Kyoto University,
Graduate School of Economics
Research Project Center Discussion Paper Series

Intergenerationally Equitable Discounting and its
Implications for Climate Policy

Hiroaki Sakamoto

Discussion Paper No. E-09-004

Research Project Center
Graduate School of Economics
Kyoto University
Yoshida-Hommachi, Sakyo-ku
Kyoto City, 606-8501, Japan

March 2010

Intergenerationally equitable discounting
and its implications for climate policy
Hiroaki Sakamoto∗
March, 2010

Abstract
This paper investigates the properties of intergenerationally equitable discounting by using an axiomatically well-founded welfare function which
was originally developed by Epstein (1983), and more recently extended
by Bommier and Zuber (2008). In stead of seeking for the appropriate
value of social rate of time preference, intergenerational equity is incorporated at axiomatic level. I show that the intergenerational-equity-consistent(IE-consistent) discount rate can be higher or lower than the standard notime-preference case without appealing to uncertainty. The relationshipbetween IE-consistent discount rates and risk of world extinction is also
examined with an emphasis on the case where the hazard rate is endogenously determined. With an application to climate change, I show that
endogenous hazard rate can increase the discount rate, which implies relatively less stringent carbon abatement as the optimal climate policy.

Keywords: Intergenerational Equity, Discounting, Climate Change, Uncertainty,
Endogenous Hazard Rate
JEL Classification: D91, Q54, Q56∗

Graduate School of Economics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto,606-8501, JAPAN. Email: hsakamoto@eco.mbox.media.kyoto-u.ac.jp

1

1 Introduction
The first comprehensive study of economic implications of climate change waslaunched by William R. Cline. In his pioneering work, Cline (1992) conducted
the cost-benefit analysis of pollution abatement policies, and Cline (1993) summarized the findings that aggressive abatement is worthwhile even though the
future is much richer, because potential massive damages warrant the costs. On
the other hand, William D. Nordhaus, who has been studying the economics
of climate change for decades, tackled this problem based on his Dynamic Integrated Model of Climate and the Economy (DICE), and reached a quite different
conclusion. The main message of his series of influential studies is that any radical reduction of carbon emissions in the near future should be avoided despite
the serious threats to the global economy posed by climate change. In fact,
the most well-known policy implication given by Nordhaus (1994) is that controls on carbon-intensive economic activities should be put into effect in an increasing but gradual manner, starting several decades from now. More recently,
Stern (2007) tried to revive the spirit of Cline’s approach with insights fromnew scientific findings and concluded that strong and immediate actions to curb
global carbon emissions can be justified based on their framework. Although theframework and assumptions adopted by Stern were severely criticized by many
economists such as Dasgupta (2007), Weitzman (2007), and Nordhaus (2007),
his contribution to the literature rekindled the debate over appropriateness of theexisting approaches.
One of the most controversial issues in the literature is how to incorporate
the idea of intergenerational equity into the policy evaluation framework. Since
the impact of climate change easily stretches across generations, it is inappropriate to evaluate economic impacts of alternative climate policies solely based on
the view point of the present generation. In the literature, such intergenerational
consideration has been boiled down to the choice of one particular parametervalue: social rate of time preference. Following Sidgwick (1907) and Pigou(1920), some economists have been arguing that the interest of future generations should be taken into account by setting the rate of social time preferenceas small as possible (Ramsey, 1928). But others including Nordhaus (2008)
claim that the discount rate must be based on actual behavior in markets rather
than any idealized philosophy about the treatment of future generations.
More generally, not only the rate of social time preference, but also the discount rate as a whole is important in evaluating climate policies. The discount2

rate is defined as∂W (x)
ρt (x) :=∂ct/∂W (x)
− 1,
∂ct+1

(1.1)

where x = (c1 , c2 , c3 , . . .) is a consumption path and W (x) is a welfare function.
The commonly used form of welfare function is the additive and separable one:
)t−1
∞ (
∑
1
W (x) =v(ct ).
(1.2)
1+δ
t=1
Given this specification, the discount rate is computed as
ρt (x) =v 0 (ct )(1 + δ) − 1
v 0 (ct+1 )≈ δ + ηt (x)gt (x)whereηt (x) := −v 00 (ct )ct,
v 0 (ct )gt (x) :=(1.3)
ct+1 − ct.
ctHere, ηt (x) is the elasticity of the marginal utility of consumption. Higher valueof ηt (x) implies less value of additional consumption for rich people. In (1.3),
this parameter is multiplied by gt (x), the consumption growth rate. This meansthat when consumption is expected to grow, the value of future consumption is
discounted at higher rate. Hence, the second term of (1.3) as a whole represents
the aversion to consumption inequality.
The rate of social time preference is captured by the other term, δ, and thisis the parameter that plays a crucial role in evaluating climate policies. Since
the planning horizons of climate policies are typically very long, even a small
difference in δ could make a huge difference. Actually, most of the discrepancy
among the conclusions in the past studies can be explained by the differencein their choice of this parameter. Cline (1992) and Stern (2007) set δ ≈ 0 in
consideration of intergenerational equity, and consequently reached almost thesame conclusions. On the other hand, Nordhaus (1994) chose δ = 0.03 to ensure
consistency between the model and real-world economic data, leading to very
different policy implications. As an illustration, the optimal paths of emission
reduction rate based on a simplified version of DICE model for each case areplotted in Figure 1 below. Obviously, there is a huge difference between thetwo cases. If we stick with the choice of δ = 0, which is usually considerednecessary to meet the requirement of intergenerational equity, seemingly a bit
too ambitious carbon abatement is required compared with the case of δ = 0.03.

31Delta = 0
Delta = 0.03

0.9

Carbon Abatement Rate

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2015

2035

2055

2075

2095

Figure 1: Optimal Abatement Rates for δ = 0 and δ = 0.03
The question is whether there is any way of reconciling those two approaches
without violating the requirement of intergenerational equity. To answer thisquestion, it seems necessary to reconsider assumptions adopted in the standard
argument on discounting. In particular, the functional form of welfare function
should be examined. As is clear from (1.1), discount rate largely depends on thespecification of welfare function. The discussion over discounting in the literature implicitly premises that the welfare function can appropriately be specified
as (1.2), and this is the very reason why the intergenerational equity immediately
implies the choice of δ = 0. If the essence of intergenerational equity is not all
captured by the standard model, other forms of welfare functions could provide
different ways of incorporating concerns for future generations.
The origin of the additive and separable discounted welfare function is traced
back to Samuelson (1937), and its axiomatic foundation was later given by
Koopmans (1960). Koopmans (1960) demonstrated that the set of axioms assumed behind the use of the standard welfare function necessarily implies impatience of the underlying preference. In line with this argument, Diamond (1965)
showed that impartial treatment among generations is not compatible with very
weak axioms imposed on the orderings over infinite utility streams. This typeof unavoidable impatience was also reconfirmed by Epstein (1983) in the context of preference over uncertain consumption streams. More recently, however,Bommier and Zuber (2008) derived a recursive form of welfare function which4satisfies the axiom of anonymity, and showed that preference for catastrophe
avoidance justifies higher discount rates.
In this paper, I investigate the properties of intergenerationally equitable discounting by using an axiomatically well-founded welfare function which was
originally developed by Epstein (1983), and later extended by Bommier andZuber (2008). In stead of choosing δ = 0 in the standard welfare function, I
incorporate intergenerational equity at axiomatic level. This approach makes it
possible to derive the intergenerationally equitable discount rates in a more flexible manner. Moreover, the risk of world extinction, which is positive but usuallytiny, is also incorporated. As Yaari (1965) suggested, exogenous hazard rate ofworld ending can be a basis for higher discounting. I examine this argument
under a fairly general framework of intergenerationally equitable discounting.
In particular, I take a close look at the case where the risk of world extinction is
endogenously determined. This is of particular relevance to climate change because, as is discussed by Weitzman (2009), increasing temperature could causesome climatic catastrophe in the future.
The paper is structured as follows. Section 2 derives a class of welfare functions which represent a ranking over possible outcomes in a way consistent withintergenerational equity. Based on the result of section 2, section 3 provides theintergenerationally equitable discount rates and analyzes their properties. The
difference from the commonly-used discount rate is studied. The relationshipbetween uncertainty and intergenerationally equitable discounting is also investigated. Section 4 applies this welfare function to the evaluation of climate policies and derives some implications for optimal carbon abatement paths. Section
5 concludes.

2 Welfare Function with Intergenerational Equity
Consider a society with a sequence of generations or cohorts. Let ct ∈ Rn be thequantity of n-dimensional goods consumed by t-th generation. With X := {x =(c1 , c2 , c3 , . . .) : ct ∈ [0, c̄]n ∀t}, we denote the set of bounded consumption
streams. We restrict our attention to the case where the world ends at some pointin the future and consumption will never occur thereafter. To be more precise,the outcome set in our analysis is given by
X ∗ := {x = (c1 , c2 , c3 , . . .) ∈ X : ct = 0 ∀t ≥ T for some T < ∞}.
It is worth noting that X ∗ includes practically every outcome which could5be realized in the real world. Since the end-time T can be arbitrarily large, consumption can be positive up until farthest in the future one could think of. The
assumption of “world end” is introduced here just for the instrumental purposeof excluding positive consumption in the infinite future. Or perhaps we could
reasonably assume the world actually ends in some very distant future. As isindicated by Sackmann et al. (1993), the Earth is likely to have a dire consequence in the next several billion years as the Sun gradually goes through its
evolutionary life-cycle.
Also notice that the introduction of world end is consistent with the assumption of bounded consumption set. If we suppose the world lasts forever, it is
not very reasonable to assume a finite upper bound on consumption set because
the economy can grow without a limit. Once the possibility of world ending
is embraced, however, the infinitely large level of consumption can be readily
ruled out at least from a practical point of view. As long as the upper bound c̄ istaken to be sufficiently large, we could safely say that any realistic consumption
stream would not hit the bound before the world ends.
Let M (X ∗ ) be the set of probability measures on the measurable space(X ∗ , R(X ∗ )), where R(X ∗ ) denotes the Borel σ-algebra of X ∗ . The social preference or ranking % is a binary relation defined on M (X ∗ ). As usual, indifference and strict preference are denoted by ∼ and  respectively. Let px ∈ M (X ∗ )denote the probability measure that assigns probability 1 to an outcome x ∈ X ∗ .
For each c1 ∈ [0, c̄]n and p ∈ M (X ∗ ), we define (c1 , p) as a probability measurewith which the consumption level c1 occurs with certainty for the first generation
while consumption for later generations is realized according to p.
To derive a welfare function which represents the ranking %, we first assume% satisfies the following basic axioms:
Monotonicity: If x > x0 , then px  px0 .
Stationarity: There exists ĉ1 ∈ [0, c̄]n such that for all p, q ∈ M (X ∗ ),(ĉ1 , p) % (ĉ1 , q) if and only if p % q.
Independence: For all c1 , c01 ∈ [0, c̄]n and p, q ∈ M (X ∗ ), (c1 , p) % (c1 , q)
if and only if (c01 , p) % (c01 , q) .
Expected Utility: There exists a continuous function U : X ∗ → R such∫∫that X ∗ U (x)dp ≥ X ∗ U (x)dq if and only if p % q, and U is unique up to
positive affine transformations.

6

These axioms are almost the same as the ones in Epstein (1983). The only difference is that the sensitivity axiom is now replaced by the monotonicity axiom.
This stronger assumption gives a more precise structure to the form of welfare
function.
In order to make sure that every generation is treated equally, we need to introduce some axiom of intergenerational equity. Following Bommier and Zuber(2008), let π : N → N be a finite permutation over the set of natural numbersand Π denote the set of all such permutations. For each x = (c1 , c2 , . . .) ∈ X ∗and π ∈ Π, define x(π) asx(π) := (cπ(1) , cπ(2) , . . .) ∈ X ∗ .
Our axiom of intergenerationl equity is then given byIntergenerational Equity: For any x ∈ X ∗ and π ∈ Π, px ∼ px(π) .
Note that this axiom ensures intergenerational equity in a bit stronger sense thanthe one in Bommier and Zuber (2008). Their anonymity axiom only requires
impartial treatment among those generations who born before the world end. In
our axiom, on the other hand, equal treatment between pre-world-ending generations and post-world-ending generations are required as well.
The following proposition is a straightforward extention of theorem 1 in Epstein (1983):
Proposition 1. The ranking % sasisfies the axioms above if and only if the von
Neuman-Morgenstern utility function U can be expressed in the form
U (x) =

∞
∑
t=1

t−1
∏

v(ct )(1 − Kv(cτ ))∀x ∈ X ∗ ,

(2.1)

τ =1

where v : [0, c̄]n → R+ is an increasing function with v(0) = 0 and K < 1/v(c̄).
Proof. See Appendix A.1.
The functional form (2.1) is basically the same as the one derived by Bommier and Zuber (2008). The corresponding welfare function is now given by∫
W (p) =
U (x)dp∀p ∈ M (X ∗ ).
X∗

Since the concern for future generations is already incorporated at axiomaticlevel, the ranking over alternative outcomes provided by this function meets the7requirement of intergenerational equity. Hence, the discount rate derived fromthis welfare function is always consistent with intergenerational equity. We callit IE-consistent discount rate.
Notice that the parameter K can be positive or negative, and such choice does
not change the fact that the ranking is consistent with intergenerational equity.
This point makes a sharp contrast to the additive and separable form of welfare
function, where impartial treatment among generations automatically implies
δ = 0. Bommier and Zuber (2008) linked K with the aversion to catastrophe.
I will present another interpretation of this new parameter shortly. It should
also be emphasized here that K is bounded above by 1/v(c̄). Bearing in mind
that v(c) is increasing in c and c̄ can be a very large number, this boundedness
implies that K must be very small if it is positive.

3 IE-consistent Discount Rates
We are now ready to derive the IE-consistent discount rate and investigate its
properties. In what follows, we restrict our attention to the case with n = 1 forsimplicity. Also, we assume v 00 (c) < 0 for any c ∈ [0, c̄] and write
UT (x) :=

T
∑
t=1

t−1
∏

v(ct )(1 − Kv(cτ )) ,

τ =1

for each T ∈ N.

3.1 Deterministic Case
We first analyze the case where consumption streams are non-stochastic. In
this case, it will be useful to use ρ0t (x) := ηt (x)gt (x) as a benchmark. Thisis the discount rate which is considered to be intergenerationally equitable forthe standard welfare function. We are interested in if and in what condition the
IE-consistent discount rate can be higher or lower than this benchmark rate.
Before presenting the result, notice that the welfare function for non-stochastic
consumption streams is equivalent to the von Neuman-Morgenstern utility function:
∫
∞
t−1
∑
∏
W (px ) =U (z)dpx =v(ct )(1 − Kv(cτ )) .
(3.1)
X∗

t=1

τ =1

Given this welfare function, a bit tedious computation yields the following result:8

Proposition 2. Given a deterministic consumption path x ∈ X ∗ , the IE-consistent
discount rate ρdet (x) is given byρdet (x) =v 0 (ct ) 1 − Kv(ct+1 )− 1,
v 0 (ct+1 ) 1 − Kv(ct )0and thus, as long as ct < ct+1 , ρdet (x) is higher than ρt (x) if and only if K is
negative.

Proof. See Appendix A.2.
The first thing to be noted here is that higher discount rates are justifiable even when the impartial treatment among generation is required. Hence,
rephrased in the context of standard welfare function, the proposition says that
positive value of δ in equation (1.3) does not necessarily imply the violation ofintergenerational equity. On the other hand, however, negative value of δ is also
justifiable. In that case, the discount rate is even lower than the benchmark rate,
which is usually considered as the lowest discount rate. While the choice ofδ = 0 can also be supported as intergenerationally equitable discounting, suchzero time-preference is not required in general.
This point should be clarified by noting
ρdet (x) ≈ δt (x; K) + ηt (x)gt (x),
where

[

]
1 − Kv(ct+1 )δt (x; K) := log.
1 − Kv(ct )Here, δt (x; K) is a counterpart to the time preference δ in the standard discountrate. This means, in the IE-consistent discount rate, the “time preference” canbe positive or negative and even vary over time. The parameter K plays animportant role, which is illustrated in Figure 2 below. This figure displays the
trajectories of δt (x; K) for different values of K in each case of increasing anddecreasing consumption path.
To obtain some interpretation of K, observe∂ρdev 0 (ct ) v(ct+1 ) − v(ct )
t (x)
=− 0
,
∂K
v (ct+1 ) (1 − Kv(ct ))2and hence

∂ρde
t (x)
Q0
∂K

⇔

9

ct Q ct+1 .

Figure 2: δt (x; K) for different K
When consumption increases over time, higher value of K implies lower valueof the discount rate. If the level of consumption is expected to become lower inthe future, on the other hand, higher value of K means higher value of discountrate. In other words, higher (lower) value of K scales down (scales up) theaversion to consumption inequality. Therefore, one possible interpretation of
K is that it is another parameter which controls the aversion to consumption
inequality among generations.
It should also be noticed that K provides a linkage between discount rates
and the level of consumption enjoyed by present generation. As is clear from
(1.3), the discount rate in the standard model does not depend on ct as long as
both the elasticity of the marginal utility of consumption, ηt , and the growth rateof consumption, gt , are fixed. In the IE-consistent discount rate, however, notonly the growth rate of, but also the current level of consumption matters.
To better illustrate this point, suppose both ηt and gt do not depend on today’s
consumption ct . Then∂δt (x; K)
Kv 0 (ct+1 ) de=
ρ (x),∂ct
1 − Kv(ct+1 ) t
which is negative if and only if K < 0. Thus negative value of K implies thatthe discount rate declines as consumption level rises. In other words, “people
become less envy as they become richer.” When K is positive, on the other hand,

10higher level of today’s consumption implies a higher discount rate for consumption of future generations. This could be rephrased as “people become more
averse to inequality of consumption when they become wealthier.” If the formercase (i.e., K < 0) sounds more reasonable than the latter, the welfare function (3.1) provides a justification for higher discount rate as long as increasing
consumption is expected.

3.2 Uncertain World End
The discussion in the previous subsection suggests the IE-consistent discountrate can be higher than the benchmark rate without appealing to any kind of
uncertainty. This argument alone, however, might not be able to fill the gap
between the cases of δ = 0 and δ = 0.03. As is illustrated in Figure 2,
the term δt (x; K) can not be arbitrary large. In fact, it is bounded above bylog(v(ct+1 )/v(ct )) when the level of consumption grows. Hence, the relationship between uncertainty and the IE-consistent discount rate is still worth investigating. As Yaari (1965) showed and Bommier and Zuber (2008) reiterated,uncertainty provides another basis for higher discount rates. If society is subject
to an exogenous risk of extinction in each period, it is reasonable to discountfuture consumptions even when impartial treatment among generations is required. One of the interesting results in this context is that once extinction risk
is incorporated, the role of K is reversed.
In order to investigate this point in more detail, let f : N → [0, 1] be aprobability density function of extinction date. This extinction risk is assumed
to be exogenous at this stage of the analysis. Then the corresponding sequence{ht }∞
t=1 of hazard rates is constructed recursively by/ T∏−1hT = fT(1 − hs ),s=1∏0where s=1 (1 − hs ) := 1. We assume ht ∈ (0, 1) for all t ∈ N and let H be
the set of all such sequences of hazard rates. When uncertainty exists only in theextinction date, the welfare function is given by
W (p) ==∞∑UT (x)fT

T =1
∞ ∑
T
∑
T =1 t=1v(ct )

t−1∏(1 − Kv(cτ )) hT

τ =1

T
−1
∏

(1 − hs ),

s=1

11

where p ∈ M (X ∗ ) is the probability measure which corresponds to the density f . Our next proposition clarifies the relationship between the IE-consistent
discount rate, extinction risk, and the parameter K:
Proposition 3. Given a consumption path x ∈ X ∗ and a sequence {ht }∞t=1 ∈ H
exof hazard rates of extinction date, the IE-consistent discount rate ρt (x) is givenbyρext (x) =)v 0 (ct ) 1 − Kv(ct+1 ) (
−1
1
+
Φ
(x;
K)
−1
t
v 0 (ct+1 ) 1 − Kv(ct )

(3.2)

where
Φt (x; K) :=

(
) ∏
T
∞
∑
hT 1 − htht
T =t+1

1 − hT{[1 − Kv(cτ )] (1 − hτ )} > 0,

τ =t+1and thus uncertainty of extinction date increases discount rates.
Proof. See Appendix A.3.
The role of uncertain extinction date is all captured by the term Φt (x; K) in(3.2). Since Φt (x; K) is always positive, introduction of uncertainty of this kindincreases the discount rate in general. Hence, even higher value of discount rate
is justifiable compared with the deterministic case studied above. This argument
will be highlighted by considering the case where the hazard rate is constantover time. If ht = h ∈ (0, 1) for all t ∈ N, then Φt (x; K) may be written as
Φt (x; K) =∞∑(1 − h)T −t

T∏(1 − Kv(cτ )),τ =t+1

T =t+1

which is decreasing in h. Thus, as depicted in Figure 3 below, higher constanthazard rate implies higher discount rate.
The parameter K is as important here as in the non-stochastic case and,interestingly, it works in quite a different way. Notice first
de
−1
ρex
t (x) ≈ Φt (x; K) + ρt (x),and recall that ρdet (x) is decreasing in K when the level of consumption is expected to rise. The first term, Φ−1t (x; K), on the other hand, is increasing in K
because Φt (x; K) is decreasing in K. Thus, while negative value of K increases
the discount rate through the deterministic version of the IE-consistent discount12

0Figure 3: Φ−1t (x; K) for different K with h > hrate, it suppress the role of uncertainty in the first term. In particular, if K is
negative and (1 − Kv(cτ ))(1 − hτ ) > 1 for all τ > T for some T > t, thenΦ−1t (x; K) = 0, which means the impact of uncertainty completely disappears.
This could easily happen when the extinction risk is very small and consumption
grows over time.
While proposition 3 clarifies higher discount rates are justifiable when thehazard rate is positive, it is not obvious what would happen when the hazard
rate at one point of time increases while the one at another point decreases. In
particular, it will be of interest to see the impact on the discount rate when a partof extinction risk can be transferred from present generation to future generation.
More formally, given a sequence {ht }∞t=1 ∈ H of hazard rates, define another∞sequence {h̃t }t=1 ∈ H ashτ − γ for τ = th̃τ = hτ + γ for τ = t + 1hotherwise,τfor some γ > 0. Here {h̃t }∞t=1 represents a sequence of hazard rates after a partof risk is transferred from generation t to generation t + 1.
ex
Letting ρext (x) and ρ̃t (x) be the discount rates respectively corresponding∞
to {ht }∞t=1 and {h̃t }t=1 , we have the following proposition:

13

Proposition 4. There exists γ > 0 such that

Moreover, ifexρ̃ext (x) < ρt (x).

(3.3)

1 − ht+1 1 − ht
≥ ht
ht+1
ht

(3.4)

then (3.3) holds for any γ > 0.
Proof. See Appendix A.4.
Proposition 4 says the IE-consistent discount rate should be decreased whenthe extinction risk is transferred to future generations. Put differently, higher extinction risk in the future does not justify higher discount rate when such higher
risk is attributed to some “risk transfer” from present generation. In such a case,consumption for future generation should rather be discounted at lower rate.
Note that the condition (3.4) is not very restrictive. In fact, it is satisfied by awide rage of hazard rates. To be more specific, suppose hτ ≤ 0.5 for τ = t, t+1.
Then
1 − ht+1 1 − ht
≥ 1 > ht .
ht+1
htHence, if the hazard rate is lower than or equal to 0.5 in both periods, the condition is easily met. Since world extinction risk is usually far smaller than halfa chance, we could say any reasonable transfer of extinction risk into the future
decreases the discount rate.
This result has an interesting implication for such problems as climate change.
There exist some ways of reducing carbon concentration in the atmospherewhich seemingly transfer the extinction risk from present generation to future
generations. The carbon capture and sequestration technology is one example.
By capturing emitted carbon dioxide and storing them deep into the ocean, we
will be able to avoid climatic catastrophe in the near term. Such technology,
however, may not be completely reliable, and the vast amount of carbon stored
in the ocean may leak out at some point in the future. The resulting abrupt increase of carbon concentration will be likely to increase the risk of world endingat that point. Although the overall impact of such technology is ambiguous,
our result indicates that lower discount rates should be used if it increases theextinction risk in future.

14

3.3 Endogenous Uncertainty
In the case of climate change, it seems more reasonable to assume the extinction
risk is endogenously determined. With an application to climate change in mind,
I introduce endogeneity of extinction risk as follows. First, it is natural to assume
that if the level Mt of carbon concentration in the atmosphere rises, the hazard
rate of world extinction increases:
ht (x) = h(Mt )with h0 (M ) ∈ (0, +∞).

(3.5)

While the carbon dioxide in the atmosphere decays at rate of ξ ∈ (0, 1), humaninduced carbon emission Et accumulates in the atmosphere:
Mt+1 = (1 − ξ)Mt + κEtwith κ > 0.

(3.6)

Finally, we assume carbon emission is proportional to the level of consumption:
Et = σt ct

with σt > 0,

(3.7)

where σt is an index of emission intensity of consumption. We here assumeσt+1 ≤ σt , meaning that the energy efficiency improves over time. Note that
under these assumptions, both hazard rate ht (x) and probability density ft (x)are determined by the past consumption level (c1 , c2 , . . . , ct−1 ).
Before presenting the main result of this subsection, I provide the following
lemma:
Lemma 1. Suppose a sequence {ht (x)}∞t=1 ∈ H of hazard rates is endogenously
determined by (3.5), (3.6), and (3.7). Then, given a consumption path x ∈ X ∗ ,the IE-consistent discont rate ρent (x) can be written asexρent (x) = ρt (x) + Θt (x),

where

/
∂ft+1 (x) ∂W (x)Θt (x) = Ut+1 (x)
∂ct
∂ct+1
(
)/
∞
∑
∂fT (x)∂fT (x)∂W (x)ex+
UT (x)− (1 + ρt (x))
.
∂ct
∂ct+1
∂ct+1
T =t+2

Proof. See Appendix A.5.

15

Notice that ρext (x) in the lemma is nothing but the IE-consistent discount rate
when hazard rate is exogenous. Hence, we can say the endogeneity of extinction
risk increases the discount rate if Θt (x) > 0 and vice-versa. Since the first termin Θt (x) is positive, the real question is whether the second term can be negative
or not. This may be rephrased as
Υt (x, T ) :=∂fT (x)∂fT (x)− (1 + ρexR 0,t (x))
∂ct
∂ct+1

(3.8)

for each T ≥ t + 2.
Inequality (3.8) can be interpreted as follows. Now that the hazard rates
are endogenously determined by the past consumption history, both ct and ct+1
change the hazard rates h(MT ), and hence fT (x), for T ≥ t + 2. The impacts ofct and ct+1 on h(MT ) are different in general, and such difference determines thesign of Υt (x, T ). If ct increases future hazard rates more than ct+1 , which happens when today’s production technology is more pollution-intensive than theex
one in the future, then Υt (x, T ) is likely to be positive, and thus ρent (x) > ρt (x).
This is consistent with the result of previous subsection. Increase of future hazard rates (not transfer) usually raises the IE-consistent discount rate. Hence,
when today’s consumption increases future hazard rates more than tomorrow’s
consumption does, future consumption will be even more discounted than in thecase of exogenous hazard rate.
Does this mean endogeneity of extinction risk provides yet another justification for higher discount rates? Our next proposition indicates that it is not always
the case. To see this, we restrict our attention to the case where the atmosphericcarbon concentration is stabilized over the planning horizon. Notice that along
a stabilized path with Mt = M for all t, the corresponding hazard rate is also
constant over time at the level of h(M ).
Proposition 5. If
σt
1
< 1,
de
1 + ρt (x) σt+1

(3.9)

then there exists h∗ > 0 such that for any stabilized path with h(M ) < h∗ ,exρent (x) < ρt (x).

Proof. See Appendix A.6.
This proposition says that the IE-consistent discount rate under endogenous
hazard rates can be lower than in the case of exogenous hazard rates. Note that16

the conditions in the proposition may be decomposed into three different factors:
time effect, technology effect, and risk effect. Time effect, which is captured by
the term 1/(1 + ρdet (x)), purely comes from the fact that today’s consumption ispreferred to tomorrow’s consumption even when there is no uncertainty. Recall
that ρdet (x) is the IE-consistent discount rate in the deterministic case. Inequality (3.9) indicates that when the discount rate for non-stochastic consumption
streams is high enough in the first place (i.e., time effect is large), considerationof endogenous risk decreases the discount rate compared with the case of exogenous hazard rates. This effect is governed by the specification of the individual
utility function v and the parameter K.
Technology effect, on the other hand, is represented by the term σt /σt+1 .
Since σt is an index of pollution intensity, the ratio σt /σt+1 represents how fast
alternative clean technologies develop. If cleaner technologies become available
and consumption with fewer emission is possible at t + 1, the emission intensityof ct+1 decreases, which means σt+1 < σt . When σt /σt+1 is relatively large(i.e., technology effect is large), the condition (3.9) is less likely to be satisfied.
If technological development is not so promising, however, the condition is more
likely to be satisfied, and thus the discount rate is more likely to be lower. Thiseffect is determined by social circumstances regarding environmental technologies.
The last factor, risk effect, is naturally represented by the stabilized hazard
rate h(M ). According to the proposition above, when time effect is so large
or technology effect is so small that the condition (3.9) is satisfied, sufficiently
small hazard rate implies lower discount rate. If the stabilized hazard rate is relatively large (i.e., risk effect is large), however, even when time effect is large andtechnology effect is small, endogenous hazard rate can imply higher discountrate. This effect is largely determined by the relationship between the level ofpollution and the risk of climatic catastrophe, relevance of which belongs to thefield of natural science.
The overall effect of endogenous risk depends on the assumptions about thewelfare function (ρde ), the prospect of technological development (σ), and thephysical characteristic of the problem (h). Relative importance among thesefactors can not be determined without conducting quantitative analysis. Hence,
if we are to derive more concrete implications for climate policy, it is necessary
to consult with some climate-economy modeling, which is to be explored in thenext section.

17

4 Implications for Climate Policy
In this section, we further investigate the implications of intergenerationally equitable discounting for climate policy by conducting a numerical simulation.
Our model is a variant of DICE model with tiny risk of world extinction whichis endogenously determined.

4.1 The ModelFirst, technology is specified in the form of Cobb-Douglas production function
Yt =ψt
K  N 1−
1 + dt t twhere¯ t − τ ∗ )2
dt = d(Z(4.1)

where Yt is output, Kt is capital, Nt is population, and Zt is temperature. Notice
we are assuming that increasing temperature has a negative impact on production. Carbon emission Et is produced according to the equation
Et = σt (1 − µt )Yt ,(4.2)

where µt is control rate or abatement rate of the emission. Emitted carbon dioxide is accumulated in the atmosphere through the equation
Mt+1 = (1 − ξ)Mt + κEt .

(4.3)

We simplify the climate module used by Nordhaus (2008) and assume instead
Zt+1 = τ0 + τ1 Zt + τ2 log(Mt )+ τ3 Ft + τ4 log(1 + τ5 Ft )(4.4)

where Ft is aerosol emission, which is exogenously given. The dynamics ofcapital accumulation is governed by
Kt+1 = (1 − υ)Kt + It

(4.5)

where It is investment. The resource constraint is given as
Yt = Ct + It + At

where Aj,t = αt (µt )ζ Yt ,(4.6)

where Ct is consumption and At represents abatement cost of carbon emission.

18

1.0002e-05

1700
Hazard Rate (BAU)
Carbon Concentration (BAU)

1.00018e-05

1550

Hazard Rate

1.00014e-05

1400

1.00012e-05
1250
1.0001e-05
1.00008e-05

1100

Carbon Concentration

1.00016e-05

1.00006e-05
950
1.00004e-05
1.00002e-05

800
2015

2035

2055

2075

2095

Figure 4: Hazard Rate along BAU path
A key feature of our model is the specification of welfare function:
W (x) =∞ ∑T∑v(ct )

T =1 t=1

t−1∏(1 − Kv(cτ )) h(MT )

τ =1

where
v(ct ) = ω0 +T
−1∏

(1 − h(Mt ))

(4.7)

s=1

(ω1 + ct )1−ω2
1 − ω2

with ct = Ct /Nt .

(4.8)

The parameters ω0 , ω1 , and ω2 are introduced to control the value of ηt (x) andmake sure v(0) = 0. Following Fisher and Narain (2003), the hazard rate function is specified as
[
]
2
−1 ,
(4.9)
h(Mt ) = h̄ + (1 − h̄)
1 + e−β(Mt −M1750 )where h̄ is the extinction risk which does not depend on the climatic condition.
The relationship between the hazard rate and the concentration along the Business as Usual (BAU) scenario of DICE model is illustrated in Figure 4.
The optimal carbon abatement path {µt }∞t=1 is defined as a part of the solution to the problem
max W (x)Ct ,µt ,Itsubject to (4.1) through (4.9),

19

1Delta = 0
Delta = 0.03
IE (no risk)
IE (exogenous)
IE (endogenous)

Carbon Abatement Rate (K = 0)

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2015

2035

2055

2075

2095

Figure 5: Optimal Carbon Abatement Paths (K = 0)
given K1 , M1 , and Z1 . The model is calibrated based on Ikefuji et el. (2010),
and is solved with GAMS. The data and calibrated parameter values used in thisanalysis are all listed in Table 1 of Appendix B.

4.2 Result
The result of our simple simulation is well summarized in Figure 5 above. In thisfigure, the optimal abatement paths for each case of deterministic consumption,exogenous hazard rate, and endogenous hazard rate are plotted. The exogenous
hazard rate was chosen such that the probability density in the initial period iscomparable to the case of endogenous one. The parameter K was chosen to be 0
here. As is illustrated in the Figure 6 and Figure 7 of Appendix C, small changes,
either positive or negative, of the value of K do not affect the qualitative natureof the result.
First, in the case of no-risk of world extinction, the optimal abatement rateis the same as the one in the benchmark case with δ = 0. This is just whatis expected from the theoretical analysis in the preceding section. When K =0, the IE-consistent discount rate is exactly the same as the benchmark rate.
Manipulation of K alone does not change the result much. Once the exogenous
risk of world extinction is incorporated, however, the optimal carbon reduction
rate becomes a bit lower and gets closer to the case of δ = 0.03. Again, thisresult is consistent with the analysis in the previous section. Uncertainty over20

extinction date always increases the discount rate.
What is more of interest is the optimal abatement path under endogenous
risk. The figure above reveals that endogeneity of hazard rate can work as adownward driving force for the optimal carbon abatement rate. This is an indication that the IE-consistent discount rate is higher than the one in the case ofexogenous hazard rate. Hence, endogenous risk of climatic catastrophe makes
it even easier to justify higher rates of discounting. Having said that, there still
remains a huge gap between the normative approach with intergenerational equity and the descriptive approach with δ = 0.03. Therefore, the commonly useddiscount rate of δ = 0.03 is not readily justifiable as long as impartial treatment
among generations is required. Although the proposition in section 3.3 cannot
directly be applied to this model, our simulation result implies that time effect issmall, technology effect is large, or risk effect is large in the context of climatechange.
These implications of course depend on several assumptions. The scale ofabatement rate is not completely independent of the choice of various parameters. Our simulation result should at best be considered as a qualitative analysis
in its nature. Moreover, relatively large uncertainty exists in the specification ofsome exogenous trend and functions. The endogenous hazard function h(M ),for instance, does not have rigorous scientific background. To discuss the appropriateness of this functional form is beyond the scope of a brief paper.

5 Conclusion
In this paper, I investigated the properties of intergenerationally equitable discounting by using a welfare function which was originally developed by Epstein(1983), and more recently extended by Bommier and Zuber (2008). In stead ofseeking for the appropriate value of social rate of time preference, intergenerational equity was incorporated at axiomatic level. I showed that the IE-consistent
discount rate can be higher or lower than the standard no-time-preference casewithout appealing to uncertainty. The relationship between IE-consistent discount rates and risk of world extinction was also examined with an emphasis on
the case where the hazard rate is endogenously determined. With an application to climate change, I showed that endogenous hazard rate can increase thediscount rate, which implies relatively less stringent carbon abatement as the
optimal climate policy.

21

A Proofs of Propositions
A.1 Proof of Proposition 1
First, let Ũ : X ∗ → R be a von Neuman-Morgenstern utility function withwhich the expected utility form represents the ranking %. Define U : X ∗ → R
by
U (x) := −Ũ (0) + Ũ (x)∀x ∈ X ∗ ,so that U (0) = 0. Since Ũ is unique up to positive affine transformation, U isanother von Neuman-Morgenstern utility function for the same social ranking.
Next the independence axiom implies∫∫
U (c1 , x)dp ≥U (c1 , x)dq ⇔ (c1 , p) % (c1 , q)
X∗

X∗

⇔
⇔

(0, p) % (0, q)∫∫
U (0, x)dp ≥X∗

U (0, x)dq,

X∗for any p, q ∈ M (X ∗ ) and c1 ∈ [0, c̄]n . This means there must be mappings
a : [0, c̄]n → R and b : [0, c̄]n → R++ such thatU (c1 , x) = a(c1 ) + b(c1 )U (0, x),(A.1)

for any x ∈ X ∗ and c1 ∈ [0, c̄]n . On the other hand, the stationarity axiom
implies∫∫U (0, x)dp ≥U (0, x)dq ⇔ (0, p) % (0, q)
X∗

X∗

⇔

(ĉ1 , p) % (ĉ1 , q)

⇔

p%q
∫
∫
U (x)dp ≥⇔X∗

U (x)dq,X∗

for any p, q ∈ M (X ∗ ), and hence there exist constant numbers â ∈ R andb̂ ∈ R++ such that
U (0, x) = â + b̂U (x),(A.2)
for any x ∈ X ∗ . Define mappings v : [0, c̄]n → R and B : [0, c̄]n → R++ asv(c1 ) := a(c1 ) + âb(c1 )B(c1 ) := b̂b(c1 ).

22

Then (A.1) and (A.2) mean
U (c1 , x) = v(c1 ) + B(c1 )U (x),(A.3)

for any x ∈ X ∗ and c1 ∈ [0, c̄]n . Notice that v(0) = 0 and U (c1 , 0) = v(c1 )because U (0) = 0. Recursive application of equation (A.3) yields
U (x) =∞∑v(ct )t=1

t−1∏

B(cτ ),

(A.4)

τ =1

∏
for each x = (c1 , c2 , . . .) ∈ X ∗ where 0τ =1 B(cτ ) := 1.
Take a pair of non-zero consumption bundles c, c0 ∈ [0, c̄]n \{0} and consider
the following three alternative outcomes z, z 0 , z 00 ∈ X ∗ :z = (c, c0 , 0, 0, . . .)z 0 = (c0 , c, 0, 0, . . .)z 00 = (0, c, c0 , 0, . . .).
Then the axiom of intergenerational equity requires pz ∼ pz0 and pz ∼ pz00 .
Observe first that since both c and c0 are non-zero vectors, the monotonicity
axiom implies v(c) > 0 and v(c0 ) > 0. Hence∫∫pz ∼ pz 0 ⇔U (x)dpz =U (x)dpz0
X∗

X∗

0

⇔

U (z) = U (z )⇔v(c) + B(c)v(c0 ) = v(c0 ) + B(c0 )v(c)1 − B(c0 )1 − B(c)=.
v(c)
v(c0 )⇔

This means there exists a constant K ∈ R such that K = (1 − B(c))/v(c) or
B(c) = 1 − Kv(c),(A.5)

for any c, c0 ∈ [0, c̄]n \ {0}. Notice here that since B(c) > 0 and v(c̄) ≥ v(c) for
any c ∈ [0, c̄],1 − B(c)
1
1
K=
<
≤
,
v(c)
v(c)
v(c̄)
meaning 1/v(c̄) is the upper bound of K.

23

On the other hand, noting v(0) = 0 and U (z) > 0,∫∫pz ∼ pz00 ⇔U (x)dpz =U (x)dpz00
X∗

X∗

00

⇔

U (z) = U (z ) = v(0) + B(0)U (z)⇔B(0) = 1,

and thus
B(0) = 1 − Kv(0).

(A.6)

Finally, apply (A.5) and (A.6) to (A.4) and obtain
U (x) =∞∑v(ct )t=1

t−1∏(1 − Kv(cτ )),τ =1

for each x = (c1 , c2 , . . .) ∈ X ∗ .

A.2 Proof of Proposition 2
Let B(c) := 1 − Kv(c) and note that
k
t−1
∞
∏
∏
∂U (x)B 0 (ct ) ∑0v(ck+1 )B(cτ )= v (ct )B(cτ ) +∂ct
B(ct ) k=t
τ =1
τ =1
}
{ t
k
∞
∏
∑
v 0 (ct ) ∏=v(ck+1 )B(cτ )
B(cτ ) − K
B(ct ) τ =1
τ =1
k=t




k
t
∞


∏
∑
∏
v 0 (ct )=(1 − Kv(ct+1 ))v(ck+1 )B(cτ )
B(cτ ) − K{z}B(ct ) 

|
τ =1
τ =1
k=t+1
= B(ct+1 )
{ t+1
}
∞
k
∑
∏
v 0 (ct ) ∏=B(cτ ) − Kv(ck+1 )B(cτ )
B(ct ) τ =1
τ =1
k=t+1
{ t+1
}
∞
k
∑
∏
v 0 (ct ) B(ct+1 ) v 0 (ct+1 ) ∏= 0
B(cτ ) − Kv(ck+1 )B(cτ )v (ct+1 ) B(ct ) B(ct+1 ) τ =1
τ =1
k=t+1
|
{z
}
∂U (x)=∂ct+1
0v (ct ) B(ct+1 ) ∂U (x)= 0.
v (ct+1 ) B(ct ) ∂ct+1

24

Hence/∂U (x) ∂U (x)ρde−1t (x) =
∂ct
∂ct+1
v 0 (ct ) B(ct+1 )= 0−1v (ct+1 ) B(ct )v 0 (ct ) 1 − Kv(ct+1 )= 0− 1.
v (ct+1 ) 1 − Kv(ct )A.3 Proof of Proposition 3
Notice first



0






t
∂UT (x)  v 0 (ct ) ∏=B(cτ )
∂ct
B(ct ) τ =1v 0 (ct ) B(ct+1 ) ∂UT (x) 0v (ct+1 ) B(ct ) ∂ct+1

for T ≤ t − 1
for T = t

(A.7)

for T ≥ t + 1.

Hence∞
∂W (x) ∑ ∂UT (x)
=
fT
∂ct
∂ct
T =t

= ft

∞
∑
∂Ut (x)∂UT (x)
+
fT
∂ct
∂ct
T =t+1t∞v 0 (ct ) ∏v 0 (ct ) B(ct+1 ) ∑∂UT (x)=ft
B(cτ ) + 0
fT
B(ct ) τ =1
v (ct+1 ) B(ct ) T =t+1
∂ct+1
∞
) ∑
∂UT (x)v 0 (ct ) B(ct+1 ) (
−1
fT
,
= 0
1 + Φt (x; K)v (ct+1 ) B(ct )∂ct+1
T =t+1|{z}∂W (x)
=
∂ct+1

where

∞
∑

∂UT (x)Φt (x; K) :=fT∂ct+1
T =t+125/ 0tv (ct+1 ) ∏ft
B(cτ ).
B(ct+1 ) τ =1

(A.8)

Therefore/∂W (x) ∂W (x)ρex−1t (x) =
∂ct
∂ct+1)
v 0 (ct ) 1 − Kv(ct+1 ) (= 0
1 + Φ−1t (x; K) − 1.
v (ct+1 ) 1 − Kv(ct )For more precise form of Φt (x), apply (A.7) recursively and obtain∂UT (x)v 0 (ct ) B(ct+1 ) ∂UT (x)= 0∂ctv (ct+1 ) B(ct ) ∂ct+1
T
v 0 (ct ) ∏=B(cτ )
B(ct ) τ =1
for T ≥ t + 1. This, together with (A.7) for T = t, means
T∂UT (x)v 0 (ct ) ∏=B(cτ )∂ct
B(ct ) τ =1

∀T ≥ tPlugging this into (A.8) yields/ 0Ttv (ct+1 ) ∏v 0 (ct+1 ) ∏Φt (x; K) =fT
B(cτ )
ft
B(cτ )
B(ct+1 ) τ =1
B(ct+1 ) τ =1
T =t+1
∞
∑

=

∞
T
∑
fT ∏

B(cτ )ft τ =t+1
T =t+1∏ −1∞
T∑
hT Ts=1(1 − hs ) ∏=B(cτ )∏
ht t−1s=1 (1 − hs ) τ =t+1
T =t+1
(
) ∏
T
T
∞
∏
∑
hT 1 − ht(1 − hs )B(cτ )
=
h
1
−
h
t
T
s=t+1
τ =t+1
T =t+1()∞
T∑ hT 1 − ht∏={[1 − Kv(cτ )] (1 − hτ )} ,ht 1 − hT τ =t+1
T =t+1

which is decreasing in K. This means that higher value of K increase the term
(1 + Φ−1t (x; K)).

26

A.4 Proof of Proposition 4Notee t (x; K) =
Φ

∞
∑
h̃T
T =t+1=h̃t(

1 − h̃t
1 − h̃T)

T{}∏
B(cτ )(1 − h̃τ )τ =t+1h̃t+1(1 − h̃t )B(cτ )
h̃t

(
) ∏
∞
T
ht 1 − h̃t 1 − h̃t+1 ∑ hT 1 − ht+
{B(cτ )(1 − hτ )}h̃t 1 − ht 1 − ht+1 T =t+2 ht 1 − hT τ =t+1|{z}=: φ(h)h̃t+1
ht+1(1 − h̃t )B(cτ ) − φ(h)(1 − ht )B(cτ )
ht
h̃t
(
) ∏
∞
T
∑
hT 1 − ht+ φ(h){B(cτ )(1 − hτ )}
h
1
−
h
t
T
τ =t+1
T =t+1|{z}= Φt (x; K)()
ht+1 1 − h̃t+1 h̃t+1= φ(h)Φt (x; K) + 1 −B(ct+1 )(1 − h̃t )h̃t+1 1 − ht+1
h̃t=Since h̃t+1 = ht+1 + γ > ht+1 , the second term is positive for any γ > 0. As forthe first term, observe
ht 1 − h̃t 1 − h̃t+1−1
h̃t 1 − ht 1 − ht+1
ht (1 − h̃t )(1 − h̃t+1 ) − h̃t (1 − ht )(1 − ht+1 )=h̃t (1 − ht )(1 − ht+1 )
}
)
(
{
1
− (1 − ht ) − γ
ht γ (1 − ht+1 )
ht,=h̃t (1 − ht )(1 − ht+1 )φ(h) − 1 =which is positive if{
γ < (1 − ht+1 )}1− (1 − ht ) .
ht

27

(A.9)

Hence there is positive value of γ such that φ(h) > 1. For such γ,
(
)
h
1
−
h̃
h̃t+1
t+1
e t (x; K) = φ(h)Φt (x; K) + 1 − t+1
Φ
B(ct+1 )(1 − h̃t )h̃t+1 1 − ht+1
h̃t
> Φt (x; K),
and thus)v 0 (ct ) 1 − Kv(ct+1 ) (e −1 (x; K) − 11+Φtv 0 (ct+1 ) 1 − Kv(ct ))v 0 (ct ) 1 − Kv(ct+1 ) (< 0
1 + Φ−1t (x; K) − 1v (ct+1 ) 1 − Kv(ct )ρ̃ext (x) == ρext (x),
as desired. Moreover, since h̃t = ht − γ > 0, condition (A.9) is always satisfied
as long as{}1
ht ≤ (1 − ht+1 )− (1 − ht )ht
1 − ht+1 1 − ht⇔
≥ ht .
ht+1
ht

A.5 Proof of Lemma 1
Since ∂fT (x)/∂ct = 0 for all T ≤ t,)
∞ (∂W (x) ∑ ∂UT (x)∂fT (x)=fT (x) + UT
∂ct
∂c
∂ct
t
T =1
=

∞
∑
∂UT (x)
T =t∂ct

fT (x) +∞∑T =t+1UT∂fT (x).
∂ct(A.10)

Thus it is immediate from the proof of proposition 3 that∞∞∑∑
∂UT (x)∂UT (x)fT (x) = (1 + ρex(x))
fT (x)
t
∂c
∂c
t
t+1
T =t
T =t+1( ∞)∞∑ ∂UT (x)∑∂fT (x)= (1 + ρexfT (x) +UTt (x))∂ct+1∂ct+1
T =t+1
T =t+2− (1 + ρext (x))= (1 + ρext (x))∞∑

T =t+2∞∑UT∂fT (x)∂ct+1∂W (x)∂fT (x)UT− (1 + ρext (x))∂ct+1∂ct+1
T =t+228

Putting this back into (A.10) yields∂W (x)∂W (x)= (1 + ρext (x))
∂ct
∂ct+1
∞
∞
∑
∑
∂fT (x)∂fT (x)ex
UT+
UT− (1 + ρt (x))
.
∂ct
∂ct+1
T =t+2
T =t+1
Therefore/∂W (x) ∂W (x)ρen−1t (x) =
∂ct
∂ct+1
= ρex
t (x) + Θt (x),
where

(

Θt (x) =∞∑∞∑∂fT (x)∂fT (x)ex
UT (x)− (1 + ρt (x))
UT (x)∂ct∂ct+1
T =t+1
T =t+2)∂W (x)∂ct+1/∂ft+1 (x) ∂W (x)= Ut+1 (x)
∂ct
∂ct+1
)/
(
∞
∑
∂W (x)∂fT (x)∂fT (x)ex− (1 + ρt (x)).
+
UT (x)
∂ct
∂ct+1
∂ct+1
T =t+2

A.6 Proof of Proposition 5First notice
MT = (1 − ξ)MT −1 + κσT −1 cT −1
T −1= (1 − ξ)

M1 +

T −1
∑

(1 − ξ)T −1−k κσk ck

k=1

and hence∂hT (x)∂MT= h0 (MT )
∂ct
∂ct
0
= h (MT )κσt (1 − ξ)T −t−1 .

29

Thus

)
(
T −1
T −1
∑
∂fT (x) ∏∂hT (x)∂hl (x) hT (x)=−(1 − hs (x))∂ct∂ct∂ct (1 − hl (x))
s=1
l=t+1
(
)
T −1
∑
h0 (MT )(1 − ξ)T −t−1h0 (Ml )(1 − ξ)l−t−1= κσt fT (x)−hT (x)(1 − hl (x))
l=t+1

For a spabilized path with Mt = M for all t, let h := h(M ) and h0 := h0 (M ).
Then
(
)
T −1
∑
∂fT (x)h= h0 κσt (1 − h)T −1 (1 − ξ)T −t−1 −(1 − ξ)l−t−1∂ct(1 − h) l=t+1{()}
h0 κσt (1 − h)T −1hh
T −t−1=ξ+(1 − ξ)
−
.
ξ
1−h
1−h
Hence

∂fT (x)= h0 κσt (1 − ξ)T −t−1 .
h→0
∂ct
lim

(A.11)

Similarly,∂fT (x)h0 κσt+1 (1 − h)T −1
=
∂ct+1
ξ
and thus

{(

h
ξ+
1−h

)
T −t−2(1 − ξ)h−1−h∂fT (x)= h0 κσt+1 (1 − ξ)T −t−2 .
h→0 ∂ct+1
lim}.

(A.12)

Notice that since Φt−1 (x; K) ≥ 0 for any h,)v 0 (ct ) 1 − Kv(ct+1 ) (
−1
1
+
lim
Φ
(x;
K)
−1
t
h→0
v 0 (ct+1 ) 1 − Kv(ct )v 0 (ct ) 1 − Kv(ct+1 )≥ 0−1v (ct+1 ) 1 − Kv(ct )lim ρext (x) =h→0= ρdet (x),(A.13)

which is positive because ρdet (x) > σt /σt+1 − 1 and σt ≥ σt+1 . Also notice
Ut+1 =
≤

t+1
∑
k=1
T
∑
k=1

v(ck )

k−1∏(1 − Kv(cτ ))τ =1v(ck )

k−1∏(1 − Kv(cτ )) = UT

τ =1

30

(A.14)

for T ≥ t + 2. Therefore, if
1−

σt+1(1 + ρdet (x)) < 0,σtthen (A.11), (A.12), (A.13) and (A.14) imply∂W (x)
h→0∂ct+1∂ft+1 (x)= Ut+1 lim
h→0
∂ct
(
)
∞
∑
∂fT (x)∂fT (x)ex− (1 + lim ρt (x)) lim+
UT limh→0 ∂ct+1
h→0
h→0
∂c
t
T =t+2lim Θt (x)= h0 κσt Ut+1[0
+ h κσt 1 −] ∑∞1 σt+1ex(1 + lim ρt (x))(1 − ξ)T −t−1 UT
h→0
1 − ξ σtT =t+2≤ h0 κσt Ut+1
] ∑[∞1 σt+1de0(1 + ρt (x))(1 − ξ)T −t−1 UT+ h κσt 1 −1 − ξ σt|{z
} T =t+2<00≤ h κσt Ut+1[0
+ h κσt 1 −] ∑∞1 σt+1de(1 + ρt (x))(1 − ξ)T −t−1 Ut+1
1 − ξ σtT =t+2()
0h κσt Ut+1
σt+1
=
1−
(1 + ρdet (x))
ξσt< 0,which means there exists h∗ > 0 such that Θt (x) < 0 for all h < h∗ .

31

B Assumptions for Numerical Simulation
B.1 Exogenous Trends
The exogenous trends for population, total factor productivity, carbon intensity
are specified as follows:n= (1 − δ n )gtn ,Nt+1 = (1 + gtn )Nt with gt+1
ψ
ψt+1 = (1 + gtψ )ψt , with gt+1
= (1 − δ ψ )gtψ ,σ= (1 − δ σ )gtσ .
σt+1 = (1 − gtσ )σt with gt+1

The population trend is based on the projection provided by the United Nations
Population Division1 . I directly input the projected population for the first threenperiods and then chose g2025
and δ n such that the model matches the future prediction thereafter. The parameters governing productivity and carbon intensity
are the same as Nordhaus (2008).
The marginal abatement cost is assumed to decline overtime. On this point,
I follow Nordhaus (2008) and specify αt as
αt =

]
σt [
m ν + (1 − g m )t (1 − ν) ,ζwhere ν ∈ (0, 1).

B.2 Data & Parameter Values
The emission data of carbon dioxide was taken from International Energy Annual of US Energy Information Administration2 . Following Ikefuji et al. (2010),
I used sulfur emission as a representative index of aerosols in the atmosphere.
The sulfur emission data is based on the work of Stern (2005)3 . The GDP data ofWorld Economic Outlook Databases of International Monetary Fund4 was used
for the initial output. The data of carbon concentration comes from SCRIPPS
CO2 of Mauna Loa Observatory5 . All of the parameter values used in this analysis is listed in Table 1 below.
1

http://esa.un.org/unpd/wpp2008/index.htm
http://www.eia.doe.gov/environment.html
3
http://www.sterndavidi.com/datasite.html
4
http://www.imf.org/external/index.htm
5
http://scrippsco2.ucsd.edu/data/atmospheric_co2.html
232

Symbol

Value

Description

M1750
M2005
Z2005
K2005
N2025
ψ2005
σ2005
m
n
g2025
ψ
g2005
σ
g2005
gm
δn
δψ
δσ
ν

υ
d¯
τ∗
ξ
κ
τ0
τ1
τ2
τ3
τ4
τ5
α
ζ
ω0
ω1
ω2
h̄
β

596
809
14.482
91
6468
3.189
0.187
0.9
0.07980.092−0.0724
0.05
0.656
0.001
−0.02960.5
0.3
0.1
0.00284
13.71
0.05240.47−4.5610.786
1.206−0.001−0.201
0.024
0.651
2.8
12.5590.01
1.2
5 × 10−6
2.75 × 10−11

Unit

Carbon concentration (1750)
GtC
Carbon concentration (2005)
GtC
oMean temperature (2005)
C
Capital stock (2005)
trill US$
Population (2025)
millionTotal factor productivity (2005)—
Emission-output ratio (2005)—
Parameter in cost function—
Population growth rate—
Growth rate of TFP (2005)—
De-carbonization rate (2005)—
Convergence rate of carbon abatement cost—
Convergence rate of population growth—
Convergence rate of TFP—Convergence rate of de-carbonization (2005)—
Ratio of initial to final abatement cost—
Capital’s share of income—
Capital depreciation rate per decade—
Damage coefficient—oDesirable mean temperature
C
Carbon depreciation rate per decade—
Carbon retention rate per decade—Parameter in temperature equation—
Parameter in temperature equation—
Parameter in temperature equation—
Parameter in temperature equation—
Parameter in temperature equation—
Parameter in temperature equation—
Parameter in cost function—
Parameter in cost function—
Parameter in utility function—
Parameter in utility function—
Parameter in utility function—
Parameter in hazard rate function—
Parameter in hazard rate function—

Table 1: Data & Parameter Values

33

C Results for Different K

1
Delta = 0
Delta = 0.03
IE (no risk)
IE (exogenous)
IE (endogenous)

Carbon Abatement Rate (K > 0)

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2015

2035

2055

2075

2095

Figure 6: Optimal Carbon Abatement Paths (K = 1 × 10−6 )1Delta = 0
Delta = 0.03
IE (no risk)
IE (exogenous)
IE (endogenous)

Carbon Abatement Rate (K < 0)

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2015

2035

2055

2075

2095

Figure 7: Optimal Carbon Abatement Paths (K = −1 × 10−6 )34

References
Bommier, A. and A. Zuber (2008). “Can preference for catastrophe avoidance
reconcile social discounting with intergenerational equity?” Social Choiceand Welfare, 31, 415–434.
Cline, W.R. (1992). The Economics of Global Warming, Institute for International Economics.
——— (1993). “Give greenhouse abatement a fair chance,” Finance and Development, 30, 3–5.
Dasgupta, P. (2007). “Commentary: The Stern Review’s Economics of Climate
Change,” National Institute Economic Review, 199, 4–7.
——— (2008). “Discounting Climate Change,” Journal of Risk and Uncertainty, 37, 141–169.
Diamond, P. (1965). “The Evaluation of Infinite Utility Streams,” Econometrica, 33(1), 170–177.
Epstein, L.G. (1983). “Stationary Cardinal Utility and Optimal Growth under
Uncertainty,” Journal of Economic Theory, 31, 133–152.
Fisher, A.C. and U. Narain (2003). “Global Warming, Endogenous Risk, andIrreversibility,” Environmental and Resource Economics, 25, 395–416.
Ikefuji, M., J.R. Magnus, and H. Sakamoto (2010). “Climate Change, Economic Growth, and Health,” mimeo.
Koopmans, T.C. (1960). “Stationary Ordinal Utility and Impatience,” Econometrica, 28, 287–171.
Nordhaus, W.D. (1994). Managing the Global Commons, The MIT Press.
——— (2007). “The Stern Review on the Economics of Climate Change,”
Journal of Economic Literature, 45, 3, 686–702.
——— (2008). A Question of Balance: Weighting the Options on Global
Warming Policies, Yale University Press.
Pigou, A.C. (1920). The Economics of Welfare, Macmillan.

35

Ramsey, F.P. (1928). “A Mathematical Theory of Saving,” The Economic Journal, 38(152), 548–559.
Sackmann, I., A.I. Boothroyd, and K.E. Kraemer (1993). “Our Sun. III. Present
and Future,” Astrophysical Journal, 418, 457–468.
Samuelson, P. (1937). “A Note on Measurement of Utility,” The Review of
Economic Studies, 4, 155–161.
Sidgwick, H. (1907). The Methods of Ethics, 7th edition, McMillan.
Stern, D.I. (2005). “Global Sulfur Emissions from 1850 to 2000,” Chemosphere, 58, 163–175.
Stern, N. (2007). The Economics of Climate Change, Cambridge University
Press.
Weitzman, M.L. (2007). “A Review of the Stern Review on the Economics of
Climate Change,” Journal of Economic Literature, 45(3), 703–724.
——— (2009). “On Modeling and Interpreting the Economics of Catastrophic
Climate Change,” The Review of Economics and Statistics, 91, 1–19.
Yaari, M.E. (1965). “Uncertain Lifetime, Life Insurance, and the Theory of theConsumer,” The Review of Economic Studies, 32, 137–150.

36