Discussion Paper No. 825

WEITZMAN MEETS NORDHAUS:
EXPECTED UTILITY AND
CATASTROPHIC RISKIN A STOCHASTIC,ECONOMY-CLIMATE MODEL
Masako Ikefuji
Roger J. A. Laeven
Jan R. Magnus
Chris Muris

December 2011

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Weitzman meets Nordhaus:
Expected utility and catastrophic risk in astochastic economy-climate model∗
December 15, 2011

Masako IkefujiInstitute of Social and Economic Research, Osaka University

Roger J. A. Laeven
Department of Econometrics & Operations Research, Tilburg University

Jan R. Magnus
Department of Econometrics & Operations Research, Tilburg University

Chris Muris
Department of Economics, Simon Fraser University∗

We are very grateful to Graciela Chichilnisky, John Einmahl, Reyer Gerlagh, Johan Eyckmans, Sjak Smulders, Peter Wakker, Aart de Zeeuw, Amos Zemel, and seminar and conference participants at Kyushu University, Tilburg University, EAERE 2011
in Rome, and the Tilburg Sustainability Center for helpful comments. This research
was funded in part by the JSPS under grant C-22530177 (Ikefuji) and by the NWO
under grant Vidi-2009 (Laeven). E-mail addresses: Ikefuji@iser.osaka-u.ac.jp (Ikefuji),
R.J.A.Laeven@uvt.nl (Laeven), Magnus@uvt.nl (Magnus), CMuris@sfu.ca (Muris).

1

Corresponding author:
Roger J. A. Laeven
Department of Econometrics and Operations Research
Tilburg University
P.O. Box 90153
5000 LE Tilburg
The Netherlands
e-mail: R.J.A.Laeven@uvt.nl
Abstract: We specify a stochastic economy-climate model, adapting Nordhaus’ deterministic economy-climate model by allowing for Weitzman-type
stochasticity. We show that, under expected power utility, the model isfragile to heavy-tailed distributional assumptions and we derive necessary
and suﬃcient conditions on the utility function to avoid fragility. We solve
our stochastic economy-climate model for two cases with compatible pairs ofutility functions and heavy-tailed distributional assumptions. We further develop and implement a procedure to learn the input parameters of our modeland show that the model thus specified produces robust optimal policies.
The numerical results indicate that higher levels of uncertainty lead to lessabatement and consumption, and to more investment.
JEL Classification: D61; D81; Q5.
Keywords: Economy-climate models; Catastrophe; Expected utility; Heavy
tails; Power utility.

2

1

Introduction

An economist, when asked to model decision making under risk or uncertainty for normative purposes, would typically work within the expectedutility framework with constant relative risk aversion (that is, power utility). A statistician, on the other hand, would model economic catastrophes
through probability distributions with heavy tails. Unfortunately, expected
power utility is fragile with respect to heavy-tailed distributional assumptions: expected utility may fail to exist or it may imply conclusions that are‘incredible’.
Economists have long been aware of this tension between the expectedutility paradigm and distributional assumptions (Menger, 1934), and thediscussions in Arrow (1974), Ryan (1974), and Fishburn (1976) deal explicitlywith the trade-oﬀ between the richness of the class of utility functions and thegenerality of the permitted distributional assumptions. Compelling examples
in Geweke (2001) corroborate the fragility of the existence of expected power
utility with respect to minor changes in distributional assumptions.
The combination of heavy-tailed distributions and the power utility family may not only imply infinite expected utility, but also infinite expectedmarginal utility, and hence, via the intertemporal marginal rate of substitution (the pricing kernel), lead to unacceptable conclusions in cost-benefit
analyses. For example, with heavy-tailed log-consumption and power utility,
the representative agent should postpone any unit of current consumption
to mitigate future catastrophes. The latter aspect was recently emphasizedby Weitzman (2009) in the context of catastrophic climate change. Weitzman also argues that attempts to avoid this unacceptable conclusion will
necessarily be non-robust.
In this paper we study the question of how to conduct expected utility analysis in the presence of catastrophic risks, in the context of extreme
climate change. Our paper is built on four beliefs, which will recur in our
analysis:
Catastrophic risks are important. To study risks that can lead to catastrophe is important in many areas, e.g., financial distress, traﬃc accidents,
dike bursts, killer asteroids, nuclear power plant disasters, and extreme climate change. Such low-probability high-impact events should not be ignoredin cost-benefit analyses for policy making.
A good model ‘in the center’ is not necessarily good ‘at the edges’. Models
are approximations, not truths, and approximations may not work well if we3move too far away from the point of approximation. In our context, thewidely adopted family of power utility functions, often appropriate when
one considers large inputs remote from zero, may not work well for decision
making under heavy-tailed risks with non-negligible support beyond the usual
domain of inputs.
The price to reduce catastrophic risk is finite. Are we willing to spend
everything to avoid children being killed at a dangerous street? Or dikes toburst? Or a power plant to explode? Or a killer asteroid to hit the Earth? Or
climate to change rapidly? No, we are not. To assume the opposite (that asociety would be willing to oﬀer all of its current wealth to avoid or mitigate
catastrophic risks) is not credible, not even from a normative perspective. In
our context, there is a limit to the amount of current consumption that the
representative agent is willing to give up in order to obtain one additional
certain unit of future consumption, no matter how extreme and irreversible
climate change may be. In other words: the expected pricing kernel is finite.
Light-tailed risks may result in heavy-tailed risk. When x is normally distributed (light tails) then 1/x has no moments (heavy tails). Also, when x is
normally distributed then ex has finite moments, but when x follows a Student distribution then ex has no moments. In the context of extreme climatechange: temperature has fluctuations but one would not expect heavy tails
in its distribution. This does not, however, imply that functions of temperature cannot have heavy tails. For example, it may well be reasonable to use
heavy-tailed distributional assumptions to model future (log) consumption.
There is an important literature on stochastic economy-climate models(see, for example, Keller et al., 2004, Mastrandrea and Schneider, 2004,
and the references therein). The integrated assessment models of climate
economics are, however, predominantly deterministic and rarely incorporate
catastrophic risk (Ackerman et al., 2010). To allow for uncertainty and extreme climate change, we start by specifying a stochastic economy-climate
model, adapting Nordhaus’ (2008) deterministic dynamic integrated climate
and economy (DICE) model by allowing for stochasticity in the spirit of
Weitzman (2009). We show formally that, under expected power utility, themodel is indeed fragile to heavy-tailed distributional assumptions, and we
derive necessary and suﬃcient conditions on the utility function to avoid
fragility, thus ensuring that expected utility and expected marginal utility(hence the expected pricing kernel) are finite, also under heavy-tailed distributional assumptions.
Next, we solve a two-period version of the model, first with power utility
and light-tailed distributional assumptions. Since the assumption of expected
power utility is incompatible with heavy-tailed distributional assumptions,
we then restrict attention to utility functions that satisfy the derived compat4

ibility conditions, and solve our stochastic economy-climate model with thewell-known exponential utility function and also with the less well-known(but more suitable) ‘Pareto utility’ function, under both light-tailed andheavy-tailed distributional assumptions.
Completing the resulting model requires specifying a number of model
parameters as inputs. These parameters cannot ‘simply’ be determined byconventional statistical inference based on historical data. We discuss how
to set the model parameters in a process towards agreement, using experts’priors on parameter values, and learning about parameters from resulting
optimal model output. The key to this learning and agreement process isthe translation of model parameters that are relatively diﬃcult to interpret
into quantities that allow a more straightforward interpretation. Contrary
to Weitzman’s (2009) claim, we find that our optimal policies thus derived
are quite robust with respect to minor and reasonable changes to the input
parameters. This means that the policymaker is not left empty-handed whenit comes to cost-benefit analyses under catastrophic risk.
Our numerical analysis indicates that allowing for heavy-tailed distributional assumptions in extreme climate change modeling, inducing bothheavy-tailed downside risk and heavy-tailed upward potential, leads to a reduction of current abatement and consumption and to an increase in current
investment, when compared to a deterministic analysis. The increase in current investment may be interpreted via precautionary savings. While thediﬀerences are clearly visible, they are not unlimited.
The paper is organized as follows. In Section 2 we propose a simplified
version of Nordhaus’ economy-climate model. In Section 3 we introduce uncertainty in the spirit of Weitzman. This is the first new feature added to
the Nordhaus model. Section 4 discusses expected (marginal) utility anduncertainty in a more general setting, deriving results on the trade-oﬀ between permitted distributional assumptions and the existence of expected(marginal) utility. In Section 5 we specialize the model to two periods only,
and add a second new feature to the Nordhaus model: scrap value functions.
In Section 6, we present (partial) results for power utility, which is incompatible with heavy tails, and for exponential and ‘Pareto’ utility, which arecompatible with heavy tails. Section 7 discusses how to learn the parametersof our model and calibrate policy using information such as the probabilityof catastrophe, and reports on robustness tests. Section 8 concludes. There
are two appendices: Appendix A provides the Kuhn-Tucker conditions andAppendix B contains proofs of the propositions.

52

Nordhaus simplified

Our framework is a simple economy-climate model in the spirit of Nordhaus
and Yang (1996) and Nordhaus (2008). Everybody works. In period t, thelabor force Lt together with the capital stock Kt generate GDP Yt througha Cobb-Douglas production function
Yt = At Ktγ L1−γt(0 < γ < 1),

where At represents technological eﬃciency and γ is the elasticity of capital.
Capital is accumulated throughKt+1 = (1 − δ)Kt + It

(0 < δ < 1),

where It denotes investment and δ is the depreciation rate of capital. Production generates carbon dioxide (CO2) emissions Et :Et = σt (1 − µt )Yt ,where σt denotes the emissions-to-output ratio for CO2, and µt is the abatement fraction for CO2. The associated CO2 concentration Mt accumulates
through
Mt+1 = (1 − ϕ)Mt + Et
(0 < ϕ < 1),
where ϕ is the depreciation rate of CO2 (rate of removal from the atmosphere). Temperature Ht develops according to
Ht+1 = η0 + η1 Ht + η2 log(Mt+1 )(η1 > 0, η2 > 0).

In each period t, the fraction of GDP not spent on abatement or ‘damage’ iseither consumed (Ct ) or invested (It ) along the budget constraint(1 − ωt )dt Yt = Ct + It .
The temperature-impact function dt depends only on temperature and satisfies 0 < dt ≤ d¯t , where d¯t represents the optimal temperature for the economy.
Deviations from the optimal temperature cause damage. We specify dt as
dt =

d¯t
1 + ξHt2(ξ > 0).

For very high and very low temperatures dt approaches zero. The optimalvalue of dt occurs at Ht = 0 (the temperature in 1900, as in Nordhaus) whendt = d¯t . Hence, ‘net’ output dt Yt is a fraction, not of Yt as in Nordhaus, but
6

of d¯t Yt , the output achievable under optimal climate conditions. A fraction
ωt of dt Yt is spent on abatement, and we specify the abatement cost fractionasωt = ψt µθt(θ > 1).
If µt increases then so does ωt , and a larger fraction of GDP will be spent
on abatement. These equations capture the essence of the Nordhaus (2008)
DICE model.
The model includes stock variables Lt , Kt , Mt , and Ht , fractions ωt andµt , and scale variables At , dt , σt , and ψt , all measured at the beginning of
period t; and flow variables Yt , Ct , It , and Et , all measured in period t (notin year t). Notice that Lt is a stock, not a flow. As in Nordhaus (2008) one
period is ten years. We choose the exogenous variables such that Lt > 0,
At > 0, σt > 0, and 0 < ψt < 1. The policy variables must satisfy
Ct ≥ 0,It ≥ 0,0 ≤ µt ≤ 1.

(1)

With these restrictions all variables will have the correct signs and all fractions will lie between zero and one.
Given a utility function U we define welfare in period t as
Wt = Lt U (Ct /Lt ).
If the policy maker has an infinite horizon, then he/she will maximize total
discounted welfare,
W =

∞
∑

Wt
(1 + ρ)tt=0(0 < ρ < 1),

where ρ denotes the discount rate. Letting x denote per capita consumption,
the utility function U (x) is assumed to be defined and strictly concave forall x > 0. There are many such functions, but a popular choice is
U (x) =x1−α − 1
1−α(α > 0),(2)

where α denotes the elasticity of marginal utility of consumption. This is theso-called power function. Many authors, including Nordhaus (2008), selectthis function and choose α = 2 in which case U (x) = 1 − 1/x. Also popularis α = 1; see Kelly and Kolstad (1999) and Stern (2007).
Our interest is in maximizing welfare W with respect to the policy bundles(Ct , It , µt ) for t = 0, 1, 2, . . . . We choose the exogenous variables Lt , At , σt ,and ψt as in Nordhaus (2008), and we let d¯t ≡ 1 and α = 2. Calibrating the7Table 1: Comparison of stocks in Nordhaus (DICE) and our (SICE) models
2005
2055
2105
2155
DICE SICE DICE SICE DICE SICE DICE SICE
K
137
137
353
354
707
711
1317 1324
M
809
809
1048
988
1270 1233
1428 1430
H
0.7
0.7
1.8
1.5
2.7
2.4
3.3
3.2
parameters and initial values (presented in our background document, see
Ikefuji et al., 2011b), our GAMS code1 produces optimal values over sixty
periods that are very close to the values obtained in Nordhaus, as shown in
Table 1. Hence it appears that our simplified version of the DICE model(hereafter, SICE = simplified DICE) works as well as the original version.

3Enter Weitzman and uncertainty

Weitzman (2009) recently noted, in a highly stylized setting, that heavytailed uncertainty and power utility are incompatible, since this combinationof uncertainty and preferences implies an infinite expected pricing kernel. In
order to avoid this, Weitzman introduces a lower bound on consumption. He
then argues that this lower bound is related to a parameter that resembles
the value of a statistical life, and proves that the expected pricing kernel
approaches infinity as the value of this parameter approaches infinity (the‘dismal theorem’). Weitzman further argues that this ‘VSL-like’ parameteris hard to know, and interprets this result as follows:“. . . reasonable attempts to constrict the length or the heavinessof the ‘bad’ tail (or to modify the utility function) still can leave
us with uncomfortably big numbers whose exact value depends
non-robustly upon artificial constraints or parameters that wereally do not understand.” (Weitzman, 2009, p. 11)
We agree with Weitzman that incompatible pairs of utility functions anddistribution functions exist, in the sense that the expected pricing kernel
or other important policy variables become infinite. In fact, we derive in
Section 4 necessary and suﬃcient conditions on the utility functions for the
expected pricing kernel to exist, also under heavy tails. But we object to thedismal theorem for two reasons. As we demonstrate formally in Section 4
and numerically in Section 6, the dismal theorem is based on an incompatible (invalid) model specification; it is avoided when the economic model
1

Available at http://www.janmagnus.nl/items/risk.pdf

8

(utility function) is compatible with the statistical model (heavy tails). Furthermore, as we show in Section 7, more eﬀort can be made to know an
input parameter that is ‘hard to know’, and we develop and implement alearning-and-agreement procedure for precisely this purpose.
We introduce uncertainty in the simplified Nordhaus model in a Weitzmanlike manner. There is much uncertainty in the economics of climate change(Manne and Richels, 1992; Nordhaus, 1994; Weitzman, 2009). There is model
uncertainty, parameter uncertainty, and uncertainty about the possible reduction of parametric variability over time (updating); see Kelly and Kolstad(1999) and Leach (2007). We model uncertainty through stochasticity. In
the literature, stochasticity is typically introduced through the damage function (Roughgarden and Schneider, 1999; Mastrandrea and Schneider, 2004)or through a random shock in temperature (Kelly and Kolstad, 1999; Leach,2007). We follow this literature by introducing stochasticity through thetemperature-impact function dt , more precisely through d¯t , the impact under optimal temperature. We are uncertain about the optimal temperature,
because we are uncertain about the correctness of the functional form of dt ,about the values of the parameters, and about the underlying temperatureequation. We capture these three sources of uncertainty by writing2d¯t = e−τ /2 eτ ϵt ,where ϵt denotes a random error with mean zero and variance one. Thisimplies that ‘net GDP’ is given bye−τ /2 Zt
dt Yt =,
1 + ξHt22

Zt = At Ktγ L1−γeτ ϵt ,
t(3)

so that random noise enters the Cobb-Douglas production function in theusual ‘linear’ way when we write log(Zt /Lt ) = log At + γ log(Kt /Lt ) + τ ϵt .
Introducing uncertainty in this way is in the spirit of Weitzman and it isconvenient for our purpose. An alternative, possibly more intuitive, specification is to let stochasticity enter through a random multiplicative shock intemperature, by setting d¯t = 1 and randomizing the parameter ξ. This leads
to an alternative specification of the temperature-impact function,
[
]
2
2 −1
,
u = eτ /2 e−τ ϵt ,dalt
t = 1 + uξHt−1

as opposed to our original specification dt = [u(1 + ξHt2 )] . For our analysis,≈large values of u are relevant. For large u we have approximately daltt
2 −1[uξHt ] , whose behavior is similar to dt . Hence, for our purpose the two
specifications are conceptually equal.
9If ϵt follows a normal distribution N(0, 1), then the moments of d¯t exist,
2
and we have E(d¯t ) = 1 and var(d¯t ) = eτ − 1. Since the distribution ofd¯t is heavily skewed, more uncertainty (higher τ ) implies more probability
mass of d¯t close to zero, and a higher probability of damage. If, however,
we move only one step away from the normal distribution and assume that
ϵt follows a Student distribution with any (finite) degrees of freedom, thenthe expectation is infinite (Geweke, 2001). This fact predicts that expectedwelfare may be very sensitive to distributional assumptions: random noisewith finite moments (Student distribution) may turn into random variables
without moments (d¯t , dt Yt ).
We need sensible values for the uncertainty parameter τ . The stochasticity, as given in (3), captures uncertainty about GDP that is due to uncertainty
about climate change. Historical variation in GDP may therefore serve as
an initial upper bound proxy for τ . Barro (2009) calibrates the standard
deviation of log GDP to a value of 0.02 on an annual basis. Over a 10year horizon this would correspond to about 0.06, under normality. Barro,
however, only considers rich (OECD) countries, which means that for our
purposes this value needs to be scaled up. In addition to the value of τ weneed to consider the question whether or not the uncertainty introduced isindeed heavy-tailed. A (partial) answer to this question is contained in a recent paper by Ursúa (2010) who claims that the growth rate of GDP indeed
features heavy tails.
4.5

Normal
t(10)

4
0.1

3.5

density

3
2.5
2
0.3

1.5
1
0.7
0.5
0
0

0.5

1
d1

1.5

Figure 1: Density of d¯t for τ = 0.1, 0.3, and 0.7
102

In Figure 1 we plot the density of d¯t for three values of τ : 0.1, 0.3, √and 0.7,
both when ϵt follows a N(0, 1) distribution (solid line) and when ϵt = 4/5 u,where u follows a Student distribution (as adopted in Weitzman, 2009) with
10 degrees of freedom (which implies a ‘tail index’ that is broadly consistent
with the empirical analysis in Ursúa, 2010). Notice that E(ϵt ) = 0 andvar(ϵt ) = 1 in both cases. When τ = 0.1, we see that almost 100% of the
distribution of d¯t lies in the interval (0.5, 2.0), both for the N(0, 1) distribution
and for the t(10) distribution. When τ = 0.3, 97.8% (97.2% for the Student
distribution) lies in the interval (0.5, 2.0); and, when τ = 0.7, only 64.9%(67.2% for the Student distribution) lies in this interval. We conclude thatτ = 0.7 may serve as a credible upper bound for the uncertainty range, andhence we report our results for τ = 0.0, 0.3, 0.5, and 0.7.

4

Expected utility and heavy tails

Since the axiomatization of expected utility (EU) by Von Neumann and Morgenstern (1944) and Savage (1954), numerous objections have been raised
against it. Most of these relate to empirical evidence that the behavior ofagents under risk and uncertainty does not agree with EU. Despite importantdevelopments in non-expected utility theory, EU remains the dominant normative decision theory (Broome, 1991; Sims, 2001), and the current paper
stays within the framework of EU. Our results presented below corroborate
the fact that expected utility theory may reliably provide normatively appealing results, also in the presence of catastrophic risks.
We formulate our decision under uncertainty problem in Savage (1954)
style, independent of the specific model considered in this paper, so that theresults in this section are generally applicable. We fix a set S of states ofnature and we let A denote a σ-algebra of subsets of S. One state is the true
state. We also fix a set C of consequences (outcomes, consumption) endowedwith a σ-algebra F. Since we are only interested in monetary outcomes, we
may take C = R+ . A decision alternative (policy bundle) X is a measurable
mapping from S to C, so that X −1 (A) ∈ A for all events A ∈ F. We assume
that the class of all decision alternatives X is endowed with a preference
order ⪰.
Definition 4.1 We say that expected utility (EU) holds if there exists a
measurable and strictly increasing function U : C → R on the space of consequences, referred to as the utility function, and a probability measure P on A,
such that the∫ preference order ⪰ on X is represented by a functional V of theform X 7→ S U (X(s)) dP = V (X). Thus, the decision alternative X ∈ X ispreferred to the decision alternative Y ∈ X if, and only if, V (X) ≥ V (Y ).
11

In the Von Neumann and Morgenstern (1944) framework, utility U is
subjective, whereas the probability measure P associated with A is objective
and known beforehand (decision under risk). In the more general frameworkof Savage (1954) adopted here, the probability measure itself can be, butneed not be, subjective (decision under uncertainty).
Definition 4.2 We say that a risk ϵ : S → R is heavy-tailed to the left
(right) under P if its moment-generating function is infinite: E (eγϵ ) = ∞for any γ < 0 (γ > 0).
Examples of heavy-tailed risks are the Student, lognormal, and Pareto
distributions. Heavy-tailed risks provide appropriate mathematical models
for low-probability high-impact events, such as environmental catastrophes.
Proposition 4.1 If EU is to discriminate univocally among all possible alternative outcome distributions, the utility function must be bounded.
Proposition 4.1 states that the EU functional is finite for all outcome
distributions if, and only if, the utility function is bounded. Moreover, theaxiomatization of EU is valid for all outcome distributions if, and only if,
the utility function is bounded. The implications are non-trivial: boundedness of the utility function must hold not just in exotic situations but also
in more familiar and economically relevant settings involving high levels of
uncertainty. (See Moscadelli, 2004, regarding operational risk.)
In what follows we do not require the utility function to be bounded. We
simply assume that the class of feasible outcome distributions is restricted(though the restriction may be void) in such a way that the utility function
permits discrimination among them. Only a combination of utility function
and outcome distribution that leads to finite expected utility is covered by
the axiomatic justification of EU. Recall that RRA(x) = −xU ′′ (x)/U ′ (x) andARA(x) = −U ′′ (x)/U ′ (x), and let
α∗ = inf RRA(x),β ∗ = sup ARA(x).

x>0

x>0

Now consider a representative agent with time-additive EU preferences andtime-preference parameter ρ > 0. We normalize (without loss of generality)
the agent’s consumption by setting C0 = 1, and we define the pricing kernel(intertemporal marginal rate of substitution) as
P (C1∗ ) =U ′ (C1∗ ),(1 + ρ)U ′ (1)

12

(4)

where C1∗ is optimal consumption at t = 1. Consumption C1 is commonly restricted to a budget-feasible consumption set which is subject to uncertainty(ϵ1 ). We assume that the budget restriction takes the general form
C1∗ (ϵ1 ) ≤ B exp(Aϵ1 ),B, A > 0,(5)which need not be best-possible. (In our economy-climate model of Section 2,
2
B = e−τ /2 Y1 /(1 + ξH12 ) and A = τ .) The expectation E(P ) represents theamount of consumption in period 0 that the representative agent is willingto give up in order to obtain one additional certain unit of consumption inperiod 1.
The following result states that the expectation of the pricing kernel isfinite for all outcome distributions whenever the concavity index (ArrowPratt index, index of absolute risk aversion) ARA(x) is bounded.
Proposition 4.2 Assume that EU holds and that the budget feasibility restriction (5) applies.
(a) If α∗ > 0 and ϵ1 is heavy-tailed to the left under P, then E(P ) = ∞;(b) If β ∗ < ∞ and α∗ = 0, then E(P ) < ∞ for any ϵ1 .
If the EU maximizer has decreasing absolute risk aversion and increasing
relative risk aversion, as is commonly assumed, a complete and elegant characterization of boundedness of the expected pricing kernel can be obtained,
as follows.
Proposition 4.3 Assume that EU holds and that the budget feasibility restriction (5) applies. Assume furthermore that RRA(x) exists and is nonnegative and non-decreasing for all x ≥ 0 and that ARA(x)∫ γ is non-increasing
for all x > 0. Then, E(P ) < ∞ for any ϵ1 if and only if 0 ARA(x) dx < ∞for some γ > 0.
∫γ
Notice that, when 0 ARA(x) dx = ∞ for some γ > 0, both α∗ > 0
and α∗ = 0 can hold. If α∗ > 0 then we do not need the full force of
Proposition 4.3; it is suﬃcient that ϵ1 is heavy-tailed to the left. Then
E(P ) = ∞ by Proposition 4.2(a). If α∗ = 0 then heavy-tailedness alone isnotfind an ϵ1 such∫ that E(P ) = ∞. When∫ γ suﬃcient, but we can alwaysγ∗
ARA(x)
dx
=
∞
then
β
=
∞.
But when 0 ARA(x) dx < ∞, both0β ∗ < ∞ and β ∗ = ∞ can occur.
The above propositions provide necessary and suﬃcient conditions on theutility function to ensure that expected utility and expected marginal utility(hence the expected pricing kernel) are finite, also in the presence of heavy
tails. These compatibility results are generally applicable to standard multiperiod welfare maximization problems. The importance of the results lies in13the fact that (i) if expected utility is infinite, the axiomatic justification of
EU is not valid, and (ii) if the expected pricing kernel is infinite, then theamount of consumption in period 0 which the representative agent is willingto give up in order to obtain one additional certain unit of consumption inperiod 1 is infinite, which is not credible.

5

A two-period model

In Sections 2–4 we assumed an infinite horizon. We now specialize to two
periods. The two-period model captures the essence of our problem whileremaining numerically tractable in the presence of uncertainty.
If the policy maker has a (finite) T -period policy horizon, then we write
welfare as
T −1
∞
∑
∑
Lt U (xt )1
LT +t U (xT +t )
W =
+
,
t
T
(1 + ρ)(1 + ρ) t=0(1 + ρ)tt=0
where xt = Ct /Lt denotes per capita consumption in period t. If {x∗t } denotes
the optimal path for {xt }, then we define the scrap value as
ST =

∞
∑
LT +t U (x∗T +t )(1 + ρ)tt=0.

Maximizing W is then equivalent to maximizing
T −1∑
Lt U (xt )t=0(1 + ρ)+tST.
(1 + ρ)T

The scrap value ST will depend on the state variables at time T , in particular
KT and MT , and this functional relationship is the scrap value function:
ST = S(KT , MT ). If T is large we may ignore the scrap value ST because of
the large discount factor (1 + ρ)T . But if T is small, then we need to model
ST explicitly, thus emphasizing the fact that the policy maker has the double
objective of maximizing discounted welfare over a finite number of periods
T , while also leaving a reasonable economy for the next policy maker, basedon the remaining capital stock and CO2 concentration.
The simplest approximation to ST is the linear function
ST = ν0 + ν1 KT − ν2 MT(ν1 > 0, ν2 > 0),(6)where ν1 and ν2 denote the scrap prices of capital and pollution at the beginning of period T . This scrap value function captures the idea that the14

next government will be happier if there is more capital and less pollution
at the beginning of its policy period. But the linear scrap value function has
some problems. We shall therefore introduce nonlinear scrap value functions,whose specific form depends on the form of the utility function; see our background document Ikefuji et al. (2011b) for further details on our treatment
of scrap value functions.
The simplest version of the model occurs when T = 2 in which case wehave only two periods. We can write welfare in this case as
W = W (µ0 , C0 , µ1 , C1 , ϵ1 ) = W0 +

W1
S2
+
.
1 + ρ (1 + ρ)2

The policy restrictions (1) are explicitly imposed, so that we maximize a
restriction of expected welfare; see Appendix A. Randomness results fromd1 only, because the temperature-impact d0 at the beginning of period 0 is
known to us (we set d¯0 = 1, equal to its expectation), and d2 at the end of
period 1 does not appear in the welfare function. Hence, the only source ofrandomness is caused by the error ϵ1 . The policy maker has to choose thepolicy bundles (C0 , I0 , µ0 ) at the beginning of period 0 and (C1 , I1 , µ1 ) at the
beginning of period 1 that will maximize expected welfare.
Realizing that at the beginning of period 1 the temperature-impact d1
is observed based on the realization of ϵ1 , the policy maker will maximize
expected welfare in three steps as follows. First, he/she maximizes welfare
W = W (µ0 , C0 , µ1 , C1 , ϵ1 ) with respect to (µ1 , C1 ) conditional on (µ0 , C0 , ϵ1 )and under the restriction (1). This gives (µ∗1 , C1∗ ) and concentrated welfareW ∗ (µ0 , C0 , ϵ1 ) = W (µ0 , C0 , µ∗1 , C1∗ , ϵ1 ).
Then the expectation W (µ0 , C0 ) = E (W ∗ (µ0 , C0 , ϵ1 )) is computed, if it exists. Finally, W is maximized with respect to (µ0 , C0 ).

6

Compatibility

We now have a simplified Nordhaus model with Weitzman-type uncertainty
in a two-period framework. In Table 2 we present the optimal values ofthe policy and other selected variables obtained from maximizing expectedwelfare. (Our background document contains the complete tables.) The
results allow for uncertainty, consider the short run (two periods) rather
than the long run (sixty periods), and also take scrap values into account.
We need values for the exogenous variables Lt , At , σt , and ψt ; theseare given in our background document. We note that Y0 = 556.67 and15

16
178.10218.78832.51864.060.88241.0339189.67256.16832.74864.070.88261.0341

178.12∞
832.51864.060.88241.0339190.01∞
832.74864.090.88261.0341172.80210.81
834.32868.530.88441.0403179.25216.73
834.42869.390.88451.0413203.60242.08834.80872.450.88501.0450179.23∞
834.42869.380.88451.0413174.78210.90832.44863.940.8823
1.0337

StocksK1 172.49
K2 211.66
M1 834.25
M2 868.31
H1 0.8843
H2 1.0400

178.66 204.33
219.32 255.70
834.34 834.69
869.18 872.74
0.8844 0.8848
1.0411 1.0452

0.09100.1135424.33∞131.46∞202.89∞834.79872.360.88491.04490.08610.1175400.67∞155.12∞Student(10)0.3
0.7Table 2: Three utility functions with normal and Student(10) distributions
Power
Exponential
ParetoNormalNormalStudent(10)Normal
τ
0.0
0.3
0.7
0.0
0.3
0.7
0.3
0.7
0.0
0.3
0.7
Policy instrumentsµ0 0.0933 0.0920 0.0874 0.1175 0.1166 0.1135 0.1166 0.1135 0.0924 0.0910 0.0859µ1 0.1137 0.1141 0.1142 0.1473 0.1515 0.1697 0.1516 0.1700 0.1124 0.1135 0.1175
C0 431.07 424.90 399.24 428.74 425.43 413.86 425.41 413.52 430.76 424.31 399.97
C1 547.20 549.08 550.63 551.45 584.64 527.76∞∞ 548.53 552.59 563.14
I0
124.72 130.89 156.56 127.00 130.32 141.90 130.35 142.24 125.03 131.48 155.83∞∞ 150.56 154.23 171.09
I1
151.51 157.03 184.45 149.95 156.68 190.03

d0 = 0.9985 are constant over diﬀerent scenarios and functions, and that thevalues of µ0 , C0 , I0 , E0 , ω0 , K1 , M1 , and H1 are optimal values. In contrast,µ1 , C1 , I1 , Y1 , E1 , ω1 , d1 , K2 , M2 , and H2 are optimal functions of ϵ1 . What
we present in the tables are their expectations.
We consider three utility functions (power, exponential, Pareto) and two
distributions (normal, Student). First, power utility as given in (2) for α = 2:
U (x) = 1 − 1/x. The following proposition states that if the random errors
ϵt are generated by a normal N(0, 1) distribution, then the expectation of
welfare exists for power utility, but if we move one step away from normality
and assume a Student distribution with any finite degrees of freedom, thenthe expectation does not exist. It illustrates the consequences of violating
the conditions of Proposition 4.1.
Proposition 6.1 With power utility, expected welfare exists under normality but not under a Student distribution.
It follows that the much-used power utility function is incompatible withexpected utility theory with heavy tails, not because utility theory itself is
at fault but because power utility is inappropriate when tails are heavy.
Motivated by the conditions derived in Section 4 and by the fundamental
insight that the economic model and the statistical model must be compatible, and because we wish to leave distributional assumptions unrestricted
at this stage, we consider two alternative utility functions: the exponentialfunction and the ‘Pareto’ function. Other choices are permitted but may
require restrictions on distributional assumptions. The exponential utility
function is given by
U (x) = 1 − e−βx(β > 0)(7)

with ARA(x) = β and RRA(x) = βx, and the Pareto utility function by()k
λU (x) = 1 −(k > 0, λ > 0)(8)x+λwith ARA(x) = (k + 1)/(x + λ) and RRA(x) = (k + 1)x/(x + λ). The
Pareto function was proposed in Ikefuji et al. (2011a), where it is also shownthat this function enjoys a combination of appealing properties especiallyrelevant in heavy-tailed risk analysis. We choose the parameters as follows:
exponential function (β = 25), and Pareto function (k = 1.5, λ = 0.02).
This choice of parameters is determined by the point x∗ , where we want thethree utility functions to be close. Suppose we want the functions to be close
at x∗ = 0.08, which is approximately the value of C0 /L0 and C1 /L1 . Then,
given that α = 2, we find β = 2/x∗ = 25, and, for any k > 1, λ = (k −1)x∗ /2.
17

The power function is unbounded from below, hence violates the conditions of Proposition 4.1, and has constant and positive RRA, hence violates the conditions of Proposition 4.3. Both the exponential and the Pareto
function are bounded from above and below, hence satisfy the conditions of
Proposition 4.1. The exponential function has constant and positive ARA,
hence satisfies the conditions of Proposition 4.3, while the RRA is unbounded
for large x. In contrast, the RRA in the Pareto function is bounded between
0 and k +1, and it further satisfies RRA(0) = 0 and ARA(0) < ∞, hence satisfies the conditions of Proposition 4.3. Notice that the fact that RRA(0) = 0(as is the case for the exponential and the Pareto utility functions) does not
imply that the representative agent is risk-neutral at x = 0. In particular,
we have ARA(0) = β for the exponential function and ARA(0) = (k + 1)/λfor the Pareto function.

6.1

Results for power utility

The first panel of Table 2 gives the results for power utility. For τ = 0 there
is no uncertainty. For τ > 0 there is uncertainty, and all policy variables
are aﬀected when τ increases. More uncertainty results in less abatement,
less consumption, and more investment in period 0, and to more abatement,
consumption, and investment in period 1. In period 1, the changes in abatement and consumption are negligible. The increase in I0 with τ can be
explained by precautionary savings. The restriction on I1 can be viewed asa penalty for negative investment. To avoid this penalty, the policy maker
can increase the budget in period 1 by investing more in period 0. As theamount of uncertainty increases, the probability of negative investment increases, ceteris paribus. In response, the policy maker increases investment
at the expense of abatement and consumption in period 0. The increase in
I0 leads to higher output in period 1, which explains the increases in I1 , K2 ,M2 , and H2 . The decrease in µ0 leads to higher emissions in period 0, andincreases carbon concentration and temperature in period 1. An additional
reason why investment in period 1 increases with uncertainty is that positive
shocks translate into possibly unlimited upward shocks in I1 , but negative
shocks will never cause I1 to drop below zero.

6.2

Results for exponential and Pareto utility

Turning now to the alternative utility functions, we first maximize (deterministic, hence τ = 0) welfare over sixty periods (600 years) for both exponential
and Pareto utility. A selection of the resulting optimal values is shown in
Table 3. When we compare the results with those in Table 1, we see that the18

Table 3: Comparison of stocks in Exponential and Pareto models
2005
2055
2105
2155
Expo Pareto Expo Pareto Expo Pareto Expo Pareto
K
137
137
286
343
388
666
456
1220
M
809
809 1012
993 1328
1258 1727
1512
H
0.7
0.7
1.6
1.5
2.6
2.5
3.7
3.3
optimal stock values from the Pareto function closely resemble the optimal
stock values from the power function, but not those from the exponentialfunction. In contrast to power and Pareto, where RRA flattens out, theRRA for the exponential distribution continues to increase, and hence thegrowth rate of marginal utility continues to increase as well. As x increases,
consumption will therefore increase, and investment and abatement will decrease. As a result, C/Y is relatively large for exponential utility. The low
growth rate of capital (for exponential utility) leads to a low growth rateof output. However, since more consumption leads to less abatement, thegrowth rate of CO2 concentration is high even when the amount of production is low. Consequently, M and H are high compared to power and Pareto.
When x < x∗ , RRA (Pareto) is close to RRA (exponential), so that more isconsumed and less invested when the Pareto function is used instead of thepower function. But when x > x∗ , RRA (Pareto) is close to RRA (power).
The optimal path of K is slightly lower and the optimal paths of M and H
are slightly higher for Pareto than for power utility.
Since exponential utility is calibrated to be close to power utility at
x = x∗ , the two-period results for the two utility functions do not diﬀer
greatly; see the second panel of Table 2. This is especially true for τ = 0,
where only the abatement fraction µ is higher for exponential utility, andtherefore temperature H is lower. When τ increases, I0 increases less andI1 increases more for exponential than for power. Moreover, as the uncertainty parameter τ increases, M2 does not change much in the exponential
case, while it increases in the power case. The eﬀect of uncertainty on themarginal scrap values is therefore larger in the exponential case than in the
power case. As in the first panel (power utility), more uncertainty results inless abatement, less consumption, and more investment in period 0, and to
more abatement, and investment in period 1.
Suppose next that the underlying distribution has heavier tails: Student
instead of normal. Under power utility, expected welfare does not exist any
more. But under bounded utility, expected welfare always exists. Although
the eﬀect of the excess kurtosis on expected welfare is large and discontinuous,
the eﬀect on the optimal values is relatively small. This is good, because the19Student distribution with 10 degrees of freedom is in fact quite close to thenormal distribution as Figure 1 reveals, and hence it would be unreasonable ifa ‘small’ change in distributional assumptions would lead to a large possibly‘discontinuous’ change in optimal policies.
All variables move in the same direction as before when τ increases. Notice that some variables (C1 , I1 , and K2 ) have infinite expectations eventhough expected welfare is finite. This is no surprise because these variables2
are unbounded and depend on d¯1 = e−τ /2 eτ ϵ1 . When ϵ1 follows a Student
distribution, E(d¯1 ) = ∞ and this property carries over to the other threevariables.
Finally we consider the third utility function, Pareto utility. The optimalvalues are presented in the third panel of Table 2. We would expect that
Pareto and power are relatively close in the observed data range. This isindeed the case as a comparison of the first and third panels reveals. Thereis little diﬀerence between the two panels in the case of no uncertainty, andalso when τ increases. The eﬀect of excess kurtosis is again small, as it shouldbe.

Figure 2: µ∗1 as a function of ϵ1 : Pareto versus power utility
The important diﬀerence between power and Pareto utility is only revealed when low levels of per capita consumption become relevant, that is,in near-catastrophe cases. This is clarified in Figure 2, where we present µ∗1
as a function of ϵ1 for τ = 0.3. The expected value of µ∗1 is 0.1141 for power
20

utility under normality (first panel), and 0.1135 for Pareto utility under either normality or Student(10) (third panel). This is not very diﬀerent. But
for values of ϵ1 further away from 0 the diﬀerence is large.

7

Learning, agreement and robustness

7.1

Learning and agreement

To complete the model we need to specify our input parameters. We now
show, in a stylized setting, how this can be achieved in a process towards
agreement, using experts’ priors. The key to this learning and agreement
process is the translation of model parameters that are relatively diﬃcult
to interpretinto quantities that allow a more straightforward interpretation,
hence are easier to specify.
Our parameters cannot be estimated using conventional methods andhistorical data, but experts will have prior ideas about these parameters.
Diﬀerent experts will have diﬀerent priors. Model output can be generated
on the basis of various priors. Then, in an iterative procedure, one learns
about the parameter values from experts’ opinions and model output, andan agreeable intersection of model parameters may be reached.

input 1

output 1

output 2

input 2

Figure 3: The decision making processThis process is illustrated in Figure 3. In the left panel, we visualize thecontributions of two experts. One expert states that the value of input 2should be bounded as indicated by the two vertical lines. The other expert
provides a lower and upper bound for the value of input 1, depending on
the value of input 2. The horizontally-shaded area gives the combinations

21

of inputs that are acceptable to both experts. The right panel is more complicated. We first visualize the contributions of two policy makers regarding
two output variables. This is the vertically-shaded area, giving the combinations of outputs that are acceptable to both policy makers. Next we map
the left panel onto the right panel. For every acceptable combination of inputs the model provides one combination of outputs, that is, one point in
the right panel. The horizontally-shaded area in the right panel is the imageof the horizontally-shaded area in the left panel. We now have two areas in
the right panel: the vertically-shaded area and the horizontally-shaded area.
If the two areas do not intersect, then the experts and policy makers must
adjust their priors in an iterative process of learning. Once the areas do intersect, agreement is possible. The black triangle then contains all points forwhich both inputs and outputs are acceptable. Agreement must be reachedon the three policy variables (µ0 , C0 , I0 ), and we recall that expected welfare
is maximized in three steps as described at the end of Section 5, yielding the
optimal policy (µ∗0 , C0∗ , I0∗ ).
7.1.1

InputsOur analysis requires prior beliefs about various inputs, in particular: form of
the utility function (Pareto or otherwise), degree of risk-aversion (k, λ), discount rate (ρ), form of the distribution (Student or otherwise), and volatility(τ ). If agreement is to be reached, then the policy makers must be willing toadjust their individual priors on each of these inputs, based on the experts’opinions and the generated output.
We want a distribution which allows heavy tails (Ursúa, 2010), such as
the Student distribution (Weitzman, 2009). Given our treatment of stochasticity, power utility is not compatible with the Student distribution, because
the required expectations don’t exist. Also, exponential utility has the disadvantage that RRA increases without bound. Pareto utility provides a usefulcompromise: it exhibits exponential-like features when per capita consumption is small, and power-like features otherwise (Ikefuji et al., 2011a). We
therefore confine ourselves to Pareto utility, assume that ϵ1 follows a Student
distribution, and take the following parameter values as our benchmark:k = 1.5,

λ = 0.02 τ = 0.3,

df = 10,

ρ = 0.1605.

Note that the value of λ is linked to k through λ = 0.04(k − 1), as explainedin Section 6. Priors regarding risk aversion are based on our backgrounddocument. The symbol df denotes the degrees of freedom in the Student
distribution, and the discount rate of 0.1605 per decade corresponds to anannual discount rate of 0.015.
22

Table 4: Parameter calibration based on Pareto utility and Student distribution
AgreementabcdParameter values
τ
0.3
0.3
0.5
0.3
df
10
25
10
10
k
1.5
1.5
1.5
2.0
Policy instruments, beginning of period 0µ00.09100.09100.08880.1192C0424.33424.31413.71438.01
I0
131.46131.47142.08117.73
Capital stock and expectations
K1
179.23179.25189.86165.50µ10.11350.11350.11540.1604H2
1.0413
1.0413
1.04291.0309Probabilities of catastrophe πℓ
πa 5.0E−03 3.3E−03 5.2E−02 5.1E−03
πb 2.3E−05 5.9E−07 1.4E−03 2.6E−05
πc 2.5E−07 5.0E−11 2.8E−05 2.6E−07
Values of statistical subsistence Vℓ = VSSℓ /C0
Va 2.8E+01 2.9E+01 4.8E+00 2.4E+01
Vb 1.3E+04 2.8E+05 3.1E+02 1.1E+04
Vc 1.9E+06 4.2E+09 2.0E+04 1.6E+06

eRobustness
f

g

0.5
25
1.5

0.7
10
1.5

0.5
10
2.0

0.0887413.50142.290.0861400.67155.120.1163427.56128.19190.060.11541.0430202.890.11751.0449175.96
0.16551.0323

5.3E−02
5.5E−04
8.2E−07

1.4E−01
1.2E−02
4.9E−04

5.2E−02
1.5E−03
3.0E−05

4.1E+00
5.3E+02
3.6E+05

2.9E+00
5.2E+01
1.4E+03

4.1E+00
2.6E+02
1.7E+04

Our benchmark is column a in Table 4. The model outputs are withincredible bounds: policy variables at the beginning of period 0 (µ0 , C0 , I0 );stock variables at the beginning of period 1 (K1 and also M1 = 834.42 andH1 = 0.8845); and expectations (E(µ1 ), E(H2 ), and also E(M2 ) = 869.38). If
we consider temperature H2 as a function of ϵ1 we find relatively low volatility in comparison to the confidence intervals proposed by the IPCC (2007,
Chapter 10). The reason for this is twofold. First, the IPCC determines confidence intervals by considering multiple deterministic climate models, not asingle stochastic one as we do. Second and more importantly, the IPCC confidence intervals are based on non-mitigation scenarios, while our model takes
policy eﬀects into account. For both reasons, the volatility in temperaturefound by the IPCC is higher than what we find.

23

7.1.2

Outputs

In addition to the ‘direct’ outputs of our model we also have ‘derived’ outputs, in particular the probability of catastrophe. These derived outputs arefunctions of the direct outputs and they represent important policy variables
on which prior information is available. Hence, they also require agreement.
We propose to define catastrophe as the event C1∗ ≤ C for some given
value C > 0. The probability of catastrophe is then given by π = Pr(C1∗ ≤C). We shall consider three diﬀerent values of C: C a , C b , and C c , corresponding to three levels of catastrophe, labeled A, B, and C. Catastrophe
A occurs when 20% of the world population live in extreme poverty, andcatastrophes B and C occur when 50% and 80% of the world population
live in extreme poverty, respectively. The definitions and priors proposed inthis subsection are based on background material provided in Ikefuji et al.
(2011b).
We must agree on acceptable values for the probability π of catastrophe.
We have studied acceptable risks in various situations, and we conclude thatan acceptable probability for an economy-climate catastrophe in the next
10-year period is in the range 10−5 –10−6 . Given the definition of catastrophe
we propose: πa = 0.1, πb = 0.001, and πc = 0.00001 as reasonable values.
In the benchmark model we find πa = 0.005, πb = 0.00002, and πc =0.0000003, which is much lower than the acceptable values. Given the associated costs, it seems unnatural that policies would be chosen that mitigate
the probability of a global economy-climate catastrophe far beyond acceptable levels. What can we do about this? One possibility is to make the tails
heavier or lighter, that is, to adjust the degrees of freedom. If we set df = 25
then π becomes even smaller. In general, π becomes smaller as the tails
become lighter (df increases), as one would expect. For df = ∞ (the normal distribution) we find πa = 2.3E−03, πb = 5.3E−10, and πc = 1.5E−24.
Interestingly, the policy variables are hardly aﬀected (column b), not evenwhen df = 200 or df = ∞. If we set df = 3, which is the minimum value
where var(ϵ1 ) exists, then πa = 0.008, a little higher than for df = 10, but
not enough. So, adjusting the degrees of freedom hardly changes the results.
Perhaps the fact that the heaviness of the tail (degrees of freedom) has
little eﬀect on the optimal policy is caused by the Pareto utility function.
Maybe this function does not distinguish well between diﬀerent tails? Infact, this is not so. It follows from Figure 1 (and Section 6) that τ has much
more impact than df. Hence the Pareto function does distinguish between
diﬀerent tails.
Perhaps we should then adjust the value of τ . In our benchmark we set
τ = 0.3 as a reasonable starting point. We could revise τ upwards. We
24argued in Section 3 and Figure 1 that τ = 0.7 is an upper bound to thevolatility. Let us therefore consider the case τ = 0.5. A larger value of τmeans more volatility and hence one would expect less consumption and moreinvestment. This is indeed what happens (column c). Also, the probabilities
are aﬀected and are now much closer to our prior ideas.
We can also adjust the curvature k (and λ). If k increases, then agents
become less risk-averse and, as expected, there is more consumption and lessinvestment (column d ). The probabilities are not much diﬀerent from ourbenchmark in a, but the values of µ0 and µ1 are very high and the capitalstock accumulation rate is only 1.9% per year, which is too low.
Finally, we could adjust the discount rate ρ. This is an important issue(see, for example, Gollier, 2002, 2008, and the references therein), with possibly significant (yet not ‘discontinuous’) impact on the optimal policies. It
is, however, beyond the scope of this study.
Based on these comparisons it seems that policy c should be recommended. There is, however, one other derived output which is often discussed, namely the value of statistical life. If we agree on the definition ofcatastrophe, then we can also define the ‘value of a statistical subsistence’(VSS) as the amount of consumption in period 0 that the government iswilling to trade oﬀ in order to change the probability of catastrophe; see
Ikefuji et al. (2011b) for further details. The VSS is similar to the valueof statistical life (VSL), except that it refers to the condition of just having
enough food to stay alive (more than $1/day) rather than to life. We propose VSSa = C0 , VSSb = 10C0 , and VSSc = 100C0 as reasonable orders ofmagnitude. The VSS (and the VSL) is a diﬃcult concept to measure, andthe VSS priors (Ikefuji et al., 2011b) may be unreliable. As such it should
not carry too much weight as a derived output. Still we notice that the VSSs
of our preferred policy c are much closer to our reasonable values than the
VSSs in columns a, b, and d.

7.2

Robustness

If we believe that column c is the best, then we should do some further robustness checks, starting from column c rather than column a. We have done
extensive robustness checks and some representative results of this analysis
is reported in columns e–g of Table 4; see Ikefuji et al. (2011b) for further
results. If we adjust the degrees of freedom (column e), then not much happens. There is little to choose between columns c and e. The optimal policy(µ∗0 , C0∗ , I0∗ ) is hardly aﬀected, which is a good thing, because it means thatour policy is not too sensitive to changes in the heaviness of the tail (degrees
of freedom). In column f we consider τ = 0.7. Here the probabilities of25catastrophe seem to be too large. For example, we have πc = 0.0005 and it isdoubtful if the government would find this acceptable. The choice of volatility τ does, however, aﬀect the policy, and hence is important. In column g
we adjust the curvature of the Pareto utility function. The probabilities are
hardly aﬀected but there will be more consumption, less investment, and inparticular more (perhaps too much) abatement. On the basis of these andother robustness checks we conclude that policy c is robust against smallchanges in the underlying assumptions and parameter values.

8

Concluding remarks

Our strategy in this paper has been to specify a stochastic economy-climate
model, building on Nordhaus’ deterministic economy-climate model while allowing for Weitzman-type stochasticity. We show that, under expected power
utility, the model is fragile with respect to distributional assumptions. Basedon general results regarding the relationship between the richness of the classof utility functions and the generality of the permitted distributional assumptions, we restrict ourselves to utility functions that are compatible with our
distributional assumptions. Thus we avoid the unacceptable conclusion thatsociety should sacrifice an unlimited amount of consumption to reduce theprobability of catastrophic climate change by even a small amount. Afterreaching agreement on the model parameters, a sensitivity analysis showsthat our completed model and the resulting optimal policies are quite robust
and sensibly sensitive.
The fragility of the model under expected power utility to heavy-tailed
distributional assumptions is not unexpected. Weitzman (2009) summarizes
this fragility and the perceived non-existence of a robust solution in a ‘dismal theorem’. While we agree with Weitzman that incompatible pairs ofutility functions and distribution functions exist, our objections to the dismal theorem are twofold. First, the result is implied by using an incompatible (invalid) model specification. A key ingredient in Weitzman’s model isthe power utility function. This popular utility function is characterized byconstant relative risk aversion (CRRA). The assumption of CRRA, hence
RRA(0) > 0, is not appropriate when dealing with extremely low levels of
consumption, and it is exactly the behavior at these low consumption levels
that leads to the dismal theorem. As we have demonstrated formally in Section 4 and numerically in Section 6, Weitzman’s result is avoided when theeconomic model (utility function) is compatible with the statistical model(heavy tails).
Second, more eﬀort can be made to know an input parameter that is ‘hard26

to know’, and we have described a (stylized) learning-and-agreement procedure for precisely this purpose in Section 7. Although it is diﬃcult to state
upper and lower bounds for the ‘VSL-like’ input parameter of Weitzman,
we can still obtain reasonable constraints on diﬃcult-to-know parameters ofinterest indirectly. The economic model translates the parameter of interest
into output variables with an easier interpretation (such as the optimal policies and the probability of catastrophe). Bounds on these output variables,
together with the economic model, imply bounds on the input parameter ofinterest.
Much of the analysis in our paper is not limited to extreme climate change.
A similar analysis could apply in other policy making settings involving catastrophic risks, such as the development of new financial incentive schemes tomitigate the risk of extreme systemic failures and resulting financial economic
crises, or policies concerning medical risks (pandemic flu and vaccination
risks).
Let us finally admit four limitations of our paper, and indicate possible generalizations. First, in Section 6 we have focussed our attention on
bounded utility functions, so as to avoid having to restrict distributional
assumptions. In general, one could assume more structure on stochasticity(yet still allow for heavy tails) and broaden the constraints on utility. In
particular, unbounded utility (such as HARA with 0 < α ≤ 1) is also permitted under additional assumptions on stochasticity. Second, for simplicity
and clarity of presentation, we have restricted our numerical analysis to only
two periods. Conceptually, much of our analysis will remain intact when
considering more than two periods. Third, to account for the fact that thepolicy maker has the double objective of maximizing current consumption,
while also leaving a reasonable economy for the next policy maker, we haveused scrap values in our analysis. We ignore, however, stochasticity in the
scrap value function after the second period. The development of a numerically tractable economy-climate model with multi-period stochasticity andstochasticity in scrap values after the last period is a subject for further
research. Finally, the equations making up our stochastic economy-climate
model are of a simple and stylized nature, and each one of them, including thespecification of stochasticity, leaves room for generalizations and extensions.

27

AppendicesA

Kuhn-Tucker conditions under positive investment

Consider the economy-climate model of Section 2 in a two-period set-up. Let
U be a general well-behaved utility function and let S (1) and S (2) be general
well-behaved scrap value functions. At the beginning of period 1 our welfare
function, conditional on (C0 , µ0 , ϵ1 ), is
W = L1 U (C1 /L1 ) + ν1 S (1) (K2 ) − ν2 S (2) (M2 ).
We have four constraints: C1 ≥ 0, I1 ≥ 0, µ1 ≥ 0, and µ1 ≤ 1, but only
two of these can be binding as we shall see. Hence, we define the Lagrangian
L = L(C1 , µ1 ) as
L = L1 U (C1 /L1 ) + ν1 S (1) (K2 ) − ν2 S (2) (M2 ) + κ1 I1 + κ2 (1 − µ1 ),and we findand∂L
= U ′ (C1 /L1 ) − (ν1 g1 + κ1 )
∂C1
(
)
∂L
= −(ν1 g1 + κ1 )ψ1 θµθ−1
1 d1 + ν2 g2 σ1 Y1 − κ2 ,
∂µ1

where
g1 = g1 (C1 , µ1 ) =∂S (1) (K2 ),∂K2

g2 = g2 (µ1 ) =∂S (2) (M2 ).
∂M2This leads to the Kuhn-Tucker conditions:κ1 = U ′ (C1 /L1 ) − ν1 g1 ≥ 0,
I1 = (1 − ψ1 µθ1 )d1 Y1 − C1 ≥ 0,and)(
κ2 = −U ′ (C1 /L1 )ψ1 θµθ−1
1 d1 + ν2 g2 σ1 Y1 ≥ 0,
µ1 ≤ 1,
together with the slackness conditions κ1 I1 = 0 and κ2 (1 − µ1 ) = 0.
Under the assumption that I1 > 0 we have κ1 = 0 and we distinguishbetween two cases, as follows.
28

Case (1): κ2 > 0. We have µ1 = 1 and g2 = g2 (1), and we solve two equations
in two unknowns:
U ′ (C1 /L1 ) = ν1 g1 ,g1 = g1 (C1 , 1),

under the restrictions:C1
ν2 g2 σ1
≤ d1 <.
(1 − ψ1 )Y1
ν1 g1 ψ1 θCase (2): κ2 = 0. We solve four equations in four unknowns:
U ′ (C1 /L1 ) = ν1 g1 ,µθ−1
1 d1 =

g1 = g1 (C1 , µ1 ),ν2 g2 σ1,
ν1 g1 ψ1 θ

g2 = g2 (µ1 ),under the restrictions:C1 ≤ (1 − ψ1 µθ1 )d1 Y1 ,µ1 ≤ 1.

The following two points are worth noting. First, we see that the restrictionsµ1 ≥ 0 and C1 ≥ 0 are automatically satisfied, so that they do not need to
be imposed. Second, we see that U ′ (C1 /L1 ) = ν1 g1 in both cases. This fact
will be used in the proof of Proposition 6.1.

B

Proofs of the propositions

Proof of Proposition 4.1: See Menger (1934, p. 468) in the context of St.
Petersburg-type lotteries, and also Arrow (1974) and Gilboa (2009, pp. 108109). Menger (implicitly) assumes boundedness from below and demonstrates that boundedness from above should hold, and it is straightforward
to generalize his result to an a priori unrestricted setting.
Proof of Proposition 4.2: Let α∗ > 0. The EU maximizer is then morerisk-averse in the sense of Arrow-Pratt than an agent with power (CRRA)
utility of index α∗ . It follows from (4) thatU ′′ (C1∗ )P ′ (C1∗ )== −ARA(C1∗ ).
P (C1∗ )U ′ (C1∗ )29

Since ARA(x) = RRA(x)/x ≥ α∗ /x, we then have
( ∫
)
(∫
)
1
1
1
1
E(P ) =
E exp −
d log P (x) =E exp
ARA(x) dx
1+ρ
1+ρ
C1∗
C1∗
(∫
)
∫
1
1
exp
(α∗ /x) dx dF (ϵ1 )≥1 + ρ C1∗ ≤1
C1∗
∫
∗ ∫
B1−α
1
∗
∗ −α∗
dF (ϵ1 ) ≥=(C1 )e−τ α ϵ1 dF (ϵ1 ) = ∞,1 + ρ C1∗ ≤1
1 + ρ C1∗ ≤1withe−τ /2 Y1
B1 =,
1 + ξH12
using (5) and the fact that ϵ1 is heavy-tailed to the left. This proves part (a).
Intuitively, if agent 1 is more risk-averse in the sense of Arrow-Pratt thanagent 2, and if it is optimal to postpone all consumption for agent 2, thenthis will also be optimal for agent 1.
Next let α∗ = 0 and β ∗ < ∞. The EU maximizer is then less risk-aversein the sense of Arrow-Pratt than an agent with exponential (CARA) utilityof index β ∗ . Since α∗ = 0, we have 0 ≤ ARA(x) ≤ β ∗ and hence∫∫
E(P ) =P dF (ϵ1 ) +P dF (ϵ1 )C1∗ ≤1C1∗ >1
(∫
)
∫
1
1
≤
exp
β ∗ dx dF (ϵ1 )∗1 + ρ C1∗ ≤1
C1( ∫ C∗
)
∫
1
1
+
exp −
ARA(x) dx dF (ϵ1 )1 + ρ C1∗ >1
12∗eβ Pr(C1∗ ≤ 1) + Pr(C1∗ > 1)
≤
< ∞.
1+ρ
Proof
of Proposition 4.3: To prove the ‘only if’ part, we assume that∫γ
ARA(x)
dx is infinite for every γ > 0, and then show that there exist
0(S, A, P) and ϵ1 defined on it such that E(P ) = ∞. We note that β ∗ = ∞.
Define a function g : (0, 1] → [1, ∞) by(∫ 1)
g(y) = exp
ARA(x) dx .
y

Then,
1
E(P ) ≥
1+ρ∫C1∗ ≤1g(min(C1∗ , 1)) dF (ϵ1 ).

30

∗Recall from (5) that C1∗ ≤ B1 eτ ϵ1 , and let ϵ∗1 be such that B1 eτ ϵ1 = 1, so that∗
0 < B1 eτ ϵ1 ≤ 1 if and only if ϵ1 ≤ ϵ∗1 . Define u : (−∞, ∞) → [0, ∞) by{g(B1 eτ ϵ1 ) − 1 if ϵ1 ≤ ϵ∗1 ,u(ϵ1 ) =0
if ϵ1 > ϵ∗1 .
Since ARA(1) > 0, g is monotonically decreasing and we obtain∫∫∗g(B1 eτ ϵ1 ) dF (ϵ1 )g(min(C1 , 1)) dF (ϵ1 ) ≥
∗
∗
C1 ≤1
ϵ1 ≤ϵ1∫(u + 1) dF (ϵ1 ) = E(u) + Pr(ϵ1 ≤ ϵ∗1 ).
=ϵ1 ≤ϵ∗1

Strict monotonicity of g implies its invertibility. Hence we can choose u to
be any non-negative random variable whose expectation does not exist (forexample, the absolute value of a Cauchy distribution), and then define ϵ1
through B1 eτ ϵ1 = g −1 (u + 1). With such ∫a choice of ϵ1 we have E(P ) = ∞.
γTo prove the ‘if’-part we assume that 0 ARA(x) dx is finite. This implies∫1
that 0 ARA(x) dx is finite, so that
(∫
)
∫
1
1
exp
ARA(x) dx dF (ϵ1 )E(P ) =
1 + ρ C1∗ ≤1
C1∗)( ∫ C∗∫1
1
ARA(x) dx dF (ϵ1 )exp −+1 + ρ C1∗ >1
1)(∫ 1
Pr(C1∗ ≤ 1)
Pr(C1∗ > 1)≤
ARA(x) dx +
exp
< ∞,
1+ρ
1+ρ
0
using the fact that α∗ = RRA(0) = 0.
Proof of Proposition 6.1: We shall prove the proposition both for the
linear scrap and the non-linear scrap case. In both cases the inequality
constraints (1) are imposed. Sincee−τ /2 Y1
B1 =,
1 + ξH122

d1 Y1 = B1 e
we obtainτ ϵ1,

C1∗ ≤ C1∗ + I1∗ = (1 − ω1∗ )d1 Y1 ≤ B1 eτ ϵ1 ,I1∗ ≤ C1∗ + I1∗ ≤ B1 eτ ϵ1 ,(1 − δ)K1 ≤ K2∗ ≤ (1 − δ)K1 + B1 eτ ϵ1 ,
31

(9)

and

M2∗ ≤ (1 − ϕ)M1 + σ1 Y1 .

We distinguish between three cases.
Linear scrap under normality. Linear scrap implies that S (1) (K2 ) = K2 andS (2) (M2 ) = M2 . Since E(eτ ϵ1 ) exists under normality, it follows that C1∗ , I1∗ ,K2∗ , and M2∗ all have finite expectations, and therefore that E(W ∗ ) exists if
and only E(1/C1∗ ) exists. For notational convenience we do not distinguishbetween the random variable ϵ1 and its realization. With this slight abuse ofnotation, we write
∫
∫
∫ ∞
∗
∗
∗
(1/C1∗ ) dF (ϵ1 )(1/C1 ) dF (ϵ1 ) +(1/C1 ) dF (ϵ1 ) =E(1/C1 ) =−∞I1∗ =0∫I1∗ >0∫e−τ ϵ1dF (ϵ1 ) +(1/C1∗ ) dF (ϵ1 )
∗
∗
∗
1
−
ω
I1 =0
I1 >0
1
∫
1
(1/C1∗ ) dF (ϵ1 ).
E(e−τ ϵ1 ) +≤∗(1 − ψ1 )B1
I1 >0∫
Since E(e−τ ϵ1 ) is finite, it suﬃces to show that I ∗ >0 (1/C1∗ ) dF (ϵ1 ) is finite.
1
Now, it follows from Appendix A that, under the assumption that I1∗ > 0,
U ′ (C1∗ /L1 ) = L21 /C1∗ 2 = ν1 g1∗ = ν1 , because g1∗ = 1. Hence,= (1/B1 )

∫

1/2
1/2
ν1
ν1
∗
∗
Pr(I1 > 0) ≤< ∞.
(1/C1 ) dF (ϵ1 ) =L1
L1
I1∗ >0Nonlinear scrap under normality. Nonlinear scrap implies that
( )−p
(
)q
K0 K2
M0 M2
(2)
(1)
S (K2 ) = −,S (M2 ) =p K0
qM0
where p > 0 and q > 1. Since(K2∗ )−p ≤ ((1 − δ)K1 )−pand(M2∗ )q ≤ ((1 − ϕ)M1 + σ1 Y1 )q ,∗we see that E(W ∗ ) exists if and
1 ) exists. As in the linear scrap∫ only E(1/C∗
case, it suﬃces to show that I ∗ >0 (1/C1 ) dF (ϵ1 ) is finite. Since
1∂S (1) (K2 )g1 = g1 (K2 ) =
=
∂K2
32

(

K0
K2

)p+1
,

it follows from Appendix A that, under the assumption that I1∗ > 0,
(
)p+1
(
)p+1
K0
K0
′
∗
2
∗2
∗
,
U (C1 /L1 ) = L1 /C1 = ν1 g1 = ν1
≤ ν1
K2∗
(1 − δ)K1
and hence that∫)(p+1)/2
1/2 (
ν1
K0
∗
(1/C1 ) dF (ϵ1 ) ≤< ∞.
L1 (1 − δ)K1
I1∗ >0Student distribution. From (9) we have 1/C1∗ ≥ e−τ ϵ1 /B1 . Under a Student
distribution, the right-hand side has no finite expectation, and hence theleft-hand side has no finite expectation either. In the non-linear scrap case,this is suﬃcient to prove the non-existence of E(W ∗ ) because S (1) (K2∗ ) andS (2) (M2∗ ) are both bounded. In the linear scrap case, M2∗ is bounded, but
K2∗ is not. Now, since
C1∗ ≤ B1 eτ ϵ1 ,K2∗ ≤ (1 − δ)K1 + B1 eτ ϵ1 ,we obtain
L1 (1 − L1 /C1∗ ) + ν1 K2∗ ≤ L1 − (L21 /B1 ) e−τ ϵ1 + ν1 (1 − δ)K1 + ν1 B1 eτ ϵ1 ≡ G(ϵ1 ).
Since G is monotonically increasing from −∞ to +∞, there exists a uniqueϵ∗1 defined by G(ϵ∗1 ) = 0. Hence, G(ϵ1 ) ≤ 0 for all ϵ1 ≤ ϵ∗1 and∫∗∗E |(L1 (1 − L1 /C1 ) + ν1 K2 )| ≥|G(ϵ1 )| dF (ϵ1 )ϵ1 ≤ϵ∗1∫∗
2≥ −L1 − ν1 (1 − δ)K1 + (L1 /B1 )e−τ ϵ1 dF (ϵ1 ) − ν1 B1 eτ ϵ1 = ∞.
ϵ1 ≤ϵ∗1

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