Optimum Growth and Carbon Policies with Lags in the Climate System

Abstract

We study the optimal carbon tax in an economy in which climate change, stemming from polluting non-renewable resource, affects the economy’s growth potential. Our main contribution is to introduce and explore the natural time lag of the climate system between emissions and damages to capital accumulation in an endogenous growth setting. This allows us to investigate how optimal climate policy, and its interplay with climate dynamics, affect long-run growth and the transition of the economy towards it. Without pollution decay, a higher speed of emissions diffusion steepens the growth profile of the economy. With pollution decay, this leads to lower short-run but higher long-run economic growth during transition. Poor understanding of the emissions diffusion process leads to suboptimal carbon taxes, resource extraction and growth.

Appendices

Appendix 1: Time Lags in the Climate System

In this part of the Appendix we present the mathematic modeling of distributed time lags in the climate system and its properties. For the analysis we rely largely on MacDonald (1978). Take first the case of instantaneous diffusion of emissions. In this case the usual assumption is that the use of a pollutant, $R_{t}$, increases the harmful stock of emissions $P_{t}$ at a rate

$$\begin{aligned} \dot{P}_{t}=\phi R_{t}+\theta \left( \bar{P}-P_{t}\right) ,\qquad P_{0}\ge 0 \text { given}, \end{aligned}$$

(25)

with $\phi >0$ representing the carbon intensity of the polluting energy resource, $\theta \ge 0$ the carbon decay parameter, and $\bar{P} \in (0,P_0)$ the pre-industrial level of carbon concentration in the atmosphere. Now let’s introduce a distributed lag in the model in order to relax the usual assumption of instantaneous pollution accumulation and let this process depend on the history of resource use

$$\begin{aligned} \dot{P}_{t}=\int _{-\infty }^{t}G_{t-s}\phi R_{s}ds+\theta \left( \bar{P}-P_{t}\right) ,\qquad P_{0}\ge 0\text { given}. \end{aligned}$$

(26)

With this formulation (25) becomes an integro-differential equation. The function $G_{x}$ represents the memory of the system (or the delaying function) with $\int _{0}^{\infty }G_{x}dx=1$. Function $G_{x}$ could be also interpreted as the probability density function of the inherent time lag of the particular system so that the mean time lag $\bar{T}$ for a given memory function would read $\bar{T}=\int _{0}^{\infty }xG_{x}dx$ . With a special choice of the memory function one can replace (26) with a set of linear differential equations. For this purpose it is a standard approach to exploit the properties of the exponential functions by using the exponential distribution

$$\begin{aligned} G_{x}=\kappa e^{-\kappa x},\qquad \kappa >0. \end{aligned}$$

(27)

The parameter $\kappa$ measures the speed of emissions diffusion, or speed of adjustment, and is the reciprocal of the mean time lag $\bar{T}$ from the same memory function, i.e. $\kappa =\bar{T}^{-1}$. We can then define the lagged history of carbon emissions as $Z_{t}\equiv \int _{-\infty }^{t}G_{t-s} \phi R_{s}ds$, and by using the Leibniz rule of integration to get the familiar equivalent system of differential equations, with $P_0$ and $Z_0$ given:

$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{P}_{t}=Z_{t}+\theta \left( \bar{P}-P_{t}\right) ,\\ \dot{Z}_{t}=\kappa \left( \phi R_{t}-Z_{t}\right) . \end{array}\right. } \end{aligned}$$

(28)

It can be proven that the corresponding system is globally stable for the relevant range of the parameters $\kappa ,\phi , \theta >0$ and $\bar{P} >0$. Since in general the initial value of $Z_{t}$ cannot be defined, $Z_{0}$ is chosen such that $lim_{\kappa \rightarrow \infty }P_{t=0}=P_{0}$, as expected, i.e. $Z_{0}=0$. The solution for the climate system given the rate of resource extraction for each time period, $R_t$, now reads

$$\begin{aligned} P_{t}=P_{0}+\left( \bar{P}-P_0\right) \left( 1-e^{-\theta t}\right) +\int _{0}^{t}\kappa \frac{e^{-\theta (t-s)}-e^{-\kappa (t-s)}}{ \kappa -\theta }\phi R_{s}ds. \end{aligned}$$

(29)

The limiting cases for $\kappa \rightarrow 0$ and $\kappa \rightarrow \infty$ follow readily from the last equation. From (29), the marginal increase in the stock of carbon in period $\nu$ from a marginal unit of emissions in period t reads:

$$\begin{aligned} \frac{dP_{\nu }}{d\left( \phi R_{t}\right) }\equiv f_{\nu t}=\kappa \frac{e^{-\theta (\nu -t)}-e^{-\kappa (\nu -t)}}{\kappa -\theta }>0,\qquad \text { for all }\nu \ge t. \end{aligned}$$

(30)

Appendix 2: Social Optimum

The social planner chooses the share $\epsilon _{t}$, and resource extraction $R_{t}$ in order to maximize lifetime utility $\int _{0}^{\infty }U_{t}e^{-\rho t}dt$, with $U_{t}=\log (C_{t})$ if $\sigma =1$ and $U_{t}=\frac{C_{t}^{1-\sigma }}{1-\sigma }$ otherwise, subject to Eqs. (3)–(6). The Hamiltonian of the social planner reads

$$\begin{aligned} H_{t}=&\frac{C_{t}^{1-\sigma }}{1-\sigma } +\lambda _{Ct}\left[ A\left( \epsilon _{t}K_{t}\right) ^{\alpha }R_t^{1-\alpha }-C_{t}\right] +\lambda _{Kt}K_{t}\left[ B\left( 1-\epsilon _{t}\right) -D\left( P_{t}\right) \right] \\&-\lambda _{St}R_t +\lambda _{Pt}\left[ Z_{t}+\theta \left( \bar{P}-P_{t}\right) \right] +\lambda _{Zt}\kappa \left[ \phi R_t-Z_{t}\right] , \end{aligned}$$

with $\lambda _{Ct},\lambda _{Kt},\lambda _{St},\lambda _{Pt},\lambda _{Zt}$, the shadow prices of the consumption good, $C_{t}$, capital stock, $K_{t}$, stock of non-renewable resources, $S_{t}$, stock of pollution, $P_{t}$, and the lagged history of emissions, $Z_{t}$. Assuming an internal solution, the first order conditions w.r.t. the $C_{t},\epsilon _t, R_{t}$, i.e. $\partial H_{t}/\partial (\cdot )=0$ imply

$$\begin{aligned} C_{t}^{-\sigma }&=\lambda _{Ct}, \end{aligned}$$

(31)

$$\begin{aligned} \alpha \frac{C_{t}}{K_{t}}&=\frac{\lambda _{Kt}}{\lambda _{Ct}} B\epsilon _{t}, \end{aligned}$$

(32)

$$\begin{aligned} (1-\alpha )\frac{C_{t}}{R_t}&=\frac{\lambda _{St}}{\lambda _{Ct}}-\phi \kappa \frac{\lambda _{Zt}}{\lambda _{Ct}}. \end{aligned}$$

(33)

Moreover $\partial H_{t}/\partial (\cdot )=\rho q_{t}-\dot{q}_{t}$ for every state variable, $K_{t},S_{t},P_{t},Z_{t}$, with $q_{t}$ its shadow price. This leads to

$$\begin{aligned} \left( \widehat{\lambda _{Kt}K_{t}}\right)&=-\alpha \frac{\lambda _{Ct}C_{t}}{\lambda _{Kt}K_{t}}+\rho , \end{aligned}$$

(34)

$$\begin{aligned} \hat{\lambda }_{St}&=\rho , \end{aligned}$$

(35)

$$\begin{aligned} \hat{\lambda }_{Pt}&=D'(P_t)K\frac{\lambda _{Kt}}{\lambda _{Pt}}+\theta +\rho , \end{aligned}$$

(36)

$$\begin{aligned} \hat{\lambda }_{Zt}&=-\frac{\lambda _{Pt}}{\lambda _{Zt}}+\kappa +\rho , \end{aligned}$$

(37)

Finally, the relevant transversality conditions read

$$\begin{aligned} \lim _{t\rightarrow \infty }\lambda _{St}S_{t}e^{-\rho t}&=0, \end{aligned}$$

(38)

$$\begin{aligned} \lim _{t\rightarrow \infty }\lambda _{Kt}K_{t}e^{-\rho t}&=0, \end{aligned}$$

(39)

$$\begin{aligned} \lim _{t\rightarrow \infty }\lambda _{Pt}P_{t}e^{-\rho t}&=0, \end{aligned}$$

(40)

$$\begin{aligned} \lim _{t\rightarrow \infty }\lambda _{Zt}Z_{t}e^{-\rho t}&=0. \end{aligned}$$

(41)

From (35) we get that $\hat{\lambda } _{St}=\rho$, i.e. the Hotelling rule for the extraction of the non-renewable resource. Equation (32) shows indifference about allocating capital between the two activities: producing the consumption and the investment good. When $\sigma =1$ we can combine (32), with $\lambda _{Ct}C_{t}=1$, from (31), and (34), to get $\dot{\epsilon }_{t}=B\epsilon _{t}^{2}-\rho \epsilon _{t}$. With the use of the transversality condition (39), we get that the capital share jumps immediately to its steady state value, $\bar{\epsilon }=\rho /B$.

Appendix 3: Asymptotic Constancy of Capital Share and Resource Depletion Rate

In this part of the Appendix we derive the asymptotic constancy of the capital share $\epsilon$, and the resource depletion rate $u=R/S$, in the general case of $\sigma \ne 1.$ In the main text we explained that the economy at hand is always in transition while it reaches a balanced growth path at the limit when resources get asymptotically depleted and pollution reaches its steady state value. The transversality condition (39) implies that $\widehat{\lambda _{Kt}K_{t}}-\rho <0$ while (34) with (32) can be rewritten as $\widehat{\lambda _{Kt}K_{t}}=-B\epsilon _{t}+\rho$. We combine these two conditions to get that $\lim _{t\rightarrow \infty }\epsilon _{t}>0$. From (5), asymptotic constancy of $\hat{K}_{t}$ implies $\lim _{t\rightarrow \infty }\hat{\epsilon }_{t}\le 0$. Since $\epsilon _{t}$ is strictly positive, the last inequality implies that $\lim _{t\rightarrow \infty }\hat{\epsilon }_{t}=0$. From the transversality condition for the stock of resources, (38), we get that $\hat{\lambda }_{St}+\hat{S}_{t}-\rho <0$. We substitute $\hat{\lambda }_{St}=\rho$ and $\hat{S}_{t}=-u_{t}$ to get $\lim _{t\rightarrow \infty }u_{t}>0$. We then log-differentiate the production function for the consumption good, (3), with constant $\epsilon$ at the limit. Asymptotic constancy of $\hat{C}_{t}$ then demands that $\lim _{t\rightarrow \infty }\hat{u}_{t}\le 0$. The last two conditions indicate that $\lim _{t\rightarrow \infty }\hat{u}_{t}=0$, i.e. u asymptotically constant and positive, i.e. $g_R$ asymptotically constant and negative. The upper bound $\epsilon _{t},u_{t}<1$ follows from the essentiality of the resource and capital in the production function.

Appendix 4: The Social Cost of Carbon

We defined as $X_t=-\phi \kappa \lambda _{Zt}/\lambda _{Ct}$ the marginal externality damage from burning an additional unit of polluting non-renewable resource. We will now prove that this is equivalent to expression (8). From (29), it follows that $P_{\nu }\ge P_{s}e^{-\theta (\nu -s)}$, for each $\nu \ge s$. We combine the previous inequality with the transversality condition (40) to get that $0=\lim _{\nu \rightarrow \infty }\lambda _{P\nu }P_{\nu }e^{-\rho \nu }\ge \lim _{\nu \rightarrow \infty }\lambda _{P\nu }P_{s}e^{-\theta (\nu -s)}e^{-\rho \nu }$ or that

$$\begin{aligned} \lim _{\nu \rightarrow \infty }\lambda _{P\nu }e^{-\left( \rho +\theta \right) (\nu -s)}=0,\qquad \text {for all}\qquad \nu \ge s. \end{aligned}$$

(42)

Following the same procedure for the transversality condition (41) we get

$$\begin{aligned} \lim _{s \rightarrow \infty }\lambda _{Zs }e^{-(\rho +\kappa )(s -t)}=0,\qquad \text {for all}\qquad s \ge t. \end{aligned}$$

(43)

We multiply Eq. (36) with $e^{-(\rho +\theta )(\nu -s)}$ to get that $\dot{\lambda }_{P\nu }e^{-(\rho +\theta )(\nu -s)}-(\rho +\theta )\lambda _{P\nu }e^{-(\rho +\theta )(\nu -s)}=\eta D'(P_v)\lambda _{K\nu }K_{\nu }e^{-(\rho +\theta )(\nu -s)}$, i.e. $d\left[ \lambda _{\nu }e^{-(\rho +\theta )(\nu -s)}\right] /d\nu =\eta D'(P_v)\lambda _{K\nu }K_{\nu }e^{-(\rho +\theta )(\nu -s)}$. Using (42) we can then calculate the definite integral from $\nu =s$ to $\nu \rightarrow \infty$ as

$$\begin{aligned} -\lambda _{Ps}=\int _{s}^{\infty }\eta D'(P_v) \lambda _{K\nu }K_{\nu }e^{-(\rho +\theta )(\nu -s)}d\nu , \end{aligned}$$

(44)

while the same procedure for (37) gives

$$\begin{aligned} \lambda _{Zt}=\int _{t}^{\infty } \lambda _{P s }e^{-\left( \rho +\kappa \right) (s -t)}ds . \end{aligned}$$

(45)

Substituting $\lambda _{Pt}$ from (44) into (45) and using (31) we get that

$$\begin{aligned} X_t=-\kappa \phi \frac{\lambda _{Zt}}{\lambda _{Ct}}= \frac{\kappa \phi }{\lambda _{Ct}}\int _{t}^{\infty }\left( \int _{s}^{\infty }\eta D'(P_v) \lambda _{K\nu }K_{\nu }e^{-\left( \rho +\theta \right) (\nu -s)}d\nu \right) e^{-\left( \rho +\kappa \right) (s-t)}ds. \end{aligned}$$

(46)

Using (31) and (32) we get expression (8). If $\sigma =1$, $X_t$ rewrites as in (9), all along the optimal path. If $\sigma \ne 1$, on the asymptotic BGP, due to constancy of the $\epsilon$ share, it holds, by combining (34) with (32), that $\lim _{t\rightarrow \infty }\widehat{\lambda _{Ct}C_{t}}=\lim _{t\rightarrow \infty }\widehat{\lambda _{Kt}K_{t}}=-B {\epsilon _\infty }+\rho$, constant. Accordingly, the double integral in (46) gives at the limit $\frac{\chi \eta }{(B \epsilon _\infty +\theta )(B \epsilon _\infty +\kappa )}\lambda _{Kt}K_{t}$, and with (32), (46) can be written as

$$\begin{aligned} \lim _{t\rightarrow \infty }X_t/C_t=\kappa \frac{\alpha \eta \phi \chi }{B \epsilon _\infty \left( B \epsilon _\infty +\theta \right) \left( B \epsilon _\infty +\kappa \right) }. \end{aligned}$$

(47)

In “Appendix 7” we will calculate the steady state value $\epsilon _\infty =\left( B \sigma \right) ^{-1}(\rho +\alpha \left( \sigma -1\right) (B-\eta (\delta +\chi P_\infty ))$ in the general case of $\sigma \ne 1$. Substituting the result in (47) leads to Eq. (23) in the main text.

Appendix 5: Decentralized Equilibrium with Log-Utility

For ease of exposition we define $\psi \equiv p_{Rt}/\left( p_{Rt}+\tau _{t}\right)$, the fraction of the producers’ price in the total price for the non-renewable resource. Log-differentiating this expression gives $\hat{\psi }_{t}=\left( 1-\psi \right) \left( \hat{p}_{Rt}-\hat{\tau }_{t}\right)$, where we define $\hat{V}_{t}=g_{Vt}\equiv \dot{V}_{t}/V_{t}$ the growth rate of variable $V_{t}$. The equations that characterize the decentralized economy can be found by log-differentiating the production function (3) with $Y_{t}=C_{t}$, the FOC (10) for the capital share and resource demand, together with the aggregate version of the Keynes–Ramsey rule, (14), the Hotelling rule, (15), the no-arbitrage condition, (16), and the aggregate capital accumulation, (5). Solving the occuring system in $g_{C},g_{K},g_{R},g_{\epsilon },g_{\psi },g_{pK},g_{pR},r$ in the case of $\sigma =1$ leads to the following dynamic equation for $\epsilon$:

$$\begin{aligned} \dot{\epsilon }_t=B\epsilon _{t}^{2}-\rho \epsilon _{t} \end{aligned}$$

(48)

with a solution given by $\epsilon _{t}=\left( \frac{B}{\rho }\left( 1-e^{\rho t}\right) +\frac{e^{\rho t}}{\epsilon _{0}}\right) ^{-1}$. Combining the demand for capital in (10) and the aggregate version of the transversality condition (17) for capital pins down the initial level of the capital shares $\epsilon _{0}=\rho /B$ . Substituting this back to the solution we get that $\epsilon _{t}=\bar{\epsilon }=\rho /B$ for every t.

Appendix 6: Dynamics of the Optimal Per-unit Tax

To see why any term in the optimal per-unit tax that grows with the interest rate has no effect on the extracting behavior of the economy proceed as follows. Given Assumptions 1 and 2, apply the optimal tax on the FOC for the non-renewable resource, (10), with $C_t=Y_t$ in equilibrium: $R_t^o=\left( 1-\alpha \right) \left( \frac{p_{Rt}}{C_t} +\frac{\tau _t^o}{C_t}\right) ^{-1}$; $R_t^o$ is the optimal path of resource extraction. Substituting $p_{Rt}=p_{R0}e^{\int _0^t r_s ds}$, from (15), and $C_t=C_0 e^{\int _0^t(r_s -\rho )ds}$, from (14), gives $R_t^o=(1-\alpha )\left( \frac{p_{R0}}{C_0}e^{\rho t}+\frac{\tau _t^o}{C_t}\right) ^{-1}$. Now consider a different tax $\tau _t^o+\Delta e^{\int _0^1 r_s ds}$ with $\Delta > -p_{R0}$. It is straightforward to verify that in this case $R_t^o=(1-\alpha )\left( \frac{p_{R0}+\Delta }{C_0}e^{\rho t}+\frac{\tau _t^o}{C_t}\right) ^{-1}$. Since $\tau _t^o/C_t$ is constant, and the resource is essential, we can use the feasibility constraint $\int _0^\infty R_t dt{=}S_0$ to calculate in both cases the same optimal resource extraction path as $R_t^o=(1-\alpha )\left[ \frac{\tau _t^o}{C_t}\left( 1+e^{\rho t}\left( e^{\frac{S_0 \rho }{1-\alpha } \frac{\tau _t^o}{C_t}}-1 \right) ^{-1}\right) \right] ^{-1}$.

Appendix 7: Social Optimum with Non-logarithmic Utility

This part of the Appendix provides the dynamic system used in the simulation for Sect. 6, i.e. we treat the asymptotic balanced growth path and stability in the general case of $\sigma \ne 1.$ It will be convenient to modify the dynamic system of the social planner in variables that converge to a constant at the limit. In “Appendix 3” we proved asymptotic constancy of $\epsilon$, and $u=R/S$. Moreover, we showed that $\tilde{\tau }^o_{t}=\frac{\tau _{t}^{o}}{C_{t}}=-\phi \kappa \frac{\lambda _{Zt}}{\lambda _{Ct}C_{t}}$ reaches also a constant value at the limit. From (44) and (32) with $\lim _{t\rightarrow \infty }\widehat{\lambda _{Kt}K_{t}}=-B \epsilon _\infty +\rho$ same holds for $\tilde{\gamma _{t}}=\frac{\lambda _{Pt}}{\lambda _{Ct}C_{t}}$. Furthermore, we define, as in “Appendix 5”, $\psi _{t}\equiv p_{Rt}/(p_{Rt}+ \tau _{t}^{o})=(\lambda _{St}/\lambda _{Ct})/(\lambda _{St}/\lambda _{Ct}-\kappa \phi \lambda _{Zt}/\lambda _{Ct})$, the fraction of the producers’ price in the total price paid by consumers. It follows from the asymptotic constancy of $\tau _{t}^{o}/C_{t}$, the Hotelling rule, $\hat{p}_{Rt}=r_{t}$, and the Keynes-Ramsey rule, $\sigma \hat{C}_{t}=r_{t}-\rho$, that the term $\tau _{t}^{o}/p_{Rt}$ grows at $-((\sigma -1)r_{t}+\rho )/\sigma <0$, for $\sigma \ge 1$, which we conventionally assume, so that $\lim _{t\rightarrow \infty }\psi _{t}=1$. To get the dynamic system in $\{\epsilon _{t},u_{t},\tilde{\tau } _{t},\tilde{\gamma }_{t},\psi _{t},S_{t},P_{t},Z_{t}\}$ we proceed as follows.¹⁷ By log-differentiating (3) with $u_{t}=R_{t}/S_{t}$, using (5) we get

$$\begin{aligned} \hat{C}_{t}-\alpha \left( \hat{\epsilon }_{t}+B\left( 1-\epsilon _{t}\right) -\eta \left( \delta +\chi P_{t}\right) \right) -(1-\alpha )\left( \hat{u}_{t}-u_{t}\right) =0. \end{aligned}$$

(49)

With our definitions, Eq. (33) can be written as $(1-\alpha )C_{t}/(u_{t}S_{t})=p_{Rt}+\tau _{t}^{o}$. We log-differentiate this expression with $\hat{p}_{Rt}=\hat{\lambda }_{St}-\hat{\lambda }_{Ct}=\rho +\sigma \hat{C}_{t}$ from (31), the Hotelling rule $\hat{\lambda }_S=\rho$, and $\tilde{\tau }_{t}=\tau _{t}^{o}/C_{t}$, to get

$$\begin{aligned} \hat{u}_{t}-u_{t}+\psi _{t}\left( \rho +\left( \sigma -1\right) \hat{C}_{t}\right) +\left( 1-\psi _{t}\right) \hat{\tilde{\tau }}_{t}=0. \end{aligned}$$

(50)

Furthermore, we log-differentiate the definition for $\psi _{t}$ which gives

$$\begin{aligned} \hat{\psi }_{t}-\left( 1-\psi _{t}\right) \left( \rho +(\sigma -1)\hat{C}_{t}-\hat{\tilde{\tau }}_{t}\right) =0. \end{aligned}$$

(51)

From (31), (32) and (34) we get

$$\begin{aligned} \hat{\epsilon }_{t}+(\sigma -1)\hat{C}_{t}+\rho -\epsilon _{t}B=0. \end{aligned}$$

(52)

Finally, we substitute the growth rate of the shadow prices in the log-differentiated version of the definitions for $\tilde{\tau }_{t}=\frac{\tau _{t}^{o}}{C_{t}}=-\kappa \phi \frac{\lambda _{Zt}}{\lambda _{Ct}C_{t}}$ and $\tilde{\gamma _{t}}=\frac{\lambda _{Pt}}{\lambda _{Ct}C_{t} }$ from (31), (36), and (37) to get

$$\begin{aligned}&\hat{\tilde{\tau }}_{t}-(\sigma -1)\hat{C}_{t}-\kappa \left( 1+\phi \frac{\tilde{ \gamma }_{t}}{\tilde{\tau }_{t}}\right) -\rho =0, \end{aligned}$$

(53)

$$\begin{aligned}&\hat{\tilde{\gamma }}_{t}-(\sigma -1)\hat{C}_{t}-\frac{\alpha \eta \chi }{B\epsilon _{t}\ \tilde{\gamma }_{t}}-\theta -\rho =0. \end{aligned}$$

(54)

We then combine Eqs. (49)–(54) to get the relevant dynamic system in $\{\epsilon _{t},u_{t},\tilde{\tau }_{t},\tilde{\gamma }_{t},\psi _{t},S_{t},P_{t},Z_{t}\}$, as

$$\begin{aligned} \dot{\epsilon }_{t}&=\epsilon _{t}\left( \kappa \frac{(\sigma -1)(1-\alpha )(1-\psi _{t})}{\sigma }\left( 1+\phi \frac{\tilde{\gamma }_{t}}{\tilde{\tau }_{t}} \right) -\frac{(\sigma -1)\alpha \Theta _{t}+\rho }{\sigma }+B\epsilon _{t}\right) , \end{aligned}$$

(55)

$$\begin{aligned} \dot{u}_{t}&=u_{t}\left( -\frac{(\sigma -1)\alpha \Theta _{t}+\rho }{\sigma }-\kappa \frac{(1+\alpha (\sigma -1))(1-\psi _{t})}{\sigma }\left( 1+ \phi \frac{ \tilde{\gamma }_{t}}{\tilde{\tau }_{t}}\right) +u_{t}\right) ,\end{aligned}$$

(56)

$$\begin{aligned} \dot{\tilde{\tau }}_{t}&=\tilde{\tau }_{t}\left( \kappa \frac{\left( 1+\alpha (\sigma -1)\right) (1-\psi _{t}) +\sigma \psi _{t}}{\sigma }\left( 1+ \phi \frac{\tilde{\gamma }_{t}}{\tilde{\tau }_{t}}\right) +\frac{(\sigma -1)\alpha \Theta _{t}+\rho }{\sigma }\right) , \end{aligned}$$

(57)

$$\begin{aligned} \dot{\tilde{\gamma }}_{t}&=\tilde{\gamma }_{t}\left( -\kappa \frac{(\sigma -1)(1-\alpha )(1-\psi _{t})}{\sigma }\left( 1+ \phi \frac{\tilde{\gamma }_{t}}{\tilde{\tau }_{t}}\right) +\frac{\alpha \eta \chi }{B\epsilon _{t}\ \tilde{\gamma } _{t}}+\frac{(\sigma -1)\alpha \Theta _{t}+\rho }{\sigma }+\theta \right) , \end{aligned}$$

(58)

$$\begin{aligned} \dot{\psi }_{t}&=\psi _{t}(\psi _{t}-1)\kappa \left( 1+ \phi \frac{\tilde{\gamma }_{t}}{\tilde{\tau }_{t}}\right) , \end{aligned}$$

(59)

along with (1) and (6), with $\Theta _{t}=B-\eta \left( \delta +\chi P_{t}\right)$, and $R_{t}=u_{t}S_{t}$. The steady state values read

$$\begin{aligned} \epsilon _\infty&=\frac{\rho +\alpha (\sigma -1)\left( B-\eta \left( \delta +\chi P_\infty \right) \right) }{B\sigma },\end{aligned}$$

(60)

$$\begin{aligned} u_\infty&=B \epsilon _\infty , \end{aligned}$$

(61)

$$\begin{aligned} \tilde{\tau }_\infty&=\kappa \frac{\alpha \eta \phi \chi }{B \epsilon _\infty \left( B \epsilon _\infty +\theta \right) \left( B\epsilon _\infty +\kappa \right) }, \end{aligned}$$

(62)

$$\begin{aligned} \tilde{\gamma }_\infty&=-\frac{\alpha \eta \chi }{B\epsilon _\infty \left( B\epsilon _\infty +\theta \right) }, \end{aligned}$$

(63)

$$\begin{aligned} \psi _\infty&=1, \end{aligned}$$

(64)

$$\begin{aligned} S_\infty&=0, \end{aligned}$$

(65)

$$\begin{aligned} P_\infty&=P_{0},\quad \text { if }\theta >0 \text { and }P_{0}+\phi S_{0},\quad \text { if } \theta =0, \end{aligned}$$

(66)

$$\begin{aligned} Z_\infty&=0. \end{aligned}$$

(67)

The eigenvalues of the jacobian matrix of the corresponding system, calculated at the steady state values, are $\{-\theta ,-\kappa ,-\frac{\rho +\alpha (\sigma -1)\Theta _\infty }{\sigma },-\frac{\rho +\alpha (\sigma -1)\Theta _\infty }{\sigma },\frac{\rho +\alpha (\sigma -1)\Theta _\infty }{\sigma },\frac{\rho +\alpha (\sigma -1)\Theta _\infty }{\sigma },\frac{\rho +\theta \sigma +\alpha (\sigma -1)\Theta _\infty }{\sigma },\frac{\rho +\kappa \sigma +\alpha (\sigma -1)\Theta _\infty }{\sigma }\}$, with $\Theta _\infty =B-\eta (\delta +\chi P_\infty )$, and $P_\infty =P_{0}+\phi S_{0}$, if $\theta =0$, or $\bar{P}=P_{0}$, if $\theta >0$, i.e. four negative and four positive eigenvalues, implying a saddle-path stability around the steady state. The growth rate of consumption can be calculated by (49) to be

$$\begin{aligned} g_{Ct}=\frac{\alpha \Theta _{t}-\rho }{\sigma }-\frac{\kappa (1-\alpha )(1-\psi _{t})}{\sigma }\left( 1+ \phi \frac{\tilde{\gamma }_{t}}{\tilde{\tau }_{t}}\right) , \end{aligned}$$

(68)

with a steady state value of

$$\begin{aligned} g_{C\infty }=\frac{\alpha \left( B-\eta \left( \delta +\chi P\infty \right) \right) -\rho }{\sigma }. \end{aligned}$$

(69)

1.1 Linearized Version of the Model for $\sigma \ne 1$

In order to simulate the model we use the standard linearization technique: the linearized version of our autonomous dynamic system in $\mathbf{x}_t = \{\epsilon _t, u_t, \tilde{\tau }_t,\tilde{\gamma }_t,\psi _t, S_t, P_t, Z_t\}^{\top }$ can be obtained by using the jacobian matrix $\mathbf {J}$ evaluated at the steady states presented above as $d(\mathbf {x}_t-{\mathbf{x}}_{\infty })/dt \approx \mathbf{J}(\mathbf{x}_t-{\mathbf{x}}_{\infty })$, with $\mathbf {x_\infty }=\{\epsilon _\infty , u_\infty , \tilde{\tau }_\infty , \tilde{\gamma }_\infty , \psi _\infty , S_\infty , P_\infty , Z_\infty \}^{\top }$, which allows for an easy solution of the system of linear homogenous equations. For the simulation we use as initial conditions $\{\psi _0=0.65, S_0=6000$ GtC, $P_0=830$ GtC, $Z_0=0\}$, while we calculate the initial level of the control variables $\{\epsilon _0, u_0, \tau _0, \gamma _0\}$ such that the constants of integration associated with the unstable roots (positive eigenvalues) are zero. The parameters chosen are $\{\rho =0.015, \sigma =1.5, \alpha =0.9, \theta = 0, \delta =0.05, B=0.106, \chi =1.7 \times 10^{6}$$/GtC, $\phi =1, \eta =1, \kappa _{\text {low}}=0.02,\kappa _{\text {high}}=0.04\}$. Below we justify our choices for the numerical exercise.

1.2 Choice of Initial Conditions and Parameters

We choose 2010 to be our $t=0$. An approximation for $\psi _0$, the share of the before tax price to the total price paid by consumers, can be taken from the IEA Monthly Oil Statistics, IEA (2015), to be around 0.65 as an average for the period 2006–2015. $S_0=6000$ GtC follows estimates from the 2010 version of the DICE model. $P_0=830$ GtC was retrieved by data from the European Environmental Agency.¹⁸ $Z_0=0$ is chosen according to our discussion in “Appendix 1”. The values of $\rho$, $\sigma$ and $\delta$ are standard in the literature of endogenous growth. We normalize $\phi =1$, so that a unit of resource use equals a unit of emissions; we also set $\eta =1$ in the numerical exercise. In this model real GDP equals to $C_t+p_{It}I_t$. Worldwide gross capital formation, $p_{It}I_t/GDP_t$, is on average 0.24 for the period 1960–2014, World Bank Indicators, 2015. Accordingly $C_t/GDP_t=0.76$. Energy expenditure as share of GDP in the US for the period 1949–2011 is in the range of 0.04–0.1, EIA (2015). We choose then a value of 0.08. Using now the FOC (10) for energy we can calculate $\alpha =1- \frac{(p_{Rt}+\phi \tau _t)R_t/GDP_t}{C_t/GDP_t} \approx 0.9$. The value of natural disasters reported in the period 2005–2015 amounts on average to ${139 \times 10^9}$ $/year (EM-DAT The International Disasters Database 2015). Using the value for the stock of capital from the 2010 version of the DICE model (DICE2010) and $P_0=830$ GtC, we get that ${\chi =1.7 \times 10^{6}}$ $/GtC. B was chosen so that the initial level of the interest rate is about $5\%$ (as in DICE2010) and that it satisfies the condition $\alpha (B-\eta (\delta +\chi (P_0+\phi S_0))>\rho$, from (69), equivalent to the one assumed in Sect. 4.2. Parameter $\kappa$ is the reciprocal of the mean time lag. We choose a low value to reflect a time lag of 50 years, and a high value to reflect a time lag of 25 years.