DBJ Research Center on Global Warming
Discussion Paper Series No. 55 (2/2016)

Modified Ramsey Rule,
Optimal Carbon Tax and Economic Growth

Morio Kuninori
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 the
Research Institute of Capital Formation or Development Bank of Japan.

2Modified Ramsey Rule, Optimal Carbon Tax and Economic Growth
Morio Kuninori
Hosei University
kuninori@hosei.ac.jp
Masayuki Otaki
The University of Tokyo
ohtaki@iss.u-tokyo.ac.jp

Abstract
In contrast to the overlapping-generations model, it is allowable to discount the future
utility in a dynasty model without the ethical difficulty related to intergenerational
conflicts. Much precedent research uses Ramsey-type optimal growth theory in order toestimate the social discount rate. However, one must note almost all the formulations
neglect the existence of negative intertemporal externalities. This problem is vital when
one analyzes the global warming problem mainly caused by the excess concentration ofcarbon dioxide (CO2). This is because an adjoining effect of capital accumulation exists
besides the improvement of product capacity, which is reflected in the rate of interest (orequivalently, the marginal productivity of capital). That is, one cannot neglect anegative externality to the future productivity that originates from the excess emissionsof CO2. Accordingly, following the optimal growth theory, the effective social discountrate should be heightened by a proportional carbon tax to suppress future excess
consumption/emissions than in the case of the existing analyses, which exclude such anintertemporal external diseconomy.

Keywords: Modified Ramsey Rule; Social Discount Rate; Optimal Carbon Tax; Negative
Externality by the Emissions of CO2
JEL Classifications: Q54; Q58; D61; D62

3

1.

Introduction

Some elaboration is necessary to estimate the effective social discount rate in a dynastic
social planning problem because such a rate is unobservable. Many preceding studies
rely on Ramsey-type optimal growth theory for the estimation (Ramsey, 1928).
Whenever the rate of time preference exceeds the rate of interest, the optimal
consumption proportionately decreases because current consumption is moreadvantageous than the future. Accordingly, the following well-known formula isobtained:cc  r  ,(1)

where  is the rate of time preference, and r is the (real) rate of interest. andcdenotecthe elasticity of marginal (instantaneous) utility1 and the increase rate of consumption,
respectively.
In discounting the far-distant future events by the Ramsey rule in Equation (1), twoapproaches are noted (Arrow, et al. 1996). One is the descriptive approach, which
typically focuses on the opportunity cost of capital whose level is observable as the rateof return on alternative investments in the market (Nordhaus, 2013). The advocates ofthis approach employ the opportunity cost of capital as the discount rate since theyclaim that investments in reducing climate change must compete with alternative
investments in the market2.
The other approach is dubbed the prescriptive approach, where the advocates
maintain the view that the market interest rates fail to indicate the trade-offs in
consumptions across generations and that the discount rate should be derived from an
ethical point of view. These advocates include Cline (1992) and Stern (2007). Proponents
of both approaches use the same Ramsey rule of Equation (1) in interpreting theirdiscount rates.
Nevertheless, such arguments are erroneous from the rigorous view point of standard
economic theory. Even though the utility function is cardinal and measurable in termsof money, there is no theoretical basis for discounting future utilities by the real interest
rate. Whenever the real interest rate differs from the rate of time preference, whichmust be genetically applied to discounting future utilities, a social planner misevaluates
Three kinds of interpretation of the elasticity of marginal utility are known (Stern,2008). They are intratemporal distribution, intertemporal distribution and attitudes torisk, respectively. Directly related to our argument is the intertemporal distribution andwe postulate that we can set the appropriate values for  .
2 See also Nordhaus (2007) and Weitzman (2007).
1

4

the future utilities.
For example, suppose that the real rate of interest exceeds the rate of time preference,and that a social planner applies the real rate of interest to the social discount rate.
This implies that the social planner underestimates the future utilities, and hence theundertaken consumption stream is excessively concentrated within the near futurecompared with the optima path.
In sum, neither approach stands on theoretically firm logic in the sense that they bothregard the interest rate as the social discount rate3.What is important in this discussionis that the intertemporal utility should be evaluated by the consumer’s effective rate oftime preference, which represents how much future consumption deserves in terms ofthe current consumption with taking the effect of the interest rate to optimal path into
consideration4.
In addition, as we argue later, the existing approaches assume that there is no
negative externality postulated in the rule5. According to Equation (1), except for astationary state, the rate of time preference is lower than the rate of interest as far as
consumption grows. Those who infer that the time preference in a social planning might
be lower than the market interest rate focus on refining the estimation of the second
term of Equation (1) (i.e., 

ct 6
).
ct

In addition, the formula in Equation (1) does not provide the solutions for the rates oftime preference  or elasticity of marginal consumption  endogenously, leaving eachadvocate to assign appropriate values as parameters according to one’s observations orvalue judgement.
4 A serious misunderstanding exists concerning the role of rate of time preference andinterest rate in existing approaches. An increase in the interest rate implies a lower (nothigher) social discount rate because future consumption becomes more advantageous.
That is, one should note that the social discount rate must be high enough if one wishes
suppressing excess consumption/emissions. We will intensively discuss this property in
Section 3.3.
Introduction of uncertainty into discounting leads us to a theory of declining discountrates (Arrow, et al., 2013). For example, uncertainty in the growth rates of future
consumption (see Gollier, 2012) and uncertainty of future discount rates such as therates of return on investment (see Weitzman, 1998 and Gollier-Weitzman, 2010) will
typically cause the discount rates to decline as time goes by.
5 See Section 3 to find the limitations of the standard Ramsey rule. See also Section 4
for the scope of its validity in relation to technological change. Examples of otherapproaches where the social discount rate is endogenously derived are found in Otaki(2013, 2015).
6 Such an assertion made mainly by advocates of the prescriptive approach is3

unacceptable from the view of economic growth theory. A large increase rate,
5ct, is not actHowever, when one analyzes the efficient allocation of public bads such as theemissions of CO2, one finds that Equation (1) alienates the effect caused by suchnegative intertemporal externalities. This calls for an extension of the theory.
Suppose that unit current capital investment not merely strengthens the production
capacity of an individual firm but also, via the aggravation of global warming, if notoffsetting the former positive effect, lowers overall productivity. Let this rate be denoted . In this case, the social rate of return is less than the rate of interest.
This study obtains the following modified formula:cct  rt  * t   t(2)where  * is the optimal carbon tax rate that is derived in the Section 3, and  tdenotes income per capital. Thus, one finds that the optimal carbon tax, which reflects
the true productivity of capital, retards the increase of consumption, and hinders excess
emissions.
This article is organized as follows. Section 2 reviews theoretical problems, which
preceding research commonly holds. Section 3 derives the modified Ramsey rule as inEquation (2) and discusses its implications. Meanwhile, since the modified Ramsey’srule is only a portion of an optimal growth theory with a negative intertemporalexternality such as the concentration of CO2, stability of the overall economy andequilibrium stationary growth rate are examined by a general equilibrium model.
Section 4 considers how emissions-reducing technology affects the welfare of world
economy. Section 5 concludes with brief remarks.

2. Critique of the Cost-Benefit Approach
In this section, we briefly review the existing research, and clarify the serious
misunderstandings as to which rate should be applied as the social discount rate.
All existing studies of this issue are based on Ramsey-type models. Accordingly, theultimate objective function is the integral of the discounted (including zero discountrate) instantaneous utility such that

U t   u  x  e   t 

t

d .

(3)

cause but a consequence of a low rate of time preference  . This fact implies that a low
rate of time preference corresponds to a low social discount rate. However, a low social
discount rate evaluates future consumption more highly; thus, this delays the
mitigation of CO2 emissions.
6What one should note is that the correct social discount rate is not the interest rate in
the market but the rate of time preference.
Meanwhile, we do not exclude the effect associated with a change in interest rate.
Such an effect is, however, indirect in the sense that the interest rate appears only inthe budget constraint, not directly in the utility function. As such, the rate of time
preference should normatively be equivalent to the social discount rate. Thus, incontrast to Weitzman (2007), Nordhaus (2007), and Stern (2008) --- which strive to infer
the effective interest rate through Equation (1) in order to avail this rate for cost-benefit
analysis concerning investments for reducing emissions of CO2, one must concentrateon how to estimate the effective social discount rate by using the observable andmeasurable economic variables.
As the foregoing discussion illustrates, the preceding studies, which commit
themselves theoretically by defining the effective interest rate as the social discountrate, are erroneous in general. We consider whether there is a case in which theircost-benefit approach becomes valid.
Since the interest rate is not related to the individual’s time preference, this rate willaffect the economic decision via the intertemporal budget constraint. In terms of theforegoing discussion, as long as the interest rate is the social discount rate, this rateshould be equalized with the rate of time preference via a dynamic optimization undersome specific instantaneous utility function.
Since the cost-benefit analysis, which applies the rate of interest to the discount rate,
assumes that the marginal monetary gain obtained by a project is not affected by thelevel of the gain in itself, the marginal utility of consumption (or marginal utility ofmoney) should be constant over time. Accordingly, if the interest rate can become equal
to the social discount rate, the corresponding instantaneous utility function isu  ct  e t  ct e t .

(4)

The Hamiltonian, H , in this case isH  ct e t  t  rKt  ct  .
The optimization conditions are

7

(5)

H
 e  t  t  0  t  t ,ct
H t  rt ,
K t

(6)

H
 K t  rKt  ct ,twhere r denotes the interest rate and K t is the total amount of capital. By the two top
equations in (6), we can confirm that the interest rate coincides with the rate of time
preference, and thus one can avail the interest rate as the social discount rate instead of
the rate of time preference.
Even though the discounting utility by the interest rate is admissible, one must notethat the second term of the right-hand side of Equation (1),

ct, never appears, because
ct

this term corresponds to the effect related to the intertemporal substitution effect(consumption smoothing effect) that appears only in the case of decreasing marginalutility; and such a case never emerges in Equation (4).
To summarize, none of the cost-benefit approaches, that apply the market interest
rate to the social discount rate, will support the formulation given Equation (1).

3. The Model
Section 2 reveals that all preceding analyses contain two logical inconsistencies. One isthat although the Ramsey rule in Equation (1) is upheld insomuch as the instantaneous
utility function is concave, those cost-benefit analyses using the interest rate as thesocial discount rate are validated only when the utility function is linear. The other flawof the preceding approach is that their models do not contain the intertemporal negative
externality caused by emissions of CO2. As such, we construct an optimal growth modelwith a dynamic negative externality assuming a concave instantaneous utility function.
3.1

Derivation of the Modified Ramsey Rule

Since Equations (1) and (2) are derived from the local maximization obtained on the
optimal path, it is sufficient to consider the optimality condition between two sequential
periods. Let the global aggregate production function, f , be denoted as7
We assume that each economy in the world has the same production and utility
function, as well as the same endowments. Accordingly, such functions can be78

yt  f  Kt , Et  , Et    Kt  ,

f
f

 0,
 0,
 0.
K
E
K

(7)

yt denotes the volume of current output, Kt , Et are the capital stock and theaccumulated emissions of CO2 that prescribe the environmental condition, respectively,and  is the function that represents the emissions accumulated achieved through theuse of capital.8
Consequently, the social planning problem that one must solve is

 





1
max It u f Kt* ,   Kt*   I t t 
u f  Kt t ,   Kt t    I t*t t  ,1  t









(8)

subject to

Kt t  Kt*  It  t ,(9)where u is a strictly concave utility function. K t* is the optimal capital stock at the
beginning of period t that has been determined beforehand by past decisions; I t*t is thegiven optimal investment during period t  t ; and  denotes the instantaneous rate oftime preference.
Since the productivity slowdown brought by the CO2 emissions is treated as anegative externality for each firm, the profit-maximization condition requiresft  1  r t 
K

(10)

where r is the instantaneous interest rate. In addition, let us assume that a proportional
carbon tax is incurred on the productivity of capital, which aims at internalizing thenegative externalities originating from CO2 emissions. We denote the instantaneous tax
rate as . The optimal tax rate * is obtained by calculating the marginal negative
productivity of capital brought about by the emissions. Thus, one obtains

f 
f

t   *t ,   KKf K   * .
f  K(11)

aggregated.
8 We do not exclude the possibility that an economy-wide carbon neutral technologythat effectively absorbs emitted CO2 might become available; and/or that
non-carbon-emitting energy resources are fully used in order to avoid the increase of
CO2 concentrations. For more detail, see Section 4.
9Thus, the optimal instantaneous tax rate * is equal to the product of the elasticity ofcapital income with respect to accumulated CO2 and the elasticity of emissions withrespect to existing capital.
Furthermore, one must note the following relationships:

 K c o  K   o  t   o  t  , o  c   o 
t   o  t  .
 t t 

(12)

Differentiating Equation (8) with respect to I t , and taking the relationships inEquations (10), (11) and (12) into consideration, one obtains

1  t  1  r t  t  o  K   o  t  * u 'c u ' c*t t*t o  K  o  t    c 1   r  * t  o  c   t  1  tt   
ct

 

(13)

Dividing both sides of Equation (13) by t , one can ascertain that  r  * c o  t 

.
ct
t

(14)

Taking the limit t  0 of Equation (14), one finally obtains the following
representation9:cct

  rt  * t   t .
3.2

(15)

Implications of the Derived FormulaFirst, we shall consider the economic meaning of the formula in Equation (15). The mostprominent feature of the formula is that the social rate of return from unit capital
accumulation is lower than the real interest rate r . This is because additional capital
reduces productivity via the accumulation of CO2 emissions. This negative effectappears in the second term of the right-hand side of Equation (15). Since  * isdetermined optimally in order that it completely internalizes such a diseconomy, thesecond term of Equation (15) corresponds to the exact carbon price that has to be paid
9

For example, if the form of the production function isyt    f  kt   ,   Kt      Kt    kt    , 0      1,

  Kt , t    Kt  , Kt  10  kt    d ,,and the utility function belongs to the constant relative risk aversion (CRRA) family, theformula in Equation (15) becomes an exact solution.
10

for the efficient intertemporal resource allocation. Thus, the social rate of return ofcapital is lowered by  * t in comparison with the market interest rate. Accordingly,
the rate of time preference  is strictly smaller than the rate of interest, r , in thestationary state wherect 0 holds. Such a constant consumption path is dubbed
ct

sustainable by Dasgupta and Heal (1974) and Solow (1986) 10 . This definition ofsustainability seems to be characterized by the faire intergenerational opportunities for
consumption, although the analysis is formulated as an optimization problem ofrepresentative economic agents with infinite longevity. In Section 3.5, it will be shownthat a sustainable economy in the sense of Dasgupta, Heal and Solow is also azero-growth economy to the extent that there is no technological progress in CO2
reduction, as will be discussed in Section 4.
3.3

The Effective Social Discount Rate: Is Discounting a Vice?

In this subsection, an effective social discount rate under the modified Ramsey rule
given by Equation (15) is derived. For simplicity, it is assumed that the instantaneous
utility function belongs to the constant relative risk aversion (CRRA) family, themarginal utility elasticity of which is denoted  . The effective social discount rate,  ,E

is defined as the rate by which the discounted future utility is equalized with thediscounted value of current utility. That is11,| u  ct t  | e t | u  ct  | e  t .
E

(16)

Equation (16) can be transformed intoe  t  1 E| u  ct t  | e t  | u  ct  || u  ct  |(17)Dividing both sides of the equation by t , rearranging the terms, and taking the limitt  0 , an elementary calculation and the modified Ramsey rule lead us totE  1 |1  1 |   |1  1 | r  * t  .

(18)

Equation (18) has a prominent feature. The optimal carbon tax, which is levied on
Note that Solow (1992) is skeptical about the arbitrariness of the concept ofsustainability.
11 Note that the instantaneous utility function takes negative values when   1 .
1011

capital income, raises the effective social discount rate. Neither supporters of thedescriptive nor the prescriptive (ethical) approach find such a paradoxical result.
Supporters of the descriptive approach are apt to regard the optimal interest rate as
higher than it is in actuality. This is because they might be inclined to consider that theclimate change is not serious enough in general and that people should demand a highsocial discount rate, which is applied to the emissions-saving investment.
The ethical approach emphasizes the converse consequence. However, the perspectivethat a high interest rate means a high social discount rate is commonly shared despiteof the profound rift between the policy recommendations of the two approaches.
The seemingly counterintuitive fact that a high interest rate corresponds to a lower
effective social discount rate has firm roots in the correct interpretation of Ramsey’srule in Equation (15). Equation (15) does not determine the interest rate, but describes
the dynamic path of consumption. If the rate of time preference is larger than theinterest rate, people enjoy much greater current consumption than future, and viceversa. Accordingly, a higher interest rate prompts a longer time horizon and preference
toward future consumption; thus, the corresponding effective discount rate becomes
lower as is apparent in Equation (18).
In other words, the carbon tax provides the true rate of return for capital and lowers
the effective interest rate. This allows a high social discount rate, which enables to the
reduction of consumption and capital investment over time. It is natural that values ofthese macroeconomic variables should be reduced whenever the world economy iscaught in the situation in which excess consumption and investment causes excessconcentration of greenhouse gases. To summarize, it is quite a natural consequence thatthe social discount rate moves in the opposite direction to the rate of interest.
Finally, one must note that, in contrast with an overlapping-generations model, ahigher social discount rate does not mean the exploitation of future resources in adynasty model. Accordingly, no ethical problem is provoked by applying a high social
discount rate.
It should be emphasized that the Ramsey rule, while important, in general, is only
one component of an optimal growth theory. The dynamics of emissions and/or
accumulation processes of capital should be simultaneously analyzed. That is, one needs
a full model of optimal growth that contains the negative and intertemporal external
economy caused by CO2 emissions. This is the topic we shall analyze in subsequent
subsections.

12

3.4

Modifying the Ramsey Rule to Analyze the Global Warming Problem

The Ramsey rule   r  cis a special case of our formula (2) which corresponds to theccase where   0 and r  const. One should note that since the production function*of each industry is linear against capital, the aggregation is feasible. Thus, we obtainthe macroeconomic production function F asyt  rKt .

(19)

Equation (19) implies that no scarce production resource other than capital exists. This
fact demonstrates that there is a serious drawback in the use of the standard Ramseymodel to analyze the global warming problem.
This is because the scarcity that originates from the quality of our atmosphere(measured by the mass of CO2 it contains), is the vital issue in this problem. As such,
unless we assume some optimistic exogenous emissions-absorbing technologicalprogress, as will be analyzed in Section 4, it is a plausible theoretical formulation thatthe aggregated production function suffers from diminishing returns to scale owing to
the Marshallian negative externality originating from CO2 emissions. That is,theoretically, the congestion that emerges from polluting the atmosphere intensively is
the very core of the global warming problem.
3.5

The Limit of Economic Growth: The Case of a Linear Homogenous Individual
Production Function with Marshallian Diseconomy

We heretofore assume that the instantaneous rate of interest, r , is constant over time.
However, as Equation (10) shows, this varies together with capital accumulation. Let us
assume that

2 f
 2 f d f d 2

0,

0.
K 2
K dK  dK 2

(20)This assumption implies that the marginal product of capital of individual firms isconstant or decreasing and that the marginal external diseconomy increases withcapital accumulation. Accordingly, the relationship between capital stock, K , and theinstantaneous social rate of interest, r , becomes the downward sloping curve illustrated
in Figure 1. This method is a simple application of endogenous growth theory, which
originates from Romer (1990).
The feasibility condition of the economy is depicted as

Kt  rt  * t  Kt  ct .
13

(21)

Equations (15) and (21) describe the law of motion of this economy. Average productivityof capital,  t , is a decreasing function of capital, K t 12, and the inside of the bracket of theright-hand side of Equation (21) is positive. If these effects dominate the effect of
decreasing marginal product of capital, the locus of  c, K  , which satisfies Kt  0 ,becomes upward-sloping as Curve KK illustrated in Figure 2.
On the other hand, the locus of  c, K  that satisfies ct  0 is obtained by Equation(11). To the extent that the effective interest rate, r  * , is a monotonously decreasing
function of capital, K , by Equation (15), some K exists that satisfies c  0 ; thus the*locus of c  0 becomes horizontal as illustrated by curve CC in Figure 2.
It is clear from Figure 2 that if the world interest rate is located above curve KK ,capital accumulation advances. Otherwise, disinvestment should be hastened by the
carbon tax because too many emissions have already accumulated in such a situation.
Ultimately, the world economy converges to Point E . This demonstrates that there is alimit to the world economy’s growth; and this converges to zero economic growth when asignificant negative intertemporal externality such as global warming exists.
In addition, one can ascertain that other paths such as AA or BB are not optimal
as follows. Let us consider path AA . On path AA , people initially experience excess
consumption compared with the optimal path. Such inertia does not cease even when
the level of capital falls below the critical level K , because a shortage of capital lowers*

the effective social discount rate (raises the effective interest rate) as discussed in
Section 2.3. Accordingly, capital depletion advances. The economy eventually reaches abankruptcy point such as E and no consumption is possible thereafter. Since theBprinciple of optimality proposed by Bellman requires that an initial optimal policy
should be also optimal with respect to subsequent periods, and since stationary state

E B is Pareto inferior to Point E , it is evident that path AA is not optimal.
Path BB is also non-optimal. This path is characterized by excess capital
accumulation and shortage of consumption. Even when the economy crosses curve KK ,F  Kt , F '  0, F ''  0 ,Kt
d t F ' Kt  F

0
dKt
Kt2

12

Since  t 

holds.
14

where investment temporarily ceases, the decrease in consumption does not stopbecause the effective social discount rate is high enough (the interest rate is low
enough). Accordingly incessant capital accumulation occurs and the consumption level
asymptotically approaches zero. Accordingly, again from the principle of optimality,
such a path is not optimal.
What one must note is that even though the production function of each economy is alinear function on capital stock, which means that

2 f
=0,
K 2

(22)

Equation (14) still holds, and thus the properties of Figure 2 are preserved. This meansthat our atmosphere is not a renewable resource for human beings overall, andtherefore, even though the anthropologic technological system in the short run contains
no non-renewable resources, the gradation of the quality of atmosphere creates abottleneck in the production/consumption process in the long run. It is apparent that a
constant growth path of consumption levels is unfeasible as long as the aggregate
production exhibits decreasing returns to scale and the interest rate is endogenously
determined.

4. The Formula  r cand Technological ProgresscPreserving the relevance of the formula   r  cin the context of the global warmingcproblem, we need to introduce some exogenous technological progress that reduces theconcentration of CO2 at a constant rate  . In such a case, for example, the production
function of firm  becomes1Yt  F  Kt ,   Kt    Kt  Kt e  t   , Kt   Kt d .
0



(23)

The inside of the bracket indicates the external effect associated with theconcentration/reduction of CO2, which corresponds to the exogenous total factor
productivity (TFP) for each firm. Accordingly, as long as the TFP is kept constant, theworld economy can achieve a steady growth rate because the macroeconomic production
function becomes linear.
By Equation (23) such a rate is represented as

15

dKt e  t
K0 t   ,
dt
Kt

(24)


F  Kt   Kt e  t   r *  const. .
Kt

(25)

Applying Equations (24) and (25) to formula (2) at the stationary state, one obtainscc  r*  .

(26)

In addition, from the budget constraint,

Kt   Kt  r * Kt  ct
 Kt 

ct
K c
  t  t
r 
Kt ct

(27)

*

holds. Thus, Equation (26) at the steady state equilibrium becomes  r *  .

(28)

In other words, Equation (28) implies that the feasibility of future increases in
consumption relies entirely on autonomous technological progress in the reduction of
the stock of CO2.
The stability of the stationary state can be proved as follows: Sincert   Kt e  t  ,(29)

logarithmically differentiating Equation (29) with respect to time, one obtainsrt     gt  .
rtwhere gt 

(30)

Kt
. Again, from the budget constraint and the optimality condition (15), one
Ktobtains

Kt  gt K t   rt  *  K t   K t  ct
1  *  rt  gt
ctc
 Kt 
 gt 
 t   1  1  *  rt   **
1   rt  gt
 1   rt  gt  ct
 gt   1  *  rt  gt  gt  1  *     gt  rt   1  1  *  rt  gt   1  *  rt    .
(31)
16

When linear approximation around the stationary state is applied to the right-hand sideof Equation (31), one can ascertain thatgt  2   gt     r *  1      1  *  rt  r *  .
Thus, our economy can be described by two differential equations on(32)

 gt , rt  as

expressed in Equations (30) and (32). There is ambiguity in the sign of the second termof (32). This is because growth rate is affected by two opposite factors from the interest
rate. One factor is that a high interest lowers the social discount rate and stimulates
consumption. This effect retards capital accumulation. The other dynamic is rather
indirect; the heightened interest rate raises real income and accelerates capital
accumulation as a residue of real income minus consumption. Let the latter effect beassumed to dominate the former, as assumed in Section 3.5. Then, the second term of
the right-hand side of Equation (32) becomes negative.
The phase diagram of this system is illustrated in Figure 3. Accordingly, the economy
has a saddle-point property, and the planned economy can achieve these stable paths,
which converge to the stationary equilibrium, E . Thus, equilibrium growth rate of
consumption in the steady state is equal to  , and the formula (28) is upheld.

5. Concluding Remarks
A modified Ramsey rule with a negative intertemporal externality (e.g., caused by theexcess emissions of CO2) has been developed. According to the rule, a proportional
carbon tax rate to income per capital is required to achieve the efficient intertemporal
allocation of emissions of CO2. Even under a sustainable equilibrium where theaggregate consumption level is kept constant, such taxation is unavoidable because of
the dis-incentivizing further emissions. It is notable that such a negative effect is nottaken into consideration in the preceding articles analyzing the global warming problem
despite of its importance.
Second, the optimal rate of the proportional carbon tax is derived explicitly. The rateis equal to the product of the elasticities of output to emissions and emissions to capital.
That is, the optimal carbon tax rate per unit capital income is equal to how many
outputs are curtailed by one percent capital accumulation owing to the negative
externality caused by emissions.
In addition, one should note that a higher interest rate implies a lower discount rate
in optimal growth theory. The causality, which the Ramsey rule asserts, is that17

consumption increases with the difference between the interest rate and the rate of time
preference. Accordingly, consumption increases with the interest rate over time because
individuals find more advantage in postponing consumption. This implies that thesocial discount rate is lowered by an increase in the interest rate; thus, a higher interest
rate aggravates the global warming problem. Both descriptive and prescriptive (ethical)
approaches, which are noted in Arrow et al. (1996), are prone to overlook thisrelationship. In reality, applying a proportional carbon tax is the proper diagnosis for
mitigating the global warming problem. This is because a higher social discount rate
suppresses future consumption, or equivalently, future CO2 emissions.
Finally, it is proved that, to the extent that there is no technological progress which
lessens the stock of CO2 in the atmosphere and overall emissions control assures theviability of human beings, there is a limit of economic growth and the world economy
will reach a zero-growth era eventually. This is because a quality atmosphere is a scarceand unreproducible resource of production; thus, the aggregate production function
comes to exhibit decreasing to scale.

18

References
Arrow, K.J., W.R. Cline, K-G. Maler, M. Munasinghe, R. Squitieri, and J.E. Stiglitz(1996)“Intertemporal

Equity,Discounting,andEconomic

Efficiency,

Intergovernmental Panel on Climate Change, Working Group III, pp. 127-144.
Arrow, K., M. Cooper, C. Gollier, B. Green, G. Heal, R. Newell, W. Nordhaus, R. Pindyck,
W. Pizer, P. Portney, T. Steiner, R.S.J. Tol and M. Weitzman (2013)“Determining Benefits and Costs for Future Generations,” Science, Vol. 341, pp.
349-350.
Cline, W.R. (1992) The Economics of Global Warming, Washington DC: Institute forInternational Economics.
Dasgupta, P. and G. Heal (1974) “The Optimal Depletion of Exhaustible Resources,”

Review of Economic Studies, Vol.41, pp. 3-28.
Gollier, C. (2012) Pricing the Planet’s Future: The Economics of Discounting in an

Uncertain World, Princeton, NJ: Princeton University Press.
Gollier, C. and M. Weitzman (2010) “How Should the Distant Future be Discounted
when Discount Rates are Uncertain?,” Economics Letters, Vol. 107, pp.350-353.
Nordhaus, W. (2007) “A Review of the Stern Review on the Economics of Climate
Change,” Journal of Economic Literature, Vol. 45, pp. 686-702.
Nordhaus, W. (2013) The Climate Casino: Risk, Uncertainty, and Economics for a

Warming World, New Haven, CT: Yale University Press.
Otaki, M. (2013) “Endogenous Social Discount Rate, Proportional Carbon Tax, andSustainability: Do We Have the Right to Discount Future Generations’Utility?,” Environmental Systems Research, 2:1, pp. 1-8.
Otaki, M. (2015) “Local Altruism as an Environmental Ethic in CO2 Emissions Control,”Atmospheric and Climate Sciences, 5:4, pp. 433-440.
Ramsey, F.P. (1928) “A Mathematical Theory of Saving,” Economic Journal, Vol. 38, pp.
543-559.
Romer, P.M. (1990) “Endogenous Technological Change,” Journal of Political Economy
98, pp. s71-s102.
Solow, R.M. (1986) “On the Intergenerational Allocation of Natural Resources,” The

Scandinavian Journal of Economics, Vol.88, pp.141-149.
Solow, R.M. (1992). “Sustainability: an Economist's Perspective, National Geographic

Research and Exploration, Vol. 8, pp. 10-21.
Stern, N. (2007) The Economics of Climate Change: The Stern Review, Cambridge, UK:
Cambridge University Press.
19

Stern, N. (2008) “The Economics of Climate Change,” American Economic Review:

Papers & Proceedings, Vol. 98, pp. 1-37.
Weitzman, M.L. (1998) “Why the Far-Distant Future Should be Discounted at Its
Lowest Possible Rate,” Journal of Environmental Economics and Management,
Vol. 36, pp. 201-208.
Weitzman, M.L. (2007) “A Review of the Stern Review on the Economics of Climate
Change,” Journal of Economic Literature, Vol. 45, pp. 703-724.

20

21

22

23