Climatic Tipping Points and Optimal Fossil-Fuel Use

Abstract

The economics of climate change is characterized by many uncertainties regarding, for instance, climate dynamics, economic damages and potentially irreversible climate catastrophes. Using an optimal growth model of a fossil-fuel-driven economy subject to climate externalities and potentially irreversible climatic regime shifts, this paper contributes to the understanding of how the risk of such events impacts on optimal fossil-fuel use, carbon taxes and fossil-fuel prices over time. We show that in excess, to an increase in the expected present value of marginal damages and an increase in the probability of triggering the event, there also exists a third opposing effect. This effect comes from the fall in value of remaining fossil-fuel reserve, which results from the potential regime shift which may (or may not) occur sometime in the future, and implies that optimal fossil-fuel policy shifts towards using more resources early on. This effect is related to the idea of the green paradox. We prove the existence of this effect and show under which circumstances it can become quantitatively important. Numerically, the green-paradox effect seems to be somewhat smaller than, but comparable in size to, the increase in expected marginal damages. In general, our findings highlight the importance of considering the supply-side impacts on climate policy in response to catastrophic climate events.

Appendices

Appendix 1: Solution to the Post-shift Problem

We will here outline the steps of solving the post-shift problem described in the Bellman Eq. (8).

Under assumptions (4a)–(4c), the first-order condition with respect to $K'$ implies the constant consumption and savings rates in (12).

The first-order condition with respect to E is

$$\begin{aligned} U'(C)\left[ \hat{D}(T)F_E(K,E)\right] = \beta \left[ \hat{V}_R(K',R',T')-\hat{\sigma }\lambda \hat{V}_T(K',R',T')\right] . \end{aligned}$$

(26)

This contains the derivatives of the value function with respect to R and T. Differentiating the the maximized version of the right-hand side of (8) with respect to R we get

$$\begin{aligned} \hat{V}_R(K,R,T) = \beta \hat{V}_R(K',R',T') \end{aligned}$$

(27)

implying that the scarcity value of the resource increases at the rate $\frac{1}{\beta }$ between periods.

Differentiating the the maximized version of the right-hand side of (8) with respect to T, substituting repeatedly for $\hat{V}_T$ in future time periods and assuming that $\hat{V}_T$ is bounded (follows from finiteness of the resource for “reasonable” $\hat{D}$) and using also assumptions (4a)–(4d) and (12) we get

$$\begin{aligned} \hat{V}_T(K,R,T)=\sum _{n=0}^\infty \beta ^nU'(C^{(n)}) \hat{D}'\left( T^{(n)} \right) F\left( K^{(n)}, E^{(n)}\right) = -\frac{\hat{\gamma }}{1-\alpha \beta }\frac{1}{1-\beta } \equiv -\hat{{\varGamma }}.\nonumber \\ \end{aligned}$$

(28)

Hence, the marginal externality cost of emissions is constant in the post-shift environment.

The first-order condition with respect to E (26) can then be rewritten as

$$\begin{aligned} E=\frac{\nu }{(1-\alpha \beta )\left( \hat{V}_R(K,R,T) +\beta \hat{\sigma }\lambda \hat{{\varGamma }} \right) }. \end{aligned}$$

(29)

Since all fossil fuel will be used up asymptotically, condition (2) will hold with equality. Moving (29) forward and using (27) we have that

$$\begin{aligned} R=\sum _{n=0}^\infty E^{(n)}=\frac{\nu }{1-\alpha \beta }\sum _{n=0}^\infty \frac{1}{\beta ^{-n}\hat{V}_R\left( K,R,T\right) +\beta \hat{\sigma }\lambda \hat{{\varGamma }}}. \end{aligned}$$

(30)

This implicitly gives the value of $\hat{V}_R(K,R,T)$ as a function of the state variables. We note that this expression does not contain the state variables K and T. This implies separability. Furthermore, $\hat{V}_K$ is

$$\begin{aligned} \hat{V}_K(K,R,T)=U'(C)\hat{D}(T)F_K(K,E)=\frac{\alpha }{1-\alpha \beta }\frac{1}{K}. \end{aligned}$$

(31)

This depends only on K. From (30) we know that $\hat{V}_R$ only depends on R and from (3.1.1) we know that $\hat{V}_T$ is constant. This all implies that $\hat{V}$ is additively separable and can be written as

$$\begin{aligned} \hat{V}(K,R,T)=\frac{\alpha }{1-\alpha \beta }\ln K+\hat{W}(R)-\hat{{\varGamma }}T, \end{aligned}$$

where $\hat{W}(R)$ fulfills

$$\begin{aligned} R=\frac{\nu }{1-\alpha \beta }\sum _{n=0}^\infty \frac{1}{\beta ^{-n}\hat{W}'(R) +\beta \hat{\sigma }\lambda \hat{{\varGamma }}}. \end{aligned}$$

Appendix 2: Solution to the Pre-shift Problem

We will here go through the details of solving the pre-shift Bellman Eq. (14). Here, it can be shown that the constant consumption/savings rate (12) still applies. The constant consumption/savings rate reduces complexity by reducing the dimensionality of the problem. It is fairly straightforward to verify that the value function can then be written

$$\begin{aligned} V(K,R,T) = \frac{\alpha }{1-\alpha \beta } \ln K + W(R,T), \end{aligned}$$

where W(R, T) fulfills

$$\begin{aligned} W(R,T) =&\max _E \;\; \frac{1}{1-\alpha \beta }\ln \left( D\left( T\right) E^\nu \right) +\mathbf {C}+ \beta \left( 1-\pi (T')\right) W(R-E,T+\sigma \lambda E) \\&\beta \pi (T')\left( \hat{W}\left( R-E\right) -\hat{\varGamma }\left( T +\sigma \lambda E+\hat{P}\right) \right) , \end{aligned}$$

with $\mathbf {C} =\frac{(1-\alpha \beta )\ln (1-\alpha \beta ) +\alpha \beta \ln (\alpha \beta )}{1-\alpha \beta }$. Note that we have also used the form of the post-shift value function (9) here.

Returning to the original Bellman equation, the first-order condition with respect to E is

$$\begin{aligned} U'(C)D(T)F_E(K,E)= & {} \beta \left[ \left( 1-\pi (T')\right) V_R'+\pi (T')\hat{V}_R\right] \nonumber \\&-\beta \lambda \sigma \left[ \left( 1-\pi (T')\right) V_T'+\pi (T') \hat{V}_T'\right] +\beta \sigma \lambda \pi '(T')\left[ V' -\hat{V}' \right] \nonumber \\ \end{aligned}$$

(32)

Differentiating the maximized value function with respect to R we get

$$\begin{aligned} V_R(K,R,T)=\beta \left[ \left( 1-\pi (T')\right) V_R'+\pi (T')\hat{V}_R'\right] . \end{aligned}$$

(33)

Differentiating the maximized value function with respect to T we get

$$\begin{aligned} V_T(K,R,T)= & {} U'(C)D'(T)F(K,E) + \beta \left[ \left( 1-\pi (T') \right) V_T' +\pi (T')\hat{V}_T'\right] \nonumber \\&-\beta \pi '(T')\left[ V' -\hat{V}' \right] \end{aligned}$$

(34)

We can simplify the first term in the right-hand side of this equation by using the exponential form of the damage function (4d), logarithmic utility (4a) and the savings rule (12)

$$\begin{aligned} U'(C)D'(T)F(K,E)= U'(C)\frac{D'(T)}{D(T)}D(T)F(K,E)=-\frac{\gamma }{1-\alpha \beta }. \end{aligned}$$

From (28) we have that $\hat{V}_T^{(n)}=-\hat{{\varGamma }}$ for all n and we can thus substitute forwards in (34), assuming as before that $V_T^{(N)}$ is bounded, we can write $V_T'$ from (32) as

$$\begin{aligned} V_T'= & {} -\frac{\gamma }{1-\alpha \beta }-\frac{\gamma }{1-\alpha \beta } \sum _{n=1}^\infty \beta ^n\left( 1-\pi \left( T^{(n+1)}\right) \right) {\varOmega }_2^{n+1}-\hat{{\varGamma }}\sum _{n=1}^\infty \beta ^n\pi \left( T^{(n+1)}\right) {\varOmega }_2^{n+1} \\&-\sum _{n=1}^\infty \beta ^n\pi '\left( T^{(n+1)}\right) \left( V^{(n+1)}-\hat{V}^{(n+1)}\right) {\varOmega }_2^{n+1}. \end{aligned}$$

Returning to the first order condition with respect to E in (32), the left-hand side can be simplified to

$$\begin{aligned} U'(C)D(T)F_E(K,E)=\frac{D(T)F_E(K,E)}{(1-\alpha \beta )D(T)F(K,E)} =\frac{\nu }{1-\alpha \beta }\frac{1}{E}. \end{aligned}$$

Substituting this, $\hat{V}_T'=-\hat{{\varGamma }}$ and (33) in (32) and using that $\left( 1-\pi \left( T'\right) \right) {\varOmega }_2^{n+1}={\varOmega }_1^{n+1}$ gives

$$\begin{aligned} \frac{\nu }{1-\alpha \beta }\frac{1}{E}= & {} V_R+\beta \sigma \lambda \sum _{n=0}^\infty \beta ^n\pi ' \left( T^{(n+1)}\right) \left( V^{(n+1)}-\hat{V}^{(n+1)}\right) {\varOmega }_1^{n+1}\\&+\,\beta \sigma \lambda \frac{\gamma }{1-\alpha \beta } \sum _{n=0}^\infty \beta ^n\left( 1-\pi \left( T^{(n+1)}\right) \right) {\varOmega }_1^{n+1} \\&+\,\beta \sigma \lambda \hat{{\varGamma }}\sum _{n=0}^\infty \beta ^n\pi \left( T^{(n+1)}\right) {\varOmega }_1^{n+1}. \end{aligned}$$

Using definitions (18) and (19) we arrive at (17).

Appendix 3: Sign of ${\varTheta }$

Proposition 4

Consider a shift in at least one parameter. Then for any set of state variables (K, R, T) we have that

$$\begin{aligned} V(K,R,T)\ge \hat{V} \left( K,R,T+\lambda \sigma \hat{P}\right) \end{aligned}$$

Assuming, furthermore, that there is a strictly positive probability of a regime shift in some future period and that $\pi '(T)>0$ in at least one such period, then ${\varTheta }>0.$ If, instead $\pi '(T)=0$ for all T, then ${\varTheta }=0$.

The part of the damages that depend on the risk of triggering a regime shift is thus weakly positive and strictly positive whenever emissions affect the probability of a shift that has negative welfare effects. This is intuitive and a formal proof is available from the authors upon request.

Appendix 4: The Size of $\tilde{\varGamma }$

We can define

$$\begin{aligned} {\varGamma }=\frac{\gamma }{1-\alpha \beta }\frac{1}{1-\beta } \end{aligned}$$

which is equal to $\tilde{{\varGamma }}$ when $\pi =0$ and gives the marginal damages if there is no risk of a shift. Comparing ${\varGamma }$, $\tilde{{\varGamma }}$ and $\hat{{\varGamma }}$, we have the following proposition

Proposition 5

It is always the case that

$$\begin{aligned} \tilde{\varGamma }\in \left[ {\varGamma },\hat{{\varGamma }}\right] \end{aligned}$$

For a shift in $\sigma$ or $\hat{P}$, $\tilde{\varGamma }={\varGamma }=\hat{\varGamma }$. For a shift in $\gamma$ (with $\hat{\gamma }>\gamma$) $\tilde{{\varGamma }}$ increases over time and if $\pi$ goes to one, $\tilde{\varGamma }$ goes to $\hat{\varGamma }$.

This proposition is quite intuitive since the marginal damage from emissions in the current period in each future period is either the pre-shift marginal damages or the post-shift marginal damages. Hence, the expected marginal damages accumulated over all future periods should lie between the marginal damages when there is no shift (${\varGamma }$) and the post-shift marginal damages ($\hat{\varGamma }$). The proof is relatively straightforward and available from the authors upon request.

Appendix 5: Proof of Proposition 1

The third term in (17), $V_R$, is the shadow value of the fossil-fuel resources. A higher shadow value translates into less fossil-fuel use. As shown below, the introduction of a potential regime shift decreases the shadow value.

In general all three terms in (17) will depend on the entire future path and we will now analyze the net effects of changes in all of these variables on optimal fossil-fuel use.

Let variables for the situation when the regime shift does not cause any parameter changes be denoted by variables without tildes and the variables for the case when the regime shift does cause changes by tildes. Solving both problems will give sequences

$$\begin{aligned} \left\{ E^{(n)},R^{(n)},V_R^{(n)},\hat{V}_R^{(n)},{\varGamma }^{(n)},\pi ^{(n)}\right\} \text { and }\left\{ \tilde{E}^{(n)}, \tilde{R}^{(n)},\tilde{V}_R^{(n)}, \tilde{\hat{V}}_R^{(n)}, \tilde{{\varGamma }}^{(n)},\tilde{{\varTheta }}^{(n)}\tilde{\pi }^{(n)}\right\} \end{aligned}$$

where the sequences are conditional upon that the regime shift has not happened yet (or in the case of $\hat{V}_R^{(n)}$ and $\tilde{\hat{V}}_R^{(n)}$ that it just happened). In both cases, fossil-fuel use will be given by

$$\begin{aligned} E^{(n)}=\frac{\nu }{1-\alpha \beta }\frac{1}{V_R^{(n)}+ {\varTheta }^{(n)} +\beta \sigma ^{(n)}\lambda {\varGamma }^{(n)}} \end{aligned}$$

or the corresponding expression with tildes on all variables. In both cases we also have

$$\begin{aligned} V_R^{(n)}=\beta \left[ \left( 1-\pi ^{(n+1)}\right) V_R^{(n+1)} +\pi ^{(n+1)} \hat{V}_R^{(n+1)}\right] \end{aligned}$$

(35)

and the corresponding expression with tildes. Note that in the case where the regime shift does not cause any changes, the probabilities do not matter since the values of the variables with or without hats will be the same. From Eq. (11) we know that

$$\begin{aligned} \hat{V}_R\ge \tilde{\hat{V}}_R \text { for given }R \text { and that both of them are decreasing in } R. \end{aligned}$$

(36)

and the inequality will be strict if at least one of $\hat{\gamma }\ge \gamma$ and $\hat{\sigma }\ge \sigma$ are strict.

We will now show that assuming that $\tilde{V}_R\ge V_R$ (for a given initial R) leads to a contradiction. We can start by noting that for all n, $\tilde{{\varTheta }}^{(n)}\ge {\varTheta }^{(n)}=0$ and $\tilde{\sigma }^{(n)}\tilde{{\varGamma }}^{(n)}\ge \sigma ^{(n)}{\varGamma }^{(n)}$. We must also have that

$$\begin{aligned} R=\sum _{n=0}^\infty E^{(n)}=\sum _{n=0}^\infty \tilde{E}^{(n)} \end{aligned}$$

since all fossil fuel will be used in all realizations including the ones where the regime shift happens arbitrarily far into the future. Assume now that $\tilde{V}_R\ge V_R$. This implies that $\tilde{E}\le E$ and consequently that $\tilde{R}'\ge R'$. If $\pi '(T)>0$ for all T then $\tilde{{\varTheta }}>0$ and we have that $\tilde{E}<E$. If at least one of $\hat{\gamma }\ge \gamma$ and $\hat{\sigma }\ge \sigma$ are strict then $\hat{V}_R'>\tilde{\hat{V}}_R'$. Combined, this implies that under the assumptions made, $\tilde{\hat{V}}_R'<\hat{V}_R'$. Combining $\tilde{V}_R\ge V_R$ and $\tilde{\hat{V}}_R'<\hat{V}_R'$ and using (35) we get $\tilde{V}_R'>V_R'$. This in turn implies that $\tilde{E}'<E'$ and $\tilde{R}''>R''$. Using (36) this implies that $\tilde{\hat{V}}_R''<\hat{V}_R''$. Using (35), $\tilde{V}_R'>V_R'$ and $\tilde{\hat{V}}_R''<\hat{V}_R''$ implies that $\tilde{V}_R''>V_R''$. This in turn implies that $\tilde{E}''<E''$. Going on we can show that assuming $\tilde{V}_R\ge V_R$ implies that

$$\begin{aligned} \sum _{n=0}^\infty \tilde{E}^{(n)}<\sum _{n=0}^\infty E^{(n)} \end{aligned}$$

since this inequality holds for each term in the sums. This contradicts that both sums should be equal to R and consequently proves that $V_R\ge \tilde{V}_R$ can not hold.

Appendix 6: Numerical Model with Imperfect Substitutes

We assume energy can be produced from either a fossil fuel $E_1$ or clean technology $E_2$ and that both sectors use labor $N_i$ in production and differ by technology levels $A_i$

$$\begin{aligned} E_1&= A_1 N_1 g(R) \\ E_2&= A_2 N_2 \end{aligned}$$

The fossil-fuel sector is further constrained (as in the original model) by the amount of available resources but also by extraction costs $g(R)\equiv R_0/R$. Total energy production is thus a compound of $E_1$ and $E_2$ and assumed to be given by a CES function where $\rho <0$ implies that the two energy sources are complements and $\rho >0$ substitutes.

$$\begin{aligned} E=\left( \kappa _1E_1^\rho +\kappa _2E_2^\rho \right) ^\frac{1}{\rho }. \end{aligned}$$

Final-goods production is given by

$$\begin{aligned} Y=D(T)AK^\alpha N_{0}^{1-\alpha -\nu }E^\nu . \end{aligned}$$

Finally, the total amount of available labor N can be freely allocated between the two energy sectors and final goods production and must thus satisfy

$$\begin{aligned} N = N_0 + N_1 + N_2. \end{aligned}$$

Below we sketch out the post- and pre-shift problems for this economy. Code for numerical simulations were done in Matlab and are available from the authors upon request. Parameter values are as in the original model with the addition of $A_1=7693$, $A_2=1311$, $N=1$, $\kappa _1=0.6$ and $\kappa _2=0.4$ (borrowed from Golosov et al. (2014)). We simulate the model for $\rho =0.2$ and $\rho =-0.2$ and use a value of $\hat{\gamma }=5 \gamma$ in order to make the dynamic of the paths in the graphs become more visible (see Fig. 5).

Post-shift

The constant savings rate applies. The value function can be written

$$\begin{aligned} \hat{V}(K,R,T)=\frac{\alpha }{1-\alpha \beta }\ln K-\hat{\varGamma }T +\hat{W}(R), \end{aligned}$$

where $\hat{W}$ fulfills the Bellman equation

$$\begin{aligned} \hat{W}(R)&= \max _{E_1,E_2} \frac{1-\alpha -\nu }{1-\alpha \beta }\ln \left( N_0\right) +\frac{\nu }{1-\alpha \beta }\ln E-\beta \hat{\sigma }\hat{\lambda }\hat{\varGamma }E_1 \\&\quad \, +\beta \hat{W}(R-E_1) +{\varPhi }\\ \text { s.t. } N_0&= N-\frac{E_1}{A_{1}g(R)}-\frac{E_2}{A_{2}},\; E=\left( \kappa _1 E_1^\rho +\kappa _2E_2^\rho \right) ^\frac{1}{\rho }, \end{aligned}$$

where

$$\begin{aligned} {\varPhi }\equiv \frac{\ln (A)+(1-\alpha \beta )\ln (1-\alpha \beta )+ \alpha \beta \ln (\alpha \beta )}{1-\alpha \beta }. \end{aligned}$$

(37)

The first-order condition with respect to $E_2$ gives

$$\begin{aligned} \frac{1-\alpha -\nu }{\nu }=\frac{\kappa _2E_2^{\rho -1}}{\kappa _1E_1^\rho +\kappa _2E_2^\rho } A_{2}\left( N-\frac{E_1}{A_{1}g(R)}-\frac{E_2}{A_{2}}\right) . \end{aligned}$$

(38)

The right-hand side is strictly decreasing in $E_2$ for a given $E_1$ (for values such that $N_0$, $E_1$ and $E_2$ are all positive). Hence this defines $E_2$ as a function of $E_1$. Call this function $E_{2}^*(E_1,R)$. The Bellman equation can then be written

$$\begin{aligned} \hat{W}(R)&= \max _{E_1} \frac{1-\alpha -\nu }{1-\alpha \beta }\ln \left( N_0\right) +\frac{\nu }{1-\alpha \beta }\ln E-\beta \hat{\sigma }\hat{\lambda }\hat{\varGamma }E_1 \\&\quad \, +\,\beta \hat{W}(R-E_1)+{\varPhi }\\ \text { s.t. } N_0&= N-\frac{E_1}{A_{1}g(R)}-\frac{E_{2}^*(E_1,R)}{A_{2}}, \\ E&= \left( \kappa _1 E_1^\rho +\kappa _2\left( E_{2}^*(E_1,R)\right) ^\rho \right) ^\frac{1}{\rho }\\ E_1&\in \left( 0,g(R)A_{1}N\right) . \end{aligned}$$

This equation can be solved numerically on a grid in terms of R.

Pre-shift

The pre-shift value function can be written

$$\begin{aligned} V(K,R,T)=\frac{\alpha }{1-\alpha \beta }\ln K+\tilde{W}(R,T), \end{aligned}$$

where $\tilde{W}(R,T)$ fulfills the Bellman equation

$$\begin{aligned} \tilde{W}(R,T)&= \max _{E_1,E_2} \frac{1-\alpha -\nu }{1-\alpha \beta }\ln (N_0)+\frac{\nu }{1-\alpha \beta }\ln E-\frac{\gamma }{1-\alpha \beta }T + {\varPhi }\\&\quad \, +\,\beta \left( 1-\pi (T')\right) \tilde{W}(R-E_1,T') \\&\quad \, +\,\beta \pi (T')\left[ -\hat{\varGamma }(T'+P)+\hat{W}(R-E_1)\right] \\ \text { s.t. } N_0&= N-\frac{E_1}{A_{1}g(R)}-\frac{E_2}{A_{2}}, \\ E&= \left( \kappa _1E_1^\rho +\kappa _2E_2^\rho \right) ^\frac{1}{\rho }, \\ T'&= T+\sigma \lambda E_1. \end{aligned}$$

As in the post-shift problem, the first-order condition with respect to $E_2$ implies (38) that implicitly defines $E_2$ as a function $E_{2}^*(E_1,R)$. Futhermore, temperature is given by

$$\begin{aligned} T_t=\sigma \lambda \sum _{n=0}^{t-1}E_{1,n}=\sigma \lambda (R_0-R_t). \end{aligned}$$

We can thus write the Bellman equation in a value function that only depends on R. Define this value function as

$$\begin{aligned} W(R)\equiv \tilde{W}(R,\sigma \lambda (R_0-R)). \end{aligned}$$

The Bellman equation can then be written

$$\begin{aligned} W(R)&= \max _{E_1} \frac{1-\alpha -\nu }{1-\alpha \beta }\ln (N_0)+\frac{\nu }{1-\alpha \beta }\ln E-\frac{\gamma }{1-\alpha \beta }\sigma \lambda (R_0-R) + {\varPhi }_t \\&\quad \, +\beta \left( 1-\pi (T')\right) W(R-E_1) \\&\quad \, +\beta \pi (T')\left[ -\hat{\varGamma }(T'+P)+\hat{W}(R-E_1)\right] \\ \text { s.t. } N_0&= N-\frac{E_1}{A_{1}g(R)}-\frac{E_{2}^*(E_1,R)}{A_{2}}, \\ E&= \left( \kappa _1E_1^\rho +\kappa _2\left( E_{2}^*(E_1,R)\right) ^\rho \right) ^\frac{1}{\rho }, \\ T'&= \sigma \lambda (R_0-R+E_1), \\ E_1&\in \left( 0,g(R)A_{1}N\right) . \end{aligned}$$

This equation can be solved numerically. Note that there are suppressed constraints $E_1\ge 0$ and $E_1\le R$. These should not bind in the optimal solution, but may need to be taken into consideration in the numerical solution.