DBJ Research Center on Global Warming
Discussion Paper Series No. 56 (2/2016)

Properties of the Social Discount Rateand Negative Intertemporal Externality
in the Utility or Production Function

Masayuki Otaki

The aim of this discussion paper series is to stimulate discussion andcomment from academics and policymakers through a preliminary draft
form. This paper is included by permission of the author(s) as a tentative
material before publication in peer-reviewed journals or books. Copyrightand all related rights are maintained by the author(s). Views expressed inthis paper are those of the author(s) and do not reflect the views of theResearch Institute of Capital Formation or Development Bank of Japan.

2Properties of the Social Discount Rate and Negative Intertemporal Externality in theUtility or Production Function
Masayuki Otaki1
Institute of Social Sciences, the University of Tokyo
Abstract
This study examines the role of the social discount rate, as applied to the efficient
allocation of carbon dioxide (CO2) emissions, which provokes serious intertemporal
negative externalities such as global warming. Two approaches exist in introducing thenegative intertemporal externality brought about by such emissions into economictheory. One is the utility-function approach, and the other is the production-function
approach. The former is well-suited to the case in which climate change directly
threatens the living standard of human, for example, through a drastic degradation of
the environment. The latter approach becomes relevant to case in which global warming
is expected to lead to productivity stagnation particular to the additional costs ofadaptation.
This study shows that both approaches achieve the most efficient allocation in thevicinity of stationary state when a social planner applies a social discount rate close tozero, because the marginal substitution rate of carbon emissions to consumption iscorrectly reflected in the relative price. In other words, a positive social discount rate
underestimates the negative capitalized value of emissions and hence prompts excess
consumption and emissions, regardless of which of the two approaches is chosen.

Keywords: Utility function with a negative externality; Production function with anegative externality; Social discount rate; Efficiency in emissions allocation
JEL Classifications: D61; ;D62; Q30; Q51

The author is grateful to Morio Kuninori for incisive and intensive discussions.
Possible remaining errors are, needless to say, attributed to the author himself. This
study is subsidized by MEXT/JSPS KAKENHI Grant Number 60183761.
131. Introduction
Global warming can be regarded as a type of intertemporal congestion problem, which
originates from the abuse of our non-renewable atmosphere. Excess emissions of CO2,
provoked by unlimited combustion of fossil fuels are a crucial factor in global warming.
Theoretically speaking, this can be seen as an application of the club goods theory asfollows. The deterioration in the quality of the atmosphere due to excess emissions
decreases the service that the atmosphere can provide us. For each individual emitter
the CO2 contribution is negligible, and congestion is as entirely external for them.
However, as the foregoing discussion suggests, this congestion phenomenon is internal
for the overall global economy. It is particularly serious and complicated is to resolve
owing to its dynamic nature. With continued emissions, the concentration of CO2 in theatmosphere gradually increases over time, leading to more severe global warming.
Accordingly, determining methods of evaluating future damage becomes vital, making
the issue of the social discount rate highly significant.
In addition, in a market economy, the social discount rate is closely related to carbontax. This is because the optimal carbon price (tax) should be equalized with thecapitalized value of carbon by the social discount rate. One will find how dear thecarbon price (tax rate) is and sensitive to the social discount rate.
There is an important distinction over which functions is more such seriously affectedby this negative externality: the utility function or the production function. One canargue that such distinction is meaningless and that the negative externality affects bothfunctions. Nevertheless, there are two schools of thought in theoretical environmental
economics. One is the production function approach. The typical examples are Nordhaus(2014) and Kuninori and Otaki (2016). The other is the utility function approach (e.g.,
Uzawa (2009), Otaki (2013, 2015)). It seems an acute issue to analyze whether theimplications concerning the optimal social discount rate are dependent on the approachchosen.
This study is an attempt to clarify the following two theoretical issues. One is toexplore the property of optimal social discount rate if it exists. The other is to consider
whether such properties are affected by the choice of the foregoing two models.
The rest of the study is organized as follows: Chapter 2 constructs an optimal growthmodel in which the intertemporal negative externality originating from excess CO2
emissions affects the utility function. The properties of optimal social discount rate andcarbon tax are also considered. Chapter 3 transforms the model with the negative
externality in the production function, and compares with the results obtained in theformer section. Section 4 provides some brief concluding remarks.
42. Utility-Function Approach Model
Utility function approaches assumes that the negative intertemporal externality
originating from excessive CO2 emissions directly affects people’s utility. It is assumed
that a strictly concave instantaneous utility function, ut , is represented byu  t   u  ct ,  Et  e t , u1 , u2  0 ,(1)where ct , Et denote consumption and accumulated emissions, respectively.

 is thesocial discount rate applied to emissions. ui is the partial derivative with respect to thei -th argument of the function.
It is assumed that emissions are partially absorbed mainly by oceans and forests, andalso increases proportionately with consumption. Accordingly, the carbon cycle can be
described by the following differential equation. That is,cEt   Et  t ,(2)

where  is the absorption ratio of CO2, and  denotes the efficiency of production
measured by the amount of goods produced by per emitted CO2 tonnage.
The maximization problem can be formatted asmax  u  t  dt ,ct0subject to (2).

(3)

The corresponding Hamiltonian, H tU , of this problem is defined asc
(4)
H tU  u  ct ,  Et  e t  tU     Et   t  .
It should be noted that the state variable, Et , is interchanged with  Et . The necessary
and sufficient conditions of this problem are Equation (2) andu1 tU U,   tU e t , ttU      tU  u2 ,(5)lim tU Et e  t  0.

t The optimal paths are illustrated by bold arrows in Figure 1. In a situation with excess
emissions situation, which corresponds to the lower arrow directing to north-east,
optimal consumption is monotonously decreasing and converges to a zero-growtheconomy unless some technological progress, which absorbs emissions, is developed.
Next, we analyze the properties of the optimal social discount rate in the vicinity ofthe stationary state. Combining the above two equations in (5) and letting   0 , oneU5

obtains the following important relationship between optimal consumption andemissions. That is,u11.
u2     (6)The left-hand side of Equation (6) corresponds to the tangency of an upward-sloping
indifference curve II on  c, E  plane as illustrated by Figure 2. By Equation (2), E cholds as indicated by Line FF , which represents the feasible allocation in thestationary equilibrium; thus the stationary state is located at the intersection of curve

II and FF . Because the indifference curve I 0 I 0 , which corresponds to zero social
discount rate achieves the best allocation, it is evident that nearly-zero discounting(note that the utility integral diverges when zero-discounting is strictly applied)
improves economic welfare in the vicinity of the stationary state. To summarize,
Theorem 1

Welfare is maximized when the social discount rate is preferable as low as possible inthe vicinity of stationary state in the utility function approach.
Moreover, additional important information exists in Equation (6). Let a market
economy be considered instead of a planned economy. Whenever perfect competition
prevails in goods market, the tangency of the indifference curve is equal to the relative
price of carbon to consumption goods. Hence, Equation (6) represents the effective
carbon price (i.e., tax) in terms of consumption goods. Such a price becomes equal to thecapitalized value of goods discounted by the effective social discount rate,      2. A
society that becomes more tolerant to emissions and more highly discounts future
utilities, whilst possibly not, would imply a lower carbon tax in a market economy.
2

From the equations in (5), the carbon price becomes equal to1
U       tU t .

Meanwhile it is evident from Figure 1 that

Uis strictly positive when an economy isUin the situation of excess emissions compared with the stationary state. Accordingly, theoptimal carbon price in transition process should be always higher than in thestationary state. In other words, the desired carbon price becomes minimal at thestationary state.
6

Thus, one obtains
Theorem 2

A higher social discount rate in a planned economy is equivalent to a lower carbon tax ina market economy.
It should be noted that the Earth’s absorption rate,  , is an important factor for
determining the carbon price. A large value of  permits a lower carbon price because
emitted CO2 does not stay long in the atmosphere. Conversely, if  does not significantly
differ from zero, the solution to global warming depends on human beings’ ethic, asOtaki (2015) proves.
In addition, technological progress is also meaningful in introducing a carbon tax
with less friction. Whenever some emissions-saving technology is developed andprevails, this is represented by heightening parameter  in the model. As indicated inEquation (6), such a progress lowers the relative price of carbon because the production
process produces fewer emissions.

3. Production Function Approach
Under a production function approach, the negative effect of excess emissions isattributed to the production function rather than to the utility function directly. Thisapproach assumes that global warming does not affect the basic living and survival
conditions of human beings, but more indirectly, hinders prosperity via loweringproductivity.
Let the strictly concave production function, F , be denoted asyt  F  kt ,  Et  , F1  0, F2  0 ,(7)where yt is total output of gods, and kt denotes input of goods for production. CO2
emissions originating from anthropologic combustion of fossil fuels lowers productivity;
thus, the second argument of F is positive because of the negative externality toproduction. Then, the differential equation that represents the carbon cycle becomesEt   Et yt  Et 1F  kt ,  Et  .

(8)The optimization problem to be solved ismax  v  ct e t dt , subject to Eqs. (8) and (9),ct

07(9)where v is a strictly concave instantaneous utility function. The corresponding
Hamiltonian, H P , is
1H P  v  F  kt ,  Et   kt  e t  tP     Et   F  kt ,  Et   .
(10)The necessary-sufficient conditions are Equation (8) andv' tP F1,  P  tP e t ,  F1  1 tF2  P  v ' F2 ,  tlim tP Et e t  0.

tP     (11)t The dynamics of this economy are described by two differential equations (8) and (11).
Figure 3 is its phase diagram with the optimal paths shown by bold arrows.
By evaluating Equation (8) and the top two equations in (11) at the stationary state,
one obtains

dE F1  11

.
dcF2
    (12)The left-hand side of Equation (12) is the slope of an isoquant. Figure 4 illustratesstationary state in production function approach model. Curve F F is the feasible set,PP

which corresponds to Equation (8) evaluated at  P  0 . Curves IQ, IQO are isoquantsin  c, E  plane. The same logic as in the case of the utility-function approach can beapplied; thus, the relative price of carbon becomes equal to the tangency of isoquant,1, in the production-function approach. This is because the tangency
    corresponds to the factor price whenever perfect competition prevails. Accordingly,Theorem 3

Regardless of whether one relies on the utility-function or production-function approach,
the optimal carbon in stationary state is equal to1. In addition, the    

relationship between the carbon price and the social discount rate is invariant.
Theorem 3 implies that there is no significant difference in the calculation methods of8the carbon price in the vicinity of a stationary equilibrium. In this sense, whether one
adopts a utility-function or production-function approach is immaterial.

4. Concluding Remarks
This study considered the welfare implications of the social discount rate and calculated
the optimal carbon price using an optimal growth model with an intertemporal negativeexternality. In addition, it examined whether results differ when the negative
externality exists in the utility or the production function. The results obtained are as
follows:
First, the optimal carbon price in the vicinity of stationary state is independent of theorigin of the negative externality: be it either the utility function or the production
function. In addition, the adjustment processes toward a stationary state are alike:consumption monotonously decreases and the concentration of CO2 is graduallyreduced.
Second, as the social discount rate approaches zero, resource allocation in the vicinityof stationary equilibrium improves. Nevertheless, applying a lower discount rate raises
the carbon price in a market economy.
Third, the absorption ratio of CO2 by the Earth is also a crucial factor in determining
the carbon price. If this ratio is high, the carbon price is lower because emitted CO2 does
not remain in the atmosphere for a long time. However, it should be noted that theacidification of the oceans may not lead a stable equilibrium in the long run.
Finally, increased carbon-efficiency of production lowers the carbon price. This isbecause such a technological progress enables an economy to produce more goods inexchange for lower emission. As Kuninori and Otaki (2016) shows, it is only when suchtechnological progress continuously accumulates that sustainable growth is feasible.

9References
Kuninori, M., and Otaki, M. (2016). Modified Ramsey Rule, Optimal Carbon Tax andEconomic Growth. Forthcoming in Atmospheric and Climate Sciences.
Nordhaus, W.D. (2014). Dice-2013R Model of as of November 15, 2013.
http://www.econ.yale.edu/~nordhaus/homepage/Web-DICE-2013-April.htm
Accessed at 14th February 2016.
Otaki, M. (2013). The Endogenous Social Discount Rate, Proportional Carbon Tax, andSustainability: Do We Have the Right to Discount Future Generations’ Utility?

Environment Systems Research 2, 1-8.
Otaki, M. (2015). Local Altruism as an Environmental Ethic in CO2 Emissions Control.

Atmospheric and Climate Sciences 5, 433-440.
Uzawa, H. (2009). Economic Theory and Global Warming. Cambridge, UK: Cambridge
University Press.

Appendix: Derivation of Equation (12)From the top and middle equations in (11) evaluated at tP  0 , one obtains

v ' F2 v '  F1  1

 F2 F1

 F1  1       F  2
 F1

 F1  1 F2

        F2 1.
   10

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