## 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.

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## Notes

Throughout this paper we will somewhat loosely refer to such events as regime shifts. Further, as in Lenton et al. (2008) we denote a “tipping point” as a critical threshold at which a tiny perturbation can qualitatively alter the state or development of a system and the term “tipping element” as a large-scale component of the Earth system that may pass a tipping point.

If gas is lumped together with oil as a finite resource stock then this covers over 50% of current energy consumption (IEA 2014).

We have also solved the model and derived the analytical results of this paper, using explicit representations of the carbon-cycle dynamics such as the one used by Golosov et al. (2014). We believe however that the Matthews et al. (2009) representation of the carbon-climate interaction makes the results of the present paper more clear and we have thus chosen to stick with this form.

Alternatively, in difference form this relationship can be written as \(T_{t+1}=T_t + \lambda \sigma _t E_{t}\). Note also that throughout the rest of the paper we normalize \(T_0\) setting it to zero.

For other examples of CCR type relationships in IAMs, see Anderson et al. (2014).

For further empirical motivation of potential global tipping points see e.g. Lenton et al. (2008).

As it will turn out, a technically interesting feature, which results from modeling it as a pulse, is that it will only affect the optimal amount of fossil-fuel use through its effect on the probability of the shift occurring and not via the marginal damages resulting from further emissions.

This applies only to variables. In the case of general functional forms of one variable, the prime represents the first order derivative (e.g. \(U'(C) = dU/dC\)). This is also standard notation and should hopefully not cause confusion. Partial derivatives will be denoted by subscripts with the differentiation variable e.g. \(F_K(K,L) = \partial F(K,L)/\partial K\).

The damage function \(\hat{D}\) is parameterized by \(\hat{\gamma }\). If the tipping point involves a greenhouse-gas pulse \(\hat{P}\), this will already be part of the temperature state

*T*and will not explicitly show up in the post regime shift solution.We go through the computations of the pre-shift problem in detail in Appendix “Solution to the pre-shift problem”.

Specifying a hazard rate function is an empirically daunting task. Efforts have been made by e.g. Cai et al. (2012) to inform their functional form using expert elicitation. Since we are primarily interested in making qualitative statements we make no such effort here.

While it may seem reasonable to expect that the different energy sources should be good substitutes, some studies indicate that this may not be the case (Stern 2012). We therefore consider both cases.

## References

Allen MR, Frame DJ, Huntingford C, Jones CD, Lowe JA, Meinshausen M, Meinshausen N (2009) Warming caused by cumulative carbon emissions towards the trillionth tonne. Nature 458(7242):1163–1166

Alley RB, Marotzke J, Nordhaus WD, Overpeck JT, Peteet DM, Pielke RA Jr, Pierrehumbert RT, Rhines PB, Stocker TF, Talley LD, Wallace JM (2003) Abrupt climate change. Science 299(5615):2005–2010

Anderson EW, Brock WA, Hansen LP, Sanstad A (2014) Robust analytical and computational explorations of coupled economic-climate models with carbon-climate response. RDCEP Working Paper No. 13–05. http://ssrn.com/abstract=2370657

Barrage L (2014) Sensitivity Analysis for Golosov, Hassler, Krusell, and Tsyvinski (2014): Optimal taxes on fossil fuel in general equilibrium, Econometrica supplemental materia. Tech Rep 82. http://www.econometricsociety.org/ecta/supmat/10217_extensions

Cai Y, Judd KL, Lontzek TS (2012) DSICE: a dynamic stochastic integrated model of climate and economy. RDCEP Working Paper No. 12

Clarke HR, Reed WJ (1994) Consumption/pollution tradeoffs in an environment vulnerable to pollution-related catastrophic collapse. J Econ Dyn Control 18(5):991–1010

Cropper ML (1976) Regulating activities with catastrophic environmental effects. J Environ Econ Manag 3:1–15

Engström G, Gars J (2015) Climate tipping points and optimal fossil fuel use. Beijer discussion paper, No. 259. http://www.beijer.kva.se/pubinfo.php?pub_id=726

Gerlagh R (2011) Too much oil. CESifo Econ Stud 57:79–102

Golosov M, Hassler J, Krusell P, Tsyvinski A (2014) Optimal taxes on fossil fuel in general equilibrium. Econometrica 82(1):41–88

Hassler J, Krusell P, Olovsson C (2012) Energy-saving technical change. Working Paper 18456, National Bureau of Economic Research. http://www.nber.org/papers/w18456

IEA (2014) Key world energy statistics. http://www.iea.org/publications/freepublications/publication/KeyWorld2014

IPCC (2007) Climate change 2007: the fourth assessment report of the intergovernmental panel on climate change. Cambridge UniversityPress, Cambridge

Kriegler E, Hall JW, Held H, Dawson R, Schellnhuber HJ (2009) Imprecise probability assessment of tipping points in the climate system. Proc Natl Acad Sci USA 106:50415046

Lemoine D, Traeger C (2014) Watch your step: optimal policy in a tipping climate. Am Econ J Econ Policy 6(1):137–166

Lenton TM, Held H, Kriegler E, Hall JW, Lucht W, Rahmstorf S, Schellnhuber HJ (2008) Tipping elements in the Earth’s climate system. Proc Natl Acad Sci USA 105(6):1786–1793

Matthews DH, Gillett NP, Stott PA, Zickfeld K (2009) The proportionality of global warming to cumulative carbon emissions. Nature 459:829833

Matthews DH, Solomon S, Pierrehumbert R (2012) Cumulative carbon as a policy framework for achieving climate stabilization. Philos Trans R Soc A 370:43654379

McGlade C, Ekins P (2015) The geographical distribution of fossil fuels unused when limiting global warming to 2 [deg] c. Nature 517(7533):187–190

Nordhaus WD (1994) Managing the global commons: the economics of the greenhouse effect. MIT Press, Cambridge

Nordhaus WD (2007) A question of balance. Yale University Press, New Haven

Nordhaus WD, Boyer J (2000) Warming the world: economic models of global warming. MIT Press, Cambridge

Pindyck R (2013a) The climate policy dilemma. Rev Environ Econ Policy 7(2):219–237

Pindyck RS (2013b) Climate change policy: What do the models tell us? J Econ Lit 51(3):860–872

Polasky S, de Zeeuw A, Wagener F (2011) Optimal management with potential regime shifts. J Environ Econ Manag 62(2):229–240

Quaas MF, Van Soest D, Baumgärtner S (2013) Complementarity, impatience, and the resilience of natural-resource-dependent economies. J Environ Econ Manag 66(1):15–32

Ren B, Polasky S (2014) The optimal management of renewable resources under the risk of potential regime shift. J Econ Dyn Control 40:195–212

Rezai A, van der Ploeg F (2015) Robustness of a simple rule for the social cost of carbon. Econ Lett 132:48–55

Schaefer K, Zhang T, Bruhwiler L, Barrett AP (2011) Amount and timing of permafrost carbon release in response to climate warming. Tellus B 63(2):165–180

Sinn HW (2008) Public policies against global warming: a supply side approach. Int Tax Public Financ 15(4):360–394

Smith JB, Schneider SH, Oppenheimer M, Yohe GW, Hare W, Mastrandrea MD, Patwardhan A, Burton I, Corfee-Morlot J, Magadza CHD, Fussel HM, Pittock AB, Rahman A, Suarez A, van Ypersele JP (2009) Assessing dangerous climate change through an update of the Intergovernmental Panel on Climate Change (IPCC) “reasons for concern”. Proc Natl Acad Sci 106(11):4133–4137

Smulders S, Tsur Y, Zemel A (2014) Uncertain climate policy and the green paradox. In: Moser E, Semmler W, Tragler G, Veliov VM (eds) Dynamic optimization in environmental economics, dynamic modeling and econometrics in economics and finance, vol 15. Springer, Berlin, pp 155–168

Stern N (2007) The economics of climate change: the Stern review. Cambridge University Press, Cambridge

Stern DI (2012) Interfuel substitution: a meta-analysis. J Econ Surv 26(2):307–331

Tol RSJ (1999) The marginal costs of greenhouse gas emissions. Energy J 20:61–80

Tsur Y, Zemel A (1998) Pollution control in an uncertain environment. Econ Dyn Control 22:967

van der Ploeg R (2016) Second-best carbon taxation in the global economy: the Green Paradox and carbon leakage revisited. J Environ Econ manag 78(1):85–105

van der Ploeg F, de Zeeuw A (2013) Climate policy and catastrophic change: be prepared and avert risk. OxCarre Working Papers No. 118, Oxford Centre for the Analysis of Resource Rich Economies, University of Oxford

van der Ploeg F, Withagen C (2012) Is there really a green paradox? J Environ Econ Manag 64(3):342–363

van der Ploeg F, Withagen C (2015) Global warming and the green paradox: a review of adverse effects of climate policies. Rev Environ Econ Policy 9:285. doi:10.1093/reep/rev008

Winter RA (2014) Innovation and the dynamics of global warming. J Environ Econ manag 68(1):124–140

## Acknowledgments

We thank John Hassler, Per Krusell, Rick van der Ploeg, Aart de Zeeuw and Torsten Persson for valuable comments on earlier drafts as well as seminar participants at the Complex Systems workshop, Stockholm, The Ulvön Conference on environmental economics, the Department of economics at Gothenburg University and the WCERE conference in Istanbul. We also wish to acknowledge funding support from the Ragnar Söderberg Foundation, the Erling-Persson Family Foundation and from a European Project supported within the Ocean of Tomorrow call of the European Commission Seventh Framework Programme.

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## 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

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

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

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

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

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

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

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

### 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

where *W*(*R*, *T*) fulfills

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

Differentiating the maximized value function with respect to *R* we get

Differentiating the maximized value function with respect to *T* we get

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)

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

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

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

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

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

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

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

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

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

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

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

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

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 \(A_i\)

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.

Final-goods production is given by

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

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

where \(\hat{W}\) fulfills the Bellman equation

where

The first-order condition with respect to \(E_2\) gives

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

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

**Pre-shift**

The pre-shift value function can be written

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

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

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

The Bellman equation can then be written

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.

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Engström, G., Gars, J. Climatic Tipping Points and Optimal Fossil-Fuel Use.
*Environ Resource Econ* **65**, 541–571 (2016). https://doi.org/10.1007/s10640-016-0042-2

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DOI: https://doi.org/10.1007/s10640-016-0042-2