Discussion Papers In Economics
And Business

Modeling pollution and economic growth: the effect of alethal threshold

Asuka Oura, Yasukatsu Moridera and Koichi Futagami
Discussion Paper 15-06

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANModeling pollution and economic growth: the effect of alethal threshold

Asuka Oura, Yasukatsu Moridera and Koichi Futagami
Discussion Paper 15-06

April 2015

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANModeling pollution and economic growth: the eﬀect of alethal threshold∗Asuka Oura†, Yasukatsu Moridera‡, Koichi Futagami§.

Abstract
The accumulation of pollution negatively impacts human health. Extreme increases
in pollution, in particular, may have lethal implications for human beings—and, indeed,
all living organisms. This paper thus devises a new model of economic growth that takes
into account these lethal eﬀects of accumulated pollution via a pollution threshold toshow two key results. First, if an abatement technology is relatively ineﬃcient, there
exists a stationary steady state in which consumption and pollution stop growing.
Second, if the abatement technology is suﬃciently eﬃcient, there exists a path along
which pollution decreases at an accelerating rate until finally reaching zero. In thiscase, consumption grows at a constant rate.

Keywords: Endogenous growth, Pollution disutility, Pollution abatement
JEL Classification Numbers: O44, Q52∗

We are especially grateful to participants of the 2013 Japanese Economic Association Autumn Meetingat Kanagawa University and the Economic Theory and Policy Workshop at Aoyama Gakuin University for
their helpful comments and encouragement. This research benefited from a Grant-in-Aid for JSPS Fellows(26-3695). Of course, all remaining errors are ours.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. E-mail: pge801oa@student.econ.osaka-u.ac.jp‡
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. E-mail: rge032my@student.econ.osaka-u.ac.jp§
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. E-mail: futagami@econ.osaka-u.ac.jp1

1

Introduction

According to the World Health Organization (2014), about 3.7 million people worldwide
died as a result of outdoor air pollution in 2012. The most common sources of outdoor
air pollution include particulates, ozone, nitrogen dioxide, and sulfur dioxide, which arehighly prevalent in urban areas of developing countries. In Japan, corporate industrial waste
during a period of rapid economic growth led to the so-called Big Pollution Diseases of Japan;
Itai-itai disease, Minamata disease, Niigata Minamata disease and Yokkaichi Asthma. For
example, Minamata disease, traced to the release of methyl mercury in industrial wastewater,
was responsible for many human deaths. Many developing economies have recently begun toencounter similar problems. In addition to these direct harms, large volumes of greenhousegas emissions contribute to global climate change, which also may have disastrous eﬀects
on human lives. Indeed, human health and lives may be strongly adversely impacted if
pollution levels exceed a given set of thresholds. If economic growth proceeds with anunlimited accumulation of pollution, global society may find itself headed down a path todestruction.
In this paper, by using a simple endogenous growth model with AK technology, we consider a situation in which human beings cannot sustain their existence over a given pollution
threshold. In other words, we incorporate an upper limit on pollution into the individual
utility function. Therefore, the unlimited societal accumulation of pollution is analyticallydisallowed. Moreover, the existence of the pollution threshold introduces a strong nonlinearity into the model, complicating its dynamics. However, we show that there exists
a unique optimal path, depending on the eﬃciency of the abatement technology available
to reduce the pollution level. If the abatement technology is not suﬃciently eﬃcient—that
is, it exhibits decreasing returns with respect to input—then sustained growth is not possible. The economy reaches a steady state with finite levels of output and consumption;
the pollution level stays below the threshold and may even be zero. On the other hand,
if the abatement technology is suﬃciently eﬃcient—that is, exhibits constant returns with2

respect to input—then sustained growth is possible while keeping the pollution level under
the threshold.
Following new developments in economic growth theories in the 1990s, many studies
have examined this problem.1 For instance, Huang and Cai (1994) emphasize the possibility
that the consumption level grows at a constant rate along the optimal path. Because theyignore any biological limitations on the accumulation of pollution, the level of pollution is
also able to grow at a constant and positive rate on this “optimal” path. By focusing on
the disutility of pollution, Michel and Rotillon (1995) show that unlimited economic growth
and a continuous decrease in pollution are compatible. However, they neither explicitly
examine the stability of the long-run equilibrium nor derive the transitional path. Furthermore, the above-mentioned lethal eﬀects of pollution were not taken into consideration inany previous studies. In our analysis of the relationship between economic growth and environmental quality, the eﬃciency of the abatement technology is also an important factor. In
their dynamic model, Brock and Taylor (2010) integrate technological progress not only in
production but also in abatement, showing that achieving faster growth in the eﬃciency ofabatement technology than in production is key for maintaining economic growth withoutcausing additional damage to the ecosystem.
The remainder of the paper is organized as follows. The model is presented in Section 2,
and Section 3 analyzes the equilibrium in which the marginal cost of abatement increases.
Section 4 then considers the equilibrium in which the marginal cost of abatement remains
constant. Section 5 concludes.

2

The model

Consider a closed economy with constant population normalized to one. The utility of therepresentative agent, U , depends on per capita consumption at time t, ct , and on the level

1

See Ricci (2007) for a survey.

3

of pollution at time t, pt :∫ ∞U=exp(−ρt){ln ct + σ ln(p̄ − pt )}dt,(1)0where ρ > 0 is the time preference rate, σ < 1 is a weight denoting the overall utility
attached to environmental quality, and p̄ is a biological threshold for pollution above which
no humans can survive.
The production function of final goods at time t, yt , is given by:

yt = Akt ,(2)

where kt is capital and a constant parameter A represents productivity. We assume thatthe pollution level is a stock variable. The emission of environmental pollutants is the byproduct of production and is proportional to output; that is, αAkt , where α measures thedegree of emissions emerging from the production sequence. We also assume that the stock ofpollution decays at a fixed rate, δ. We further assume that the stock of pollution is reduced
by using the abatement good, mt . The level of pollution is assumed to be nonnegative inorder to provide the traction needed to analyze the properties of the equilibrium.2 Thus thelevel of the pollution stock is given by:p˙t = αAkt − δpt − mt .

(3)

The abatement good is produced using final goods, with the required input given by thefollowing technological transformation:c(mt ) = θmγt ,2(4)

In our setting, the marginal rate of substitution between consumption and pollution is positive and finite
when the level of pollution, pt , is zero. Thus it would be preferable to reduce the pollution level to below
zero. However, even if we allow the pollution level to take negative values, the equilibrium properties do notchange.

4where θ > 0 and γ ≥ 1. θ stands for the eﬃciency level of the abatement technology, andγ represents the degree of returns; when γ is greater than one, the abatement technology
exhibits increasing marginal costs. On the other hand, when γ is equal to one, the abatement
technology exhibits constant marginal costs.3 Then capital accumulates according to thefollowing equation:k̇t = yt − ct − c(mt ).

(5)

Based on this analytical structure, we consider the social planner’s problem in the next
section.

3

Optimal path

The social planner maximizes the welfare level (1) by choosing consumption ct and theabatement good mt , subject to the production function (2) and the transition equations of
the pollution (3) and capital stocks (5). The current value Hamiltonian of this problem is:

Ht = ln ct + σ ln(p̄ − pt ) + λkt (Akt − ct − θmγt ) + λpt (αAkt − δpt − mt ) + νt pt ,where λkt and λpt denote the co-state variables associated with capital accumulation (5) andpollution accumulation (3), respectively. νt is the Lagrangian multiplier associated with the
non-negative pollution constraint.
The first-order condition with respect to consumption ct is:1− λkt = 0.
ct

(6)

The first-order conditions with respect to the abatement good and pollution stock are givenby:mt (−γθmγ−1
λkt − λpt ) = 0, mt ≥ 0, −γθmtγ−1 λkt − λpt ≤ 0,t
3Michel and Rotillon (1995) examine only the case of γ = 1.

5(7)

νt pt = 0, pt ≥ 0, −νt ≤ 0.

(8)

Condition (6) requires the marginal values of consumption to be equal to the shadow valueof income, λkt . The complementary slackness condition, (7), indicates that if there is somepositive input in abatement, the shadow cost of additional pollution reduction is equal to theshadow value of pollution reduction. If the shadow cost is above the shadow value, there isno input into abatement. Complementary slackness condition (8) ensures that the pollution
stock remains positive. The dynamics of the co-state variables are given by:

λ˙kt = ρλkt − (Aλkt + αAλpt ),)(−σ˙− δλpt + νt .
λpt = ρλpt −p̄ − pt(9)(10)

Finally, the transversality conditions for this problem are:lim exp(−ρt)λkt kt = 0, lim exp(−ρt)λpt pt = 0.

t→∞

t→∞

Properties of the solution depend crucially on the degree of returns, γ. We first consider
the case of increasing marginal cost, γ > 1. We next examine the case of constant marginal
cost, γ = 1. We assume that the following inequality holds throughout our analysis:A − ρ − αθA > 0.

3.1

(A1)

An abatement technology with increasing marginal cost

In this section, we consider the case in which the abatement technology is relatively ineﬃcientand thus characterized by increasing marginal cost, γ > 1. In this case, the marginal cost ofabatement would be zero if the input into abatement were zero; as this entails that it is always
beneficial to use some final goods for abatement, therefore, mt never becomes zero. Thenequations (3)-(7), (9), and (10) constitute the following dynamic system for kt , pt , ct , mt ,6and νt :k̇t = Akt − ct − θmγt ,(11)p˙t = αAkt − δpt − mt ,(12)c˙t = ct (A − ρ − αAγθmγ−1
),
t
(
)
mt
νt ct m1−γ
σct m1−γ
γ−1
t
t
ṁt =
A + δ − αAγθmt −+.
γ−1(p̄ − pt )γθ
γθ

(13)
(14)

We can characterize the equilibrium path of the economy using the equations above alongwith complementary slackness condition (8) and the initial values of pt and kt . Then we
prove the following proposition:
Proposition 1 Suppose that inequality (A1) holds and γ > 1. Then there exists a uniquestationary steady state in which all variables become constant. In particular, the level ofpollution in the steady state becomes positive if p̄ > p̂ and zero if p̄ ≤ p̂, where p̂ ≡
(
) 1 (
)
A(γ−1)
A−ρ γ−1ρσ+ γ . Furthermore, as long as γ > 1, the threshold level p̂(A−ρ)(ρ+δ) αγθAγ
is decreasing in the level of abatement technology γ.
Proof. When the level of pollution is positive, equations (8), (11)-(14), and νt = 0 constitutethe following four-dimensional dynamic system for kt , pt , ct , and mt :k̇t = Akt − ct − θmγt ,(15)p˙t = αAkt − δpt − mt ,

(16)

),
c˙t = ct (A − ρ − αAγθmγ−1t()
mt
σct m1−γ
γ−1
t
ṁt =
A + δ − αAγθmt −.
γ−1(p̄ − pt )γθ

(17)
(18)

Denote the long-run levels of capital, pollution, consumption, and the abatement good byk ∗ , p∗ , c∗ , and m∗ , respectively. As k̇ = ṗ = ċ = ṁ = 0 holds in the steady state, the steady7

state values are as follows:() 1
A − ρ γ−1
m =
,
αγθA
(A − ρ)(ρ + δ)p̄ + σA(αθm∗γ − m∗ )∗p =,σδA + (A − ρ)(ρ + δ)
(A − ρ)(ρ + δ){p̄δ − αθm∗γ + m∗ }∗,c =α{σAδ + (A − ρ)(ρ + δ)}c∗ + θm∗γ
∗
k =
.
A
∗

(19)
(20)
(21)
(22)

In equation (20), the necessary and suﬃcient condition for p∗ to be positive is,(A − ρ)(ρ + δ)p̄ + σA(αθm∗γ − m∗ ) > 0.

(23)

By inserting equation (19) into condition (23) and solving with respect to p̄, then the condi() 1 ()
A(γ−1)
A−ρ γ−1ρσtion for p∗ to be positive becomes p̄ > (A−ρ)(ρ+δ)+. Here the left-handαγθAγγside is what we define as p̂.
On the other hand, when the level of pollution is zero, equations (8), (11)-(14), and pt = 0
constitute the following dynamic system for kt , pt , ct , and mt :pt = 0,(24)

mt = αAkt ,(25)k̇t = Akt − θ(αAkt )γ − ct ,()
c˙t = ct A − ρ − αγθA(αAkt )γ−1 .

(26)

8

(27)

k̇t = c˙t = 0 must be satisfied in the steady state. Then the steady state values are derived
as follows:
(
m

∗∗

=

A−ρ
αγθA

1
) γ−1

,

(28)

p∗∗ = 0,

(29)

(
) 1
γA + ρ − A A − ρ γ−1
c =
,
αγA
αγθA
(
) 1
1
A − ρ γ−1∗∗.
k =αA αγθA
∗∗

(30)
(31)

Next, to show that p̂ is decreasing in γ, we diﬀerentiate p̂ with respect to γ. Then we
can show that:
2−γ

∂ p̂
σ(A − ρ) γ−1
=−
γ
1
∂γ
(ρ + δ)γ γ−1 (αθA) γ−1(

ρA(γ − 1) + ρ
+
ln γ
γ−1
(γ − 1)2

)
< 0.

(32)

□
According to the household utility function (1), it is clear that the utility from consumption is
independent of the upper bound on pollution, p̄. By contrast, the disutility from pollution isnegatively related to the upper bound, p̄: the greater the marginal disutility from pollution,
the lower the upper bound. Therefore, if p̄ ≤ p̂, then when p = 0 the marginal disutility islarge enough to keep pollution at zero. If p̄ > p̂, the marginal disutility from pollution whenp = 0 is suﬃciently small that it is optimal for the economy to maintain a positive levelof pollution. In addition, if p̄ > p̂, the levels of pollution, consumption, and capital in thesteady state are increasing in p̄, as a higher value for p̄ entails a low marginal disutility ofpollution. Considering pollution reduction eﬃciency, the marginal cost increases with γ sothat it become more diﬃcult to maintain pollution at a high level.
The following lemma defines the conditions for the acceptable amount of pollution, p̄,under which the level of pollution can become zero in the transition.

9

Lemma 1 Suppose inequality (A1) and γ > 1. If p̄ < p̂, once the level of pollution becomes
zero, it is optimal for the economy to keep it at zero along the equilibrium path. Otherwise,
it is not optimal to maintain zero pollution, and it becomes positive in the long run.
□

Proof. See Appendix A.1.
We analyze the local stability of the economy in the following proposition.

Proposition 2 The equilibrium path is uniquely determined both in the case of the steadystate with a positive level of pollution and in the case of the steady state without pollution.
Proof. Firstly, we analyze the case in which the level of pollution is positive in the steadystate. Since the level of pollution is always positive, the transition dynamics are expressedby equations (15)-(18). We denote the Jacobi matrix of the linearized system around thesteady state as Jp>0 . Then we have that:

A


 αA

Jp>0 = 

 0

0



∗γ−1

0

−1

−γθm

−δ

0

−1

0

0

−αγ(γ − 1)θAc∗ m∗γ−2σc∗ m∗2−γγ(γ−1)θ(p̄−p∗ )2

∗2−γ

∗

∗(1−γ)σmm− γ(γ−1)θ(p̄−p−αγθAm∗γ−1 + σc(p̄−p
∗)
∗ )γθ





.




(33)

Thus the characteristic equation becomes:

λ4 + a1 λ3 + a2 λ2 + a3 λ + a4 = 0.

(34)

where the coeﬃcients are

a1 = −2ρ,

(35)

ασAc∗
(ρ + δ)m∗−,(γ − 1)(p̄ − p∗ )p̄ − p∗ρ(ρ + δ)m∗ασAc∗
a3 = Aδ(2ρ + δ − A) −
+
(A
−
δ)
,
(γ − 1)(p̄ − p∗ )p̄ − p∗αAc∗a4 =(σδA − (A − ρ)(ρ + δ)).
p̄ − p∗a2 = (A − δ)(2ρ + δ − A) − Aδ −

10

(36)
(37)
(38)

According to the Routh-Hurwitz theorem,4 the number of roots of equation (34) with positive
real parts is equal to the number of variations of sign in the following scheme:

1, a1 ,

a1 a2 − a3
a21 a4, a3 −, a4 .
a1
a1 a2 − a3(39)

We know that a1 < 0, a3 > 0, and a4 > 0 under assumption (A1) and the signs of a1 aa21−a3
a2 a4
and a3 − a1 a12 −a
are ambiguous. Considering the following four cases in Table 1 is enough.
3

Table 1: Summary of signs in (39)1Case 1 +Case 2 +Case 3 +Case 4 +

a1
−
−
−
−

a1 a2 −a3
a1

+
+
−
−

a2 a

4
a3 − a1 a12 −a
3
+
−
+
−

a4
+
+
+
+

a2 a

4
If a1 aa21−a3 > 0, then a1 a2 −a3 < 0 because a1 is negative. Thus, the sign of a3 − a1 a12 −abecomes3

positive. Therefore case 2 in Table 1 does not arise. We find that the number of variationsof sign in (39) is two in the other cases listed in Table 1. The characteristic equation (34)
has only two roots with positive real parts. Because the number of predetermined variables
is two, the equilibrium path is uniquely determined.
Second, we analyze the case in which the level of pollution is zero in the steady state. In
this case, the dynamic system around the steady state is expressed by equations (26) and(27). We denote the Jacobi matrix of the linearized system around the steady state as Jp=0 .
Then we have that
Jp=0 = 4ργ−1 ∗∗γ−2−αγθA(γ − 1)(αA)

See Gantmacher (1960) p.194.

11

k

−1 
.
0

(40)

The determinant of Jp=0 is given by |Jp=0 | = −αγθA(γ − 1)(αA)γ−1 k ∗γ−2 < 0. Because
the number of predetermined variables is one in this case, the equilibrium path is uniquely□determined.

We have now established that if p̄ < p̂, any pollution must be fully abated within a finite
period. The dynamic system follows equations (15)-(18) until the level of pollution becomeszero. After this point, it is optimal for the economy to keep pollution at zero, creating a
final steady state with no pollution. If p̄ > p̂, the level of pollution never becomes zero andconverges to a steady state with a positive level of pollution.

4

An abatement technology with constant marginal
cost

In this section, we consider the case in which the abatement technology is characterized byconstant marginal cost. In contrast to the case of γ > 1, the marginal cost of abatement
is positive and finite when the input into abatement is zero. Thus if the marginal cost ofadditional pollution reduction is larger than the marginal value when input into abatement
is zero, it is not beneficial to devote any final goods to abatement—that is, mt = 0 becomes
optimal. However, for analytical simplicity, we assume the following condition, which ensuresthat it is beneficial to reduce pollution at the initial time:δp0 ≤ αAk0 .

(A2)This states that the pollution level emitted is greater than that absorbed at the initial time.
By substituting γ = 1 and mt > 0 into equations (4) and (7), we can rewrite the necessary

12

conditions (8), (11)-(14) as follows:k̇t = Akt − ct − θmt ,(41)

p˙t = αAkt − δpt − mt ,(42)c˙t = (A − ρ − αθA)ct ,(43)

pt = p̄ −σct.
(A + δ − αθA)θ + νt ct(44)

Then we can describe the equilibrium path of the economy using the above equations together
with the complementary slackness condition (8) and the initial values of pt and kt .
Lemma 2 When γ = 1, under assumption (A1), the level of pollution must be zero in thelong run regardless of the level of upper bound p̄. In addition, once the level of pollution
becomes zero, it is optimal to keep its level at zero.
□

Proof. See Appendix A.2.

Lemma 2 indicates that there exists a unique time at which the level of pollution becomeszero. Let us define this time as T . Since the dynamics depend on the level of pollution, we
separate the analyses into two regimes: one corresponds to the case in which the level ofpollution is still positive (t < T ), and the other corresponds to the case in which the level ofpollution is zero (T ≤ t).

4.1

Dynamics without pollution

First, we analyze the equilibrium path when T ≤ t. We use the normalization c̃t = ct /kt ,p̃t = pt /kt , m̃t = mt /kt and define the growth rate of consumption as g ≡ A − ρ − αθA (seeequation (43)). Since pollution is always completely eliminated, by substituting pt = 0 andp˙t = 0 into equations (42) and by using the normalization above, we can rewrite equations

13

(41) and (42) as follows:c̃˙t = c̃t (c̃t − ρ),(45)m̃t = αA.

(46)

Given the ratio of consumption and capital at time T , c̃T , the path can be uniquely determined. If c̃T > ρ, the growth rate of consumption per unit of capital accelerates so that it
violates the TVC. If c̃T < ρ, then c̃T shrinks and finally approaches zero. The only path thatis optimal is realized by c̃T = ρ. Along this optimal path, ct , kt , and mt grow at a constantrate, g. This entails that the economy remains on the balanced growth path only when theratio of consumption to capital at time T , c̃T , becomes ρ. The ratio of abatement good tocapital at time T , m̃T , simultaneously becomes αA. Otherwise, the economy embarks ona destructive path along which the level of consumption shrinks over time or the level ofcapital converges to zero.

4.2

Dynamics with a positive level of pollution

Next, let us characterize the dynamics when the level of pollution is still positive. Since
pt > 0 as long as t < T , by substituting νt = 0 into equations (44), we can rewrite equations(41)-(44) as follows:
{(
)
}
σg
c̃˙t =
1+c̃t − δθp̃t − ρ c̃t ,
ρ+δ+g
)
}
{(
σg
σg
p̃t −
c̃t − {δθp̃t + (1 − αθ)A} p̃t ,p̃˙t =1+
ρ+δ+g
θ(ρ + δ + g)
σgc̃t,m̃t = αA − δ p̃t +θ(ρ + δ + g)

14

(47)
(48)
(49)

where m̃t > 0.5 Thus, we can draw the dynamics of p̃t and c̃t on the p̃ − c̃ plane. Fromequation (47), we can derive the relation between c̃t and p̃t that satisfies c̃˙t = 0. As long as
c̃t > 0, the c̃˙t = 0 locus is expressed as follows:c̃t =δθ(ρ + δ + g)
ρ(ρ + δ + g)p̃t +.
ρ + δ + (1 + σ)gρ + δ + (1 + σ)g(50)

Since the coeﬃcient on p̃t in equation (50) is positive and constant, the c̃˙t = 0 locus is astraight line and increasing in p̃t . Moreover, we can find that the coeﬃcient on c̃t in brackets
in equation (47) is positive (namely, c̃˙t /c̃t is increasing in c˜t ). Therefore, c̃t increases overtime above the c̃˙t = 0 locus, whereas it decreases below the c̃˙t = 0 locus.
In order to analyze the dynamics of the pollution level, we make the following assumption
regarding the initial condition:

p̃0 <

σg
.
θ(ρ + δ + g + σg)(A3)

Under assumption (A3), since the coeﬃcient on c̃t in the first bracket is negative at theinitial time, then p̃t decreases. It follows that p̃t always decreases over time because the firstbracket of equation (48) is also decreasing in p̃t . Hence, the phase diagram in (p̃t , c̃t ) spacecan be drawn as in Figure 1.

4.3

Optimal path for the economy

As mentioned above, in order to reach the unique balanced growth path, consumption percapital c̃t should reach ρ exactly when pollution is eliminated. To analyze the optimal path,
we run dynamics (47) and (48) backwards in time. Inserting t = T and p̃T = 0 in equationσρg(48), we find that p̃˙T = − θ(ρ+δ+g)< 0. It follows that the level of pollution p̃t becomes zero5

σct
The derivation of equation (49) is as follows. Since equation (44) and νt = 0 yield pt = p̄ − (A+δ−αθA)θ
,

σgct
σ ċt
= (A+δ−αθA)θ
. Then by substituting this equation into equation (42), we
it follows that ṗt = (A+δ−αθA)θ
σgc̃t
σgct
, which leads to m̃t = αA − δ p̃t + θ(ρ+δ+g).
obtain mt = αAkt − δpt + θ(ρ+δ+g)

15

c̃t
c̃˙t = 0c̃˙t > 0
ρ
c̃0

(1)

c̃˙t < 0ρσg
1 + ρ+δ+g

0

p̃0

p̃t

Figure 1: Phase diagram in (p̃t , c̃t ) space

p̃t
p̃0

0

T

t

T

t

c̃t
ρ

0

Figure 2: Dynamics of p̃t and c̃t

in a finite amount of time. Then we can find a unique path toward a balanced growth path.
As a result, there exists a unique c̃0 that allows the economy to reach that unique path.
We have now completed our analysis of an economy when the abatement technology ischaracterized by constant marginal cost. The optimal path of each variable (p̃t , c̃t ) is drawn
in Figure 2. Unlike in the case of an abatement technology characterized by increasing
marginal cost, economic growth is compatible with a high-quality environment in the long
run.

16

5

Conclusion

By setting an upper limit on pollution in the utility function, we have shown that economic
growth is compatible with environmental conservation only when the abatement technology
is suﬃciently eﬃcient. Otherwise, the economy will stagnate even if abatement is conducted
eﬃciently. To briefly consider policy implications, our results imply that highly polluted
countries with high abatement costs (e.g., developing countries) must invest not only inundertaking abatement activities but also in developing better abatement technologies.

A

Appendix

A.1

Proof of Lemma 1

Assume that the level of pollution becomes zero in the long run when p̄ > p̂. Then there
exists a time point after which the level of pollution remains at zero. Define this time point
as s. As long as s ≤ t, the economic variables follow equations (11)-(14) with ṗt = pt = 0.
Thus, the dynamic system describing the economy consists of kt and ct and becomes asfollows:k̇t = Akt − θ(αAkt )γ − ct ,()
c˙t = ct A − ρ − αγθA(αAkt )γ−1 .

(A.1)
(A.2)

(
) 1
A−ρ γ−1
1
It can be immediately shown that there exists a unique steady state where kt = αA
αγθA
1
(
) γ−1
A−ρ
and ct = γA+ρ−A. By substituting these variables into equation (14), we can deriveαγAαγθA
the value of νt in the long run as σ(p̂−p̄). Since we assume p̄ > p̂, it follows that the long-run
p̄p̂
value of νt should be negative. However, due to equation (8), νt must be nonnegative whenpt = 0, giving rise to a contradiction. Hence, it is not optimal for the economy to maintain
zero pollution in the long run when p̄ > p̂. On the other hand, when p̄ < p̂, the value of νt

17

in the long run is nonnegative. Therefore, in this case the necessary conditions are optimal
and also suﬃcient for the level of pollution to become zero.

A.2□

Proof of Lemma 2

Assume that the level of pollution again becomes positive after it becomes zero. That is,there are at least two time points at which the level of pollution decreases to zero. Define thetime point T1 as the earlier time at which pollution is completely eliminated and T2 (̸= T1 ) as
the later one. As long as pt > 0 the dynamics of the economy are summarized by equations(41)-(44) with νt = 0. That is,k̇t = Akt − ct − θmt ,(A.3)

ṗt = αAkt − mt ,(A.4)

ċt = gct ,(A.5)

pt = p̄ −σct .
(ρ + δ + g)θ

(A.6)

These equations should be satisfied when t = T1 as well as when t = T2 . Therefore, by
substituting pT1 = 0 and pT2 = 0 into equation (A.6 ), we obtain:

cT1 = cT2 =(ρ + δ + g)θp̄
.
σ

(A.7)

On the other hand, due to (A.5 ), ct grows at a constant rate g except when ct = 0. Since
T1 < T2 , the following inequality must be satisfied:

cT1 < cT2 .

This implies that inequality (A.8 ) contradicts equation (A.7 ).

18

(A.8)

□

References[1] Brock, W. A., and Taylor, M. S. (2010). The green Solow model.Journal of Economic
Growth 15(2): 127-153.
[2] Gantmacher, F. R. (1960). Theory of Matrices. 2V. New York: Chelsea Publishing Company.
[3] Huang, C.-H., and Cai, D. (1994). Constant-returns endogenous growth with pollution
control.Environmental and Resource Economics 4(4): 383-400.
[4] Michel, P., and Rotillon, G. (1995). Disutility of pollution and endogenous
growth.Environmental and Resource Economics 6(3): 279-300.
[5] Ricci, F. (2007) Channels of transmission of environmental policy to economic growth:
A survey of the theory.Ecological Economics 60(4): 688-699.
[6] World Health Organization. (2014). Ambient (outdoor) air quality and health. Fact sheet
no. 313, March 2014. Geneva: WHO.

19