DBJ Research Center on Global Warming
ECONOMIC ANALYSIS No. 13 (December 2009)

Global Warming, Imputed Prices,and Sustainable Development
Hirofumi Uzawa

RESEARCH INSTITUTE OF CAPITAL FORMATIONDEVELOPMENT BANK OF JAPAN
1-9-3, OTEMACHI, CHIYODA-KU, TOKYO 100-0004, JAPANE-MAIL: rcgw@dbj.jp

Abstract
In this paper, we prove in terms of a simple dynamic model of global warming thatsustainable time-paths of the atmospheric accumulations of carbon dioxide are obtained
as the standard market equilibrium under the system of proportional carbon taxes, where
the carbon taxes are levied at the rate that is proportional to the per capita national
income of each country, with the discounted present value of the impact coefficient of
global warming as the coefficient of proportion.
Keywords
global warming
imputed pricevirtual capital market
sustainable development
proportional carbon taxes

1. Introduction

Global warming involves international and intergenerational equity and justice. Although
global warming is largely caused by the emission of carbon dioxide and other greenhousegases associated with economic activities mostly in developed countries and by the disruptionof forests − particularly tropical rain forests − again mostly resulting from the industrial
activities of the developed countries, it is the people in developing countries who have to bear
the burden. While the current generation may enjoy the fruits of the economic activities thatcause global warming, it is the people in all future generations who will have to suffer from asignificant increase in atmospheric instability as a consequence.
The problems of global warming are genuinely dynamic. We inherit an excessconcentration of atmospheric carbon dioxide from past human activities and the choices wemake today concerning the use of fossil fuels and related activities significantly affect all
future generations through the phenomenon of global warming that is brought about by theatmospheric accumulations of carbon dioxide due to the combustion of fossil fuels today.
Thus, we explicitly have to take into account the changes in the levels of welfare of all future
generations caused by the increases in the atmospheric accumulations of carbon dioxide
today. The degree of uncertainty involved with these phenomena, however, is too great to be
subject to any objective analysis. We take instead a simplistic approach that all future paths
are presumed to be known for certain and derive a number of propositions concerning policy
and institutional arrangements to bring about the sustainable processes of economic
development. The concept of sustainability as introduced in this paper, however, is defined insuch a manner that the basic nature of policy choices and institutional arrangements to bringabout the sustainable economic development remains valid regardless of the specific patternof future paths that is presumed to be known for certain at the beginning of the analysis.
1

The dynamic analysis of global warming in this paper is developed primarily along
the lines suggested by a large number of contributions concerning the dynamic theory of
optimum capital accumulation and scientific analysis of global warming, as in detail referred
to in Uzawa (2003, 2005). We are particularly concerned with the processes of economic
development that result in the sustainable time-path of the atmospheric accumulations ofcarbon dioxide. One of the main conclusions obtained in this paper is the proposition thatsustainable processes of the atmospheric accumulations of carbon dioxide are obtained as thestandard market equilibrium under the system of proportional carbon taxes, where the carbon
taxes are levied at the rate that is proportional to the per capita national income of eachcountry, with the discounted present value of the impact coefficient of global warming as thecoefficient of proportion.
Our analysis is focused upon the role of the imputed price of the atmosphericaccumulations of carbon dioxide in the processes of sustainable economic development. The
imputed price π νt of the atmospheric accumulations of CO 2 for each country ν at aparticular time t expresses the extent to which the marginal increase in the atmosphericaccumulations of CO 2 at time t induces a marginal decrease in units of market prices in thewelfare level of country ν , including those of all future generations. The imputed price π νt
is at the sustainable level at time t, if it remains stationary at time t; i. e.,π̇ νt = 0 at time t,where π̇ νt refers to the time derivative with respect to the time of the virtual capital market
for the atmospheric accumulations of carbon dioxide at time t. A time-path of theatmospheric accumulations of CO 2 is defined sustainable for country ν , if the imputed
price π νt for country ν is at the sustainable level at all time t.
Because the sustainability of the imputed price is defined with respect to thefictitious time of the virtual capital market at time t, the sustainability of the imputed price2

does not necessarily imply the stationarity of the imputed price, as inadvertently stated in
Uzawa (2003, 2005). All the policy and institutional conclusions obtained in Uzawa (2003,
2005), however, remain valid with regard to the concept of the sustainability of time-pathsof the atmospheric accumulations of carbon as introduced in this paper.

2. Accumulation of Carbon Dioxide in the Atmosphere − A Simple Dynamic Model

We denote by Vt the amount of CO 2 accumulated in the atmosphere at time t. The quantity
Vt is measured in actual tons (in weights of carbon content) or in terms of the density of CO 2
in the atmosphere. We also adopt as the origin of the measurement the stable pre-Industrial
Revolution level of 600 GtC (approximately corresponding to the density of 280 ppm). The
current level of 760 GtC (380 ppm) is expressed as Vt = 160 GtC.
The premises of the dynamic model introduced in this paper are primarily based upon
the scientific findings on global warming and related topics, as reported in Dyson andMarland (1979), IPCC (1991, 1996, 2002, 2007), Keeling (1968, 1983) Ramanathan (1985),
Stern (2007), Takahashi (1980), Uzawa (199l, 2003, 2005) , among others.

Changes of Carbon Dioxide Accumulations over TimeThe atmospheric accumulations of CO 2 change over time due to natural and anthropogenic
activities. Let us first consider changes in the atmospheric level of CO 2 due to naturalactivities.
A certain portion of atmospheric accumulations of CO 2 is absorbed by the oceans(roughly estimated at 50%) and by living land plants as well. In the simple dynamic model3

postulated in the present paper, the exchange of CO 2 between the atmosphere and theterrestrial bio-sphere is not taken into consideration. It is explicitly discussed in the moregeneral model introduced in Uzawa (2003, 2005).
Roughly 75-90 GtC of carbon are annually exchanged between the atmosphere andthe surface oceans. The mechanisms by which atmospheric CO 2 is absorbed into the surface
oceans are rather complicated. They depend partly upon the extent to which the surface
waters of the oceans are saturated with CO 2 , on the one hand, and partly upon the extent towhich CO 2 is accumulated in the atmosphere in excess of the equilibrium level, on the other.
The studies made by several meteorologists and oceanologists, in particular by [1, 7, 8, 9, 11]suggest that the rate of ocean uptake is closely related to the atmospheric accumulations of
CO 2 in excess of the stable, pre-Industrial Revolution level of 280 ppm.
As an approximation of the first order, we assume that the amount of atmospheric
CO 2 annually absorbed by the oceans is given by µ Vt , where Vt is the atmosphericaccumulations of CO 2 measured in actual tons of CO 2 , with the pre-Industrial Revolution
level of 600 GtC as the origin of measurement, and the rate of absorption µ is a certain
constant. The rate of absorption µ is rather difficult to estimate. The studies made by
Takahashi and others referred to above suggest that the rate of absorption µ would have amagnitude of 2-4%. Indeed, according to the estimates made by Ramanathan (1985), CO 2
remains airborne 7 to 17 years, and about a half is absorbed by the surface ocean, again
resulting in the magnitude of µ = 2-4%.
In the simple dynamic model introduced in this paper, the anthropogenic change in
the atmospheric accumulations of CO 2 is exclusively due to the combustion of fossil fuels inconnection with industrial, agricultural, and urban activities. We denote by at the annual rateof increase in the atmospheric level of CO 2 due to anthropogenic activities. The magnitude
of at is currently estimated around 6 GtC per annum; that is, at = 6 GtC.
The dynamic equation for the atmospheric level of CO 2 is now formally written as4

V̇t = at − µ Vt ,

(1)

dVt
denotes the rate of change in Vt over time.
dt
The rate of anthropogenic change in the atmospheric accumulations of CO 2 , at , iswhere V̇t =primarily determined by the combustion of fossil fuels, to be closely related to the levels of
production and consumption activities conducted during the year observed.
There exist n countries in the world. Each country is generically denoted by ν(ν =1, . . ., n ) . We are concerned with the processes by which various relevant economic
variables are determined at each time t. However, for the time being, all variables aredenoted without explicit reference to time t.

Specifications for Utility Functions
We assume that the level of the utility uν of each country ν depends upon the atmosphericaccumulations of CO 2 at time t, V . That is, the utility function for each country ν isexpressed in the following manner:uν = uν (cν , V ),where cν = (cνj ) is the vector of goods consumed in country ν on the per capita basis and V
is the atmospheric accumulations of CO 2 measured in tones of CO 2 accumulated in theatmosphere, both at time t.
Remark: To simplify the exposition, our discussion is carried out as if the size of
population of each country ν is one. However, when we discuss the policy and institutional
implications, the standard case of numerous individuals in each country will be explicitly
brought out.
For each country ν , we assume that utility function uν (cν , V ) is strongly separablewith respect to cν and V in the sense originally introduced in Goldman and Uzawa (1964);
that is,5uν (cν , V ) = ϕν (V ) uν (cν ) .
The function ϕν (V ) expresses the extent to which people in country ν feel against
the effect of global warming, which is referred to as the impact index of global warming forcountry ν . We assume that the impact index function ϕν (V ) satisfies the following
conditions:ϕν (V ) > 0, ϕν '(V ) < 0, ϕν"(V ) < 0 for all V .
The relative rate of the marginal change in the impact index ϕν (V ) due to themarginal increase in the atmospheric accumulation of CO 2 is given byϕν '(V ),τ (V ) = − νϕ (V )
which is referred to as the impact coefficient of global warming for country ν . In whatν

follows, we assume that the impact coefficient of global warming τ ν (V ) is identical for all
countries. That is,τ ν (V ) = τ (V ) for all ν .
The impact coefficient τ (V ) satisfies the following conditions:τ (V ) > 0, τ '(V ) − [τ (V )]2 > 0 .
The impact index function ϕ (V ) of the following form is often postulated:ϕ (V ) = (Vˆ − V )β , 0  V < Vˆ ,(2)

where V̂ is the critical level of the atmospheric accumulation of CO 2 and β is the sensitivity
parameter, 0 < β < 1. If the level of atmospheric accumulations of CO 2 were higher thanthe critical level V̂ , irrevocable damage would be inflicted on the global environment. In
Uzawa (199l), where the concept of the impact coefficient of global warming is originallyintroduced, the critical level V̂ is assumed to be twice that prevailing before the Industrial
Revolution, i. e., V̂ =560 GtC.
For the impact index function ϕ (V ) of the form (2), the impact coefficient τ (V ) is
given by6β.
Vˆ − VWe assume that the neoclassical assumptions are satisfied for utility functions uν (cν ):τ (V ) =(U1')

uν (cν ) is defined, positive, continuous, and continuously twice-differentiable for allcν  0.
(U2')

Marginal utilities are positive for the consumption of goods cν = (cνj ):ucνν (cν ) > 0 for all cν  0.

(U3')

uν (cν ) is strictly quasi-concave with respect to cν  0; that is, for any pair of vectorsof consumption, c0ν , c1ν , such that uν (c0ν ) =uν (c1ν ) and c0ν ≠ c1ν ,uν [(1 − λ )cν0 + λ c1ν ] < (1 − λ ) uν (cν0 ) + λ uν (c1ν ) for all 0 < λ < 1.
(U4')

uν (cν ) is homogeneous of order one with respect to cν :uν (λ cν ) = λ uν (cν ) for all λ  0, cν  0 .

The Euler identity will play a crucial role in the following analysis.
uν (cν ) = ucνν (cν ) cν for all cν  0 .

Specifications for Production Possibility Sets
The conditions concerning the production of goods in each country ν are specified by the
production possibility set T ν .
Those factors of production that are essentially needed in the production of goods are
generically denoted by( = 1,, L ) . The endowments of factors of production available

in each country ν are expressed by a vector K ν = (K ν ) .
In the dynamic situation, the endowments of factors of production available in eachcountry ν at each time are endogenously determined in the sense that they are the outcomesof the choices made in the past by the members of the economy. In this paper, however, we
assume that the endowments of factors of production are regarded as given, focusing our7

attention upon the atmospheric accumulations of CO 2 and their implications upon the naturaland human environments.
The conditions concerning the production of goods in each country ν are specified by
the production possibility set T ν that summarizes the technological possibilities andorganizational arrangements for country ν , with the quantities of factors of production
available in country ν as given. The main conclusions of this paper, however, remain valid
under the general circumstances where the quantities of factors of production change overtime.
In country ν , the minimum quantities of factors of production that are needed to
produce goods by the vector of production x ν = ( x νj ) with the CO 2 emission at the level aνare specified by a vector-valued function f ν ( xν , aν ) = ( f ν ( xν , aν )) . The production
possibility set T ν is then given by
T ν = {( x ν , aν ): ( x ν , aν )  0, f ν ( x ν , aν )  K ν } .
We assume that marginal rates of substitution between the production of goods andthe emission of CO 2 are smooth and diminishing, there always exist trade-offs between the
production of goods and the emission of CO 2 , and the conditions of constant returns to scaleprevail. That is, we assume that(T1) f ν ( x ν , aν ) are defined, positive, continuous, and continuously twice-differentiable forall ( x ν , aν )  0.
(T2) fxνν ( x ν , aν ) > 0, faνν ( x ν , aν )  0 for all ( x ν , aν )  0.
(T3) f ν ( x ν , aν ) are strictly quasi-convex with respect to ( x ν , aν ) for all ( x ν , aν )  0.
(T4) f ν ( x ν , aν ) are homogeneous of order one with respect to ( x ν , aν ) ; that is,f ν (λ xν , λ aν ) = λ f ν ( xν , aν ) for all λ  0, ( x ν , aν )  0.
Postulates (T1-3), as specified above, imply that the production possibility set T ν is a
closed, convex set of vectors ( x ν , aν ) .

8Market Equilibrium and Global Warming
We consider the situation where carbon taxes at the rate of θ ν are levied upon the emissionof CO 2 in each country ν .
Market equilibrium for the world economy is obtained if we find the prices of goods
p at which total demand is equal to total supply:∑c = ∑ x .
ν

ν

ν

ν

Total CO 2 emissions, a, are given bya = ∑ aν .
νSupply and demand conditions are specified in the following manner.
(i) Supply conditions in each country ν are obtained by maximizing net profitspx ν − θ ν aνover ( x ν , aν ) ∈ T ν ; that is, subject to the constraints thatf ν ( x ν , aν ) K ν .
(ii) Demand conditions in each country ν are obtained by maximizing utility functionuν ( c ν , V ) = ϕ (V ) uν ( c ν )subject to budget constraints:pcν = yν , cν  0,
where yν is per capita national income of country ν :The optimum levels of production and CO 2 emissions in country ν are determined by
the following marginality conditions:p = rν f xνν ( xν , aν )(3)

θ ν = r ν [ − faνν ( x ν , aν )],(4)

where r ν is the vector of rentals prices of factors of production in country ν .
Because the technologies are assumed to be subject to constant returns to scale, (T4),
we have thatp x ν − θ ν aν = r ν [ fxνν ( x ν , aν ) x ν + faνν ( x ν , aν ) aν ] = rν f ν ( xν , aν ) = rν K ν .
9That is, the net evaluation of output is equal to the sum of the rentals payments to all factorsof production.
When all factors of production are owned by the individual members of the country ν, then national income yν of country ν is equal to the sum of the rentals payments r ν K ν andthe tax payments θ ν aν made by the producers for the emission of CO 2 in country ν ; that is,yν = rν K ν + θν aν = ( p xν − θν aν ) + θν aν = p xν .
That is, national income yν of country ν is equal to the value of outputs in units of market
prices, px ν .
The optimum vector of consumption cν is obtained by solving the following
marginality conditions:ϕ (V ) ucνν (cν ) = λν p ,where λν is the Lagrange unknown associated with the budgetary constraints.

10

3. Deferential Equilibrium

One of the basic problems we face in the dynamic analysis of global warming is to see how
the carbon tax rate θ ν in each country ν is determined. Our analysis will be carried out interms of the concept of deferential equilibrium, originally introduced in [14].
Deferential equilibrium is obtained if, when each country chooses the levels of
economic activities today, it takes into account the negative impact upon the future levels of
its own utilities brought about by the CO 2 emissions of that country today.
Consider the situation in which a combination ( x ν , aν ) of production vector xν and
CO 2 emissions aν is chosen in country ν . Suppose CO 2 emissions aν in country ν areincreased by a marginal amount. This would induce a marginal increase in the aggregate
amount of CO 2 emissions in the world, effecting a marginal increase in the atmospheric levelof CO 2 . The resulting marginal increase in the degree of global warming would induce a
marginal decrease in country ν 's utility. Deferential equilibrium is obtained if this marginal
decrease in country ν 's future utility due to the marginal increase in CO 2 emissions incountry ν today is taken into consideration in determining the levels of consumption,
production, and CO 2 emissions today.
The marginal decrease in country ν 's utility due to the marginal increase in CO 2
emissions in country ν is given by the partial derivative, with minus sign, of utility functionuν (cν , V ) of country ν with respect to atmospheric accumulations of CO 2 , V ; that is,∂ uν= −ϕ ' (V ) uν (cν ) = τ (V )ϕ (V ) uν (cν ),∂Vwhere τ (V ) is the impact coefficient of global warming.
−

We assume that future utilities of country ν are discounted at the social rate ofdiscount δ that is exogenously given. We also assume that the rate of discount δ is apositive constant and is identical for all countries in the world. As atmospheric carbon
dioxide is annually absorbed at the rate µ , the imputed price ψ ν of the atmospheric11

accumulations of CO 2 of each country ν is given by the discounted present value of themarginal decrease in country ν 's utility due to the marginal increase in CO 2 emissions incountry ν ; that is,
1
∂ uν
1
[−
]=
τ (V )ϕ (V ) uν (cν ) .
(5)
δ +µ ∂Vδ +µDeferential equilibrium is obtained if this marginal decrease in country ν 's futureψν =utility due to the marginal increase in CO 2 emissions today in country ν is balanced with themarginal increase in country ν 's utility due to the resulting marginal increase in the level ofproduction.
If the level of CO 2 emissions in country ν is aν , then the net level of country ν 's
utility is given by

ϕ (V ) uν (cν ) − ψ ν aν .
The choice of the levels of consumption, production, and CO 2 emissions for eachcountry ν under the deferential behavioristic postulates then may be viewed as the optimum
solution to the following maximum problem:
Find the combination (cν , x ν , aν ) of consumption vector cν , production vector xν ,and CO 2 emissions aν that maximizes the net level of country ν 's utilityϕν (V ) uν (cν ) − ψ ν aν
subject to the constraints thatpcν = pxν , cν 0(6)f ν ( x ν , aν ) K ν .

(7)

The maximum problem is solved in terms of the Lagrangian form:Lν (cν , x ν , aν ; λν , λν r ν )= ϕν (V ) uν (cν ) − ψ ν aν + λν ( pxν − pcν ) + λν rν [K ν − f ν ( xν , aν )] ,where V and ψ ν are given. The variables λν , λν r ν are the Lagrangian unknowns associatedwith constraints (6) and (7), respectively.
The optimum conditions are:12ϕν (V ) uνcν (cν ) = λν p
λν p = λν r ν fxνν ( x ν , aν )
ψ ν = λν r ν [ − faνν ( x ν , aν )].
When expressed in units of market prices, the optimum conditions are

α νϕν (V ) uνcν (cν ) = p(8)p = r ν fxνν ( x ν , aν )(9)

θ ν = r ν [ − faνν ( x ν , aν )],(10)

where α ν is the coefficient to convert values in units of the utility to those in units of market
prices1> 0,
λνand θ ν is the imputed price of atmospheric accumulations of CO 2 in units of market pricesαν =θν = α νψ ν ,which, by substitution into (5), implies
1τ (V )α νϕν (V ) uν (cν ) .
δ +µ
By multiplying both sides of relations (8) by cν ,

θν =

(11)

α νϕν (V ) uνcν (cν )cν = pcν .
By making use of the Euler identity and noting the budget constraints (6), we obtainα νϕν (V ) uν (cν ) = yν ,(12)where yν is per capita national income of country ν :yν = pxν .
In view of (11) and (12), we obtainτ (V ) νy .
δ +µMarginality conditions (9) and (10) imply that net profits
θν =

(13)

p x ν − θ ν aν
are maximized over the technological possibility set ( x ν , aν ) ∈ T ν , where carbon tax rate θ ν
is given by (13).
13Hence, the conditions for deferential equilibrium of global warming are identical withthose for the standard market equilibrium when carbon taxes with rate θ ν given by (13) are
levied; that is,(i) For each country ν , the consumption vector cν maximizes country ν 's utility functionuν (cν , V ) = ϕν (V ) uν (cν )subject to the budgetary constraints
pcν = yν , cν  0,
where yν is per capita national income of country ν , and the level of atmosphericaccumulations of CO 2 , V , is assumed to be given.
(ii) For each country ν , the combination ( x ν , aν ) of production vector x ν and CO 2
emissions aν maximizes net profitspxν − θν aνover the technological possibility set ( x ν , aν ) ∈ T ν .
Differential equilibrium for the world economy then is obtained if we find the pricesof goods p at which total demand is equal to total supply:∑c = ∑ x
ν

ν

ν

ν

and total CO 2 emissions a are given bya = ∑ aν .
νThe preceding discussion may be summarized by the following proposition.

Proposition 1. Deferential equilibrium corresponds precisely to the competitive market
equilibrium under the system of proportional carbon taxes, where, in each country ν , thecarbon taxes are levied at the rate θ ν that is proportional to the per capita national income

τ (V )of the impact coefficient ofδ +µglobal warming τ (V ) as the coefficient of proportion; that is,yν of each country ν , with the discounted present value

14τ (V ) νy ,
δ +µwhere τ (V ) is the impact coefficient of global warming, δ is the social rate of discount, andθ ν=µ is the rate at which atmospheric CO 2 is annually absorbed by the oceans.

4. Imputed Price and Sustainable Development

In the dynamic analysis of global warming, a crucial role is played by the concept of theimputed price of the atmospheric accumulations of CO 2 . The imputed price ψ νt of theatmospheric accumulations of CO 2 in units of utility for country ν at time t expresses theextent to which the marginal increase in the atmospheric accumulations of CO 2 at time t
induces the marginal decrease in the welfare level of country ν , including those of all future
generations. The calculation of the magnitude of the welfare gains may be made along thelines similar to those discussed in the previous sections.
Suppose ψ tν is the price in units of utility charged to the emission of CO 2 in country

ν at time t. The resulting pattern of consumption, production, and the level of CO2 emissions
in country ν are obtained as the optimum solution for the following maximum problem:
Find the combination (cν , x ν , aν ) of consumption vector cν , production vector xν ,and CO 2 emissions aν that maximizes the net level of country ν 's utilityϕν (V ) uν (cν ) − ψ ν aν
subject to the constraints thatpcν = pxν , cν 0f ν ( x ν , aν ) K ν .
The maximum problem is solved exactly in the same manner as for the case of15

deferential equilibrium. The marginal decrease mνt in the welfare level of country ν at time t
induced by the marginal increase in the atmospheric accumulations Vt of CO 2 at time t isgiven by∂[ϕν (Vt ) uν (cνt )] = τ (Vt )ϕν (Vt ) uν (cνt ),∂ Vtwhich, in units of market prices, may be expressed asmνt = −

(14)α νt mνt = τ (Vt )α νt ϕν (Vt ) uν (cνt ) = τ (Vt ) yνt ,(15)where yν is per capita national income of country ν at time t.
Let us denote by π νt the imputed price of the atmospheric accumulations of CO 2 inunits of market prices for country ν at time t; that is, π νt = α νt ψ νt . Then, by noting (15), we
have from the definition of the imputed price that
∞

∞

t

t

π νt = ∫ ατν mτν e −(δ + µ )(τ − t )dτ = ∫ τ (Vτ ) yτν e −(δ + µ )(τ − t )dτ .

(16)

By differentiating both sides of (16) with respect to time t, we obtain the following
differential equation:π˙ νt = (δ + µ )π νt − τ (Vt ) yνt .

(17)

Differential equation (17) is nothing but the Euler-Lagrange differential equation in
the calculus of variations or the Ramsey-Keynes equation in the theory of optimum economic
growth.
To clarify the intrinsic meaning of the imputed price of the atmosphericaccumulations of CO 2 , we introduce a virtual capital market at time t that is perfectlycompetitive and the atmospheric accumulations of CO 2 are transacted as an asset. The
imputed price π νt at time t is identified with the market price. Consider the situation in which
the unit of such as asset is held for a short time period [ t, t + ∆t ] , ( ∆t > 0 ). The gains
obtained by holding such an “asset” are composed of “capital gains” ∆π νt = π νt + ∆t − π νt and“earnings” τ (Vt ) yνt ∆t , minus “depreciation charges” µπ νt ∆t ; that is,∆π νt + τ (Vt ) yνt ∆t − µπ νt ∆t .
On other hand, the cost of holding such an “asset” for the time period [ t, t + ∆t ] is16

“interest” payment δπ νt ∆t , where the social rate of discount is identified with the “ market
rate of interest”. Hence, on the virtual capital market, these two amounts become equal; thatis,∆π νt + τ (Vt ) yνt ∆t − µπ νt ∆t = δπ νt ∆t ,which, by dividing both sides by ∆t and taking the limit as ∆t → 0 , yields differential
equation (17).
It may be noted that the time derivative in the differential equation (17) refers to thefictitious time of the virtual capital market at time t and remains valid only at time t. For any
future time τ ( τ > t), equation (17) has to be modified by taking into account the changing
circumstances that are relevant in the determination of market equilibrium.
We define that the imputed price π νt of the atmospheric accumulations of CO 2 forcountry ν is at the sustainable level at time t, if π̇ νt = 0 at time t. That is,τ (Vt ) νy at time t.
δ+ µ tA time-path (Vt ) of the atmospheric accumulations of CO 2 is defined sustainable forπ νt = θ νt , θ νt =country ν , if the imputed price π νt for country ν is at the sustainable level at all time t. It isdefined sustainable, if it is sustainable for all countries in the world.
The discussion above may be summarized as the following proposition:

Proposition 2. Sustainable time-paths (Vt ) of the atmospheric accumulations of CO 2 are
obtained as the competitive market equilibrium under the system of proportional carbon
taxes, where, in each country ν , the carbon taxes are levied at the rate θ ν that is
proportional to the per capita national income yν of each country ν , with the discounted

τ (V )of the impact coefficient of global warming τ (V ) as the coefficient ofδ +µproportion; that is,present valueτ (V ) νy ,δ +µwhere τ (V ) is the impact coefficient of global warming, δ is the social rate of discount, and
θ ν=

17

µ is the rate at which atmospheric CO 2 is annually absorbed by the oceans.

5. A Hypothetical CaseA hypothetical case of the incidences of the proportional carbon taxes with the coefficient ofproportion 0.01 is presented in terms of the statistical data of 2005 (in US$). The data
include all radiative forcing agents (RFA).
Country

National Income

RFA Increase

Imputed Price

Carbon Taxes

per capita

per Capita (Ct)per Ctper capita

United States

42,000

5.90

420

2,500

Canada34,000

6.20

340

2,100

United Kingdom

32,000

3.00

320

950

France31,000

2.20

310

680

Germany

31,000

3.20

310

980

Italy

28,000

2.20

280

600

Netherlands

35,000

3.60

350

1,200

Sweden32,000

1.90

320

610

Denmark

34,000

3.20

340

1,100

Indonesia

3,100

1.70

30

50

Japan

31,000

2.70

310

840

Korea

21,000

2.60

210

560

Malaysia

11,000

1.90

110

210

Philippines

3,200

0.30

30

8

Singapore

40,000

3.20

400

1,300

Thailand

6,900

1.20

70

80

India

2,200

0.30

20

7

China

4,100

1.10

40

40

Australia

33,000

7.10

330

2,300

New Zealand

23,000

3.50

230

790

Sources UNFCCC, World Development Indicators
18

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