, with $p_i$ being the consumer price of the resource or the substitute. Note that the demand functions $D_i$ and $D_i^*$ , $i=x,y$ can vary between the two regions. We also suppose that $r=\rho$, i.e. the world market interest rate equals the rate of time preference as in Sect. 2.3.

With the possibility of different taxes $\tau _y, \tau _x, \tau _y^*$ and $\tau _x^*$ the first-best equilibrium is characterized by six equations, comprising two stock conditions and four transition equations.

$$\begin{aligned}&\int _0^{T_1}D^*_y(\lambda ^*_ye^{rt}+c^*_y+\tau _y^*)dt+\int _0^{T_ 1}D_y(\lambda _y^*e^{rt}+c^*_y+\tau _y)dt = Y_0^*, \end{aligned}$$

(34)

$$\begin{aligned}&\lambda _ye^{rT_1}+c_y^*+\tau _y^*=\lambda _xe^{rT_1}+c_x+\tau _x^*, \end{aligned}$$

(35)

$$\begin{aligned}&\lambda _y^*e^{rT_1}+c_y^*+\tau _y=\lambda _xe^{rT_1}+c_x+\tau _x \end{aligned}$$

(36)

$$\begin{aligned}&\int _{T_1}^{T_2^*}D^*_x(\lambda _xe^{rt}+c_x+\tau _x^*)dt+ \int _{T_1}^{T_2}D_x(\lambda _xe^{rt}+c_x+\tau _x)dt = X_0, \end{aligned}$$

(37)

$$\begin{aligned}&\lambda _xe^{rT_2^*}+c_x+\tau ^*_x=b^*-\sigma ^*, \end{aligned}$$

(38)

$$\begin{aligned}&\lambda _xe^{rT_2}+c_x+\tau _x=b-\sigma , \end{aligned}$$

(39)

Equations (35) and (36) are equivalent; one of them can be thus dropped. Following Hoel (2011), we differentiate the equation system (34) to (39) which gives:

$$\begin{aligned} M \left( \begin{array}{c} d T_1 \\ d \lambda ^*_y \\ d T_2^* \\ d T_2 \\ d \lambda _x \\ \end{array}\right)&= \left( \begin{array}{c} -B_y^* \\ -1 \\ 0 \\ 0 \\ 0 \\ \end{array} \right) d\tau _y^* + \left( \begin{array}{c} -B_y \\ -1 \\ 0 \\ 0 \\ 0 \\ \end{array} \right) d \tau _y + \left( \begin{array}{c} 0 \\ 1 \\ -B_x^* \\ -1 \\ 0 \\ \end{array} \right) d\tau _x^* + \left( \begin{array}{c} 0 \\ 1 \\ -B_x \\ 0 \\ -1 \\ \end{array} \right) d \tau _x \nonumber \\&\quad + \left( \begin{array}{c} 0 \\ 0 \\ 0 \\ -1 \\ 0 \\ \end{array} \right) d\sigma ^* +\left( \begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ -1 \\ \end{array} \right) d\sigma , \end{aligned}$$

(40)

where

$$\begin{aligned} B_y^*=\int _0^{T_1}D_y^{*\prime }\left( \lambda _y^*e^{rt}+c^*_y+\tau _y^*\right) dt, \quad B_y=\int _0^{T_1}D_y^\prime \left( \lambda _y^*e^{rt}+c^*_y+\tau _y\right) dt \end{aligned}$$

and

$$\begin{aligned} B_x^*=\int _{T_1}^{T^*_2}D_x^{*\prime }\left( \lambda _xe^{rt}+c_x+\tau _x^*\right) dt, \quad B_x=\int _{T_1}^{T_2}D_x^\prime \left( \lambda _xe^{rt}+c_x+\tau _x\right) dt, \end{aligned}$$

and:

(41)

Adjusting this general framework to our baseline case of no taxes and no subsidy in the foreign country ($\tau _{x}^{*}=\tau _{y}^{*}=\sigma ^*=0)$, a uniform tax in the home country ($\tau _{x}=\tau _{y}=\tau >0)$ and a positive subsidy ($\sigma >0)$ we get

$$\begin{aligned}&\int _{0}^{T_{1}}D^{*}(\lambda _{y}^{*}e^{rt}+c_{y}^{*})dt+\int _{0}^{T_{1}}D(\lambda _{y}^{*}e^{rt}+c_{y}^{*}+\tau _{y})dt =Y_{0}^{*}, \end{aligned}$$

(42)

$$\begin{aligned}&\text {(total demand for cheap oil equals supply)}\nonumber \\&\lambda _{y}^{*}e^{rT_{1}}+c_{y}^{*} =\lambda _{x}e^{rT_{1}}+c_{x},\end{aligned}$$

(43)

$$\begin{aligned}&\text {(continuous world market price at transition)}\nonumber \\&\int _{T_{1}}^{T_{2}^{*}}D^{*}(\lambda _{x}e^{rt}+c_{x})dt+\int _{T_{1}}^{T_{2}}D(\lambda _{x}e^{rt}+c_{x}+\tau )dt =X_{0},\end{aligned}$$

(44)

$$\begin{aligned}&\text {(total demand for expensive oil equals supply)}\nonumber \\&\lambda _{x}e^{rT_{2}^{*}}+c_{x} =b^{*}, \end{aligned}$$

(45)

$$\begin{aligned}&\text {(continuous consumer price at transition in foreign)}\nonumber \\&\lambda _{x}e^{rT_{2}}+c_{x}+\tau =b-\sigma ,\\&\text {(continuous world market price at transition in home)}\nonumber \end{aligned}$$

(46)

We have then 5 equations and 5 unknowns ($\lambda _{y}^{*},\lambda _{x}$, $T_{1},T_{2}^{*},T_{2}).$ They all depend on the tax $\tau$. From these Eqs. (42) to (46) we derive the direction of change of our Green Paradox variables of interest.

The effect of an increased tax $\tau$ on the resource rents is:

$$\begin{aligned} \frac{d\lambda ^*_y}{d\tau }&=\frac{-B_x-B_y+\frac{D_x+D_x^*}{\lambda _x} (\lambda _y^*-\lambda _x)\frac{-B_y}{D_y+D_y^*}+\frac{D_x}{r\lambda _xe^{rT_2}} +Q(\lambda ^*_y-\lambda _x)r\frac{B_y}{D_y+D_y^*}}{A+\frac{D_x^*+D_x}{ \lambda _x}\left( (\lambda ^*_y-\lambda _x)\frac{A}{D_y+D_y^*} +1\right) +Q\left( (\lambda ^*_y-\lambda _x)r\frac{-A}{D_y+D_y^*}+1\right) } \end{aligned}$$

(47)

$$\begin{aligned} \frac{d\lambda _x}{d\tau }&=\frac{-B_x+B_y\frac{D_y+D_y^*}{L}+\frac{D_x}{ \lambda _xe^{rT_2}}}{-(D_y+D_y^*)\frac{A}{L}-\frac{D_x+D_x^*}{\lambda _x}+Q} \end{aligned}$$

(48)

where:

$$\begin{aligned} B_y&= \int _{0}^{T_1}D_y^\prime (\lambda ^*_y e^{rt}\!+\!c^*_y\!+\!\tau ) \ dt\!<\!0\, (\text {derivative of the demand function with respect to } \tau ),\\ B_x&= \int _{T_1}^{T_2}D_x^\prime (\lambda _x e^{rt}\!+\!c_x\!+\!\tau ) \ dt\!<\!0\, (\text {derivative of the demand function with respect to }\tau ),\\ A&= \int _{0}^{T_1}e^{rt} D_y^\prime (\lambda ^*_ye^{rt}+c^*_y+\tau ) \ dt + \int _{0}^{T_1^*}e^{rt} D_y^{*\prime } (\lambda ^*_y e^{rt}+c^*_y+\tau ^*) \ dt<0 \\&(\text {derivative of the demand function with respect to }\lambda ^*_y),\\ Q&= \int _{T_1}^{T_2}e^{rt} D_x^\prime (\lambda _xe^{rt}+c_x+\tau ) \ dt + \int _{T_1^*}^{T_2^*}e^{rt} D_x^{*\prime } (\lambda _xe^{rt}+c_x+\tau ^*) \ dt<0 \\&(\text {derivative of the demand function with respect to }\lambda _x),\\ L&= A(\lambda ^*_y-\lambda _x)r-(D_y+D_y^*)<0, \end{aligned}$$

and abbreviating $D_y(\lambda ^*_y e^{rT_1}+ c^*_y+ \tau )$ with $D_y$, $D_y^*(\lambda ^*_y e^{rT_1^*}+c_y^*+\tau ^*)$ with $D_y^*$, $D_x(\lambda _x e^{rT_2}+c_x+ \tau )$ with $D_x$ and $D_x(\lambda _x e^{rT_x^*}+ c_x+\tau ^*)$ with $D_x^*$.

Thus, both $\frac{d\lambda ^*_y}{d\tau }<0$ and $\frac{d\lambda _x}{d\tau }<0$.

The effect of an increase in $\tau$ on the first switching point $T_1$ equals:

$$\begin{aligned} \frac{dT_1}{d\tau }= \frac{-B_x-\frac{D_x+D_x^*}{r\lambda _x}\frac{B_y}{A}+ \frac{D_x}{r\lambda _xe^{rT_2}}+\frac{QB_y}{A}}{-(D_y+D_y^*)-\frac{D_x+D_x^* }{r\lambda _x}\left( (\lambda ^*_y-\lambda _x)r-\frac{D_y+D_y^*}{A} \right) +Q\left( \lambda ^*_y-\lambda _x)r-\frac{D_y+D_y^*}{A}\right) }, \end{aligned}$$

(49)

where:

$$\begin{aligned} M=D_y+D_y^*, \end{aligned}$$

Whereas the denominator is unambigously negative, the nominator is also negative but for the first and second terms $\left( -B_x+\frac{D_x}{ r\lambda _x e^{rT_2}}\right) >0$. Thus, in order to have $\frac{dT_1}{d \tau }<0$ , it needs to hold that:

$$\begin{aligned} \left| -B_x+\frac{D_x}{e^{rT_2}r\lambda _x}\right| >\left| -\frac{(D_x+D_x^*)B_y }{Ar\lambda _x}+\frac{QB_y}{A}\right| . \end{aligned}$$

(50)

The effect of an increase in $\tau$ on the second switching point of the home region is:

$$\begin{aligned} \frac{dT_2}{d\tau }=\frac{-B_x+(D_y+D_y^*)\left( \frac{-B_y+e^{-rT_2}A}{L} \right) -\frac{D_x^*}{e^{rT_2}}+\frac{Q}{e^{rT_2}}}{(D_y+D_y^*)\frac{ r\lambda _xA}{L}+D_x+D_x^*-Qr\lambda _x}, \end{aligned}$$

(51)

with

$$\begin{aligned} L=D_y+D_y^*-r(\lambda ^*_y-\lambda _x)A. \end{aligned}$$

Whereas the denominator is unambigously positive, the sign of the nominator depends on the parameter values.

For $\frac{dT_2}{d\tau }>0$ it must hold that:

$$\begin{aligned} \left| -B_x+(D_y+D_y^*)\left( \frac{-B_y}{L}\right) \right| >\left| -(D_y+D_y) \left( \frac{A}{Le^{rT_2}}\right) -\frac{D_x^*}{e^{rT_2}}+\frac{Q}{e^{rT_2}} \right| \end{aligned}$$

The effect of an increase in $\tau$ on the second switching point of the foreign region is:

$$\begin{aligned} \frac{dT_2^*}{d\tau }=\frac{-B_x-B_y\frac{D_y+D_y^*}{X}+\frac{D_y^*}{ r\lambda _x e^{rT_2}}}{-Ar\lambda _x \frac{D_y+D_y^*}{X}+D_x+D_x^*-Qr\lambda _x }>0, \end{aligned}$$

(52)

with:

$$\begin{aligned} X=D_y+D_y^*-Ar(\lambda ^*_y-\lambda _x). \end{aligned}$$

This expression is unambigously positive.

1.2 Proofs for Propositions 1 and 2

Proof of Proposition 1

The home country increases its subsidy $\sigma$. Suppose then that the scarcity rent of the low cost resource would increase, i.e. $\frac{d\lambda _y^*}{d \sigma }>0$. Then the first switching point $T_1$ occurs later and the low resource cost extraction period is prolonged because demand decreases in the intervall $[0, T_1)$. It holds that $(\lambda _y^*-\lambda _x)e^{rT_1}=c_x-c_y^*+\tau$. Hence, if $T_1$ increases and $\lambda _y$ increases, then also $\lambda _x$ must increase. As a consequence, both $T_2$ and $T_2^*$ have to decrease, since at the transition to the backstop it has to hold that:

$$\begin{aligned} \lambda _x e^{rT_2}&=b-\sigma -c_x-\tau , \end{aligned}$$

(53)

$$\begin{aligned} \lambda _x e^{rT_2^*}&= b^*-c_x. \end{aligned}$$

(54)

Yet, this is impossible since then there is not enough demand for the high cost resource. Hence, both $T_1$ and $\lambda _y^*$ have to decrease. Moreover $(\lambda _y^*-\lambda _x)$ has to increase, hence, also $\lambda _x$ decreases. From (54) we know that $T_2^*$ has to increase then. The increased demand for the high cost resource from the part of the foreign region is matched by less demand in the home region. The home region’s non-renewable resource usage period is shortened and $T_2$ goes down. $\square$

Proof of Proposition 2

The home country increases its tax $\tau$. Suppose then that the scarcity rent $\lambda _x$ of the high cost resource increases. Then both $T_2^*$ and $T_2$ would fall as the backstop becomes competitive earlier.

It holds that $(\lambda _y^*-\lambda _x)e^{rT_1}=c_x-c_y^*+\tau$. If $\lambda _y^*$ would decrease then $T_1$ goes up. An increase in $T_1$ while $T_2^*$ and $T_2$ decrease is impossible because there is not enough demand for the expensive resource.

An increase in $\lambda _y^*$ can have two effects: the term $(\lambda _y^*-\lambda _x)$ can decrease or increase if the increase in $\lambda _y^*$ is larger than the rise in $\lambda _x$. In the latter case $T_1$ increases; yet, this is not possible for the reasons described above. In the former case $T_1$ needs to decrease. Due to the shorter low cost resource usage period and the higher price there is not enough demand for the low cost resource. An equilibrium is hence impossible.

If $\lambda _x$ now decreases in response to a rise in $\tau$, $T_2^*$ has to increase due to (54). Also $\lambda _y^*$ needs to decrease, otherwise there was not enough demand for the low cost resource (due to a drop in $T_1$).

For an equilibrium to exist, both scarcity rents have to decrease in response to a tax increase. However, $T_1$ can increase or decrease: in case the high cost resource rent decreases more than the low cost rent, i.e. $\frac{d\lambda _x}{d\tau }>\frac{d\lambda _y^*}{d\tau }$, $T_1$ goes down. $\square$

1.3 Change in $T_1$ in Response to a Tax Increase

1.3.1 Analytical Example with Linear Demand

In this appendix we demonstrate the ambiguity of the sign in the change of the date of exhaustion of the cheap resource following an increase in the tax rate. To that end we assume linear demand, identical across the two countries:

$$\begin{aligned} y_c(p)&=x_c(p)=\alpha -\beta p,\\ y_c^*(p^*)&=x_c^*(p^*)=\alpha -\beta p^*, \end{aligned}$$

with $\alpha >b^*>b$. In an equilibrium we must have that:

$$\begin{aligned}&\lambda _x e^{rT_1}+c_x=\lambda _y^* e^{rT_1}+c_y^*,\end{aligned}$$

(55)

$$\begin{aligned}&\lambda _x e^{rT_2}=b-c_x-\tau ,\end{aligned}$$

(56)

$$\begin{aligned}&\lambda _x e^{rT_2^*}=b^*-c_x,\end{aligned}$$

(57)

$$\begin{aligned}&\int _0^{T_1}\left( y_c(t)+y_c^*(t)\right) dt=Y_0^*,\end{aligned}$$

(58)

$$\begin{aligned}&\int _{T_1}^{T_2}x_c(t) dt+\int _{T_1}^{T_2^*}x_c^*(t) dt=X_0. \end{aligned}$$

(59)

Let us, for the sake of notation, define:

$$\begin{aligned} v=rT_2-rT_1, \quad w=rT_2^*-rT_1, \quad z=rT_1. \end{aligned}$$

Then, inserting the linear demand functions into Eqs. (58) and (59) and solving the integrals, (55) to (59) boil down to:

$$\begin{aligned}&(2\alpha -2\beta c_y^*-\beta \tau )z-2\beta \left\{ (b-c_x-\tau )e^{-v}+ c_x-c_y^*\right\} (1-e^{-z})=rY_0,\end{aligned}$$

(60)

$$\begin{aligned}&\qquad (\alpha -\beta c_x-\beta \tau )v-\beta (b-c_x-\tau )(1-e^{-v})+ (\alpha -\beta c_x)w\nonumber \\&\qquad -\beta (b^*-c_x)(1-e^{-w})=rX_0,\end{aligned}$$

(61)

$$\begin{aligned}&\qquad (b-c_x-\tau )e^{-v}=(b^*-c_x)e^{-w}. \end{aligned}$$

(62)

For a given $\tau$, the unknowns in this system of three equations are $v,w$ and $z$.

Next, we compare the outcome under a no-tax regime with a tax regime where the home country never uses its own (expensive) resource. For $\tau =0$ we find $\tilde{v}$ and $\tilde{w}$ from (61) and (62) by inserting $\tau =0$. Then $\tilde{z}$ follows from (60) upon insertion of $\tilde{v}$.

In the second regime we impose $\hat{v}=0$. Then $\hat{w}$ follows from (61), with $v=\hat{v}=0$. The corresponding tax $\hat{\tau }$ follows from (62). Finally, $\hat{z}$ is found from (60) with $\hat{\tau }$ and $\hat{v}=0$ inserted. We now need to compare $\tilde{z}$ and $\hat{z}$. In the zero tax regime we have, using (62):

$$\begin{aligned} (2\alpha -2\beta c_y^*)\tilde{z}-2\beta \left\{ (b^*-c_x)e^{-\tilde{w}}+c_x-c_y^*\right\} (1- e^{-\tilde{z}})=rY_0. \end{aligned}$$

(63)

In the tax regime we have, using (62):

$$\begin{aligned} (2\alpha -2\beta c_y^*-\beta \hat{\tau })\hat{z}-2\beta \left\{ (b^*-c_x)e^{- \hat{w}}+c_x-c_y^*\right\} (1-e^{-\hat{z}})=rY_0. \end{aligned}$$

(64)

Evidently, it holds that:

$$\begin{aligned} 2\alpha -2\beta c_y^*>2\alpha -2\beta c_y^*-\beta \hat{\tau }. \end{aligned}$$

It also holds that:

$$\begin{aligned} (b^*-c_x)e^{-\tilde{w}}+c_x-c_y^*>(b^*-c_x)e^{-\hat{w}}+c_x-c_y^*, \end{aligned}$$

because $\hat{\tau }>0$ and $\hat{w}>\tilde{w}$. Moreover, $\hat{\tau }, \hat{w}$ and $\tilde{w}$ are determined by (61) and (62) and do not depend on $Y_0$. Note also that $1-e^{-z}$ is increasing in $z$. Hence, $\hat{z}<\tilde{z}$ is only possible for small $Y_0^*$, which results in small $z$ and hence makes $(1-e^{-{z}})$ small and the right hand side of (63) and (64) less important.

For high $Y_0$ is must be the case that $\hat{z}>\tilde{z}$. Hence, for high $Y_0$, $T_1$ increases with a higher tax. However, for low $Y_0$ the effect may be reversed, due to (64).

The intuition is straightforward: The high cost producers are faced with an immense drop in demand as the home country basically ceases to demand the expensive resource due to $\hat{\tau }$. They have to adjust their scarcity resource more than the low cost resource producers who still face demand from both regions, even though the home region’s demand is reduced. Hence, $\frac{d\lambda _x}{d\tau }>\frac{d\lambda _y^*}{d\tau }$ and the switching point $T_1$ occurs earlier.

1.3.2 Numerical Example with Linear Demand

We illustrate the analytical possibility of a weak Green Paradox as a consequence of a tax increase by a numerical example. For the linear demand case presented in the previous For the linear case presented in the previous subsection Analytical Example with Linear Demand in Appendix 1, we use the parameter values displayed in Table 1.

Table 1 Parameter values for the numerical exercise for $\frac{dT_1}{d \tau }<0$

For the first scenario of the subsection Analytical Example with Linear Demand in Appendix 1, when $\tau =0$, the resulting switching points amount to $T_1=0.01126$, $T_2=269.68938$ and $T_2^*=470.40049$ (i.e. $\tilde{w}=0.470389$, $\tilde{v}=0.269678$ and $\tilde{z}=0.00001126$ ).

Setting $v=0$ results in a positive $\hat{\tau }=16.19972$. Whereas the foreign region switches now much later, namely at $T_2^*=647.11838$, the first switching point occurs much earlier at $T_1=0.004894$ (i.e. $\hat{w}=0.647113486$ and $\hat{z}=0.000004894$).

A Green Paradox is likely to occur if $Y_0^*$ is low and $X_0$ relatively high. Furthermore, the world interest rate has to be relatively low, since it lowers $z$ and hence the importance of the term $(1-e^{-z})$ of Eqs. (63) and (64). Also the parameters of the linear demand function play a role: a Green Paradox is likely to occur if $\alpha$ is relatively small, but $\beta$ relatively large. Extraction costs, as long as they are not equal to zero, do not seem to play a major role: for $c_y^*=0.5$ and $c_x=1$, the switching points do not change much, only $\hat{\tau }$ decreases significantly.

1.3.3 Climate Damage and Social Cost of Carbon

We define $h(t)=\psi (x_c+x_c^*)+\psi ^*(y_c+y_c^*)$. The total damage from atmospheric $\hbox {CO}_2$ in both regions and constant marginal damages $\nu$ equals:

$$\begin{aligned}&\int _0^\infty e^{-\rho t}2 \nu (S_1(s)+S_2(s)) ds\end{aligned}$$

(65)

$$\begin{aligned}&\quad =\int _0^\infty 2\nu (S_1(s)+S_2(s))d \left( -\frac{1}{\rho }e^{-\rho s}\right) \end{aligned}$$

(66)

$$\begin{aligned}&\quad =\frac{2\nu }{-\rho }(S_1(t)+S_2(t))e^{-\rho t}\left| _0^\infty \right. +\int _0^\infty \frac{2\nu }{\rho }e^{-\rho s}\left( \dot{S}_1(s)+\dot{S}_2(t)\right) \end{aligned}$$

(67)

$$\begin{aligned}&\quad =\frac{2\nu }{\rho }(S_1(0)+S_2(0)) +\int _0^\infty \frac{2\nu }{\rho } e^{-\rho s}\left( \alpha h(s)+(1-\alpha )h(s)-\delta S_2(s)\right) \end{aligned}$$

(68)

$$\begin{aligned}&\quad =\frac{2\nu }{\rho }(S_{10}+S_{20}) +\int _0^\infty \frac{2\nu }{\rho }e^{-\rho s} h(s) ds-\int _0^\infty \frac{2\nu \delta }{\rho }e^{-\rho s}S_2(s) ds. \end{aligned}$$

(69)

We need to solve for the last term in Eq. (69):

$$\begin{aligned} \int _0^\infty e^{-\rho s}S_2(s) ds&=\int _0^\infty S_2(s)d\left( -\frac{1}{\rho }e^{-\rho s}\right) \end{aligned}$$

(70)

$$\begin{aligned}&=-\frac{1}{\rho }S_2(s)e^{-\rho s}\left| _0^\infty \right. +\int _0^\infty \frac{1}{\rho }\dot{S}_s(s)e^{-\rho s}ds\end{aligned}$$

(71)

$$\begin{aligned}&=\frac{1}{\rho }S_2(0)+\int _0^\infty \frac{1}{\rho }e^{-\rho s}((1-\alpha )h(s)-\delta S_2(s)) ds\end{aligned}$$

(72)

$$\begin{aligned}&=\frac{1}{\rho }S_{20}+\int _0^\infty \frac{1}{\rho }e^{-\rho s}(1-\alpha )h(s) ds-\frac{\delta }{\rho }\int _0^\infty e^{-\rho s}S_2(s) ds. \end{aligned}$$

(73)

Bringing the right term of Eq. (73) on the left of Eq. (71), we solve for $\int _0^\infty e^{-\rho s}S_2(s) ds$:

$$\begin{aligned} \left( 1+\frac{\delta }{\rho }\right) \int _0^\infty e^{-\rho s}S_2(s)ds&=\frac{1}{\rho }S_2(0)+\frac{1}{\rho }\int _0^\infty e^{-\rho s}(1-\alpha )h(s)ds,\end{aligned}$$

(74)

$$\begin{aligned} \int _0^\infty e^{-\rho s}S_2(s)ds&=\frac{1}{\rho + \delta }S_2(0)+\frac{1}{\rho + \delta }\int _0^\infty e^{-\rho s}(1-\alpha )h(s)ds. \end{aligned}$$

(75)

Inserting this into (69), total damages equal:

$$\begin{aligned}&\frac{2\nu }{\rho }(S_{10}+S_{20})+\frac{2\nu }{\rho }\int _0^\infty e^{-\rho s}h(s)ds\nonumber \\ {}&\qquad \ -\frac{2\nu \delta }{\rho }\left\{ \frac{1}{\rho + \delta }S_2(0)+\frac{1}{\rho + \delta }\int _0^\infty e^{-\rho s}(1-\alpha )h(s)ds.\right\} \end{aligned}$$

(76)

$$\begin{aligned}&\quad =\frac{2\nu }{\rho }(S_{10}+S_{20})-\frac{2\nu \delta }{\rho }\frac{1}{\rho +\delta }S_{20}+\left( \int _0^\infty e^{-\rho s}h(s)\right) \left\{ \frac{2\nu }{\rho }-\frac{2\nu \delta }{\rho (\delta +\rho )}(1-\alpha )\right\} \end{aligned}$$

(77)

Disregarding the given initial carbon stock in the atmosphere, the first two terms of Eq. (77) disappear. Tables 2 and 3 summarize the parameter values used in the numerical exercises and the calibration exercise respectively.

$$\begin{aligned} \left\{ 2\nu \left( \frac{\alpha }{\rho }+\frac{(1-\alpha )}{\rho + \delta }\right) \right\} \int _0^\infty e^{-\rho s}h(s)ds. \end{aligned}$$

(78)

Table 2 Parameter values for the numerical exercises

Table 3 Parameter values for the calibration exercise

Appendix 2: Numerical Analysis and Calibration

We assume the home (foreign) region’s utility function to be of a standard CRRA form:

$$\begin{aligned} A\frac{(x_c+y_c+z_c)^{1-\gamma }-1}{1-\gamma },\quad \left( A^*\frac{(x_c^*+y_c^*+z_c^*)^{1-\gamma ^*}-1}{1-\gamma ^*}\right) , \end{aligned}$$

(79)

where $A (A^*)$ is a scaling parameter. From the first order condition we know that the home (foreign) region’s demand equals (with $i=y,x$) (Table 2):

$$\begin{aligned} D_i(p_i)=\left( \frac{A}{p_i+\tau }\right) ^{\frac{1}{\gamma }},\quad \left( D_i^*(p_i)=\left( \frac{A^*}{p_i+\tau ^*}\right) ^{\frac{1}{\gamma ^*} }\right) \end{aligned}$$

As noted above, we use logarithmic utility in most of the numerical exercises and set $\gamma =\gamma ^*=1$. Yet, the calibration requires us to use the flexibilities provided by the functional forms we assumed. Table 3 summarizes the parameter values used.