Discussion Paper No. 762

OPTIMAL EMISSION TAX
WITH ENDOGENOUS LOCATION CHOICEOF DUOPOLISTIC FIRMS

Masako Ikefuji
Jun-ichi Itaya
Makoto Okamura

December 2009

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Optimal emission tax with endogenous
location choice of duopolistic ﬁrms∗
Masako Ikefuji†, Jun-ichi Itaya‡, Makoto Okamura§
December 9, 2009

Abstract
This paper explores optimal environmental tax policy under which
duopoly ﬁrms strategically choose the location of their plants in a simple three-stage game. We examine how the relationship between the
optimal emission tax and the choice of location of duopoly ﬁrms affects the welfare of the home country. We characterize the relationshipbetween the optimal emission tax and the ﬁxed cost, depending on the
degree of environmental damage from production. Finally, we show
the existence of asymmetric equilibrium in which either ﬁrm chooses
relocation of its plant even if the duopoly ﬁrms are identical ex ante.
keywords: Environmental policy, Relocation, Welfare
JEL classiﬁcation: H 23; L 13;∗

Ikefuji gratefully acknowledge ﬁnancial support from Fondazione Eni Enrico Mattei(FEEM). The majority of the work on the paper was done while the author was aﬃliatedwith FEEM. Needless to say, we are responsible for any remaining errors.
†
Corresponding Author : Institute of Social and Economic Research, Osaka University,
6-1, Mihogaoka, Ibaraki, Osaka 567-0047, Japan, ikefuji@iser.osaka-u.ac.jp‡
Graduate School of Economics and Business Administration, Hokkaido University,
Sapporo 060-0809, Japan, itaya@econ.hokudai.ac.jp§
Economics Department, Hiroshima University, 1-2-1 Kagamiyama, Higashihiroshima,
Hiroshima 739-8526, Japan, okamuram@hiroshima-u.ac.jp

1

Introduction

The environmental policy in a game theoretic model has been receiving muchattention in environmental economics. In order to reduce local pollution, thegovernment imposes a tax on emissions, which may inﬂuence the location
strategies of polluting ﬁrms. This paper describes the interaction of the government and duopoly ﬁrms and examines the optimal environmental policy
when the location choice of the ﬁrms is endogenous in a three-stage game.
When the environmental policy is stringent, the domestic ﬁrms have an
incentive to relocate their plants abroad to reduce the marginal production
cost. The ﬁrms face a trade-oﬀ between relocation costs and tax payments.
The improved environment increases welfare of the country, while at thesame time the reduction of tax revenue and proﬁts of domestic ﬁrms decreases
welfare. Therefore, the government sets the environmental policy considering
the location decisions of the domestic ﬁrms.
The optimal tax of policy makers who face mobile polluting ﬁrms has
been extensively studied in the literature on international trade, tax competition among governments, and relocation decisions of ﬁrms. Among others,
Markusen et al. (1993) analyse decisions of ﬁrms on the number and location of their plants in response to environmental policies which is given exogenously. Markusen et al. (1995) examine tax competition in a two-region
model. To determine region’s optimal tax rate, they assume that plant locations are exogenous. Petrakis and Xepapadeas (2003) analyze relocation
decisions of a monopolist under an emission tax and compare welfare under
two policy regimes in which the government can commit to its policy and it
cannot in a three-stage model. They show that welfare under government
commitment to an-ex ante emission tax is higher because the government
can aﬀect the monopolist’s location decision.
Looking at the real world, many ﬁrms face the same environmental policyin one country, but nevertheless some ﬁrms shift their plants abroad and theothers remain in home country. A model of monopolist cannot explain aphenomenon in which ex-ante homogeneous ﬁrms choose diﬀerent decision
ex post.1 Our objective is to explain this asymmetric outcome found in thereal world.
The study of an asymmetric outcome, in which ﬁrms have diﬀerent choices
under the same environmental policy even though they have identical, issparse. Some papers have studied the hybrid outcomes in a framework ofindustrial economics and organization and international trade theory. Mills
1

Katsoulacos and Xepapadeas (1995) analyze emission taxes under both ﬁxed-number
oligopoly and endogenous market structure, but not location decisions of ﬁrms.

1

and Smith (1996) establish that asymmetric equilibria can exist in a duopoly
with Cournot-Nash game in which ﬁrms choose technologies. In the context
of international trade, Yomogida (2007) models a duopoly setting in whichﬁrms strategically choose whether to export or to undertake foreign direct
investments and shows the emergence of hybrid equilibrium even if they
have identical cost structure ex ante. These studies, however, assume thatthe parameter determining technological choices and tariﬀ rate are given
exogenously.
By introducing endogenous determination of tariﬀ rate, Ohkawa et al.
(2009) examine the relationship between the optimal tariﬀ policy of the hostcountry and the strategic location choice of foreign duopoly ﬁrms. Based on
their model, we assume that the government set its emission tax level taking
into account the strategical behaviour of duopoly ﬁrms and commits to thepolicy, which gives ﬁrms endogenous choice of whether to remain in domestic
country and pay for emission tax or to relocate the plant to foreign country
and pay for ﬁxed relocation cost.
In this paper we examine how the domestic government determines the
optimal environmental tax when the government can strategically set its
emission tax rate considering location decisions of duopoly ﬁrms in a simple three-stage game. Another question we address is whether asymmetricequilibrium emerges. Also we examine how the relationship between the optimal emission tax and location choice of duopoly ﬁrms aﬀects welfare ofhome country. Moreover, we describe the relationship between the optimal
emission tax and the ﬁxed cost that is one of determinants of relocation of
production.
We derive that the optimal emission tax and welfare resulted from location choices of ﬁrms depend on the degree of damage from pollution so that
there are various pattern of equilibrium set. When the degree of damage is
suﬃciently small, the optimal tax is zero irrespective of the relocation cost.
Subsidies can be optimal without environmental damage from production.
As the degree of damage from pollution increases, asymmetric equilibrium
emerges. That is, the location pattern appears, in which one ﬁrm relocatesits plant to foreign country while another remains in home country. Whenthe environmental damage is large enough, the government attempts to drive
both ﬁrms out by setting relative higher emission tax.
Since welfare of domestic country depends on consumer surplus, proﬁts ofdomestic ﬁrms, tax revenues and environmental damage, the level of welfare
varies corresponding to the relocation patterns. When both ﬁrms remain
in domestic country, the government enjoys proﬁts and tax revenue fromtwo ﬁrms as well as consumer surplus, while it suﬀers from environmental
damage. When either ﬁrm relocates its production abroad, beneﬁts from tax
2

revenue and proﬁts decrease while beneﬁts from reduction in damage andincreases in consumer surplus because of price reduction. Therefore, the sizeof these opposite force on beneﬁts aﬀects the optimal emission tax, which
results in various relocation patterns including asymmetric equilibrium.
The remainder of the paper is structured as follows. Section 2 presents
the basic model and characterizes each stage to derive equilibrium. Section 3
derives the optimal environmental tax and the relationship between resulted
welfare and the ﬁxed cost in various degrees of environmental damage. Section 4 concludes.

2

The Model

Consider two countries, country H and country F . We assume two identicalﬁrms produce and sell a homogeneous good in the market of country H. The
production cost function is assumed to be the same in both countries andgiven by C(qi ) = cqi , where c is parameter and qi is output of ﬁrm i = 1, 2.
The ﬁrms face a demand, P (q) = a − q, where a denotes the market size andq is the demand for the product. Pollution is produced per unit of output
in its production process. The damage function is given by D(x) = dx2 /2,
where d is a damage coeﬃcient and x is the total emission in which ﬁrms
locate.
Both ﬁrms are subject to the environmental policy in country H. Since
we focus on the optimal environmental tax policy under which duopoly ﬁrms
choose the location of their plants, without loss of generality, we assume that
there is no active environmental policy in country F . Following Petrakis andXepapadeas (2003), we assume that both ﬁrms can relocate their production
plants from country H to country F with the cost of relocation, f , and thatthe ﬁrm exports its product without transportation cost and sells in country
H after the relocation. Its proﬁts are assumed to remain in country F .
We consider a three-stage game by ﬁrms 1 and 2, and the government ofcountry H. In the ﬁrst stage, the government sets an emission tax rate so
as to maximize the welfare of country H. We assume that the government
can commit to an emission tax ex ante. Given the tax rate, in the second
stage, duopoly ﬁrms choose the location of their plants. In the third stage,ﬁrms compete á la Cournot in the market, given an emission tax rate andtheir location choice. We derive a Nash equilibrium of this three-stage gameby using backward induction.

3

2.1

Output decision stage

In the last stage, given an emission tax set by the government and the location
choices of each ﬁrm in the previous stages, each ﬁrm chooses its output tomaximize proﬁts. First, assume that both ﬁrms remain in country H. Firm
i’s proﬁt maximization problem is:max (a − q)qi − cqi − tqi ,qi

i = 1, 2,

where t denotes the tax per unit of emissions. From the ﬁrst-order condition
and the symmetry assumption, the optimal output of the duopoly ﬁrms are
given by:1
q1 = q2 = (A − t),(1)3
where A ≡ a − c > 0 represents the scale of market. The aggregate output
is q = 2(A − t)/3, so that the equilibrium proﬁt of ﬁrm i = 1, 2 is:
1i
πHH= (A − t)2 ≡ πHH .
9(2)

If both ﬁrms relocate their production plants to country F and supply
their products in the market of country H, ﬁrm i’s proﬁt maximization problem is given by:max (a − q)qi − cqi − f,qi

i = 1, 2.

Similarly, the optimal outputs of the duopoly ﬁrms are given by:1
q1 = q2 = A.
3

(3)

The aggregate output is q = 2A/3 and the equilibrium proﬁt of ﬁrm i = 1, 2
is:
1
(4)
πFi F = A2 − f ≡ πF F .
9Finally, if ﬁrm 1 remains in country H while ﬁrm 2 has decided to relocate
its plant in country F , the ﬁrst-order conditions of each ﬁrm are:A − t − (q1 + q2 ) − q1 = 0,
A − (q1 + q2 ) − q2 = 0.
Solving the above two equations for q1 and q2 yields the optimal outputs of
the respective ﬁrm:
1
q1 = (A − 2t) and34

1
q2 = (A + t).
3(5)

In this case, the aggregate output is q = (2A − t)/3 and the equilibrium
proﬁts of the respective ﬁrms are given by:
1
1
πHF= (A − 2t)2 ≡ πHF
9

1
2
πHF
= (A + t)2 − f ≡ πF H .
9

and

(6)

We assume that 0 ≤ t < A/2 and f < A2 /9 in order to ensure that bothﬁrms can coexist in the market.

2.2

Location choice stage

In this stage, the duopoly ﬁrms make the relocation decisions given an emission tax rate in country H and its rival’s strategies about the location of
production: strategy H (remain in country H) and strategy F (relocate incountry F ). From (2), (4), and (6), we obtain the payoﬀ matrix depicted inTable 1.
PP
PP ﬁrm 2
P
ﬁrm 1 PPPP

H
F

H
πHH ,
πHF ,

F
πHH
πF H

πF H ,πF F ,πHF
πF F

Table 1: The payoﬀ matrix
There are three Nash equilibria: HH, FF, HF (or FH). Let us compare payoﬀs to derive the conditions for Nash Equilibrium of each case. Given ﬁrm
2’s strategy H, the condition under which ﬁrm 1 has no incentive to relocate
its plant to country F is:πHH ≥ πF H⇔

4f ≥ At.
9

Given ﬁrm 2’s strategy F, the condition under which ﬁrm 1 has no incentive
to remain in country H is:
πF F ≥ πHF

4f ≤ (A − t)t.
9⇔

Given ﬁrm 2’s strategy F, the condition under which ﬁrm 1 decides to remain
in country H is:4
πHF ≥ πF F ⇔ f ≥ (A − t)t.
9
5

Finally, when ﬁrm 2 chooses strategy H, the condition under which ﬁrm 1
decides to relocate its plant to country F is:πF H ≥ πHH⇔

4f ≤ At.
9

Taken together, we can identify the following three Nash equilibria dependingon the size of f :Lemma 1 (i) If f < (4/9)(A − t)t, then both ﬁrms relocate their plants tocountry F . (ii) If (4/9)(A − t)t ≤ f < (4/9)At, then one ﬁrm remains incountry H, while another relocates to country F . (iii) If (4/9)At ≤ f , thenboth ﬁrms remain in country H.
Figure 1 shows the possibilities of three Nash equilibria. Deﬁne FHH (t) ≡4At/9 and FHF (t) ≡ 4(A − t)t/9. Hence, the curve tHH ≡ FHH (t)−1 denotes
the maximum tax rate at which both ﬁrms remain in country H and thecurve tHF ≡ FHF (t)−1 denotes the maximum tax rate at which either ﬁrm
relocates to country F . When an environmental tax is low and the ﬁxed costis high, both ﬁrm remain in country H, while when an environmental tax
is high and the ﬁxed cost is low, both ﬁrm relocate their plants to country
F . Since an emission tax increases the marginal cost of production, it is
proﬁtable for a ﬁrm with low ﬁxed cost to relocate in country F . On theother hand, since the cost of relocation becomes sunk, it is proﬁtable for aﬁrm with high relocation cost to remain in country H if an emission tax islow. When both an emission tax and the ﬁxed cost are given at the medium
levels, either of the duopoly ﬁrms relocates in country F . In this case, if one
ﬁrm relocates abroad, relocation of the rival ﬁrm reduces the market price,which in turn reduces the revenue. Hence, the rival ﬁrm decides to remain
in country H. In contrast, if one ﬁrm chooses production in country H anddecreases its output due to the tax burden, the rival ﬁrm can increase its
output, and thus, the revenue resulting from relocation of its production incountry F . In what follows, when a ﬁrm is indiﬀerent between two locations(i.e. the home country and the foreign country), we assume that this ﬁrmchooses a location that gives a higher welfare for the home country.

2.3

Environmental policy decision stage

The government of country H chooses an emission tax rate t so as to maximize its economic welfare. However, the welfare is determined by the location
choices of the duopoly ﬁrms, which depends on the environmental policy set
by the government. To derive the optimal emission tax rate, ﬁrstly, let us
6characterize the welfare of home country associated with Nash equilibrium
in location choices. The welfare of home country when both ﬁrms remain incountry H is deﬁned as:
12
WHH = CSHH + πHH+ πHH− D(q) + tq,

Rq
where CSHH ≡ 0 p(s)ds−p(q)q is consumer surplus in Nash equilibrium HH.
The welfare when one of ﬁrms remains in country H and another relocatesin country F is given by:i− D(qi ) + tqi ,WHF = CSHF + πHF

i = 1, 2.

Note that proﬁt and emission tax of only one ﬁrm that remains in country H
and damage cost from emissions of its production in country H are included.
The welfare in case that both ﬁrms relocate to country F is:
WF F = CSF F .
Since the proﬁts of relocated ﬁrms remain abroad, the welfare function does
not include the ﬁxed cost f .
Substituting (1)–(6) into the deﬁnitions of welfare above, we obtain three
reduced forms for the welfare in each equilibrium:´
2³
WHH (t; d) =− (1 + d)t2 − (1 − 2d)At + (2 − d)A2 ,
(7)
9
´
1³
WHF (t; d) =(6 − d)A2 − (6 − 4d)At − (3 + 4d)t2 ,(8)18
2
WF F (t; d) = A2 .
(9)
9
The welfare functions (7) and (8) are a concave function of t given d,whereas (9) is constant. The intuition of the ﬁgures of welfare functions (7)and (8) is as follows. Both welfare functions consist of four components:
consumer surplus, proﬁt(s) of domestic ﬁrm(s), tax revenue, and damage
cost, three of which are a function of t and damage cost is indirectly aﬀectedby the emission tax through a change in domestic production. On one hand,
given d, an increase in t reduces consumer surplus and domestic production,
thus, damage cost and the gross proﬁts monotonically and its reduction rateis accelerated as t becomes higher. On the other hand, an increase in t
raises tax revenue when the tax rate is low, and reduces it after the tax rate
exceeds A/4 in the case of HF . When the tax rate is low, the positive eﬀect
of an increase in t on tax revenue dominates the marginal reduction of othercomponents. As the tax rate becomes high, other components decrease at an7

accelerated rate and tax revenue changes to decrease after the tax rate reaches
its threshold, which leads to the concave welfare functions with respect tothe tax rate for any d.
As we will see later, the welfare-maximizing rate in the case HH is higher
than one in the case HF given d. That is, the peak of concave function WHH
is the right of the peak of concave function WHF . Because the government in
the case HH attempts to obtain higher tax revenue and lower damage cost,
the government sets a higher tax rate until the negative eﬀects of reductionof domestic production and consumer surplus dominate the positive eﬀectof a higher tax rate. In the case HF , the government bears damage costand gains tax revenue and proﬁt from only one ﬁrm in addition to consumer
surplus aﬀected by the aggregate output. In this case, the negative eﬀect ofhigher tax on consumer surplus dominates the positive eﬀect in the smaller
level of tax rate than in the case HH. Thus, given d, the optimal tax ratein the case HF is smaller than in the case HH.
Consider the eﬀect of a change in damage coeﬃcient on WHH and WHF
given t. Since damage cost is directly aﬀected by d, an increase in d must
shift the curves of WHH and WHF downwards. The government with a higher
d wants to reduce pollution more so as to require lower level of domestic
production than the government with small d . Therefore, the government
attempts to set the higher optimal emission tax, which implies that the welfare functions shift rightward. Combining these eﬀect of a change in d on
the welfare, the welfare functions shift downwards to the right as d increases.
The positional relationships among three welfare functions depending on d
are given by Lemma 6-9 (See in Appendices) and illustrated in Figures 2-5.

3

Optimal emission tax and welfare

We now want to ﬁgure out the relationship of three welfare functions (7)–(9)with respect to an emission tax rate, given values of damage coeﬃcient d.
The welfare WHH and WHF maximizing-taxes are respectively given by(2d − 1)AetHH (d) =2(1 + d)and(2d − 3)AetHF (d) =.
3 + 4d(10)

Let us compare the welfare. Subtracting (7) from (9) yields WF F − WHH =2g (t), where
9 1
g1 (t) ≡ (1 + d)t2 − (2d − 1)At + (d − 1)A2 .

(11)

When
d < √
5/4, g1 (t)¡¢ = 0 has two diﬀerent real roots, t2 and t5 (> t2 ):A 2d − 1 ± 5 − 4d /2(1 + d). Subtracting (8) from (9), we obtain WF F −81
WHF = 18
g2 (t), whereg2 (t) ≡ (4d + 3)t2 − (4d − 6)At + (d − 2)A2 .

(12)

When d < 15/7, g2 (t) = 0 has two diﬀerent√ one ¢of which is¡ real roots,negative, t1 and another is positive, t4 : A 2d − 3 ± 15 − 7d /2(1 + d).
1
Comparing (8) with (7) yields WHH − WHF = 18
g3 (t), whereg3 (t) ≡ −t2 + (4d + 2)At − (3d − 2)A2 .

(13)

In¡this case,√g3 (t) = 0 has¢ always two diﬀerent real roots, t3 and t6 (> t3 ):2
A 2d + 1 ± 4d + d + 3 .

3.1

The case of small d

Let us ﬁrst examine the case where the damage coeﬃcient is small or equivalently, the marginal damage from pollution is relatively low. Figure 2 illustrates the optimal emission tax and the resulting welfare that country H
obtains for 0 ≤ d ≤ 2/3. Since WHH is always the largest for 0 ≤ d ≤ 2/3,
the government of country H chooses an emission tax rate so as to maximize
WHH . Thus we have found:
Proposition 2 When 0 ≤ d < 1/2, the negative tax rate (subsidy) on emission is optimal: t = −A/2 for d = 0 and t = tHH (f ) for 0 < d < 1/2. Whend = 1/2, the optimal tax rate is zero. When 1/2 < d ≤ 2/3, the government
sets the optimal tax rate according to tHH (f ) in the range f ∈ [0, fe], wherefe satisﬁes e
tHH = tHH (fe), and it sets a constant tax rate, etHH , for the range2f ∈ [fe, 2A /9).
If emission has no damage on the environment, the government attempts to
attract both ﬁrms by subsidizing emission at the welfare-maximizing rate so
as to obtain the gain from proﬁts of two ﬁrms and a price reduction eﬀect.
If environmental damage is not zero, it is not optimal for the government
to subsidize pollution any longer. When the environmental damage fromemission is small enough, WHH is the largest. Therefore, the government
imposes an emission tax according to tHH (f ) that is the highest rate forﬁrms to remain in country H, from which tax revenue contributes to thewelfare of country H. When the ﬁxed cost reaches fe, the governments sets
the welfare-maximizing rate, etHH and maintains the level for f > fe.

9

3.2

The case of medium d

Next, let us look at the case when a damage coeﬃcient is not suﬃciently
large. Deﬁne the critical value d∗ such that WHH (t; d∗ ) = WHF (t; d∗ ) = WF F .
That is, when the damage coeﬃcient is d∗ , the welfare takes the same value
for each location pattern. Figure 3 illustrates how the optimal emission tax
is determined in the left panel and the relationship between the optimal
emission tax and the ﬁxed cost and the relationship between the maximized
welfare and the ﬁxed cost in the right panel when 2/3 < d ≤ d∗ . Note thatthe pattern of the maximized welfare depending on the ﬁxed cost in this caseis diﬀerent from one in the case of 0 ≤ d ≤ 2/3.
Suppose that the ﬁxed cost is f0 . If the government sets t0 , both ﬁrms
remain in country H, which gives country H the welfare WHH (t0 ). If thegovernment increases the tax and sets any rate t ∈ [tHH (f0 ), tHF (f0 )], either
ﬁrm chooses relocation so that the government gains WHF that is larger than
WHH (t0 ). However, if it sets t > tHF (f0 ), the welfare reduces to WF F . Hence,
the best strategy is to obtain WHF for country H when f0 . Since WHF isdecreasing with t, the government desires to leave a tax rate as lower as
possible so as to attain larger WHF . Thus, the optimal tax is tHH (f0 ), under
the assumption that the ﬁrm chooses the location that gives the government
the largest welfare when a ﬁrm is indiﬀerent between two locations.
Suppose that the ﬁxed cost is f3 . If the government sets t > tHH (f3 ),the welfare becomes WHF (t) or WF F , which is strictly smaller than WHH (t3 ).
Hence, the optimal tax rate is t3 = tHH (f3 ). Until the ﬁxed cost approaches
fe that satisﬁes e
tHH = tHH (fe), it is optimal to set a tax according the curvetHH (f ). Since the welfare WHH decreases with t > e
tHH , it is optimal toeemaintain an emission tax at tHH for f ≥ f . Thus we have found:
Proposition 3 For 2/3 < d ≤ d∗ , the optimal tax rate is (i) tHH (f ) iff ∈ [0, fe]; (ii) e
tHH if f ∈ [fe, 2A2 /9).
Until the degree of damage reaches the critical value at which the welfare of
all patterns of location are the same, WHF (t) is the largest for f ∈ [0, f3 ].
The government sets an emission tax to induces either ﬁrm to relocate its
plant to country F . When the tax rate is in the range, [0, tHH (f3 )], WHF
decreases with t for any f . Hence, the government sets tHH (f ), at which two
locations are indiﬀerent for each ﬁrm. In the location pattern of HF withsmall degree of environmental damage, an increase in consumer surplus ofprice reduction due to low tax rate dominates a reduction in proﬁts and tax
revenue from one ﬁrm. For f ≥ f3 , the government sets tax rate according
to the curve tHH (f ) so as to attract both ﬁrms to country H. In this range,
10

the government enjoys tax revenue and a reduction in damage cost due to anincreased emission tax corresponding to an increased ﬁxed cost unless either
ﬁrm decides to relocate its production. When the ﬁxed cost reaches fe, thegovernment ﬁxes emission tax at etHH that maximizes welfare in the case of
HH. If the ﬁxed cost is higher than fe, both ﬁrms are reluctant to relocate
their production even if the tax rate increases. However, setting tax rate toohigh reduces the proﬁt of ﬁrms. This reduction in the proﬁt dominates the
beneﬁt of the increased tax revenue. Thus, it is optimal for the government
to ﬁxed emission tax at e
tHH for f ≥ fe.
Consider next the case of d∗ < d < 5/4. Note that when d < 5/4, there
exists tax rates such that WF F < WHH and WF F < WHF from (11) and(12). As Figure 4 shows, for f ∈ [0, f4 ], where f4 satisﬁes t4 = tHH (f4 ),the government obtains the largest welfare WHF by setting tax rate tHH (f ),under which only one ﬁrm relocate to suit the government’s purpose. Oncethe ﬁxed cost exceeds f4 , WHF is smaller than WF F for f ∈ (f4 , f2 ], where f2
satisﬁes t2 = tHH (f2 ). Hence, the government attempts to keep both ﬁrms
away from home country. The government suddenly increases the tax rate
up to tHF (f ) (at the point a in Figure 4). When the ﬁxed cost reaches f2 ,the government desires both ﬁrms to stay in home country. The government
suddenly reduces the tax to tHH (f ) (at the point b). For f ∈ [f2 , fe], thegovernment sets a tax rate along the curve tHH (f ) and ﬁxes the welfaremaximizing rate etHH for f > fe due to decreasing WHH with respect to t. We
may summarize the above:
Proposition 4 For d∗ < d < 5/4, the optimal tax rate is (i) tHH (f ) iff ∈ [0, f4 ]; (ii ) tHF (f ) if f ∈ [f4 , f2 ]; (iii) tHH (f ) if f ∈ [f2 , fe]; (iv) e
tHH
if f ∈ [fe, 2A2 /9).

When the degree of damage is medium, the relationship between the optimal
tax rate and the ﬁxed cost is complicated. For relatively small ﬁxed costs,f ∈ [0, f4 ], the government attracts only one ﬁrm to bear damage from its
production and to obtain beneﬁts from tax revenue and proﬁts of one ﬁrm.
Since a higher emission tax rate reduces consumer surplus and domestic
production and this cost dominates the positive eﬀect of an increased tax on
a reduction in damage cost, the government sets the lowest tax rate such that
only one ﬁrm relocates, which is tHH (f ) from the assumption. For the middle
level of the ﬁxed cost, f ∈ [f4 , f2 ], the government sets tHF (f ) to keep bothﬁrms away from the home country. The government enjoys beneﬁts fromthe improved environment and a price reduction although it sacriﬁces tax
11revenues and proﬁts. Once the ﬁxed cost exceed f2 , the government can
set a relatively high emission tax tHH (f ) on both ﬁrms in country H toobtain increased tax revenue because the large cost of relocation becomes
heavy burden for the ﬁrms so that both ﬁrms are reluctant to relocate theirproduction under t ≤ tHH (f ). Since t > tHH (fe) reduces WHH throughreduction in aggregate production, the government maintains e
tHH for f > fe.
Let us consider the case of 5/4 < d ≤ 15/7. Figure 5 illustrates therelationships between the optimal tax, the maximized welfare and the ﬁxed
cost of two cases in which the degrees of damage is relatively large. The
upper panel shows the case of 5/4 ≤ d ≤ 2. In this case, for f ∈ [0, f4 ], thegovernment attempts to induce only one ﬁrm to relocate because WHF is thelargest. While WHF increases with t for t ∈ [0, etHF ], it decreases with t for t ∈e[tHF , t4 ], where t4 = tHH (f4 ). Hence, until the ﬁxed cost reaches feHF , which
satisﬁes e
tHF = tHF (feHF ), it is desirable for the government to set an emission
tax according to the curve tHF (f ) and maintain the welfare-maximizing rateetHF for f ∈ [feHF , feHH ], where feHH satisﬁes e
tHF = tHH (feHH ). However,when the ﬁxed cost exceeds feHH , the welfare WHF decreases with t due tothe negative eﬀect on production, thus on proﬁt of domestic ﬁrm, which
induces the government to lower the tax to tHH (f ). Note that if both ﬁrms
remain in country H, the welfare drastically decreases to WHH so that the
government avoids both ﬁrms remain in the home country. Once the ﬁxed
cost exceeds f4 , the welfare WHF is smaller than WF F . The government
attempts to drive both ﬁrms out so as to raise the tax up to tHF (f ).
The lower panel shows the case of 2 < d ≤ 15/7. For f ∈ [0, f1 ], WF F isthe largest. Hence, the government sets tHF (f ) to drive both ﬁrms out. For
f ∈ [f1 , feHF ], the government sets tax rate according to the curve tHF (f ),which yields WHF . For f ∈ [feHF , feHF ], the government sets etHF to obtainethe maximized WHF . For the cases of f ∈ [fHF , f4 ] and f ∈ [f4 , 2A2 /9], thegovernment takes the same strategies to those in the case of 5/4 ≤ d ≤ 2 as
described above. We may summarize the above argument as the following
proposition:
Proposition 5 For 5/4 ≤ d ≤ 15/7, the optimal tax rate is (i) tHF (f ) if f ∈[0, feHF ], where feHF satisﬁes e
tHF = tHF (feHF ); (ii ) e
tHF if f ∈ [feHF , feHH ],where feHH satisﬁes e
tHF = tHH (feHH ); (iii) tHH (f ) if f ∈ [feHH , f4 ]; (iv)tHF (f ) if f ∈ [f4 , 2A2 /9).
In the case of large degree of damage, WHH is the lowest. The environmental damage from the aggregate production of two ﬁrms is so large that the12government avoids to attract both ﬁrms. When damage is enough small forcountry H to allow one ﬁrm to remain and the ﬁxed cost is relative low,
the government sets tHF (f ) to attract only one ﬁrm and enjoys the beneﬁt
of tax revenue and a proﬁt from one domestic ﬁrm and reduction in price.
Until the ﬁxed cost reaches feHF , an increase in tax has positive eﬀect on
tax revenue and a reduction in damage cost, the government raises emission tax corresponding to increasing ﬁxed cost. As the burden of the ﬁxed
cost becomes heavier, a domestic ﬁrm does not choose relocation when anemission tax is relatively low, which gives the government an incentive toraise the tax in order to obtain WHF . Once the ﬁxed cost exceeds feHF , thegovernment enjoys the maximized welfare WHF . For the larger ﬁxed costs
than feHH , the welfare-maximizing tax e
tHF induces both ﬁrms to remain incountry H. Since damage cost from the production of two ﬁrms is so large,
the government raises the emission tax in order to drive only one ﬁrm out.
The welfare WHF decreases as tax rate increases, the government sets thelowest tax that induces one ﬁrm to relocate its production, according to thecurve tHH (f ). When the ﬁxed cost reaches f4 , the government sets tHF (f )to drive two ﬁrms out. In the lower case of Figure 5, for the lower ﬁxed cost,
the government attempts to drive both ﬁrms out because tax revenue is too
small to compensate large damage from pollution.2

4

Globalization and economic welfare

We may consider the location cost, f , as a degree of globalization of world
economy. That is, the more globalization grows, the lower f is. When thedamage coeﬃcient, d, is small, the location pattern of two ﬁrms, (H, H),remains unchanged even if f decreases. Given d ∈ [0, 1/2], the welfare thatthe country H gains is maximized for a nonpositive tax rate. When d is in
the range of (1/2, 2/3], the welfare, WHH , decreases as f becomes smaller.
In this case, if the government sets a welfare-maximizing tax, a ﬁrm withsmall f relocates its production to the foreign country, which reduces thewelfare. To avoid the reduction in the welfare due to relocation of ﬁrms, thegovernment sets a lower tax according to the curve, tHH (f ).
When d is large, the welfare, WF F is the largest for various f . Hence,
the governments raises the tax to gain WF F , which is independent of f .
For the governments with high sensibility to the environmental quality, theproduction of polluting ﬁrms outside gives the largest welfare regardless of
2

In the case of large damage, d > 15/7, the government sets tHF (f ) for any f to leadboth ﬁrms to relocate their production abroad, because beneﬁts from tax revenue andproﬁts no longer oﬀset huge environmental damage.

13

the degree of globalization.
When the damage coeﬃcient is medium, the relationship between globalization and economic welfare shows various patterns with respect to d.
Firstly, when d is in the range of (2/3, d∗ ], the equilibrium location patternchanges from (H, H) to (H, F ) as the globalization proceeds. If t is small, theeﬀect of tax revenue is small (tax revenue eﬀect is zero when t = 0) so that theeﬀect of a reduction in environmental damage due to relocation of one ﬁrm
dominates reductions in tax revenue and proﬁts. As a result, WHF > WHH .
When the relocation cost decrease cross the f˜, the welfare-maximizing tax,e
tHH induces ﬁrms to relocate abroad. Hence, the government lowers the tax
rate corresponding to the globalization grows so as to attract both ﬁrms incountry H. When the location cost decreases cross f3 , the location patternthat maximizes the welfare is (H, F ). As f decreases, the tax rate thatinduces only one ﬁrm to relocate shifts lower. Since WHF is a decreasing
function of t > 0, a reduction in the tax rate increases the welfare. When d
is in the range of (d∗ , 5/4), the equilibrium location pattern (F, F ) appears
during it changes from (H, H) to (H, F ) as f2 decreases to f4 . In the case of
2/3 < d < 5/4, the welfare decreases ﬁrst, then increases as the globalization
proceeds.
When the damage coeﬃcient is relatively large and in the range of [5/4, 2],
the equilibrium location pattern changes from (F, F ) to (H, F ) as the globalization grows. The environmental damage is too large to maximize WHH .
When the tax rate is small, the marginal cost of a domestic ﬁrm is not sohigh, which contribute to consumer surplus. In addition, under the location
pattern (H, F ), tax revenue contributes to the welfare. The beneﬁt of a small
tax on consumer surplus and a proﬁt besides tax revenue dominates the environmental damage, which leads to a larger WHF than WF F . According
to the growing globalization, the government lowers the tax so as to obtain
WHF . Since the welfare-maximizing tax rate for the location pattern (H, F )is in the range of [0, t4 ], the equilibrium welfare increases and then decreases
as the globalization proceeds. It is interesting to note that the degree of
globalization reverses the change of welfare.

5

Conclusions

This paper explores the optimal environmental tax policy under which duopoly
ﬁrms strategically choose the location of their plants in a simple three-stage
game. In the ﬁrst stage, the government sets an emission tax rate to maximize the country’s welfare. Given the tax rate, in the second stage, duopoly
ﬁrms choose the location of their plants. In the third stage, ﬁrms compete
14

á la Cournot in the market, given an emission tax rate and their location
choice. In this setting, we show how the optimal environmental tax is determined when the government can aﬀect the strategies of duopoly ﬁrms about
relocation which, in turn, depends on endogenous emission tax rate.
We derive that the optimal emission tax and welfare resulted from location choices of ﬁrms depend on the degree of damage from pollution, whichaﬀects beneﬁts from proﬁts of ﬁrms, tax revenue, and price reduction andcost from environmental damage. Under some range of damage coeﬃcient,hybrid equilibrium emerges. That is, either ﬁrm chooses relocation of its
plant even if duopoly ﬁrms are identical ex ante.

AppendicesLemma 6 For 0 ≤ d ≤ 2/3, WHH is the largest if 0 ≤ t ≤ A/2. For
0 ≤ d ≤ 1/2, WHH is maximized at t = 0. For 1/2 < d ≤ 2/3, WHH ismaximized at t = e
tHH .
Proof. When d = 0, we have WF F < WHF (0) < WHH (0) and WHF (A/2) <WF F < WHH (A/2). Since welfare-maximizing taxes, e
tHH and e
tHF are negative, WHH is largest in t ∈ [0, A/2) and is maximized at t = 0 for this range.
In the case that d = 1/2, WHH is maximized at etHH = 0. Since etHF < 0and WHF (A/2) < WF F < WHH (A/2), WHH is largest in 0 < t < A/2. In
the case that d = 2/3, WHH and WHF intersect at t = 0, hence, t3 = 0
with etHH > 0 and etHF < 0. Since WHF (A/2) < WF F < WHH (A/2), WHH islargest in 0 < t < 2/3.
Lemma 7 For 2/3 < d ≤ d∗ , WHF is largest if 0 ≤ t ≤ t3 , while WHH is
the largest if t3 < t < t5 and is maximized at t = e
tHH .
Proof. This follows directly from Lemma 6.
Lemma 8 For d∗ ≤ d < 5/4, WHF is largest if 0 ≤ t ≤ t4 , WF F is largest
in t4 < t < t2 , WHH is largest if t2 < t < t6 and is maximized at t = e
tHH .
Proof. Since g3 (t) = 0 if only if WHH = WHF , we can rewrite as:
G (t(d), d) = −t2 + (4d + 2)At − (3d − 2)A2 ≡ 0.
By the implicit function theorem, we have:′t (d) = −

15

Gd
,
Gt

(14)

where the subscripts denote partial derivatives. Because g3 (t) = 0 is quadratic
function of t and t3 is its smaller roots, the sign of Gt (·) at t = t3 must be
positive. By assumption of t < A/2, we have Gd = 4A (t − (3/4)A) < 0.
′Hence, from (14), t (d) > 0.
Next, looking at the slope of WHF at t = t3 , again by assumption of′t < A/2, ∂W HF/∂d = −(2t − A)2 < 0. Combining this with t (d) > 0, ford > d∗ , we have:
WF F = WHF (t(d∗ ), d∗ ) > WHF (t(d), d∗ ) > WHF (t(d), d) .
This shows that the intersection of WHH and WHF is below the function
WF F .
Lemma 9 For 5/4 ≤ d ≤ 2, WHF is the largest if 0 ≤ t ≤ t4 , while WF F
is the largest if t4 < t < A2 /2. For 2 < d ≤ 15/7, WF F is the largest if
0 < t ≤ t1 or t4 < t < A2 /2, while WHF is the largest if t1 < t ≤ t4 .
Proof. In this case, there is no possibility that WHH is maximum since
WHH does not intersect WF F from (11). When 5/4 ≤ d ≤ 2, WHF is largest
because t1 ≤ 0 and 0 < t4 < A/2. When 2 < d ≤ 15/7, WF F is largest for
the ranges of t ∈ [0, t1 ] or t4 < t < A/2, and WHF is largest for t1 < t < t4 .

16

References[1] Katsoulaco, Y. and A. Xepapadeas, (1995), ”Environmental policy under oligopoly with endogenous market structure,” Scandinavian Journalof Economics 97, 411–420.
[2] Markusen, J. R., E. R. Morey, and N. D. Olewiler, (1995), ”Environmental policy when market structure and plant locations are endogenous,”
Journal of Environmental Economics and Management 24, 69–86.
[3] Markusen, J. R., E. R. Morey, and N. Olewiler, (1995), ”Competition
in regional environmental policies when plant locations are endogenous,”
Journal of Public Economics 56, 55–77.
[4] Mills, D. E. and W. Smith, (1996), ”It pays to be diﬀerent: Endogenous
heterogeneity of ﬁrms in an oligopoly,” International Journal of Industrial
Organization 14, 317–329.
[5] Ohkawa, T., M. Okamura, and M. Okawa, (2009), ”Optimal tariﬀ policy
with endogenous location choice,” unpublished manuscript.
[6] Petrakis, E. and A. Xepapadeas, (2003), ”Location decisions of a polluting ﬁrm and the time consistency of environmental policy,” Resource andEnergy Economics 25, 197–214.
[7] Yomogida, M., (2007), ”Fragmentation, welfare and imperfect competition,” Journal of Japanese and International Economics 21, 365–378.

17tt HF( f )(F, F)(H, F)t HH( f )(H, H)f
Figure 1: Possibilities of three Nash equilibria

18

d=0
w

t

WHH
WHF

0
㹼

t

WFF

f

HH

max

WHH

㹼

t

0

HH

f

t

d = 2/3

w
WHH

WFF

t
WHF

㹼

t

0

t

㹼

t

HH

f

HH

max

WHH

㹼

f

f

tHF( f )ftHH ( f )Figure 2: The case of 0 ≤ d ≤ 2/3.

19

2/3 < d < d

w
WHF
WHH

t
WFF
㹼

t

HH

t3
0

0

t

t3 㹼
t

0

HH

f0

f3

f3

㹼

f

f

w

a

WHF
㹼
(
WHH t HH)㹼ff

tHF( f )ftHH ( f )Figure 3: The case of 2/3 < d ≤ d∗ .

20

d < d < 5/4
WHF

WHH
WFF

t
㹼

t

HH

t2
t4
t

t4 t3 t2 㹼
t

b

a
f4

f2

f4

f2

㹼

f

f

㹼

f

HH

max

f4
f2
㹼

WHH

a

WFF

b

t HF( f )ftHH ( f )Figure 4: The case of d∗ < d < 5/4.

21

f

5/4 < d < 2

w

WHF

WFF
t

WHH
A/2

t4

0
㹼
f HF
㹼
f HH

㹼

t

t

t4

HF

0

㹼
fHF 㹼
fHH

f4

2

2A /9

f

max

WHF

WFF

t HF( f )f4

tHH ( f )

f

f

㹼
㹼
fHF fHH f 4

2 < d < 15/7

w

WHF
t

WHH
A/2

t4
0

0

f1
㹼
f HF
㹼
f HH

t1

㹼

t

HF

t4

A/2

f1

㹼
fHF 㹼
fHH

f4

f1

㹼
fHF㹼
fHH

f4

2

2A /9

f

max

WHF

WFF

t HF( f )f4

t HH ( f )Figure 5: The cases of 5/4 ≤ d ≤ 2 and 2 < d ≤ 15/7

22f