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Managing Catastrophic Climate Risks Under ModelUncertainty AversionLoïc Berger, Johannes Emmerling, Massimo Tavoni

To cite this article:Loïc Berger, Johannes Emmerling, Massimo Tavoni (2016) Managing Catastrophic Climate Risks Under Model UncertaintyAversion. Management Science

Published online in Articles in Advance 11 Mar 2016

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MANAGEMENT SCIENCEArticles in Advance, pp. 1–17ISSN 0025-1909 (print) � ISSN 1526-5501 (online) http://dx.doi.org/10.1287/mnsc.2015.2365

© 2016 INFORMS

Managing Catastrophic Climate Risks UnderModel Uncertainty Aversion

Loïc Berger, Johannes EmmerlingFondazione Eni Enrico Mattei (FEEM) and Centro Euromediterraneo sui Cambiamenti Climatici (CMCC), 20123 Milan, Italy

{[email protected], [email protected]}

Massimo TavoniDepartment of Management, Economics and Industrial Organization, Politecnico di Milano, 20156 Milan, Italy; and

Fondazione Eni Enrico Mattei (FEEM) and Centro Euromediterraneo sui Cambiamenti Climatici (CMCC), 20123 Milan, Italy,[email protected]

We propose a robust risk management approach to deal with the problem of catastrophic climate changethat incorporates both risk and model uncertainty. Using an analytical model of abatement, we show how

aversion to model uncertainty influences the optimal level of mitigation. We disentangle the role of preferencesfrom the structure of model uncertainty, which we define by means of a simple measure of disagreement acrossmodels. With data from an expert elicitation about climate change catastrophes, we quantify the relative importanceof these two effects and calibrate a numerical integrated assessment model of climate change. The results indicatethat the structure of model uncertainty, and specifically how model disagreement varies in abatement, is the keydriver of optimal abatement and that model uncertainty warrants a higher level of climate change mitigation.

Keywords : climate change; model uncertainty; ambiguity; nonexpected utility; catastropheHistory : Received March 11, 2015; accepted September 26, 2015, by Manel Baucells, decision analysis. Published

online in Articles in Advance.

1. IntroductionThis paper applies recently developed tools from deci-sion theory and expert assessment data to discussabatement strategies in the case of climate changeunder deep uncertainty. We distinguish the notion ofrisk, which characterizes situations in which probabili-ties of a random event are perfectly known, from thebroader notion of (Knightian) uncertainty (also calledambiguity), which characterizes situations in whichsome events do not have an obvious, unanimouslyagreed upon probability assignment (Ghirardato et al.2004).1 More specifically, we focus on the notion ofmodel uncertainty that corresponds to situations inwhich different data-generating mechanisms or modelsare considered as possible or plausible by the decisionmaker (DM). Throughout the paper, we consider thenotion of model taken in its statistical sense, meaningthat it is defined as a probability distribution over asequence (or, here, over states of the world). Differentmodels may exist, for example, because too little infor-mation is available, because different predictions exist(depending on different data sets, different techniques,etc.), or because the decision is based on the advice of

1 This definition comprises the definition of deep uncertainty givenby Lempert et al. (2006). In this paper, we use the terms “Knightianuncertainty,’’ “ambiguity,’’ and “deep uncertainty’’ interchangeably.

experts who provide different assessments of proba-bilities for a given event (as will be the case in ourapplication). We present the risk management problemas an intertemporal problem of optimal abatementunder the possibility of a catastrophic climate event.As is the case with a vast range of economic problems,the climate change case illustrates particularly well asituation in which the probabilistic model is neitherexplicitly given nor can it be perfectly approximated orinferred with the available data and current scientificmethods. Choosing among different climate policies inthis situation is therefore essentially an exercise in riskmanagement that has to be performed in a situation ofdeep uncertainty (Kunreuther et al. 2013). Therefore,it requires a robust decision-making approach that isless sensitive to initial assumptions, is valid for a widerange of futures, and keeps options open (Lempertand Collins 2007), rather than a formal approach thatmaximizes the expected utility mechanically.

More than ever before, it is now widely believedthat our entire planet is undergoing climate change,and that this change is largely due to human activities(IPCC 2013). What is less clear is how this process,which is taking place over a very long time horizon,will unfold. Based on the available observations and onthe current state of knowledge, scientific experts haveconstructed models in order to simulate and quantify

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mailto:[email protected]



Berger, Emmerling, and Tavoni: Managing Catastrophic Climate Risks2 Management Science, Articles in Advance, pp. 1–17, © 2016 INFORMS

the impact of human activity on the climate systemand vice versa. However, a large degree of uncertaintysurrounds these models. These uncertainties arise fromboth the underlying climate science (and the extremecomplexity of the climatic system) and our inability toperfectly capture the way our socioeconomic systemwould respond and adapt to climate change (Healand Millner 2014). This is particularly the case whenwe consider situations with potential catastrophicconsequences, such as the collapse of the Atlanticthermohaline circulation or the melting of the Antarcticice sheet. Such catastrophic events have not beenencountered in recent history,2 and their likelihood ofoccurrence is therefore extremely difficult to assess.From a decision maker’s perspective, becoming awareof such occurrences leads to the expansion of theset of admissible states of the world. Therefore, thestate space and the associated probabilities need tobe adjusted for making decisions when consideringsuch “unforeseen events” (Karni and Viero 2013). Inrecent years, a few studies have been undertaken toestimate these probabilities, consisting generally ofexperts’ assessments.3 Since climate science is currentlyunable to determine which of these estimates is thebest or what the best combination of them is, andsince these uncertainties are expected to persist evenwhen better scientific models become available,4 adecision maker confronted with this situation wouldfind himself in a situation of model uncertainty ratherthan risk.

In view of this disagreement among experts or mod-els, how should a rational policy decision maker pro-ceed? If one follows the traditional Bayesian/subjectiveexpected utility approach, one will simply aggregatethe models by averaging them into a single represen-tative model and then use the (subjective) expectedutility framework (Newbold and Daigneault 2009). Theproblem with this approach is that the decision makerconsiders the resulting aggregated model in exactlythe same way as one would consider an equivalentobjective model representing a specific risk, and model

2 These phenomena have been referred to as “tipping elements”because they imply abrupt climate change occurring “when the cli-mate system is forced to cross some threshold, triggering a transitionto a new state” (Lenton et al. 2008, p. 1786). The correspondingcritical point at which the future state of the system is qualitativelyaltered is called a “tipping point.”3 Remark that these experts’ assessments may be the result of usingdifferent climatic models, different physical parameters, differentmethodologies, or different databases.4 Even if climate scientists have recently made a great deal of progressin understanding and describing the physical mechanisms involvedin the climate change phenomenon, many uncertainties still remain.Some of them will eventually be resolved with future scientificprogress, whereas others may be in the realm of “unknowable”(Pindyck 2013a) or “unquantifiable” (Heal and Millner 2014).

uncertainty has therefore no impact on the decision-making process. This approach has, however, beenseriously challenged in situations of deep uncertainty.The most famous example is that of Ellsberg (1961),who showed through different experiments that thechoices of individuals cannot be rationalized under thetraditional Bayesian expected utility paradigm, and thatindividuals usually manifest aversion toward situationsin which probabilities are not perfectly known. Inapplied economic models, some recent contributionshave started to apply nonexpected utility frameworks(i.e., alternative models of risk preferences and beliefs,most of which replace the expected utility formulationwith alternative criteria) to the problem of climatechange. In particular, these include applications ofthe smooth ambiguity model by Lange and Treich(2008), who provide comparative statics results of therole played by ambiguity in a simple two-period para-metric model, and Millner et al. (2013) and Lemoineand Traeger (2014), both of whom propose numeri-cal models under ambiguity aversion. Other contri-butions include applications of the macroeconomictechnique of robust control (Hansen and Sargent 2008)by Athanassoglou and Xepapadeas (2012), who con-sider an analytical pollution control problem, and byRudik (2016), who applies the concept in an integratedassessment model (IAM) including learning.5 Finally, adifferent approach is taken by Drouet et al. (2015), whouse the results of the most recent assessment of theIntergovernmental Panel on Climate Change (IPCC) tonumerically disentangle model uncertainty and risksabout mitigation costs, climate dynamics, and (continu-ous) climate damages. For what concerns the inclusionof catastrophic damages into integrated assessmentmodels, our paper also extends the work of, amongothers, Gjerde et al. (1999), who show that taking intoaccount a risk of catastrophe provides a rationale forcurrent emission control, and Keller et al. (2004) andLontzek et al. (2012), who model a collapse in oceancirculation as a permanent shock to the productionfunction, and show that the optimal policy should beassociated with immediate limitations on emissions.

In this paper, we take a step further in the directionof understanding the theoretical mechanisms underly-ing the results obtained in this literature, by applyingthe most recent robust tools developed in decisiontheory (Cerreia-Vioglio et al. 2013b, Marinacci 2015).We consider an alternative to expected utility modelsthat allows us to incorporate both risk and modeluncertainty. More specifically, we study the impactof model uncertainty aversion on optimal abatementpolicy. Our contribution is severalfold. We develop

5 Note that the preferences used by Hansen and Sargent (2008) can beseen as a special case of the smooth ambiguity preferences. In theircase, the ambiguity function is of the exponential or constant absoluteambiguity aversion type (Cerreia-Vioglio et al. 2011, Marinacci 2015).

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Berger, Emmerling, and Tavoni: Managing Catastrophic Climate RisksManagement Science, Articles in Advance, pp. 1–17, © 2016 INFORMS 3

a two-period model of emission abatement with anendogenous probability of catastrophic climate change,which allows us to disentangle the contribution of pref-erences and the structure of model uncertainty on thelevel of first-period abatement. We show that a simplemeasure of the disagreement across models or expertsis a sufficient statistic for determining the structureof model uncertainty that matters for abatement. Weapply our theoretical results using an actual assessmentof a major catastrophic climatic event with data from arecent expert’s elicitation. Finally, we extend a widelyused integrated assessment model (IAM) of climatechange (dynamic integrated climate–economy (DICE)model; Nordhaus 1993) to include a tipping elementin the climate response, in a framework where well-defined probabilities are unknown. This allows us toquantify the impact of deep uncertainty on the optimallevel of emission abatement, addressing one of the crit-icisms of IAMs, which have been recently highlightedby Stern (2013), Pindyck (2013a, b, 2015), and Kun-reuther et al. (2013). Our broader finding is that in mostsituations, a robust climate strategy implies strongermitigation policies. In that sense, we show that deepuncertainty cannot be taken as an excuse for inaction,making a clear link to the precautionary principle.6 Weshow that both preferences over model uncertainty,measured by ambiguity prudence, and the structure ofthe model uncertainty, measured by the decrease of thedegree of model disagreement in abatement, determinethe optimal abatement level. The data we use fromexpert elicitations indicate that it is the latter effectthat is by far the most important, given that the dis-agreement across models or experts increases in globalmean temperature.7 Finally, the reformulated integratedassessment model allows the generation of quantitativeestimates of the impact of risk and model uncertaintyaversion on optimal emission reductions. Compared tothe commonly used expected utility framework, modeluncertainty raises abatement significantly. Our broaderpolicy result corroborates the findings of the recentstrand of research that has emphasized the importanceof deep uncertainty and tipping points in quantitativeclimate policy making (Lemoine and Traeger 2014;Lontzek et al. 2012; Gjerde et al. 1999; Weitzman 2012,2009; van der Ploeg and de Zeeuw 2014; Lempert andCollins 2007; Drouet et al. 2015).

6 The precautionary principle states that “When an activity raisesthreats of harm to human health or the environment, precautionarymeasures should be taken even if some cause and effect relationshipsare not fully established scientifically” (Science and EnvironmentalHealth Network 1998).7 Climate data allow the prediction of a low level of future warmingwith more confidence than a high level of warming, as mentionedby Allen and Frame (2007, p. 582) who write, “once the world haswarmed by 4�C, conditions will be so different from anything we canobserve today that it is inherently hard to say when the warming willstop.”

Our results can be read in both positive and nor-mative terms. Although we recognize the existence ofa debate about the normative status of nonexpectedutility models, and the predominance of the expectedutility theory paradigm for normative purposes indecision making, we here follow the claim that thereis nothing irrational about violating Savage’s (1954)axioms in situations of deep uncertainty (Gilboa et al.2008, 2009, 2012; Gilboa and Marinacci 2013).8 Thenon-Bayesian framework we adopt is thus compatiblewith a normative assessment of optimal policies.

2. A Simple Model of OptimalAbatement Under Model Uncertainty

To investigate the effects of different types of uncer-tainty on emission abatement decisions, we constructa simple economic model of optimal abatement withtwo periods: today and the future. During the firstperiod, the decision maker chooses a level of abate-ment a that is undertaken at cost c4a5. This abatementreduces available disposable income in such a way thatconsumption in period 1 is given by C1 =w1 − c4a5,where w1 is the level of income of the first period. Inthe future, there are two possible categories of statesof the world. One is catastrophic: the environmentis severely affected so that the consumption in thesecond period C2 is given by w2 −Ls , where w2 is thedeterministic exogenous income, and Ls is the damage(loss) that occurs with probability �s, conditionallyon the fact that a catastrophe occurred (i.e., ∀ s ∈ S,where S represents those catastrophic or unfavorablestates). The other is one in which no catastrophe occurs,so that consumption is w2 (i.e., favorable state). Theprobability p4a5 that such a catastrophic event willoccur is endogenous and depends on the level of abate-ment chosen in the first period.9 Consumption in thesecond period can therefore take �S�+1 different values,and the abatement effort in the first period is the onlychoice variable in this model. It is conceptualized as aninvestment to reduce the risk of a catastrophic eventthat is difficult to compensate by ordinary savings(rather than an instrument used to optimally smoothconsumption over time). As in most environmental

8 In situations where information is scarce, alternative decision-theory models may, on the contrary, perform better in the sense thatthey have better explanatory power or are able to provide betterpredictions and guidelines.9 This type of model, with endogenous probability to model mitiga-tion, is referred to as a “self-protection model’’ in the risk literature.An alternative is to consider the case of adaptation or self-insurancein which the loss in the second period depends on the abatementlevel. Given the limited scope for adaptation in reducing catastrophicimpacts, we decided not to consider the latter case in this paper. Itcan, however, be shown that our main results hold and would evenbe reinforced by the presence of adaptation to the catastrophe orstandard continuous damages in our framework (Berger 2015).

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economic models under uncertainty, intertemporalutility is assumed to be time separable, and futureutility is discounted by a factor � ∈ 40117.

Model uncertainty is introduced by relaxing theassumption that all the elements of the maximizationprogram are objectively known or commonly agreedupon, so that the probability model over future con-sumption is no longer unique. We assume that a trueclimatic process is in place and generates observations,but that this true process and the probability modelrepresenting it are unknown to the decision maker. Theobservations generated by this model are, however,available and used by experts (scientists, climatologists,physicists, etc.) to construct predictive models thatbelong to a class M . The true process is assumedto belong to this class of models, and elements ofM are interpreted by the DM as possible alternativemodels that could be selected by nature to generateobservations. These possible models have a “Waldean’’interpretation in the sense that the class M is regardedas a datum of the decision problem (Wald 1950). Thisimplies that the models have to be consistent withobjectively available information (note, however, thatthe information must be incomplete; otherwise Mwould be a singleton). This set therefore contains allthe information the DM considers as “credible” in thesense that “states that are not given any weights byany of the relevant probability distributions are simplyirrelevant” (Gajdos et al. 2008, p. 34). We assume thatthere are n different models (or experts). These differentmodels P� are indexed by a parameter � ∈ 81121 0 0 0 1 n9,so that M = 8P�9�∈8110001n9. Each P� describes a possibledistribution (i.e., a possible risk) of second-period con-sumption: C24a1 �5 (in what follows, only the probabilityof catastrophe will depend on �). We also assume thatthe decision maker has a prior probability measure overthe set of possible models; that is, � has a probabilitydistribution q = 4q11 q21 0 0 0 1 qn5, so that � takes value �with probability q�. This second-order distribution �reflects model uncertainty in the sense that the DMdoes not know which of the models P� is the trueor the most accurate one and associates a subjectiveweight q� with each of them.10

In what follows, we consider different criteria fordecision making under climate model uncertainty,and compare them in terms of optimal abatement.In Appendix B, we discuss the case of uncertaintyabout the economic impacts of a climate catastrophe

10 As mentioned earlier, a parallelism can be made between theuncertainty that follows this decomposition into model (or epistemic)uncertainty and risk (also called aleatory or physical uncertainty)and what is generally referred to in the decision theory literatureas ambiguity (i.e., situations in which “a decision maker does nothave sufficient information to quantify through a single probabilitydistribution the stochastic nature of the problem he is facing” (Cerreia-Vioglio et al. 2013a, p. 975).

(i.e., the size of Ls), showing that our results carry over(and are even strengthened) in this case. Althoughdifferent existing models of ambiguity aversion couldhave been adapted to the presence of objective infor-mation (Marinacci 2015), we focus on the smoothmodel of model uncertainty aversion for its abilityto characterize the notion of ambiguity neutrality, itsmathematical convenience, and because it encompassesmany of the alternative criteria as special or limit cases.Nonetheless, we explore alternative criteria, such asmaxmin, in the online supplemental material (availableat http://dx.doi.org/10.1287/mnsc.2015.2365), showingthat they entail qualitatively similar results.

The traditional approach for addressing a problem inwhich the true distribution is unknown is to considerthat agents use their probabilistic beliefs over the sourceof uncertainty in an expected utility maximizationframework. Cerreia-Vioglio et al. (2013b) are the first toprovide a decision theoretic derivation of this type ofdeep uncertainty presented in two layers. In particular,they enrich the standard Savage framework (Savage1954) in the presence of objective information, and showthat preferences satisfying Savage’s axioms may berepresented, in the context of our abatement model, by

WSEU = v4w1 − c4a55+�E�Ev4C24a1 �550 (1)

In this expression, v is the per-period von Neumann–Morgenstern utility function reflecting both the decisionmaker’s attitude toward risk and desire to smoothconsumption over time;11 E� is the expectation opera-tor taken over prior distribution �, that is, E�X4�5=∑n

i=1 qiX4i5; and E is the expectation operator oversecond-period consumption in the different states of theworld, conditional on model �. This representation iscalled classical subjective expected utility (SEU) becauseit incorporates key objective pieces of information inSavage’s subjective framework. In the context of thispaper, the second-period expected utility for a givenmodel � may be written as

Ev4C24a1 �55 ≡ p4a1�5∑

s∈S

�sv4w2 −Ls5

+ 41 − p4a1�55v4w251 (2)

where we denote by p4a1�5 the probability of catas-trophe as a function of abatement for model �. Foreach prior distribution q, there exists an equivalent pre-dictive distribution C24a1 �5 such that E�Ev4C24a1 �55=

Ev4C24a1 �55, and it is therefore clear that the reducedform of representation (1) is nothing but the original

11 The two features could easily be disentangled using Kreps andPorteus (1978) and Selden (1978) preferences. For the sake of exposi-tional clarity and simplicity, we only consider this specification inthe quantification part of the paper (see §4).

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“Savagian’’ subjective expected utility. The decisionproblem under uncertainty is then reduced to a simpledecision problem under risk where the beliefs aresubjective. On the other hand, when M is a singleton(i.e., when there is only one model everyone agrees on),there is no longer model uncertainty, so that the riskysecond-period consumption is C24a5 and we are backto the classical von Neumann–Morgenstern expectedutility model. These different representations of theproblem are observationally equivalent to someone whois not aware of the presence of objective informationderiving from different experts’ models.

In what follows, we consider the subjective expectedutility representation as a benchmark. The economicproblem of finding the level of abatement a∗

SEU thatmaximizes program (1) is easy to solve.12 This level isimplicitly given by equalizing the marginal cost andbenefit of abatement:

v′4w1 − c4a∗

SEU55c′4a∗

SEU5= −�¡E�Ev4C24a

∗SEU1 �55

¡a0 (3)

Although the classical subjective expected utilityframework has the advantage of being easily tractable,it is unable to take into account different attitudestoward different types of uncertainty that surroundthe economics of climate change. We now introducedifferent attitudes toward different types of uncertainty.To investigate the relationship between risk aversionand model uncertainty aversion, we consider a crite-rion in which the function representing the agent’spreferences toward model uncertainty is smooth andhence everywhere differentiable. By letting v representattitude toward risk and h represent attitude towardmodel uncertainty, we can write the smooth criterion(Marinacci 2015) to be maximized as

WSmooth = v4w1 −c4a55

+�4v�h−154E�4h�v−154Ev4C24a1�55550 (4)

This expression can be written equivalently as

WSmooth = v4w1 − c4a55+�v4CE4ce4a1 �5551 (5)

where ce and CE both represent a certainty equivalent:

ce4a1�5≡ v−14Ev4C24a1 �55 and

CE4ce4a1 �55≡ h−14E�h4ce4a1 �550(6)

The first, ce4a1 �5, corresponds to the certainty equiva-lent of wealth in the second period, if the abatement

12 The maximization programs we consider in this paper are assumedto be convex. Sufficient conditions for concavity of (1) are that thecost function is increasing (c′4a5 > 0) and convex (c′′4a5 > 0) in thelevel of abatement, and that the probability function is decreasingand convex (p′4a5 < 0, p′′4a5 > 0). More generally, sufficient conditionsfor concavity of (4) may be found in Proposition 3 in Berger (2015).

level is a and the expert’s model considered is P�.Under model uncertainty, � itself takes on differentvalues, and so does the certainty equivalent ce4a1�5,which is computed conditionally on �. A second-ordercertainty equivalent of these first-order certainty equiv-alents is then defined as CE by combining all models� ∈ 81121 0 0 0 1n9. The SEU representation (1) is thenobtained in the special case in which the two certaintyequivalents are evaluated using the same function v,so that the attitudes toward risk and model uncertaintyare exactly the same.

The smooth model uncertainty criterion is mathe-matically equivalent to the two-period version of theambiguity model developed by Klibanoff et al. (2009).The significant difference is that their model, as the vastdecision theoretic literature dealing with ambiguity(see Gilboa and Marinacci 2013 for an excellent sur-vey), has been developed in a purely subjective setupand therefore does not explicitly incorporate objectiveinformation à la Wald (1950). In particular, their repre-sentation is recovered by letting �≡ h � v−1 representthe ambiguity attitude. Klibanoff et al. (2005) associate� being a concave function to ambiguity aversion andcall the ratio −4�′′4x5/�′4x55 the coefficient of absoluteambiguity aversion at x, a given level of expectedutility. From representation (4), ambiguity aversionwould correspond to h being more concave than v or,equivalently, model uncertainty aversion being strongerthan risk aversion. Unsurprisingly, in the special case inwhich the DM manifests the same attitude toward riskand model uncertainty, the problem is reduced to theone considered by a classical subjective expected utilitymaximizer defined by representation (1). In this case,the decision problem may be reduced to a problemunder risk.13 The great flexibility of this decision rule,which is based on the smoothness of function h, impliesdifferent conditions in the comparative statics analysisof optimal abatement. In particular, one conditionneeded to sign the direction of the change result-ing from higher aversion toward model uncertaintythan toward risk is the notion of ambiguity prudence(Gierlinger and Gollier 2008, Berger 2014). This con-cept, which is closely related to the notion of riskprudence introduced by Kimball (1990), correspondsto a condition under which the individual is willingto save more because of the presence of ambiguity.14

Equivalently, it expresses the sensitivity of the optimalchoice to the combination of model uncertainty and

13 Note also that the maxmin criterion presented in the onlinesupplemental material is recovered from this formulation in thespecial case of infinite model uncertainty aversion.14 Formally, an agent is said to be ambiguity prudent if the introduc-tion of ambiguity through a mean-preserving spread in the space ofconditional second-period expected utility raises the agent’s optimallevel of saving (Berger 2014).

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risk. The notion of ambiguity prudence in this contextcorresponds to decreasing absolute ambiguity aversion,which is the analogue of the widely accepted notion ofdecreasing absolute risk aversion (DARA). Formally,the notions of constant, decreasing, and increasingabsolute ambiguity aversion are defined depending onthe monotonicity properties of the ratio −4�′′4x5/�4x55.In what follows, we, respectively, use the abbreviationsCAAA, DAAA, and IAAA to denote these cases.15

Equipped with the ambiguity prudence property, whichwe investigate further in the next section, we nowcompare the optimal level of abatement chosen by adecision maker under the smooth criterion with the onechosen under classical subjective expected utility. First,let us recall the definition of comonotonic variablesbefore summarizing the result in Lemma 1, which isreminiscent of Alary et al. (2013) and Berger (2015).16

Definition 1. Consider two random variables Xand Y that are strictly monotonic transformations of asingle random variable �; that is, 4X1Y 5= 4f 4�51g4�55.The random variables X and Y are anticomonotonic if fis increasing and g is decreasing in � and comonotonicif both f and g are increasing or decreasing in �.

Lemma 1. Assume that model uncertainty aversion ishigher than risk aversion. In the optimal abatement modelcharacterized by the maximization of program (4), DAAA(IAAA) is sufficient to raise (decrease) the optimal abatementif Ev4C24a

∗SEU 1 �55 and ¡Ev4C24a

∗SEU 1 �55/¡a are anticomono-

tonic (comonotonic).

Proof. See Appendix A.1. �Lemma 1 tells us that the total effect on abatement

not only depends on the ambiguity prudence conditionbut also on a second factor that concerns the waythe second-period expected utility and the marginalbenefit of abatement interact when different experts ormodels are considered. The question of whether thecomonotonicity condition of Lemma 1 holds in practiceis not trivial. However, the intuition is relatively simple.Consider the case of two models with different proba-bility curves p4a1�5 that do not cross. When the morepessimistic one (e.g., the one with lower Ev4C24a

∗SEU 1 �55)

15 Note that DAAA encompasses the most widely used functionalforms of power and exponential � (the former is usually referred toas “constant relative ambiguity aversion” (CRAA) and the lattercorresponds to CAAA). It is stronger than requiring �′′′ > 0, but thisshould not be surprising, given that future utility in program (4) isrepresented by the � certainty equivalent of the expected utilitiesrather than by its expected � valuation.16 In a recent contribution, Millner et al. (2013) also proposed amodel of abatement with an endogenous probability in a one-periodframework similar to Alary et al. (2013). When two periods areconsidered, however, the comonotonicity condition only concernsthe second period, and general conclusions may be drawn for themore realistic cases in which the effort exerted in the first periodalso reduces the ambiguity.

believes that the probability of catastrophe decreasesfaster in abatement (e.g., a higher ¡Ev4C24a

∗SEU 1 �55/¡a),

the anticomonotonicity condition holds, and the condi-tion of ambiguity prudence—stating that the DM ismore willing to invest for the future when this futurebecomes more uncertain—is sufficient. In this case, itis equivalent to saying that the disagreement acrossmodels falls in abatement. For example, this would bethe case if experts agreed that a high level of climateprotection would give us good chances of avoiding aclimate catastrophe but disagreed on the probabilitiesin the case of limited mitigation and thus higher globalwarming. To gain more intuition, we now disentanglethe role of preferences from the structure of modeluncertainty and study the two effects separately.

2.1. The Ambiguity Prudence EffectIn terms of attitudes toward risk and model uncer-tainty, the ambiguity prudence condition turns outto be nontrivial, as is summarized in the followingproposition.

Proposition 1. Decision makers exhibit DAAA if andonly if their preferences toward risk, captured by function u,and model uncertainty, captured by function h, are such that

h′′′

h′−

v′′′

v′≥

(

−h′′

h′+

v′′

v′

)(

−h′′

h′− 2

v′′

v′

)

0 (7)

Similarly, a decision maker exhibits CAAA if inequality(7) holds with an equality, and IAAA if inequality (7) isreversed.

Proof. See Appendix A.2. �Intuitively, Proposition 1 tells us that a necessary and

sufficient condition for DAAA is that the difference indownside model uncertainty and risk aversion (theleft-hand side of (7)) is sufficiently high. To gain furtherinsight about this condition, consider the followingexamples.

Example 1. When the isoelastic CRRA–CRMUA17

specifications with relative risk aversion parameter �and relative model uncertainty aversion parameter �≥

� (so that the DM is ambiguity averse) are considered,the ambiguity aversion function is given by �4U5=

41/41−�55641−�5U 741−�5/41−�5, the coefficient of absoluteambiguity aversion is 4�−�5/441 −�5U5, and the DMexhibits DAAA when �< 1, CAAA when �= 1, andIAAA when �> 1.

17 A utility function has the constant relative risk aversion (CRRA)property if it takes the form v4x5 = x1−�/41 − �5, where � is thecoefficient of relative risk aversion (note that when � = 1, it col-lapses to v4x5= ln x). Constant relative model uncertainty aversion(CRMUA) is defined similarly for function h, in the sense thath4x5= x1−�/41 −�5, where � represents the coefficient of relativemodel uncertainty aversion.

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Figure 1 (Color online) Different Models or Experts p4a1 �5 as Functions of the Abatement Level a, Under Constant (Column A), Increasing (Column B), orDecreasing (Column C) Degree of Model Uncertainty in Abatement

A

p (a, � ) p (a, � ) p (a, �)

a

p (a, 1)

p (a, 2)

p (a, 3)

p (a, 1)

p (a, 2)

p (a, 3)

p (a, 1)p (a, 2)p (a, 3)

B

a

C

a

�2(a ) �2(a ) �2(a )

a a a

Example 2. When the CARA-CAMUA18 specifica-tions are used with absolute coefficients of risk aversionand model uncertainty aversion, respectively, � and �,the ambiguity function is �4U5= −4−U5�/�, so that theDM always exhibits IAAA.

2.2. The Convergence of Agreement EffectTo study the structure of model uncertainty, let ussimplify the notation in expression (2) above, and letw2 − L with L > 0 be the certainty equivalent con-sumption in the second period when the economyis hit by a catastrophe.19 Remember that in this case,each model P� describes a possible risk of second-period consumption, which is fully characterized byC24a1 �5∼ 6w2 −L1p4a1 �53w211−p4a1 �57. An illustrationof possible different models is depicted in the first rowof Figure 1.

To further characterize the change in the optimalabatement decision, we now define the notion of degreeof model disagreement. It is a measure of the variabilityacross models (or, equivalently, of the disagreementamong experts).

Definition 2. For any set of probability functions8p4a1�59�∈8110001n9 characterizing models 8P�9�∈8110001n9, thedegree of model disagreement is given by

�24a5 2= Var�6p4a1�571

for any given level of abatement a.

18 A utility function exhibits constant absolute risk aversion (CARA) ifit has the form v4x5= −e−�x , where � is the coefficient of absolute riskaversion. Cconstant absolute model uncertainty aversion (CAMUA)is defined analogously, so that h4x5= −e−�x , where � is the coefficientof absolute model uncertainty aversion.19 More precisely, this certainty equivalent is implicitly defined by∑

s∈S �sv4w2 −Ls5= v4w2 −L5.

The degree of model disagreement is illustrated inthe lower row of Figure 1. As can be seen, it can be(but is not limited to) constant (column A), increasing(column B), or decreasing (column C) in the levelof abatement. In what follows, we will focus on thecase where the degree of model disagreement is amonotonic function of abatement.20 We will refer to“convergence of agreement,” a situation in which thedegree of model disagreement is decreasing. Usingthis simple metric, we can now relate the results ofLemma 1 to the structure of model uncertainty. First,note that, conditional on the true model being P�, wecan write

Ev4C24a∗

SEU 1�55

=v4w25−p4a1�56v4w25−v4w2 −L57 (8)

¡Ev4C24a∗SEU 1�55

¡a=−pa4a1�56v4w25−v4w2 −L571 (9)

where pa ≡ ¡p/¡a. Lemma 1 therefore tells us that a DMexhibiting DAAA will always choose to abate moreif p4a1�5 and pa4a1�5 are anticomonotonic in �, as isthe case in column C of Figure 1, for example. In thiscase, the degree of model uncertainty will intuitivelybe decreasing in abatement since abatement decreasesthe probability of catastrophe more strongly in morepessimistic models. The sufficient condition of Lemma 1is, however, very restrictive, and Proposition 2 belowtells us that the comonotonicity property does not

20 Remark that this assumption is less restrictive than the onerequiring the set of models 8P�9 to be monotonic in � for all levels ofabatement equivalent to the one used in Alary et al. (2013) and Berger(2015). In particular, our assumption does not require probabilitycurves to not cross each other.

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necessarily have to hold for all the models considered.Instead, a simple and weaker condition on the degreeof model disagreement can be used to determine thedirection of the change induced by ambiguity aversion:

Proposition 2. The degree of model disagreement �24a5is decreasing ( increasing) in abatement if and only ifCov�4p4a1�53 pa4a1 �55≤ 4≥5 0.

Proof. See Appendix A.3. �With this intuition in mind, we can now introduce

our main result, which does not require the relativelystrong condition of comonotonicity.

Proposition 3. In the optimal abatement problem undermodel uncertainty,

(i) a decision maker exhibiting CAAA always chooses toabate more (less) than an SEU maximizer if the degree ofmodel disagreement decreases (increases) with abatement;

(ii) a decision maker exhibiting strict DAAA (IAAA)always chooses to abate strictly more (less) than an SEUmaximizer if the degree of model disagreement decreases(increases) or is constant in abatement.

This proposition tells us that if higher abatementleads to a reduction in the degree of model disagree-ment, a positive incentive is generated to abate morein the first period. Intuitively, the degree of model dis-agreement will be decreasing in abatement if abatementon average decreases the probability of a catastrophemore strongly in pessimistic models. This structuraleffect has, however, to be added to the ambiguityprudence effect to determine the direction of the totalchange in the abatement level. Ultimately, whetherexperts’ disagreement decreases in abatement, and theextent to which the model structure effect interplayswith ambiguity prudence in determining the optimallevel of mitigation, can be determined only numerically.In the next section we bring the model to the data andanalyze the direction and magnitude of both effects.

3. Empirical Evidence andExpert Judgments

The question of whether the degree of model dis-agreement is increasing or decreasing in the level ofabatement is essentially an empirical one. In this sec-tion, we use the results of a recently published studyto assess whether the conditions obtained from ourtheoretical model are met in practice. In particular, westudy separately the two effects of ambiguity prudenceand convergence of agreement we described in theprevious section.

We use the data of Zickfeld et al. (2007). Their studypresents the results from interviews with 12 leadingclimate scientists about the risk of a collapse of theAtlantic meridional overturning circulation (AMOC,also called thermohaline circulation) due to global

climate change.21 Specifically, the authors elicited theexperts’ probabilities22 that a collapse of the AMOC willoccur or will be irreversibly triggered as a function ofglobal mean temperature increase realized by the year2100. These probabilities are reproduced and approxi-mated in a least-squares sense using a power functionof the type P4T 5= k1T

k2 , where T represents the changein global mean temperature, and k1 and k2 are thebest-fit coefficients, in the upper panel of Figure 2. Notethat, for T = 01 the probability of catastrophe P4T 5 is setto zero for all experts. As can be seen, eight experts23

assessed a nonzero probability of this catastrophicevent. For an increase of 2�C in 2100 relative to 2000,four experts assessed the probability of a collapse as≥5%, whereas for a warming of 4�C, three expertsassigned a probability of ≥40%. Finally, if the increasein global warming reaches 6�C, the probability is 90%for two experts, ≥50% for four experts, and ≥10%for six experts. These curves represent the differentprobability functions p4a1 �5 we introduced in §2, giventhat the abatement of greenhouse gas (GHG) emissionslowers expected global mean temperature. Althoughthe link between cumulative emissions and temperatureincrease has been shown to be robustly described by alinear relationship (Matthews et al. 2009, IPCC 2013),the magnitude of the so-called carbon-climate responsedescribing this relationship remains uncertain. In ourframework, we have so far neglected this additionalsource of uncertainty. In the online supplemental mate-rial, we allow for different values of carbon-climateresponse and show that our results are robust to thisalternative source of uncertainty.

In Figure 3, we provide the distorted probabilityfunctions for different values of model uncertaintyaversion. Formally, this notion is defined as follows.

21 The AMOC is a major current in the Atlantic Ocean that transportsheat energy from the tropics and Southern Hemisphere toward theNorth Atlantic. Changes in this ocean circulation could have animportant impact on many aspects of the global climate system,including changes in the carbon cycle. A collapse of the AMOC isdefined in Zickfeld et al. (2007, p. 249) as a “reduction in AMOCstrength by more than 90% relative to present-day.” Such an eventmay potentially have catastrophic consequences such as changes insea level in the North Atlantic up to 1 meter (Zickfeld et al. 2007,Figure 7) and reductions in crop production or water availabilitywith consequent impacts (IPCC 2007, Table 12.4). The list of scientistsselected for this study can be found in Zickfeld et al. (2007). It includesexperts with different scientific backgrounds (observationalists,palaeoclimatologists, modelers), geographic origins, and schoolsof thought. These experts were selected based on different criteria(authors’ knowledge of the field, review of recent publications, advicefrom scientists in the field).22 Expert elicitation is a tool for systematically gathering and pro-jecting scientific information on complex policy problems that isincreasingly recognized to play a valuable role for informing climatepolicy decisions (Kriegler et al. 2009).23 Note that expert 5 did not answer the question.

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Figure 2 (Color online) (Upper Panel) Experts’ Probabilities as aFunction of Global Mean Temperature in 2100; (Lower Panel)The Degree of Scientific Model Disagreement 4� 25

–10

0

10

20

30

40

50

60

70

80

90

100

Pro

babi

lity

of A

MO

C–c

olla

pse

(%)

Exp 1Exp 2Exp 3Exp 4Exp 6Exp 7Exp 9Exp 10Exps 8, 11, 12

1.01.52.02.53.03.54.04.55.05.56.00

0.05

0.10

0.15

Temperature increase in 2100 (K)

1.01.52.02.53.03.54.04.55.05.56.0

�2

Definition 3. The distorted probability p4a5 is theprobability that would be equivalently consideredunder expected utility, and that is defined as

p4a5v4w2 −L5+ 41 − p4a55v4w25

= 4v �h−158E�4h � v−158p4a1 �5v4w2 −L5

+ 41 − p4a1 �55v4w25990 (10)

Note that for an individual exhibiting an equal atti-tude toward risk and model uncertainty, the distorted

Figure 3 (Color online) Distorted Probabilities of the Catastrophe

0

10

20

30

40

50

60

70

80

90

100

� = 200� = 40� = 20� = 8� = 4� = 2� = 1� = 0

1.01.52.02.53.03.54.04.55.05.56.0–10


Note. Specifications: v , CRRA (�= 1); and h, CRMUA (�).

probability corresponds to the predictive probability ofcatastrophe: p4a5≡ E�p4a1 �5. On the contrary, underthe smooth model uncertainty aversion criterion, theDM aggregates the different models depending on thedegree of model uncertainty aversion relative to the riskaversion and acts like an expected utility maximizerconsidering only the distorted probability p4a5. In par-ticular, if aversion to model uncertainty is stronger thanthat to risk, it must be that p4a5≥ p4a5 ∀a, leading anyambiguity-averse DM to overweight more pessimisticmodels. Since estimates of the potential loss L are hardto obtain, we follow van der Ploeg and de Zeeuw(2014) in assuming a 20% loss of GDP. This order ofmagnitude is rather speculative and is used essen-tially for illustrative purposes in the context of climatechange, but it is based on the findings of Barro (2015),who shows that, historically, catastrophes (defined aslosses of at least 10% of GDP) averaged approximately20% of GDP. Finally, regarding the weights of differ-ent experts, we consider a uniform prior distribution,given that we do not have any information about the“qualifications” of the different experts.24 We can nowstudy separately the effect resulting from the degree ofmodel disagreement (convergence of agreement effect)and the one resulting from the attitude toward modeluncertainty (ambiguity prudence effect).

Let us begin with the former. The lower panel ofFigure 2 tells us that the degree of model disagreementfor the AMOC collapse is decreasing in the level ofabatement. To compute the distorted probabilities inFigure 3, we use a utility function v of the type CRRAwith a parameter of relative risk aversion �= 1 (i.e.,log utility), and a function h of the CRMUA form, witha model uncertainty aversion parameter �. From theproperties of the CRRA–CRMUA functions discussedabove, the DM exhibits CAAA, so that there is noambiguity prudence effect. The total effect on abatementcan therefore be entirely attributed to the decrease in thedegree of model disagreement. For �= 1, the individualis ambiguity neutral and maximizes expected utility byconsidering only the probability depicted in blue. When�= 0, the DM is ambiguity loving and seems to beconsidering more optimistic experts, whereas when �increases, more weight is attached to more pessimisticexperts, and the probability of catastrophe increases forany fixed level of abatement. What Figure 3 indicates isthat not only is the distorted probability of catastrophehigher when �> 1, but so is the slope of the distorted

24 This view is supported by Zickfeld et al. (2007, p. 239) whowrote “the process of choosing experts for inclusion in this study isfundamentally different from the process of sampling to estimatesome uncertain value such as a physical quantity, or polling thepublic to predict the results of an election. The route to scientific truthis not a matter of voting. One of the outliers among the respondentsmay be correct, and those who appear to be in close agreement mayall be wrong.”

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Figure 4 (Color online) (Upper Panel) Artificial Experts’ Probabilitiesas a Function of Global Mean Temperature in 2100;(Lower Panel) The Degree of Model Disagreement 4� 24a55

0

10

20

30

40

50

60Exp 1Exp 2Exp 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

3.5

3.6

3.7

3.8

1.01.52.02.53.03.54.04.55.05.56.0

×10–3

–10

1.01.52.02.53.03.54.04.55.05.56.0


�2

Pro

babi

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of A

MO

C–c

olla

pse

(%)

probability functions, therefore making abatementmarginally more desirable. This change in the marginalbenefit of abatement induces the DM to opt for a higherabatement level.

To isolate the ambiguity prudence effect, we artifi-cially construct three different probability laws repre-senting experts’ assessments of the AMOC collapse(upper panel of Figure 4). The probability laws are con-structed in such a way that, by considering a uniformprior over experts, an EU maximizer chooses exactly thesame amount of abatement as in the case consideringthe data from the probabilities of Zickfeld et al. (2007),presented in Figure 2. Since these probability laws areperfectly parallel, the degree of model disagreement�24a5 is constant in abatement (lower panel of Figure 4).The effect of higher aversion toward model uncertaintythan toward risk on optimal abatement in this casetherefore depends exclusively on the DM’s ambiguity-prudence attitude. In particular, from our model weknow that DMs who exhibit IAAA abate less, DMswho exibit CAAA abate exactly the same amount, andDMs who exhibit DAAA abate more. To quantify theimportance of this effect, we present the distorted prob-abilities with different specifications of the functions vand h in Figure 5. We consider the CRRA–CRMUAspecification described above, spanning rather extremevalues of relative risk aversion � and model uncer-tainty aversion �. As before, when � = �, the DMacts as an expected utility maximizer and considers

Figure 5 (Color online) (Upper Panel) Distorted Probabilities ofCatastrophe for Different Specifications of v , CRRA 4�5, and h,CRMUA 4�5; (Lower Panel) Difference 4ã4a55 Between DistortedProbability and Probability Law Considered Under SEU

1.01.52.02.53.03.54.04.55.05.56.0

0

10

20

30

40

50

60

–10

5.925.945.965.986.00

Detail at T = 6 K

35.0

35.5

36.0

2.5

3.0

3.5

� = 0.1, � = 20� = 1, � = 20� = 4, � = 20� = �

1.001.021.041.061.08

Detail at T = 1 K

1.01.52.02.53.03.54.04.55.05.56.0

Temperature increase in 2,100 (K)

00.20.40.60.81.0

∆(a

)

the average probability of catastrophe, depicted insolid blue. When model uncertainty aversion is higherthan risk aversion (�> �), a DM exhibiting DAAA(respectively, CAAA and IAAA) considers the dis-torted probability represented by the red dotted line(respectively, the blue dash-dotted line and browndashed line). The lower panel of Figure 5 shows thatthe difference between the distorted probabilities andthe simple average, ã4a5 2= p4a5− p4a5, is, respectively,constant, decreasing, or increasing in abatement whenCAAA, DAAA, or IAAA is considered. This givesan ambiguity-averse individual manifesting DAAAan incentive to abate more in order to prevent therealization of the bad state in the future, since theabsolute slope of the probability curve (and hence themarginal benefit of abatement) is always higher inthis situation than under expected utility or underCAAA. However, although the direction of the effect isthe same as predicted by our model, its magnitudeappears to be small, with discernible difference only forvery high values of model uncertainty aversion.25 Thisprovides an empirically grounded assessment of therelative importance of the structure versus the attitudetoward model uncertainty, showing that the formereffect (namely the convergence of model agreement)has a bigger impact on the optimal climate policy

25 It may be shown that this result of almost no ambiguity-prudenceeffect is robust to a reasonable range of GDP losses.

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decision. To provide a quantitative assessment of thecombined effects of model uncertainty on optimalabatement, we now apply our framework to a generalequilibrium model of climate change economics.

4. Quantification Using an IntegratedAssessment Model

To quantify the theoretical predictions, we implementthe model developed in this paper using the dataof §3 in the most widely used integrated assessmentmodel for the analysis of climate change, the DICEmodel (Nordhaus 1993). The DICE (dynamic integratedclimate and economy) model is a numerical optimalgrowth model à la Ramsey, which integrates emissionsand their mitigation in the production function, andwhich provides climate change feedback on the econ-omy through climate and impact modules.26 We extendthe DICE model by reformulating it as a stochasticcontrol problem and by implementing the endoge-nous possibility of a climate catastrophe based onthe estimated experts’ probability functions. Section 2in the online supplemental material provides a moredetailed description of the model. Following the expertelicitation of Zickfeld et al. (2007), we consider thecase where the uncertainty is resolved at one singlepoint in time, in the year 2100. That is, after the year2100, either the catastrophe has hit the economy, lead-ing to the crossing of a tipping point, or not. In thecatastrophic state, an irreversible damage occurs inthat an additional 20% of baseline GDP is lost for theremaining time horizon. This means that we extendthe DICE damage function that expresses the economicimpacts of climate change D in percent of GDP as thefollowing random variable:

D�4T 5∼ 6�1T +�2T�3 +L1p�4T 53�1T +�2T

�3141−p�4T 5571(11)

where T is the change in global mean temperaturerelative to the preindustrial level, and p� is the prob-ability of suffering an additional catastrophic loss L,as given by expert �.27 The term �1T +�2T

�3 on theright-hand side of expression (11) represents the stan-dard DICE damage function. For example, the defaultcalibration of �1 = 01�2 = 0000267, and �3 = 2 yields astandard damage estimate of 5.4% of GDP for a 4.5�Ctemperature increase. The loss due to a catastrophicevent (L= 20%) adds to the standard damage functionand occurs with a probability that depends on the

26 See Nordhaus and Sztorc (2013) and Nordhaus (2014) for a descrip-tion of all assumptions, equations, and data used for the latestversion of the model, DICE2013R.27 An adjustment factor of 0.7�C, representing the global meanincrease in temperature between the years 1900 and 2000 (Hansenet al. 2006), is used since the DICE standard damage functionconsiders 1900 as a reference for temperatures.

temperature increase attained in the year 2100. Finally,although so far we have assumed that the elasticity ofintertemporal substitution was equal to the inverse ofthe degree of relative risk aversion (an assumption thatis maintained throughout the literature; see Klibanoffet al. 2009), we disentangle these two very differentnormative characteristics of the decision maker toobtain a more realistic representation of preferences. Todo so, we modify DICE’s utility function and adaptthe generalized model of Hayashi and Miao (2011) todisentangle the three concepts of risk aversion, intertem-poral elasticity of substitution, and model uncertaintyaversion. As before, we represent risk aversion by thefunction v and model uncertainty aversion by h. Theagent’s intertemporal welfare at time t is representedby the following recursive form:

Wt = u−1641 −�5u4ct5+�u4Rt4Wt+1˜4�55571 (12)

where u characterizes the attitude toward consumptionsmoothing over time,28 � is the discount factor, Ct isthe consumption at time t, and Rt4Wt+14�55 representsthe double certainty equivalent. We define the doublecertainty equivalent as follows:

Rt4Wt+14�55 2= h−14Et1�4h � v−154Etv4Wt+14�55550 (13)

In this expression, Et1 � is the expectation operatortaken at time t over models, and Et is the expectationoperator taken at time t over future consumption,conditional on �. For the implementation, we use athreefold isoelastic specification of the different func-tions: � is the inverse of the elasticity of intertemporalsubstitution, � is the constant relative risk aversion(CRRA) parameter, and � is the degree of constantrelative model uncertainty aversion (CRMUA). Themain purpose of the threefold disentanglement is toallow varying risk preferences while keeping constantthe certainty equivalent discount rate, as defined bythe Ramsey rule.

The enhanced DICE model allows us to quantify theimpact of model uncertainty aversion on abatement inlight of the insights of the theoretical model proposedin §2. In particular, it shows how the combined effects ofambiguity prudence and the convergence of agreementeffect impact the optimal abatement decisions. To doso, we compute the level of additional abatement, i.e.,the extra reduction of cumulative emissions, underthe possibility of a climate catastrophe, relative to thatin the standard version of DICE without catastrophic

28 Similar to what is proposed by Epstein and Zin (1989) and Weil(1990), we consider the particular case in which u is isoelastic, with aparameter � representing the inverse of the elasticity of intertemporalsubstitution.

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Figure 6 (Color online) Additional Abatement Based on the Modified Version of DICE with the Possibility of AMOC Collapse for Different Values ofRelative Risk Aversion 4�5 (Left) and Relative Model Uncertainty Aversion 4�5 (Right)

0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20

15

20

25

30

35

40

45

50

Relative risk aversion (�)

Add

ition

al a

bate

men

t (%

)

15

20

25

30

35

40

45

50

Add

ition

al a

bate

men

t (%

)

� = 10� = 4� = 1� = 0

Relative model uncertainty aversion (�)

� = 10� = 4� = 1� = 0

climate change.29 Figure 6 illustrates the results in termsof additional abatement realized during the period2010–2100 for different parametrizations. For �=�= 0(that is, for a risk- and model-uncertainty-neutral policymaker), the only difference with respect to the standardDICE is the presence of catastrophe, which is evaluatedas its expected future consumption loss. In this case,optimal abatement increases by 13.5% in the sense thatthe cumulative emissions are further reduced from3,813 to 3,301 GtCO2, as reported in Table 1.

The additional abatement rises to 15%, 23%, and42% when both risk and model uncertainty aversionsare increased simultaneously (�=�) to 1, 4, and 10,respectively (black crosses in Figure 6). This situationof ambiguity neutrality corresponds to the Epstein–Zin–Weil version of the model. Differentiating thecoefficients of risk aversion and model uncertaintyaversion enables us to see that the additional abatementlevel is increasing in the risk aversion parameter �,though at a decreasing rate (left panel of Figure 6).For what concerns the model uncertainty aversionparameter �, the additional abatement level mono-tonically increases. In terms of magnitude, the resultssuggest that the effect of model uncertainty aversion isapproximately one-fourth to approximately one-half

29 Unless stated explicitly otherwise, we keep the standard specifica-tions of the latest version of the DICE model (see Nordhaus andSztorc 2013, Nordhaus 2014) unaltered. This, for example, meansthat the inverse of the elasticity of intertemporal substitution is fixedto � = 1045 and that the pure rate of time preference equals 105% peryear. In the standard scenario without the possibility of a catastrophe,the global temperature increase by 2100 is approximately 3.1°C, andthe global cumulative CO2 emissions, also called the cumulativecarbon budget, amount to 3,813 gigatonnes of carbon dioxide (GtCO2)for the period 2010–2100 (see the last row of Table 1). We will referto additional abatement for any further relative reduction of thiscarbon budget.

of the effect of risk aversion (as can also be inferredfrom Figure 6): starting from the case of �=�= 1 withadditional abatement of 15% of emissions, increasing� to 10 roughly doubles the effort to 32%, whereasincreasing � to 10 increases abatement to 19%. More-over, the results of the enhanced DICE model confirmwhat we observed in the previous section concerningthe relative importance of preferences and structure ofmodel disagreement: since the experts’ disagreementdecreases in abatement, abatement always increases inthe degree of model uncertainty aversion, even when�≥ 1.

Table 1 provides additional details of the scenarioruns. In the third column, we report the social costof carbon in 2015.30 It is estimated to be $17.7 perton of CO2 in the standard version of DICE withoutcatastrophe and increases to $20.4 when the possibil-ity of a catastrophe is taken into account in a risk-and model-uncertainty-neutral environment. It furtherincreases to $27 when the relative risk and modeluncertainty aversion parameters �= �= 10 are consid-ered. Additional results concerning the temperatureincrease reached in 2100 are presented in the fourthcolumn of Table 1. As can be seen, the introduction ofa potential catastrophic climate change event reducesthe admitted global temperature increase in 2100 (com-pared to the preindustrial level) from 3.1�C in thestandard version of DICE, to 2.5�C (when �= �= 10).Both the risk and model uncertainty aversion param-eters lead to a reduction of the optimal temperature

30 Put simply, this important concept measures the market price ofemissions of GHGs. More formally, the social cost of carbon at time tis defined as the ratio of the marginal impact of emissions on welfareover the marginal welfare value of a unit of aggregate consumption(see Nordhaus 2014 for more details).

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Table 1 Global Cumulative Emissions for the Period 2010–2100, Social Cost of Carbon in 2015, Temperature Increases (with Respect toPreindustrial Levels), and Average and Distorted Probabilities of Catastrophe Obtained with the Modified Version of DICE Under thePossibility of AMOC Collapse

Cum. emissions in Additional optimal Social cost of carbon Temperature increase Average prob. of Distorted prob. of2010–2100 (GtCO2) abatement (%) in 2015 ($/tCO2) ($) in 2100 (°C) catastrophe p4T ∗5 (%) catastrophe p4T ∗5 (%)

�= 0�= 0 3,301 1304 2004 2.91 6.6 6.6�= 1 3,244 1409 2007 2.89 6.5 6.4�= 4 3,045 2001 2108 2.82 6.2 5.7�= 10 2,634 3009 2401 2.67 5.5 3.9

�= 1�= 0 3,290 1307 2005 2.91 6.5 6.6�= 1 3,230 1503 2008 2.89 6.4 6.4�= 4 3,022 2007 2109 2.81 6.1 5.7�= 10 2,585 3202 2404 2.65 5.4 4.0

�= 10�= 0 3,179 1606 2101 2.87 6.4 7.3�= 1 3,096 1808 2105 2.84 6.2 7.2�= 4 2,799 2606 2302 2.73 5.8 6.5�= 10 2,195 4204 2700 2.50 4.8 4.8

Standard optimal 3,813 0 1707 3.1 0 0version of DICE

increase.31 Finally, the two last columns of Table 1present the average p4T ∗5 and distorted p4T ∗5 probabil-ities of the AMOC collapse that is ultimately admittedby the DM. These values are computed at the optimaltemperature endogenously calculated by the model.As expected, these probabilities are decreasing in both� and � since the temperature is decreasing in bothparameters. We also remark that p4T ∗5 < p4T ∗5 as longas � > �, so that the DM always overestimates theprobability of catastrophe when model uncertaintyaversion is stronger than risk aversion (and vice versa).Overall, these results from the stochastic IAM confirmthat model uncertainty plays an important role in quan-titative terms when the convergence of agreement effectis important. Depending on the parametrization ofpreferences, the possibility of catastrophes leads to anadditional mitigation effort of the cumulative emissionsin the baseline scenario in the range of 13%–49%.

To analyze the robustness of our results, we performan extensive sensitivity analysis with respect to themost relevant model parameters and specifications. Inparticular, we take into account a different timing ofthe catastrophic event, different values for the equi-librium climate sensitivity, different utility discountrates, and different values of the economic losses of thecatastrophe. Although the full set of results is availablein the online supplemental material, a summary of theresults can be found in Table 2. We focus on the effectof model uncertainty aversion, while maintaining an

31 Additional graphs concerning the social cost of carbon and thestochastic evolution of global temperature can be found in the onlinesupplemental material.

intermediate value of �= 4 for the parameter of riskaversion.

Overall, the results show that the qualitative effectof model uncertainty aversion is robust throughoutthe specifications considered. For a lower value of theeconomic loss from the catastrophe (10% of GDP), avery low value of the climate sensitivity, or a com-parable high utility discount rate of 3%, the effectof model uncertainty is attenuated but still leads toa notable increase in optimal abatement. If, on theother hand, the values are set to the other side ofthe spectrum, model uncertainty raises precautionarymitigation effort significantly, and more than propor-tionally. For example, an impact L of 30% as opposedto 20% raises the social cost of carbon by approximately$7/tCO2. Finally, regarding the timing of learning andthe potential occurrence of the catastrophic event, wefind that the increase in abatement is higher for earlieroccurrences and that it diminishes over time.

5. ConclusionThis paper aims at understanding and quantifyingthe impact of model uncertainty aversion on optimalabatement decisions, for the policy-relevant case ofcatastrophic climate change. This attempt stems fromthe recognition that, although it is now fully recognizedthat the presence of these uncertainties represents anessential datum of the climate change issue, the waythey are treated and integrated in the models used tomake predictions or to design public policies remainsunsatisfactory. By evaluating the optimal strategy forresponding to an uncertain threat, the model we presentin this paper has the advantage of treating policyanalysis like a robust risk management problem.

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Table 2 Sensitivity Analysis

Increased Cum. emissions Social cost Temperature increaseabatement for (�= 10) during of carbon (�= 10) (�= 10)

�= 0 → 10 (%) 2010–2100 (GtCO2) in 2015 ($/tCO2) ($) in 2100 (�C)

Impact LL= 10 % 103 31442 1906 2.96L= 30 % 1908 11953 3000 2.37

Climate sensitivityECS = 105 204 41820 805 1.96ECS = 405 906 11540 4205 3.16

Discount rateprstp= 00001 1003 922 8300 1.86prstp = 0003 303 41306 901 3.20

Time of resolutiona

2075 805 21091 2608 2.322100 504 21410 2302 2.412125 305 2616 2101 2.46

Standard version 801 21799 2302 2.734�= 41 �= 105

Notes. Differences in relative abatement are given in percentage points comparing the model run with �= 10 to �= 0, keeping �= 4 andeverything else constant. Cumulative emissions, social cost of carbon, and temperature increases are reported for the high modeluncertainty aversion case (�= 10). ECS, equilibrium climate sensitivity; prstp, initial rate of social time preference per year.

aFor these cases, cumulative emissions and temperature are reported for/until the year 2075.

In particular, we consider situations in which theactions we take today (such as choosing the levelof abatement) affect the probability of incurring ahigh-damage event (of catastrophic nature) in thefuture. The selection of optimal policies in this sense isessentially an exercise in risk management. However,the particularity of this exercise is that it is carried outunder partial ignorance: the decision makers we studyadmit that they do not know the exact relationshipbetween their action and the probability of catastrophe.Rather, what the decision makers have available to helpthem make a choice is a collection of models or experts’estimates of this relationship. In contrast with purelyrisky situations in which the probabilities are known,the situations we study are therefore deeply uncertainor ambiguous. The ambiguity results precisely fromthe combination of risk and model uncertainty, and thedecision maker’s attitude toward ambiguity naturallyresults from the composition of attitudes toward thesetwo distinct sources of uncertainty.

We compare this robust decision-making approachwith the standard expected utility approach and showthat the latter is not capable of differentiating distinctattitudes toward different types of uncertainty: it implic-itly treats a situation in which experts have differentdogmatic beliefs exactly the same way as a situationof pure risk. Rather, if the policy maker is ambiguityaverse by being more sensitive to model uncertaintythan to risk, we show that the policy maker will under-take more abatement effort if the combination of theambiguity prudence effect and the convergence ofthe agreement effect is positive. The former conditionis directly related to a condition about the functions

representing preferences, whereas the latter is a charac-teristic of the available expert elicitation or model data.The intuition behind this result is that the desirabil-ity of preventive efforts is measured not only by thereduction in the expected damages, but also by thevalue of the associated reduced uncertainties. A degreeof model disagreement that is decreasing in abatementeffort is asking for a policy limiting global warming torelatively lower levels because it gives a precautionarypolicy maker an extra incentive for a more stringentmitigation policy, in the spirit of the precautionaryprinciple. Finally, in contributing to answering the needto integrate the treatment of deep uncertainties and ofpossible catastrophic events in integrated assessmentmodels, we apply our insights to the DICE model andshow that robust precautionary climate policies requirea significantly higher abatement level. Although therisk-neutral consideration of a catastrophic risk leadsto a comparably low increase in abatement effort, thisincrease is magnified for reasonable degrees of bothrisk and model uncertainty aversion.

Although the proposed framework allows us togenerate a set of original insights, many limitationsremain. For example, we abstracted from the possi-bility of learning. Although it is unclear how muchwe can actually learn about these extreme climaticoutcomes and what the implications are of learningon optimal abatement (IPCC 2014), several insight-ful applications emphasizing the role of learning inintegrated assessment models with tipping elementshave been recently proposed (Rudik 2016, Lemoine andTraeger 2014). Our framework also requires calibratingparameters for which few estimates exist, such as the

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model uncertainty aversion, potentially limiting itspractical use. Nonetheless, we believe that the flexibil-ity of the model uncertainty decision framework, theresults about the importance of the structure of modeluncertainty, and the simplicity of the proposed metricof model disagreement, are widely applicable and canbe fruitfully extended to other policy objectives anddata-generating processes such as additional tippingelements, climate engineering, technological change, oreven non-climate-related policy issues.

Supplemental MaterialSupplemental material to this paper is available at http://dx.doi.org/10.1287/mnsc.2015.2365.

AcknowledgmentsThe authors are grateful to Massimo Marinacci, LaurentDrouet, Christian Gollier, Antony Millner, Richard Peter,Nicolas Treich, three anonymous referees, and an associateeditor for helpful comments and discussions. The researchleading to these results received funding from the EuropeanUnion Seventh Framework Programme FP7/2007-2013 [Grant308329] (ADVANCE).

Appendix A. Proofs

A.1. Proof of Lemma 1The proof directly follows from Proposition 4 in Berger (2015).The condition to observe a higher (lower) abatement due toambiguity aversion may be written as

E�

[

�′4Ev4C24a1 �555¡Ev4C24a1 �55

¡a

]

≥ 4≤5�′4�−14E��4Ev4C24a1 �55555E�

¡Ev4C24a1 �55

¡a0 (A1)

Analogously to the risk theory literature, it can moreoverbe shown that CAAA is equivalent to �′4�−14E�4U 555 =

E�′4U 5, strict DAAA is equivalent to �′4�−14E�4U 555< E�′4U 5, and strict IAAA is equivalent to �′4�−14E�4U 555 >E�′4U 5. By letting A≡ Ev4C24a1 �55 and B ≡ ¡Ev4C24a1 �55/¡a,we can rewrite condition (A1) as Cov�4�

′4A53B5≥ 4≤50, orCov�4A3B5≤ 4≥50, since �′ is decreasing under ambiguityaversion. In the case of strict DAAA, we can use the chain ofinequalities

E�6�′4A5B7≥ E��

′4A5E�B >�′4�−14E��4A555E�B1 (A2)

so that the left-hand side (LHS) of (A1) is greater than theright-hand side (RHS) if Cov�4A3B5≤ 0; in the case of strictIAAA, we can use the chain of inequalities

E�6�′4A5B7≤ E��

′4A5E�B <�′4�−14E��4A555E�B (A3)

to show that the RHS of (A1) is greater than the LHS ifCov�4A3B5 ≥ 0. From Kimball (1951), it follows that thiscovariance is negative (positive) if A and B are anticomono-tonic (comonotonic). �

A.2. Proof of Proposition 1Considering the ambiguity aversion function �4U5 =

4h�v−154U5, where U represents the expected utility com-puted in the presence of risk (i.e., U ≡ Ev4x5), it is easy tocompute the index of absolute ambiguity aversion as

−�′′4U5

�′4U5= −

v′h′′ −h′v′′

4v′53

v′

h′=

1v′

[

−h′′

h′+

v′′

v′

]

1 (A4)

where, in a slight abuse of notations, we let h≡ h4v−14U55and v ≡ v4v−14U55. As expected, this ratio is positive ifmodel uncertainty aversion is higher than risk aversion.Analogously to risk theory literature, DAAA means that−�′′′4U5/�′′4U5≥ −�′′4U5/�′4U5, which is the case if andonly if

h′′′

h′+ 2

(

−v′′

v′

)2

≥v′′′

v′+

(

−h′′

h′

)2

+

(

−h′′

h′

)(

−v′′

v′

)

0 �(A5)

A.3. Proof of Proposition 2The result is obtained by decomposing �24a5= E�6p4a1 �5

27−p4a1 �52, where p4a5 ≡ E�p4a1 �51 and deriving this expres-sion with respect to a: ¡�24a5/¡a = 2E�6p4a1�5pa4a1�57 −

2p4a1 �5pa4a1 �5= 2Cov�4p4a1�53 pa4a1 �55. �

Appendix B. Economic Impact UncertaintyImpact (or socioeconomic) uncertainty results from our“imperfect understanding of the impacts of climate change onhuman societies and of how these societies will respond”(Heal and Millner 2014, p. 121). In the context of our abate-ment model, imagine that there is a scientific consensus onthe link between the probability of a catastrophic event andthe temperature increase (or abatement levels) given by aparticular probability function p4a5. Even in this far-from-realistic situation of limited scientific uncertainty, there wouldstill be room for model uncertainty to play a significantrole because of the remaining uncertainty concerning theeconomic impacts of a climate catastrophe. What, for example,would be the economic loss associated with a sea level riseof one meter? Would it be possible to construct protectivedikes to save the most vulnerable places, and if so, at whatcost? Alternatively, what would be the cost associated withrelocation and reconstruction? All of these costs correspondto what we have called the economic loss associated with thecatastrophic event and are far from being perfectly known.32

Different experts or studies may disagree on the total impactof a possible catastrophe, and this disagreement amongeconomic models may potentially affect the decision madeby a policy maker.

In our simple optimal abatement problem under impactuncertainty about the economic impacts Ls , the second-periodexpected utility for a given model P� is now written as

Ev4C24a1�55=p4a5∑

s∈S

�s4�5v4w2 −Ls5+41−p4a55v4w251 (B1)

32 Stern (2007), for example, estimates the total loss for a high climatechange scenario with nonmarket impacts and the risk of a catastropheto be between 2.9% and 35.2% of GDP per capita in 2200 (seeFigure 6.5 in Stern 2007 for more details).

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where �s4�5 denotes the probability according to expert � ofthe loss Ls . The expected marginal benefit of abatement canbe obtained as

¡Ev4C24a1 �55

¡a= −pa4a5

[

v4w25−∑

s∈S

�s4�5v4w2 −Ls5

]

0 (B2)

Given that the probability of catastrophe is assumed to bedecreasing in abatement, it is clear that expressions (B1)and (B2) will always be anticomonotonic, which leads us tothe following result.

Proposition 4. In the optimal abatement problem under modeluncertainty about impacts, an agent considering the smoothcriterion and exhibiting CAAA or DAAA always chooses to abatemore than an expected utility maximizer .

Proof. The result directly follows from Lemma 1. �On the contrary, if the DM exhibits IAAA, it is impossible

to unambiguously sign the final effect of model uncertaintyaversion since it will depend on which of the two effects(degree of model disagreement or ambiguity prudence)dominates.

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http://www.econ.yale.edu/~nordhaus/homepage/documents/DICE_Manual_103113r2.pdf

http://www.econ.yale.edu/~nordhaus/homepage/documents/DICE_Manual_103113r2.pdf

http://www.sehn.org/wing.html

managing catastrophic climate risks under model uncertainty … · 2018. 9. 2. · more...

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