DBJ Research Center on Global Warming
Discussion Paper Series No. 57 (3/2016)

Modified Ramsey Rule and Optimal Carbon Price

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Modified Ramsey Rule and Optimal Carbon Price

Masayuki Otaki
Institute of Social Sciences, the University of Tokyoohtaki@iss.u-tokyo.ac.jp

Abstract
The Ramsey rule is regarded as a convenient vehicle for estimating the social discountrate in general. Carbon pricing is treated as another theory of environmental economics.
This study clarifies the theoretical relationship between the Ramsey rule and optimalcarbon price, which has been overlooked in the existing research. It succeeds in deriving
the optimal carbon price from the modified Ramsey rule in stationary state.

Keywords: Modified Ramsey Rule; Optimal Carbon Price; Social Discount Rate; CarbonCycle
JEL Classifications: H23; D61; D71

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1. Introduction
Many studies utilize the Ramsey rule for estimating appropriate social discount rate
(e.g., Cline [1]; Stern [2]; Nordhaus [3]). However, as Kuninori and Otaki [4] point out,
the causality presumed in existing studies is incorrect. A natural interpretation ofoptimal growth theory leads us to the following proper causality of the Ramsey rule.
That is, consumption gradually increases when the rate of interest exceeds the rate oftime preference because an individual is rewarded by higher utility from postponing
consumption.
Since the Ramsey rule shows optimal consumption and/or investment path, if an
original model is devised to include a negative intertemporal externality, it is evident
that optimal carbon price can be calculated based on this formula. This study analyzes
the relationship between the Ramsey rule and optimal carbon price and calculates sucha price in reality.
It should be noted that the Ramsey rule evaluated in stationary state implies thetangency condition of optimality, that is, equalization of the marginal rate ofsubstitution in consumption or production with the relative price of carbon. Unless sucha condition is upheld in the long run, there is a room for improving utility. Accordingly,
as far as the stationary state is optimal, the foregoing tangency condition can beinduced from the Ramsey rule; thus, an integrated view is obtained on how thetraditional estimation of the social discount rate based on the Ramsey rule is related tothe optimal carbon pricing.
The rest of this paper is organized as follows. Section 2 reveals the relationshipbetween the modified Ramsey rule and the optimal carbon price using Kuninori andOtaki (2016) and Otaki (2016). One will find that the characteristic of the carbon cycle
equation plays a crucial role in determining the carbon price. Section 3 provides brief
concluding remarks.

2. Modified Ramsey Rule and the Carbon Price
Based on Kuninori and Otaki [4] and Otaki [5], this section derives the optimal carbon
price from the modified Ramsey rule.
2.1 Negative Intertemporal Externality in Utility FunctionThroughout this study, it is assumed that there is a negative intertemporal externality
in the world economy originating from excess emissions of carbon dioxide (CO2). In thissubsection let us consider the case in which such a negative externality directly affects
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people’s utility. The concave instantaneous utility function, u , is assumed to be𝑢 ≡ 𝑢(𝑐𝑡 , −𝐸𝑡 ), 𝑢1 > 0, 𝑢2 > 0 ,(1)

ct and Et denote consumption and accumulated emissions (measured by CO2 tonnage),respectively. ui is the i -th partial derivative of the utility function.
The dynamics of the carbon cycle are assumed to becEt   Et  t ,(2)where  denotes the absorption ratio of CO2 by the Earth.  denotes the efficiency ofproduction measured by the amount of production (consumption) that can be produced
by unit emissions1.
The optimization problem is represented as follows:max  u  ct ,  Et  e t dt , subject to (2).
ct0(3)

The corresponding Hamiltonian, H tU , is𝑐𝑡
𝐻𝑡𝑈 ≡ 𝑢(𝑐𝑡 , −𝐸𝑡 )𝑒 −ρt + 𝜆𝑈𝑡 [−𝛼[−𝐸𝑡 ] − 𝛽 ].

(4)The necessary and sufficient conditions for the maximization are Equation (2) andu1 tU U, t  tU e ttU       tU  u2,(5)

lim ttUEt e  t  0.

t 

Differentiating the top equation in (5) logarithmically, yields

tUc
E cc t  cE t ,Uct
Et
tc  u11
E  u12cc  ,cE , u12  0u1u1(6)

Substituting Equation (6) into the middle equation in (5), it can be ascertained thatthe following relation holds:1Another type of equation of the carbon cycle is examined in Subsection 2.3.
5tUu      2U u1
tuE c      2  cE t   cc t
Et ct  u1(7) 1 dcE c    |u const . cE t   cc t .
Et ct  dEThe bottom equation in (7) is the modified Ramsey rule when a negative externality
exists in the utility function. The economic implication is as follows: the left-hand side ofthe equation represents the effective social discount rate. The effectiveness implies thatthe absorption ratio,  , is added to the discount rate. This is because emitted CO2
becomes harmless with the ratio,  , at every moment and a social planner can bepermitted to discount future damages originating from global warming at a higher rate
than the rate of time preference,  .
The first bracket of the right-hand side of the modified Ramsey rule (the bottomequation in (7)) corresponds to the net marginal benefit in terms of utility.

1 dc| dE u const .

denotes the marginal substitution rate of consumption to emissions, which represents
the utility obtained from unit emissions. When  is large, people can enjoy highconsumption with less emissions. Accordingly, ceteris paribus, the obtained marginalutility decreases due to the saturation of consumption. cE

Etis the marginal disutility
Etfrom emissions; summing them, the net marginal utility originating from emissions isobtained.
The second term of the right-hand side of the modified Ramsey rule (the bottomequation in (7)) indicates the optimal path of consumption. The optimal consumption
decreases proportionately to the difference between the effective social discount rate
and the net marginal utility obtained by emissions. In other words, as people become
more impatient or marginal utility from emissions more attractive, the optimal
consumption stream is concentrated in the near future; and thus, decreases as time goesby. The dynamics of the economy are fully described by the differential equations (2) and(7).
Thus, we have established the modified Ramsey rule wherein a negativeintertemporal externality originating from CO2 emissions directly affects people’s utilityfunction. Using Equation (7), let the optimal carbon price be calculated in the stationary
6state as follows:E c  0 holds, which yields
E c
1 dcdc  |u const .  |u const .        .
 dEd E 

By the definition of stationary state,(8)

Because the left-hand side of Equation (8) is the marginal substitution rate ofconsumption to emissions reduction (i.e., the tangency of an indifference curve in the E, c  plane), the right-hand side of Equation (8) indicates the inverse of the optimalcarbon price in terms of unit emissions. That is, people should pay

1times of   

money for consumption in exchange for the social cost incurred by unit emissions. This
result coincides with that of Otaki [5].
2.2

Negative Intertemporal Externality in Production Function

In this subsection, we derive the modified Ramsey rule wherein a negativeintertemporal externality affects the production function. Let the strictly concave
production function, F , be denoted asyt  F  kt ,  Et  , F1  0, F2  0 ,(9)where yt is total output of goods, and kt denotes input of goods for production. It is
assumed that anthropogenic combustion of fossil fuels concentrates CO2 emissions andlowers productivity. Accordingly, the partial derivative of the production function ispositive. In such a case, the differential equation that describes the carbon cycle
becomesEt   Et yt  Et 1

F  kt ,  Et  .

(10)

The optimization problem to be solved ismax  v  ct  e t dt , subject to Equations. (8) and (9),ct0(11)

where v is a strictly concave instantaneous utility function. The corresponding
Hamiltonian, H tP , is𝑝

1𝑝

𝐻𝑡 ≡ 𝑣(𝐹(𝑘𝑡 , −𝐸𝑡 ) − 𝑘𝑡 )𝑒 −𝜌𝑡 + 𝜆𝑡 [−𝛼[−𝐸𝑡 ] − 𝛽 𝐹(𝑘𝑡 , −𝐸𝑡 )] .
The necessary-sufficient conditions are Equation (10) and7(12)v' 

tP F1,  P  tP e t ,
  F1  1 t
F2  P  v ' F2 ,  tlim tP Et e t  0.

tP     (13)t Combining the middle equation in (13) with the top equation, yieldsv tP F1
tPc 
 P   t  t  F1  1 t
ct  ttP F2     P  F1  1 t 
𝜌+𝛼 =where(14)

𝐹2
𝛾̇ 𝑡
𝑐̇𝑡
+ −𝜂 ,
𝛽[𝐹1 − 1] 𝛾𝑡𝑐𝑡

t
is defined ast t d dc
 ln .
 t dt dy(15)The bottom equation in (14) corresponds to the modified Ramsey rule. The left-handside of this equation represents the effective social discount rate as in Equation (8). The
first term on the right-hand side of Equation (14) is the marginal substitution rate ofemissions to intermediate goods, which means the marginal benefit of emissions inconjunction with economizing intermediate goods. The second term on the right-handside of Equation (14) denotes the increase rate of marginal propensity to consume. If
this rate is heightened, this saves more artificial production resources; and thereby
improves the efficiency of production. The third term on the right-hand side of Equation(14) summarizes these effects. Future consumption increases monotonously if themarginal benefit of emissions,F2
 t , exceeds the rate of tolerance to current  F1  1  tconsumption,    . Hence, it can be ascertained that Equation (14) is the modified
Ramsey rule wherein an intertemporal negative externality exists within theproduction process.
The carbon price relative to the unit emission reducing cost at the stationary state
can be solved by lettingc   0 . This yields
c 8

F2
  .
  F1  1(16)

The profit-maximization condition implies

F21 , F1  1 p(17)where p is the carbon price (i.e. goods price) in terms of unit emissions reducing cost.
Accordingly, by combining Equations (16) and (17), the optimal carbon price, p , issolved asp1.
    (18)It is evident from the discussion in Subsection 2.1, that the optimal carbon price isinvariant with where the negative externality exists.
2.3 Kuninori-Otaki’s [4] Case
Kuninori and Otaki [4] is the first study that finds some modification is necessary for
the Ramsey rule to be tenable for analyzing the global warming problem. Their model
differs from the foregoing two models in this paper on the point that it includes a capital
accumulation process and that a negative externality emerges from capital
accumulation. This is because the authors assume that the overall production capacity
is determined by capital stock and excess emissions stem from the production process.
Although the model is transformed in order to include capital accumulation, one canascertain the similar formula concerning the optimal carbon price derived as follows.
Kuninori and Otaki [4] assume the following negative externality in the concave
production function:𝑦𝑡 = 𝐹(𝐾𝑡 , 𝐸𝑡 ), 𝐾𝑡 = Ψ(𝐸𝑡 ), 𝐹1 > 0, 𝐹2 < 0, Ψ’ > 0 .

(19)The maximization problem ismax  v  yt  I t  e t dt , subject to I t  Et .
It0(20)The corresponding Hamiltonian H is defined asK

HtK  v F    Et  , Et   It e t  tK I t .
The necessary-sufficient conditions for optimality are9(21)u '  tK , tK  tK e t ,tK  tK  v '  F1  ' F2  ,(22)

lim tK Et e t  0.
t Combining the top and middle equations in (21), the following modified Ramsey rule isobtained. That is,𝑐̇𝑐̇𝑡𝑡𝐸𝜕𝐹𝐹ρ = [𝐹1 Ψ ′ + 𝐹2 ] − 𝜂 𝑐𝑡 ≡ 𝐹1 Ψ ′ − Ψ ∗ 𝜋 − 𝜂 𝑐𝑡 , Ψ ∗ ≡ − 𝐹𝜕𝐸 , 𝜋 ≡ 𝐸 .

(23)

where  * is the optimal carbon tax rate, and  denotes the average productivity ofemissions. This corresponds to the modified Ramsey rule in Kuninori and Otaki (2016).
From the static profit maximization condition, as far as one wishes to achieve thefirst-best allocation by a market economy,ddF
 py  E   p  1  0dEdE(24)holds. The optimal carbon price of a consumption goods measured by unit socialemissions cost incurred by capital accumulation in the stationary state satisfies  F1 '  * dF
11 *  p*  ,dE p(25)

where p* is the optimal relative carbon price (or goods price). Comparing Equation (25)with Equation (18), it is found that Kuninori and Otaki [4] correspond to the case inwhich   0,   1 . This is evident from the structure of the model that Kuninori andOtaki [4] regards the absorption rate by the earth,  , as zero and set the parameter ofthe carbon efficiency,  , to unity. Accordingly, Equation (25) is a special case ofEquation (18); thus, in this case too, we succeed in driving the optimal carbon price fromthe corresponding modified Ramsey rule.
To summarize our assertion, since each modified Ramsey rule represents the optimal
consumption/emissions path, it is clear that it is possible to calculate the optimal carbon
price in the stationary state which is the destination of the optimal path.

3. Concluding Remarks
This study analyzed the relationship between the modified Ramsey rule and theoptimal carbon price levied on consumption goods. Regardless of the origin of anintertemporal negative externality, the modified Ramsey rule solves the optimal carbon
price in the stationary state, p , as

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p

1.
    (26)When the rate of time preference,  , is high and people are impatient, the carbon
price becomes lower because such an economy deeply discounts future damages. If theearth’s absorption capacity of emission is higher and  takes a larger value, the carbon
price is lowered. This is because emitted CO2 does not stay within the atmosphere for along time and is less harmful. Finally, the larger emissions efficiency, which corresponds
to a high value of  , the less emissions are necessary for producing goods. Accordingly,
it is acceptable to reduce the carbon price in such a case.

References[1] Cline, W.R. (1992) The Economics of Global Warming. Washington DC: Institute forInternational Economics.
[2] Stern, N. (2007) The Economics of Climate Change: The Stern Review. Cambridge,
UK: Cambridge University Press.
[3] Nordhaus, W. (2013) The Climate Casino: Risk, Uncertainty, and Economics for aWarming World. New Haven, CT: Yale University Press.
[4] Kuninori, M., and Otaki, M. (2016) Modified Ramsey Rule, Optimal Carbon Tax, andEconomic Growth. Atmospheric and Climate Sciences, Forthcoming.
[5] Otaki, M. (2016) Properties of the Social Discount Rate and Negative Intertemporal
Externality in the Utility or Production Function. Low Carbon Economy,Forthcoming.

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