Discussion Paper Series No.
No.４５
Simple Theoretical Analysisof
The Environmental Kuznets Curve

Masahiro Yabuta
June,
June, 2, 2003

Simple Theoretical Analysisof
The Environmental Kuznets Curve

June 2003

Masahiro Yabuta*
Yabuta**Faculty of Economics, Chuo University. Financial support from Chuo University
Grant for Special Research 2002-2003 is gratefully acknowledged. This paper
presents a theoretical base for analysis of Environmental Kuznets Curve (EKC) instatic as well as dynamic frameworks. This work is in preparation for a statistical andeconometric verification of EKC, which will be presented in a future joint paper with
K. Nakamura and R. Isa.

1

Simple Theoretical Analysis of
The Environmental Kuznets Curve
By Masahiro Yabuta
Faculty of Economics, Chuo University

yabuta@tamacc.chuo-u.ac.jp

Abstract
Theoretical analysis of Environmental Kuznets Curve (EKC) is performed by extending thestatic model of Andreoni-Levinson (2001) to its dynamic version. The prototype model is also
revised towards featuring some different aspects of pollution abatement and production process.
Then the contributors which constitute various interrelations between pollution and income are
examined. Theoretical possibility of an inverted U-shaped EKC is tested, which shows initialdeterioration but later environmental improvement along the economic development path.
Even if environmental aggravation is seen with continuous growth of consumption or
income, it becomes clear that the expenditure on pollution abatement increases, and then an
environmental improvement can be found. Such a policy becomes possible because people's
consciousness to pollution control is relatively high so that people ask for an environmental
improvement as part of improvement in the living standard. Such an induced governmental
control on pollution must lead to a feasible technological progress in pollution abatement. Thus,
strengthening of environmental policy reflects people's environmental consciousness.

Keywords: Environmental Kuznets Curve, dynamic optimal control, environmental
consciousness, environmental policy

JEL Classification: O130

2

1. Introduction
It is very natural to ask whether it is possible to attain economic prosperity without
deterioration of environment. As far as some environment-related indicators such as SO2, NO2 aswell as garbage per capita are concerned, at an early stage of economic development and in theprocess of rapid industrialization, almost all economies have experienced their dramatic increase,leading to terrible environmental problems. In the course of time, governments have changed theirpolicy stance to incorporate pollution control. For example, after the mess of environmental
problems in the 1960’s, particularly related to air and water pollution, the Japanese government
enacted a new fundamental law of countermeasure for pollution and established an Environmental
Protection Authority (EPA). Corresponding to a changing pattern of people’s preference between
income growth and environment, they might support these reforms. In fact, after a series ofreforms on environmental policies or systems in the early 1970’s, we have experienced halved
economic growth but remarkable improvement in health and environment.
What kind of relation is there between environmental improvement and economic growth?
And what is the major reason to cause such a changing pattern of environmental policies? The
main purpose of this paper is to give answers to these questions mainly from theoretical viewpoints.
It seems natural to accept that the environmental situation in general deteriorates with economic
growth at low-income levels, but it will reach a turning point and then further growth leads toenvironmental improvement. The inverted U-shaped relationship between income and theenvironment is known as the environmental Kuznets curve (EKC). Grossman and Krueger (1993)
have found that for some air and water pollutants, the inverted U-shaped patterns could be
empirically observed. A large number of papers, including Cole, Rayner and Bates (1997), based on
empirical studies have followed the Grossman and Krueger paper and the common conclusions arethat a meaningful EKC can be observed only for local pollutants like NO2 and CO but for global
pollutants like CO2, and for some pollutants like CFCs, no EKC relation can be confirmed. Possibly,
there may be no general relation between pollutants and economic growth.
There is considerable literature on EKC based on theoretical viewpoints, including Andreoni
and Levinson (2001), Levinson (2002) and Lieb (2002). Their analytical frameworks are basicallystatic ones, which incorporate a specific type of utility function and pollution abatement function.

3

On the other hand, Selden and Song (1995), Stokey (1998) and Kelly (2003) have given dynamic
model frameworks for EKC. Selden and Song (1995), a revised version of Foster (1973), haveshown that even at an early stage of economic development, rapid increases in pollution abatement
can occur due to some contributors, including technological change of abatement and consumers’
demand for pollution abatement. They have argued that although capital accumulation can slow
reduction in pollution, high marginal efficacy of pollution abatement can reduce pollution as awhole, leading to an inverted U-shaped EKC. Kelly (2003) has proved that in a dynamic model with
pollution stock externality, the conditions for an inverted U-shaped EKC can emerge which requireconvexity of cost function of pollution control and normality of environmental goods. Pollution
control provides benefit, but it costs additional spending to the society. Consequently, it may beargued that an inverted U-shaped EKC occurs when marginal costs of pollution control rise byless than marginal benefits.
Although the studies concerning EKC have given consistent explanations, there is no clear
understanding that ties static frameworks to dynamic ones. Our primary purpose in this paper is toshow the possibility of inconsistency of the conditions for EKC to have inverted U-shape bothfrom static as well as dynamic viewpoints. Moreover, in the dynamic model framework, we shallcompare different cases with various behavior or policies towards pollution abatement andeconomic growth.
The plan of this paper is as follows. In Section 2, a basic version of the EKC model isdemonstrated which contains the temporal optimization controls of consumption and pollution,and proves an inverted U-shaped EKC to exist in a static framework. To give a static model on
which we base some different types of dynamic models, we shall feature a framework of utility
functions as well as pollution functions, which are employed by Andreoni and Levinson (2002). In
Section 3, the basic model is developed towards a dynamic version and it is proved that as far as the
dynamic optimal control of consumption and pollution is concerned, the necessary conditions inthe static version of the model for existence of the inverted U-shaped EKC would be contradictedin the dynamic one. In Section 4, two extensions of the dynamic model are investigated; one is astock-related pollution case and the other a variable cost case of pollution abatement. In the latter
case, no inverted U-shaped EKC emerges. However, in the former case, an inverted U-shaped EKC
may occur mainly because the pollution abatement to curb environmental deterioration eventually

4serves for improvement in the consumer’s welfare. In Section 5, after remarking on the relationshipbetween pollution control and economic development, the importance of strengthening the policyon environmental quality is examined.

2. A Basic Static ModelAt the beginning of the arguments concerning EKC, it is useful to mention a simple formuladeveloped by Andreoni and Levinson (2001) and Levinson (2002). With this simple static formula,they have proved that the inverted U-shaped EKC occurs corresponding to a consumer’s preference
between consumption and pollution. The steps to compose their model are given as follows. The
utility function U of consumer is
(1)

=

,
(
C, )

with consumption C and pollution P. Let the utility function be specified as
(1)’

= C−

.

Pollution function is given by
(2)

= (
C,
E),
,E < 0,C> 0where E denotes the effort to abate pollution and (2) is assumed to be specified as(2)’= C− Cα Eβ .

In (2)’, α and β are positive parameters and the second term represents pollution abatement.
β can be interpreted as a parameter concerning the efficiency of investment for pollution
abatement. It is clear in (2)’ that a larger β will lead to a decrease in pollution.
The marginal rate of substitution (MRS) is given by=

β C because from (1)’ and (2)’,
α E

U is simply CαEβ. Moreover, for a simple manipulation, we shall assume that the price ofconsumption goods and the abatement cost of pollution are both unity. Then we have theconsumer’s budget constraint with income, M:(3)= C+ E.

Consumer is assumed to be able to control both C and E. Accordingly, maximization behavior of
consumer in this static framework will lead to an optimal consumption, C* as well as an optimal
effort of pollution abatement, E* given by

5

α
α +β

(4) C*=

,E*=

β
α +β

.

By substituting (4) into the specified pollution function, we have the reduced form of pollutionfunction as
(5)

α
*( )=
α +β

α

 α   β 

− 
 
α + β  α + β 

β
α +β

.

It is possible to confirm the shape of EKC from (5) because it shows a direct relationship between

P and M. Differentiation of (5) with respect to M leads to

*
=
(6)

α

β

 α   β 
α
− 
 
 (
α + β ) α + β −1
α + β  α + β   α + β α α   β *)= −(α + β )(
α + β −1

 

2
α + β  α + β 

2

β

.

α + β −2

Hence it can be verified that as long as α+β is greater than unity in (6), the second derivative
becomes negative and the first derivative will be negative for a sufficiently large income, meaningthat an inverted U-shaped EKC exists as depicted in Figure 1.

P*
α＋β＝１

α＋β>1

M

0

Figure 1.
1.

An Inverted UU-shaped EKC

6

The inverted U-shaped performance is only induced by parameters affecting the marginal
rate of substitution between consumption and effort to reduce pollution. It is clear that thisdepends on a specification of the pollution function. For example, if

is sufficiently large, it willlead to a large pollution abatement and environmental improvement irrespective of income.
Lieb (2002) gives the static model analysis of EKC and proves that satiation in consumption
leads to the downturn of the EKC because, with satiation, an increment of income does notincrease utility and hence it should be devoted to pollution abatement. This means that, withoutsatiation, pollution increases monotonically. His research has made some good points. However itshould be noted that his model is not dynamic but static. Hence it does not show a dynamic optimal
relationship between income and pollution. Moreover it seems difficult to accept that, after satiationin consumption, EKC proves to be downward sloping. We do not think that the major reason ofpollution abatement comes from the consumer behavior towards their utility maximization.

3. DynDynamic
amic Optimization Models and the EKC
Main tasks in this section are to develop the dynamic counterparts of EKC model from thestatic versions, and to show their dynamic properties. Moreover, attention will be paid to the
dynamic conditions for possible optimal policy for transition towards steady state. It is assumed
that the consumers have the same utility function and pollution function specified in the former
section. For the dynamic version, the production function is replaced by capital stock, K, in thebudget constraint (3);=(7)( , ), > 0
,

<0
,

< 0.

The capital accumulation dynamics is defined by
(8)

& =

− E− C

where it is assumed that there is no capital depreciation and that both the real price ofconsumption goods and the cost of effort for pollution abatement are unity.
Accordingly, it is natural to accept that the static model given in the former section will be
extended towards the dynamic optimization version, given by

7

(9)

→∫

∞

−ρ

0

∞

=∫ (
Cα Eβ )− ρ
0

& =

(,
C− Cα Eβ )− E− C, (0)=

0where society should control C as well as E for the evolution of capital accumulation. A problem ofpollution control has been already studied by Foster (1973) in the context of the neo-classicalgrowth model. Whereas in the Foster’s model, pollution is additively separable in the capital stockand effort on pollution abatement, and is assumed not to affect production, the dynamic version (9)does not include K explicitly in the pollution function and P in the production function.
The current-value Hamiltonian is given by(10) H = Cα Eβ + λ(,
C− Cα Eβ )− E− C .

The static and dynamic necessary conditions for optimality are given by
(11)

∂H
∂
= αCα −1Eβ + λ∂C∂(
1− αCα −1Eβ )− 1= 0,

(12)

∂H
∂
= β Cα Eβ −1+ λ∂E∂(− β Cα Eβ −1)− 1= 0,

and(13) λ& = −∂H∂+ ρλ = − λ∂∂+ ρλ = λ ρ −∂∂,where (13) is the dynamic condition.
To give a clearer image of the dynamic process, we shall assign some specific properties to
production function, (7). The first one is the case where MP=0 and the second one is related to M
given by a separable form. The case of MP=0 occurs when any pollutants, even if they are emitted
from production process, never affect the production level. In the second case, it is assumed that
production decreases in proportion to the deterioration of pollution.

∂
3-1. A Simple case
case:
∂

β
= 0, then E= C and
α

β

β 
= C−   Cα + β .
α In this case, the dynamic equations are summarized by the following system:

8

C
[ρ −
α + β −1

&=
C
(14)

& =

]
'
()
.

α + β ( )− 
C α where M’ is identical with MK in (7) because MP=0. The steady state of the system (14) is given by(C*, K*), which satisfies
(15)

*=

 α 
−1
C*= '(ρ ),α + β 

−1
('
(
ρ ).

It is easy to find that the Jacobi matrix of (14) evaluated at the steady state (15) is

(16)


 0
=
− α + β

α

−

C

"
α + β −1  .


'


where M” is identical with MKK in (7) and negative. Therefore, the trace and the determinant of theJacobi matrix (16) are given by

=
(17)

D

>0
'
=−

α +β
C "> 0α(
α + β −1)<0

α + β > 1.
α + β <1

In the case of α + β > 1, the equilibrium point of the system (14) is unstable. For the
optimality of dynamic control towards the equilibrium point, the equilibrium point should be astable (saddle) point. This implies that α + β < 1 must be assumed. In an ordinary optimization
framework, the sum of parameters,and, must be less than one. This means that there nevercan be the inverted U-shaped EKC along the optimal path towards the long-run equilibrium where
income continues to increase. As far as the simple model is concerned, it may be said that the‘inverted U-shaped’ case will not be dependent on how consumers react in their decision making.
Figure 2 illustrates a relation between environmental deterioration and economic growth.
The phase angles are given by pairs of arrows in the first quadrant of Figure 2. In Figure 2, we
assume that the investment for pollution abatement is not sufficiently efficient so that the overall
pollution is increasing in the process of economic growth. This is denoted by P(C) in the second
quadrant in Figure 2. The first quadrant in Figure 2 illustrates the phase diagram of (14) when the9system has a stable saddle point denoted by E. The fourth quadrant in Figure 2 denotes the
production function. By taking the relations among P, C and K depicted by the first, second andfourth quadrant in Figure 2 into consideration, it is proved that an optimal control towards a
steady state, from A to E, provides a corresponding relation between pollution and growth,
denoted by the path from A to E in the third quadrant in Figure 2.
Although the simple case can not lead to the inverted U-shaped EKC, it may be possible toshow that whenever a society turns its policy stance towards more environment-friendly
development, introducing a damping effort on pollution can lead to a sharp decline in consumption
as well as pollution. This is shown by the path from a to E via b and c in Figure 2. This can be asemblable pattern of U-shaped EKC.

C
P(C)

& =0
C

& = 0(β=0)

E0& = 0(β>0)
E

b
c
a
A

P

K*
a
A
b

c

E

M

Figure 2. The phase diagram (A simple case)

K

10

γ

=

3-2. Separable production function case:case−,0< γ < 1,0<

Under these restrictions, the production function is separable, increasing at a diminishingrate in capital stock, and decreasing at a constant rate in pollution. We shall assume that comparedto the effect of pollution on the overall welfare, its direct effect on production is very small,indicated by a sufficiently small b. Clearly, it must be ensured that

and

∂
∂

∂
∂

γ −1

=γ

>0

= − < 0. Substituting the latter equation into (11) and (12) leads toβα(18) E= (+1 ) C.
In this case, the dynamic equations (14) change to:(19)

1− ΛCα + β −1)C
& = 1+ (ρ −γC
α + β −1
(
)(
1+ )
& =

γ

γ −1

α +β
Λ

− (
1− Cα + β −1)+
C
α
α 


，

β

β
where Λ ≡ α  (1+ )  . In this case, it is easy to show that the dynamic property of the systemαis slightly different from the simple case where∂∂= 0. For the Jacobi matrix of (19) at thesteady state, the trace and the determinant are given by

=
(20)

D

∂
∂

>0
Λ
α +β
α + β ) Cα + β −1)+
1+ (1− ΛCα + β −1) (
1− (
α
α
=−
α + β −1
(
)(
1+ )

.

∂2
C 2
∂

Because Trace J is positive, the condition for the eigenvalues of characteristic equation to be
anticlastic, meaning that Determinant J should be negative is necessary for the steady state to bestable saddle point. The sign of Determinant J depends on the terms in the numerator as well as in
the denominator. For a sufficiently small b, however, this can be positive irrespective of

1− ΛCα + β −1. Hence, we reach a conclusion similar to that in the former simple case that thesystem has the stable optimal path only in the case of the sum of parameters,

and, less than

11

one. In this case, under no circumstances does the existence of a controllable optimal policy lead tothe inverted U-shaped EKC. As shown in Section 2, this is not compatible with the static
optimization case.
As for the case of large b, the conditions for the stable dynamic system are slightly different.
Noting the non-negativity of the pollution level along the optimizing process clearly lead to thecondition given by (21)
(21)

 Λ

C)≥ 0.
= C1− Cα + β −1 = ( αWhat we would like to know is the possibility of the stable saddle point case even if
greater than unity. To obtain a sufficient condition for this, it is convenient to assume

+

+

is

–1>0

as well asα = 1, that isβ > 0. As long as these conditions are incorporated into analysis, thesecond term in the numerator of Determinant J of (20) can be negative but the first term is
positive when b is sufficiently large1. Therefore, it is likely that when the effect of pollution on
production, b, is large enough, the system can have an optimal control policy related toconsumption as well as investment for pollution abatement. It is of interest to consider themeaning of the largeness of(22). Differentiating (21) with respect to C yields

Λ α + β −1 2Λ
α + β) C
α + β )(
α + β −1
= 1− (, 2 = −() Cα + β −2.
α
α
C
C

Λ
Hence, it is easy to see that the conditions, 1− (α + β ) Cα + β −1 < 0 and α + β − 1> 0 yield anαunambiguous relationship between consumption and pollution. Then, equation (22) implies thatpollution is decreasing at a diminishing rate in consumption because dP/dC>0 and d2P/dC2<0.
The initial consumption in the under-developed economies must be so small that the first derivativein (22) will be positive; implying that any increase in consumption should entail environmental
deterioration in the embryonic stage of the economic development. In the process of capital
accumulation, however, an increase in K or C, and a decrease in P together can occur because
people are presumed to prefer pollution abatement to consumption as consumption increases.

1Λ
As for the second term of numerator of determinant J in (20), we need to assume 1− (α + β ) Cα + β −1 to be

negative.

α12

We shall offer some of these conclusions as a précis. A social planner should manage andcoordinate the expenditure for pollution abatement adequately towards the optimal growth path. In
the optimal process, when pollution has a negative impact on production and people thinkenvironment as more important, the policy that holds down pollution in the process of economic
growth becomes the optimal policy. This may lead to an inverted U-shaped EKC along A’-B’-C’
depicted by the second quadrant in Figure 3, corresponding to an optimization path from A to E
via B depicted by the first quadrant in Figure 3. Point B can be a turning point after which thepolicy to abate pollution may be preferable to growth-oriented one for the society. This is the casefor a sufficiently large b and

2.

& =0
C

C

& =0

P(C) in (21)

E’

E

B’

B

A’

P

A
0

K*

K

FigureFigure 3. The phase diagram (A separable production function case)

4. Some Extended Dynamic Versions
This section will be devoted to more theoretical frameworks related to EKC. We shall follow
some alternative procedures, and then compile conditions related to the possibility of an inverted

2 It is possible for the system to have multiple equilibriums. Although the curve & = 0is upward-sloping for a low
C as in the first quadrant in Figure 3, it can bend towards the northwest and possibly be downwards-sloping for a
sufficiently large C. Hence, it is easy to see that there can be an upper equilibrium point where the system isunstable.

13

U-shaped EKC to occur.

4-1 StockStock-related pollution case:
case
Pollution function (2) is now specified as(23) P=C – C E + a K , a > 0 .
This means that the accumulated capital stock also causes pollution through the production process.
On the other hand, we shall feature a simple form of production function given by, γ<1

M = M(K)＝K(24)

The condition on γ implies a diminishing marginal productivity3.
The optimization formula in this case is given by

→∫

(25)

∞

∞

Cα Eβ −
=∫ (

−ρ

0

0

& =

γ

γ

)− ρ

− E− C.

Because the current-value Hamiltonian is simply defined as

H = Cα Eβ −

γ

+λ

γ

− E− C ,we have the following necessary conditions for optimality: the static ones are∂H∂H= αCα −1Eβ − λ = 0, = β Cα Eβ −1− λ = 0,
∂C
∂E

(26)

together leading to a simple allocation between consumption and effort for pollution abatement,

E=

βC. Moreover, the dynamic condition is given byα(27)

λ& = λ ρ − γ

γ −1

+ γ

γ −1

.

Differentiating the first equation in (26) with respect to time and taking (27) into consideration
yields3This case is very close to Forster’s model except for the specification of consumer’s utility function.

14

(28)



C

&=
(
C
ρ −γ
α + β − 1
& = γ −  α + β C
 α 



γ

γ −1
)+
β

β 
α   Cα + β −1 .
α 

γ −1

At the equilibrium point of this system, the Jacobi matrix becomes

(29)







γ −1
γ −2 
)
γ(
γ −1
C 
γ

γ −2
−
) +
γ −1
β

−γ (
  β β1αβ+−
 β  α + β −1  .
=  α   Cα + β −1α  Cα  α 




α +β
γ γ −1−αThis gives the trace and determinant related to (29) as:

=(
1−
(30)

Φ

)
γ γ −1=Dγ 2(γ −1))+ (1− γ )1− )− γ(
α + β −1Φ(()α + β −1
Φ
Φ
β

β 
Φ ≡ α   Cα + β −1 > 0α 
It is easy to prove from the first equation in (28) that the positivity of equilibrium productionneeds an additional condition:
(31)

1−

Φ

> 0,

and hence, Trace J in (30) must be positive. As mentioned before, the condition under which thesystem (28) has the optimal controls on C as well as E is for Determinant J in (30) to have negativesign. Table 1 summarizes possible stock-related pollution cases where this is likely to occur.

Table 1 A summary of stability conditions

Case

＋ －1

a

Determinant Ji＋

Sufficiently large (a is close to Φ )

－

ii

＋

Small enough

＋－

15

iii

0< a < Φ－－In the case when pollution occurs in the production process so that capital accumulation
becomes one of the major sources of pollutants, optimal control towards steady state exists.
Unlike in the simple model version, even if＋

is larger than unity, there can be a saddle pointcase whenever a is sufficiently large and close to Φ , meaning that production process causes alarge-scale pollution to society (Case (i) in Table 1). What we would like to know is about anevolution process of pollution control along the optimal path as well as the possibility of theinverted U-shaped EKC in Case (i). This can be investigated by the following procedure. The slopes& = 0and & = 0curves are:
of C

(32)

C

& =0 =
C(
1− γ )1−Φ(Φ(
α + β −1
)

)
C

C

,

& =0 =αγ γ −1>0.
α +βMoreover it is notable that the stability condition, i.e. the negativity of Determinant J in (30),include
(33)

C
& =0
C

<

C

& = 0.

Therefore, except for the difference in shape, the dynamic properties as well as the phase are
essentially the same as the ones for the simple case, which is depicted as in Figure 2. The time
derivative of (23) yields

Φ &(34) & = 1− (α + β) C
+ γ

α

γ −1

&.

Because the dynamic optimization process with a positive income growth towards steady stateentails both increase in consumption and cumulative capital, a sufficient condition for pollution to
decrease is clearly assured when the expression in the square brackets in (34) is negative and smallenough. So long as a is large enough, evaluation of both Φ and the marginal productivity ofcapital in the neighborhood of (C,K) = (0,0) leads to an initial state with increases in pollution. In
the early stages of economic development, pollution will increase. However, increases in C provide
increases in Φ , whereas the marginal productivity of capital declines due to diminishing returns.
The result can be decreases in pollution because the first term in (34), which turns negative

16

eventually, dominates the second term., which is positive. This will lead to the possibility of an
inverted U-shaped EKC4.

4-2. Variable cost function of pollution abatement:
abatementThis subsection is devoted to a small revision of the simple case, with only the abatement
cost of pollution being considered. We shall incorporate a cost function,, in real terms related

to pollution abatement and then we have a slightly different version of (8):
(35)

& =

− E− C, = (C)= Cδ , > 0,δ > 0;.

In the capital accumulation process, C continues to grow and the social cost of efforts to reduce
pollution will increase sharply and this leads to a hard restriction on capital accumulation. The
marginal social cost of pollution abatement is increasing if δ > 1but decreasing with respect toconsumption for 1> δ > 0.
The optimization formula for this case is given by

→∫

(36)

∞

∞

Cα Eβ )− ρ
=∫ (

−ρ

0

0

& =

δ

.

( )− C E− C

The optimal trajectory satisfies
α −1 βδ −1(37) αC E − λ (δC E+ 1
)= 0,
α

β −1

(38) βC E− λ Cδ = 0,and(39) λ& = −∂H+ ρλ = − λ∂'( )+ ρλ = λ ρ −'( ),where λ is the shadow price of capital. It is notable that (37) and (38) together yield



β



−δ
1

C ,
(40) E= 
α − βδ )
 (and substituting (40) into the pollution function (2) yields5

4

Selden and Song (1995) also investigate the dynamic properties of Foster’s model and prove that an inverted
U-shaped EKC can occur mainly due to rapid increases in pollution abatement. They enumerate three contributors
by which pollution can be abated: a high marginal efficacy of abatement, ∂P/∂E and a large direct effect of growthon pollution, (∂P/∂K)×(dK/dt), and a large increment of the marginal disutility of pollution, ∂2U/∂P2. As far as
our case here is concerned, the second one seems to be the most important.
5 It is hardly acceptable that in the process of economic growth, an increase in consumption leads to a decrease in

17

(41)

β
 
 α + β (1−δ )−1β C= C 1− .
  (− βδ )
αBecause this is a slightly modified version of the simple case in section 3-1, we have the following
dynamic equations:

&=
C
(42)

&=

C
[ρ −
α + β(1− δ )− 1]'(),
α + β(1− δ )
C( )− 
 α − βδ and clearly making δ= 0 in (42) leads to the same system as (14). Therefore, it is easy to provethat the condition for the equilibrium to be stable is give by(43)

α + β(1− δ )< 0.
(
α − βδ )α + β (1− δ )− 1

Taking the restrictions imposed on (40) into consideration, we find that (43) is consistent with thefollowing condition to be compared with (17):(44)

α + β(1− δ )< 1.
As far as utility arises from consumption and pollution abatement, even if we incorporate thesocial cost into the model as the blockage of capital growth, it seems impossible to show theexistence of an inverted U-shaped EKC. All the cases show the positive correlation between
income and pollution. Of course, it would be easy to say that this is a normal or usual fact in thereal world. However, it is also notable that incorporating the social cost of pollution abatement
into the model system leads to the lower consumption as well as lower pollution along the optimal
growth path in comparison with the case of no social cost.

4-3. Conditions for an ‘inverted UU-shaped EKC’
EKC’
It is useful to give a synopsis of the model analysis concerning conditions under which an
inverted U-shaped EKC can be observed. Table 2 summarizes the cases considered in Sections 3
and 4. Table 2 shows that an inverted U-shaped EKC can occur when there is a restrictive influenceof pollution upon production and/or when a changing pattern of production via capitaleffort to abate pollution. Hence, we shall assume that 1− δ > 0in this connection as well as α − βδ > 0forthe positivity of E.

18

accumulation has an impact on pollution. The major reason why pollution can be reduced along the
optimal control path with economic growth is clear. This is because the social planner, who
controls consumption as well as pollution abatement so as to coordinate the economy on optimal
growth path, tends to prefer the latter to the former, leading to more expenditure towards
abatement of pollution. After all, preference of the society to be more environment-friendly may be
important to curb pollution.

Table 2. Possibilities of inverted UU-shaped EKC

Case

Inverted

Production

Pollution

Social marginal cost of

U-shapedfunctionfunctionfunctionpollution abatement

EKC

∂ 


∂ 

∂ 


∂ 

()

3-1

Impossible

0

0

1

3-2

Possible

－

0

1

4-1

Possible

0

+

1

4-2

Impossible

00

Variable

5. Concluding Remarks
Remarks
Theoretical analysis of EKC has been performed. Even if environmental aggravation is seen
with continuous growth of consumption or income, it has been shown clearly that the expenditure
on pollution abatement increases, and then an environmental improvement follows. Such a policy
becomes possible because people's consciousness to pollution control is relatively high so thatpeople ask for an environmental improvement as part of the improvement in the living standard.
In the early stage of economic development, it is usually observed that governments do not pay
attention to environmental degradation and people also tend not to think about environmental
situation but about their income because their living standards are too low. This means that there
must be a switching pattern of the environmental policy before income is sufficiently large when

19

they can see an environmental problem ahead. Such an induced governmental control on pollution
must lead to a feasible technological progress in pollution abatement. Such strengthening of theenvironmental policy is reflecting the change in people's environmental consciousness6. Possibly,
the most prevailing explanations for an inverted U-shaped EKC are (i) an increasing demand forenvironmental quality, (ii) technological improvements to make production cleaner, (iii)
de-industrialization of the economy, i.e. a structural change from manufacturing sector to service
sector7. Among these explanations, our model analyses have emphasized the first one which leads
to an increasing awareness of deterioration of environmental quality.
Even if people demand a better environmental quality, governments do not always act so as
to maximize people’s welfare. This occurs partly because there is no suitable socio-politicalorganization and mainly because there is little political freedom. Boyce (2002), for example, hastested the consistency of the impacts of literacy, political rights as well as civil liberties on somepollutants. His research almost supports the hypothesis that improvements in environmental
quality can be attained via better literacy and higher political and civil liberties. Bimonte (2002) has
also pointed out that expanding the real freedom can make development sustainable because
participation can lead to people’s individual preference to be justified towards social welfare.
Because political as well as social factors can influence environmental quality to a considerable
extent, it may be misleading to reduce these factors to a simple relationship, like an inverted
U-shaped EKC between per capita income and environmental quality. How environmental andsocio-economic aspects are interrelated is still an open question.

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In connection with this, a changing pattern of policy stance is analyzed by Stocky (1998). In that model, twocontrol variables are chosen, consumption and an emission standard. The government sets the latter in order tomaximize the social welfare. Because the marginal cost of pollution is assumed to increase along the growth path,the government is forced to change their environmental policy and to tighten the emission standard. This leads to asingle-peaked, non-smooth, EKC.
7 For example, Hanley, et al. (2001), gives good text-based explanations.

20

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