Discussion Papers In Economics
And Business
The impacts of pollution control policies on the pollution
in small open economies
Koichi Futagami and Yasuhiro Nakamoto

Discussion Paper 19-08

May 2019

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe impacts of pollution control policies on the pollution insmall open economies∗
Koichi Futagami† and Yasuhiro Nakamoto‡
March 1, 2019

Abstract
We examine the eﬀects of tax policies on pollution in a small open economy. There
are two pollution causes: consumption activities of households and production activities
of firms. In this setup, we examine how tax policies aﬀect pollution and the small open
economy. Considering that pollution control policies are undertaken only for temporary
periods in countries, we mainly focus on duration of governments’ pollution control policies; that is, permanent and temporary policies. The main finding is that the government
in the small open economy need to tackle pollution problems from a long-term perspectivein order to decrease the level of pollution.
Keywords: Pollution, Pollution control policies, Temporary and permanent policies, Small
open economy
JEL Classification Code: F41, H21, Q52∗

We are grateful to Minoru Nakada, Masayuki Sato, Akihiko Yanase and participants at annual meeting ofSociety for Environmental Economics and Policy Studies, and the Japan Society of International Economics.
†
Graduate School of Economics, Osaka University, 1–7 Machikaneyama, Toyonaka 650–0043 Japan. Email: futagami@econ.osaka-u.ac.jp‡
Corresponding author: Faculty of Informatics, Kansai University, 2-1-1 Reizanji-cho,Takatsuki-shi, Osaka,569-1095, Japan, e-mail: nakamoto@kansai-u.ac.jp

1

Introduction

Since economic activity itself creates environmental pollution, the control and regulation ofpollution is a significant and diﬃcult issue in almost all countries. Even if developed countries have superior technologies to combat pollution to some extent, pollution which comefrom diﬀerent pollutants has been serious problems for even these countries. Needless to say,other countries experience much severer impacts of pollution. If governments address pollution problem seriously, the control and regulation of pollution impedes the economic growthand production activities. For example, when the regulation of industrial pollution reduces
the level of production, export countries might lose opportunities to acquire foreign assets.
Because the acquisition of foreign assets enables export countries to achieve rapid growth oftheir economies, the enforcement of pollution control policy is a very serious problem. As
a result, many countries are forced to struggle with the trade-oﬀ between the regulation ofpollution and economic development.
For instance, the activities of multinational oil companies in the Niger Delta region of
Nigeria have caused substantial land, water, and air pollution. However, the regulation ofpollution has not been a priority issue in Nigeria, because oil exploration and production
enable Nigeria to maintain its rapid growth. Similarly, China would not have achieved its
dramatic economic growth over the preceding decades if it had regulated pollution from theinitial stage of its development.
Since some governments focus on improving levels of economic activities, they are more
likely to be short-sighted. This means that pollution control policies are not determined froma longer-term perspective but rather from a shorter-term one. Therefore, in some countries,
pollution control policies are undertaken only for a temporary period. From a theoreticalpoint of view, after the temporary policies are repealed, the pollution level returns to its
original level; hence, at least, the temporary enforcement of pollution control policies does
not seem to lead to any serious problem in the long run.
However, we can find some examples that show the repeal of pollution control policies has

2

a considerably negative impact on the environment.1 We here highlight two cases.2 First,
a high-occupancy vehicle (HOV) policy is known to reduce air pollution, carbon emissions,traﬃc congestion on roads, and the need for parking space because HOVs generally refer tovehicles carrying two or more persons on a single journey. Because the major thoroughfare in
Jakarta, Jalan Sudirman, suﬀered from heavy traﬃc congestion, the government introduced
the HOV policy (“three-in-one” policy) in 1992, which required all private cars to carry atleast three passengers during peak hours. However, in March 2016, the Indonesian government abruptly canceled the HOV policy. Thereafter, in Jakarta, travel delays became 46%
worse during the morning rush hour and 87% worse during the evening rush hour. Furthermore, Hanna et al. (2017) demonstrated that the pollutants per daily trip from motor
vehicles dramatically increased due to heavy traﬃc congestion after the HOV restrictions
were eliminated. Next, we introduce the case of Australia. The Australian government imposed a carbon tax through the Clean Energy Act in 2011. Although the legislation achieved
a reduction in the country’s carbon emissions, it was repealed in 2014 to lower costs for Australian businesses and to ease the cost of living pressure on households. As a result, carbon
emissions and electricity demand in Australia have jumped dramatically. Specifically, carbon
dioxide emissions from electricity increased by 6.4 million tonnes in 2015. Furthermore, according to the Department of the Environment and Energy of the Australian Government,
the emissions level continued to increase after the repeal of the carbon tax, and might exceed
the original level before the introduction of the emissions control policy.3
What would occur for such short-run pollution control policies? To tackle this question,
we examine how pollution control policies, represented by tax policies, aﬀect the pollution in
1

This may also apply to the United States in near future. Since January 2017, the current US administration

has sought to upend the environmental protection policies enforced by the previous administration, because
excessive regulations are harming US industry and thus, reducing regulation will be good for the economic
activities. At the moment, we cannot correctly evaluate the negative impact on the environment by therepeal of the environmental protection policy; however, as for the health problem of US residents, Cutler andDominici (2018) have already concluded that the policy rollbacks will lead to an additional 80,000 U.S. deaths
per year and respiratory problems in more than 1 million people, as a result of damage that would be diﬃcult
for a future administration to undo.
2
We are grateful for the useful comments of Minoru Nakada about these cases.
3
For example, see Figure 3 in http://www.environment.gov.au/system/files/resources/9437fe27-64f4-4d16b3f1-4e03c2f7b0d7/files/aust-emissions-projects-chart-data-2016.xlsx

3

a small open economy. The reason that we focus on the small open economy is that while
the small open economy relies too much on less stringent pollution controls, the pollution
emission in the economy has a larger contribution to the world pollution emission year by year,
implying that the small open economy needs to take responsibility for own large pollution
emission. Thus, we construct a dynamic general equilibrium model of the small open economy
with pollution, and incorporate pollution control policies into this model.
Our economy in this study has the following three characteristics. First, we suppose thepollution evolved over time; that is, economic activities, such as consumption and production,
generate flows of pollution, and thus, the pollution accumulates over time. In addition, acertain part of this stock is broken down at every time owing to the assimilative capacity ofthe environment.
Second, we consider two pollution causes that generate the flow of pollution. The firstcause is the production activities of firms. Owing to increases in demand for food, shelter, andhousing, more commodities are produced. Therefore, firms need to dispose of more waste,
which contains toxic chemicals and pollutants, leading to water, air, and soil pollution. For
instance, many industries drain the waste in the fresh water that goes into rivers, canals, andlater the sea. Furthermore, to produce huge amounts of energy through fossil fuels like coaland oil, and nuclear fission and fusion, thermal power stations produce substantial amountof ash in the atmosphere, and nuclear plants dispose of left-over radioactive materials that
contain harmful and toxic chemicals.
The other pollution cause we consider is the consumption activities of households. As forsoil pollution, households often use highly toxic fertilizers and pesticides to remove insects
and bacteria from their crops. In addition, each household produces tonnes of garbage eachyear, such as aluminum and plastic produced in modern technology. This garbage becomespart of landfills and causes soil pollution. With regard to water pollution, households produce
chemical waste water and sewage, and release it into rivers. Finally, considering air pollution,transportation has assumed an important role in our lives and cars, trains, shipping vessels,
and airplanes all burn a lot of fossil fuels to operate.
Third, in this study, a government plays an important role in decreasing the level ofpollution. The reason is that our model includes the pollution control policies that directlyreduce economic activities as the pollution causes. As discussed previously, we are interested
4

in the duration of policy implementation. In summary, some countries want to avoid pollution control policies that impede economic activities, so that they have strong incentives to
conduct pollution control policies not from a long-term perspective, but only for a certain
period. Therefore, by comparing the eﬀects of temporary policies with permanent ones, weconsider the dynamic movement of pollution. Then, we find that the pollution level decreases
during the pollution control policy, and that after the policy is repealed, the pollution level
always exceeds the original level in our small open economy. Importantly, as in the abovementioned cases in Australia and Indonesia, our numerical examples show a serious problem
that countries may face after the repeal of pollution control policy.
Our study is related to existing investigations in dynamic models with pollution as wellas international macrodynamic models. As for dynamic models with pollution, many studies
incorporate pollution into macrodynamic models. Earlier studies, such as Lopez (1994) andSelden and Song (1995), construct dynamic models with pollution generated by production
but not by consumption. On the other hand, John and Pecchenino (1994), John et al. (1995),
and McConnell (1997) develop dynamic models with pollution generated by consumption
rather than production. The latter authors mainly examine an inverted U-shaped curveof pollution intensity over time. In addition, many researchers focus on the relationshipbetween economic growth and pollution, and examine whether sustainable endogenous growthis achieved or not (e.g., Huang and Cai, 1994; Bovenberg and Smulders, 1995; Michel andRotillon, 1995; Smulders and Gradus, 1996; Chevé, 2000; Gupta and Barman, 2009; Greiner,2005). The purposes of these studies are not the same as ours. In other words, they do not
focus on the eﬀects of temporary implementation of pollution control policies.
The structure of our model is closely related to international macrodynamic models, suchas those of Sen and Turnovsky (1990), Turnovsky (1997), Schubert and Turnovsky (2002),
and Nakamoto and Futagami (2016), in that the models in these studies focus on temporary
policies in small open economies. These studies find that when policies temporarily change
under the assumption of perfect foresight, the small open economy does not return to its
original steady state after the policy implementation is completed. This insight on public
policies is completely diﬀerent from the results of dynamic macroeconomic models of closed
economies. The temporary implementation of public policies in closed economies does not
aﬀect the qualitative impact on the long-run economy, that is, the long-run equilibrium
5coincides with the original steady state. However, unlike our study, these studies do notinclude pollutants in their dynamic models, and furthermore, their attention is not to examine
the eﬀects of pollution control policies.
Finally, our motivation is like that of Rondeau and Bulte (2007) and Nakamoto andFutagami (2016) in that these studies cast some doubt on the usefulness of environmental
protection policies. Rondeau and Bulte (2007) and Nakamoto and Futagami (2016) focus on
renewable natural resources, such as forestry, fish, and wildlife stocks, while the current study
is interested in pollution accumulation generated by two diﬀerent pollutants. In addition,
Rondeau and Bulte (2007) make use of a single-country partial equilibrium model with theinteraction between habitat and open-access resource. Nakamoto and Futagami (2016) andthe present study make use of dynamic models in a small open economy.
The remainder of this paper is organized as follows. In the next section, we present our
set-up. Section 3 shows the dynamic system under our set-up. Section 4 shows the uniquenessof the steady-state equilibrium with the saddle-path stability. Section 5 examines the eﬀects
of pollution abatement policies on the economy. Section 6 gives numerical examples. Section
7 summarizes our findings.

2

Baseline Model

We consider a small open economy in which the world interest rate, r, is constant. The
population is constant and normalized to unity. Denoting the time index by t, we express
the production function, which satisfies constant returns to scale with respect to capital (k̂t )and labor (lt ), as follows:
yt = F (k̂t , lt ) = lt f (kt ),where yt is output and kt ≡ k̂t /lt denotes capital intensity. The intensive form of the production function, f (kt ), is monotonically increasing and strictly concave in kt , and satisfies the
Inada conditions. Taking account of competitive factors and final goods markets, the real
rent r and real wage rate wt are determined byr = f ′ (kt ),wt = f (kt ) − kt f ′ (kt ),

6

(1a)

respectively. Since the real rent is constant in the small open economy, from (1a), the capitalintensity is constant and hence, the wage rate is fixed as well (kt = k̄ and wt = w̄). As aresult, the production function is simply given by
yt = lt f (k̄).

2.1

(1b)

Pollution

Following the existing papers with the stock of pollution (e.g., Huang and Cai, 1994; Chevé,2000), we suppose that the net flow of pollution caused by consumption and output adds tothe stock of pollution, and the ratio θ of the pollution stock is broken down owing to theassimilative capacity of the environment. Then, the accumulation equation of pollution is
given byṗt = αc G(ct ) + αy N (lt f (k̄)) − θpt ,(2)

where pt is the level of the pollution stock, ct is the level of consumption, and θ is the natural
decay rate of the pollution stock. The functions G(ct ) and N (lt f (k̄)) represent the flow ofpollution. We assume thatG′ (ct ) > 0, G′′ (ct ) > 0, N ′ (lt f (k̄)) > 0, N ′′ (lt f (k̄)) > 0,lim G′ (ct ) = 0, lim G′ (ct ) = ∞, lim N ′ (lt f (k̄)) = 0, lim N ′ (lt f (k̄)) = N ′ (f (k̄)).

ct →0

ct →∞

lt →0

lt →1

(3a)
(3b)

Assumption (3a) means that the flow of pollution is monotonically increasing and strictly
convex. In other words, the more active the economy becomes, the higher the flow level ofpollution is. In what follows, assumption (3b) corresponds to the Inada conditions. Furthermore, αc and αy in (2) are non-negative parameters that specify each pollution flow.
For instance, when αc > 0 and αy = 0, we consider the case in which the pollution flow is
generated by consumption, not output.

2.2

Households

The welfare level of the household is given by
U0 ≡

∫ ∞

{
}
u(ct ) − v(pt ) + Γ(Lt ) e−ρt dt,

0

7

(4)

where ρ(> 0) is the rate of time preference and Lt (= 1 − lt ) is the leisure time. We assume
that u′ (ct ) > 0, u′′ (ct ) < 0, v ′ (pt ) > 0, v ′′ (pt ) > 0, Γ′ (Lt ) > 0, and Γ′′ (Lt ) < 0. We
further assume each of the following Inada conditions limct →0 u′ (ct ) = ∞, limct →∞ u′ (ct ) = 0,
limpt →0 v ′ (pt ) = 0, limpt →∞ v ′ (pt ) = ∞, limLt →0 Γ′ (Lt ) = ∞, and limLt →1 Γ′ (Lt ) = Γ′ (1).
The accumulation of the foreign assets holding, bt , evolves asḃt = rbt + (1 − τ y )lt f (k̄) − (1 + τ c )ct + zt ,(5)

where zt is the lump-sum transfer from the government. We suppose that τ y is the rate ofincome tax and τ c is the rate of consumption tax. τ j (j = y and c) are constant over time.
These rates satisfy 0 ≤ τ y < 1 and τ c ≥ 0. The government keeps the following balanced
budget over time:
τ c ct + τ y lt f (k̄) = zt .

(6)

We here make use of consumption and income taxes as the methods of pollution control
by the government. Then, one may wonder the use of these tax policies, because taxing
pollution/emission directly is more eﬃcient. However, because policy makers place a great
deal of weight on economic activities in own countries, it is diﬃcult to impose green taxes
called as pollution or emission taxes upon polluting firms directly in even a large economy
like the United State (e.g., see the environmental policy of the Donald Trump administrationand the Kyoto Protocol without the United States ratification). Because many developed
countries continue to evade their responsibility to provide adequate financial resources, people
and governments in other countries will not accept green taxes actively. Next, when the green
taxes are imposed on polluting firms in less developed countries, its policy may be directly
linked to the survival problem of residents because people may lose their job and may be outof work for a long time without adequate support by governments. On the contrary, imposing
the green taxes on polluting firms in developed countries may change life-style, but it does
not lead to the serious risk of death under social security system. As a result, imposing green
taxes is more diﬃcult in less developed countries than developed ones, implying that it may
be needed to consider alternative ways of controlling pollution.
Based on the fact that the consumption and income taxes are not imposed on polluting
firms directly and the primary purpose of their tax policies is not to reduce pollution, it iseasier for people and governments in many countries to accept these taxes to control certain8

pollutants. For example, when the pollution flow is generated by consumption, we adopt therate of consumption tax τ c as the pollution prevention policy, but not the rate of income tax
τ y . In other words, increasing the rate of consumption tax directly has a negative impact
on the demand for the commodities of consumption, which means that the pollution cause
will be indirectly reduced. On the other hand, when the pollution flow is caused by output,
we adopt the rate of income tax τ y as the pollution prevention policy, but not the rate of
consumption tax τ c . An increase in the rate of income tax directly has a negative impact on
working, which implies that it will lead to a lower level of pollution indirectly.

3

Dynamic system of small open economy

The representative household maximizes its lifelong utility (4) subject to the evolution ofpollution (2) and the budget constraint (5). We set the current-value Hamiltonian:[ ][ ]H = u(ct ) − v(pt ) + Γ(Lt ) − µt (2) + λt (5) ,(7)

where µt and λt show the shadow values for equations (2) and (5), respectively.
The first-order conditions of the maximization problem are
Hct : u′ (ct ) − µt αc G′ (ct ) = λt (1 + τ c ),(8a)

Hlt : λt (1 − τ y )f (k̄) = Γ′ (Lt ) + µt αy N ′ (lt f (k̄))f (k̄),(8b)

Hpt : −v ′ (pt ) + µt θ = µ̇t − ρµt ,(8c)

Hbt : λt r = −λ̇t + ρλt .

(8d)

The transversality conditions are given bylim µt pt e−ρt = 0,

lim λt bt e−ρt = 0.

t→∞

t→∞

(8e)

In what follows, since the rate of time preference ρ and the interest rate r are both fixed,
we require r = ρ for our system to have a finite interior steady-state value for the shadow
value of foreign assets. Therefore, assuming that r = ρ, from equation (8d), we obtain theconstant level of shadow value λt :λt = λ̄,9(9)where λ̄ should be determined endogenously.
Substituting (9) into (8a) and totally diﬀerentiating it, we can obtain the following:ct = c(µt , λ̄, τ c ),
where

(10)

∂ct
αc G′ (ct )∂ct
1 + τc∂ctλ̄=−(< 0),=−(< 0),= − (< 0),
c
∂µt
θc
θc
∂τ
θc
∂ λ̄
θc = µt αc G′′ (ct ) − u′′ (ct )(> 0).

Next, substituting (9) into (8b), we totally diﬀerentiate (8b) and obtain
lt = l(µt , λ̄, τ y ),

(11)

where
αy N ′ (lt f (k̄))f (k̄)
∂lt
∂lt
∂lt
(1 − τ y )f (k̄)λ̄f (k̄)=−(< 0),(> 0),(< 0),
=
=−
y
∂µt
θl
θ
∂τ
θl
∂ λ̄
l
θl = −Γ′′ (Lt ) + αy µt N ′′ (lt f (k̄))f (k̄)2 (> 0).
Finally, substituting (10) and (11) into the evolution of the pollution stock (2), we canrewrite (2) as follows:ṗt = αc G(c(µt , λ̄, τ c )) + αy N (l(µt , λ̄, τ y )f (k̄)) − θpt ,(12a)

and the dynamic equation of the shadow price of the pollution stock in (8c) isµ̇t = µt (ρ + θ) − v ′ (pt ).

4

(12b)

Steady-state equilibrium and stability

We denote the steady-state levels of variables by asterisk. Then, we observe that the steadystate levels of the pollution stock and shadow price of the pollution stock are determined byṗt = 0 in (12a) and µ̇t = 0 in (12b) as follows:µ∗ (ρ + θ) = v ′ (p∗ ).

(13a)

αc G(c(µ∗ , λ̄∗ , τ c )) + αy N (l(µ∗ , λ̄∗ , τ y )) = θp∗ .

(13b)µ̇t = 0 :ṗt = 0 :10

Using Figure 1, we now confirm the uniqueness of the steady-state equilibrium and thesaddle-path stability. First, we depict the µ̇t = 0 locus. From (13a), we observe that∂µ∗v ′′ (p∗ )=(> 0).
∂p∗ µ̇t =0
ρ+θ

(14a)

Furthermore, we observe that when the level of the shadow price of the pollution stock µ∗
approaches zero, the level of the pollution stock approaches zero as well. When µ∗ goes toinfinity, the level of the pollution stock becomes infinite. Therefore, as observed in Figure 1,
the µ̇t = 0 locus is an upward-sloping curve that goes through the origin.
We next examine the ṗt = 0 locus defined by (13b). We show the slope of the ṗt = 0 locus
given λ̄, τ c and τ y :θ∂µ∗=(< 0).
∗
∗
∂c
∂l∗
′
∗
∂p ṗt =0 αc G (c ) ∂µ∗ + αy N ′ (l∗ f (k̄))f (k̄) ∂µ
∗

(14b)

Furthermore, when the shadow value of the pollution stock µ∗ goes to zero, we observe thatlimµ∗ →0 c̃ = (u′ )−1 (λ̄(1 + τ c ))(> 0) from (8a) and that limµ∗ →0 L̃ = (Γ′ )−1 (λ̄(1 − τ y )f (k̄))
from (8b), because lt < 1; hence, ˜l = 1 − (Γ′ )−1 (λ̄(1 − τ y )f (k̄))(> 0). In summary, the stock()
of pollution takes a finite level to satisfy limµ∗ →0 p∗ = αc G(c̃) + αy N (˜lf (k̄)) /θ. On theother hand, as the level of shadow value of the pollution stock µ∗ approaches infinity, thelevel of the pollution stock goes to zero. This result is because the levels of consumption andlabor supply are zero, as easily observed from (8a) and (8b). As a result, the ṗt = 0 locus
is downward sloping, as in Figure 1, where the value of the intercept at the horizontal axis
(p−axis) is finite. Therefore, since the µ̇t = 0 locus crosses the ṗt = 0 once, the steady-state
equilibrium is uniquely determined, as shown in Figure 1.
We now examine the stability around the steady state. As for the µ̇t = 0 locus, theright-hand side of this locus holds that µ̇t = µt (ρ + θ) − v ′ (pt )(< 0) under a fixed level ofshadow value of the pollution stock µt , because of v ′′ (pt ) > 0. Alternatively, the opposite
inequality holds in its left-hand side. Next, the upper side of the ṗt = 0 locus shows thatṗt = αc G(c(µt , λ̄, τ c )) + αy N (l(µt , λ̄, τ y )) − θp̄(< 0) given the level of the pollution stockp̄. This is because ∂G/∂µt < 0 and ∂N/∂lt < 0. Below the ṗt = 0 locus, we observe thatṗt = αc G(c(µt , λ̄, τ c )) + αy N (l(µt , λ̄, τ y )) − θp̄(> 0). Therefore, as confirmed in Figure 1, thesteady-state equilibrium becomes saddle-point stable. The stable root β s and the unstable11

one β u of our system in (12a) and (12b) are given by:4
√
√
∂pt
∂pt
ρ − ρ2 − 4(−θ(ρ + θ) + v ′′ (pt ) ∂µ)ρ+
ρ2 − 4(−θ(ρ + θ) + v ′′ (pt ) ∂µ
)
t
t
s
u
β =
(< 0), β =(> 0),
2
2
(15)
where
∂ct
∂lt
∂ ṗt
= αc G′ (c∗ )+ αy N ′ (l∗ f (k̄))(< 0).
∂µt∂µt∂µt
The abovementioned results are summarized as follows.
Proposition 1 The steady-state equilibrium is uniquely determined and satisfies the saddlepath stability.
Denoting an element of an eigenvector corresponding to the stable root β s as ∆s , weobtain the linearly approximated system of (12a) and (12b) along the stable path:
pt = p∗ + (p0 − p∗ )eβ t ,(16a)µt = µ∗ + ∆s (p0 − p∗ )eβ t ,

(16b)

s

s

where ∆s is given by∆s =v ′′ (p∗ )(> 0).
ρ + θ − βs(16c)

To determine the fixed level of shadow value (9), we make use of the accumulation equationof the foreign asset holding. Approximating (5) linearly around the steady state, we can derive
the following equation:ḃt = r(bt − b∗c ) + Ω∆s (p0 − p∗ )eβ t ,
s

(17a)

where
Ω = f (k̄)
4

Our linearized system is given by
  
µ̇
ρ+θ
 t = ṗt∂ ṗt /∂µt

∂lt
∂ct
−
.
∂µt ∂µt

(17b)



−v ′′ (p∗ )µt − µ∗.
−θpt − p∗This has the trace and the determinant as follows:
Tr = ρ(> 0), Det = −θ(ρ + θ) + v ′′ (p∗ )

12

∂ ṗt
(< 0).
∂µt

Solving the dynamic equation (17a), we obtain:5
bt − b∗ = Θs (p0 − p∗ )eβ t ,
s

(17c)

where Θs is defined by
Θs =

Ω∆s
v ′′ (p∗ )
=−
s
β −r
(ρ − βs )(ρ + θ − βs )

(
)
∂lt
∂ct
f (k̄)−∂µt ∂µt

Setting t = 0 in (17c), we obtainb∗ − b0 = Θs (p∗ − p0 ).

(17d)Finally, in order to express the steady-state equilibrium, we take account of the economy
starting at time Tj and corresponding to a policy set (τjc , τjy ). Moreover, the viable steadystate is associated with the initially given levels of the pollution stock and the foreign assets
pTj and bTj at time Tj . Following (13a), (13b), (17d), and the following ḃt = 0 equation, weshow a viable steady state j with (τjc , τjy ) as follows:5µ∗j (ρ + θ) = v ′ (p∗j ).

(18a)

αc G(c(µ∗j , λ̄∗j , τjc )) + αy N (l(µ∗j , λ̄∗j , τjy )) = θp∗j ,(18b)

rb∗j = c(µ∗j , λ̄∗j , τjc ) − l(µ∗j , λ̄∗j , τjy )f (k̄).

(18c)

b∗j − b0 = Θsj (p∗j − p0 ).

(18d)

The eﬀects of pollution control policies

In this section, we analyze how the pollution control policy aﬀects the level of the pollution
stock as well as the other economic variables. We must mention the following two points.
5

Solving (17a) yields:()∗
sΩ∆s (p0 − p∗ )∗rt
s p0 − pbt = b + b0 − b −e+ΩΛeβ t .
βs − rβs − r∗Then, we obtain (17c). The parentheses on the right-hand side must be zero owing to the intertemporalsolvency condition; that is,b0 − b∗ =ΩΛs (p0 − p∗ ).
βs − ρ

13

First, when investigating the long-run eﬀects of policies on the pollution stock, we deal withtwo pollution causes separately; that is, assuming that αc > 0 and αy = 0 (αc = 0 and
αy > 0), we examine the policy eﬀect in the case in which the pollution flow is generated
by consumption (output). In what follows, as for the pollution control policies, we supposethat when the pollution flow is generated by consumption, the pollution control policy is
to increase the rate of consumption tax. This assumption is because increasing the rateof consumption tax directly decreases the demand for the consumption commodities, whichmeans that the level of pollution cause will be reduced. This dampens an increase in thelevel of the pollution stock. Next, we suppose that when the pollution flow is generated byoutput, the pollution control policy is to raise the rate of income tax. This assumption is
because an increase in the rate of income tax directly restrains the household from working,and hence, it will lead to a lower level of pollution flow. This dampens an increase in thelevel of the pollution stock.
We now consider the diﬀerence of temporary policies in closed and small open economies.
In closed economy models, a temporary policy change aﬀects the transitional path over time,and after the temporary change is removed, the economy gradually returns to the original
steady state. Therefore, we conclude that the temporary policy change does not have any
qualitative impacts on the long-run economy. On the other hand, in small open economy
models, when the policy is temporarily conducted, the position of the long-run steady state
does not coincide with the original one.
To consider these policies, we suppose that the economy at the initial time Tj = T0 is in
the steady state in which τjc = τ0c and τjy = τ0y . Then, from (18a) − (18d), we denote theoriginal steady-state equilibrium at time T0 as follows:6
p0 = p∗0 = P (λ̄0 , τ0c , τ0y ), Pλ̄ < (>)0 if αc > 0 and αy = 0, (αc = 0 and αy > 0), Pτ c < 0, Pτ y < 0,(19a)
µ0 = µ∗0 = M (λ̄0 , τ0c , τ0y ),Mλ̄ < (>)0 if αc > 0 and αy = 0 (αc = 0 and αy > 0), Mτ c < 0, Mτ y < 0,(19b)b0 = b∗0 = B(λ̄0 , τ0c , τ0y ),(19c)

λ̄0 = λ̄∗0 = Λ(τ0c , τ0y ), Λτ c < 0 if αc > 0 and αy = 0, Λτ y > 0 if αc = 0 and αy > 0. (19d)6

See Appendix A.

14

Suppose that the government announces the temporary changes of the policy instrumentsfrom the original levels (τ0c , τ0y ) to a new level (τ1c , τ1y ) at the initial period. In this case, theunanticipated change leads to an initial jump of the shadow value of foreign assets. Afterthe initial change of policy instruments at time 0, some duration, t ∈ [0, T ), has passed,
and thereafter, the policy instruments return to the original level after time T . Importantly,
because of the assumption of perfect foresight, the household can initially anticipate thepolicy return at time T . This assumption means that the level of the shadow value of foreign
assets does not change over time, including time T , except for the initial jump at time 0.
In other words, the level of this shadow value continues to be fixed so as to sustain theintertemporal solvency condition. Finally, since only this eﬀect remains in the long run, thetemporary policy changes have long-run impacts on the level of the pollution stock.

5.1

Consumption as the pollution cause

Assuming that the pollution flow is generated only by consumption (i.e., αc > 0 and αy = 0),
we now examine the eﬀects of changes in pollution control policies on the pollution stock in
the long run. Conducting a comparative static analysis, we obtain the following proposition.
Proposition 2 Assume that αc > 0 and αy = 0. Then, a permanent increase in the rateof consumption tax decreases the steady-state level of the pollution stock, but a temporary
increase leads to a more severe level of the pollution stock in the steady state.
Proof. See Appendices B and C.
We now consider the dynamic behavior of the economy shown in Proposition 2. When
αc > 0 and αy = 0, Figure 2 illustrates the eﬀects of an increase in the rate of consumption
tax, where E0 is the original steady state. Note that PER (or TEM) in this figure shows thecases in which the pollution control policies change permanently (or temporarily).
First, we observe that the enforcement of increasing the rate of consumption tax does notchange the µ̇t = 0 locus. On the other hand, the ṗt = 0 locus is influenced by a change inthe rate of consumption tax.
When the pollution flow is caused by consumption alone (i.e., αc > 0 and αy = 0), from(18b), the permanent increase in the rate of consumption tax always shifts the ṗt = 0 locus

15

downward given τ y and µ∗ :αc G′ (c∗j )∂p∗j=∂τjc ṗt =0, µ∗ =constant | {z
θ }
j

(+)

(

)
∂c∗j
Λτ c + c (< 0).
∂τj∂ λ̄∗j| {z } |{z}
∂c∗j

(#1)

(20)

(#2)

This derivative with respect to the ṗt = 0 locus consists of two parts. The term (#1) shows
the indirect impact that is caused by a change in the shadow value of foreign assets λ̄, andthe term (#2) indicates the direct impact. The indirect impact takes a positive value, while
the direct impact takes a negative value; however, the direct impact dominates the indirect
one as follows:7

(
)
∂c∗j
1
f (k̄) ∂c∗j ∂lj∗(< 0),
Λτ c + c =−∂τjBλ̄ − Θsj Pλ̄r ∂τ c ∂ λ̄∗j∂ λ̄∗j
|
{z
}|
{z
}
∂c∗j

(−)

(21)

(+)

where we make use of ∂c∗j /∂τ c < 0 in (10) and ∂lj∗ /∂ λ̄∗j > 0 in (11).
Figure 2(a) shows that when the initial economy is at the point E0 , the permanent increasein the rate of consumption tax initially leads to the downward jump of the shadow value ofthe pollution stock at the point I. Thereafter, the level of the pollution stock monotonicallydecreases toward the new steady-state equilibrium E1 .
In what follows, from (17c) and (18d), we characterize changes in foreign assets overtime. We here note that Θsj has a negative sign. Figure 2(b) depicts a negative relationshipbetween foreign assets and the pollution stock. When the rate of consumption tax permanently increases, the level of the pollution stock monotonically decreases, as shown in Figure2(a). Thus, the level of foreign assets monotonically increases from E0 to E1 , as shown in
Figure 2(b). The reason is that increasing the rate of consumption tax reduces the level ofconsumption, which increases the level of foreign assets.
We now turn to the case in which the rate of consumption tax temporarily increases.
In this case, the key element to understand the dynamic behavior of the economy is the7This can be shown as follows:][()) ∂c∗j ()∂c∗j −Bτ c + Θsj Pτ c∂c∗j (
∂c∗j
1
s
s
=
=
−Bτ c + Θj Pτ c +Bλ̄ − Θj Pλ̄ ,+Bλ̄ − Θsj Pλ̄∂τ cBλ̄ − Θsj Pλ̄ ∂ λ̄∗j
∂τ c
∂ λ̄

∂c∗j
∂c∗j
Λ c+
∗ τ
∂τjc
∂ λ̄j
where we use∂c∗j s∂c∗j s∂c∗j∂c∗jf (k̄) ∂c∗j ∂lj∗
Θ P c−
Θj Pλ̄ = 0, and −Bτ c +
Bλ̄ = −
.
∗ j τ
∗
c
c
∂τ
∂τ
r ∂τ c ∂ λ̄∗j
∂ λ̄j
∂ λ̄j
This proves (21).

16dynamic behavior of the shadow value of foreign assets λ̄. Because of the temporary change
in the pollution control policy, the level of the shadow value of foreign assets initially jumps.
Thereafter, even if the rate of consumption tax returns to the original level after the periodof implementation t = [0, T ], the level of the shadow value of foreign assets does not goback to its original level. This outcome is because the household can initially anticipate thepolicy change at time T ; that is, the level of the shadow value of foreign assets stays there
permanently in order to sustain the intertemporal solvency condition. Therefore, the indirect
eﬀect (#1) in (20) remains in the long run. This result implies that the temporary increasein the rate of consumption tax leads to the upward shift of the ṗt = 0 locus in the long run:∂p∗j
αc G′ (c∗j ) ∂c∗j=Λτ c (> 0).
∂τjc ṗt =0, µ∗ =constant | {zθ } ∂ λ̄∗jj
| {z }
(+)

(22)

(+)

As shown in Figure 2(a), the upward shift of the ṗt = 0 locus in the long run leads to a higher
level of the pollution stock in the long run.
The reason that the temporary pollution control policy increases the level of the pollution
stock in the long run is that the direction of initial jump of the shadow value of foreign assets
is downward, as observed by Λτ c < 0 of (19d). To understand its consequence, it is helpful
to make use of (8b) under αy = 0:λ̄∗j (1 − τ y )f (k̄) = Γ′ (1 − ¯lj∗ ).

(23)

As easily confirmed in the above equation (23), the constant level of the shadow value of
foreign assets leads to the fixed level of labor supply ¯lj∗ . Then, the initial change of labor
supply determines the direction of jump of the shadow value of foreign assets. An increasein the rate of consumption tax has a negative impact on the level of consumption at the
beginning. Since the household tries to maintain its welfare level, it decreases the level oflabor supply and increases the level of leisure time. Therefore, from (23), a lower level ofmarginal utility of leisure leads to the downward jump of the shadow value of foreign assets
at the initial time. Importantly, after the rate of consumption tax returns to the original
rate, the level of consumption increases to the original level; however, the lower level of the
shadow value of foreign assets remains. As a result, from (10), the lower level of the shadow
value of foreign assets leads to a higher level of consumption than its original level. Finally,17

noting that only this eﬀect is kept in the long run, we find that the level of the pollution
stock increases.
In Figure 2, we confirm the dynamic behavior of the economy in which the initial economy
starts from the original steady state E0 . When the government raises the rate of consumption
tax at the initial time, the ṗt = 0 locus moves downward to TEM: ṗt = 0 (t < T ), and the
shadow value of the pollution stock jumps downward toward the point I ′ . In duration [0, T ),the level of the pollution stock monotonically decreases toward I ′′ along the unstable path.
When the economy reaches the point I ′′ , the level of the shadow value of the pollution stock
begins to increase when the economy is still on the unstable path. When the government
returns to the original policy at time T , the economy just reaches the point I ′′′ . Importantly,
even if the government returns the rate of consumption tax to its original rate, the ṗt = 0
locus does not go back to its original position, because the level of the shadow value of foreign
assets does not return to its original level. As a result, after this policy reversion, the economy
moves along the stable path to reach the new steady-state equilibrium E2 . At the new steadystate, the level of the pollution stock is greater than its original level.
Making use of (17c) and (17d), we next consider the dynamic behavior of foreign assets.
From Figure 2(b) we observe that in the period [0, T ), the level of foreign assets monotonicallyincreases from E0 toward Q, based on the fact that p0 > pT . After the economy follows
the stable saddle path to reach the steady-state equilibrium E2 , the level of foreign assetsmonotonically decreases from Q to E2 .

5.2

Output as the pollution cause

In this subsection, we consider that the flow of pollution is caused by output alone (i.e.,
αc = 0 and αy > 0). The results of comparative static analysis can be summarized as follows.
Proposition 3 Assume that αc = 0 and αy > 0. Then, a permanent increase in the rateof income tax decreases the steady-state level of the pollution stock, but a temporary increase
leads to a more severe level of the pollution stock.
Proof. See Appendices B and C.
Figure 3 shows the eﬀects of an increase in the rate of income tax on the whole economy,
where αc = 0 and αy > 0. An increase in the rate of income tax aﬀects the ṗt = 0 locus,
18

but not the µ̇t = 0 locus. The movement of ṗt = 0 and µ̇t = 0 loci is essentially the sameas that in the case of the consumption tax rate. From (18b), when the rate of income tax
permanently increases, the ṗt = 0 locus moves downward given τ c and µ∗j :∂p∗jαy N ′ (lj∗ f (k̄))f (k̄)=θ∂τjy ṗt =0, µ∗ =constant |
{z
}

(

j

(+)

)
∂lj∗
Λτ y + y (< 0).
∂τ∂ λ̄∗j| {z } |{z}
∂lj∗

(#3)

(24)

(#4)

The term (#3) shows the indirect impact on the pollution stock through a change in the
shadow value of foreign assets and the term (#4) expresses the direct impact. As for (#3)
and (#4), we observe that the direct impact dominates the indirect one, as follows:8
(
)
∂lj∗
∂lj∗
1
1 ∂lj∗ ∂c∗jy+Λ=(< 0).
τ∂τ yBλ̄ − Θsj Pλ̄ r ∂τ y ∂ λ̄∗j∂ λ̄∗j|{z} | {z }
(−)

(25)

(+)

where ∂c∗j /∂ λ̄∗j < 0 (see (10)) and ∂lj∗ /∂τ y < 0 (see (11)).
We now consider the case in which the rate of income tax increases permanently in
Figure 3(a). Suppose that the economy starts at the initial steady-state equilibrium E0 .
Because of the increase in the rate of income tax, the shadow value of the pollution stock
initially jumps downward toward the point I. Thereafter, the level of the pollution stockmonotonically decreases until the new steady-state equilibrium E1 . The dynamic behavior of
the pollution stock and its shadow value are qualitatively the same as that in the case of thepermanent increase in the rate of consumption tax in Figure 2(a). Figure 3(b) shows that therelationship between the foreign assets and the pollution stock is positive, which is opposite
to the relationship depicted in Figure 2(b). This result is because an increase in the level of
output generates a higher level of pollution flow and a larger household income. This results
in a positive relationship between the pollution stock and foreign assets. Therefore, Figure
8

We can arrange for the terms (#3) and (#4), as follows:

∂lj∗
∂lj∗
∂lj∗
y +
Λ
=
τ
∂τ y
∂ λ̄∗j∂ λ̄∗j(−Bτ y + Θsj Pτ y
Bλ̄ − Θsj Pλ̄

)

∂lj∗
1
+ y =
∂τ
Bλ̄ − Θsj Pλ̄[))∂lj∗ (∂lj∗ (sy + Θj P τ y +−B
Bλ̄ − Θsj Pλ̄
τ
∗
y
∂τ
∂ λ̄j

where we confirm the following:∂lj∗ s∂lj∗ s∂lj∗∂lj∗
1 ∂lj∗ ∂c∗jy +
Θ P y−
Θ
P
=
0,
and
−
B
B
=
.
τ
j
λ̄
λ̄
∗ j τ
∗
∂τ y
∂τ y
r ∂τ y ∂ λ̄∗j
∂ λ̄j
∂ λ̄j
Then, we obtain (25).

19]

3(b) shows that when the rate of income tax permanently increases, the level of foreign assetsmonotonically decreases toward the new steady-state equilibrium E2 .
We now consider the impact of a temporary increase in the rate of income tax. When
this rate increases, the level of the shadow value of foreign assets jumps at the same time.
Hereafter, the level of the shadow value of foreign assets remains there a long time, even if
the rate of income tax returns to its original rate. Since only this eﬀect remains in the longrun, the ṗt = 0 locus does not return to the original position. In detail, the ṗt = 0 locus in
the long run locates over the original position, as confirmed below:
αy N ′ (lj∗ f (k̄))f (k̄) ∂lj∗∂p∗j=
Λτ y (> 0).
∂τ y ṗt =0, µ∗ =constant |θ∂ λ̄∗j{z}
j| {z }
(+)

(26)

(+)

The long-run position is shown by TEM : ṗt = 0(T < t) in Figure 3(a). From this figure,
we confirm that the upward shift of the ṗ = 0 locus in the long run generates a more severe
pollution stock in the new steady-state equilibrium E2 . This result is the same as that in thecase of a temporary increase in the rate of consumption tax.
On the other hand, the direction of initial jump of the shadow value of foreign assets isthe opposite to that in the case of a consumption tax increase. In other words, the shadow
value of foreign assets jumps upward, because Λτ y > 0 (see (19d)). The upward direction of
the shadow value of foreign assets is determined by the following equation (8a) under αc = 0:λ̄∗j (1 + τ c ) = u′ (c̄∗j ).

(27)

Note that the level of consumption is fixed. Increasing the rate of income tax leads to alower level of income at the beginning. As a result, such a lower level of income generates adownward jump of consumption at the initial period, and for a long time thereafter, the levelof consumption does not change. The downward shift of consumption leads to an upward
jump of the shadow value of foreign assets from (27). Noting that this eﬀect remains in thelong run, we find that a greater level of the shadow value of foreign assets leads to a greaterlevel of labor supply, as confirmed in (11). As a result, the long-run level of the pollution
stock increases.
In Figure 3, we observe the dynamic behavior of the entire economy by the temporary
increase in the rate of income tax. The transition processes of the pollution stock and its
20

shadow value in Figure 3(a) are qualitatively the same as that in Figure 2(a). In summary,after the shadow value of the pollution stock jumps from E0 to I ′ , the level of the pollution
stock monotonically decreases in the period (0, T ] from I ′ to I ′′′ through the point I ′′ , andthereafter, when the rate of income tax returns to its original level, the level of the pollution
stock increases from the point I ′′′ to the new steady-state equilibrium E2 .
Looking at Figure 3(b), we need to mention the dynamic movement of foreign assets,
because its movement diﬀers from that in Figure 2(b). In detail, the level of foreign assetsmonotonically decreases from E0 to Q during the policy enforcement, and after the rate ofincome tax goes back to the original one, the level of foreign assets increases toward E2 .
Finally, the level of foreign assets in the new steady-state equilibrium E2 is greater than theoriginal level.

6

Numerical Examples

As stated in introduction, some countries experienced severe pollution after the repeal of
temporary pollution control policies. These interesting cases are theoretically captured byour main findings in Propositions 2 and 3; that is, the temporary enforcement of policy
leads to a higher level of pollution in the long run. Given our theoretical findings, thenumerical analyses in our model play a pivotal role in giving richer policy implications.
Specifically, when countries become environmentally friendly from a short-term perspective,
the numerical analyses will show how severe the pollution problem in the countries becomes
after the repeal of the pollution control policy, which is the theoretically unsolved issue.
Then, the purposes of this section of the paper are as follows. First, we confirm the results
obtained in Propositions 2 and 3 by conducting numerical analyses. Second, under a longer
duration of policy enforcement and a higher rate of consumption or income tax, we examine
the quantitative impacts on the pollution stock as well as other economic variables of thewhole economy in the long run. To identify the eﬀects of the pollution control policies, we
focus on the following two cases, (αc , αy ) = (1, 0) and (0, 1), throughout this section.
We here specify the production function, the utility functions, and the pollution functions.
The production of the single homogeneous good is given by the production process f (k̄) =Ak̄ α , where 0 < α < 1 and A > 0, so that the return to capital is r = Aαk̄ α−1 and the21

constant level of capital stock is k̄ =1
( Aα ) 1−α. Next, the utility functions are given byrL1−γtand Γ(Lt ) = 1−γ, where δ > 0, ϵ > 0, and γ > 0. As for thec1+g(k̄))1+ntpollution functions, we assume that G(ct ) = 1+g
and N (lt f (k̄)) = (lt f1+n
in which g and

c1−δ
p1+ϵ
t
t
u(ct ) = 1−δ, v(pt ) = 1+ϵn are positive parameters.
Based on these functions, we make use of the following parameters:
Production and pollution parameters: α = 0.35, A = 0.5, g = n = 1, θ = 0.05.
Taste parameters: δ = 0.5, ϵ = 0.5, γ = 0.5,
Price and tax rates: r = (ρ =)0.08, τ0c = τ0y = 0,
Initial values of foreign assets: b0 = 0.35.
Some parameters used in our simulations are standard ones: α = 0.35 and r = 0.08. Moreover, setting A = 0.5 yields k̄ = 3.33. As for the pollution parameters, because the parameters
g and n, represented by the elasticities of pollution, are freely chosen, we set g = n = 1. Furthermore, the choice of θ is 0.05, which means that the 5% of the pollution stock is broken
down at every time. To grasp the intuition obtained in the numerical simulation easily, we
assume the same values of elasticity parameters in the utility function with consumption
and leisure δ = γ = 0.5. This assumption is because these elasticities play a critical role inchanging the levels of pollution causes (consumption and labor supply).
Next, noting that there are two stock variables, we provide the initially given value of
foreign assets by b0 = 0.35; however, we do not provide the initial level of the pollution stock.
The reason is that the initial economy is at the steady state, that is, the initial economy mustsatisfy the equations (18a) − (18d). Considering that b0 = b∗0 (= 0.35), we notice that thevalue of the left-hand side in (18d) is zero. If the initial level of the pollution stock is freely
given, two values of λ̄∗0 and µ∗0 at the initial economy must be determined by three equations(18a) − (18c) given b∗0 = b0 and p∗0 = p0 . Therefore, the number of equations is more than thenumber of variables. Thus, by setting (p∗0 , µ∗0 , λ̄∗0 ) to satisfy three equations (18a) − (18c), we
determine the initially unique values of (p∗0 , µ∗0 , λ̄∗0 ). Then, Table 1 shows the initial economy
in our simulation.99

We omit λ̄∗0 in Table 1 because λ̄∗0 is uniquely given by the level of consumption or labor supply using(8a) and (8b) as follows:λ∗0 = u′ (c∗0 ), when αc = 0, and αy = 1.

22

Finally, we set the initial rates of tax by τ0c = τ0y = 0, while the tax rates of consumption
and income, τ1c and τ1y , respectively, are set between 4% and 16%. Furthermore, when the
tax rate increases temporarily, we assume that the implementation period T is 2, 4, or 6.

6.1

Consumption as the pollution cause

We now assume that the pollution flow is generated by consumption (αc = 1 and αy = 0).
Then, Table 2(a) shows the quantitative eﬀects of the permanent increase in the rate of
consumption tax and Table 2(b) shows those of the temporary increase at T = 2, 4, or 6.
When the initial economy starts from p0 = 0.3468, as in Table 1, we observe from Table 2(a)that the long-run levels of pollution are lower than the original level, showing that when thepolicy is permanently enforced, the steady-state level of the pollution stock is less than its
original level. For instance, when the rate of consumption tax is 4%, the level of consumption
decreases by 1.6% in the long run, and hence, the long-run level of the pollution stock decreaseby around 5%. This relationship supports the result of Proposition 2. In addition, we findfrom Table 2(a) that the greater the rate of consumption tax, the lower the long-run level ofthe pollution stock. For instance, when the rate of consumption tax is set at 12%, the levelof the pollution stock decreases by about 10%, which is nearly double the percentage change
under a 4% consumption tax rate.
We turn to the eﬀects of the temporary increase in the rate of consumption tax, whichis shown in Table 2(b). Then, the findings in the numerical simulation are summarized asfollows. First, we confirm that the long-run levels of the pollution stock are greater thanthe original level, as stated in Proposition 2. For example, when the rate of consumption
tax is temporarily set at 4% where T = 2, the long-run level of the pollution stock increases
by around 1.7%. Furthermore, at the cancellation time of the policy (t = T = 2), Table
2(b) shows that the level of the pollution stock, pT decreases by around 0.8%. This result
corresponds to point I ′′′ of Figure 2(a).
Second, Table 2(b) shows that the larger the rate of consumption tax, the greater the
long-run level of the pollution stock. For example, when the rate of consumption tax is set

λ∗0 =Γ′ (1 − l0∗ ), when αc = 1, and αy = 0.
f (k̄)

where τ0c = τ0y = 0 at the initial economy.

23

at 12% under T = 2, the long-run level of the pollution stock increases by around 4.6%
compared with its original level, which is more than double its percentage change under a
4% consumption tax rate.
Third, we observe from Table 2(b) that a longer period of policy implementation leads
to a lower level of pollution stock at the cancellation time of policy and a higher level ofthe pollution stock in the long run. For instance, let us consider that the period of policy
implementation increases threefold, that is, from T = 2 to T = 6. Then, we observe that thechange from T = 2 to T = 6 further decreases the level of the pollution stock by 0.08% at
the cancellation time of the policy and increases the long-run level of the pollution stock by
0.14%.
As for this relationship between the level of the pollution stock and the implementation
duration of the policy, Figure 4 may aid our understanding further. We assume that theimplementation duration of the temporary policy is given by T = 2, 4, and 6, and that the rateof consumption tax is set at the range in which τ c ∈ [0.04, 0.16]. Then, we notice that TEM(2)and TEM(2)L mean that the implementation duration of policy, T is 2. Moreover, TEM(2)indicates the stage of the economy just when the policy has been repealed (i.e., t = T = 2),
and TEM(2)L shows the long-run economy (i.e., t = ∞). Furthermore, note that the broken
straight-horizontal line at p0 = 0.3468 means the initial value of the pollution stock. We
observe from this figure that the longer the implementation period, the larger the eﬀects ofa temporary increase. As for the quantitative impacts, Figure 4 shows that the steady-statelevels of the pollution stock are almost the same irrespective of the implementation period
under our setting.10
Finally, the characterization of foreign assets in Figure 2(b) can be observed in Tables
2(a) and 2(b). From Table 2(a), we observe that the long-run level of foreign assets is higher
than its original level, which corresponds to the relationship between points E0 and E1 in
Figure 2(b). As for the quantitative impacts, for example, when the rate of consumption
tax is set at 4%, the long-run level of foreign assets increases by around 2.7%. Furthermore,
Table 2(a) shows that a higher rate of consumption tax leads to a greater level of foreign
assets. Specifically, setting the rate of consumption tax as 12% makes the steady-state level
10

When the implementation period is very short (e.g., T = 0.1), the steady-state levels of pollution are very

close to the original level; however, we omit this case, because we want to focus on the interesting cases.

24

of foreign assets increase by around 6%, which is more than double the percentage change of
foreign assets under a 4% consumption tax rate.
Figure 2(b) shows that the level of foreign assets at time T is greater than the originallevel (see the point Q) but the long-run level of foreign assets is lower than its originallevel (see the point E2 ). Looking at Table 2(b), for instance, we find that when the rate of
consumption tax is given at 4% under T = 2, the long-run level of foreign assets, b∗2 , is given
by 0.3499(< b0 = 0.35) and the level of foreign assets at the cancellation time of policy, bT ,is given by 0.354(> b0 ).11

6.2

Output as the pollution cause

Assuming that αc = 0 and αy = 1, we consider the case in which the pollution cause is
output. Table 3(a) shows the quantitative eﬀects of the permanent increase in the rate ofincome tax. For example, when the rate of income tax is set at 4%, the long-run level of thepollution stock decreases by around 7%, as a result of the decrease in the level of labor supply
by about 2.8%. Looking at the case at τ y = 0.12, surprisingly, we find that the long-run levelof the pollution stock dramatically decreases by about 20%, where the level of labor supply
reduces by 9.5%.
In what follows, we pay attention to the eﬀect of the temporary increase in the rate ofincome tax, which is shown in Table 3(b). The points that we mention are the same as thosein the last subsection. First, Table 3(b) shows that the long-run levels of the pollution stock
are larger than the original level, which supports our finding in Proposition 3. For instance,
when the rate of income tax is set at 4% where T = 2, the long-run level of the pollution
stock increases by 4.3%. In addition, such pollution control policy decreases the level of thepollution stock by below 1% at the cancellation period (T = 2). This result corresponds tothe point I ′′′ in Figure 3(a). As for the quantitative eﬀects, we can argue from Table 3(b) thatthe temporary increase in the rate of income tax does not decrease the level of the pollution
stock largely, and furthermore, leads to non-negligible serious contamination after the repealof the temporary enforcement of tax policy.
11

Table 2(b) shows that when τ c = 0.04 and 0.08, b∗2 = 0.35 under T = 6, which means that the long-run

levels of foreign assets are lower than the original level; however, because the quantitative eﬀects are verysmall, we give b∗2 = 0.35 as a result of rounding oﬀ the value to four decimal places.

25Second, Table 3(b) shows that a higher rate of income tax leads to a greater level ofpollution stock in the long run. For example, setting τ y = 0.12 at T = 2, we confirm from
Table 3(b) that the long-run level of the pollution stock increases by around 10%, which ismore than double the percentage change under a 4% income tax rate.
Third, Table 3(b) and Figure 5 show that the longer the period of the implementation
period, the greater the long-run level of the pollution stock. For instance, let us assume thatthe rate of income tax is set as 4%. Then, we find that when the period of implementation
lengthens from T = 2 to T = 6, the long-run level of the pollution stock increases fromp∗2 = 0.3149 to 0.3305, which further increases by around 5%.
Finally, we confirm the levels of foreign assets by increasing the rate of income tax. Whenthe rate of income tax is permanently set as 4%, the level of foreign assets decreases by around
5.7 %, shown as point E1 in Figure 3(b). Turning to the temporary increase of the rate ofincome tax, Table 3(b) shows that the levels of foreign assets are lower than the initiallygiven level at the cancellation time of the policy, while its long-run levels are greater. This
characterization of the dynamic behavior of foreign assets shows points Q and E2 of Figure3(b). For example, when the rate of income tax is given by 4% under T = 2, the long-run
level of foreign assets decreases by 3.8%, and the long-run level of foreign assets does notincrease significantly below 0.1%.

7Conclusion

In this study, we have constructed a dynamic model of a small open economy with two
pollution causes. Assuming that the pollution flow is generated by consumption or output
and that the pollutants accumulate in the economy, we examine the impacts of pollution
control policies on the pollution stock. When the pollution cause is consumption, we pickup the consumption tax rate as a pollution control policy, because increasing the rate of
consumption tax has a negative impact on the household’s consumption directly, which will
reduce the level of pollution cause. Alternatively, when the pollution cause is output, we
suppose that the income tax rate is a pollution control policy. We make this assumption
because an increase in the rate of income tax directly has a negative impact on working,and it will lead to a lower level of pollution flow. Then, our main finding is that when the26pollution control policy is permanently conducted, the level of the pollution stock decreases
in the long run, and when the pollution control policy is temporarily conducted, it increases
in the long run. This result is independent of the pollution causes. Our numerical examples
support this finding.
The numerical simulation shows that the temporary enforcement of pollution control
policy may have little merit in small open economies. When the implementation durationof the pollution control policy is longer, and/or when the pollution control policy is stricter,we observe from our simulation that the temporary enforcement of pollution control policy
leads to a greater amount of pollution stock. In summary, when small open economies become
environmentally friendly from a short-term perspective, they experience more severe pollution
after the repeal of the pollution control policy.
According to our findings, governments must tackle pollution problems from a long-term
perspective, not a short-term perspective. This argument corresponds to the case of Japan
at the initial stage of development. Shimaoka et al. (2016, Chapter 5) mention that whenindustrial pollution was widespread in Japan at the initial stage of development, the Japanese
government regarded stringent industrial pollution policies as indispensable means for economic development. In fact, during the economic boom in Japan from 1960 to 1980, water
quality made some remarkable improvements, owing to significant reduction of serious pollution by heavy metals stemming from regulations on industrial wastewater.12

12

According to the Ministry of the Environment in Japan, the rates of non-conformity for health-related

pollutants (cadmium, lead, and creatinine) based on old environmental quality standards has significantly
decreased. Specifically, in 1971, the rate of non-conformity was indicated at around 1.4% for lead, around
1.1% for creatinine, and around 0.7% for cadmium; however, in 1982, the rates of non-conformity for thesepollutants were all below 0.1%.

27

Appendix A.
In this section, for simplicity, we abbreviate the arguments in each function as Gc ≡ G′ (c∗ )and Nl ≡ N ′ (l∗ f (k̄)), so as not to confuse readers. First, totally diﬀerentiating (18a), we
obtainp∗j = p(µ∗j ).
Making use of (18a), we obtain

(A.1)

∂p∗j
ρ+θ
(> 0).
=
∗
∂µj
v ′′

Substituting (A.1) into (18b) and totally diﬀerentiating it, we show thatµ∗j = M (λ̄∗j , τ c , τ y ),
where

(A.2)

∂c∗j
∂lj∗
∗
α
G
+
α
N
f
(
k̄)
c
c
y
l
∗
∂µj
∂ λ̄j
∂ λ̄∗jMλ̄ ≡=,θµ∂ λ̄∗j∂c∗∂µ∗j
αc Gc ∂τjc
Mτ c ≡=(< 0),
∂τjc
θµ
∂l∗

∂µ∗j

αy Nl f (k̄) ∂τjy
y
Mτ ≡
=
(< 0)
θµ
∂τjy
Note that we make use of the following:
θµ =

∂c∗j
∂lj∗
θ(ρ + θ)
−
α
G
−
α
N
f
(
k̄)
(> 0).
c cy lv ′′ (p∗j )
∂µ∗j
∂µ∗j

Therefore, substituting (A.2) into (A.1) yieldsp∗j = P (λ̄∗j , τjc , τjy ),
where
Pλ̄ ≡
Pτ c ≡Pτ y ≡∂p∗j∂ λ̄∗j

=

(A.3)

ρ+θ
M ,
v ′′ (p∗j ) λ̄

∂p∗j
ρ+θ
= ′′ ∗ Mτ c (< 0),c∂τjv (pj )∂p∗j

ρ+θ= ′′ ∗ Mτ y (< 0),v (pj )
∂τjy

Substituting (A.2) into (10) and (11) and further substituting these into (18c), we obtainrb∗j + l(M (λ̄∗j , τjc , τjy ), λ̄∗j , τ y )f (k̄) = c(M (λ̄∗j , τjc , τjy ), λ̄∗j , τjc ).
28

(A.4)

Totally diﬀerentiating (A.4), we obtain the following:b∗j = B(λ̄∗j , τjc , τjy ),(A.5)

where derivatives with respect to each argument are
[
(
)
]
∂b∗j
∂lj∗
∂lj∗
∂c∗j
∂c∗j
1
−f (k̄)
M
M ,
Bλ̄ ≡
=
+
+
+
r
∂µ∗j λ̄∂µ∗j λ̄∂ λ̄∗j∂ λ̄∗j∂ λ̄∗j
[
]
∂b∗j
∂c∗j
∂c∗j
∂lj∗
1
Bτ c ≡=
Mτ c + c ,−f (k̄) ∗ Mτ c +
∂τjc
r
∂µj
∂µ∗j
∂τj
)
]
[
(
∂b∗j
∂lj∗
∂c∗j
∂lj∗
1
=
Bτ y ≡−f (k̄)Mτ y + y +Mτ y .
r
∂µ∗j
∂τ
∂µ∗j
∂τjy
Finally, when substituting (A.3) and (A.5) into (18d), we derive the following:
λ̄∗j = Λ(τjc , τjy ),whereΛτ c ≡
Λτ y ≡(A.6)∂ λ̄∗j−Bτ c + Θsj Pτ c=(< 0),∂τjc
Bλ̄ − Θsj Pλ̄∂ λ̄∗j

y =

∂τj

(A.7)

−Bτ y + Θsj Pτ y(> 0).
Bλ̄ − Θsj Pλ̄(A.8)

We now examine the signs of the denominator and numerator in (A.7) and (A.8) undereach pollution cause, where we omit variables in each function for simplicity.
First, we calculate Bλ̄ − Θsj Pλ̄ as follows:
{(
)(
)
∗
∗
∗
∗
∗
∂l
∂c
∂p
∂c
∂l
1
j
j
j
j
j
Bλ̄ − Θsj Pλ̄ =−f ∗ +
θ ∗ −αc Gc ∗ − αy Nl f ∗
rθµ
∂µj
∂µj
∂µj
∂ λ̄j
∂ λ̄∗j
|
{z
}
(
+

∂p∗j
v ′′ ρ
1−|{z} ∂µ∗ (ρ − βs )(ρ + θ − βs )
j
(#A1)

)(

(#A1)

∂lj∗
∂c∗j
−
f
∂µ∗j
∂µ∗j

)(

αc Gc

∂c∗j
∂ λ̄∗j+ αy Nl f∂lj∗∂ λ̄∗j)},(A.9)

in which (#A1) is given by
(
)(
) (
)(
)
∂lj∗
∂c∗j
∂c∗j
∂lj∗
∂c∗j
∂lj∗
∂c∗j
∂lj∗
(#A1) = −f ∗ +− αc Gc ∗ − αy Nl f ∗ +−f ∗αc G c ∗ + αy N l f ∗ ,
∂µj
∂µj
∂µ∗j
∂µj
∂ λ̄j
∂ λ̄∗j
∂ λ̄j
∂ λ̄j
(
)
∂lj∗ ∂c∗j∂c∗j ∂lj∗= (αy Nl f + αc Gc f )−(< 0).
∂ λ̄∗j ∂µ∗j
∂ λ̄∗j ∂µ∗j29

Therefore, (A.9) can be rewritten as
{(
)
(
)
∂lj∗
∂c∗j
∂p∗j
∂lj∗ ∂c∗j∂c∗j ∂lj∗1(A.9) =
−f ∗ +θ ∗ + f (αy Nl + αc Gc )
−
rθµ
∂µj
∂ λ̄j
∂ λ̄∗j∂ λ̄∗j ∂µ∗j
∂ λ̄∗j ∂µ∗j|{z} |
{z
}
(+)

∂p∗j
v ′′ ρ− ∗∂µj (ρ − βs )(ρ + θ − βs )
|

(

∂c∗j
∂lj∗
−
f
∂µ∗j
∂µ∗j

)(

(−)

αc Gc

{z

∂c∗j
∂ λ̄∗j+ αy Nl f∂lj∗∂ λ̄∗j)}.

(A.10)

}

(#A2)
∂c∗

∂l∗

∂c∗

∂l∗

As for (#A2), because ∂µj∗ < 0, ∂µj∗ < 0, ∂ λ̄j∗ < 0 and ∂ λ̄j∗ > 0, we cannot specify the
j
j
j
j
(
)
)
( ∂c∗
∂lj∗
∂c∗j
∂lj∗
j
sign of ∂µ∗ − f ∂µ∗ as well as αc Gc ∂ λ̄∗ + αy Nl f ∂ λ̄∗ in (#A2).
j

j

j

j

On the other hand, assuming that αc > 0 and αy = 0, we show that(#A2) = −∂p∗j ∂c∗j ∂c∗j
αc Gc v ′′ ρ(< 0),(ρ − βs )(ρ + θ − βs ) ∂µ∗j ∂µ∗j ∂ λ̄∗j

(A.11)

which means that the sign of (A.9) is negative. Next, when assuming that αc = 0 and αy > 0,
we confirm the negative sign of (#A2),∂p∗j ∂lj∗ ∂lj∗
αy Nl f 2 v ′′ ρ(#A2) =(< 0),(ρ − βs )(ρ + θ − βs ) ∂µ∗j ∂µ∗j ∂ λ̄∗j

(A.12)

implying that (A.9) has a negative sign. As a result, we find that (A.9) has a negative sign
if αc > 0 and αy = 0 or αc = 0 and αy > 0.
The pollution cause is consumption:∂l∗

We assume that αc > 0 and αy = 0, so that ∂µj∗ = 0. Hence, the denominator of (A.7) has ajnegative sign.
Based on the negative sign of the denominator of (A.7), our focus is to show the positive
sign of the numerator in (A.7).
(
)
∗
∗
∂c
∂c
∂p∗j
1
j
j
s
s
c
c
c
− Bτ + Θj Pτ = −Mτ + c + Θj ∗ Mτ c ,r ∂µ∗j
∂τ
∂µj
[
(
)]
(
)
∗
∗
∂p
∂c
1 ∂c∗j
ρ(ρ
+
θ)
j
j
=−
− (αc Gc )2 1 −+ θc θ ∗ − αc Gc ∗,rθc θµ ∂τ c
(ρ − βs )(ρ + θ − βs )
∂µj
∂µj
(
)
∂p∗j
ρ(ρ + θ)(αc Gc )2
1 ∂c∗j+ θθc ∗ > 0.
(A.13)=−
rθc θµ ∂τ c (ρ − βs )(ρ + θ − βs )
∂µj
|
{z
} |
{z
}
(+)

(+)

30

Finally, we show that Λτ c < 0.
The pollution cause is output:∂c∗

We now assume that αc = 0 and αy > 0, and hence, ∂µj∗ = 0. Noting that the denominatorjof (A.8) has a negative sign, we show the negative sign of the numerator in (A.8).
[
)]
(
∂lj∗
∂p∗j
∂lj∗
1
s
s
y
y
y
− Bτ + Θj Pτ = − −f (k̄)
Mτ + y+ Θj ∗ Mτ y .
r
∂µ∗j
∂τ
∂µj
)]
[
(
(
)
∗
∂l
∂p
f ∂lj∗
ρ(ρ
+
θ)
j
j
,
=
(αy Nl f )2 1 −+ θl θ ∗ − αy Nl f ∗rθµ θl ∂τ y(ρ − βs )(ρ + θ − βs )
µj
∂µj
[
]
∗
∂p
f ∂lj∗ρ(ρ+θ)
j=(αy Nl f )2
+ θθl ∗ < 0.
(A.14)rθµ θl ∂τ y(ρ − βs )(ρ + θ − βs )∂µj
| {z } |
{z
}
(−)

(+)

Hence, we show that Λτ y > 0.

31

Appendix B.
We now examine the eﬀects of a temporary change in each tax rate (τjc and τjy ) on thelong-run level of pollution. As stated in the Section 5, the government announces changes ofthe policy instruments from the original level τ0i to a new level τ1i until time T where i = c
and y. After time T , the government reverts permanently to its original level τ0i . Since thehousehold can initially anticipate the policy change at time T , the new information arrives
only at the initial time, which means that the fixed level of the shadow value λ̄ jumps to the
new steady state at the initial economy and remains there permanently.
When a temporary change in tax rate arises, we divide the transitional dynamics into two
separate ones, called Periods 1 and 2, as follows.
Period 1: 0 ≤ t < T
During Period 1 (i.e., the period of policy implementation τ1i ), the economy moves along
the unstable transitional path:
pt = p∗1 + D1 eβ1 t + D2 eβ1 t ,us(B.1a)µt = µ∗1 + D1 ∆s1 eβ1 t + D2 ∆u1 eβ1 t ,(B.1b)

bt = b∗1 + Θs1 D1 eβ1 t + Θu1 D2 eβ1 t ,(B.1c)su

su

where Θu1 is defined byΘu1 =v ′′ (p∗1 )(ρ − β1u )(ρ + θ − β1u )

(
)
∂l1∗
∂c∗1
f (k̄) ∗ −.
∂µ1 ∂µ∗1

Note that β1s and β1u represent the stable and unstable roots, respectively, under (τ1c , τ1y , p0 , b0 ).
Given the initial levels of foreign assets and the pollution stock (b0 , p0 ), the steady-statelevels, written by p∗1 , µ∗1 , and b∗1 , are determined under the new level of tax rate τ1i (i = c ory) where we make use of (A.6):
p∗1 = P (λ̄1 , τ1c , τ1y ) = P (Λ(τ1c , τ1y , b0 , p0 ), τ1c , τ1y ),(B.2a)µ∗1 = M (λ̄1 , τ1c , τ1y ) = M (Λ(τ1c , τ1y , b0 , p0 ), τ1c , τ1y ),(B.2b)b∗1 = B(λ̄1 , τ1c , τ1y ) = B(Λ(τ1c , τ1y , b0 , p0 ), τ1c , τ1y )(B.2c)

λ̄1 = Λ(τ1c , τ1y , b0 , p0 )(B.2d)

32

In this period, we notice that λ̄0 ̸= λ̄1 , because it jumps at the initial period.
Period 2: T ≥ t
During Period 2 (i.e., the period of policy return τ0i ), the economy follows the stable path
as follows:
pt = p∗2 + D1′ eβ2 t ,(B.3a)µt = µ∗2 + D1′ ∆s eβ2 t ,(B.3b)

bt = b∗2 + Θs2 D1′ eβ2 t ,(B.3c)sss

The steady-state levels of p∗2 , a∗2 , and b∗2 are determined byp∗2 = P (λ̄∗2 , τ0c , τ0y ) = P (Λ(τ0c , τ0y , bT , pT ), τ0c , τ0y ),(B.4a)µ∗2 = M (λ̄∗2 , τ0c , τ0y ) = M (Λ(τ0c , τ0y , bT , pT ), τ0c , τ0y )(B.4b)b∗2 = B(λ̄∗2 , τ0c , τ0y ) = B(Λ(τ0c , τ0y , bT , pT ), τ0c , τ0y ),(B.4c)λ̄∗2 = Λ(τ0c , τ0y , bT , pT ).

(B.4d)

Notice that the level of shadow value does not change λ̄1 = λ̄2 , because the household
anticipates the removal of the tax rate under the assumption of perfect foresight.
We denote the policy changes by dτ i ≡ τ1i − τ0i (i = c, y). The approximation of thesteady-state changes of pollution with the diﬀerentials yieldsp∗2 − p∗1 = P (λ̄∗2 , τ0c , τ0y ) − P (λ̄∗1 , τ1c , τ1y ),= −Pτ c dτ c − Pτ y dτ y , ,(B.5a)

p∗1 − p∗0 = P (λ̄∗1 , τ1c , τ1y ) − P (λ̄∗0 , τ0c , τ0y ),= Pλ̄ (Λτ c dτ c + Λτ y dτ y ) + Pτ c dτ c + Pτ y dτ y .

(B.5b)

Notice that the equality λ̄1 = λ̄2 holds.
From (B.5a) and (B.5b), we confirm the eﬀects of a temporary change in each tax rateon the pollution as follows:p∗2 − p∗0 = Pλ̄ (Λτ c dτ c + Λτ y dτ y )(B.6)

This shows the eﬀect of the temporary change in the environmental policies on the level ofthe pollution stock.
33

Appendix C.
We now examine the eﬀects of a permanent increase in the pollution abatement policyon the pollution stock in the long run, where for simplicity we omit the arguments in eachfunction as before.
First, assuming that αc > 0 and αy = 0, we consider the case in which the pollution flow
is generated by consumption. Using (B.5b), we show how the permanent increase in the rateof consumption tax aﬀects the level of the pollution stock as follows:dp∗j= Pλ̄ Λτ c + Pτ c ,
dτ c
(
[
(
)
)]
Θsj (ρ + θ)Mτ c
Θsj (ρ + θ)Mλ̄
ρ+θ() Mλ̄ −Bτ c +=+ Mτ c Bλ̄ −,Θs (ρ+θ)Mv ′′v ′′v ′′ Bλ̄ − j v′′ λ̄ρ+θ() (−Mλ̄ Bτ c + Mτ c Bλ̄ ) ,Θs (ρ+θ)M
v ′′ Bλ̄ − j v′′ λ̄
[(
]
)
ρ+θ
f (k̄) ∂lj∗()=−
Mτ c (< 0).
Θs (ρ+θ)Mr∂ λ̄∗jv ′′ Bλ̄ − j v′′ λ̄
{z
}
|
{z
}|
(+)
=

(C.1)

(−)

Hence, dp∗j /dτ c < 0.
In what follows, we assume that αc = 0 and αy > 0, so that the pollution flow is caused
by output. Similarly, when using (B.5b), we show how the permanent increase in the rate ofincome tax aﬀects the level of the pollution stock:dp∗j= Pλ̄ Λτ y + Pτ y ,
dτ y
[
(
)
(
)]
Θsj (ρ + θ)Mτ yΘsj (ρ + θ)Mλ̄
ρ+θ
y
y
(
)
M
−B
+
=
+
M
B
−
,
τ
τ
λ̄
λ̄
Θs (ρ+θ)Mv ′′v ′′v ′′ Bλ̄ − j v′′ λ̄ρ+θ() (−Mλ̄ Bτ y + Mτ y Bλ̄ ),Θs (ρ+θ)M
v ′′ Bλ̄ − j v′′ λ̄[]Mτ y ∂c∗j
ρ+θ
(
)
(< 0)
=
Θs (ρ+θ)Mr ∂ λ̄∗jv ′′ Bλ̄ − j v′′ λ̄
{z
}
|
{z
} | (+)=(−)

Therefore, we find that dp∗j /dτ y < 0.

34

(C.2)

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36

Table 1: The original steady state where τ c = 0 and τ y = 0
b0

p0

c0

l0

µ0

αc = 1 and αy = 00.35

0.34680.18620.20764.5301αc = 0 and αy = 1

0.35

0.30170.20160.22784.2251

Table 2(a): Steady state E1 where αc = 1 and αy = 0
τc

b∗1

p∗1

c∗1

l1∗

0.04
0.08

0.35930.32920.18320.2026

4.41370.365

0.31830.17950.19724.33950.12

0.37090.31067

0.17590.19194.26µ∗1

Table 2(b): Steady state E2 where αc = 1 and αy = 0 at T = 2, 4, or 6

T =2

T =4

T =6

τc

b∗2

p∗2

c∗2

l2∗

µ∗2

bT

pT

0.04

0.34990.35260.18630.20264.5555

0.354

0.346

0.08

0.34970.35790.18640.1972

4.5796

0.35730.34590.12

0.34950.36270.18640.19194.60220.36040.34510.04

0.34990.35290.18620.20264.55720.35530.34470.08

0.34990.35830.18620.19724.584

0.35950.34310.12

0.34910.36340.18630.19194.60790.36290.34090.04

0.35

0.35310.18620.20264.5586

0.35640.34310.08

0.35

0.35860.18620.1972

4.5866

0.36120.34040.12

0.34990.364

0.18620.19194.61350.366

0.3376

Table 3(a): Steady state E1 where αc = 0 and αy = 1
τy

b∗1

p∗1

c∗1

l1∗

µ∗1

0.04

0.33020.2792

0.19520.22144.06480.08

0.31440.26230.18810.21373.93950.12

0.29670.24450.18080.206

3.8037

Table 3(b): Steady state E2 where αc = 0 and αy = 1 at T = 2, 4, or 6

T =2

T =4

T =6

τy

b∗2

p∗2

c∗2

l2∗

µ∗2

bT

pT

0.04

0.35020.31490.19520.233

4.29050.33670.29980.08

0.35030.32590.1881

0.23984.34480.3256

0.29780.12

0.35070.33760.18080.24714.40050.31430.29530.04

0.35020.3216

0.19520.23134.33520.3276

0.29530.08

0.3501

0.33820.1881

0.23674.42410.30950.28980.12

0.35010.35510.18080.24284.50930.29130.28380.04

0.35010.33050.19520.229

4.39440.31730.29130.08

0.35010.35510.18810.23264.53020.29150.28260.12

0.35

0.37970.18080.23724.65590.26650.273237

Figure 1: Phase diagramµtµ̇t = 0µ∗ṗt = 0
p0

pt

p∗

38

Figure 2: The pollution cause is consumption(a)TEM : ṗt = 0(T < t)

PER : ṗt = 0(0 < t)
TEM : ṗt = 0(t < T )µtµ̇t = 0(TEM : t < T )(TEM : t < T )
µ∗2
E2

I ′′′

µT
µ0

E0

I ′′

I′
I

E1

µ∗1

p∗1

pT

p∗2

p0

(b)

pt

bt
b∗1

E1
Q

b∗T

b0
E0

b∗2

E2

p∗1

pT

p0

p∗2

39

pt

Figure 3: The pollution cause is output(a)

TEM : ṗt = 0(T < t)

PER : ṗt = 0(0 < t)
TEM : ṗt = 0(t < T )µtµ̇t = 0(TEM : t < T )(TEM : t < T )
µ∗2

E2

I ′′′

µT
µ0

E0
I′

I ′′

I

E1

µ∗1

p∗1

(b)

pT

p∗2

p0

pt

bt
b∗2

E2

E0

b0

bT
b∗1

E1

p∗1

Q

pT

p∗2

p0

40

pt

Fig 4: Pollution cause is consumption

0.37

TEM(2)LTEM(4)LTEM(6)LTEM(2)
TEM(4)
TEM(6)
P0

0.365

Level of pollution stock

0.36

0.355

0.35

0.345

0.34

0.335

0.33
0.04

0.06

0.08

0.1

0.12

0.14

0.16

Consumption tax rate τ(c)Figure 5: Pollution cause is output

0.42

TEM(2)LTEM(4)LTEM(6)LTEM(2)
TEM(4)
TEM(6)
P0

0.4

Level of pollution stock

0.38

0.36

0.34

0.32

0.3

0.28

0.26
0.04

0.06

0.08

0.1
Income tax rate (τy)

41

0.12

0.14

0.16