* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Playing the Climate Dominoes: Tipping Points and the Cost of
Climate change mitigation wikipedia , lookup
German Climate Action Plan 2050 wikipedia , lookup
Climate governance wikipedia , lookup
Surveys of scientists' views on climate change wikipedia , lookup
Global warming hiatus wikipedia , lookup
Effects of global warming on human health wikipedia , lookup
Attribution of recent climate change wikipedia , lookup
2009 United Nations Climate Change Conference wikipedia , lookup
Public opinion on global warming wikipedia , lookup
Climate change, industry and society wikipedia , lookup
Climate sensitivity wikipedia , lookup
Global warming wikipedia , lookup
Solar radiation management wikipedia , lookup
Climate change in the United States wikipedia , lookup
Carbon pricing in Australia wikipedia , lookup
Climate change and poverty wikipedia , lookup
Low-carbon economy wikipedia , lookup
General circulation model wikipedia , lookup
Climate change in New Zealand wikipedia , lookup
Mitigation of global warming in Australia wikipedia , lookup
Climate change in Canada wikipedia , lookup
Years of Living Dangerously wikipedia , lookup
Effects of global warming on Australia wikipedia , lookup
Economics of global warming wikipedia , lookup
Economics of climate change mitigation wikipedia , lookup
Climate engineering wikipedia , lookup
Politics of global warming wikipedia , lookup
Instrumental temperature record wikipedia , lookup
Citizens' Climate Lobby wikipedia , lookup
Carbon emission trading wikipedia , lookup
Business action on climate change wikipedia , lookup
Climate change feedback wikipedia , lookup
Playing the Climate Dominoes: Tipping Points and the Cost of Delaying Policy Derek Lemoinea & Christian P. Traegerb a b Department of Economics, University of Arizona Department of Agricultural and Resource Economics, University of California Berkeley Abstract: Greenhouse gas emissions can trigger irreversible regime shifts in the climate system, known as tipping points. Multiple tipping points affect each other’s probability of occurrence (a “domino effect”) and they reinforce each other’s economic impacts. Today’s optimal mitigation policy has to consider all possible future interactions between tipping points, economic activity, and policy responses over long time horizons. We find that the presence of three commonly discussed tipping points nearly doubles the currently optimal carbon tax, reducing peak warming along the optimal path by approximately 1 degree Celsius. Our study also provides the first quantification of the expected cost of delaying optimal policy in the face of tipping points, which is comparable to the annual output of several large nations. This novel assessment of the policy implications of tipping points bridges the increasingly severe divide between scientific concerns about abrupt climate change and economic analyses. Furthermore, we demonstrate that even the recognition of an already-triggered tipping point can have substantial economic value, complementing the scientific literature’s focus on the recognition of early warning signals. 1 Introduction The threat of climate tipping points plays a major role in calls for aggressive emission reductions to limit warming to 2 degrees Celsius [1, 2, 3, 4]. The few preceding quantitative economic studies analyze optimal policy in the presence of a single type of tipping point that directly affects economic output [5, 6, 7] or in the presence of a single tipping point that modifies either the carbon cycle or the climate’s sensitivity to emissions [8]. However, climate scientists have become particularly concerned with a possible “domino effect” arising through interactions among tipping points [9, 10, 11, 12, 13]. For instance, reducing the effectiveness of carbon sinks amplifies future warming, which in turn makes further tipping points more likely. Here, we solve for the optimal policy under interacting tipping points in a stochastic integrated assessment model with anticipated Bayesian 2 MODEL 2 learning. Optimality means that resources within and across periods are distributed to maximize the expected stream of global welfare from economic consumption over time under different risk states. The anticipation of learning acknowledges that future policymakers will have new information about the location of temperature thresholds and can also react to any tipping points that may have already occurred. We then calculate the welfare cost of delaying optimal climate policy. 2 Model We integrate three tipping points into the DICE integrated assessment model [14, 15]. DICE combines an economic growth model with a simplified climate module (Schematic 1, left). The policymaker decides how to allocate output between consumption, investment in capital, and emission reductions. Unabated emissions accumulate in the atmosphere where they change the radiative balance, induce further warming feedbacks, increase global average surface temperature, and thereby cause economic damages. In addition to integrating tipping points, uncertainty, and learning, we also modify DICE’s damage relationship to avoid double-counting: around half of DICE’s damages arise from an ad hoc adjustment meant to reflect the unmodeled possibility of tipping points [15], and we eliminate this adjustment in our setting with explicit tipping points. The bold arrows in Figure 1’s left schematic illustrate how our tipping points alter the climate system. The first tipping point (a) makes temperature more sensitive to CO2 emissions. It reflects the possibility that warming mobilizes large methane stores locked in permafrost and in shallow ocean clathrates [18, 19, 20, 21]. It also reflects the possibility that land ice sheets begin to retreat on decadal timescales: the resulting loss of reflective ice could double the long-term warming predicted by models that hold land ice sheets fixed [1]. DICE characterizes the temperature response to CO2 through a climate sensitivity parameter that represents the equilibrium warming from doubling CO2 . The value of 3◦ C used in DICE is inferred from climate models that hold land ice sheets and most methane stocks constant. We represent a climate feedback tipping point as increasing climate sensitivity to 5◦ C. The second tipping point (b) increases the residence time of atmospheric CO2 . Warminginduced changes in oceans [22], soil carbon dynamics [23], and standing biomass [24] could affect the uptake of CO2 from the atmosphere. We represent such a weakening of carbon sinks as reducing the rate of atmospheric CO2 removal by 50%. The third tipping point (c) directly affects the economic damage function. This damage function encapsulates all impacts from warming, including damages from sea level rise, from habitat loss, and from a weakening of the Atlantic conveyor belt (Gulf Stream). Any of these channels is subject to potential abrupt changes that would cause substantial economic damages. For example, if the West Antarctic or Greenland ice sheets collapsed, sea levels 3 2 MODEL Ancipates future observaon, updated prior, opmal consumpon and policy responses b No Tipping CO2 Forcing a No Tipping Aerosols and non-CO2 gases Safe domain expands Temp CO2 Tipping Feedback New Emissions Consumpon Capital Investment Dynamics change irreversibly Other Tipping Scenarios … Safe domain expands Emissions Temperature c Temp t Consump on New Emissions Abatement Output Capital No Tipping CO2 t CO2 t+1 Tipping Feedback New Emissions Capital Consumpon Investment Consumpon U lity Capital t Investment Safe domain expands Temp CO2 Tipping Carbon Cycle Dynamics change irreversibly Other Tipping Scenarios … Dynamics change irreversibly Capital t+1 Other Tipping Scenarios … Figure 1: Left: A simplified schematic of our DICE-based integrated assessment model. Unabated emissions from economic production accumulate in the atmosphere. Together with other greenhouse gases, the emission stock changes the energy balance of the planet, induces various feedback processes, and increases global surface temperature. The bold arrows indicate the relationship changed by each of the warming feedback (a), carbon sink (b), and damage (c) tipping points described in the main text. Right: Schematic illustrating the decision problem underlying the optimal policy choice. The policymaker anticipates that (i) a tipping point might happen, (ii) future policymakers learn about the location of temperature thresholds by observing whether past warming triggered a tipping point, (iii) future policymakers adjust to the new climate dynamics in case of tipping, and (iv) these adjustments and their consumption and welfare effects depend on the climate and capital passed on to the future policymakers. could rise quickly and dramatically [25, 26, 27, 28]. These higher sea levels would interact with the existing pathways by which warming causes damages. By stressing adaptive capacity, higher sea levels make damages increase faster with warming. We model such a tipping point as changing the DICE damage function from the conventional assumption of a quadratic in temperature to a cubic in temperature: if this tipping point occurs, then doubling man-made warming increases damages eightfold rather than fourfold. Tipping points make the policymaker uncertain about future climate dynamics and their economic impact. We employ a recursive, stochastic, dynamic programming implementation of DICE following and extending the approach of [29, 30, 31, 8]. Solving a model with three interacting tipping points requires the solution of eight nested dynamic programming problems representing the different combinations in which tipping points might possibly occur. Moreover, these tipping points can occur at different times, at different environmental or economics states, and in different orders. Today’s optimal policy has to foresee all of these possibilities and anticipate how they affect future welfare, which in turn depends on how future policy responds to tipping at a given time and state (Figure 1, right schematic). In particular, the future abatement policy affects how warming and damages evolve within a given climate regime and also the probability of triggering further 3 OPTIMAL POLICY 4 Figure 2: Optimal carbon tax in the years 2015 and 2050, assuming no tipping point has yet occurred. The columns plot scenarios without any possible tipping points, scenarios with a single possible tipping point, scenarios with two possible tipping points, and scenarios with three possible tipping points. The upper end of each bar states the optimal carbon tax, and the length of the bar gives the (average) contribution of adding the last tipping point (the lower end of the bar corresponds to the average optimal tax when eliminating one of the noted tipping points). The horizontal lines show the average carbon tax among the scenarios with only one or two tipping points. tipping points. We calibrate identical, uniform Bayesian priors for each tipping possibility so that with year 2005 information the expected warming at which the first tipping point occurs is 2.5◦ C relative to 1900. 3 Optimal Policy Figure 2 presents the optimal tax on a ton of CO2 in 2015 (left) and 2050 (right) in settings without any potential tipping points, with only a single potential tipping point, with two potential tipping points, and with all three potential tipping points. The optimal tax in 2015 nearly doubles from $6.2 per ton of CO2 in the absence of tipping concerns to $10.8 when taking the full range of tipping possibilities and their interactions into account. The damage tipping point has the strongest individual effect, increasing the optimal emission tax by 30% ($1.8). The feedback tipping point increases the optimal emission tax by 14% ($0.8), and the carbon sink tipping point increases it by 8% ($0.5). Similar findings hold for 2050: the possibility of three interacting tipping points approximately doubles the optimal carbon tax from $15 to $30 per ton of CO2 , and the damage tipping point remains the single most important. Figure 3 presents the optimal emission and temperature trajectories, conditional on not having crossed a threshold by the corresponding date. Again, recognizing the potential for a carbon sink tipping point only slightly reduces optimal 4 TIPPING POINT INTERACTIONS 5 emissions and temperature. Recognizing the potential for a feedback tipping point reduces optimal peak annual emissions by 1 Gt C and peak temperature by 0.3◦ C, and recognizing the potential for a damage tipping point reduces optimal peak annual emissions by more than 2 Gt C and peak temperature by over 0.6◦ C. Anticipating all three potential tipping points reduces optimal peak annual emissions by almost 3 Gt C and reduces the optimal peak temperature from close to 4◦ C to just below 3◦ C. Anticipating these tipping points makes optimal emissions and temperatures decline more than half a century earlier. 4 Tipping Point Interactions Our results indicate strong interactions between the different tipping points.1 Both the optimal temperature reduction and the optimal carbon tax at a given time are larger than the sum of the adjustments obtained from the individual tipping point models. The optimal carbon tax adjustment in the joint model is 50% higher in 2015 (and almost 27% higher in 2050) than suggested by simply adding the tax adjustments of the three individual tipping point models. The primary interaction is between the temperature feedback and the damage tipping points: combining these two tipping points into a single model raises the optimal 2015 tax adjustment by 35% above the sum of the individual tax adjustments. The interaction between the temperature feedback and carbon sink tipping points has the smallest effect, raising the carbon tax by 6% in 2015 above the individual sums, and the interaction between the carbon sink and the damage tipping points implies a 22% increase of the optimal tax adjustment over the sum of the adjustments deriving from the individual tipping point models. The ranking of the relative importance of the tipping point interactions persists throughout the century, with absolute adjustments increasing and relative adjustments slightly decreasing. 5 Policy Delay Our results so far demonstrate the implications of potential tipping points for optimal mitigation policy. But actual emission policy has deviated substantially from model recommendations. Figure 4 analyzes the cost of postponing optimal policy in the presence of tipping points. It shows 1 A simple analytic theory model cannot determine whether the joint policy effect of multiple tipping points is larger or smaller than the sum of the adjustments derived from individual tipping point models. On the one hand, a given policy intervention simultaneously reduces the probability of several tipping points in the joint model. On the other hand, each additional unit of emission reductions is more costly and less beneficial than the previous one. In addition, an overall larger tipping probability makes the decision maker less willing to spend more on a risk reduction because the expected benefit from saving increases: the probability of experiencing a tipped future is higher and in a tipped world additional resources are more valuable. 5 POLICY DELAY 6 Figure 3: Optimal emissions (left) and temperature (right) in settings with a single potential tipping point (blue lines) and with all three potential tipping points (red lines). the expected welfare loss from delaying policy (left), as well as the optimal carbon tax that the policymaker would have to set in year t if beginning to optimize only at that point (right). Prior to year t, the emission pathway is defined by the Representative Concentration Pathway scenario that stabilizes total radiative forcing at 6 W m−2 by 2100 (RCP6) [32, 33] and by a 22% investment rate. RCP6 exhibits initially only moderate mitigation and then more significant stabilization efforts towards the end of the century (Figure 5 in the Supplementary Material). The initial carbon tax is not affected much by a delay of less than 50 years. However, delay is expensive. Suppose policy follows a “delay with optimal tipping response” scenario: the policymaker follows the exogenous RCP6 emission pathway until 2050 if no tipping occurs, but switches to the optimal policy immediately in case of tipping before. Such a delay costs $1.5 trillion in current dollars, approximately the GDP of Australia. If the policymaker follows this “delay with optimal tipping response” scenario not just until mid-century but until the first (potential) occurrence of a tipping point, this cost increases to $3.3 trillion, almost the GDP of Germany, even though the RCP6 emission pathway implies substantial mitigation efforts by the end of the century.2 Now suppose that the policymaker follows a “delay without optimal tipping response” scenario. First, assume that the policymaker fails to react to the first tipping point that occurs but does switch to the optimal policy in case of triggering a second tipping point. Then the welfare loss increases to $7.3 trillion, about twice of Germany’s GDP. We can also view this scenario as a proxy for observing a threshold crossing only with major delay. Second, let us assume that policymakers 2 Figure 5 in the supplementary material shows that approximately a century from today the RCP6’s emission trajectory catches up with the optimal emission trajectory (conditional on no tipping point having occurred, yet). Thus, our numbers reflect the cost of delaying (not abandoning) policy even when the decision maker never switches from RCP6 to the optimal policy. 6 CONCLUSIONS 7 Figure 4: The graph presents the welfare (left) and policy (right) consequences of delaying optimal climate policy. On the left, “Delay with optimal tipping response” depicts the welfare cost from switching to optimal policy only after period t or after a first tipping point occurs (if that happens before period t). “Delay without optimal tipping response” depicts the welfare cost from switching to optimal policy only in period t, regardless of whether tipping points occur before then. The right graph presents the optimal carbon tax upon beginning to optimize policy in year t, assuming that year t has been reached without triggering a tipping point. never switch to the optimal policy, even if all three tipping points occur. Then, the welfare loss from following the suboptimal RCP6 emission pathway increases to $13.5 trillion, almost four times the GDP of Germany. The social social cost of emitting a ton of carbon today in this suboptimal policy scenario increases by 50% to $15.3 per ton of CO2 compared to the case of optimal policy (or even the case of optimal response to a first tipping point). These results demonstrate the value of a reactive policy.3 They also emphasize the necessity of evaluating policy in a framework that anticipates not only tipping possibilities but also how policy responds to future information. 6 Conclusions Our findings show that tipping points are of major quantitative importance for climate change policy. The presence of the three potential tipping points in our Bayesian learning model nearly doubles the optimal carbon tax and reduces optimal peak warming by approximately 1◦ C. Moreover, delaying optimal policy in the face of irreversible tipping points is very costly, even in a standard economic model that exhibits a conservative damage function and discounts the future costs from climate change in the usual way. 3 The supplementary material demonstrates how optimal policy responds to tipping points that happen to occur in 2075. 6 CONCLUSIONS 8 Future studies should undertake more detailed scientific and economic analysis of individual tipping points. Our results suggest that adding the optimal tax (and temperature) adjustments across separate studies provides a reasonable first-order approximation to the effect of jointly interacting tipping points, with the sum still underestimating the effect on optimal policy. Complementing the literature on early warning signals for tipping points [34, 35], we demonstrate the value of the ability to detect tipping points even after they have been irreversibly triggered. Delaying policy is expensive, but if detecting a triggered tipping point would spur nations to collectively respond so as to reduce the chance of triggering further tipping points, then the detection capability cuts the cost of policy delay by 75%. The precise quantitative results from any integrated assessment model are sensitive to a number of assumptions such as the discount rate and the damage parameterization. However, the model allows us to gain a general understanding of how tipping point interactions amplify the cost of following suboptimal policy paths and of how they affect optimal policy and the cost of delaying policy. The underlying DICE model was employed by the U.S. government in determining the social cost of carbon for use in cost-benefit assessments of proposed regulations [16, 17]. REFERENCES 9 References [1] J. Hansen, et al., The Open Atmospheric Science Journal 2, 217 (2008). [2] V. Ramanathan, Y. Feng, Proceedings of the National Academy of Sciences 105, 14245 (2008). [3] J. Rockström, et al., Nature 461, 472 (2009). [4] National Research Council, Abrupt Impacts of Climate Change: Anticipating Surprises (National Academies Press, Washington, D.C., 2013). [5] K. Keller, B. M. Bolker, D. F. Bradford, Journal of Environmental Economics and Management 48, 723 (2004). [6] T. S. Lontzek, Y. Cai, K. L. Judd, Tipping points in a dynamic stochastic IAM, RDCEP Working Paper 12-03 , The Center for Robust Decision Making on Climate and Energy Policy (2012). [7] F. van der Ploeg, A. de Zeeuw, Climate policy and catastrophic change: Be prepared and avert risk, OxCarre Working Paper 118 , Oxford Centre for the Analysis of Resource Rich Economies, University of Oxford (2013). [8] D. Lemoine, C. Traeger, American Economic Journal: Economic Policy 6, 137 (2014). [9] T. M. Lenton, et al., Proceedings of the National Academy of Sciences 105, 1786 (2008). [10] E. Kriegler, J. W. Hall, H. Held, R. Dawson, H. J. Schellnhuber, Proceedings of the National Academy of Sciences 106, 5041 (2009). [11] A. Levermann, et al., Climatic Change 110, 845 (2012). [12] E. Kopits, A. L. Marten, A. Wolverton, Moving forward with incorporating “catastrophic” climate change into policy analysis, Working Paper 13-01 , National Center for Environmental Economics, U.S. Environmental Protection Agency (2013). [13] T. M. Lenton, J.-C. Ciscar, Climatic Change 117, 585 (2013). [14] W. D. Nordhaus, Science 258, 1315 (1992). [15] W. D. Nordhaus, A Question of Balance: Weighing the Options on Global Warming Policies (Yale University Press, New Haven, 2008). REFERENCES 10 [16] Interagency Working Group on Social Cost of Carbon, Appendix 15a. social cost of carbon for regulatory impact analysis under executive order 12866, Tech. rep., United States Government (2010). [17] M. Greenstone, E. Kopits, A. Wolverton, Review of Environmental Economics and Policy 7, 23 (2013). [18] S. A. Zimov, E. A. G. Schuur, F. S. Chapin, Science 312, 1612 (2006). [19] D. C. Hall, R. J. Behl, Ecological Economics 57, 442 (2006). [20] D. Archer, Biogeosciences 4, 521 (2007). [21] K. Schaefer, T. Zhang, L. Bruhwiler, A. P. Barrett, Tellus B 63, 165 (2011). [22] C. Le Qur, et al., Science 316, 1735 (2007). [23] T. Eglin, et al., Tellus B 62, 700 (2010). [24] C. Huntingford, et al., Philosophical Transactions of the Royal Society B: Biological Sciences 363, 1857 (2008). [25] M. Oppenheimer, Nature 393, 325 (1998). [26] M. Oppenheimer, R. B. Alley, Climatic Change 64, 1 (2004). [27] D. G. Vaughan, Climatic Change 91, 65 (2008). [28] D. Notz, Proceedings of the National Academy of Sciences 106, 20590 (2009). [29] D. L. Kelly, C. D. Kolstad, Journal of Economic Dynamics and Control 23, 491 (1999). [30] A. J. Leach, Journal of Economic Dynamics and Control 31, 1728 (2007). [31] C. Traeger, Environmental and Resource Economics Forthcoming (2014). [32] R. H. Moss, et al., Nature 463, 747 (2010). [33] T. Masui, et al., Climatic Change 109, 59 (2011). [34] M. Scheffer, et al., Nature 461, 53 (2009). [35] M. Scheffer, et al., Science 338, 344 (2012). [36] J. B. Smith, et al., Proceedings of the National Academy of Sciences 106, 4133 (2009). [37] P. Valdes, Nature Geoscience 4, 414 (2011). Supplementary Information A The RCP 6 Scenario, Optimal Policy, and Post-Threshold Policy Reaction Figure 5 compares emissions along the IPCC’s RCP6 scenario and our optimal emission trajectory (on the left). The RCP6 scenario initially delays policy, but incurs major reduction measures Figure 5: Emissions and temperature under the exogenous RCP6 scenario (dashed) and under the optimal policy (solid) when there are three potential tipping points. toward the end of the century to stabilize radiative forcing (and, thus, temperature; on the right). The optimal policy shown in Figure 5 is conditional on not having crossed a tipping point. Figure 6 depicts how policy and the climate system respond if a tipping point occurs in 2075. The decision maker’s stochastic knowledge did not allow him to anticipate the precise crossing, but he recognizes the start of the regime shift and the optimal policy responds immediately as shown in the figure. After 2075, the other two tipping points may still occur and the depicted policy anticipates this possibility. Our depicted policy representation assumes that no further tipping point occurs by 2150: the decision maker is “lucky”, but cannot count on his luck while setting policy. The top left panel of Figure 6 show that the optimal carbon tax increases once any of the three tipping points has occurred. The optimal tax increases most strongly after the damage tipping point occurs. The greater abatement effort translates into greater emission reductions than in the case where no tipping point happens to occur (top right panel). These reduced emissions in turn lead to sharply lower CO2 concentrations under the feedback and damage tipping points (bottom left panel); however, the greater persistence of CO2 in the wake of the carbon sink tipping point 11 A THE RCP 6 SCENARIO, OPTIMAL POLICY, AND POST-THRESHOLD POLICY REACTION12 Figure 6: The effect of a tipping point on post-threshold policy and on the climate system. Plotted simulations assume that all three tipping points are possible and that a particular one happens to occur in 2075 as the first tipping point. means that the post-tipping CO2 trajectory is very similar to the no-tipping CO2 trajectory, despite the reduction in emissions. The temperature trajectory is therefore also similar whether or not a carbon sink tipping point happens to occur (bottom right panel). It is sharply lower in the wake of a damage tipping point; however, the feedback tipping point’s effect on climate sensitivity outweighs the emission reductions undertaken after it occurs, so that optimal warming is greatest in the wake of a feedback tipping point. B B 13 METHODS Methods Model Summary We reformulate the DICE-2007 model of [15] into a recursive dynamic programming problem following [29] and [31]. In response to the curse of dimensionality in dynamic programming, we reduce the state space of the climate system’s representation. [31] shows that this simplification does not impair the model’s ability to reproduce the DICE model’s climate response. We do make one substantive change to DICE, besides the introduction of tipping points. [15] adjusts a coefficient in the damage function to compensate for not explicitly modeling climate catastrophes and tipping points. This smooth and reversible damage adjustment contributes about half of DICE’s damages at 1◦ C of warming. We eliminate this adjustment because we now explicitly model tipping points. We introduce tipping points as regime shifts following [8]. That paper provides more details about the modeling of the probability of tipping and of learning. In each period, the climate system might irreversibly trigger any of the possible tipping regimes. The Bayesian policymaker assesses the probability of tipping by updating its previous beliefs based on whether it has just observed a tipping point or not. Each tipping point occurs with identical, independent probability in any given state of the world, and the hazard of tipping is determined by the global average temperature increase. These assumptions induce a binomial distribution over the number of tipping points triggered between any two periods. When there are n potential tipping points, the probability of crossing k of them between times t and t+1 is: Hkn (Tt , Tt+1 ) = n! k! (n−k)! [h(Tt , Tt+1 )]k [1 − h(Tt , Tt+1 )]n−k . The function h(Tt , Tt+1 ) is the time t hazard rate for a single tipping point as a function of temperature at times t and t + 1. Currently, scientific modeling cannot suggest that some specific temperature is a more likely threshold than another similar temperature [10, 36, 37]. Our policymaker therefore believes that each tipping point’s temperature threshold is uniformly distributed between oa fixed n min{Tt+1 ,T̄ }−Tt . The upper bound T̄ and a lower bound subject to learning: h(Tt , Tt+1 ) = max 0, T̄ −T t hazard rate increases with Tt+1 because greater warming over the next interval carries a greater risk of tipping over the next interval. The hazard rate’s dependence on Tt reflects that if a tipping point is still possible, then its threshold has not been crossed yet and the policymaker has learned that its threshold is either above the current temperature or does not exist at all. We calibrate the threshold distributions so that in a model with three potential thresholds, the policymaker with year 2005 information expects that 2.5◦ C of warming relative to 1900 would trigger some tipping point. This requirement implies an upper bound T̄ of 7.8◦ C, which is among the highest values explored in the sensitivity analysis for the single-tipping runs in [8]. An expected trigger of 2.5◦ C is consistent with the political 2◦ C limits for avoiding dangerous anthropogenic interference. Further, in the “burning embers” diagram [36], the yellow (medium-risk) shading for B 14 METHODS the “risk of large-scale discontinuities” begins around 1.6◦ C of warming relative to 1900, and the red (high-risk) shading begins around 3.1◦ C of warming relative to 1900. We solve the tipping scenarios recursively, starting with a world where all tipping thresholds have been crossed. We solve the corresponding Bellman equation by value function iteration and approximate the value function using a 104 -dimensional Chebychev-basis. Each iteration uses the knitro solver to optimize abatement and consumption decisions at the Chebychev nodes. Once we have successfully approximated that value function, we use it to define the post-tipping continuation value in the scenarios where all but one tipping point has been crossed. We then approximate the value functions for those scenarios following the procedure described above. Once we have successfully approximated these value functions, they provide additional continuation values in scenarios where all but two tipping points have been crossed. We then approximate these scenarios’ value functions, proceeding in this fashion until we approximate the value function for a scenario in which no tipping point has yet occurred. Model Equations The state variables are effective capital kt , atmospheric CO2 Mt , cumulative temperature change Tt , and time t. Throughout, bold parameters indicate parameters that are altered by tipping points. Effective capital kt = Kt Lt A t is capital Kt measured per amount of labor Lt and labor augmenting technology At level. Labor and technology follow the exogenous equations gA,0 −tδA 1−e , At =A0 exp δA gA,t =gA,0 e−tδA , (Production technology) (Growth rate of production technology) Lt =L0 + (L∞ − L0 ) 1 − e−tδL , −1 L∞ tδL . e −1 gL,t =δL L∞ − L0 (Labor) (Growth rate of labor) Gross production in the economy is Ytgross = (At Lt )1−κ Ktκ , or in effective units ytgross = ktκ . Climate change damages reduce the effective gross output to the level yt 1+Dt (TT ) , where the damage function Dt (Tt ) = d1 Ttd2 (Damages) measures output loss as percentage of total production. The damage tipping point increases the B 15 METHODS coefficient d2 from 2 to 3. The remaining output is divided between effective consumption ct = Ct Lt A t , abatement expenditure, and capital investment. The equation of motion for capital is kt+1 =e −(gL,t +gA,t ) yt − ct . (1 − δk )kt + (1 − Λ(µt )) 1 + Dt (Effective capital) The function Λ(µt ) = Ψt µat 2 with a0 σt Ψt = a2 1 − etgΨ 1− a1 (Abatement cost) measures the abatement cost as a function of the abatement rate µt . The abatement rate µt measures the policy-induced emission reductions at time t as a fraction of the business-as-usual emissions. The parameter σt =σ0 exp gσ,0 1 − e−tδσ δσ (Uncontrolled emissions per output) measures the time t emission intensity of production. The CO2 concentration follows the equation of motion Mt+1 =Et + Mt b11 + b21 [b12 + (b22 + b32 b23 ) αB (Mt , t) + b32 b33 αO (Mt , t)] , (CO2 transition) where the b parameters govern DICE’s carbon transition matrix. The function αB (M, t) represents the combined biosphere and shallow ocean carbon stock as a fraction of the atmospheric stock, and the function αO (M, t) represents the deep ocean carbon stock as a fraction of the atmospheric stock. We estimate these functions using a set of emission scenarios simulated in the full version of DICE [8]. This CO2 transition equation can be reduced to the following form: Mt+1 = Et + [1 − δM (Mt , t)] Mt , , where δM (Mt , t) gives the carbon dioxide removal rate as a function of CO2 and time. Along the first 100 years of the optimal pre-threshold policy path, it averages 0.0056. The carbon sink tipping point reduces this removal rate by 50%. The emissions Et = σt (1 − µt )Ytgross + Bt (Emissions) are unabated CO2 emissions from fossil fuel use and CO2 emissions from land use change and B 16 METHODS forestry. The latter evolve exogenously according to Bt =B0 etgB . (Non-industrial CO2 emissions) A change in the atmospheric CO2 concentration causes a shift in the earth’s energy balance expressed by the radiative forcing F (Mt , t) = f ln (Mt /Mpre ) + EFt , where f is the forcing from doubled CO2 and Mpre is the preindustrial CO2 concentration. The exogenous forcing EFt includes non-CO2 greenhouse gases and aerosols. It evolves according to EFt =EF0 + 0.01(EF100 − EF0 ) min{t, 100}. (Non-CO2 forcing) Radiative forcing changes the temperature increase above 1900 levels as Tt+1 =Tt + CT f F (Mt+1 , t + 1) − Tt − [1 − αT (Tt , t)] CO Tt , s (Temperature transition) a function of the lagged temperature, the carbon concentration, and the exogenous forcing from other greenhouse gases. Ocean cooling enters through both the calibration of the heat capacity parameters CT and CO as well as through the direct cooling function αT (Tt , t) that, following [8], we interpolate from different scenarios that we simulated with the full DICE model, which explicitly keeps track of the ocean temperature. The parameter s in the temperature transition equation is climate sensitivity, or the equilibrium temperature change from doubling CO2 concentrations. The climate feedback tipping point increases this parameter from 3 to 5. The initial conditions correspond to 2015, even though DICE-2007 begins in the year 2005. We obtain these initial conditions by simulating the model for 10 years with RCP6 emissions and a 22% investment rate, as described below. The resulting values for the climate variables are similar to recent projections. Value Functions and Solution Method We solve the model using stochastic dynamic programming. The solution algorithm ensures that the decision maker always has in mind the complete set of possible futures, including the optimal future reactions to tipping points, when choosing her optimal policy in a given period. We recursively solve for different sets of value functions, starting with the case where all tipping points have occurred and working back to the present situation without any threshold yet being crossed. We will describe the solution method for a case with three potential tipping points. The methods for settings with fewer potential tipping points follow straightforwardly. 0 (k , M , T , t) the value function in the world where three tipping points (labeled Denote by V1,2,3 t t t B 17 METHODS Table 1: Parameterization of the numerical model following DICE-2007. Several values are rounded, and CT and δκ vary slightly over time in order to reproduce the DICE results with an annual timestep. Parameter Value Description A0 gA,0 δA 0.027 0.009 0.001 Production technology in 2005 Annual growth rate of production technology in 2005 Annual rate of decline in growth rate of production technology L0 L∞ δL 6514 8600 0.035 Population in 2005 (millions) Asymptotic population (millions) Annual rate of convergence of population to asymptotic value σ0 0.13 gσ,0 δσ -0.0073 0.003 Emission intensity in 2005 before emission reductions (Gt C per unit output) Annual growth rate of emission intensity in 2005 Annual change in growth rate of emission intensity a0 a1 a2 gΨ 1.17 2 2.8 -0.005 Cost of backstop technology in 2005 ($1000 per t C) Ratio of initial backstop cost to final backstop cost Abatement cost exponent Annual growth rate of backstop cost B0 gB 1.1 -0.01 Annual non-industrial CO2 emissions (Gt C) in 2005 Annual growth rate of non-industrial emissions EF0 EF100 -0.06 0.30 Forcing in 2005 from non-CO2 agents (W m−2 ) Forcing in 2105 from non-CO2 agents (W m−2 ) κ δκ d1 d2 s f Mpre CT CO 0.3 0.06 0.0019 2 3 3.8 596.4 0.03 0.3 Capital elasticity in Cobb-Douglas production function Annual depreciation rate of capital Coefficient on temperature in the damage function Exponent on temperature in the damage function Climate sensitivity (◦ C) Forcing from doubled CO2 (W m−2 ) Pre-industrial atmospheric CO2 (Gt C) Translation of forcing into temperature change Translation of surface-ocean temperature gradient into forcing b11 ,b12 ,b13 b21 ,b22 , b23 0.978,0.023,0 0.011,0.983,0.005 b31 ,b32 ,b33 0,0.0003,0.9997 Transfer coefficients for carbon from the atmosphere Transfer coefficients for carbon from the combined biosphere and shallow ocean stock Transfer coefficients for carbon from the deep ocean ρ η 0.015 2 Annual pure rate of time preference Relative risk aversion (also aversion to intertemporal substitution) k0 M0 T0 187/(A10 L10 ) 865.98 0.915 Effective capital in 2015, with 187 US$trillion of capital Atmospheric carbon dioxide (Gt C) in 2015 Surface temperature (◦ C) in 2015, relative to 1900 B 18 METHODS 1, 2, and 3) have already occurred and no further ones may occur. We obtain this value function by solving the Bellman equation 0 V1,2,3 (kt ,Mt , Tt , t) = max ct ,µt ct1−η 0 + βt V1,2,3 (kt+1 , Mt+1 , Tt+1 , t + 1) 1−η (1) subject to the equations summarized in section B. We solve the Bellman equation by function iteration using a four dimensional tensor basis of Chebychev polynomials to approximate the value function. The effective discount factor βt = exp (−ρ + (1 − η)gA,t + gL,t ) (Effective discount factor) accounts for the per effective unit of labor transformations of the Bellman equation [31]. The optimization over consumption and abatement has to satisfy the constraints ct +Ψt µat 2 Yt Yt ≤ , 1 + Dt 1 + Dt µt ≤ 1 . (Output constraint) (Non-negativity constraint for emissions) The solution of equation 1 gives us the optimal policies in a world where all tipping points have occurred, as well as the maximal welfare that can be achieved over the infinite time horizon in such a world starting from any given state. In the second step, we derive the value functions for the three worlds where two tipping points have already occurred and one is still possible. We denote these value functions by V i (kt , Mt , Tt , t), i , i, j, k ∈ {1, 2, 3} where i ∈ {1, 2, 3} labels the tipping point that is still possible. V i is short for Vj,k 0 with i 6= j < k 6= i. Given V1,2,3 derived in the first step, we solve the three Bellman equations V i (kt , Mt , Tt , t) = max ct ,µt ct1−η + βt [1 − H11 (Tt , Tt+1 )] V i (kt+1 , Mt+1 , Tt+1 , t + 1) 1−η + 0 H11 (Tt , Tt+1 ) V1,2,3 (kt+1 , Mt+1 , Tt+1 , t + 1) for V i , i ∈ {1, 2, 3}. In the third step, we let Vi (kt , Mt , Tt , t) denote the value functions in the world where tipping point i ∈ {1, 2, 3} has already occurred and the other two are still possible. Vi is short for 0 Vij,k (kt , Mt , Tt , t) with i 6= j < k 6= i. Given the value functions V i and V1,2,3 obtained in the earlier B 19 METHODS steps, we derive the value functions Vi , i ∈ {1, 2, 3} by solving the three Bellman equations ct1−η + βt 1 − 2H12 (Tt , Tt+1 ) − H22 (Tt , Tt+1 ) Vi (kt+1 , Mt+1 , Tt+1 , t + 1) Vi (kt , Mt , Tt , t) = max ct ,µt 1 − η h i + H12 (Tt , Tt+1 ) V k (kt+1 , Mt+1 , Tt+1 , t + 1) + V j (kt+1 , Mt+1 , Tt+1 , t + 1) 2 0 + H2 (Tt , Tt+1 ) V1,2,3 (kt+1 , Mt+1 , Tt+1 , t + 1) where k 6= i 6= j 6= k. Finally, we obtain the value function V01,2,3 and optimal policies in the pre-threshold regime, where no tipping point has yet occurred. This value function solves the Bellman equation V01,2,3 (kt , Mt , Tt , t) = max ct ,µt ct1−η + βt 1 − 3H13 (Tt , Tt+1 ) − 3H23 (Tt , Tt+1 ) − H33 (Tt , Tt+1 ) 1−η × V01,2,3 (kt+1 , Mt+1 , Tt+1 , t + 1) + + 3 X i=1 3 X H13 (Tt , Tt+1 )Vi (kt+1 , Mt+1 , Tt+1 , t + 1) H23 (Tt , Tt+1 )V i (kt+1 , Mt+1 , Tt+1 , t + 1) i=1 + 0 H33 (Tt , Tt+1 ) V1,2,3 (kt+1 , Mt+1 , Tt+1 , t + 1) . The approximation intervals in the no-tipping runs cover effective capital values from 3 to 6, atmospheric carbon stocks from 700 to 1800 Gt C, and temperature levels from 0.5 to 4◦ C above 1900 (plus the infinite time horizon mapped to the unit interval). We must vary the approximation intervals for the pre- and post-threshold value functions in order to accommodate changes in dynamics and curvature due to tipping points. So that the continuation values are contained within valid approximation regions, it is crucial that any value function with some number (potentially zero) of tipping points already crossed and tipping point j remaining possible is approximated over a weakly smaller interval than any value function with those same tipping points already crossed and also having tipping point j already crossed. Much of the numerical burden lies in finding combinations of approximation intervals that allow each of the required value function iteration steps to converge, where convergence is as described in [8]. Cases in which feedback and carbon sink tipping points have already been crossed typically require larger approximation intervals because the new dynamics tend to drive the carbon and temperature processes through wider regions of the state space. On the other hand, cases in which the damage tipping point has already been crossed B METHODS 20 typically require narrower approximation intervals due to the tendency of the value function to become strongly curved at high temperatures. In the setting with only a single possible tipping point, all post-threshold value functions extend the effective capital interval upwards to 7, extend the carbon interval upwards to 3000 Gt C, and extend the temperature interval upwards to 6◦ C. The pre-threshold intervals are the same as in the no-policy case when the single tipping point is a feedback or carbon sink tipping point, but the case with a damage tipping point uses an interval with an upper bound of 7 for effective capital and of 2000 Gt C for atmospheric carbon. Now consider a setting with two potential tipping points. When both have been crossed, the approximation interval extends up to 7 in the effective capital domain, extends up to 3000 Gt C in the carbon domain, and extends up to 6◦ C in the temperature domain (8◦ C if the two tipping points are the feedback and carbon sink tipping points). When only one has been crossed and the damage tipping point is not the remaining one, the approximation interval extends up to 7 in the effective capital domain, up to 2500 Gt C in the carbon domain, and up to 5◦ C in the temperature domain. The interval is the same when none have been crossed and the damage tipping point is not possible, except the temperature interval does not extend past 4◦ C. When only one tipping point has been crossed and the damage tipping point remains possible, the approximation interval extends up to 7 in the effective capital domain, up to 2000 Gt C in the carbon domain, and up to 4◦ C in the temperature domain. The interval is the same when no tipping points have been crossed and the damage tipping point is one of the two that is possible. Finally, consider the full setting with three potential tipping points. When all three have already been crossed, the approximation interval extends up to 7 in the effective capital domain, up to 3000 Gt C in the carbon domain, and up to 6◦ C in the temperature domain. When only two tipping points have been crossed, the approximation interval extends up to 7 in the effective capital domain, up to 2500 Gt C in the carbon domain, and up to 5◦ C in the temperature domain (or 2000 Gt C and 4◦ C if the damage tipping point is the one that has not been crossed). When only a single tipping point has been crossed, the approximation interval extends up to 7 in the effective capital domain, up to 2000 Gt C in the carbon domain, and up to 4◦ C in the temperature domain. This same interval applies to the case in which no tipping points have been crossed. Our results for present-day policy directly use the optimal value function and policy choices in the year 2015. The optimal policies incorporate the possibility that any of the tipping points might occur in any given year in the future, with the tipping probabilities depending on the evolution of the climate states. The plotted paths simulating policy and climate characteristics into the future, and also the plotted optimal policy values for 2050, rely on the additional assumption that no tipping point has occurred by the given time. We use the equations of motion summarized in B METHODS 21 section B and the optimal policies derived above to simulate the paths. We calculate the welfare loss from following RCP6 instead of the optimal scenario by replacing the optimal policy in a given stage with a fixed 22% investment rate and RCP6’s fixed emission trajectory, whose annual values we obtain from MAGICC 6.0 (see Figure 5in the Supplementary Material). For these exogenous policy trajectories, we sum welfare over each year up to the time when optimal policy begins, with the approximated value functions determining welfare from that time forward. In cases where the policymaker reacts to some threshold crossing, we obtain the summation for the period with RCP6 emissions by starting at the initial year and calculating perperiod utility along every possible path. In cases where the policymaker never reacts to a threshold crossing, the large number of possible tipping paths makes this approach computationally impractical if policy is delayed by more than 40 years and nearly interminable if policy is delayed by more than 70 years. In those cases with a nonresponsive policymaker, we reduce the number of necessary calculations by using value function approximations to determine welfare along those potential paths for which at least one tipping point has occurred. More specifically, we use Chebychev polynomials (as described above) to approximate welfare after having crossed each possible combination of tipping points at any potential combination of state variables, assuming optimal policy begins at a particular year of interest. This approach yields nearly identical results to the explicit calculation approach for all simulations (i.e., periods of delay up to and including 70 years) for which we were able to obtain results under both approaches. Reported results for which optimal policy never begins in the absence of triggered tipping points are implemented by delaying policy for 400 years.