Reactions to the Nordhaus Critique
Martin L. Weitzman_
February 9, 2009 .preliminary .rst draft .comments appreciated
Abstract
In this note I react to the reactions of William Nordhaus to a recent article of mine
entitled .On Modeling and Interpreting the Economics of Climate Change,. which
appeared in the February 2009 issue of the Review of Economics and Statistics.
1 Introduction
I am grateful toWilliam Nordhaus for his thoughtful, considered critique of my paper.1 Pro-
fessor Nordhaus.s views on the economics of climate change .a subject which he pioneered
and to which he has made seminal contributions .are worthy of respect. I respectfully dis-
agree with some of his comments on my paper, while agreeing with some others. Naturally,
I focus here on the points of disagreement.
At .rst I was inclined to debate (and, in my own fantasies at least, perhaps refute) some
of his criticisms line by line, so to speak. After all, this is the usual operating procedure
of scholars defending their own ideas and their own turf against would-be detractors or
encroachers. But on second thought I found myself anxious not to be drawn, by so doing,
into what might appear to an outsider as squabbling among insiders about technical details
and their interpretations. Instead, I am more keen here to emphasize in fresh language the
substantive concepts that, I think, motivate Nordhaus.s critical reactions to my paper. I am
far more committed to the simple basic ideas that underlie my theory than am I committed
to the particular mathematical form in which I have chosen to express them. I know only too
well that these core ideas could have been wrapped in a variety of alternative mathematical
shells .and the particular one that I chose was somewhat arbitrary. If I can make the
_Without blaming them for remaining de.ciencies of the paper, I am extremely grateful for the construc-
tive comments of Stephen DeCanio, John Harte, Robert Pindyck, and Richard Tol.
1Nordhaus (2009), available online at http://cowles.econ.yale.edu/P/cd/d16b/d1686.pdf, reacting to
Weitzman (2009a).
1
underlying simple basic ideas acquire greater intuitive plausibility, then I believe that time,
experience, and the .owering of di?erent forms will uncover the best way of expressing them
.and these ideas will then become more self-evidently resistant to many of the criticisms
that Nordhaus is expressing.
2 Deep Structural Uncertainty about Climate Extremes
I begin by setting up a bit of a straw man that I will label .standard.cost-bene.t analysis
(CBA) of climate change. My notion here of CBA is somewhat broader than an integrated
assessment model (IAM), which I view as an important subset of CBA. For my purposes
in this note I treat the two forms as essentially interchangeable. Of course there is no
.standard.CBA (or IAM) of climate change, but I think this is an allowable simpli.cation
in the present context.
I try now to make the case empirically that there is enormous structural uncertainty about
the economics of extreme climate change, which, if not unique, is pretty rare. I will argue
on intuitive grounds that the way in which this deep structural uncertainty is conceptualized
and formalized should in.uence substantially the outcomes of any reasonable CBA (or IAM)
of climate change. Further, I will argue that the seeming fact that this deep structural
uncertainty does not in.uence substantially outcomes from the .standard. CBA hints at
an implausible treatment of uncertainty. My argument is not intended to be airtight or
rigorous. It is an intuitive presentation based on stylized facts, but it should be su? ciently
convincing to make a plausible presumptive case that there may be a serious mistreatment
of uncertainty in the .standard. CBA of climate change. We all know that even the
most complicated computer-driven simulations are dependent upon the core assumptions
inside the model that is inside the black box that is inside the computer. My intuitive
examples are frankly aimed at sowing some seeds of doubt that the .standard. CBA of
climate change is fairly representing the deep structural uncertainties that characterize this
particular application, and therefore its conclusions might be more shaky than is commonly
acknowledged. I will try to make this opening argument by citing four aspects of the climate
science and economics that do not seem to me to be adequately captured by the .standard.
CBA. While di?erent aspects of structural uncertainty might alternatively (or additionally)
be cited, I restrict my stylized facts to these four examples, which I call .Exhibits A, B, C,
and D..
.Exhibit A. concerns the atmospheric level of greenhouse gases (GHGs) over the last
800,000 years. Ice core drilling in Antarctica began in the late 1970s and is still ongoing.
The record of carbon dioxide (CO2) and methane (CH4) trapped in tiny ice-core bubbles
2
currently spans 800,000 years.2 It is important to recognize that the numbers in this
unparalleled 800,000-year record of GHG levels are among the very best data that exist in
the science of paleoclimate. Almost all other data (including past temperatures) is inferred
indirectly by proxy variables, whereas this ice-core GHG data is directly observed.
The pre-industrial-revolution level of atmospheric CO2 (about two centuries ago) was
_280 parts per million (ppm). The ice-core data show that carbon dioxide was within a
range roughly between _180 and _280 ppm during the last 800,000 years. Currently, CO2
is at _385 ppm, and climbing steeply. Methane was never higher than _750 parts per
billion (ppb) in 800,000 years, but now this extremely potent GHG, which is thirty times
more powerful than CO2, is at _1,780 ppb. The sum total of all carbon-dioxide-equivalent
(CO2-e) GHGs is currently at _435 ppm. Even more alarming in the 800,000-year record
is the rate of change of GHGs, with increases in CO2 being below (and typically well below)
_40 ppm within any past sub-period of ten thousand years, while now CO2 has risen by
_40 ppm in just the last quarter century.
Thus, anthropogenic activity has elevated atmospheric CO2 and CH4 to levels extraor-
dinarily far outside their natural range .and at a stupendously rapid rate. The scale and
speed of recent GHG increases makes predictions of future climate change highly uncertain.
There is no analogue for anything like this happening in the past geological record. There-
fore, we do not really know with much con.dence what will happen next. The link between
GHG levels and temperature changes in the ice-core record is not unicausal, and it is not
fully understood, but this unsure link just adds more uncertainty to the picture because, any
way one categorizes what are the forcings and what are the feedbacks, GHGs are strongly
implicated in global warming. To keep atmospheric CO2 levels at twice pre-industrial-
revolution levels would require not just stable but sharply declining emissions within a few
decades from now. Forecasting ahead a century or two, the levels of atmospheric GHGs that
may ultimately be attained (unless drastic measures are undertaken) have likely not existed
for tens of millions of years and the rate of change will likely be unique on a time scale of
hundreds of millions of years.
Remarkably, the .standard.CBA of climate change takes essentially no account of the
extraordinary magnitude of the scale and speed of these unprecedented changes in GHGs
.and the extraordinary uncertainties they create for any believable economic analysis of
climate change. Perhaps even more astonishing is the fact that the .policy ramp.of gradu-
ally tightening emissions, which emerges from the .standard.CBA, attains stabilization at
levels of CO2-e GHGs that approach _700 ppm. The .standard.CBA thus recommends
2See Dieter et al (2008), from which my numbers are taken (supplemented by data from the Keeling curve
for more recent times).
3
imposing an impulse or shock to the Earth.s system by geologically-instantaneously jolting
atmospheric stocks of GHGs up to _21
2 times their highest past level over the last 800,000
years .without even mentioning what an unprecedented planetary experiment such an .op-
timal.policy would entail. This is my Exhibit A in the case that .standard.CBA does
not adequately encompass the extraordinary structural uncertainties associated with climate
change.
.Exhibit B.concerns the ultimate temperature response to such kind of unprecedented
increases in GHGs. So-called .climate sensitivity.(hereafter denoted S1) is a key macro-
indicator of the eventual temperature response to GHG changes. Climate sensitivity is
de.ned as the global average surface warming following a doubling of carbon dioxide concen-
trations. Climate sensitivity is not the same as temperature change, but, for the benchmark-
serving purposes of the simplistic example I used in the paper, I assumed that the shapes
of both PDFs are roughly similar in the long run because a doubling of anthropogenically-
injected CO2-equivalent (CO2-e) GHGs relative to pre-industrial-revolution levels is essen-
tially unavoidable within about the next half century and will plausibly remain well above
twice preindustrial levels for one or two centuries thereafter. Other things being equal, higher
values of climate sensitivity will produce higher temperatures at a more remote time in the
distant future. This begs many questions including, especially, the question of whether
enough can be learned su? ciently rapidly .relative to the tremendous systemic inertias
and lags (like the super-long time that extra CO2 remains in the atmosphere) .to under-
take politically realistic mid-course corrections (more on this later). To make a long story
short, I believe that, even after admitting it is overly simplistic, Exhibit B still constitutes
telling evidence supporting my case of insu? cient attention to structural uncertainty in the
.standard.CBA of climate change.
Twenty-two peer-reviewed studies of climate sensitivity published recently in reputable
scienti.c journals and encompassing a wide variety of methodologies (along with 22 imputed
PDFs of S1) are cited by IPCC-AR4 (2007). It might be argued that these 22 studies
are of uneven reliability and their complicatedly-related PDFs cannot easily be combined.
I e?ectively assumed that all 22 studies have equal credibility and for my purposes their
PDFs could be simplistically aggregated into one .representative.PDF. This form of meta-
analysis can be defended as an example of Bayesian model averaging, in which the di?erent
studies represent di?erent equally-credible models. In his critique, Nordhaus promotes
a non-Bayesian classical-like statistical position where the di?ering results from di?erent
studies are treated more like draws from the same underlying distribution, but with varying
measurement errors around the .true.value. I believe it is fair to say that most climate
scientists who know statistical theory think that it is more appropriate to view the di?erent
4
studies as di?erent models with di?erent methodologies giving di?erent results .rather than
to view the results as di?erent observations from the same model. Without question, a
more sophisticated analysis of how to aggregate scienti.c data from di?erent sources would
be useful, even though such a meta-analysis would have a Bayesian architecture with largely
subjective judgements on how to combine results from various studies. In any event, the
median upper 5% probability level over all 22 climate-sensitivity studies cited in IPCC-AR4
(2007) is 6.4_C .and this stylized fact alone is telling. Glancing at Table 9.3 and Box 10.2
of IPCC-AR4, it is apparent that the upper tails of these 22 PDFs tend to be su? ciently
long and heavy with probability that one is allowed from a simplistically-aggregated PDF of
these 22 studies the rough approximation P[S1>10_C]_1%. The actual empirical reason
why these upper tails are long and heavy with probability dovetails nicely with the theory of
my paper: inductive knowledge is always useful, of course, but simultaneously it is limited
in what it can tell us about extreme events outside the range of experience .in which case
one is forced back onto depending more than one might wish upon the prior PDF, which of
necessity is largely subjective and relatively di?use. As a recent Science commentary put
it: .Once the world has warmed by 4_C, conditions will be so di?erent from anything we
can observe today (and still more di?erent from the last ice age) that it is inherently hard to
say where the warming will stop..3 However one looks at how to combine di?erent results
from di?erent studies, the tail of the .representative.PDF of climate sensitivity is heavy
with probability, which is disturbing in and of itself and, I think, is not easily dismissed.
This is Exhibit B in my case that the .standard.CBA may not adequately cover the deep
structural uncertainties associated with climate change.
.Exhibit C.concerns possibly disastrous releases over the long run of bad-feedback com-
ponents of the carbon cycle that are currently omitted from most general circulation models.
The chief worry here is a signi.cant supplementary component that conceptually should
be added on to climate sensitivity S1. This omitted component concerns the potentially
powerful self-ampli.cation potential of greenhouse warming due to heat-induced releases of
sequestered carbon. One vivid example is the huge volume of GHGs currently trapped in
arctic permafrost and other boggy soils (mostly as methane, a particularly potent GHG). A
more remote (but even more vivid) possibility, which in principle should also be included,
is heat-induced releases of the even-vaster o?shore deposits of CH4 trapped in the form of
hydrates (aka clathrates) .for which there is a decidedly non-zero probability over the long
run of destabilized methane seeping into the atmosphere if water temperatures over the
continental shelves warm just slightly. The amount of methane involved is huge, although
it is not precisely known. Most estimates place the carbon content of methane hydrate
3Allen and Frame (2007).
5
deposits at about the same order of magnitude as the sum total of all of the traditional
fossil fuels that will ever have been extracted and burned by humans. Over the long run, a
CH4 outgassing-ampli.er process could potentially precipitate a cataclysmic strong-positive-
feedback warming. This real physical basis for a highly unsure but truly catastrophic
scenario is my Exhibit C in the case that conventional CBAs and IAMs do not adequately
cover the deep structural uncertainties associated with possible climate-change disasters.
Other examples of an actual real physical basis for a catastrophic outcome could be cited,
but this one will do here.
The real physical possibility of endogenous heat-triggered releases at high temperatures
of the enormous amounts of naturally-sequestered GHGs is a good example of indirect
carbon-cycle feedback e?ects that I think should be included in the abstract interpreta-
tion of a concept of .climate sensitivity. that is relevant here. What matters for the
economics of climate change is the reduced-form relationship between atmospheric stocks
of anthropogenically-injected CO2-e GHGs and temperature change. Instead of S1, which
stands for .climate sensitivity narrowly de.ned,.the example in the paper used S2, which
(abusing scienti.c terminology) stands for a more abstract .generalized climate-sensitivity-
like multiplier-parameter. that includes heat-induced feedbacks on the forcing from the
above-mentioned releases of naturally-sequestered GHGs, increased respiration of soil mi-
crobes, climate-stressed forests, and other weakenings of natural carbon sinks. The main
point here is that the PDF of S2 has a tail even heavier with probability than the PDF
of S1. Contrary to what Nordhaus states, my paper relied on three recent peer-reviewed
scienti.c studies to estimate the PDF of S2. Extraordinarily rough calculations implied
P[S2>10_C]_5% and P[S2>20_C]_1%, which presumably corresponds to a scenario where
CH4 and CO2 are outgassed on a large scale from degraded permafrost soils, wetlands, and
clathrates. The e?ect of heat-induced GHG releases on the PDF of S2 is extremely nonlinear
at the upper end of the PDF of S2 because, so to speak, .thick tails conjoined with thick
tails beget yet thicker tails..
When fed into an economic analysis, the great open-ended uncertainty about eventual
mean planetary temperature change cascades into yet-much-greater yet-much-more-open-
ended uncertainty about eventual changes in welfare. There exists here a very long chain
of tenuous inferences fraught with huge uncertainties in every link beginning with unknown
base-case GHG emissions; then compounded by huge uncertainties about how available poli-
cies and policy levers will transfer into actual GHG emissions; compounded by huge uncer-
tainties about how GHG .ow emissions accumulate via the carbon cycle into GHG stock
concentrations; compounded by huge uncertainties about how and when GHG stock concen-
trations translate into global mean temperature changes; compounded by huge uncertainties
6
about how global mean temperature changes decompose into regional climate changes; com-
pounded by huge uncertainties about how adaptations to, and mitigations of, climate-change
damages are translated into utility changes at a regional level; compounded by huge uncer-
tainties about how future regional utility changes are aggregated .and then how they are
discounted .to convert everything into expected-present-value global welfare changes. The
result of this immense cascading of huge uncertainties is a reduced form of truly stupendous
uncertainty about the aggregate expected-present-discounted utility impacts of catastrophic
climate change, which mathematically is represented by a very-spread-out very-fat-tailed
PDF of what might be called .welfare sensitivity..
Even if S2 could somehow be bounded above by some big number, the value of .welfare
sensitivity.is e?ectively bounded only by some very big number representing something like
the value of statistical civilization as we know it, or maybe even the value of statistical life
on earth as we know it. Even granted some absolute upper bound on S2 (above which
point its PDF is exactly zero) the essential point here is that the enormous unsureness
about (and enormous sensitivity of CBA to) an arbitrarily-imposed .damages function.for
high temperature changes makes the relevant reduced-form criterion of welfare sensitivity
seem almost unbelievably uncertain at high temperatures .to the point of being essentially
unbounded for practical purposes. This extension of the logic of Exhibit C bespeaks of
the .standard.CBA of climate change possibly underestimating seriously the role of deep
structural uncertainty.
.Exhibit D.concerns what I view as an unusually cavalier treatment of damages or disu-
tilities from extreme temperature changes. The .standard.CBA treats high-temperature
damages by a rather passive extrapolation of whatever speci.cation is assumed (typically
arbitrarily) to be the low-temperature .damages function.. High-temperature damages ex-
trapolated from a low-temperature damages function seem to be remarkably sensitive to
assumed functional forms (and, to a lesser degree, parameter choices). Almost any function
can be made to .t the low-temperature damages assumed by the modeler, even though these
functions may give enormously di?erent evaluations at higher temperatures. The .standard.
CBA damages function reduces welfare-equivalent consumption by a quadratic-polynomial
multiplier, calibrated to some postulated loss for low temperatures, and then does some
sensitivity analysis by tweaking the exponent. This particular choice of functional form al-
lows the economy to substitute consumption for higher temperatures relatively easily, since
the limiting elasticity of substitution between consumption and higher temperatures is one.
As one of many substantive consequences from using this particular high-substitution dam-
ages speci.cation, Nordhaus argues that serious warming is rendered less important because
we would have more output to o?set it .due to economic growth being highly correlated
7
with the atmospheric CO2 stocks that drive the higher temperatures. The strength of this
kind of a conclusion is largely an artefact of Nordhaus.s arbitrary choice of a multiplicative
functional form for his .damages function..
There was never any more compelling rationale for the .standard.CBA damages speci.-
cation than the comfort that economists feel from having previously worked with a quadratic-
polynomial loss function, and the ease of interpreting a multiplicative loss because it is
directly translatable .as if. into a fraction of output that is lost. In other words, the
multiplicative quadratic-polynomial speci.cation is extrapolated to assess climate-change
disutilities at high temperatures for no better reason than casual familiarity and conve-
nience of interpretation. This might be justi.ed as an acceptable approximation for the
disutility of small temperature changes, but it is highly questionable when it is being used
as an extrapolative device for evaluating the disutility of catastrophic climate changes.
Here is not the place to get involved in all of the details, but su? ce it to say that
very di?erent optimal policies can be produced when other, in my opinion more plausible,
functional forms are used to express the disutility of disastrously high temperatures.4 As just
one example, in the presence of fat-tailed uncertainty a multiplicative exponential damages
speci.cation is capable of inducing a far more stringent curtailment of GHG emissions than
the multiplicative polynomial speci.cation of the .standard.CBA. Or, to take a second
example, suppose that the disutility of temperature change is additively separable instead of
being multiplicatively separable (within the utility function of consumption). If welfare is the
di?erence between a CRRA utility function of consumption with coe? cient of relative risk
aversion two, and a quadratic loss function of temperature changes, it implies an elasticity
of substitution between consumption and temperature change of one half. Empirically,
simulating this additive form yields a far more stringent curtailment of GHG emissions than
what emerges from the multiplicative speci.cation of the .standard.CBA.5
The above two examples demonstrate how seemingly minor changes in the speci.cation
of high-temperature damages can dramatically alter the gradualist policy ramp outcomes
recommended by the .standard.CBA. Such fragility of policy to postulated forms of disu-
tility functions are my Exhibit D in making the case that the .standard. CBA does not
adequately cope with deep structural uncertainty .here structural uncertainty about the
speci.cation of damages.
4Examples are discussed more elaborately in Weitzman (2009b).
5 If the coe? cient of relative risk aversion is two, the above additively-separable speci.cation is mathe-
matically identical to the CES speci.cation of Sterner and Persson (2008) with elasticity of substitution 1
2 .
In their pioneering study, Sterner and Persson showed empirically .by plugging it into Nordhaus.s DICE
model .that this (their CES or my additive) welfare speci.cation yields a signi.cantly more aggressive policy
response to global warming than the multiplicative speci.cation of the .standard.CBA. For details on the
isomorphism with my additively separable formulation, see Weitzman (2009b).
8
What I would wish a reader to take away from these four exhibits is the idea that
something is suspicious about the seeming immunity of the .standard. CBA of climate
change to such stylized facts. An experiment without precedent is being performed on
planet Earth by subjecting the world to the shock of a geologically-instantaneous injection
of massive amounts of GHGs. Yet the .standard. CBA seems almost oblivious to the
extraordinarily uncertain consequences of catastrophic climate change. A reader should feel
intuitively that it goes against common sense when, in view of the above four exhibits, a
climate-change CBA does not much depend on how potential climate-change catastrophes are
being modeled and incorporated into the CBA. This intuitive feeling .that the consequences
of low-probability extreme-impact climate change are not adequately represented in the
.standard.CBA .is my opening argument. I turn next to the theory.
3 In.nity, CBA, and the .Dismal Theorem.
I begin this section by asking why it is relevant in the .rst place to have any supporting
theory at all if the four stylized-fact .exhibits.from last section are convincing. Why isn.t
this su? cient evidence that there is a problem with the .standard.CBA? My answer is that
a combined theoretical-empirical argument delivers a particularly powerful one-two punch
at the plausibility of the .standard.CBA. In this respect I believe that the whole of my
argument is bigger than the sum of its two main parts.
I come now to a critical modeling issue in the economics of catastrophic climate change.
How is the intuitively plausible dependence of climate-change CBA on extreme structural
uncertainty best expressed and examined theoretically? By di?erently conceptualizing and
di?erently answering this basic question, I think Nordhaus may be missing at least part of
his intended target. In this theoretical section of the paper, as in the last empirical section,
I emphasize the intuitive plausibility of the case I am trying to make .here concerning the
underlying fat-tailed logic driving the theory.
Last section argued that it is only common sense that policy implications (from a plan-
etary experiment, which is unique in geological history, of instantaneously injecting huge
amounts of GHGs into Earth.s atmosphere) should depend sensitively on the treatment of
low-probability extreme-impact outcomes. The main question I attempted to address in my
paper was whether such intuitive sensitivity is re.ecting some deeper underlying principle.
My answer in the paper was that, for the application to climate change, there are indeed
some basic underlying theoretical principles that point in the direction of extreme sensitivity
of CBA outcomes to the treatment of low-probability extreme-impact outcomes. The ba-
sic underlying logic is simple enough to understand intuitively and without the fancy math
9
required to state and prove a formal version.
First of all, a point of technical clari.cation is in order. To which PDF is the term .fat
tail.6 intended to apply? Unless otherwise noted, my default meaning here is the PDF
of the logarithm of the disutility of extreme temperatures .by whatever combination of
temperature-change PDFs, temperature-sensitive damages, and so forth, this reduced form
comes about. It may seem arcane, but the tail thickness of this particular reduced-form
PDF is the analytical essence of what Nordhaus and I are debating in this interchange.
Because the integral over a nonnegative probability measure is one, the PDF of the
logarithm of the disutility of extreme temperatures must decline to an asymptote of zero
probability. Thus, extreme outcomes can happen, but their likelihood diminishes to zero
as a function of how extreme is the output. The fact that extreme outcomes cannot be
eliminated altogether, but are hypothetically possible with some positive probability, is not
at all unique to climate change. Almost nothing in our world has a probability of exactly
zero or exactly one. What is worrisome is not the fact that extreme tails are long per se
(re.ecting the fact that a meaningful upper bound on disutility does not exist), but that they
are fat (with probability density). The critical question is how fast does the probability
of a catastrophe decline relative to the welfare impact of the catastrophe. Other things
being equal, a thin-tailed PDF is of less concern because the probability of the bad event
declines exponentially (or faster). A fat-tailed distribution, where the probability declines
polynomially, can be much more worrisome.
The theory outlined in the paper indicates that there is a theoretical tendency for the
relevant PDFs in the economics of climate change to have fat tails. Conceptually, the
underlying mechanism is not di? cult to grasp. Structural uncertainty essentially means that
the relevant probabilities are unsure, a formal Bayesian translation of which might be that
the structural parameters of the relevant PDFs are themselves uncertain and have their own
PDFs. The paper expressed this idea in a formal argument that the relevant reduced form
(.posterior predictive.in Bayesian jargon) of the PDF of log welfare tends to be fat tailed.
Loosely speaking, the driving mechanism is that the operation of taking .expectations of
expectations.or .probability distributions of probability distributions.spreads apart and
fattens the tails of the reduced-form compounded posterior-predictive PDF. It is inherently
di? cult to learn from .nite samples alone enough about the probabilities of extreme events
6As I use the term, a PDF has a .fat.(or .thick.or .heavy.) tail when its moment generating function
(MGF) is in.nite .i.e., the tail probability approaches zero more slowly than exponentially. The standard
example of a fat-tailed PDF is the power law (aka polynomial aka Pareto) distribution, although, for example,
a lognormal PDF is also fat-tailed, as is an inverted-normal or inverted-gamma or Student-t. By this
de.nition a PDF whose MGF is .nite has a .thin.tail .i.e., the tail probability approaches zero at least
as rapidly as exponentially. A normal or a gamma are examples of thin-tailed PDFs, as is any PDF having
.nite supports.
10
to thin down the bad tail of the PDF because, by de.nition, we don.t get many data-point
observations of such catastrophes. This mechanism provides at least some kind of a generic
story about why fat tails might be inherent in many situations.
The part of the distribution of possible future outcomes that can most readily be learned
(from inductive information of a form as if conveyed by data) concerns the relatively more
likely outcomes in the middle of the distribution. From previous experience, past observa-
tions, plausible interpolations or extrapolations, and the law of large numbers, there may be
at least some modicum of con.dence in being able to construct a reasonable picture of the
central regions of the posterior-predictive PDF. As we move towards probabilities in the pe-
riphery of the distribution, however, we are increasingly moving into the unknown territory
of subjective uncertainty, where our probability estimates of the probability distributions
themselves become increasingly di?use because the frequencies of rare events in the tails
cannot be pinned down by previous experiences. It is not possible to learn enough about
the frequency of extreme tail events from .nite samples alone to make the outcome of a
CBA independent of arti.cially-imposed bounds on the extent of possibly-ruinous disasters.
Climate-change economics generally . and the fatness of climate-change tails speci.cally
.are prototype examples of this principle, because we are trying to extrapolate inductive
knowledge far outside the range of limited past experience. To put a sharp point on this
seemingly abstract issue, the thin-tailed PDFs that Nordhaus requires implicitly to support
his gradualist .policy ramp.conclusions have some theoretical tendency to morph into being
fat tailed when he admits that he is fuzzy about the functional forms or structural parameters
of his assumed thin-tailed PDFs .at least for high temperatures.
Although the basic idea is more general, it can be illustrated concretely by the relationship
between the normal distribution and the Student-t. A normal distribution is thin tailed
because the tail probabilities in the PDF decline faster than exponentially. However, if we
do not know the parameters of the normal distribution (the mean and, more importantly,
the standard deviation), but we have n observations drawn from this normal distribution,
then the implied posterior-predictive distribution is Student-t with n degrees of freedom. A
Student-t PDF with n degrees of freedom is fat tailed because it is readily con.rmed that
the tails are polynomial of order n.
Suppose that damages from extreme climate change are somehow translated into welfare-
equivalent consumption. When one combines fat tails in the PDF of the logarithm of
welfare-equivalent consumption with a utility function that is sensitive to high damages
from extreme temperatures, it will tend to make the willingness to pay (WTP) to avoid
extreme climate changes very large. The paper gave a formal argument within a speci.c
mathematical structure, but this formal argument could have been embedded in alternative
11
mathematical structures .with the same basic message. The particular formal argument
I gave in the paper came in the form of what I called a .dismal theorem. (DT). In this
particular formalization, the limiting expected stochastic discount factor is in.nite (or, what
I take to be equivalent for purposes here, the limiting WTP to avoid fat-tailed disasters
constitutes all of output). Of course, actual real-world expected discount factors are not
in.nite and WTPs are not 100% of output. Presumably the PDF in the bad fat tail is
thinned, or even truncated, perhaps from considerations akin to what lies behind the value
of a statistical life (VSL). (After all, we would not pay an in.nite amount to eliminate
altogether the fat tail of climate-change catastrophes.) Alas, in whatever way the bad fat
tail is thinned or truncated, a CBA based upon it remains highly sensitive to the details
of the thinning or truncation mechanism, because the disutility of extreme climate change
has .essentially.unlimited liability. In this sense climate change is unique (or at least very
rare) because the conclusions from a CBA for such an unlimited-liability situation have some
built-in tendency to be non-robust to assumed tail fatness.
It is apparent that there is much disagreement, and perhaps even confusion, about how
to interpret the in.nity symbol that appears in the formulation of DT. There is a natural
tendency to sneer at results in economics that involve in.nity. This reaction is presumably
based on the idea that in.nity is a ridiculous outcome, and therefore any model that has
an in.nity symbol in it is ridiculous and dismissable. Critics argue earnestly from their
favorite examples that expected disutility from climate change cannot be in.nite, as if this
were a telling indictment of the entire fat-tailed approach. I believe this attitude represents
a misunderstanding, because in this particular case the in.nity is trying to tell us something
important about CBA applied to rare fat-tailed unlimited-liability situations such as climate
change.
In.nity in this context is a limit, not a number. The issue isn.t about whether in.nity
makes sense as a number (which it does not, here or in most other applications), but whether
it makes sense as a limit. This is a basic distinction, but I believe that much confusion
springs from a failure to focus on it when interpreting DT. The in.nite limit in DT is just a
mathematical abstraction for expressing the idea that structural uncertainty in the form of
fat tails is likely to swamp the outcome of any .standard.CBA that disregards this aspect.
An in.nite limit here is just a formal way of presenting the thought that fat-tailed CBA is
highly sensitive to the exact speci.cation of assumptions that would attenuate or eliminate
the welfare e?ects of the bad fat tail.
It cannot be emphasized emphatically enough that the key issue here is not a mathe-
matically illegitimate use of the in.nity symbol in DT, which incorrectly seems to o?er a
deceptively easy way of discrediting the model on the narrow grounds that in.nities are
12
not allowed in a legitimate theory of choice under uncertainty. It is easy to modify utility
functions, to truncate probability distributions arbitrarily, or to introduce ad hoc priors that
arbitrarily cut o?or otherwise severely dampen low values of welfare-equivalent consumption.
Introducing any of these (or many other attenuating mechanisms) formally closes the model
in the sense of replacing the in.nity symbol by some uncomfortably large, but .nite, number.
Indeed, the model of my paper was closed in just such a fashion by placing a lower bound on
post-damages consumption, which bound is de.ned indirectly by some VSL-like parameter.
This is one gimmick for removing the in.nity, but many others are possible. Unfortunately,
removing the in.nity symbol in this or any other way does not eliminate the underlying
problem because it then comes back to haunt in the form of an expected stochastic discount
factor or WTP to eliminate the uncertainty that is arbitrarily large, and whose exact value
depends sensitively upon obscure bounds, truncations, severely-dampened or cut-o? prior
PDFs, or whatever other tricks have been used to banish the in.nity symbol. One can
easily remove the irritating in.nity symbol, but one cannot so easily remove the underlying
substantive economic problem of extreme sensitivity to fat tails. The take-away message
here is that reasonable attempts to constrict the fatness of the .bad.tail can still leave us
with uncomfortably big numbers, whose exact value depends non-robustly upon arti.cial
constraints, functional forms, or parameters that we really do not understand. The only
legitimate way to avoid this potential problem is when there exists strong a priori knowledge
that restrains the extent of total damages. 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, ex-
posure 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.
Many of the critics also misinterpret the implications of my theory for CBA. According to
them, my viewpoint is essentially equivalent to a statement that CBA is worthless for climate
change. That is not my position and was not stated as such anywhere in the paper. What
I do claim is that something like DT indicates a built-in theoretical tendency to cause the
conclusions from CBA to be more fuzzy than we might prefer, because they are dependent
on essentially arbitrary decisions about how the fat tails are expressed and about how the
damages from high temperatures are speci.ed. I would make a strong distinction between
thin-tailed CBA, where there is no reason in principle that outcomes should not be robust,
and fat-tailed CBA, where even in principle outcomes are highly sensitive to functional
13
forms and parameter values. For ordinary run-of-the-mill limited exposure or thin-tailed
situations, there is at least the underlying theoretical reassurance that .nite-cuto?-based
CBA might (at least in principle) be an arbitrarily-close approximation to something that is
accurate and objective. In fat-tailed unlimited exposure situations, by contrast, there is no
such theoretical assurance underpinning the arbitrary cuto?s or attenuations .and therefore
CBA outcomes have a theoretical tendency to be sensitive to fragile assumptions about the
likelihood of extreme impacts and how much disutility they cause. Maybe this position is
interpretable to others as an argument that CBA for climate change is essentially worthless,
but I think a more nuanced interpretation is appropriate here.
On the issue of what DT implies for CBA, I occupy what I like to consider a centrist
middle ground between two extreme positions. An economist does not want to abandon
lightly the ideal that CBA should bring independent empirical discipline to any application
by being based upon empirically-reasonable parameter values. Even when fat-tailed logic
applies, climate-change CBA might, at least in principle, reveal useful information about
whether fat tails are or are not actually relevant for .reasonable.functional forms and para-
meter values at extreme temperatures. (What .reasonable.means in a context of extreme
impacts with low probabilities is unclear, which introduces a vast grey area into CBA of
climate-change catastrophes.) Simultaneously, one does not want to be obtuse by insisting
that the catastrophe logic behind fat tails makes no practical di?erence for CBA because
the parameters just need to be determined empirically and then simply plugged into the
analysis along with some extrapolative guesses about the form of the .damages function.for
high-temperature catastrophes (combined with speculative extreme-tail probabilities). The
paper stated clearly my position that some sort of a tricky balance is required between being
overawed by fat-tailed catastrophe logic into abandoning CBA altogether and being under-
awed into insisting that it is just another empirical issue to be sorted out by business-as-usual
CBA. My target is not CBA in general, but the particular false precision conveyed by the
misplaced concreteness of the .standard.CBA of climate change. By all means plug in
tail probabilities, plug in disutilities of high impacts, plug in rates of pure time preference,
and so forth, and then see what emerges empirically. Only please do not be surprised when
outcomes from fat-tailed CBA are fragile to speci.cations concerning catastrophic extremes.
The extraordinary magnitude of the deep structural uncertainties involved in climate-
change CBA, and the implied limitations that prevent CBA from reaching robust conclusions,
are highly frustrating for most economists, and in my view may even push some into a state of
denial. After all, economists make a living from plugging rough numbers into simple models
and reaching speci.c conclusions (more or less) on the basis of these numbers. What are we
supposed to tell policy makers and politicians if our conclusions are ambiguous and fragile?
14
The public has little tolerance for ambiguity and craves some kind of an answer, one way or
the other, so there is little place for even a whi? of fuzziness in what we are delivering up
to them. It is threatening for economists to have to admit that the structural uncertainties
and unlimited liabilities of climate change run so deep that gung-ho .can do. economics
may be up against limits on the ability of quantitative analysis to give robust advice in such
a grey area. But if this is the way things are with the economics of climate change, then
this is the way things are .and non-robustness to subjective assumptions is an inconvenient
truth to be lived with rather than a fact to be denied or evaded just because it looks less
scienti.cally objective in CBA.
In my opinion, we economists need to admit to the policy makers, the politicians, and the
public that CBA of climate change is unusual in being especially fuzzy because it depends
especially sensitively on what is subjectively assumed about the high-temperature damages
function, along with subjective judgements about the fatness of the extreme tails and/or
where they have e?ectively been cut o?. Policy makers and the public will just have
to deal with the idea that CBA of climate change is less crisp (maybe I should say even
less crisp) than CBAs of more conventional situations. What we can do constructively as
economists is to explain the outlines of the extraordinary structural uncertainties that are
involved, present the best CBAs and most honest sensitivity analyses that we can under
fat-tailed circumstances, and leave people to make up their own minds about what to do on
the basis of an admittedly sketchy economic analysis of a grey area that just cannot render
crisp robust answers. The moral of the dismal theorem is that under extreme uncertainty,
seemingly casual decisions about functional forms, parameter values, and tail thickness may
be dominant. We economists should not pursue a narrow, super.cially precise, analysis by
blowing away the low-probability high-impact catastrophic scenarios as if this is a necessary
price we must pay for the worthy goal of giving crisp advice. An arti.cial infatuation with
precision is likely to make our analysis go seriously askew and to undermine the credibility
of what we say by e?ectively marginalizing the very possibilities that make climate change
grave in the .rst place.
One .nal note about a point that is often overlooked. When I say there is a low
probability of a bad disaster, I mean there is a low probability. The theory I am talking
about does not support the catastrophist view that we are inevitably heading for a disaster
unless we change our ways drastically. It is likely that people will look back on climate
change from some remote future vantage and be relieved that we dodged a bullet. That is
what a low probability means. Ex post, the world dodged an atomic bullet in the Cuban
missile crisis. But were we right to be highly concerned at the time? Would an ex-ante CBA
of the Cuban missile crisis have given policy advice that might have been especially sensitive
15
to assumptions about the unknowable probabilities and disutilities of atomic war? I think
we were right then to be highly concerned about a low-probability extreme-impact situation
whose structure is very uncertain .and I think we are right now to be highly concerned
about a low-probability extreme-impact situation whose structure is very uncertain.
4 Whom or What Should We Believe?
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 CO2) were short (as is the case for SO2;
particulates, and many other airborne pollutants). Then an important part of an optimal
strategy would presumably be along the lines of .wait and see.. With strong reversibility,
an optimal climate-change policy should logically involve (among other elements) waiting to
see how far out on the bad fat tail the planet will end up, followed by midcourse corrections
if we seem to be headed for a disaster. This is the ultimate backstop rebuttal of DT given by
some critics of fat-tailed reasoning, including Nordhaus. Alas, the problem of climate change
is characterized everywhere by immensely long inertias .in atmospheric CO2 removal times,
in the capacity of the oceans to absorb heat (as well as CO2), and in many other relevant
physical and biological processes. Therefore, it is an open question whether or not we could
learn enough in su? cient time to make politically feasible midcourse corrections. When
the critics are gambling on this midcourse-correction learning mechanism to undercut the
message of DT, they are relying more on an article of faith than on any kind of evidence-based
scienti.c argument.
Take atmospheric carbon dioxide as just one speci.c example (there are many others).
The time it takes for an excess bulge of CO2 to get purged out of the atmosphere is a
weighted sum of exponential decay terms, each term representing a di?erent CO2 absorption
process transpiring on a di?erent time scale. According to a standard accepted formula,
for every unit of CO2 anthropogenically added to the atmosphere, _70% remains after 10
years, _35% remains after 100 years, _20% remains after 1,000 years, _10% remains after
10,000 years, and _5% remains after 100,000 years.7 Oceanic absorption of atmospheric
CO2, or the capacity of the oceans to take up atmospheric heat, or several other relevant
mechanisms, present a similar situation of extraordinarily long inertias relative to the time
it takes to extract and act upon meaningful signals .ltered out of a lot of noise. The
kinds of numbers cited above for the natural removal times of atmospheric CO2 do not
look to me like evidence supporting .wait and see.policies. Furthermore, the examples of
7See Archer (2007), pages 122-124, and the further references he cites.
16
possible midcourse-correction strategies cited by Nordhaus strike me as unreliable, to put it
mildly. Can we shut down all GHG emissions in a short period of time? Will we have
viable technologies for removing existing stocks of CO2 from the atmosphere? I think the
actual scienti.c facts behind the alleged feasibility of .wait and see.policies are, if anything,
additional evidence for the importance of fat-tailed irreversible uncertainty about ultimate
climate change.
The relevance of .wait and see.policies is an important unresolved issue, which in princi-
ple could decide the debate between me and Nordhaus, but my own take right now would be
that the built-in pipeline inertias are so great that if and when we detect that we are heading
for unacceptable climate change, it will likely prove too late to do anything much about it for
centuries to come thereafter (except, possibly, for lowering temperatures by geoengineering
the atmosphere to re.ect back incoming solar radiation). In any event, I see this whole
.wait and see.issue as yet another component of fat-tailed uncertainty .rather than being
a reliable backstop strategy for dealing with excessive CO2 in the atmosphere.
Nordhaus states that there are so many low-probability catastrophic-impact scenarios
around that .if we accept the Dismal Theorem, we would probably dissolve in a sea of anxi-
ety at the prospect of the in.nity of in.nitely bad outcomes.. This is rhetorical excess and,
more to the point here, it is fallacious. Most of the examples Nordhaus gives have such
miniscule thin-tailed probabilities that they can be written o?. In my paper I listed what
I consider to be the half-dozen or so serious contenders with climate change for potentially
catastrophic impacts with non-negligible probabilities: biotechnology, nanotechnology, as-
teroids, strangelets, pandemics, runaway computer systems, nuclear proliferation. It may
well be that each of these possibilities of environmental catastrophe deserves its own CBA
application of DT along with its own empirical assessment of how much probability mea-
sure is in the extreme tails. Even if this were true, however, it would not lessen the need
to reckon with the strong potential implications of DT for CBA in the particular case of
climate change. The fallacy in Nordhaus.s position here is trying to argue his case through
guilt by association.
The critics of DT promote alternative thin-tailed speci.cations that do not imply nearly
such extreme outcomes as do my speci.cations. Their thin-tailed reduced-form speci.cations
appear super.cially to be plausible, and my fat-tailed reduced-form speci.cations (I hope)
appear super.cially to be plausible. They have credentials and fans of their approach, but
I do too. So whom or what is a reader to believe? Of course the reader should weigh
the plausibility of the arguments and the reasonableness of the speci.cations on their own
merits. But it is di? cult to have prior notions about how fat are climate-change tails, or
about how sensitive are disutility functions to extreme temperatures, or about lots of other
17
relevant things. Suppose, for the sake of argument, that a policy maker believes there
is a 50% probability that my fat-tailed speci.cation is correct and a 50% probability that
the thin-tailed speci.cation of some .representative critic.is correct. Then rational policy
should lean more in the direction of my fat-tailed conclusions than in the direction of the
representative critic.s thin-tailed conclusions because of the highly asymmetric consequences
of fat tails vs. thin tails. In this sense, whether it is fair or unfair, there is not a level
playing .eld between me and the .representative critic.. If someone advises you that a
.re insurance policy protecting your house against extreme losses is unnecessary because so
few houses burn to the ground, and someone else advises you that a .re insurance policy
is necessary in your case, should you .ip a coin in deciding what to do just because both
advisers seem equally credible?
As for Nordhaus.s framing of the issue that a combination of three (implicitly unlikely,
in his mind) conditions must simultaneously be ful.lled in order to buy into what I am
calling fat-tailed logic, I think it is a subjective judgement where the burden of proof lies
here. His point is essentially correct, but the issue is how to interpret it. To cut to the
analytical core of the matter, the reduced form that Nordhaus must assume to justify his
gradualist .policy ramp. is a thin-tailed PDF in the logarithm of the disutility of high
temperatures (and, importantly, his reduced-form PDF must be thin tailed after integrating
out the uncertainty in functional forms and structural parameters). The issue of whether
a fat-tailed or a thin-tailed reduced form in the PDF of the logarithm of the disutility of
extreme temperatures emerges, or does not emerge, from some particular combination of
temperature-sensitive disutilities, temperature PDFs, wait and see policies, or anything else,
is essentially secondary. The primary issue is that a reader must decide, in the light of all
of the evidence taken together, which of us is assuming more restrictive conditions and less
plausible speci.cations overall than the other, and what implications this might have.
5 Concluding Comments
Nordhaus summarizes his critique with the idea there are indeed deep uncertainties about
virtually every aspect of the natural and social sciences of climate change .but these uncer-
tainties can only be resolved by continued careful analysis of data and theories. I heartily
endorse his constructive attitude about the necessity of further research targeted toward a
goal of resolving as much of the uncertainty as it is humanly possible to resolve. I would
just add that we should also recognize the reality that, for now and perhaps for some time
to come, the sheer magnitude of the deep structural uncertainties, and the way we express
them in our models, will likely dominate plausible applications of CBA to the economics of
18
climate change.
References
[1] Allen, Myles R. and David J. Frame. .Call O? the Quest.Science, 2007 (October
26), 318, pp. 582-583.
[2] Archer, David. Global Warming. Blackwell Publishing, 2007.
[3] Dieter L?thi, Martine Le Floch, Bernhard Bereiter, Thomas Blunier, Jean-
Marc Barnola, Urs Siegenthaler, Dominique Raynaud, Jean Jouzel, Hubertus
Fischer, Kenji Kawamura & Thomas F. Stocker. .High-resolution carbon dioxide
concentration record 650,000.800,000 years before present.. Nature, 453, 379-382 (15
May, 2008).
[4] IPCC-AR4. Climate Change 2007: The Physical Science Basis. Contribution of Work-
ing Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate
Change. Cambridge University Press, 2007 (available online at http://www.ipcc.ch).
[5] Nordhaus, William D. .An Analysis of the Dismal Theorem.. Cowles Foundation
Discussion Paper No. 1686, Yale University, January 2009.
[6] Sterner, Thomas, and U. Martin Persson. .An Even Sterner Review: Introducing
Relative Prices into the Discounting Debate.. Review of Environmental Economics and
Policy, 2008 (Winter), 2(1), pp. 61-76.
[7] Weitzman, Martin L. .On Modeling and Interpreting the Economics of Catastrophic
Climate Change..Review of Economics and Statistics, February 2009 (2009a).
[8] Weitzman, Martin L. .Some Basic Economics of Extreme Climate Change..Mimeo,
February 2009 (2009b).
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