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Halting the Climate Dominoes:

Mitigation Policy Under Interacting Tipping Points

Derek Lemoine & Christian P. Traeger

WCERE June 2014

Scientific research highlights the possibility that man-made greenhouse gas emis-

sions can trigger irreversible regime shifts in the climate system, so-called tipping

points. The literature is particularly concerned with the possibility that a first regime

shift, e.g., reinforcing the temperature response to emissions, significantly increases

the likelihood of other tipping points, starting a domino effect. However, economic

analyses have considered only one type of tipping point at a time, not allowing for

such interaction. The present paper is the first to assess climate policy in the presence

of multiple irreversible, interacting tipping points. We find that the presence of three

commonly discussed tipping points nearly doubles the currently optimal carbon tax

and reduces warming along the optimal path by approximately 1◦C. Delaying optimal

policy in the face of possible tipping points is expensive. Coordinating on optimal

policy once a first tipping point occurs can help halting the domino effect; failing to

respond optimally increases the expected cost of policy delay almost fivefold. Beyond

the search for early warning signals of future tipping points, scientific monitoring for

already triggered tipping points creates economic value by allowing mitigation policy

to reduce the chance that any one initial tipping point triggers a cascade.

The threat of climate tipping points plays a major role in calls for aggressive emission reductionsto limit warming to 2 degrees Celsius [1, 2, 3]. The few preceding quantitative economic studiesanalyze a single type of tipping point that directly affects economic output [4, 5, 6] or analyze asingle tipping point that modifies either the carbon cycle or the climate’s sensitivity to emissions[7]. However, climate scientists have become particularly concerned with a possible “domino effect”arising through interactions among tipping points [8, 9, 10, 11, 12]. For instance, either reducingthe effectiveness of carbon sinks or increasing climate sensitivity would amplify future warming,which in turn makes further tipping points more likely. Here, we solve for the optimal policy underinteracting tipping points in a stochastic integrated assessment model with anticipated Bayesianlearning. Optimality means that resources within and across periods are distributed to maximize theexpected stream of global welfare from economic consumption over time under different risk states.The anticipation of learning acknowledges that future policymakers will have new informationabout the location of temperature thresholds and can also react to any tipping points that mayhave already occurred.

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Lemoine & Traeger WCERE June 2014

We integrate three tipping points into the DICE integrated assessment model [13, 14]. DICEcombines an economic growth model with a simplified climate module (Schematic 1, left). The pol-icymaker decides how to allocate output between consumption, investment in capital, and emissionreductions. Unabated emissions accumulate in the atmosphere where they change the radiative bal-ance, induce further warming feedbacks, increase global average surface temperature, and therebycause economic damages. In addition to integrating tipping points, uncertainty, and learning, wealso modify DICE’s damage relationship to avoid double-counting: around half of DICE’s damagesarise from an ad hoc adjustment meant to reflect the unmodeled possibility of tipping points [14],and we eliminate this adjustment in our setting with explicit tipping points. The precise quanti-tative results from any integrated assessment model are sensitive to a number of assumptions suchas the discount rate. However, the model allows us to gain a general understanding of how tippingpoint interactions amplify the cost of following suboptimal policy paths, and how they affect opti-mal policy and policy delay. The underlying DICE model was employed by the U.S. governmentin determining the social cost of carbon for use in cost-benefit assessments of proposed regulations[15, 16].

The bold arrows in Figure 1’s left schematic illustrate how our tipping points alter the climatesystem. The first tipping point (a) makes temperature more sensitive to CO2 emissions. It reflectsthe possibility that warming mobilizes large methane stores locked in permafrost and in shallowocean clathrates [17, 18, 19, 20]. It also reflects the possibility that land ice sheets begin to retreat ondecadal timescales: the resulting loss of reflective ice could double the long-term warming predictedby 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 doublingCO2. The value of 3◦C used in DICE is inferred from climate models that hold land ice sheets andmost methane stocks constant. We represent a climate feedback tipping point as increasing climatesensitivity to 5◦C.

The second tipping point (b) increases the residence time of atmospheric CO2. Warming-induced changes in oceans [21], soil carbon dynamics [22], and standing biomass [23] could affect theuptake of CO2 from the atmosphere. We represent such a weakening of carbon sinks as reducing therate of atmospheric CO2 removal by 50%. The third tipping point (c) directly affects the economicdamage function. This damage function encapsulates all impacts from warming, including damagesfrom sea level rise, from habitat loss, and from a weakening of the Atlantic conveyor belt (GulfStream). Any of these channels is subject to potential abrupt changes that would cause substantialeconomic damages. For example, if the West Antarctic or Greenland ice sheets collapsed, sea levelscould rise quickly and dramatically [24, 25, 26, 27]. These higher sea levels would interact with theexisting pathways by which warming causes damages. By stressing adaptive capacity, higher sealevels make damages increase faster with warming. We model such a tipping point as changing theDICE damage function from the conventional assumption of a quadratic in temperature to a cubicin temperature: if this tipping point occurs, then doubling man-made warming increases damageseightfold rather than fourfold.

Tipping points make the policymaker uncertain about future climate dynamics and their eco-nomic impact. We employ a recursive, stochastic, dynamic programming implementation of DICEfollowing and extending the approach of [28, 29, 30, 7]. Solving a model with three interactingtipping points requires the solution of eight nested dynamic programming problems representing

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An�cipates future observa�on, updated prior, op�mal consump�on and policy responses

No Tipping

Safe domain expands

Tipping

Feedback

Dynamics change irreversibly

Temp t

CO2 t

New Emissions

Consump�on

InvestmentOther Tipping Scenarios …

Safe domain expands

Tipping

Carbon

Cycle Dynamics change irreversibly

Temp

CO2

New Emissions

Consump�on


No Tipping

Safe domain expands

Tipping

Feedback

Dynamics change irreversibly

Temp

CO2

New Emissions

Consump�on


No Tipping

Capital t+1

CO2 t+1

Capital t

Capital

Capital

Figure 1: The schematic on the left presents the simplified structure of our DICE-based integrated assess-ment model. Unabated emissions from economic production accumulates in the atmosphere. Together withother greenhouse gases, the emission stock changes the energy balance of the planet, induces various feedbackprocesses, and increases global surface temperature. The bold arrows indicate the relationship changed byeach of the warming feedback (a), carbon sink (b), and damage (c) tipping points described in the maintext. The schematic on the right illustrates the decision problem underlying the optimal policy choice. Thepolicymaker anticipates that (i) a tipping point might happen, (ii) future policymakers learn about the lo-cation of temperature thresholds by observing whether past warming triggered a tipping point, (iii) futurepolicymakers adjust to the new climate dynamics in case of tipping, and (iv) these adjustments and theirconsumption and welfare effects depend on the climate and capital passed on to the future policymakers.

the different combinations in which tipping points might possibly occur. Moreover, these tippingpoints can occur at different times, at different environmental or economics states, and in differentorders. The optimal policy today has to foresee all of these possibilities and anticipate how theyaffect future welfare, which in turn depends on how future policy responds to tipping at a giventime and state (Schematic 1, right). In particular, the future abatement policy affects how warmingand damages evolve within a given climate regime and also the probability of triggering furthertipping points. We calibrate identical, uniform Bayesian priors for each tipping possibility so thatwith year 2005 information the expected warming at which the first tipping point occurs is 2.5◦Crelative to 1900.

Figure 2 presents the optimal tax on a ton of CO2 in 2015 (left) and 2050 (right) in settingswithout any potential tipping points, with only a single potential tipping point, with two potentialtipping points, and with all three potential tipping points. The optimal tax in 2015 nearly doublesfrom $6.2 per ton of CO2 in the absence of tipping concerns to $10.8 when taking the full range oftipping possibilities and their interactions into account. The damage tipping point has the strongestindividual effect, increasing the optimal emission tax by 30% ($1.8). The feedback tipping pointincreases the optimal emission tax by 14% ($0.8), and the carbon sink tipping point increases itby 8% ($0.5). Similar findings hold for 2050: the possibility of three interacting tipping pointsapproximately doubles the optimal carbon tax from $15 to $30 per ton of CO2, and the damagetipping point remains the single most important. Figure 3 presents the optimal emission and

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Figure 2: Optimal carbon tax in the years 2015 and 2050, assuming no tipping point has yet occurred. Thecolumns 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 ofeach bar states the optimal carbon tax, and the length of the bar gives the (average) contribution of addingthe last tipping point (the lower end of the bar corresponds to the average optimal tax when eliminating oneof the noted tipping points). The horizontal lines show the average carbon tax among the scenarios withonly one or two tipping points.

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 optimalemissions and temperature. Recognizing the potential for a feedback tipping point reduces optimalpeak annual emissions by 1 Gt C and peak temperature by 0.3◦C, and recognizing the potentialfor a damage tipping point reduces optimal peak annual emissions by more than 2 Gt C and peaktemperature by over 0.6◦C. Anticipating all three potential tipping points reduces optimal peakannual emissions by almost 3 Gt C and reduces the optimal peak temperature from close to 4◦Cto just below 3◦C. Anticipating these tipping points makes optimal emissions and temperaturesdecline more than half a century earlier.

Our results indicate strong interactions between the different tipping points.1 Both the optimaltemperature reduction and the optimal carbon tax at a given time are larger than the sum of theadjustments obtained from the individual tipping point models. The optimal carbon tax adjustmentin the joint model is 50% higher in 2015 (and almost 27% higher in 2050) than suggested by simplyadding the tax adjustments of the three individual tipping point models. The primary interactionis between the temperature feedback and the damage tipping points: combining these two tipping

1A 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.

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points into a single model raises the optimal 2015 tax adjustment by 35% above the sum of theindividual tax adjustments. The interaction between the temperature feedback and carbon sinkweakening tipping points has the most moderate effect raising the carbon tax by 6% in 2015above the individual sums, and the interaction between the carbon sink and the damage tippingpoints implies a 22% increase of the optimal tax adjustment over sum of adjustments deriving fromthe individual tipping point models. The ranking of the relative importance of the tipping pointinteractions persists throughout the century, with absolute adjustments increasing and relativeadjustments slightly falling.

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).

Our results so far demonstrate the implications of potential tipping points for optimal mitiga-tion 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 showsthe expected welfare loss from delaying policy (left), as well as the optimal carbon tax that thepolicymaker would have to set in year t if beginning to optimize only at that point (right). Prior toyear t, the emission pathway is defined by the Representative Concentration Pathway scenario thatstabilizes total radiative forcing at 6 W m−2 by 2100 (RCP6) [31, 32] and by a 22% investment rate.RCP6 exhibits initially only moderate mitigation and then more significant stabilization efforts to-wards the end of the century (Figure 5 in the Supplementary Material). The initial carbon tax isnot affected much by a delay of less than 50 years. However, delay is expensive. Suppose policyfollows a “delay with optimal tipping response” scenario: the policymaker follows the exogenousRCP6 emission pathway until 2050 if no tipping occurs, but switches to the optimal policy immedi-ately in case of tipping before. Such a delay costs $1.5 trillion in current dollars, approximately theGDP of Australia. If the policymaker follows this “delay with optimal tipping response” scenarionot just until mid-century but until the first (potential) occurrence of a tipping point, this costincreases to $3.3 trillion, almost the GDP of Germany, even though the RCP6 emission pathwayimplies substantial mitigation efforts by the end of the century.

Now suppose that the policymaker follows a “delay without optimal tipping response” scenario.

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Figure 4: The graph presents the welfare and policy consequences of delaying optimal climate policy. Theleft graph presents the welfare loss from delaying optimal policy. “Delay with optimal tipping response”depicts the welfare cost from switching to optimal policy only after period t or after a first tipping pointoccurs (if that happens before period t). “Delay without optimal tipping response” depicts the welfare costfrom 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 thatyear t has been reached without triggering a tipping point.

First, assume that the policymaker fails to react to the first tipping point that occurs but doesswitch to the optimal policy in case of triggering a second tipping point. Then the welfare lossincreases to $7.3 trillion, about twice of Germany’s GDP. We can also view this scenario as a proxyfor observing a threshold crossing only with major delay. Second, let us assume that policymakersnever switch to the optimal policy, even if all three tipping points occur. Then, the welfare lossfrom following the suboptimal RCP6 emission pathway increases to $13.5 trillion, almost four timesthe GDP of Germany. The social social cost of emitting a ton of carbon today in this suboptimalpolicy scenario increases by 50% to $15.3 per ton of CO2 compared to the case of optimal policy (oreven the case of optimal response to a first tipping point). These results demonstrate the value of areactive policy. It also emphasizes the necessity of evaluating policy in a framework that anticipatesnot only tipping possibilities, but also how policy responds to future information.2

Our findings show that tipping points are of major quantitative importance for climate changepolicy. The presence of the three potential tipping points in our Bayesian learning model nearlydoubles the optimal carbon tax and reduces optimal peak warming by approximately 1◦C. Futurestudies should undertake more detailed scientific and economic analysis of individual tipping points.Our results suggest that adding the optimal tax (and temperature) adjustments across separatestudies provides a reasonable first-order approximation to the effect of jointly interacting tippingpoints, with the sum still underestimating the effect on optimal policy. Further, complementingthe previous literature’s focus on the potential for early warning of tipping points [33, 34], we

2The supplementary material demonstrates how optimal policy responds to tipping points that happen to occur

in 2075.

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demonstrate the value of the ability to detect tipping points even after they have been irreversiblytriggered. Delaying policy is expensive, but if detecting a triggered tipping point would spurnations to collectively respond so as to reduce the chance of triggering further tipping points, thenthe detection capability cuts the cost of policy delay by 75%.

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. Rockstrom, et al., Nature 461, 472 (2009).

[4] K. Keller, B. M. Bolker, D. F. Bradford, Journal of Environmental Economics and Manage-

ment 48, 723 (2004).

[5] 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).

[6] F. van der Ploeg, A. de Zeeuw, Climate policy and catastrophic change: Be prepared andavert risk, OxCarre Working Paper 118 , Oxford Centre for the Analysis of Resource RichEconomies, University of Oxford (2013).

[7] D. Lemoine, C. Traeger, American Economic Journal: Economic Policy 6, 137 (2014).

[8] T. M. Lenton, et al., Proceedings of the National Academy of Sciences 105, 1786 (2008).

[9] E. Kriegler, J. W. Hall, H. Held, R. Dawson, H. J. Schellnhuber, Proceedings of the National

Academy of Sciences 106, 5041 (2009).

[10] A. Levermann, et al., Climatic Change 110, 845 (2012).

[11] E. Kopits, A. L. Marten, A. Wolverton, Moving forward with incorporating “catastrophic”climate change into policy analysis, Working Paper 13-01 , National Center for EnvironmentalEconomics, U.S. Environmental Protection Agency (2013).

[12] T. M. Lenton, J.-C. Ciscar, Climatic Change 117, 585 (2013).

[13] W. D. Nordhaus, Science 258, 1315 (1992).

[14] W. D. Nordhaus, A Question of Balance: Weighing the Options on Global Warming Policies

(Yale University Press, New Haven, 2008).

[15] Interagency Working Group on Social Cost of Carbon, Appendix 15a. social cost of carbon forregulatory impact analysis under executive order 12866, Tech. rep., United States Government(2010).

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[16] M. Greenstone, E. Kopits, A. Wolverton, Review of Environmental Economics and Policy 7,23 (2013).

[17] S. A. Zimov, E. A. G. Schuur, F. S. Chapin, Science 312, 1612 (2006).

[18] D. C. Hall, R. J. Behl, Ecological Economics 57, 442 (2006).

[19] D. Archer, Biogeosciences 4, 521 (2007).

[20] K. Schaefer, T. Zhang, L. Bruhwiler, A. P. Barrett, Tellus B 63, 165 (2011).

[21] C. Le Qur, et al., Science 316, 1735 (2007).

[22] T. Eglin, et al., Tellus B 62, 700 (2010).

[23] C. Huntingford, et al., Philosophical Transactions of the Royal Society B: Biological Sciences

363, 1857 (2008).

[24] M. Oppenheimer, Nature 393, 325 (1998).

[25] M. Oppenheimer, R. B. Alley, Climatic Change 64, 1 (2004).

[26] D. G. Vaughan, Climatic Change 91, 65 (2008).

[27] D. Notz, Proceedings of the National Academy of Sciences 106, 20590 (2009).

[28] D. L. Kelly, C. D. Kolstad, Journal of Economic Dynamics and Control 23, 491 (1999).

[29] A. J. Leach, Journal of Economic Dynamics and Control 31, 1728 (2007).

[30] C. Traeger, Environmental and Resource Economics Forthcoming (2014).

[31] R. H. Moss, et al., Nature 463, 747 (2010).

[32] T. Masui, et al., Climatic Change 109, 59 (2011).

[33] M. Scheffer, et al., Nature 461, 53 (2009).

[34] M. Scheffer, et al., Science 338, 344 (2012).

[35] J. B. Smith, et al., Proceedings of the National Academy of Sciences 106, 4133 (2009).

[36] P. Valdes, Nature Geoscience 4, 414 (2011).

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Supplementary Material

A Model Summary

We reformulate the DICE-2007 model of [14] into a recursive dynamic programming problem fol-lowing [28] and [30]. In response to the curse of dimensionality in dynamic programming, we reducethe state space of the climate system’s representation. [30] shows that this simplification does notimpair the model’s ability to reproduce the DICE model’s climate response. We do make one sub-stantive change to DICE, besides the introduction of tipping points. [14] adjusts a coefficient inthe damage function to compensate for not explicitly modeling climate catastrophes and tippingpoints. This smooth and reversible damage adjustment contributes about half of DICE’s damagesat 1◦C of warming. We eliminate this adjustment because we now explicitly model tipping points.

We introduce tipping points as regime shifts following [7]. That paper provides more detailsabout the modeling of the probability of tipping and of learning. In each period, the climate systemmight irreversibly trigger any of the possible tipping regimes. The Bayesian policymaker assesses theprobability of tipping by updating its previous beliefs based on whether it has just observed a tippingpoint or not. Each tipping point occurs with identical, independent probability in any given stateof 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 triggeredbetween any two periods. When there are n potential tipping points, the probability of crossing kof them between times t and t+1 is: Hn

k (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 temperatureat times t and t+ 1. Currently, scientific modeling cannot suggest that some specific temperatureis a more likely threshold than another similar temperature [9, 35, 36]. Our policymaker thereforebelieves that each tipping point’s temperature threshold is uniformly distributed between a fixed

upper bound T and a lower bound subject to learning: h(Tt, Tt+1) = max{

0, min{Tt+1,T}−Tt

T−Tt

}

. The

hazard rate increases with Tt+1 because greater warming over the next interval carries a greaterrisk of tipping over the next interval. The hazard rate’s dependence on Tt reflects that if a tippingpoint is still possible, then its threshold has not been crossed yet and the policymaker has learnedthat 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 wouldtrigger some tipping point. This requirement implies an upper bound T of 7.8◦C, which is amongthe highest values explored in the sensitivity analysis for the single-tipping runs in [7]. An expectedtrigger of 2.5◦C is consistent with the political 2◦C limits for avoiding dangerous anthropogenicinterference. Further, in the “burning embers” diagram [35], the yellow (medium-risk) shading forthe “risk of large-scale discontinuities” begins around 1.6◦C of warming relative to 1900, and thered (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 thresholdshave been crossed. We solve the corresponding Bellman equation by value function iteration andapproximate the value function using a 104-dimensional Chebychev-basis. Each iteration uses theknitro solver to optimize abatement and consumption decisions at the Chebychev nodes. Once we

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have successfully approximated that value function, we use it to define the post-tipping continuationvalue in the scenarios where all but one tipping point has been crossed. We then approximatethe value functions for those scenarios following the procedure described above. Once we havesuccessfully approximated these value functions, they provide additional continuation values inscenarios 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 scenarioin which no tipping point has yet occurred.

B Model Equations

The state variables are effective capital kt, atmospheric CO2 Mt, cumulative temperature changeTt, and time t. Throughout, bold parameters indicate parameters that are altered by tipping points.

Effective capital kt =Kt

LtAtis capital Kt measured per amount of labor Lt and labor augmenting

technology At level. Labor and technology follow the exogenous equations

At =A0 exp

[

gA,0

δA

(

1− e−tδA)

]

, (Production technology)

gA,t =gA,0e−tδA , (Growth rate of production technology)

Lt =L0 + (L∞ − L0)(

1− e−tδL)

, (Labor)

gL,t =δL

[

L∞

L∞ − L0etδL − 1

]−1

. (Growth rate of labor)

Gross production in the economy is Y grosst = (AtLt)

1−κKκt , or in effective units

ygrosst = kκt .

Climate change damages reduce the effective gross output to the level yt1+Dt(TT ) , where the damage

functionDt(Tt) = d1T

d2

t (Damages)

measures output loss as percentage of total production. The damage tipping point increases thecoefficient d2 from 2 to 3. The remaining output is divided between effective consumption ct =

Ct

LtAt,

abatement expenditure, and capital investment. The equation of motion for capital is

kt+1 =e−(gL,t+gA,t)

[

(1− δk)kt + (1− Λ(µt))yt

1 +Dt− ct

]

. (Effective capital)

The function

Λ(µt) = Ψtµa2t with Ψt =

a0σta2

(

1−1− etgΨ

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

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emissions. The parameter

σt =σ0 exp

[

gσ,0δσ

(

1− e−tδσ)

]

(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 + b32b23)αB(Mt, t) + b32b33 αO(Mt, t)]

]

, (CO2 transition)

where the b parameters govern DICE’s carbon transition matrix. The function αB(M, t) representsthe 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 atmosphericstock. We estimate these functions using a set of emission scenarios simulated in the full versionof DICE [7]. 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 thefirst 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)Ygrosst +Bt (Emissions)

are unabated CO2 emissions from fossil fuel use and CO2 emissions from land use change andforestry. The latter evolve exogenously according to

Bt =B0etgB . (Non-industrial CO2 emissions)

A change in the atmospheric CO2 concentration causes a shift in the earth’s energy balanceexpressed by the radiative forcing F (Mt, t) = f ln (Mt/Mpre) + EFt, where f is the forcing fromdoubled CO2 and Mpre is the preindustrial CO2 concentration. The exogenous forcing EFt includesnon-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 (Mt+1, t+ 1)−f

sTt − [1− αT (Tt, t)]CO Tt

]

, (Temperature transition)

a function of the lagged temperature, the carbon concentration, and the exogenous forcing fromother greenhouse gases. Ocean cooling enters through both the calibration of the heat capacityparameters CT and CO as well as through the direct cooling function αT (Tt, t) that, following [7],we interpolate from different scenarios that we simulated with the full DICE model, which explicitlykeeps track of the ocean temperature. The parameter s in the temperature transition equation is

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climate sensitivity, or the equilibrium temperature change from doubling CO2 concentrations. Theclimate 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. Weobtain these initial conditions by simulating the model for 10 years with RCP6 emissions and a22% investment rate, as described below. The resulting values for the climate variables are similarto recent projections.

C Value Functions and Solution Method

We solve the model using stochastic dynamic programming. The solution algorithm ensures that thedecision maker always has in mind the complete set of possible futures, including the optimal futurereactions to tipping points, when choosing her optimal policy in a given period. We recursively solvefor different sets of value functions, starting with the case where all tipping points have occurredand working back to the present situation without any threshold yet being crossed. We will describethe solution method for a case with three potential tipping points. The methods for settings withfewer potential tipping points follow straightforwardly.

Denote by V 01,2,3(kt,Mt, Tt, t) the value function in the world where three tipping points (labeled

1, 2, and 3) have already occurred and no further ones may occur. We obtain this value functionby solving the Bellman equation

V 01,2,3(kt,Mt, Tt, t) = max

ct,µt

c1−ηt

1− η+ βtV

01,2,3(kt+1,Mt+1, Tt+1, t+ 1) (1)

subject to the equations summarized in section B. We solve the Bellman equation by functioniteration using a four dimensional tensor basis of Chebychev polynomials to approximate the valuefunction. 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 [30]. Theoptimization over consumption and abatement has to satisfy the constraints

ct+Ψtµa2t

Yt1 +Dt

≤Yt

1 +Dt, (Output constraint)

µt ≤ 1 . (Non-negativity constraint for emissions)

The solution of equation 1 gives us the optimal policies in a world where all tipping points haveoccurred, as well as the maximal welfare that can be achieved over the infinite time horizon in sucha world starting from any given state.

In the second step, we derive the value functions for the three worlds where two tipping pointshave already occurred and one is still possible. We denote these value functions by V i(kt,Mt, Tt, t),where i ∈ {1, 2, 3} labels the tipping point that is still possible. V i is short for V i

j,k, i, j, k ∈ {1, 2, 3}

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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 annualtimestep.

Parameter Value Description

A0 0.027 Production technology in 2005gA,0 0.009 Annual growth rate of production technology in 2005δA 0.001 Annual rate of decline in growth rate of production technology

L0 6514 Population in 2005 (millions)L∞ 8600 Asymptotic population (millions)δL 0.035 Annual rate of convergence of population to asymptotic value

σ0 0.13 Emission intensity in 2005 before emission reductions (Gt C per unitoutput)

gσ,0 -0.0073 Annual growth rate of emission intensity in 2005δσ 0.003 Annual change in growth rate of emission intensity

a0 1.17 Cost of backstop technology in 2005 ($1000 per t C)a1 2 Ratio of initial backstop cost to final backstop costa2 2.8 Abatement cost exponentgΨ -0.005 Annual growth rate of backstop cost

B0 1.1 Annual non-industrial CO2 emissions (Gt C) in 2005gB -0.01 Annual growth rate of non-industrial emissions

EF0 -0.06 Forcing in 2005 from non-CO2 agents (W m−2)

EF100 0.30 Forcing in 2105 from non-CO2 agents (W m−2)

κ 0.3 Capital elasticity in Cobb-Douglas production functionδκ 0.06 Annual depreciation rate of capitald1 0.0019 Coefficient on temperature in the damage functiond2 2 Exponent on temperature in the damage functions 3 Climate sensitivity (◦C)

f 3.8 Forcing from doubled CO2 (W m−2)Mpre 596.4 Pre-industrial atmospheric CO2 (Gt C)CT 0.03 Translation of forcing into temperature changeCO 0.3 Translation of surface-ocean temperature gradient into forcing

b11,b12,b13 0.978,0.023,0 Transfer coefficients for carbon from the atmosphereb21,b22, b23 0.011,0.983,0.005 Transfer coefficients for carbon from the combined biosphere and

shallow ocean stockb31,b32,b33 0,0.0003,0.9997 Transfer coefficients for carbon from the deep ocean

ρ 0.015 Annual pure rate of time preferenceη 2 Relative risk aversion (also aversion to intertemporal substitution)

k0 187/(A10L10) Effective capital in 2015, with 187 US$trillion of capitalM0 865.98 Atmospheric carbon dioxide (Gt C) in 2015T0 0.915 Surface temperature (◦C) in 2015, relative to 1900

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with i 6= j < k 6= i. Given V 01,2,3 derived in the first step, we solve the three Bellman equations

V i(kt,Mt, Tt, t) = maxct,µt

c1−ηt

1− η+ βt

{

[1−H11 (Tt, Tt+1)]V

i(kt+1,Mt+1, Tt+1, t+ 1)

+H11 (Tt, Tt+1)V

01,2,3(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 tip-

ping point i ∈ {1, 2, 3} has already occurred and the other two are still possible. Vi is short for

V j,ki (kt,Mt, Tt, t) with i 6= j < k 6= i. Given the value functions V i and V 0

1,2,3 obtained in the earliersteps, we derive the value functions Vi, i ∈ {1, 2, 3} by solving the three Bellman equations

Vi(kt,Mt, Tt, t) =maxct,µt

c1−ηt

1− η+ βt

{

[

1− 2H21 (Tt, Tt+1)−H2

2 (Tt, Tt+1)]

Vi(kt+1,Mt+1, Tt+1, t+ 1)

+H21 (Tt, Tt+1)

[

V k(kt+1,Mt+1, Tt+1, t+ 1) + V j(kt+1,Mt+1, Tt+1, t+ 1)]

+H22 (Tt, Tt+1)V

01,2,3(kt+1,Mt+1, Tt+1, t+ 1)

}

where k 6= i 6= j 6= k.Finally, we obtain the value function V 1,2,3

0 and optimal policies in the pre-threshold regime,where no tipping point has yet occurred. This value function solves the Bellman equation

V 1,2,30 (kt,Mt, Tt, t) = max

ct,µt

c1−ηt

1− η+ βt

{

[

1− 3H31 (Tt, Tt+1)− 3H3

2 (Tt, Tt+1)−H33 (Tt, Tt+1)

]

× V 1,2,30 (kt+1,Mt+1, Tt+1, t+ 1)

+3

∑

i=1

H31 (Tt, Tt+1)Vi(kt+1,Mt+1, Tt+1, t+ 1)

+3

∑

i=1

H32 (Tt, Tt+1)V

i(kt+1,Mt+1, Tt+1, t+ 1)

+H33 (Tt, Tt+1)V

01,2,3(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 above1900 (plus the infinite time horizon mapped to the unit interval). We must vary the approximationintervals for the pre- and post-threshold value functions in order to accommodate changes in dy-namics and curvature due to tipping points. So that the continuation values are contained withinvalid approximation regions, it is crucial that any value function with some number (potentiallyzero) of tipping points already crossed and tipping point j remaining possible is approximated overa weakly smaller interval than any value function with those same tipping points already crossed

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and also having tipping point j already crossed. Much of the numerical burden lies in finding com-binations of approximation intervals that allow each of the required value function iteration stepsto converge, where convergence is as described in [7]. Cases in which feedback and carbon sinktipping points have already been crossed typically require larger approximation intervals becausethe new dynamics tend to drive the carbon and temperature processes through wider regions of thestate space. On the other hand, cases in which the damage tipping point has already been crossedtypically require narrower approximation intervals due to the tendency of the value function tobecome strongly curved at high temperatures.

In the setting with only a single possible tipping point, all post-threshold value functions extendthe effective capital interval upwards to 7, extend the carbon interval upwards to 3000 Gt C, andextend the temperature interval upwards to 6◦C. The pre-threshold intervals are the same as inthe no-policy case when the single tipping point is a feedback or carbon sink tipping point, but thecase with a damage tipping point uses an interval with an upper bound of 7 for effective capitaland of 2000 Gt C for atmospheric carbon.

Now consider a setting with two potential tipping points. When both have been crossed, theapproximation interval extends up to 7 in the effective capital domain, extends up to 3000 Gt Cin the carbon domain, and extends up to 6◦C in the temperature domain (8◦C if the two tippingpoints are the feedback and carbon sink tipping points). When only one has been crossed and thedamage tipping point is not the remaining one, the approximation interval extends up to 7 in theeffective capital domain, up to 2500 Gt C in the carbon domain, and up to 5◦C in the temperaturedomain. The interval is the same when none have been crossed and the damage tipping point isnot possible, except the temperature interval does not extend past 4◦C. When only one tippingpoint has been crossed and the damage tipping point remains possible, the approximation intervalextends up to 7 in the effective capital domain, up to 2000 Gt C in the carbon domain, and up to4◦C in the temperature domain. The interval is the same when no tipping points have been crossedand 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 alreadybeen crossed, the approximation interval extends up to 7 in the effective capital domain, up to 3000Gt C in the carbon domain, and up to 6◦C in the temperature domain. When only two tippingpoints 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 Cand 4◦C if the damage tipping point is the one that has not been crossed). When only a singletipping point has been crossed, the approximation interval extends up to 7 in the effective capitaldomain, up to 2000 Gt C in the carbon domain, and up to 4◦C in the temperature domain. Thissame 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 inthe year 2015. The optimal policies incorporate the possibility that any of the tipping points mightoccur in any given year in the future, with the tipping probabilities depending on the evolutionof the climate states. The plotted paths simulating policy and climate characteristics into thefuture, and also the plotted optimal policy values for 2050, rely on the additional assumption thatno tipping point has occurred by the given time. We use the equations of motion summarized insection 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

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the optimal policy in a given stage with a fixed 22% investment rate and RCP6’s fixed emissiontrajectory, whose annual values we obtain from MAGICC 6.0 (Figure 5). For these exogenouspolicy trajectories, we sum welfare over each year up to the time when optimal policy begins, withthe approximated value functions determining welfare from that time forward. In cases where thepolicymaker reacts to some threshold crossing, we obtain the summation for the period with RCP6emissions by starting at the initial year and calculating per-period utility along every possiblepath. In cases where the policymaker never reacts to a threshold crossing, the large number ofpossible tipping paths makes this approach computationally impractical if policy is delayed bymore than 40 years and nearly interminable if policy is delayed by more than 70 years. In thosecases with a nonresponsive policymaker, we reduce the number of necessary calculations by usingvalue function approximations to determine welfare along those potential paths for which at leastone tipping point has occurred. More specifically, we use Chebychev polynomials (as describedabove) to approximate welfare after having crossed each possible combination of tipping pointsat any potential combination of state variables, assuming optimal policy begins at a particularyear of interest. This approach yields nearly identical results to the explicit calculation approachfor all simulations (i.e., periods of delay up to and including 70 years) for which we were able toobtain results under both approaches. Reported results for which optimal policy never begins inthe absence of triggered tipping points are implemented by delaying policy for 400 years.

Figure 5: Emissions and temperature under the exogenous RCP6 scenario (dashed) and under the optimalpolicy (solid) when there are three potential tipping points.

D Post-Threshold Policy

Figure 6 depicts how policy and the climate system respond if a tipping point happened to be crossedin 2075. Prior to 2075, no tipping point happens to occur, and after 2075, the other two tippingpoints may still occur. The top left panel show that the optimal carbon tax increases once any ofthe three tipping points has occurred. The optimal tax increases most strongly after the damage

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Figure 6: The effect of a tipping point on post-threshold policy and on the climate system. Plottedsimulations assume that all three tipping points are possible and that a particular one happens to occur in2075 as the first tipping point.

tipping point occurs. The greater abatement effort translates into greater emission reductions thanin the case where no tipping point happens to occur (top right panel). These reduced emissionsin 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 tippingpoint 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 ornot a carbon sink tipping point happens to occur (bottom right panel). It is sharply lower in thewake of a damage tipping point; however, the feedback tipping point’s effect on climate sensitivityoutweighs the emission reductions undertaken after it occurs, so that optimal warming is greatestin the wake of a feedback tipping point.

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