DBJ Research Center on Global Warming
ECONOMIC ANALYSIS No. 15 (December 2009)

Global Warming, Proportional Carbon Taxes,
and International Fund for Atmospheric Stabilization

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 that sustainable
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 arelevied at the rate that is proportional to the per capita national income of each country, with thediscounted present value of the impact coefficient of global warming as the coefficient ofproportion.

Keywords
dynamic model of global warming
imputed pricesustainable development
proportional carbon taxes

1. Introduction

The atmospheric concentration of greenhouse gases, particularly carbon dioxide, has been
increasing since the times of the Industrial Revolution, with an accelerated rate in the last several
decades. According to the IPCC reports (1991a, 1991b, 1992, 1996a, 1996b, 2002, 2007), it isestimated that, if the emission of carbon dioxide and other greenhouse gases and the disruptionof tropical rain forests were to continue at the present pace, global average air surface
temperature toward the end of the twenty-first century would be 3−6 ˚C higher than the level
prevailing before the Industrial Revolution, resulting in drastic changes in climatic conditions
and accompanying disruption of the biological and ecological environments.
The problems of global warming are genuinely dynamic. From past human activities weinherit an excess concentration of atmospheric CO 2 , and the choices we make today concerning
the use of fossil fuels and related activities significantly affect all future generations through thephenomenon of global warming that is brought about by the atmospheric concentrations of CO 2
due to the combustion of fossil fuels today. Thus, we explicitly have to take into account thedecreases in the welfare levels of all future generations caused by the increases in theatmospheric accumulations of CO 2 today.
In this paper, the economic analysis of global warming is developed within thetheoretical framework of dynamic analysis of global warming, as introduced in Uzawa (1991,
1993, 2003, 2005). We are particularly concerned with the policy arrangements of a proportional
carbon tax scheme under which the tax rate is made proportional to the levels of the per capita
national income of the countries where greenhouse gases are emitted. In the first part of thispaper, we consider the case in which the oceans are the only reservoir of CO 2 on the earth,whereas, in the second part, we explicitly take into consideration the role of the terrestrial forests
in moderating processes of global warming by absorbing the atmospheric accumulation of CO 2 ,on the one hand, and in affecting the level of the welfare of people in the society by providing a1

decent and cultural environment, on the other.
Finally, an International Fund for Atmospheric Stabilization is proposed to narrow thedivergence in economic performance between developed countries and developing countries thathas been steadily widening in the last several decades. Many institutional and policy measuresthat have been devised internationally or bilaterally have not had much impact in narrowing the
gap between these two groups of countries. The International Fund for Atmospheric Stabilization
is an institutional framework in which it is possible to combine an international arrangement tostabilize atmospheric equilibrium with a redistributive scheme to help developing countries toaccelerate processes of economic development.

2. Atmospheric Concentrations of Carbon Dioxide

We denote by Vt the amount of CO 2 that has 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 (=10 9 tons in carbon content), approximately corresponding to thedensity of 280 ppm (= part per million). The current level of 760 GtC (approximately 380 ppm)
is expressed as Vt = 160 GtC.
The premises of the dynamic model introduced in this paper are primarily based on thescientific findings on global warming and related subjects referred to in Uzawa (1991, 2003). A
detailed description of the model was described in Uzawa (2003, 2005).

2

Changes of CO 2 Accumulations to Natural Causes
The atmospheric accumulations of CO 2 change over time as the result of natural andanthropogenic factors. A certain portion of atmospheric concentrations of CO 2 is absorbed by
the oceans (roughly estimated at 50%) and to a lesser extent by living terrestrial plants. In thesimple dynamic model postulated in this section, the exchange of carbon between theatmosphere and the terrestrial biosphere is not taken into consideration.
Approximately 75-90 GtC of carbon are annually exchanged between the atmosphere andthe surface oceans. The studies made by several meteorologists and oceanologists — inparticular, Keeling (1968, 1983) and Takahashi et al. (1980) — 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.
In the simple dynamic model introduced in this section, we assume that the amount of
atmospheric CO 2 annually absorbed by the oceans is given by µVt , where Vt is the atmosphericconcentrations of CO 2 measured in actual tons of CO 2 with the pre-Industrial Revolution levelof 600 GtC as the origin of measurement. The rate of absorption µ would have a magnitude
2—4%. In what follows, we assume that µ = 0.04.
In the simple dynamic model, we assume that the anthropogenic change in atmospheric
CO 2 is exclusively caused by the combustion of fossil fuels in connection with industrial,
agricultural, and urban activities.
The change in the atmospheric level of CO 2 is given byV̇t = at − µVt ,where at is the annual rate of increase in the atmospheric level of CO 2 due to anthropogenic
activities and µVt is the amount of atmospheric CO 2 annually absorbed by the oceans.
The rate of anthropogenic change in the atmospheric level of CO 2 , at , is determined by
the combustion of fossil fuels and is closely related to the levels of production and consumption
activities conducted during the year observed. In what follows, the time variable t is often
omitted for the sake of expository brevity.
33. The Simple Dynamic Model of Global Warming

We postulate that each greenhouse gas is measured so as to equate the greenhouse effect withthat of CO 2 . We postulate that the welfare effect of global warming is measured in relation to
the stock of greenhouse gases accumulated in the atmosphere.
There are a finite number of individual countries in the world that share the earth's
atmosphere as social common capital. Each country is generically denoted by ν = 1, . . ., n.
The behavioral characteristics of individual countries are expressed in the aggregate by
two representative economic agents−the consumers, who are concerned with the choice ofeconomic activities related to consumption, on the one hand, and the producers, who are in
charge of the choice of technologies and levels of productive activities, on the other.

Specifications of the Utility Functions
We assume that the utility function for each country ν is expressed in the following manner:uν = uν (cν , aνc , V ) ,where cν = (cνj ) is the vector of goods consumed in country ν on the per capita basis, aνc is theamount of CO 2 emissions released during the processes of consumption, and V is theatmospheric concentration of CO 2 measured in tons of carbon in CO 2 .
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 also assume that utility function uν (cν , aνc , V ) is strongly
separable with respect to (cν , aνc ) and V in the sense originally introduced by Goldman andUzawa (1964); that is,4uν = ϕν (V ) uν (cν , aνc ) .
As in the case of the models of social common capital introduced in Uzawa (2005), thefunction φν (V ) expresses the extent to which people in country ν are adversely affected by
global warming. It is referred to as the impact index of global warming. We assume that the
impact index function φν (V ) of global warming for each country ν satisfies the following
conditions:φν (V ) > 0, φν'(V ) < 0, φν"(V ) < 0 for all V > 0.
The impact coefficient of global warming for country ν is the relative rate of themarginal change in the impact index due to the marginal increase in the atmosphericaccumulation of CO 2 ; that is,φν '(V ).
φν (V )
We assume that the impact coefficients of global warming τ ν (V ) are identical for allτ ν (V ) = −countries ν :τ ν (V ) = τ (V ) for all ν .
The impact coefficient function τ (V ) satisfies the following conditions:τ (V ) > 0, τ '(V ) > 0 for all V > 0.
The impact index function φ (V ) of the following form is often postulated:φ (V ) = (Vˆ − V ) β , 0 < V < Vˆ ,where V̂ is the critical level of the atmospheric accumulation of CO 2 and β is the sensitivity
parameter (0 < β < 1). The critical level V̂ of the atmospheric accumulation of CO 2 is usuallyassumed to be twice the level prevailing before the Industrial Revolution; that is, V̂ = 600 GtC.
The impact coefficient τ (V ) is given byβτ (V ) = ˆ.
V −V
Utility functions uν (cν , aνc ) satisfy the following neoclassical conditions:(U1) Utility function uν (cν , aνc ) is defined, positive, continuous, and continuously twicedifferentiable for all (cν , aνc ) 0.
5

(U2) Marginal utilities are positive both for the consumption of goods cν and CO 2 emissions aνc:uνcν (cν , aνc )0 , uνaν (cν , aνc )0 for all (cν , aνc ) 0.
c

ν

ν

ν
c

(U3) Utility function u (c , a ) is strictly quasi-concave with respect to (cν , aνc ) 0.
(U4) Utility function uν (cν , aνc ) is homogeneous of order 1 with respect to cν :uν ( tcν , taνc ) = tuν (cν , aνc ) for all (cν , aνc ) 0.
The Euler identity holds:uν (cν , aνc ) = uνcν (cν , aνc ) cν + uνaν (cν , aνc ) aνc for all (cν , aνc ) 0.
c

The Consumer Optimum
The world markets for produced goods are assumed to be perfectly competitive and prices ofgoods are denoted by a nonzero, nonnegative vector p ( p ≥ 0). Carbon taxes at the rate θ ν arelevied upon the emission of CO 2 in each country ν . Carbon tax rate θ ν is assumed to benonnegative: θ ν  0.
Suppose national income of country ν in units of world prices is given by yν . Then, theconsumers in country ν would choose consumption vector cν and CO 2 emissions aνc thatmaximize country ν ' s utility functionuν (cν , V ) = φν (V ) uν (cν , aνc )subject to the budget constraints
pcν + θν aνc = yν , (cν , aνc ) 0.
The optimum combination of consumption vector cν and CO 2 emissions aνc ischaracterized by the following marginality conditions:α νφν (V ) uνcν (cν , aνc )  p(mod. cν )(1)

α νφν (V ) uνaν (cν , aνc )  θν(mod. aνc ) ,(2)cwhere α ν is the inverse of the marginal utility of income yν of country ν .
6

If we note the Euler identity for the utility function uν (cν , aνc ) of each country ν , we havethe following relation:

α νφν (V ) uν (cν , aνc ) = yν .

(3)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 that summarizes the technological possibilities and organizationalarrangements for country ν ; the endowments of factors of production available in country ν aregiven.
We assume that there are a finite number of factors of production that are essentiallyneeded in the production of goods. They are generically denoted by f ( f = 1,

, F ) . Without

loss of generality, we may assume that the factors of production needed in productive activities
are the same for all countries involved.
The endowments of factors of production available in each country ν are expressed by avector K ν = (K νf ), where K ν ≥ 0.
In each country ν , the minimum quantities of factors of production required to produce
goods by the vector of production xν with CO 2 emissions at the level aνp are specified by avector-valued function:f ν ( xν , aνp ) = ( f fν ( xν , aνp )) .
We assume that marginal rates of substitution between the production of goods and theemission of CO 2 are smooth and diminishing, that there always exist trade-offs between them,and that the conditions of constant returns to scale prevail. That is, we assume(T1) f ν ( xν , aνp ) are defined, positive, continuous, and continuously twice-differentiable for all( xν , aνp )  0 .
(T2) f xνν ( xν , aνp ) > 0, f aνν ( xν , aνp )  0 for all ( xν , aνp )  0 .
p7

(T3) f ν ( xν , aνp ) are strictly quasi-convex with respect to ( xν , aνp ) for all ( xν , aνp )  0 .
(T4) f ν ( xν , aνp ) are homogeneous of order 1 with respect to ( xν , aνp ) for all ( xν , aνp )  0 .
The following Euler identity holds:f ν ( xν , aνp ) = f xνν ( xν , aνp ) xν + f aνν ( xν , aνp ) aνp for all ( xν , aνp )  0 .
p

The production possibility set of each country ν , T ν , is composed of all combinations( xν , aνp ) of vectors of production xν and CO 2 emissions aνp that are possibly produced with theorganizational arrangements and technological conditions in country ν and the given
endowments of factors of production K ν of country ν . Hence, it may be expressed as{}T ν = ( xν , aνp ): ( xν , aνp )  0, f ν ( xν , aνp )  K ν .
Postulates (T1—3) imply that the production possibility set T ν is a closed, convex set of
J + 1-dimensional vectors ( xν , aνp ) .

The Producer Optimum
The producers in country ν would choose those combinations ( xν , aνp ) of vectors of productionxν and CO 2 emissions aν that maximize net profitspxν − θν aνpover ( xν , aνp ) ∈ T ν .
Conditions (T1−3) postulated above ensure that, for any combination of price vector p
and carbon tax rate θ ν , the optimum combination ( xν , aνp ) of vector of production xν and CO 2
emissions aνp always exists and is uniquely determined.
To see how the optimum levels of production and CO 2 emissions are determined, let us
denote by rν ( rν  0) the vector of imputed rental prices of factors of production. Then the
optimum conditions arep  rν f xνν ( xν , aνp )(mod. xν )(4)θν  rν [− f aνν ( xν , aνp )](mod. aνp )

(5)

p

8

f ν ( xν , aνp ) K ν(mod. rν ) .

(6)

The first condition (4) means the familiar principle that the choice of production
technologies and the levels of production are adjusted so as to equate marginal factor costs with
output prices.
The second condition (5) similarly means that CO 2 emissions are controlled so that themarginal loss due to the marginal increase in CO 2 emissions is equal to carbon tax rate θ ν whenaνp > 0 and is not larger than θ ν when aνp = 0.
The third condition (5) means that the utilization of factors of production does not exceed
the endowments, and the conditions of full employment are satisfied whenever rental price rfν ispositive.
We have assumed that the technologies are subject to constant returns to scale (T4), andthus, in view of the Euler identity, conditions (2) − (4) imply thatpxν − θν aνp = rν [ f xνν ( xν , aνp ) xν + f aνν ( xν , aνp ) aνp ]
p

ν

ν

ν

ν
p

ν

ν

= r f (x , a ) = r K .
That is, the net evaluation of output is equal to the sum of the rental payments to all factors of
production.
Suppose all factors of production are owned by individual members of the country ν .
Then, national income yν of country ν is equal to the sum of the rental payments rν K ν and thetax payments θν aν , aν = aνc + aνp , made for the emission of CO 2 in country ν , all on the percapita basis; that is,yν = rν K ν + θν aν = ( pxν − θν aνp ) + θν aν
= pcν + θν aνc .
Hence, national income yν of country ν is equal to the sum of the expenditures, thus
conforming with the standard practice in national income accounting.

9Market Equilibrium and Global Warming
We consider the situation in which carbon taxes at the rate of θ ν are levied on the emission of
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 , aν = aνc + aνp .
ν(i) Demand conditions in each country ν are obtained by maximizing utility functionuν = φν (V ) uν (cν , aνc )subject to budget constraints
pcν + θν aνc = yν ,where yν is the national income of country ν .
(ii) Supply conditions in each country ν are obtained by maximizing net profitspxν − θν aνpover ( xν , aνp ) ∈ T ν .
(iii) Total CO 2 emissions in the world, a, are given as the sum of CO 2 emissions in all
countries; that is,a = ∑ aν , aν = aνc + aνp .
ν

10

4. Imputed Price and Sustainable Development

In the dynamic analysis of global warming, a crucial role is played by the concept of the imputed
price of the atmospheric accumulations of CO 2 . The imputed price ψ νt of the atmosphericaccumulations of CO 2 in units of utility for country ν at time t measures the extent to which themarginal 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.
Suppose ψ tν is the price in units of utility charged to the emission of CO 2 in country νat time t. The resulting pattern of consumption, production, and the level of CO 2 emissions incountry ν are obtained as the optimum solution for the following maximum problem:
Find the combination (cν , xν , aνc , aνp , aν ) of consumption victor cν , production vector xν ,and CO 2 emissions, aνc , aνp , aν , that maximizes the utility of country νu*ν = φν (V ) uν (cν , aνc ) − ψ ν aν
subject to the constraints thatpcν = pxν(7)f ν ( xν , aνp ) K ν(8)aν = aνc + aνp ,(9)where V and ψ ν are given.
The maximum problem is solved in terms of the Lagrangian formLν (cν , xν , aνc , aνp , aν ; λν , λν rν , λνθν ) = φν (V ) uν (cν , aνc ) − ψ ν aν
+ λν ( pxν − pcν ) + λν rν [K ν − f ν ( xν , aνp )] + λνθν ( aν − aνc − aνp ) ,where the variables λν , λν rν , λνθν are the Lagrangian unknowns associated with constraints (7),(8), and (9), respectively.
The optimum conditions are

α νφν (V ) uνcν (cν , aνc )  p(mod.cν )(10)

α νφν (V ) uνaν (cν , aνc )  θν(mod. aνc )

(11)

c

11

p  rν f xνν ( xν , aνp )(mod. xν )(12)θν  rν [− f aνν ( xν , aνp )](mod. aνp )(13)f ν ( xν , aνp ) K ν(mod. rν ) ,

(14)

p

1
.
λν
Marginality conditions (13) and (14) imply that net profits

where α ν is the inverse of marginal utility of income, i. e., α ν =pxν − θν aνp
are maximized over the technological possibility set ( xν , aνp ) ∈T ν .
The marginal decrease mνt in the welfare level of country ν at time t induced by themarginal increase in the atmospheric accumulations Vt of CO 2 at time t is given by∂[ϕν (Vt ) uν (cνt )] = τ (Vt )ϕν (Vt ) uν (cνt ),∂ Vtwhich, in units of market prices, may be expressed asmνt = −α νt mνt = τ (Vt )α νt ϕν (Vt ) uν (cνt ) = τ (Vt ) yνt ,

(15)

(16)

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 in unitsof market prices for country ν at time t; that is,π νt = α νt ψ νt .
Then, by noting (16), 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 .

(17)

By differentiating both sides of (17) with respect to time t, we obtain the following
differential equation:π˙ νt = (δ + µ )π νt − τ (Vt ) yνt .

(18)

Differential equation (18) is nothing but the Euler-Lagrange differential equation in thecalculus 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 atmospheric accumulations of
CO 2 , we introduce a virtual capital market at time t that is perfectly competitive and the12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 an asset is held
for a short time period [ t, t + ∆t ] , ( ∆t > 0 ). The gains obtained by holding such an “asset” arecomposed 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 ] is“interest” payment δπ νt ∆t , where the social rate of discount is identified with the “market rate ofinterest”. Hence, on the virtual capital market, these two amounts become equal; that is,∆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(18).
It may be noted that the time derivative in the differential equation (18) 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 (18) has to be modified by taking into account the changing
circumstances due to the accumulation of capital.
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 times 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 1. 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,
13where, in each country ν , the carbon taxes are levied at the rate θ ν that is proportional to theτ (V )ofδ +µ
the impact coefficient of global warmingτ (V ) as the coefficient of proportion; that is,per capita national income yν of each country ν , with the discounted present valueτ (V ) νy ,δ +µwhere τ (V ) is the impact coefficient of global warming, δ is the social rate of discount, and µθ ν=

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

5. Global Warming and Forests

The economic analysis of global warming, as developed in the previous sections, may be
extended to examine the role of terrestrial forests in moderating processes of global warming, on
the one hand, and in affecting the level of the welfare of people in the society by providing adecent and cultural environment, on the other.
In the simple, dynamic analysis of global warming introduced in the previous sections,
we have assumed that the combustion of fossil fuels is the only cause for atmospheric instability
and that the surface ocean is the only reservoir of carbon on the earth's surface that exchanges
carbon with the atmosphere. In this section, we consider the role of terrestrial forests,
particularly tropical rain forests, in stabilizing the processes of atmospheric equilibrium.
Terrestrial forests are regarded as social common capital and managed by social
institutions with an organizational structure similar to that of private enterprise except for themanner in which prices of the forests themselves and products from the forests are determined.
We assume that the amount of atmospheric CO 2 absorbed by the terrestrial forest per hectare ineach country ν is a certain constant on the average to be denoted by γ ν ( γ ν 0). Then the basic
dynamic equation concerning the change in the atmospheric concentrations of CO 2 may be
14modified to take into account the amount of atmospheric CO 2 absorbed by terrestrial forests. We
haveV̇t = at − ∑ γ νt Rνt − µVt ,

(19)

ν

where at is total CO 2 emissions in the world;at = ∑ aνt ,ννt

R is the acreages of terrestrial forests of country ν , µ is the rate at which atmospheric CO 2 isabsorbed by the oceans. We assume that the carbon sequester rate for temperate forests is around
7.5 tC/ha/yr and for tropical rain forests, it is assumd at 9.6-l0.0 tC/ha/yr.
The change in the acreages of the terrestrial forests Rν in each country ν is determined
first by the levels of reforestation activities and secondly by various economic activities carriedout in country ν during the year in question−particularly by agricultural and lumber industries
and by processes of urbanization. We denote by zνt the acreages of terrestrial forests annuallyreforested and by bνt the acreages of terrestrial forests in country ν annually lost due toeconomic activities. Then the acreages of terrestrial forests Rνt in each country ν are subject tothe following differential equations:Ṙνt = zνt − bνt .

Specifications for Utility Functions
We assume that the utility level uν of each country ν is influenced by the acreages of terrestrial
forests Rν in country ν , in addition to the atmospheric concentrations of CO 2 , V . That is, theutility function for each country ν is expressed in the following manner:uν = uν (cν , aνc , Rν , V ) ,where cν is the vector of goods consumed in country ν , aνc is the amount of CO 2 emitted by the
consumers in country ν , Rν is the acreages of terrestrial forests in country ν , and V is theatmospheric concentrations of CO 2 accumulated in the atmosphere, all at time t.
15

For each country ν , we assume that utility function uν (cν , aνc , Rν , V ) is strongly separablewith respect to (cν , aνc ) , Rν , and V :uν (cν , aνc , Rν , V ) = φν (V )ϕν ( Rν ) uν (cν , aνc ) .
As with the case discussed in the previous sections, the function φν (V ) expresses theextent to which people in country ν are adversely affected by global warming, which is referred
to as the impact index of global warming. Similarly, the function φν ( Rν ) expresses the extent towhich people in country ν are positively affected by the presence of the terrestrial forests incountry ν , which is referred to as the impact index of forests. We assume that the impact
indices, φν (V ) and ϕν ( Rν ) , satisfy the following conditions:φν (V ) > 0, φν'(V ) < 0, φν"(V ) < 0 ; ϕν ( Rν ) > 0, ϕν'( Rν ) > 0, ϕν"( Rν ) < 0 .
The impact coefficients of global warming and forests are, respectively, defined byφν '(V ) ν ν ϕν'( Rν ), τ (R ) = ν ν .
φν (V )ϕ (R )
We assume that the impact coefficients of global warming τ ν (V ) are identical for allντ (V ) = −countries ν :τ ν (V ) = τ (V ) for all ν .
The impact coefficient functions, τ (V ) and τ ν ( Rν ) , satisfy the following conditions:τ (V ) > 0, τ '(V ) > 0; τ ν ( Rν ) > 0, τ ν '( Rν ) > 0.
We assume that, for each country ν , the utility function uν (cν , aνc ) satisfies theconditions (U1)−(U4), as introduced in Section 3.

The Consumer Optimum
The consumers in country ν would choose the vector of consumption cν and CO 2 emissions aνcthat maximizes country ν ' s utility functionuν (cν , aνc , Rν , V ) = φν (V )ϕν ( Rν ) uν (cν , aνc )subject to the budget constraints
16

pcν + θν aνc = yν ,where yν is national income of country ν in units of world prices.
The optimum combination of consumption cν and CO 2 emissions aνc is characterized by
the following marginality conditions:α νφν (V )ϕν ( Rν ) uνcν (cν , aνc )  p(mod. cν )(20)

α νφν (V )ϕν ( Rν ) uνaν (cν , aνc )  θν(mod. aνc ) ,

(21)

c

where α ν is the inverse of the marginal utility of income yν of country ν .
The linear homogeneity hypothesis for the utility function uν (cν , aνc ) implies thatα νφν (V )ϕν ( Rν ) uν (cν , aνc ) = pcν + θν aνc = yν .

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 in exactly the same manner as in the previous sections.
In each country ν , the minimum quantities of factors of production needed to produce
goods by the vector of production xν with the use of the natural resources of the forests by theamount bν and the CO 2 emission at the level aνp are specified by a vector-valued functionf ν ( xν , bν , aνp ) = ( f νf ( xν , bν , aνp )) .
Similarly, the minimum quantities of factors of production needed to engage inreforestation activities at the level zν are specified by a vector-valued function gν ( zν ).
We assume that marginal rates of substitution between the production of goods, the useof the natural resources of forests, reforestation activities, and the emission of CO 2 are smoothand diminishing, trade-offs always exist between them, and the conditions of constant returns toscale prevail. That is, we assume that(T'1) f ν ( xν , bν , aνp ) and gν ( zν ) are defined, positive-valued, continuous, and continuously
twice-differentiable for all ( xν , bν , aνp ) 0 and zν 0, respectively.
17

(T'2) f xνν ( xν , bν , aνp ) 0, f bνν ( xν , bν , aνp ) < 0, f aνν ( xν , bν , aνp ) < 0 for all ( xν , bν , aνp ) 0
p

ν
zν

ν

g ( z ) 0 for all zν 0.
(T'3) f ν ( xν , bν , aνp ) and gν ( zν ) are strictly quasi-convex with respect to ( xν , bν , aνp ) and zν ,respectively.
(T'4) f ν ( xν , bν , aνp ) and gν ( zν ) are homogeneous of order 1 with respect to ( xν , bν , aνp ) and zν ,respectively.
The production possibility set T ν is given by{}

T ν = ( xν , zν , bν , aνp ): ( xν , zν , bν , aνp )  0, f ν ( xν , bν , aνp ) + gν ( zν )K ν ,where K ν is the vector of endowments of fixed factors of production in country ν .
Postulates (T'1-T'3), as specified above, imply that the production possibility set T ν is a
closed convex set of ( xν , zν , bν , aνp ) .

The Producer Optimum
Suppose that prices of goods are given by p and the imputed price of forests in each country νby π ν , whereas carbon taxes at the rate θ ν are levied on the emission of CO 2 in each country ν .
Forests are regarded as social common capital, and there are no markets on which either
the ownership of forests or the entitlements for the products from forests are transacted. Hence,
prices of forests are generally not market prices, but rather imputed prices. The imputed price ofthe ownership of a particular forest is the discounted present value of the stream of the marginalutilities of the forest and the expected value of the entitlements for the natural resources in forest
in the future.
The producers in country ν would choose those combinations ( xν , zν , bν , aνp ) of vectorsof production x ν , levels of reforestation zν , use of resources of forests bν , and CO 2 emissionsaνp that maximize net profitspxν + π ν ( zν − bν ) − θν aν
18

over ( xν , zν , bν , aνp ) ∈ T ν .
Marginality conditions for the producer optimum arep  rν f xνν ( xν , bν , aνp )(mod. xν )(22)

π ν  rν [− f bνν ( xν , bν , aνp )](mod. bν )(23)θν  rν [− f aνν ( xν , bν , aνp )](mod. aνp )(24)

π ν  rν gνzν ( zν )(mod. zν )(25)

f ν ( xν , bν , aνp ) + gν ( zν )  K ν (mod. rν ) ,

(26)

p

where rν is the vector of imputed rental prices of factors of production.
The meaning of these conditions is simple. Condition (22) means that the choice of
production technologies and the levels of production are adjusted so as to equate marginal factor
costs with output prices.
Condition (23) means that the use of resources of forests is determined so that themarginal gain due to the marginal increase in the use of resources of forests is equal to imputed
price of forests π ν when bν > 0 and is not larger than π ν when bν = 0.
Condition (24) means that CO 2 emissions are determined so that the marginal gain due tothe marginal increase in CO 2 emissions is equal to carbon tax rate θ ν when aνp > 0 and is notlarger than θ ν when aνp = 0.
Condition (25) means that the level of reforestation activities is determined so that themarginal gain due to the marginal increase in the use of resources of forests is equal to theimputed price of forests π ν when zν > 0 and is not larger than π ν when zν = 0.
Condition (26) means that the employment of factors of production does not exceed theendowment, and conditions of full employment are satisfied whenever rental price is positive.
We have assumed that the technologies are subject to constant returns to scale, and thus,
in view of the Euler identity, conditions (22)− (26) imply thatpxν + π ν ( zν − bν ) − θν aνp= rν [ f xνν ( xν , bν , aνp ) xν + f bνν ( xν , bν , aνp )bν + f aνν ( xν , bν , aνp ) aνp + gνzν ( zν ) zν ]p19

= rν [ f xνν ( xν , bν , aνp ) xν + gνzν ( zν ) zν − f bνν ( xν , bν , aνp )bν − f aνν ( xν , bν , aνp ) aνp ]
p

ν

ν

ν

ν

ν
p

ν

ν

ν

ν

= r [ f ( x , b , a ) + g ( z )] = r K .
That is, the net evaluation of output is equal to the sum of the rental payments to all factors of
production.

6. Imputed Prices of Capital in General

The imputed price, in units of the utility, of each kind of capital at time t, ψ t , is the discounted
present value of the marginal increases in total utility in the future due to the marginal increasein the stock of that kind of capital at time t. When we denote by rτ the marginal increase in the
total utility at future time τ , the imputed price at time t, ψ t , is given by∞ψ t = ∫ rτ e −(δ + µ )(τ − t )dτ .

(27)

t

By differentiating both sides of (27) with respect to time t, we obtain the following
differential equation:ψ˙ t = (δ + µ )ψ t − rt .

(28)

We suppose that capital is transacted as an asset on a virtual capital market that isperfectly competitive and the imputed price ψ t is identified with the market price at time t.
Consider the situation in which the unit of such an asset is held for the short time period[ t, t + ∆t ] (∆t > 0). The gains obtained by holding such an asset are composed of "capitalgains" ∆ψ t = ψ t + ∆t − ψ t and "earnings" rt ∆ t ; that is,∆ψ t + rt ∆ t .
On other hand, the costs of holding such an asset for the time period [ t, t + ∆t ] consistof "interest payments" δψ t ∆ t and "depreciation charges" µψ t ∆ t , where the social rate ofdiscount δ is identified with the market rate of interest; that is,20δψ t ∆t + µψ t ∆ t .
On the virtual capital market, these two amounts become equal; that is,∆ψ t + rt ∆ t = δψ t ∆t + µψ t ∆ t .
By dividing both sides of this equation by ∆t and taking the limit as ∆t → 0 , w obtain relation(28).
We define that the imputed price ψ t is at the sustainable level at time t, if it remains
stationary at time t; i. e.,ψ̇ t = 0 at time t,where it may be reminded that ψ̇ t refers to the time derivative with respect to the time of thevirtual capital market at time t.
From the basic differential equation (28), the imputed price ψ t is at the sustainable levelat time t, if, and only if,rt
at time t,
δ+ µwhere rt is the marginal increase in total utility due to the marginal increase in the stock ofψt =capital of that kind at time t.

7. Imputed Prices of Atmospheric Concentrations of CO 2 and Forests

Consider the situation in which a combination (cν , aνc , xν , zν , bν , aνp ) of vectors of consumption
and production, cν , x ν , level of reforestation activities zν , CO 2 emissions aνc , and aνp ,aν = aνc + aνp , is chosen in country ν . Imputed prices of atmospheric concentrations of CO 2 andforests are defined as follows.
Suppose CO 2 emissions in country ν , aν , are increased by a marginal amount. This
would induce a marginal increase in the aggregate amount of CO 2 emissions in the world,21

causing a marginal increase in the atmospheric level of CO 2 . The resulting marginal increase in
the degree of future global warming would cause a marginal decrease in country ν 's utility.
The marginal decrease in country ν 's utility due to the marginal increase in CO 2
emissions today in country ν is given by the partial derivative, with minus sign, of utility ofcountry νuν = φν (V )ϕν ( Rν ) uν (cν , aνc )with respect to atmospheric accumulations of CO 2 , V ; that is,∂uν= τ (V )φν (V )ϕν ( Rν ) uν (cν , aνc ) ,∂V
where τ (V ) is the impact coefficient of global warming.
−

We assume that future utilities of country ν are discounted at the social rate of discountδ that is assumed to be positive and identical for all countries in the world. We have assumed
that the rate at which atmospheric carbon dioxide is annually absorbed by the oceans is a certain
constant µ . Hence, for each country ν , the sustainable imputed price ψ ν of the atmosphericaccumulations of CO 2 , in units of utility of country ν , is given by the discounted present valueof the marginal decrease in utility of country ν due to the marginal increase in CO 2 emissions incountry ν today ; that is,τ (V ) νφ (V )ϕν ( Rν ) uν (cν , aνc ) .
δ +µHence, the sustainable imputed price θν of the atmospheric accumulations of CO 2 for country νψν =, in units of world prices, is given by

τ (V ) νy ,δ +µwhere α ν is the inverse of the marginal utility of country ν .
θν = α νψ ν =(29)

Similarly, the imputed prices of forests, in units of world prices, are defined as follows.
Suppose the acreages of forests of country ν , Rν , are increased by a marginal amount. This
would induce a marginal increase in the level of the utility of country ν , on the one hand, and amarginal increase in the utility of country ν in the future due to the marginal decrease in theatmospheric level of CO 2 induced by the absorbing capacity of forests in country ν , on the22other.
The first component is the marginal utility with respect to the acreages of forests ofcountry ν , Rν , in units of world prices. It is given by∂ (α ν uν )= τ ν ( Rν )α νφν (V )ϕν ( Rν ) uν (cν , aνc ) = τ ν ( Rν ) yν .
ν∂RThe second component is the marginal increase in country ν 's utility in the future due to themarginal decrease in the atmospheric level of CO 2 induced by the absorbing capacity of forests
in country ν . It is given by

τ (V ) νy .
δ +µHence, the sustainable imputed price π ν of forests of country ν , in units of world prices,γ ν θν = γ νis given by the discounted present value of the sum of these two components:

τ (V ) ν1
π ν = [τ ν ( Rν ) + γ ν
]y .
δ +µ
δ
The discussion above may be summarized in the following proposition.
Proposition 2. The optimum conditions for the sustainable time-path of atmosphericconcentrations of CO 2 corresponds precisely to the optimum conditions for the competitivemarket equilibrium under the following system of proportional carbon taxes for the emission of
CO 2 and tax-subsidy measures for the reforestation and depletion of resources of forests:(i) In each country ν , the carbon taxes are levied with the rate θ ν that is proportional to theper capita national income yν :τ (V ) νy ,δ +µwhere τ (V ) is the impact coefficient of global warming, δ is the social rate of discount, and µθν =is the rate at which atmospheric CO2 is annually absorbed by the oceans.
(ii) In each country ν , tax-subsidy arrangements are made for the depletion of resources andreforestation of forests with the rate π ν that is proportional to the national income yν , to begiven byτ (V ) ν1
π ν = [τ ν ( Rν ) + γ ν
]y ,
δ +µ
δ
where τ ν ( Rν ) , γ ν are, respectively, the impact coefficient and carbon sequester rate for forests23

in country ν .

8. International Fund for Atmospheric Stabilization

The divergence in economic performance between developed countries and developing countries
has steadily widened in the last several decades, and various institutional and policy measuresthat have been devised internationally or bilaterally have not had much impact in narrowing the
gap between these two groups of countries. The introduction of the proportional carbon tax
system as envisioned here, in spite of the implicit recognition of the equity aspect in its design,
may tend to worsen the relative position of developing countries, at least in the short-run. It
would be desirable, therefore, to supplement the carbon tax system with the international
redistributive scheme that would have significant impact in narrowing the gap between thestages of economic development of various countries involved.
The International Fund for Atmospheric Stabilization is an institutional framework inwhich it is possible to combine an international arrangement to stabilize atmospheric equilibriumwith a redistributive scheme to help developing countries to accelerate processes of economic
development.
The International Fund for Atmospheric Stabilization presupposes that each country
adopts the proportional carbon tax system under which emissions of carbon dioxide and othergreenhouse gases are charged a levy evaluated at the imputed prices proportional to the percapita level of national income and a charge (or a subsidiary payment) is made for the depletion(or the afforestation) of terrestrial forests, again based upon the evaluation at the imputed pricesof terrestrial forests that are proportional to the per capita level of national income, as in detail
discussed in the present paper.
24

The tax revenues from the proportional carbon tax system are principally put into thegeneral revenue account of each government, preferably to be partly earmarked for the purposesof restoring the natural and ecological environments, and for encouraging private economic
agents to develop those technological and institutional knowledge that are crucial in restoring
equilibrium conditions in the global environment.
Each country then transfers a fixed portion, say 5%, of the net revenue from the carbon
tax system to the International Fund for Atmospheric Stabilization . The total amount transferredto the International Fund for Atmospheric Stabilization from individual countries then would beallocated to developing countries according to a certain predetermined schedule, properly taking
into account the per capita levels of national income and the size of population. Developing
countries may use the amounts transferred from the International Fund for Atmospheric
Stabilization for the purposes which they think appropriate, preferably for compensating those
who would suffer from the phenomena of global environmental disequilibrium and incur thehardships by the implementation of the carbon tax system, for restructuring industrialorganizations and social infrastructure, and for introducing substitutional energy sources andenergy-saving technologies.
It is difficult to imagine that the International Fund for Atmospheric Stabilization orsimilar international arrangements on the global scale may be instituted in any immediate future.
Whether such international arrangements may be effectively implemented or not depends to asignificant extent upon the degree of awareness on the part of the general public concerning theenormous burden and costs future generations will have to suffer from the phenomena of global
warming and other global environmental disequilibrium.
The strenuous effort by a large number of geo-scientists, ecologists, and other scientists
to clarify the mechanism of global warming and to identify the specific implications of global
warming and other environmental issues for ecological, biological, social, and cultural life on
Earth has had a significant impact to the awareness and consciousness of the general public and25the national governments. The numerous conferences and symposia organized by various
international organizations, such as the 1991 Rio Conference and the Intergovernmental Panel on
Climate Change, particularly the Kyoto Protocol of 1997, have substantially altered theperception of the international community as regards the plausibility and danger of global
warming and other atmospheric disequilibria.
All these help the national governments involved to search for those policy andinstitutional arrangements that will make the practical implementation of the International Fundfor Atmospheric Stabilization or similar international agreements feasible from economic, social,
and political points of view. It would not be too optimistic to expect to have the International
Fund for Atmospheric Stabilization or a similar framework to be instituted within a foreseeable
period, though not in the immediate future.

9. A Hypothetical CaseA hypothetical case of the incidences of the proportional carbon taxes under the system of
proportional carbon taxes for the emission of CO 2 and tax-subsidy measures for thereforestation and depletion of resources of forests are presented, all in terms of the statistical
data of 2005 (in US$).

26

Table 1
Incidences of proportional carbon taxes with the coefficient of proportion 0.01
including all radiative forcing agents (RFA)Country

National Income
per capita

RFA Increase Imputed Price Carbon Taxes
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

Norway48,000

1.60

480

760

Finland

31,000

2.00

310

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, etc.
27

Table 2
Incidences of Tax-Subsidy Measures for the Reforestation
and Depletion of Resources of Forests

Forest andNet Annual

ImputedWoodlands

Reforestation

Price[Million ha][1000 ha][Per ha]Country

Assessment
Total

Per Capita[Million Dollars] [Dollars]

United States

303

159

42,0006,627

22Canada310

0

34,00000United Kingdom

3

10

32,000321

5France16

41

31,0001,264

21

Germany

11

031,000

0

0

Italy

10

106

28,0002,929

50

Netherlands

0

1

35,00035

2

Sweden28

11

32,000351

39

Norway9

17

48,000808

175

Finland23

5

31,000153

29

Denmark

1

3

34,000102

19Indonesia

88-1,871

9,300-17,120

-77

Japan

25

-2

31,000

-62

0

Korea

6

-7

21,000-149-3

Malaysia

21-140

33,000-4,679-179

Philippines

7

-157

9,600

-1,507-18

Singapore0040,0000

0Thailand

15-59

20,700-1,220

-19

India

68

29

2,200

64

0

China

197

4,058

4,100

16,678

13

Australia

164

-193

33,000-6,319

-310

8

17

23,000388

94

New Zealand

Source) World Resources Institute, etc.
28

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