GHG Targets as Insurance Against Catastrophic Climate Damages

A critical issue in climate-change economics is the specification of the so-called "damages function" and its interaction with the unknown uncertainty of catastrophic outcomes. This paper asks how much we might be misled by our economic assessment of climate change when we employ a conventional quadratic damages function and/or a thin-tailed probability distribution for extreme temperatures. The paper gives some numerical examples of the indirect value of various GHG concentration targets as insurance against catastrophic climate-change temperatures and damages. These numerical examples suggest that we might be underestimating considerably the welfare losses from uncertainty by using a quadratic damages function and/or a thin-tailed temperature distribution. In these examples, the primary reason for keeping GHG levels down is to insure against high-temperature catastrophic climate risks.

known now. They must be conceptualized instead as random variables (RVs), yet to be drawn from some probability density function (PDF). How bad will it get? An answer must ultimately be expressed in the language of tail probabilities. This paper concentrates on the appropriate way to represent uncertain global warming and uncertain damages. The "damages function"is a notoriously weak link in the economics of climate change, because it is di¢ cult to specify a priori and because, as will be shown, the results from a cost-bene…t analysis (CBA) or an integrated assessment model (IAM) can be quite sensitive to its speci…cation at the upper end of extreme impacts. Another notoriously weak link in the economics of climate change is the estimation of tail fatness for the PDF of extreme warmings. These problems are especially acute at catastrophically high temperatures, because huge uncertainties surround any estimates of extreme damages or probabilities of climate-change disasters. Ideally, one wants analytically tractable forms that capture adequately the economic reality of global warming.
The existing literature on CBAs and IAMs of climate change mostly concentrates on super-moderate quadratic damages and on super-thin-tailed point-mass PDFs. 1 This paper investigates what might happen to an economic analysis of climate change with a signi…cantly more reactive damages function than the quadratic and with PDFs having tails of varying degrees of fatness. The paper attempts to give some extremely rough ballpark estimates of the di¤erences in steady-state temperature PDFs and damages as a function of greenhouse gas (GHG) target concentration levels. These di¤erences vary greatly according to the speci…cation, but on the whole they are substantial enough to suggest that in some situations -especially when catastrophic damages interact with fat-tailed uncertainty -we might be underestimating welfare losses considerably. The critical question here is: How fast does the probability of a catastrophe decline relative to the welfare impact of the catastrophe? Even tiny probabilities can be o¤set by negative welfare impacts that are big enough. In such conditions the fact that the tiny probabilities are themselves unknown is, other things being equal, more troubling than if they were known precisely. With the examples being considered in this paper, the primary reason for keeping target GHG levels down is to insure against high-temperature catastrophic climate risks. In situations where fat-tailed PDFs are combined with a reactive damages function, the welfare di¤erences between various target GHG levels are typically very large and there is a much stronger case for keeping down GHG target levels than when tails are thin or damages are quadratic.
Climate change is so complicated, and it involves so many sides of so many di¤erent disciplines and viewpoints, that no analytically-tractable model or paper can aspire to illuminate more than but a facet. Because the climate change problem is so complex, there is frequent reliance on sophisticated numerical computer simulations. These can be indispensable, but sometimes they do not provide a simple intuition for the processes they are modeling. In this paper I go to the opposite extreme by focusing on relatively tractable comparative-steadystate solutions. What I am presenting here is a kind of "stress test" approach to grasping intuitively the robustness of modeling highly uncertain extreme damages. This paper is mostly about conceptualizing the problem of high-temperature catastrophic damages and giving some rough sense of the magnitudes involved via particular numerical examples. It is less about giving decisive numerical values for actual practical policy advice, although some policy implications will be apparent. The beauty of this toy model approach is that the formulas I will use are su¢ ciently simple and transparent that readers can easily plug in different speci…cations or attach the model to other frameworks and make their own inferences. A drawback of my toy model approach is that it might be missing some critical dynamic interactions that are unable to be captured by the crudeness of what is largely an exercise in comparative steady states. So any conclusions of this paper are at most suggestive and may need to be modi…ed in the light of performing detailed numerical simulations from more complicated dynamic computer models. Still, I think there is an important role for baby models such as this one, which give some direct intuition that may be considerably more transparent than what emerges from detailed simulations of more complicated formulations.

Uncertain Equilibrium Warmings
There are so many sources of uncertainty in climate change that a person almost does not know where or how to begin cataloging them. For speci…city, I focus on the uncertainty of so-called "equilibrium climate sensitivity." This is a relatively well-de…ned and relatively well-studied example of known unknowns, even if the uncertainties themselves are uncertain. However, it should be clearly understood that under the rubric of "equilibrium climate sensitivity"I am trying to aggregate together an entire suite of uncertainties, including some non-negligible unknown unknowns. So climate sensitivity is to be understood here as a prototype example or a metaphor, which is being used to illustrate much more generic issues in the economics of highly uncertain climate change. The insights and results of this paper are not intended to stand or fall on the narrow issue of accurately modeling uncertain climate sensitivity per se. Whatever its source, greater uncertainty generally strengthens the case I am trying to make in this paper.
The economics of climate change consists of a very long chain of tenuous inferences fraught with big uncertainties in every link: beginning with unknown base-case GHG emissions; then compounded by big uncertainties about how available policies and policy levers will transfer into actual GHG emissions; compounded by big uncertainties about how GHG ‡ow emissions accumulate via the carbon cycle into GHG stock concentrations; compounded by big uncertainties about how and when GHG stock concentrations translate into global average temperature changes; compounded by big uncertainties about how global average temperature changes decompose into regional climate changes; compounded by big uncertainties about how adaptations to, and mitigations of, climate-change damages are translated into utility changes at a regional level via a "damages function"; compounded by big uncertainties about how future regional utility changes are aggregated into a worldwide utility function and what should be its overall degree of risk aversion; compounded by big uncertainties about what discount rate should be used to convert everything into expected-present-discounted values. The result of this lengthy cascading of big uncertainties is a reduced form of truly enormous uncertainty about the form of an integrated assessment problem whose structure wants badly be transparently understood and stress tested for catastrophic outcomes.
Let welfare W stand for expected present discounted utility, whose theoretical upper bound is B. Let D B W be expected present discounted disutility. Here D stands for what might be called the "diswelfare"of climate change. Unless otherwise noted, my default meaning of the term "fat tail"(or "thin tail") will concern the upper tail of the PDF of ln D, resulting from whatever combination of probabilistic temperature changes, temperaturesensitive damages, discounting, and so forth, by which this comes about. Empirically, it is not the fatness of the tail of temperature PDFs alone or the reactivity of the damages function to high temperatures alone, or any other factor alone, that counts, but the combination of all such factors. It may seem arcane, but the tail thickness of the reduced-form PDF of ln D plays an essential role in the economics of catastrophic climate change. In this paper, the default fatness that really matters concerns the bad tail of the reduced-form PDF of ln Dnot climate sensitivity per se. Of course it is extremely di¢ cult to know the thickness of the upper tail of the PDF of ln D, which is my main thesis in this paper.
"Equilibrium climate sensitivity" (hereafter denoted S) is a key macro-indicator of the eventual temperature response to GHG changes. It is de…ned as the global average surface warming that follows a sustained doubling of atmospheric carbon dioxide (CO 2 ), after the climate system has reached a new equilibrium. 2 Calculating the actual time trajectory of 2 In scienti…c jargon, S is a so-called "fast equilibrium" concept based on "fast feedbacks" (geologically speaking). The concept omits slower-acting feedbacks, such as changes in albedo, changes in biological sinks or sources, temperature-induced releases of carbon from clathrates, and the like. So-called "earth system sensitivity" includes slower-acting feedbacks and is presumably larger, perhaps signi…cantly so. For a time horizon on the scale of 150 years or so, it is not implausible that "earth system sensitivity" might be the more relevant concept. Greater details are available, e.g., in Hansen et al (2008). temperatures is a complicated task that requires sophisticated computer modeling based on general circulation models with hundreds of parameters and variables. The human mind being what it is, however, there is a need to reduce and relate such a complicated dynamic reality to an aggregate indicator, like S. This is a simplistic reduction that overlooks important spatial and temporal aspects of climate change. Nevertheless, the concept is still very useful for capturing the "big picture" -perhaps because the more complicated simulation models …nd that several aspects of climate change seem to scale approximately linearly with S. 3 As just one example of an application of this convenient reductionism, the GHG concentrations that would prevent so-called "dangerous anthropogenic interference" -however it is de…ned -are often made by back-of-the-envelope calculations based on S. But because S is uncertain, the uncertain temperature changes induced by a given GHG concentration can only be described in terms of probabilities. This paper follows very closely the spirit and assumptions (and drawbacks) of the S-reductionist approach.
The Intergovernmental Panel on Climate Change in its IPCC-AR4 (2007) Executive Summary explains S this way: "The equilibrium climate sensitivity is a measure of the climate system response to sustained radiative forcing. It is not a projection but is de…ned as the global average surface warming following a doubling of carbon dioxide concentrations. It is likely to be in the range 2 to 4.5 C with a best estimate of 3 C, and is very unlikely to be less than 1.5 C. Values substantially higher than 4.5 C cannot be excluded, but agreement of models with observations is not as good for those values." The IPCC de…nes "likely"as a probability above 66% but below 90%. In this paper I choose 70% as de…ning "likely"and I calibrate all upper-tail probability distributions so that P[S 3 C]=50% and P[S 4.5 C]=15%. 4 The upper-half tail of the probability distribution is the region S > S M , whose total probability mass is .5, where the climate-sensitivity median is S M =3 C. I use three PDFs to represent this upper-half tail of climate sensitivity: (1) the Normal distribution, which has a thin upper tail; (2) the Pareto (or Power) distribution, which has a fat upper tail; (3) the Lognormal distribution which has an upper tail on the borderline between fat and thin. There is some wiggle room in the de…nition of what constitutes a fat-tailed PDF or a thintailed PDF, but almost everyone agrees that probabilities declining exponentially or faster (like the Normal) are thin tailed, while probabilities declining polynomially or slower (like the Pareto) are fat tailed. The intermediate-tailed Lognormal is an interesting borderline case because the probabilities in its upper tail decline slower than exponentially but faster than polynomially. 5 For all three PDFs I calibrate parameters so that P[S 3]=.5 and P[S 4.5]=.15. A major goal of this paper is to experiment with di¤erent PDFs above the median value of S M =3 C. For the purposes of this paper, very little depends on the exact form of the PDF for the 50% of probability below the median. By contrast, we are forced to speculate and extrapolate wildly concerning the PDF for the 50% of probability above the median, and, as we shall see, this can have major consequences.
The notation f I (S) refers to the PDF of climate sensitivity S. The subscript I=L refers to a Lognormal PDF, the subscript I=N refers to a PDF whose upper-half tail is Normal, and the subscript I=P refers to a distribution whose upper-half tail is Pareto (or Power). I begin with the base case of the Lognormal, whose upper-half PDF here is for all S 3. As can readily be con…rmed, the parameter values in (1) have been calibrated so that P[S 3 C]=.5 and P[S 4.5 C]=.15. I also consider two other possibilities for the upper-half tail: a fat-tailed Pareto PDF and a thin-tailed Normal PDF. My upper-half-tail Pareto PDF is also speci…ed by its parameters being set so that simultaneously P P [S 3]=.5 and P P [S 4.5]=.15. It is readily con…rmed that the corresponding upper-half-tail Pareto PDF is f P (S) = 38:76 S 3:969 : ( My upper-half-tail Normal PDF is again speci…ed by its two parameters being set so that simultaneously P N [S 3]=.5 and P N [S 4.5]=.15. It is readily con…rmed that the corresponding upper-half-tail Normal PDF is The following I think that not many climate scientists would quibble about the "big picture" of the PDF of climate sensitivity given by Table 2 for low values of climate sensitivity. For what it is worth, the median upper …ve percent probability level over all 22 climate-sensitivity PDFs cited in IPCC-AR4 is 6.4 C, which …ts with the Pareto PDF above. 6 Notice that the absolute probabilities of very high values of S are quite small. Even so, the relative probabilities of high S are extremely dependent on whether the upper tail of the relevant PDF is fat, thin, or intermediate. It is tempting to say that climate sensitivity above, say, 15 C is "impossible." I would prefer to think that anything is possible under the novel experiment of (geologically instantaneously) doubling atmospheric CO 2 concentrations in a situation where so many unknowns are so highly uncertain. Take the lognormal PDF as a base case. I am not sure how anyone would distinguish operationally here between a very rare event S>15 C that is "impossible" and a very rare event S>15 C that, from Table 1, has a 1/50,000 chance of materializing. Such …ne distinctions can be ignored in most applications, and the analysis can proceed as if the event is "impossible"for all practical purposes. But in the extraordinary case of global warming, whose potential damages could engulf the entire planet, one does not have the luxury of ignoring even the lowest of low-probability events if they occur with the highest of highly-negative impacts.
The next step is to convert PDFs of equilibrium climate sensitivity S into PDFs of equilibrium temperature change T , as a function of given stable greenhouse gas (GHG) target concentrations. Let G stand for atmospheric GHGs as measured in parts per million (ppm) of carbon dioxide equivalent (CO 2 e). Climate sensitivity corresponds to the equilibrium temperature change eventually induced by a sustained doubling of CO 2 e. Let (G) represent the "forcing factor" as a function of the steady-state GHG level G, with (G) normalized by making (560) 1. An atmospheric concentration of G=560 ppm represents a doubling of the pre-industrial-revolution level of G=280. As is well known, the forcing factor increases linearly in the logarithm of CO 2 e concentrations. 7 With normalization (560) 1, the precise formula is Therefore, a given constant level of GHGs G and a given equilibrium climate sensitivity S translates into a steady-state temperature change of If f I (S) is the relevant PDF of climate sensitivity, then the relevant PDF of temperatures T for a given level of G is To anchor the upper tail of extreme warmings, I focus sharply on just two iconic (if arbitrary) values of extraordinarily high global average temperature increases: 6 C and 12 C. Six degrees of extra warming is about the upper limit of what the human mind can envision for how the state of the planet might change. It serves as a routine upper bound in attempts to communicate what the most severe global warming might signify, including the famous "burning embers" diagram of the IPCC and several other popular expositions. 8 One recent study 9 asked 52 experts for their subjective probability estimates of triggering a "tipping point of major changes" in each of …ve possible categories: (1) the Atlantic meridional overturning circulation; (2) the Greenland ice sheet; (3) the West Antarctic Ice Sheet; (4) the Amazon rainforest; (5) the El Niño/Southern Oscillation. For what it is worth, at an average temperature increase of T 6 C the expected (probability weighted) number of such expert-assessment tipping points was three (out of a possible …ve).
Twelve degrees of global warming is used here as an example of a round number (12 C=2 6 C) that transcends our ability to imagine, with any reasonable measure of accuracy, what the earth might be like for super-high temperature increases. For me, 12 C is especially iconic because of a recent study, which estimated that global average temperature increases of 11-12 C would cause conditions under which more than half of today's human population would be living in places where, at least once a year, there would be periods when death from heat stress would likely ensue after about six hours of exposure. 10 The authors of this study furthermore point out: "This likely overestimates what could practically be tolerated: our limit applies to a person out of the sun, in a gale-force wind, doused with water, wearing no clothing and not working." The massive unrest and uncontainable pressures this would bring to bear on the world's population are almost unimaginable. A temperature change of 12 C therefore represents an extreme threat to human civilization as we now know it, even if it does not necessarily mean the end of Homo sapiens as a species.
Throughout the numerical examples that follow, I arbitrarily take 18 C (3 6 C) to be an upper bound beyond which temperatures are not allowed to go -by …at. Thus, for all calculations of expected values, damages are capped at 18 C and probabilities of such damages are calculated as P[T 18 C]. In this sense 18 C might be envisioned as something like a global "death temperature." The issue of how to deal with the deep structural uncertainties in climate change would be completely di¤erent and immensely simpler if systemic inertias, like the time required for the system to naturally remove extra atmospheric CO 2 , were short, as is the case for many airborne pollutants like particulates, sulfur dioxide, and ozone. Then an important component of an optimal strategy might be along the lines of "wait and see." With strong reversibility, an optimal climate-change policy should logically involve (among other elements) waiting to learn how far out on the bad fat tail the planet might end up, followed by midcourse corrections if we seem to be headed for a disaster. Alas, the problem of climate change seems bedeviled almost everywhere by signi…cant stock-accumulation inertias -in atmospheric CO 2 , in the absorption of heat or CO 2 by the oceans, in the uptake of CO 2 by the biosphere, in albedo changes, in the wildcard behavior of methane clathrates, and in many other relevant physical and biological processes that are extremely slow to respond to attempts at reversal.
Take atmospheric carbon dioxide as a prime example. Solomon et al (2009) calculated how concentrations of CO 2 would be expected to fall o¤ over time if all anthropogenic emissions were to cease immediately, following a future 2% annual growth rate of emissions up to peak concentrations of 450, 550, 650, 750, 850 and 1,200 ppm. As the authors state: "The example of a sudden cessation of emissions provides an upper bound to how much reversibility is possible, if, for example, unexpectedly damaging climate changes were to be observed." Results di¤ered for di¤erent trajectories and scenarios, but a crude rule of thumb seemed to be that approximately 70% of the peak enhancement level over the preindustrial level of 280 ppm persevered after 100 years of zero emissions, while approximately 40% of the peak enhancement level over the preindustrial level of 280 ppm persevered after 1,000 years of zero emissions. In the Solomon et al study, were atmospheric CO 2 concentrations to peak at 800 ppm, followed forever thereafter by zero emissions, then atmospheric concentrations would be 650 ppm after 100 years and 500 ppm after 1,000 years. These numbers do not look to me like evidence supporting "wait and see"policies. The capacity of the oceans to take up atmospheric heat, and many, many other relevant mechanisms, tell a similar story of long stock-accumulation irreversibilities relative to the time it takes to …lter out and act upon meaningful signals of impending disasters. Under such conditions of limited learning relative to reversibility, the fact that the small probabilities of big disasters are themselves uncertain is not an excuse for delay. Just the opposite, if anything it is a stronger call to immediate action than if the probabilities were known precisely.
In the following Table 2, the …rst row represents steady-state atmospheric stocks of greenhouse gas concentrations G (measured in ppm of CO 2 e). The second row below it gives the median equilibrium temperature T M as a function of stabilized GHG stocks. The rows starting just below T M give the probabilities of achieving at least the steady state temperature increase represented by the entries in the table ( Table 2: Probabilities of exceeding T =6 C, T =12 C, T =18 C, for given G = ppm of CO 2 e.
The thing that seems so striking about Table 2 is how relatively rapidly the probabilities of high temperatures increase as a function of GHG concentrations -and how dependent these high temperatures can be on the assumed fatness of the upper tail of the PDF of climate sensitivity. Throughout Table 2, the target level of GHG concentrations in ‡uences strongly the probabilities of high temperatures. One can readily see in shorthand form what are the ultimate temperature consequences of moving from lower to higher steady-state GHG concentrations. Of course these ultimate temperature consequences are expressible only as probabilities. It can be quite misleading to look just at measures of central tendency, like the median. What to me is far more alarming than the moderate rise of T M as a function of G is what is happening in the upper reaches of the various PDFs, where the really catastrophic outcomes are concentrated. The higher levels of GHGs seem especially worrisome to me because they are pushing temperature probabilities towards the upper tail at an uncomfortably rapid rate.
To see things most sharply, notice at the two opposite extremes that 400 ppm of G here e¤ectively blocks temperatures from rising much above 6 C, whereas 1000 ppm of G here assigns a probability of 41% to P[T 6 C] and 1%-5% to P[T 12 C], depending on the assumed tail fatness. Notice too how the di¤erences between the three di¤erent PDFs (with three di¤erent degrees of fatness in their tails) are manifested for various GHG concentrations. Throughout most of Table 2 there is a disturbingly non-robust dependence of outcomes on the presumed fatness of the upper tail of the PDF, which we simply cannot know. The thin-tailed normal distribution e¤ectively excludes the really hotter temperatures, while the fat-tailed Pareto distribution presents a much more worrisome picture. This awkward dependence upon presumed tail fatness is more pronounced the deeper one penetrates into the extreme tail of the underlying PDF of climate sensitivity. At the higher concentrations of GHGs, say 650 ppm of CO 2 e, a temperature increase of 6 C is su¢ ciently close to the middle-body range of all three climate-sensitivity PDFs that tail fatness per se does not matter so much in determining P[T 6]. On the other hand, tail fatness always matters a lot for determining P[T 12], even for higher GHG concentrations 650, because this part of the range of temperatures is well into the extreme tail of the underlying climate-sensitivity PDFs. In more colorful language, very di¤erent tails may be appended to animals having roughly similar bodies. Table 2 suggests that the primary purpose of keeping down G may be to prevent possibilities of extreme warmings in the upper range of the PDFs, and perhaps only secondarily to keep down the median temperature (although this may be important too). In this sense Table 2 is indicating indirectly how much "insurance"society is willing to buy to ward o¤ the risk of very high temperatures by paying the "cost" of keeping GHG concentrations below various levels. I do not analyze explicitly the costs of achieving various steady-state GHG targets, being content to let them stand for themselves as proxies for less or more active mitigation measures. Only the "value"or demand side of insurance (against high-temperature extreme damages) is being presented (and that indirectly), not its "cost" or supply side. Throughout this paper, target steady-state GHG concentrations are interpreted as an imperfect proxy for "policy." If one wants a transparent summary of the temperature consequences of higher GHG concentrations, I think that Table 2 is …ne. Perhaps the analysis should be ended here, as the table speaks for itself quite eloquently in ways that an informed citizen might understand once the possible consequences of the "iconic"values T =6 C and T =12 C have been sketched out, as they were previously. But an economist is tempted to take the analysis at least one step further toward quantifying damages before succumbing to a computer simulation of a full-blown dynamic IAM with lots of opaque moving parts.
For any given G, and for any given PDF of S, we have derived a PDF of T (via (4) and (6)), some numerical values of which are displayed in Table 2. Of course such type of analysis ignores all kinds of dynamics to concentrate on the more easily understandable long run value of T associated with a sustained stationary level of G. In the coming extension to potential damages, I will follow closely the spirit and assumptions of the approach to temperature PDFs of this section. Thus, throughout this paper the simplistic methodology looks primarily at the "big picture" of a still photograph of damages in a steady state, with only the most primitive story (later) about dynamics.
I do not investigate seriously the important subject of the motion picture describing the dynamics of getting to the steady state. I only open the comparative-steady-state envelope a bit further to let in the sticky subject of high-temperature damages. There is an arti…cial timing here that compresses dynamics into statics by dealing primarily with steady states. Such an aggregate comparative-steady-state approach has proved to be a useful shortcut way for organizing thinking about eventual temperature responses to target GHG concentrations. It tells us in shorthand form what temperatures (more accurately temperature PDFs) we are eventually buying into when we set target GHG concentrations at various levels. I propose extending the same strategy a little further to at least discuss the possible present discounted damages from large temperature changes.

Uncertain Damages From Climate Change
From the very outset, the representation of damages from climate change presents some severe conceptual and practical problems. I follow most of the literature by postulating that damages from increased temperatures are manifested in reduced form as if they impair output. 11 In my version of this just-so story, all losses from climate change will be interpreted as if they literally translate into a welfare-equivalent loss of consumption. There are some genuine doubts about what it means operationally to separate welfare-equivalent consumption from welfare, but here I largely follow the existing literature. As mentioned, this paper examines only the damages side, and that very simplistically. I do not try to explicitly estimate costs of achieving various GHG targets, much less attempt to determine an optimal policy by explicitly balancing the costs of achieving a given GHG target against its bene…ts.
Even granted that it multiplicatively diminishes consumption, no one knows how to specify a "damages function" for high temperature changes. The predominant approach attempts to calculate what the world would be like for a given small increase in global average temperatures. The climate-change economist tries to quantify such things as net damages (after subtracting out adaptation costs) from changes in: agricultural productivity, life styles, population movements, rising oceans, hurricanes, and so forth. This is a constructive approach that probably represents the best we can do for small temperature changes. But I am uneasy when this approach is extended to large changes in global average temperatures. Taking the most extreme example I can imagine for making my point, suppose for the sake of argument that average global warming were to increase by the extraordinary amount of 12 C (with an extraordinarily low probability, of course). It is true that people live very well in places where the average temperature is 12 C higher than in Yakutsk, Siberia. However, I do not think that these kinds of analogies can justify using such a comparative geography approach for estimating welfare-equivalent damages from an average planetary temperature change of 12 C. There is just too much structural uncertainty to put meaningful bounds on the unprecedented almost-unimaginable changes to planetary welfare from average global temperatures increasing by 12 C. I don't think anyone knows how to evaluate the welfareequivalent "damages" from super-high average global temperatures, but global warming of 12 C has a good chance of going far beyond an absolute heat-stress limit that could extinguish many mammals on earth and impair very severely human functioning. 12 Let T represent the change in future worldwide average surface temperature, always measured in degrees Centigrade. Let e C(T ) represent "welfare equivalent"consumption as a fraction of what potential consumption would be (at that time when the RV T materializes) in the absence of any climate change. (This concept itself has some problematical aspects, which are ignored here, although my intention is that such intangibles as loss of the environment as we know it are somehow included in "welfare equivalent"consumption.) The most popular single formulation of a damages function in the literature is the quadratic form e C Q (T ) = 1= 1 + (T = ) 2 , where is a positive temperature-scaling parameter calibrated to give some "reasonable" values of e C Q (T ) for relatively small warmings, say up to T 2.5 C. Standard estimates of in the literature are more or less similar, although I hasten to add that such calibrations were intended by the authors to capture low-temperature damages and were never intended to be extrapolated to very high temperature changes, which is just what I will be doing here. For the sake of having a speci…c prototype example, I calibrate in (7) where the natural scaling factor for T is the rather large value =20.46 C. The results in terms of welfare-equivalent relative consumption levels for this quadratic case are given by e C Q in the second row of Table 3. (The third-row variable e C R will be discussed presently.) T 2 C 3 C 4 C 5 C 6 C 7 C 8 C 9 C 10 C 12 C 15 C 18 C e C Q 99% 98% 96% 94% 92% 90% 87% 84% 81% 74% 65% 56% e C R 99% 97% 91% 75% 50% 27% 13% 7% 3% 1% .2% .1% Table 3: Welfare-equivalent consumption e C Q (T ) and e C R (T ).
I do not …nd such numbers as e C Q (T ) in Table 3 at all convincing for high temperatures. At the mind-bending average global temperature change of T =18 C, the welfare-equivalent damage as a fraction of consumption at that time (when T =18 C materializes) is "only" 44%. The implied welfare-equivalent consumption damages of 35% for T =15 and 19% for T =10 also seem to me to be far too low for doing a credible analysis of the consequences of catastrophic losses from extreme climate change. My tentative conclusion is that the quadratic form (7), which was never intended to be applied for temperature changes beyond a few degrees centigrade, is not appropriate for assessing the welfare impacts of disastrously high temperature changes. The quadratic "welfare equivalent"damages function (expressed as a fraction of what potential consumption would be if T =0), which is enumerated as e C Q in the second row of Table 3, is pre-ordained to make extreme climate change look empirically negligible almost no matter what else is assumed.
Of course I have no objective way to determine magnitudes of high-temperature damages, but the last time that the world experienced episodes where global average temperatures were very roughly 10 C or so above the present was during the Eocene epoch 55-34 mya. During these warming periods the earth was ice free while palm trees and alligators lived near the North Pole. The Eocene was also the last epoch in which there were geologically rapid increases in mean global temperatures of magnitude very roughly 5 C or so above an already warm background. Such hyperthermal events occurred over an average period of very roughly 100K years or so, which is extremely gradual compared with current worstcase anthropogenically-induced trajectories. It is unknown what exactly triggered these temperature spikes, but they were accompanied by CO 2 spikes. One leading culprit is the strong-feedback release of large amounts of methane hydrates from clathrate deposits, which is a non-negligible possibility over the next century or two if current emissions trends are extrapolated. 14 The major point here is that relatively rapid changes of global average temperatures 5 C above present values are extremely rare events extraordinarily far outside the scope of human experience. As for huge temperature increases like T 12 , the planetary e¤ects are di¢ cult to imagine. To …nd a geologically instantaneous increase in temperatures of magnitude T 12 , one would perhaps have to go back hundreds of millions of years. Others are free to calibrate any welfare-equivalent consumption loss they want in the range above T 4 , as anybody's guess here is as good as mine. I don't think that a person needs accurate speci…c stories about what might happen for T >12 to imagine truly upending catastrophes undoing the planet and severely undermining the security of human civilization -at least.
I now want to "give the devil his due" by characterizing very roughly two points on a much more reactive damages function, which seems to me more plausible than the quadratic and which attributes far bigger welfare-equivalent damages to higher temperatures. Of course no one knows how to estimate welfare-equivalent damages for very high temperature changes. I anchor my "give the devil his due"damages function on two iconic (if arbitrary) global-average temperature changes: 6 C and 12 C. What these two iconic global warmings might mean for the human condition and for the rest of the planet has already been sketched out. At 6 C I propose welfare-equivalent consumption of e C R (6 C)=50% (at that time), while for 12 C I propose welfare-equivalent consumption of e C R (12 C)=1%. Some IAMs and CBAs recommend a "climate policy ramp" gradualism that would approach atmospheric CO 2 levels of 700 ppm, which would arguably make GHG CO 2 e levels be 750 ppm.
From Table 2, GHG concentrations of 750 ppm would eventually result in temperature increases 6 C with probability 19% and temperature increases 12 C with average probability 1% (depending very much on how fat-tailed is the relevant PDF). Using the proposed reactive speci…cation of damages ( e C R (6 C)=50% and e C R (12 C)=1%), I calculated for the lognormal PDF that at G=750 ppm of CO 2 e there is 19% chance of damages greater than 50% and 1% chance of damages greater than 99%. With the quadratic damages function (7) shown in Table 3, at G=750 ppm I calculated for the lognormal PDF that the probability of damages 50% is 0.1%, while the probability of damages 99% is 10 8 . With these kinds of numbers, it is no wonder that a quadratic damages function is fearless about attaining CO 2 e concentrations of 750 ppm -or even much higher! The third row of Table 3 adds a term to the denominator of (7) making it have the polynomial form e C Q (T ) = 1= 1 + (T = ) 2 + (T = ) , where (as before) =20.46 C, while I calibrated the temperature-scaling factor and the exponent so that e C R (6 C)=50% and e C R (12 C)=1%. The relevant parameter values are =6.081 and =6.754. For this case, in place of the "non-reactive"(7) we have instead a "reactive"damages function of form The two T -dependent terms in the denominator of (8) have equal impact for a temperature change of 3.65 C. Notice, however, that when temperature changes having a scaling factor of 6.081 C are exponentiated to such a high power as 6.754 in (8), the consequence is something like a tipping point where the damages function changes dramatically beginning around the "iconic"global warming of 6 C. It is readily con…rmed from Table 3 that e C R (T ) is indeed more "reactive"to higher temperature changes than e C Q (T ). As mentioned, global average temperatures are arbitrarily forbidden from going above T =18 C, which corresponds in Table 3 to e C R (18 C)=0.1%.

Welfare E¤ects of Uncertain Climate Change
From equation (5), steady-state global warmings T (given steady-state GHG levels G), are equal to the forcing function (G) (de…ned by expression (4)) times climate sensitivity S. Since S is a RV with some postulated PDF, then (for any given G) T is a RV with PDF given by (6). And then, given some postulated damages function of temperature (namely e C Q (T ) or e C R (T )), welfare-equivalent consumption in that steady state is itself a RV. I now manufacture an arti…cial example of welfare impacts by linking the uncertain-temperature methodology of Section 2 with the damages functions of Section 3.
Suppose a constant relative risk aversion utility function (of consumption) having the form where is the coe¢ cient of relative risk aversion. With r being the interest rate, being the rate of pure time preference (or "utility discount rate"), and g being the growth rate of per-capita consumption, the fundamental Ramsey equation is Following what Ramsey originally proposed, I take the rate of pure time preference (or the so-called "utility discount rate") throughout this paper to be zero (i.e., =0 in (10)). As Ramsey famously put the issue, "it is assumed that we do not discount later enjoyments in comparison with earlier ones, a practice which is ethically indefensible and arises merely from the weakness of the imagination." Several other (but far from all) famous economists concur with this Ramsey interpretation of intergenerational equity. 15 Taken together, quotations from these "famous economists"sound to me much more like a normative judgement about intergenerational ethics than a description of short-run individual behavior. I think that the Ramsey case of zero discounting of future utilities is the appropriate abstraction for a normative analysis of climate change. Ethically or morally, the Ramsey abstraction treats the utility of di¤erent generations equally, while taking full account of the fact that economic growth will make future generations richer and less needy than the present generation. My base-case CRRA coe¢ cient is =3, which corresponds to an eminently plausible degree of risk aversion that I believe is close to the best estimate of most economists. My base-case future growth rate of per capita consumption is g=2% per year. 16 These base-case values imply an interest rate of r=6% per year, and therefore the numerical results to follow cannot in any way be ascribed to assuming an unrealistically low discount rate.
Were to be changed substantially, then r and g would not mesh quite so nicely with past reality. If =2 and r=6%, then (10) with =0 implies g=3% -probably too high. If =4 and g=2%, then (10) with =0 implies r=8%, also probably too high. So I think it is fair to say that this proposed "package" of base-case point-estimate values ( =0, r=6%, =3, g=2%) looks more or less realistic, is internally consistent, and is immune from the criticism that discounting of climate change is being marginalized.
For my base case I use the lognormal PDF of S (with its intermediate tail fatness), as given by (1). I assume a particularly simplistic time scenario. Let G be the GHG target. For the next years, consumption grows at annual rate g=2% and GHG levels build up to (and stay at) G. My base case is =150 years. Then, suddenly, at time =150 years from now, consumption is reduced by a fraction corresponding to the realization of T (given G), and the assumed damages function (namely e C Q or e C R ). After this permanent shock to the level of consumption 150 years in the future, growth continues thereafter at annual rate g=2%. In other words, there are no damages whatsoever until time =150 years from now, when the sky is allowed to fall all at once. The growth rate is never impacted, either 15 Pigou: [pure time preference] "implies ... our telescopic faculty is defective." Harrod: "pure time preference [is] a polite expression for rapacity and the conquest of reason by passion." Koopmans: "[I have] an ethical preference for neutrality as between the welfare of di¤erent generations." Solow: "in solemn conclave assembled, so to speak, we ought to act as if the social rate of pure time preference were zero." (All quotes are taken from Arrow (1999).) I think it should be clear that the above citations refer to a normative or prescriptive, rather than a positive or descriptive, view of the world. 16 These values of =3 and g=2% per year are close to those that were proposed by Dasgupta (2008), and were considered fully acceptable by Nordhaus (2008, pp . 61 and 187). before or after the output shock at =150. This is a primitive formulation, but I think it is good enough to make the point that a reactive damages function and a tail of intermediate fatness su¢ ce to dominate the e¤ects of discounting, even at the very high interest rate of r=6% and with climate changes occurring a century and a half from now. At an interest rate of 6%, the relevant discount factor for goods and services a century and a half hence is exp( :06 150) = exp( 9) = 0:01%, which is a very, very low number.
Without loss of generality, present consumption at time t=0 is normalized at C(0)=1.

Let b
C represent the deterministic-equivalent value of C(0) that would give the same welfare relative to there being zero climate change. Then b C is the implicit solution of the equation where E[ ] is the expected-value operator and J=Q or J=R. Now substitute the lognormal PDF (1), (6) (for I=L), the utility function (9) (for base case =3), and the damages functions (7) and (8) into equation (11). Solving for b C then yields where J=Q or J=R.
The following table gives rounded-o¤ values of b C Q and b C R as a function of G.  Table 4 Certainty-equivalent consumption for base-case lognormal PDF, =3, T =18.
With the quadratic damages function (7), there is essentially the same welfare-equivalent consumption level of 100% independent of GHG concentrations G. This is because the expected present discounted welfare impact of quadratic damages incurred a century and a half from now, evaluated at an interest rate of r=6%, is essentially zero. Thus, with a standard quadratic damages function, in this formulation GHG concentrations of 1000 ppm of CO 2 e are essentially no worse than GHG concentrations of 400 ppm of CO 2 e when discounted at rate r=6% per year. There is only a miniscule willingness to pay (WTP) now to avoid signi…cantly higher GHG concentrations a century and a half from now. No wonder, then, that optimal IAM trajectories derived from a quadratic damages function encourage gradualist climate policy ramp CO 2 e levels approaching 750 ppm! The welfare-equivalent certainty-equivalent "as if" consumption levels b C R enumerated in Table 4 are each expressed relative to an arti…cial norm of G=280 ppm, T =0. In other words these numbers represent the WTP, in terms of certainty-equivalent consumption now and forever, to eliminate all climate change. The various "as if" consumption levels b C R (as a function of steady-state G) are di¢ cult to interpret in absolute terms, and should be compared with each other as fractions or multiples.
For example, the welfare-equivalent fractional loss of as-if-deterministic consumption accompanying a change in GHG concentrations from 550 ppm to 750 ppm by Table 4 is (:976 :579) =:976 = :407 -i.e., keeping target GHG levels at 550 ppm rather than letting them rise to 750 ppm is worth spending up to 40.7% of present certainty-equivalent consumption at G=550. Or, to take another example, the welfare-equivalent fractional loss of as-if-deterministic consumption accompanying a change in GHG concentrations from 600 ppm to 650 ppm in Table 4 is (:924 :827)=:924 = :105.
In other words, a modest coe¢ cient of relative risk aversion of 3 is big enough to make it worth spending 10.5% of consumption, now and forever, to avoid the higher (but still very small) probabilities of bad warmings 150 years from now (which are indicated in Table 2) by keeping target GHG levels at 600 ppm rather than letting them rise to 650 ppm.
Notice that the WTP to keep GHG concentrations below 500 ppm of CO 2 e are very small because such low concentrations e¤ectively wall o¤ the higher temperature changesand discounting moderate events happening 150 years from now at an e¤ective interest rate of 6% (or an e¤ective discount factor of 0.01%) takes care of the lower temperature changes. Above 550 ppm of CO 2 e, however, the danger of higher temperatures accelerates greatly the WTP now in order to avoid bad climate change outcomes a century and a half from now, overriding even a discount rate of 6% per year. Such high WTP levels are testimony to the power of combining risk aversion with fat tails and a reactive damages function. At greater and greater GHG concentrations, risk aversion to the possibility of taking a catastrophic "hit" to consumption becomes more and more the dominant force in WTP calculations. Again, the impression is that GHG policy is most accurately viewed as an insurance policy against disastrous outcomes.
I now mention brie ‡y a few results from some primitive sensitivity experiments. In order to compress these results, I report them only for 750 ppm of CO 2 e. I pick 750 ppm of CO 2 e for two reasons. First, 750 ppm of CO 2 e is an upper limit on optimal GHG concentrations that is approached by some optimizing IAMs and CBAs. Second, under non-optimal business-as-usual scenarios, concentrations of 750 ppm of CO 2 e are all too conceivable as early as the end of this century.
Generally speaking, outcomes are very dependent on how extreme tail damages and extreme tail probabilities are formulated. Results are not robust to how catastrophic outcomes are modeled and speci…ed. In this sense, the main robust …nding of the paper is non-robustness to stress tests.
As was already shown, the standard quadratic damages function never produces a significant enough welfare impact to matter very much in determining policy. The very …rst form of non-robustness to report on, therefore, is the sensitivity of results to the form of the damages function, already discussed previously in the paper. The "devil's advocate" reactive damages function paints a very di¤erent picture in Table 4 than the standard non-reactive quadratic damages function.
I next examine what happens for di¤erent values of the coe¢ cient of relative risk aversion (still keeping =0 in the background -more on this later). From Table 4, welfare-equivalent consumption for =3 is 58%. For =4, welfare-equivalent consumption at 750 ppm is 22%. For =2, welfare-equivalent consumption at 750 ppm is 88%. So quantitative values of the WTP to avoid a GHG concentration of 750 ppm vary widely with the assumed degree of risk aversion, and so too do corresponding policy recommendations. What is extremely interesting here is the strong reversal of the traditional role of . Through the Ramsey formula, a higher value of is traditionally associated with a higher value of the discount rate (here r = g), for any given growth rate of consumption g. This higher value of then translates into a lower weighting for distant-future events, like climate change. As an example, with an annual growth rate g=2%, the relevant discount factor for converting bene…ts a century and a half from now into today's currency for =4 is exp(-.02 150 4)= 6. 1 10 6 ; for =3 it is exp(-.02 150 3)=1 .2 10 4 ; and for =2 it is exp(-.02 150 2)= 2.5 10 3 . This is a very wide range for discount factors, although all of these numbers are extremely low. However, higher values of also indicate higher relative risk aversion, which can easily have an even more powerful e¤ect in the opposite direction for a reactive damages function combined with a semi-fat upper tail of the temperature-change PDF. Thus, the damages function (8) combined with even an intermediate-fatness tail like the lognormal is easily su¢ cient to reverse strongly the traditional role of , because the e¤ect of aversion to catastrophic uncertainty here easily outweighs the e¤ect of time discounting.
If the Pareto fat-upper-tail PDF (2) is substituted for the lognormal (1) in the range of climate sensitivity above the median S M =3, then welfare-equivalent consumption at 750 ppm is 27% instead of 58% in Table 4. If the normal thin-upper-tail PDF (3) is substituted for the lognormal (1) above the median S M =3, then welfare-equivalent consumption at 750 ppm is 81% instead of 58% in Table 4. The WTP to avoid the uncertain consequences of a GHG concentration of 750 ppm of CO 2 e is thus highly dependent on the assumed fatness of the upper tail of the PDF of climate sensitivity.
In Table 4, I assumed that global warming arrives at =150 years from now. If the time of arrival for global warming is =200 years from now, then welfare-equivalent consumption at 750 ppm is 89% instead of 58% in Table 4. If the time of arrival for global warming is =100 years from now, then welfare-equivalent consumption at 750 ppm is 25% instead of 58% in Table 4.
For the base case enumerated in Table 4, I projected an annual growth rate of consumption g=2%.
If the annual growth rate of consumption is instead g=1%, then welfareequivalent consumption at 750 ppm is 16%, instead of 58% in Table 4. If the annual growth rate of consumption is instead g=3%, then welfare-equivalent consumption at 750 ppm is 95%, instead of 58% in Table 4.
Finally, I examine the arti…cially imposed upper bound cuto¤ T , beyond which global average temperatures are arbitrarily not allowed to go. In Table 4, I assumed an upperbound temperature cuto¤ of T =18 C. If the upper-bound temperature cuto¤ is arbitrarily made 6 higher, so that T =24 C, then welfare-equivalent consumption at 750 ppm is 38% instead of 58% in Table 4. If the upper-bound temperature cuto¤ is arti…cially made 6 lower, so that T =12 C, then welfare-equivalent consumption at 750 ppm is 91% instead of 58% in Table 4.
In the following Table 5, I summarize brie ‡y the results of the above sensitivity experiments.
case CRRA PDF impact yr growth g temp bd T welfare alt2 4)22% P )27% 100)25% 1%)16% 24 )38% n.a. Readers can form their own judgements, but for me Table 5 seems to be indicating a disturbing lack of robustness with respect to parameter values that are extremely di¢ cult to know with any degree of accuracy. Many researchers promote alternative speci…cations that do not imply nearly such extreme outcomes as do some of my speci…cations. I do not claim that their formulations are wrong or even implausible. I merely point out that they are unlikely to be robust with respect to assumptions about extreme catastrophic climate change under uncertainty, and therefore they fail a reasonable stress test. To test parametric sensitivity with respect to the rate of pure time preference , consider the base case ( =1.5%, =2, g 2%) in Nordhaus (2008), which from (10) corresponds to a discount rate r 5.5% per year. When Nordhaus runs instead through his DICE model "my" base case ( 0, =3, g 2%), which implies r=6% per year, there is no substantive di¤erence in outcomes. 18 However, DICE is (essentially) a deterministic formulation in the spirit of an optimal control problem featuring a relatively non-reactive quadratic loss function. When lognormal uncertainty of form (1) is introduced, then there is a tremendous di¤erence between the two base cases. With quadratic losses (7), as usual b C Q 100%. Even with my reactive damages function (8), if I plug the Nordhaus base-case speci…cation ( =1.5%, =2, g=2%) into my model I get b C R 99% for G=750 ppm of CO 2 e. In other words, the WTP now to avoid altogether the consequences of G=750 at future time =150 goes from 42% to 1% for two speci…cations that would otherwise have near-identical consequences in a deterministic world. The reason for such a dramatic di¤erence is that when pure time discounting is as high as =1.5% per year, the risk aversion e¤ect is overcome by the discounting e¤ect. Once again, readers can form their own judgements about the implied robustness of policy implications under stress-test uncertainty -here with respect to various values of and .

Discussion
I think that several themes emerge from this paper.
The paper suggests that economic analysis of climate change might be very sensitive to uncertainties about such things as the fatness of PDF tails for temperature changes, the speci…cation of the damages function, cuto¤ bounds, relative risk aversion, rates of pure time preference, growth rates, concentrations of greenhouse gases, and so forth. When relatively fat-tailed PDFs are combined with a reactive damages function, then seemingly modest changes in target levels of GHGs can sometimes have very big welfare consequences. In such conditions, the primary purpose of keeping down GHGs is to prevent large damages from extreme warmings in the "bad" tail, which is a much more powerful incentive to target low GHG levels than trying to keep down the relatively modest damages from median temperatures. But the exact quantitative extent to which changes in target levels of GHGs can cause these very big welfare consequences depends sensitively on how the extremes are modeled and speci…ed. While conclusions from some plausible formulations seem relatively is hardly any di¤erence in implied damages, even for temperature changes as high as 18 C. My tentative conclusion is that their model, which uses the form exp ( (T = ) 2 ), is equally non-robust to the stress-test numerical exercises of this paper, which uses the form 1=(1 + (T = ) 2 ). 18 This is essentially "Run 3" reported on page 187 of Nordhaus (2008). immune to being represented by a measure of central tendency like the median, conclusions from some other formulations, which appear equally if not more plausible to me, are extraordinarily far from being captured by median values and seem to be highly dependent on a variety of underlying uncertainties. Thus, we might be in an unfortunate position where results from an economic analysis of climate change have a wide range of possible policy recommendations, which depend upon barely knowable assumptions well beyond the realm of ordinary experience. While I do not think that this feature nulli…es CBAs or IAMs, I do think it should make us especially cautious about the ability of economic analysis to give robust policy advice for the speci…c application of such methods to catastrophic climate change. The moral of the story may be that, under extreme tail uncertainty, seemingly casual decisions about functional forms and parameter values for catastrophic outcomes might well dominate CBA of climate change.
Another suggestion coming out of the paper is that the standard quadratic damages function simply cannot register, and therefore will not react to, the possibility of catastrophic climate change. Once the usual quadratic damages speci…cation is made, an optimal policy will not get alarmed by high values of GHG concentrations, and almost inevitably it will recommend relatively mild mitigation measures. The climate policy ramp gradualism that emerges from many IAMs may be rooted in the fact that, even when uncertainty is introduced in the form of Monte Carlo simulations, the usual quadratic damages function never really allows the model to get very far away from e¤ectively plugging median values into a deterministic climate-change CBA and then discounting away the consequences.
To summarize, there is an underlying generic problem with CBAs or IAMs of climate change that is not present in other, more standard, applications of economic analysis. In rare situations with e¤ectively unlimited downside liability, like climate change, CBAs or IAMs can be extraordinarily sensitive to speci…cations of extreme tail events. Conventional CBAs or IAMs of climate change ignore this basic message at their own peril.
Needless to say, a very large number of caveats apply to the toy model of this paper. The main omission is the lack of realistic dynamics in the model. For simplicity, the toy model of this paper essentially analyzes and compares steady states, with only the most primitive cause-and-e¤ect dynamics. I think that this simpli…cation allows some transparent insights that can get obscured by much more complicated dynamic models, but it comes at a price by omitting nuanced considerations of growth, discounting, how long it takes to approach a steady state, and so forth. A drawback of my toy model approach is that I could be missing some critical dynamic interactions that are unable to be captured by the crudeness of such a simplistic comparative-steady-state formulation. So conclusions of this paper are at most suggestive and may need to be modi…ed in the light of performing numerical simulations from much more complicated dynamic computer models. 19 That having been said, the suggestive comparative-steady-state numerical outcomes of this toy model seem to me as if they might be su¢ ciently powerful that an appropriately mu-ed version would likely survive a fully dynamic treatment. Remember, throughout this paper the default fatness that really matters concerns the bad tail of the reducedform PDF of the log of "diswelfare" (which includes damages).
Nobody knows the tail fatness of the PDF of the log of diswelfare -which is the very feature driving the lack of robustness to speci…cations of climate extremes. As a consequence, I somehow doubt very much that robustness will miraculously be restored by introducing more sophisticated dynamics. Actually, the various "stress tests" of this paper do not strike me as being particularly "stressful" at all. Therefore, I suspect rather strongly that it may be di¢ cult to dislodge altogether the verdict that a CBA of climate change is terribly sensitive to assumptions about extreme tail events -and that the primary reason for keeping GHG levels down is mainly to insure against high-temperature catastrophic climate damages.

Conclusion
If a particular type of idiosyncratic uncertainty a¤ects only one small part of an individual's or a society's overall portfolio of assets, exposure is naturally limited to that speci…c component and bad-tail fatness is not such a paramount concern. However, some very few but very important real-world situations have potentially unlimited exposure due to structural uncertainty about their potentially open-ended catastrophic reach. Climate change potentially a¤ects the whole worldwide portfolio of utility by threatening to drive all of planetary welfare to disastrously low levels in the most extreme scenarios. The comparative-steadystate toy model of this paper suggests that the results of climate change CBA can sometimes depend non-robustly on seemingly casual decisions about functional forms and parameter values associated with extreme tails. The …ndings of this paper may be a warning that the results of climate change CBA can be largely driven by the "fear factor" associated with low-probability high-impact catastrophes, which is di¢ cult to model robustly.