DBJ Discussion Paper Series, No.1006

Sustainability, Endogenous Social Discount Rate,and Proportional Carbon Tax

Masayuki Otaki(Institute of Social Science, University of Tokyo)

January

2011

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Sustainability, Endogenous Social Discount Rate, andProportional Carbon Tax

Abstract
Whether or not the utility of future generations should be discounted is themost serious problem, speciﬁcally when we consider the allocation of intergenerational public bads such as CO2 . This paper analyzes how the social optimum
is attained under the constraint of sustainability proposed by Pezzey (1997). We
ﬁnd the following equivalent relationship between the optimal policies: the condition that utility should not be discounted in social planning is equivalent toadopting the socially optimal carbon tax rate in a decentralized economy.

1Introduction“Sustainability” is used in various contexts. For example, Dasgupta and Heal
(1974), Hartwick (1977), and Solow (1986) deﬁne sustainability as the maintaining of some constant consumption level over time. However, the welfare
economics foundation for such a deﬁnition is not necessarily clear.
This article uses a utilitarian deﬁnition of sustainability proposed by Pezzey(1997). In other words, we adopt the egalitarian view that each generation
can enjoy the same utility level iﬀ an economy is sustainable. Based on thisdeﬁnition, we analyze the optimal tax that is proportionate to the emission of
CO2 . Since generations are not concerned with the utility of the generations
thereafter, and only a part of the CO2 emission is absorbed by sea or expelledfrom the earth, etc., excess emission remains in the future. This is because theproportional carbon tax is desirable to sustain the economy.
By employing the method of Negishi (1960), we can also exactly calculate
how much weight should be allotted to the utility of each generation in the social
welfare function of a planning economy. Namely, the Negishi method makes it
possible to clarify the relationship between the carbon tax rate in the market
economy and the weight of each generation in the social welfare function forintertemporal social planning.
Usually, researchers presume that the utility of future generations can bediscounted. Nevertheless, such discounting possesses are not rigorous in thewelfare economic foundation. When used with our deﬁnition of sustainability,
the Negishi method reveals that conventional discounting in social planning cannot achieve optimal taxation in a market economy, and that if each generationis certiﬁed to enjoy the same utility, its utility should be equally weighed.
This paper is organized as follows. Section 2 constructs a model in which theconsumption/emission decision is diversiﬁed across generations. We also analyze
the properties of the model in the stationary state, and calculate the optimalcarbon tax rate in a market economy and the weight of each generation in the
additive social welfare function for a planning economy. Section 3 contains brief
concluding remarks.

2ModelFor simplicity, we assume the dynamics of CO2 emission as follows:
et = αet−1 + ct ,10 < α < 1,(1)where et is the stock of CO2 in period t, ct is the consumption level, and α
denotes the remaining ratio of CO2 per period. This diﬀerence equation implies
the assumption that one unit of consumption emits unit of CO2 . Note that thissimplifying assumption can be easily relaxed.
Individuals live for one period and their well-behaved identical utility function Ut isUt ≡ u(ct , et ),uc > 0, ue < 0,(2)where each subscript represents the partial derivative on the argument. Individuals’ budget constraint is
et = αet−1 + (1 + θ)ct − τt ,(3)

where θ denotes the proportional carbon tax rate and τt is the transfer from thegovernment.
The government transfers the collected tax to individuals equally. Namely,
the budget constraint of the government isθct = τt .

(4)

In a market economy, each individual maximizes (2) subject to (3). Hence,
the following ﬁrst-order condition should hold:uc (t)+ ue (t) = 0,1+θ∀ t.

(5)

The dynamics of the market economy is fully described by two diﬀerence equations: (1) and (5).
For simplicity, we assume that the economy is initially located at some stationary state (c∗ , e∗ ). Then, it is straightforward from Figure 1 that the optimal
αtax rate θ∗ in the stationary state E ∗ is.
1−α
We now focus on the social planning problem under an egalitarian deﬁnitionof sustainability1 . Namely, we have the following:Deﬁnition 1 An economy is sustainable iﬀu(ct , et ) ≥ U ,∀t(6)holds for some given U .

1 This deﬁnition of sustainability is identical to that in Pezzey (1997), although we do notconsider some contradiction between sustainability and “optimality.”2¿From this deﬁnition, we have the maximization problem of the government
as follows:
max u(c0 , e0 ),ctu(ct , et ) ≥ U

s.t. e−1 = e,

∀ t.

(7)

The corresponding Lagrangian L is
L ≡ u(e0 − αe−1 , e0 ) +T∑]
[λt u(et − αet−1 , et ) − U .

(8)

t=1

The ﬁrst-order condition yields[]
λt uc (t) + ue (t) − αλt+1 uc (t + 1) = 0.

(9)An important property of the stationary state emerges in (9). The stream{ }T, satisﬁes the following diﬀerence equation:of Lagrangian multipliers, λ∗tt=1[[ue (∗) ]λ∗t 1 − [−] = αλ∗t+1uc (∗)⇔λ∗t+1 =ue (∗) ]]uc (∗)· λ∗t ,
α1 − [−λ∗0 = 1, (10)ue (∗)
is the marginal substitution rate between consumption and theuc (∗)
stock of CO2 .
where −

Using the method of Negishi (1960), we can prove that the maximal problem(8) with the initial condition, e−1 = e∗ , is equivalent to the maximization
problem of the following social welfare function:
maxetT∑λ∗t u(et − αet−1 , et ),e−1 = e∗ .

(11)

t=0

It is clear from (10) that utility discounting is permitted only when1−α=1ue (∗)<−.
∗1+θ
uc (∗)

Whenever such planning becomes incentive compatible for every generation,in other words the same allocation is also attained by a market economy, (5)should hold. Consequently, the necessary and suﬃcient condition for permitting
discounting programming isθ∗ > θ,

(12)where θ is the existing carbon tax rate.
Such a steady state is illustrated by point ES in Figure 1. It implies thatutility-discounting social programming yields excess emission of CO2 from theegalitarian viewpoint. To sum up the discussion, we have the following theorem:

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Theorem 1 As long as the economy is sustainable, the weight of each generation’s utility in the social welfare function should be equally allotted. Utility
discounting results in the eﬀective carbon tax rate being lower than the optimum.
Finally, we discuss the case where the time horizon for planning is inﬁnite.
1
From Theorem 1, the equal weight is not conﬁned to unity. As such, we set
Tfor the weight and take the limit T → +∞. Then, we have the correct socialwelfare function for the inﬁnite horizon case:1∑u(ct , et ).
T →+∞ T
t=0
Tlim(13)Thus, the divergence problem for the sum of utility can be avoided even if thesocial discount rate is unity.

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Concluding Remarks

This paper analyzed the theoretical relationship between the social discount rate
in a planning economy and the tax rate of CO2 emission in a market economy.
If the social planner discounts the utility of future generations, it corresponds tolowering the carbon tax rate below the optimum in a market economy. Under
the egalitarian viewpoint of sustainability, it is desirable to pay the correct priceof CO2 , which is
α + α2 + · · · =α.
1−αThis implies that the planner should not discount the utility of any generationin the centralized economy, and that the optimal tax rate must be equal to theαtotal remaining ratio
in a decentralized economy.
1−αReferences[1] Dasgupta Partha and Heal Geoﬀrey (1974) The optimal depletion of exhaustible resources. Review of Economic Studies, Symposium: 3-28.
[2] Hartwick John M. (1977) Intergenerational equity and the investing rents
from exhaustible resources. American Economic Review: 972-974.
[3] Negishi Takashi (1960) Welfare economics and existence of an equilibrium
for a competitive economy. Metroeconomica: 92-97.

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[4] Pezzey John. C. B. (1997) Sustainability constraints versus “optimality”
versus intertemporal concern, and axiom versus data, Land Economics 73:448-466.
[5] Solow Robert M. (1986) On the intergenerational allocation of natural resources. Scandinavian Journal of Economics: 141-149.

5e∗ESu(c∗ , e∗ ) = U 1u(c∗ , e∗ ) = U1+θE∗

11−αOc∗Figure 16