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Second Best Environmental Policies under Uncertainty*
Fabio Antoniouaand Phoebe Koundourib
April 2008
a;b Department of International and European Economic Studies, Athens University of Eco-
nomics and Business, Patision Str 76, Athens 10434, Greece.
Corresponding Author: Phoebe Koundouri, Department of International and European
Economic Studies, Athens University of Economics and Business, Patision Str 76, Athens
10434, Greece. e-mail: [email protected]
�Acknowledgments:We thank A. Xepapadeas and P. Hatzipanayotou for their valuable
suggestions and comments. Fabio Antoniou would like to thank Greek National State of Schol-
arships (I.K.Y) for funding his research activities. The usual disclaimer applies.
Second Best Environmental Policies under Uncertainty
Abstract
We construct a model of strategic environmental policy in an international duopoly context
with various modes of uncertainty. We assume that governments are misinformed about the
actual level of demand, or the cost of abatement and the damage caused from pollution. Under
either of these modes of uncertainty, governments select one policy instrument, an emission
standard or an emission tax, in order to regulate pollution production. In contrast to the
existing literature, using commonly used functional forms, we provide su¢ cient conditions for
governments to be optimal (maximize expected welfare) to select a tax instead of a standard.
We illustrate this when governments face an unknown demand and the pollutant is assumed to
be local. The results can be further extended for the existence of transboundary pollution, and
uncertainty in the marginal cost of abatement and the marginal damage functions.
JEL classi�cation: F12; F17; F18; Q53; Q56
Keywords: Strategic environmental policy, Local and Transboundary Pollution, Imperfect
competition, Choice of Policy Instrument, Incomplete Information
1 Introduction
During the last three decades developed countries have recognized the need for regulating pol-
luting agents, since they produce harmful and irreversible e¤ects on human health and the en-
vironment (see Stern review [19]). The US Environmental Protection Agency (EPA), founded
in 1970, and the European Environment Agency, founded in 1995, have as their main purpose
the e¢ cient and e¤ective regulation of pollution.
There are two general ways to regulate industrial pollution: (a) through the use of quantity
constraints, which translate into several forms of maximum emission standards or pollution
permits; and (b) through price constraints, which are emission taxes. There exists a large
theoretical literature on the impact of the above policy tools on the social surplus. It can be
shown that when the marginal cost of abatement and the marginal pollution damage are known
with certainty by the regulator, emission standards and taxes are equivalent policy instruments
(Baumol and Oates [3]). Initially, Weitzman [23] and then Fishelson [7], Adar and Gri¢ n [1] and
Stavins [18] argued, among others, that this equivalence does not hold when the marginal cost of
abatement and marginal pollution damage become uncertain. In particular, when uncertainties
in the marginal damage and marginal cost of abatement are uncorrelated, then the uncertainty
in the marginal damage function plays no role in the choice of the policy instrument. However,
uncertainty in the marginal cost of pollution control a¤ects signi�cantly the expected social
welfare loss associated with each instrument, depending on the slopes of marginal damage and
marginal abatement cost functions. An implicit assumption of the aforementioned papers is
that the market structure is perfectly competitive. This assumption, however, is not innocuous
as illustrated by Heuson [9], [10]. In particular, applying the Weitzman rule in the optimal
choice between taxes and standards in a closed economy under Cournot competition can be
misleading since the use of taxes increases the total output and thus lowers the market price
leading to higher welfare in contrast to the case of a standard. Therefore a positive bias in
favour of a tax appears extending the range of the parameters supporting a tax as the preferred
policy instrument when two distortions appear simultaneously.1
Another stream of the literature, the so called strategic environmental policy literature,
based among others on the papers of Conrad [5], Barrett [2], Kennedy [11], Rauscher [17], Ulph
1The original Weitzman rule suggests that a standard should be preferred to a tax when the marginal damageruns steeper than the marginal cost of abatement.
1
[21] and Neary [14], argues that when international industry competition is oligopolistic and
�rms compete in output, there is a unilateral incentive by the governments to under-regulate
pollution so that the competitiveness of the exporting �rms will be enhanced. This is a typical
example of how environmental policy instruments can be used as second best instruments for
trade purposes when other traditional tools, as subsidies or tari¤s are restricted from interna-
tional agreements.
The aim of our paper is to rank environmental policy instruments in terms of expected
welfare loss when these policies are employed to regulate pollution and to shift rents towards
domestic �rms, under the assumption that �rms possess better information than governments.
Put it di¤erently, we aim to grade emission standards and taxes in an international environ-
ment with imperfect competition and predict which policy instrument will be preferred when
governments are uncertain about demand or cost functions faced by the �rms. The analysis is
rather complex since the existence of more than one country implies more than one regulator.
Hence, the option of the optimal instrument for each regulator depends crucially on the choices
of the other regulators.
Recent papers by Lahiri and Ono [12] and Yanasee [24] deal with a similar problem and
both papers conclude that standards tend to lead to welfare superior outcomes compared to
taxes. In particular, Lahiri and Ono[12] suggest that when the number of �rms is �xed, an
emission standard is welfare superior to a pollution equivalent emission tax. However, their
model assumes complete information and the authors do not examine the asymmetric cases
in order to obtain a full payo¤ matrix and thus attain the Nash equilibrium of the game. In
other words no prediction is made about which is the optimal strategy for each government in
the policy instrument choice game. A similar conclusion is obtained by Yanasee [24] using a
dynamic game under a perfectly competitive market structure. His main argument in favor of
standards is that taxes imply greater distortions in the economies. Moreover, Ulph [22] allowing
�rms to invest strategically so that future output competition is a¤ected, compares a pollution
standard to a tax and concludes that standards yield a Pareto superior outcome compared to
taxes when the country is an exporter of the good.2
2Ulph [22], assumes that governments select their policy instruments such that international agreements aresatis�ed. Hence, governments cannot select their policies strategically. Under this assumption it can be the casethat a tax leads to a Pareto superior outcome in contrast to standards if the country is a signi�cant consumer ofthe good. Similarly to Lahiri and Ono [12], uncertainty is not taken into account and the results do not consistNash equilibrium.
2
These papers indicate that there is a strong tendency towards standards. Our model and
results strengthen this result and provide a framework in which we can claim that with no
uncertainty, using standards not only leads to Pareto superior outcomes in terms of welfare, but
it also consists a dominant strategy for each regulator and thus a Nash equilibrium strategy for
each government. Nonetheless, we claim that uncertainty should play a key role in the decision of
the optimal policy instrument. After introducing uncertainty in demand or abatement cost and
damage functions, we provide su¢ cient conditions under which it is optimal for the regulator
to select an emission tax to regulate pollution,3 contrary to the aforementioned literature.
Our result is in line with Pizer [15] where the bene�t of introducing a tax in a US climate
change model leads to 50% higher welfare than in the case where a quantity constraint is used
to regulate pollution. Additionally, Harrington et al. [8] illustrate that di¤erent countries fre-
quently face identical environmental problems through the use of di¤erent policy instruments.4
In section 2 we present the model, in section 3 we derive the Nash equilibrium when the
pollutant is assumed to be local and demand is uncertain to the governments. In section 4 we
extend the analysis for the case where the pollutant is assumed to be global, as greenhouse
gases. In section 5 we assume that uncertainty appears only in the marginal cost of pollution
control and in the marginal damage function and derive the new Nash equilibrium. Section 6
concludes the paper.
2 The Model
We consider a symmetric two country (home and foreign) international duopoly model, where
each �rm belongs to a di¤erent country and produces a homogenous good whose consumers
reside in a third country. Consumers preferences can be mapped into a quasi linear utility
function which implies a linear inverse demand of the form p = B � x�X + �, where B is the
demand intercept, � is a random variable assumed to follow a distribution with mean zero, and
x;X are the output levels for the domestic and the foreign �rm respectively.5
3This conclusion holds both in the case where governments face an uncertain demand as in Cooper andRiezman [6] and the case where governments (in contrast to �rms) cannot foresee exactly abatement cost anddamage functions, as was originally assumed by Weitzman [23].
4In their book they provide twelve case studies on six environmental problems and they illustrate that the USand several EU countries apply di¤erent policy instruments in each case.
5Throughout the paper foreign �rm�s and country�s variables and functions are neglected, since the model issymmetric. Furthermore, we assume that when � takes negative values, interior solutions for our variables arestill obtained. This could be described by an �2 distribution with zero mean.
3
Both �rms face the same technology which implies that a unit of production generates a
unit of pollution (z). However, an abatement technology (a) is assumed to exist and thus total
pollution equals production minus abatement carried out by the �rm,
z = x� a (1)
Abatement cost function is assumed to be convex of the form:
ca =1
2ga2 + au; (2)
where g is a positive coe¢ cient which determines the cost of pollution control and u is an error
term following a distribution with zero mean. Pro�t function of the domestic �rm depends on
the policy instrument chosen by the government in order to regulate pollution and is given by
the following expression:
� = (B � x�X)x� cx� ca � tz; (3)
where c is marginal cost of production (common for both �rms) and tz the tax payments due
to pollution when tax is the policy instrument chosen. The choice variables of the �rms are
output and abatement level.
Regulation of pollution by the governments takes place prior to production decisions. We
examine two di¤erent ways to regulate pollution. First, we assume that governments can use
an emission standard, z, a maximum allowed level of pollution by the �rms, which results as
a quantity constraint. The alternative policy instrument available to the governments is a tax
for each unit of emissions, t, which is considered as a price constraint. Governments choose the
optimal level of regulation by maximizing welfare which is given by:
w = � + tz � d; (4)
where tz are the revenues from the pollution tax when this is implemented and d stands for the
damage caused from pollution and has the following form:
d =1
2k(z + Z)2 + (z + Z)�; (5)
4
where the terms in parenthesis are the sum of domestic pollution plus foreign pollution weighted
by a coe¢ cient which takes the value of zero when the pollutant is local and one when it is
global. The coe¢ cient k must be positive and determines the injuriousness of the pollutant.
Similarly to abatement cost function we assume that the damage function is stochastic and
depends on a random variable, �, which also follows a distribution where its expected value
equals zero. Throughout the paper we assume that uncertainties between abatement cost and
damage functions are uncorrelated.
Following the above assumptions we understand that in the paper we will introduce various
modes of uncertainty. In the next two sections we will assume that demand is unknown to the
governments while in section 5 we will assume that marginal cost of abatement and marginal
damage are subject to uncertainty. Despite this the structure of the model will be the same
throughout the paper and is summarized in the following �gure:
{Figure 1 About Here}
Initially the governments choose between emission standards and taxes in order to regulate
pollution. Given this, the governments choose the actual policy levels. After governments�
actions, uncertainty (about demand�s intercept or marginal cost of abatement and marginal
damage) is revealed to the �rms. Hence, when �rms compete in outputs we assume that they
act in an environment with full information.6 This assumption is also adopted by Cooper and
Riezman [6], Weitzman [23], Fishelson [7] and Adar and Gri¢ n [1]. It is based on the fact that
governments in general are less informed than �rms about their demand and cost functions.
3 Choice of Optimal Policy Instrument with Demand Uncertainty and Local
Pollution
In this section we assume that governments are uncertain about the demand that the �rms
will face and that pollution is only local (i.e. = 0). Hence, marginal cost of abatement and
6Nannerup [13] introduced uncertainty about �rms�s costs in a strategic environmental policy model witha similar structure to ours. However, in contrast to our model, Nannerup allows the implementation of bothstandards and lump sum taxes, simultaneously, and thus the construction of a truth revealing contract is possible.Furthermore, Nannerup does not examine the choice of an optimal policy instrument which is the purpose of ourpaper.
5
marginal damage are perfectly observed by the government as well (i.e. u = 0 and � = 0).
In order to determine which policy instrument will be chosen we need to derive the Nash
equilibrium of the game taking into account all the possible choices of both governments. In
other words we need to complete the full payo¤ matrix of expected welfares for every possible
contingent, i.e. the domestic and the foreign government select a standard or a tax, or they
select di¤erent policy instruments.
3.1 Emission Standards
In order to derive the Bayes Nash equilibrium for this case we need to solve the problem via
backwards induction. Hence, we start solving from the �nal stage of the game where �rms
compete in outputs given that governments choose emission standards to regulate pollution. It
is important to note that, when standards are used as an instrument, �rms have as a control
variable only production since abatement must take a value that satis�es equation(1). Bearing
this in mind, we maximize domestic pro�ts with respect to output and obtain the reaction
function of output,d�
dx= 0 () x =
B � c+ � �X + gz
2 + g; (6)
where @x@X = � 12+g is the slope of the domestic �rm�s reaction function. Solving simultaneously
the domestic and the analogue foreign �rms�reaction functions we obtain equilibrium outputs
as a function of standards:
x =(B � c+ �)(1 + g) + g(2 + g)z � gZ
(1 + g)(3 + g): (7)
Given equilibrium outputs, governments select the optimal level of emission standards by
maximizing expected welfare since they do not know the exact position of demand:
dEw
dz=@E�
@x
@x
@z+@E�
@z+@E�
@X
@X
@z� @d@z= 0 () z =
g(2 + g)2 [(B � c)(1 + g)� gZ]�1
; (8)
where E� denote expected pro�ts and �1 = g f9 + 2g [8 + g(5 + g)]g+(1+g)2(3+g)2k. Equation
(8) summarizes the reaction function of the domestic regulator. We observe that @z@Z < 0 and
therefore domestic and foreign emission standards are strategic substitutes (see Bulow et al.
[4]). If the foreign regulator ratchets up its standard then the domestic one will ratchet down
6
its own standard.
Solving simultaneously (7) the domestic government�s reaction function (8) and the corre-
sponding equations for the foreign �rm and government we obtain the Bayes Nash equilibrium:8><>: x� = X� = (B�c)(1+g)(3+g)(g+k)m + �
3+g
z� = Z� = (B�c)g(2+g)2m
9>=>; ; (9)
where m = g[9+ g(11+ 3g)] + (1+ g)(3+ g)2k. These are the equilibrium levels of outputs and
standards. In order to detemine the expected welfare in the case of standards we substitute
equilibrium values given in (9) in (4) and after taking the expectation and some algebraic
simpli�cations we get:7
Ew�zZ =(2 + g)[(B � c)2(3 + g)2(g + k)�1 +m2var(�)]
2fg(3 + g)[9 + g(11 + 3g)] + (1 + g)(3 + g)3kg2 (10)
where var(�) is the mean-preserving spread (variance) of the demand intercept. We observe that
expected welfare depends positively on var(�). In other words the ex ante equilibrium welfare
is greater with uncertainty than with no uncertainty. The greater var(�) is, the greater the
gains are. This is due to the convexity of the pro�t function in terms of the demand intercept.
Another interpretation of this result is that �rms are better informed about � and thus expected
pro�ts depend positively on var(�), while damage from pollution is not a¤ected by the demand
variability since pollution is �xed at the pollution standard selected.
3.2 Emission Taxes
In contrast to the previous case we now assume that both governments use taxes to control
pollution. Now �rms have available two control variables, output and abatement level. Solving
backwards we derive the �rst order conditions for the �rms:
d�
dx= 0 () x =
B � c� t+ � �X2
(11)
d�
da= 0 () a =
t
g(12)
7All the calculations in the paper were done using Mathematica 6.
7
The output reaction function of the domestic �rm is implied by equation (11). We observe that
when taxes are used the output reaction function is steeper than the corresponding one in the
case of standards. Pro�t maximizing condition with respect to abatement is given by equation
(12) and states that marginal cost of abatement equals the tax. The terms are rearranged such
that a does not appear into the pro�t function. Equilibrium values of outputs as a function of
taxes are obtained as we solve domestic and foreign �rm�s reaction functions simultaneously:
x =B � c+ � � 2t+ T
3(13)
Moving towards governments decisions about optimal levels of taxes we maximize expected
welfare with respect to the emission tax:
dEw
dt=
@E�
@x
@x
@t+@E�
@t+@E�
@z
@z
@t+@E�
@X
@X
@t+@tz
@t� @d@z
@z
@t= 0 (14)
() t =g[3k + g(�1 + 2k)](B � c� T )
�2;
where �2 = g(9 + 4g) + (3 + 2g)2k. If 3k + g(�1 + 2k) is greater than zero then the reaction
function of the domestic regulator implies that taxes are strategic substitutes, which as we will
see later is a su¢ cient condition for the existence of an interior solution in equilibrium, otherwise
we obtain a negative pollution tax (a pollution subsidy) which from (12) implies a negative level
of abatement which is infeasible.
In order to obtain the equilibrium levels of outputs, taxes and pollution in the two countries
we need to solve simultaneously equations (1),(12), (13) and (14) and their analogues for the
foreign �rm and government:8>>>><>>>>:x�t = X
�T =
(B�c)(3+2g)(g+k)n + �
3
z�t = Z�T =
(B�c)g(2+g)n + �
3
t� = T � = (B�c)g[3k+g(�1+2k)]n
9>>>>=>>>>; ; (15)
where n = g(9 + 5g) + (3 + g)(3 + 2g)k. In contrast to the case of emission standards, when
taxes are the policy instrument, pollution is stochastic since pollution depends on production
which in turn is also random. In addition to that we observe that output in the latter case is
8
more sensitive to demand variability. This is due to the fact that marginal cost of production is
steeper when a standard is used in comparison to the case of a tax where marginal cost is �at.
Hence, �rms are more �exible in the case of taxes. This is illustrated in �gure 2:
{Figure 2 About Here}
The marginal cost of production when a tax is the policy instrument is obtained through equa-
tion (3) and equals c + t, which is represented in �gure 2 by the horizontal line TMCt. The
marginal cost of production when a pollution quota is the policy instrument chosen by the
government, is obtained again from (3) and is represented by the kinked line cAB. The kink at
point A appears because from this point onwards the standard becomes binding. We initially
assume that the equilibrium is at point E, where the marginal cost of production when a stan-
dard and a tax is used are equal and intersect the marginal revenue (MR). Two possible shocks
are represented in the demand intercept (one negative and one positive which shift marginal
revenue from MR to MR1 and MR2, respectively) and four new equilibria are obtained (E1t
and E2t for taxes, and E1s and E
2s for standards). We can obsreve that the equilibrium output
is less responsive to demand variability, when a standard is the policy instrument rather than
a pollution tax.
Subtracting the new equilibrium levels from the welfare function in (4) and taking the
expectation, we obtain the expected welfare for each country:
Ew�tT =(2 + g)(B � c)2�2
2n2� (k � 2)
18var(�): (16)
In contrast to the case of standards, expected welfare is a negative function of var(�) if k >
2. Nonetheless if k < 2, then expected welfare depends positively on the variance of the
demand intercept. This re�ects the fact that pollution is now stochastic. Subsequently, the
more stochastic the demand is, the more stochastic is pollution and has a negative e¤ect on
welfare. However, as in the previous case expected pro�ts depend positively on var(�). The
overall e¤ect of the variance on welfare is ambiguous. If the damage caused from the pollutant
is severe enough (k > 2) then expected welfare falls as uncertainty rises, while the opposite
holds when the pollutant is less harmful.
9
3.3 Asymmetric Case
Before determining the Nash equilibrium in the policy instrument choice game, we need to work
out the case where one government uses a standard and the rival selects a tax. We will only
solve for the case where the domestic government chooses a standard and the foreign selects a
tax, since the reverse problem is easily implied as the model is symmetric.
We solve again backwards, where in the last stage �rms maximize pro�ts. The domestic one
maximizes with respect to output while the foreign one with respect to output and abatement.
The reaction function of output of the domestic �rm is given by equation (6), while the �rst
order conditions for the foreign �rm are given by (11) and (12) correspondingly. Solving these
simultaneously we attain equilibrium levels of outputs as a function of a domestic emission
standard and a foreign pollution tax:
x =B � c+ � + 2gz + T
3 + 2gand X =
(B � c+ �)(1 + g)� (2 + g)T � gz3 + 2g
. (17)
Having (17) in their minds and without knowing the exact state of demand regulators in
each country select their optimal policy levels by maximizing expected welfare with respect to
the corresponding instrument:
z =(B � c+ T )2g(2 + g)
�2and T =
g[�g + (1 + g)(3 + g)k][(B � c)(1 + g)� gz]�1
. (18)
In equations (18) the domestic standard is a function of foreign tax and the foreign tax is a
function of the domestic standard. The domestic government�s reaction function is positively
sloped (i.e. z and T are strategic complements) whilst the foreign regulator�s reaction function
must be negatively sloped.8
Solving equations (17) and (18) together with (1) and (12) for the foreign �rm, we obtain the
Bayes Nash equilibrium levels of outputs, the domestic and foreign pollution and the foreign
8Hence, a foreign tax and a domestic standard must be strategic substitutes (i.e. �g + (1 + g)(3 + g)k > 0),otherwise, as we will see right after, we will not achieve an interior solution for the foreign emission tax inequilibrium. It can be checked out that stability of the system is satis�ed by the slopes of the reaction functionsgiven in (18).
10
tax: 8>>>>>>>>>><>>>>>>>>>>:
x�zt =(B�c)(3+2g)(g+k)�3
q + �3+2g
X�zt =
(B�c)(1+g)(g+k)(3+g)�4q + (1+g)�
3+2g
z�t =2(B�c)g(2+g)�3
q
Z�t =(B�c)g(2+g)2�4
q + (1+g)�3+2g
T �zT =(B�c)g[�g+(1+g)(3+g)k]�4
q
9>>>>>>>>>>=>>>>>>>>>>;; (19)
where
q = g2(3 + 2g)[9 + 2g(4 + g)] + g f54 + gf132 + g[120 + g(48 + 7g)]gg k
+(1 + g)2(3 + g)2(3 + 2g)k2;
�3 = g[3 + g(g + 3)] + (1 + g)2(3 + g)k and �4 = g(g + 3) + (1 + g)(3 + 2g)k:
A similar reasoning as the one presented above helps us to understand why domestic output
is less sensitive to unanticipated shifts in demand. We are ready to calculate domestic and
foreign welfare in the case where in the home country an emission standard is used, while in
the foreign one an emission tax is implemented. Since the model is symmetric, domestic and
foreign welfare in the case where the domestic government employs a tax and the foreign uses
a standard are the reverse of the ones above. Therefore:
Ew�zT = EW�tZ =
1
2(2 + g)
�(B � c)2(g + k)�23�2
q2� var(�)
(3 + 2g)2
�(20)
and EW �zT = Ew
�tZ =
1
2
�(B � c)2(g + k)(2 + g)�24�1
q2� (1 + g)
2(k � 2)(3 + 2g)2
var(�)
�. (21)
3.4 Form of Intervention
Having derived expected welfare for every possible combination of policy instruments in the two
countries we are now ready to derive the Nash equilibrium in the policy instrument choice game.
To do so we need to provide the optimal response of the domestic regulator for each possible
policy instrument chosen by the opponent. The optimal response of the foreign regulator for
each policy instrument choice of the rival will be identical due to symmetry. The following
lemma de�nes optimal response for the governments:
Lemma 1a)When uncertainty tends to zero choosing an emission standard dominates in terms
11
of welfare to an emission tax regardless of what the other regulator does. b) When uncertainty
is su¢ ciently high and the rival regulator chooses a standard, choosing a tax is optimal i¤
g(1+g)(3+g) < k < gf15+2g[12+g(6+g)]g
(1+g)2(3+g)2. c) When var(�) is high enough and the rival regulator
choose taxes then choosing taxes is optimal i¤ g(3+2g) < k < g(15+8g)
(3+2g)2. d) When uncertainty
is su¢ ciently high both governments choosing taxes Pareto dominates in terms of welfare to
choosing standards i¤ g(3+2g) < k <
g(3+2g)(3+g)2
.
Proof in the Appendix�
Using Lemma 1, we de�ne the Nash equilibrium both for the cases of certainty and uncer-
tainty in the following proposition:
Proposition 1a) If var(�)! 0, the Nash equilibrium suggests that both governments regu-
late pollution through emission standards. b) When demand variability is su¢ ciently high and
g(3+2g) < k <
gf15+2g[12+g(6+g)]g(1+g)2(3+g)2
then in the Nash equilibrium both governments choose an emis-
sion tax as a policy instrument. c) When variance is su¢ ciently high and g(15+8g)(3+2g)2
> k >
gf15+2g[12+g(6+g)]g(1+g)2(3+g)2
then we obtain two symmetric Nash equilibria in pure strategies where both
governments select either a standard or a tax and one in mixed strategies where each regulator
selects a standard and a tax with positive probabilities. d) If k > g(15+8g)(3+2g)2
then both governments
choose emission standards to deal with pollution problems irrespectively of var(�). Additionally,
when k � 2 , choosing standards is always an equilibrium strategy independently of the value of
g.
Proof in the Appendix�
Despite the fact that Lemma 1 and Proposition 1 might seem rather tedious, they do suggest
an important result which adds to the existing literature. Lahiri and Ono [12], Ulph [20], [22]
and Yanasee [24] suggest that quantity restrictions (as several modes of emission standards)
lead to welfare superior outcomes. However, what we claim through this proposition is that
this is not always true. In particular, when the demand in the third market is unknown to the
governments, under some parameter constellations, the use of taxes Pareto dominates in terms
of welfare, the use of standards. In addition to that we provide an exact prediction of what
policy instrument will be chosen in equilibrium given the values of k and g.
It can be shown that the derivatives of gf15+2g[12+g(6+g)]g(1+g)2(3+g)2
and g(3+2g) with respect to g are
increasing. Moreover the �rst derivative is greater than the second one and it results to a greater
12
range of values of k that support the use of pollution taxes in equilibrium as g increases. This is
attributed to the fact that when a standard is the policy instrument used, output (as g is high)
becomes less sensitive to any possible demand shocks since marginal cost of abatement and thus
marginal cost of production become steeper. As a result the expected bene�ts attributed to the
�rms sourcing from the high variability in demand are now lower. The same does not happen
when a tax is the policy instrument because marginal cost of production does not depend on
g. The expected bene�ts attributed to the �rm due to the use of a tax remain unchanged
irrespectively of the level of g. Moreover, the losses caused from the variability of pollution in
the damage function are now lower since greater g implies a steeper marginal abatement cost
and thus lower variability in pollution. Putting these results together, intuitively we understand
that taxes become attractive as a policy instrument. Unless, the injuriousness of the pollutant
is extremely high (i.e. k � 2 ) then emission taxes can emerge as an equilibrium strategy9.
In terms of pollution it can be shown that the common scenario suggests that emission
standards lead to a superior outcome (lower pollution) in comparison to taxes unless a signi�cant
negative shock a¤ects the demand (see Appendix). This result suggests that if our objective
is to achieve the lower level of pollution in equilibrium then standards should be preferred to
taxes.
4 Perfectly Trans-Boundary Pollution
In this section we investigate whether the previous analysis holds in the case where pollution
emitted by each �rm (e.g. CO2 emissions) harms equally the citizens in both exporting countries
( = 1). For simplicity, we assume that g = 1, as Ulph (1996a) and Nannerup (1998), among
others. We illustrate that the results still hold even if we have a global pollution problem. Then
we show that the assumption g = 1 is innocuous through the use of numerical simulations.
4.1 Emission Standards
The solution algorithm is the same as before. Again we solve backwards. Equilibrium output
of the domestic �rm as a function of standards coincides with the one given in equation (7)
since �rms�decisions remain una¤ected by the fact that the pollutant is global. Similarly the
9Note that Proposition 1 does not provide an exact prediction of what happens in equilibrium for intermediatevalues of the var(�). In such a case the use of taxes becomes less likely.
13
equilibrium output for the foreign �rm is obtained.
The reaction function of the domestic regulator is calculated in the same manner as in
section 3.1 and we attain:
z =18(B � c)� (9 + 64k)Z
37 + 64k. (22)
The foreign regulator�s reaction function is easily attained. If we solve simultaneously both reg-
ulators reaction functions and replace the solutions into equation (7) and the analogue equation
for the foreign �rm we obtain the subgame perfect equilibrium after setting g = 1:8><>: x�� = X�� = 8(B�c)(1+2k)23+64k + �
4
z�� = Z�� = 9(B�c)23+64k
9>=>; : (23)
If we compare equilibrium values of standards in the cases of local and global pollution,
we observe that standards are lower in the latter case. This is not peculiar since in the latter
case, marginal damage of pollution is higher for each unit of pollution as the damage function
contains the emissions from the rival �rm as well. As a result, production of both �rms in
equilibrium falls. Using the equilibrium levels of outputs and pollution given in (23) we can
derive expected welfare in the case where both governments use standards and pollution is
perfectly trans-boundary as following:
Ew��zZ =3(B � c)2[37 + 4k(37 + 64k)]
2(23 + 64k)2+3
32var(�). (24)
4.2 Emission Taxes
Since the nature of the pollutant does not a¤ect �rm�s decision in the �nal stage equilibrium
output as a function of taxes remains unchanged from equation (13). Moving backwards we
obtain the government reaction function if we maximize expected welfare with respect to the
emission tax:
t = �(B � c)(1� 8k) + (1 + 16k)T13 + 16k
. (25)
From (25), we observe that taxes are strategic substitutes in this case as well. In order
to obtain the subgame perfect equilibrium for this sub-case we need to solve the system of
equations (1) (12) (13) (25) and their analogues for the foreign �rm and governments (and
14
where possible set g = 1):
8>>>><>>>>:x��t = X��
T = (B�c)(5+8k)14+32k + �
3
z��t = Z��T = 3(B�c)7+16k +
�3
t�� = T �� = (B�c)(�1+8k)14+32k
9>>>>=>>>>; ; (26)
In order to obtain an interior solution for taxes we must restrict k such that k � 18 . It is
easy to see that in the case of transboundary pollution, the output of both �rms is reduced
in comparison to the case of local pollution. This is due to the fact that emission taxes in
both countries are now higher. As before, and in contrast to the case of local pollution, now
governments have an additional incentive to raise taxes as marginal damage becomes more
severe due to the addition of the foreign pollution to the local one. Note also that the pollution
in each country now equals the summation of emissions from both �rms. Given equilibrium
values for our variables in (26) we can derive expected welfare for this sub-case as well:
Ew��tT =3(B � c)2[13 + 32k(1 + 2k)]
8(7 + 16k)2+1
9(1� 2k)var(�). (27)
As in the corresponding case of local pollution var(�) can a¤ect either positively or negatively
the expected welfare. The di¤erence now is that it is su¢ cient to have k � 12 in order to make
the second term negative. Thit is because the damage from pollution is now higher for any
given value of k. As a result the welfare losses caused from the variability in pollution can o¤set
the bene�ts sourcing from higher expected pro�ts, even if the pollutant is not very harmful (low
k).
4.3 Asymmetric Case
We are now ready for the last step before determining the Nash equilibrium in the policy
instrument game. We assume that the domestic government selects an emission standard and
the foreign one selects a tax to control pollution. The reverse case is not presented as it can be
easily implied. The equilibrium outputs of the �rms as functions of the domestic standard and
foreign tax, remain unchanged from the ones given in (17), since the behavior of the �rms is
the same regardless of what kind of pollutant they emit.
In contrast, governments� decisions are altered when they face a global pollutant. The
15
reaction functions of the domestic and foreign regulator respectively are:
z =(B � c)(6� 8k) + (6 + 32k)T
13 + 16kand T =
2(B � c)(�1 + 8k) + (1 + 32k)z37 + 64k
. (28)
Subtracting equations (28) into equations (17), using the corresponding of (1) and (12) for the
foreign �rm and solving the system of equations simultaneously, we obtain the Bayes Nash
equilibrium, which is given by the following set of equations:
8>>>>>>>>>><>>>>>>>>>>:
x��zT =(B�c)(35+72k)
95+240k + �5
X��zT =
16(B�c)(2+3k)95+240k + �
5
z��zT =6(B�c)(7+4k)95+240k
Z��zT =12(B�c)(3�2k)
95+240k + 2�5
T ��zT =4(B�c)(�1+18k)
95+240k
9>>>>>>>>>>=>>>>>>>>>>;, (29)
The restriction on k that must be imposed in order to attain an interior solution for foreign
pollution and tax is 32 > k >118 . From the solutions given in (29) it is implied that output of
the home �rm is greater than the one of the rival. This directly results from the fact that a
change in the standard a¤ects more signi�cantly the production than a tax. We can observe
this through the corresponding reaction functions of the �rms when the government selects a
standard or sets a tax (given in equations (6) and (11) respectively). The opposite result of
course holds when the foreign government controls pollution through a quota and the domestic
one through an emission tax.
In order to derive expected welfare for the domestic and the foreign country we need to
substitute into (4) and the analogue welfare function for the foreign country the equilibrium
values given in (29) and take the expectations:
Ew��zT = EW��tZ =
1
50
�39(B � c)2[49 + 12k(15 + 32k)]
(19 + 48k)2� (4k � 3)var(�)
�(30)
and EW ��zT = Ew
��tZ =
2
25
�3(B � c)2(148 + 405k + 528k2)
(19 + 48k)2� (k � 2)var(�)
�. (31)
16
4.4 Form of Intervention
The procedure to determine the Nash equilibrium in the policy instrument choice game is similar
to the corresponding one of section 3.4. The following lemma de�nes the optimal response for
the governments, given the strategy of the rival government, and compares expected welfare
levels when standards and taxes are used, given the level of var(�):
Lemma 2 a) When uncertainty is close to zero then choosing an emission standard dominates
in terms of welfare to an emission tax, irrespective of what the other regulator does. b) When
uncertainty is su¢ ciently high and the rival regulator chooses a standard, choosing a tax is
optimal i¤ 118 < k <
5364 . c) When uncertainty is high enough and the rival regulator selects a
tax, choosing a tax is optimal for the domestic regulator i¤ 18 < k <
2364 . d) Both governments
choosing standards always Pareto dominates in terms of welfare to choosing taxes.
Proof in the Appendix�
Using Lemma 2 we obtain the following proposition:
Proposition 2a) If var(�)! 0 , the Nash equilibrium suggests that both governments regu-
late pollution through emission standards. b) When demand variability is su¢ ciently high and
18 < k <
2364 then both governments choose an emission tax in equilibrium. c) When variance is
su¢ ciently high and 5364 > k >
2364 then we obtain two asymmetric Nash equilibria in pure strate-
gies: one where the government selects a tax and the rival one chooses an emission standard,
and one in mixed strategies where each regulator selects a policy instrument with a, positive
probability. d) If 32 > k > 5364 then both governments choose emission standards to deal with
pollution problems irrespectively of var(�).
Proof in the Appendix�
In case we have a global pollution problem we observe that similar claims with the previous
section still hold. The mechanisms are very similar in both cases. If we set g = 1 there is a
range for k where taxes consist a Nash equilibrium. In comparison with the local pollution case,
what changes here is that the range of the values of k that makes taxes an equilibrium strategy
in both countries, is reduced. If we set g = 1 in Proposition 1 we observe that pollution taxes
consist a Nash equilibrium for 15 < k <5364 . This result should not surprise us, since in the latter
17
case marginal damage from pollution is more severe and the cost from the higher variability in
pollution associated with the use of taxes leads to greater losses. Therefore, taxes are preferred
only when k is su¢ ciently low.
The results presented above can be extended for cases where g 6= 1 as well. In order to
illustrate this claim, we run some numerical simulations summarized in Table 1. In panel A, we
give several values for g given speci�c values for k, B, c and var(�). In the second column the
di¤erence in the values of expected welfares is given, which determines whether it is optimal to
choose a standard or a tax when the rival regulator selects a tax. The next column presents
the di¤erences in expected welfares that determine when a standard or a tax is the optimal
instrument used, when the foreign regulator chooses a standard. We observe that as g increases
the di¤erences given in the second column decline and become negative for high values of g.
This means that when the rival regulator selects taxes it is optimal to choose taxes as well.
Similarly, as g increases the di¤erences in the third column are reduced. Even for low values of
g (i.e. g = 0:7) the sign becomes negative which implies that when pollution quotas are used
by the foreign regulator, emission taxes are the optimal response for the domestic government.
Hence for high values of g, low values of k and high variability in demand, taxes can be sustained
as an equilibrium strategy.
{Table 1 About Here}
In panel B we simulate the model for di¤erent values of var(�), given k, g, B and c. The
�rst column in panel B gives several levels of variance in an increasing order and the next two
columns are the same as columns 2 and 3 of panel A. As expected, we observe that taxes are a
dominant strategy when demand volatility becomes high. However, for medium values of var(�)
we can attain taxes as a Nash equilibrium strategy for one of the two governments.
5 Uncertain Cost of Abatement and Damage
In this section we abandon demand uncertainty (� = 0) and introduce new modes of uncer-
tainty in the cost of pollution control and damage caused from pollution. Therefore, u and
� are now assumed to be stochastic variables entering abatement cost and damage functions,
respectively. Our aim is to derive the optimal policy instrument chosen in equilibrium under the
new conditions. Weitzman [23], Fishelson [7], Adar and Gri¢ n [1] and Stavins [18] examined
18
the conditions under which standards are superior to taxes and the reverse. Yet, their analysis
is based on the decision of a single regulator that uses environmental policy to deal only with
pollution. We enrich this analysis by adding a foreign regulator and by allowing governments
to use environmental policies for trade purposes as well as pollution control. In order to check
out if the results hold for this case we need to proceed as in the previous two sections. In order
to simplify the analysis we assume that the pollutant is local ( = 0) and that g = 1.
5.1 Emission Standards
Once more we solve via backwards induction. Firms��rst order conditions given from pro�t
maximization with respect to output and equilibrium levels of outputs as functions of standards
are the same with the one given in section 3.1, if we set � = �u and g = 1. Given that, both
governments set optimal pollution quotas such that they maximize their expected welfare levels.
The reaction function of the domestic government implied by expected welfare maximization
with respect to the standard, is given by equation (8). Hence, if we solve equations (7), (8)
and the corresponding ones for the foreign �rm and government we obtain the Bayes Nash
equilibrium which coincides with the one given in (9) after setting � = �u and g = 1.
Substituting the equilibrium levels of outputs and standards into the welfare function (4)
and taking the expectation, we obtain expected domestic welfare which equals the foreign one.
Expected welfare equals the one given in equation (10) if we set g = 1 and replace var(�) with
var(u), which represents the variance of the marginal cost of abatement function. Thus,
Ew���zZ =3
32
�16(B � c)2(1 + k)(37 + 64k)
(23 + 32k)2+ var(u)
�. (32)
From (32) we perceive that expected welfare depends positively on the variance of marginal
cost of abatement since abatement cost is convex. This re�ects the fact that �rms are better
informed about their own true costs. Another important conclusion taken out from equation
(32) is that expected welfare does not depend on the variance of marginal damage (var(�)).
This means that the existence of variability in the damage function has no e¤ect on expected
welfare. This result is in line with Fishelson [7] and Adar and Gri¢ n [1]. However, as Stavins
[18] pointed out, this is true only in the case where stochastic variables � and u are uncorrelated.
19
5.2 Emission Taxes
Solving for the last period subgame between �rms, we maximize pro�ts with respect to output
and abatement. The reaction function of output remains unchanged from the one given in (11
) after setting � = 0. Therefore, the output decisions for the �rms are not a¤ected by the level
of the shock, u, in abatement cost functions. However, the �rst order condition with respect to
abatement is altered and it is given by the following equation:
a =t� ug. (33)
As the �rst order condition of the �rm with respect to production does not depend on
the level of abatement, the equilibrium level of output as a function of domestic and foreign
taxes remains the same as in equation (13) after setting � = 0. Given this, governments
maximize their expected welfare with respect to their own tax. Again the reaction function of
the government remains the same as in the case of demand uncertainty and is given by equation
(14). Hence, setting � = 0 and solving equations (1) (33) (13) (14) simultaneously together
with the corresponding foreign ones we get the new Bayes Nash equilibrium:8>>>><>>>>:x���t = X���
T = x�t (g = 1; � = 0) = X�t (g = 1; � = 0)
z���t = Z���T = z�t (g = 1; � = 0) + u = Z�t (g = 1; � = 0) + u
t��� = T ��� = t�(g = 1; � = 0) = T �(g = 1; � = 0)
9>>>>=>>>>; ; (34)
where the parenthesis attached to the equilibrium levels of section 3.2 indicate the values of �
and g at which the equilibrium levels of the variables are evaluated.
Replacing these equilibrium levels in the welfare function given in (4) and taking the expec-
tation, we obtain the expected welfare when taxes are used:
Ew���tT =3(B � c)2(1 + k)(13 + 25k)
8(7 + 10k)2� 12(k � 1)var(u). (35)
Note that when k > 1, in other words the marginal damage coe¢ cient is greater than the
coe¢ cient of marginal cost of abatement, the variability in abatement cost of the �rm has a
negative e¤ect on welfare. Otherwise, the opposite holds. The intuition behind this is the
following. Expected pro�ts depend positively on var(u) as �rms are better informed about
20
their own true costs. In contrast to that, since governments are misinformed about �rms�true
costs, they are also misinformed about the actual level of pollution and thus the damage from
pollution depends negatively on var(u). Hence, if g (in our case g = 1) is greater than k, the
�rst e¤ect dominates the second and the overall result is positive.
5.3 Asymmetric Case
Following a similar procedure as in sections 3.3 and 4.3, it is easy to obtain the Bayes Nash
equilibrium for this sub-case as well:8><>: x���zT = x�zT (g = 1; � = 0)� 2u3+2g ; X
���zT = X�
zT (g = 1; � = 0) +2u3+2g
z���zT = z�zT (g = 1; � = 0); T���zT = T �zT (g = 1; � = 0)
9>=>; . (36)
Substituting the equilibrium levels of outputs, the domestic standard and the foreign tax in the
welfare function given in (4), as well as the analogous one for the foreign country, and taking the
expectations, we attain expected welfares in the home and foreign country for the case where
the domestic regulator picks a standard and the foreign regulator a tax. Expected welfares
for the reverse case are easily implied since the model assumes that everything is symmetric
between the two countries. Thus:
Ew���zT = EW ���tZ =
3(B � c)2(1 + k)(7 + 16k)2(13 + 25k)2[95 + k(361 + 320k)]2
+6
25var(u) (37)
and EW ���zT = Ew���tZ =
�6(B � c)2(1 + k)(2 + 5k)2(37 + 64k)
[95 + k(361 + 320k)]2� 9
50(4k � 3)var(u)
�. (38)
5.4 Form of Intervention
The following lemma provides the necessary conditions for the determination of the new Nash
equilibrium in the policy instrument choice game:
Lemma 3a) When var(u)! 0 choosing an emission standard dominates in terms of welfare
the choice of an emission tax regardless of what the other regulator does. b) When uncertainty
in abatement cost is su¢ ciently high and the rival regulator chooses a standard, choosing a tax
is optimal i¤ 18 < k < 119
192 . c) When uncertainty in cost of abatement is high enough and
the rival regulator chooses taxes, choosing a tax is optimal i¤ 15 < k < 13
25 . d) When var(u)
21
is su¢ ciently high, then both governments choosing standards, Pareto dominates in terms of
welfare, to choosing taxes i¤ k > 1316 .
Employing Lemma 3 we can derive the following proposition analogous to the previous two:
Proposition 3a) If var(u)! 0 the Nash equilibrium suggests that both governments regulate
pollution through emission standards. b) When uncertainty in marginal cost of abatement is
su¢ ciently high and 15 < k <
1325 , then in the Nash equilibrium both governments choose emission
taxes as a policy instrument. c) When uncertainty in marginal cost of abatement is su¢ ciently
high and 1325 < k <
119192 , then we obtain two asymmetric Nash equilibria in pure strategies where
one government selects a tax and the rival one chooses an emission standard, and one in mixed
strategies where each regulator selects a policy instrument assigning positive probabilities. d) If
k > 119192 then both governments in equilibrium choose an emission standard to deal with pollution
problems irrespective of var(u).
Proof in the Appendix�
Proposition 3 establishes the necessary and su¢ cient conditions for taxes to consist a Nash
equilibrium strategy. Similarly to the previous cases, we observe that taxes constitute an equilib-
rium strategy only when the marginal damage cost coe¢ cient is low. An important implication
resulting by Proposition 3 is that uncertainty about the damage function does not a¤ect the
equilibrium choice of the policy instrument. In other words, the fact that regulators are misin-
formed about the exact position of the damage function should not a¤ect their decisions both
about the choice of the policy instrument and the choice of the optimal level per se. This is
a result of the fact that uncertainty enters the damage function such that it leads to parallel
shifts in marginal damage. If regulators are uncertain about the slope of marginal damage, we
expect that this will not hold anymore. Another interesting inference of this proposition is that
the range of the values of k that leads to asymmetric Nash equilibria is now limited10.
So far we simpli�ed the analysis by assuming g = 1. However, the implication discussed
above holds more generally. To illustrate these we run numerical simulations, allowing a number
10It is important to note here that due to the assumption that uncertainties in the marginal cost of abatementand the marginal damage are uncorrelated, uncertainty about the damage function does not a¤ect the regulator.According to Stavins [18] this is not true when this assumption does not hold. Following Stavins�rationale wecould expect that when uncertainties are positively correlated, the use of standards should become more likely.If uncertainties are negatively correlated then taxes should be more favorable. What correlation of uncertaintiescalls forth is the addition of an extra bene�t in favor of the one or the other policy instrument.
22
of values for g and var(u). The results are presented in Table 2. The rows and columns in Table
2 are as in Table 1. In panel B instead of var(�) we simulate the model for di¤erent values of
var(u) given the values of g; k;B and c. The values of the variables used in the two simulations
given in panel A and B are given at the bottom of each panel.
{Table 2 About Here}
From the �rst numerical simulation, where we test how the model behaves for di¤erent
values of g, we observe that when g is low, standards should be the unique equilibrium strategy.
For intermediate values of g emission taxes consist a Nash equilibrium. As g reaches higher
levels, taxes become more attractive than standards. The results of the simulation for di¤erent
values of var(u) are presented in panel B. As expected, when the variability in the marginal
cost of abatement becomes high, then selecting a tax becomes the Nash equilibrium strategy
for both regulators.
6 Concluding Remarks
The aim of this paper was to de�ne the optimal choice of the environmental policy instrument
in an international duopoly model, where governments use their policies not only to deal with
pollution but to shift rents towards their own �rms. We have shown that when the regulators
know everything, then standards should be used in equilibrium in both countries. Nonetheless,
we have also shown that after introducing various modes of uncertainty in demand and in the
cost of abatement and damage functions, this is not always true.
Our results suggest that when uncertainty is close to zero then standards are still preferred
to taxes. As uncertainty increases this result becomes weaker. In particular, we have illustrated
that if the pollutant is not very harmful to the society and at the same time the parameter
that determines the marginal cost of abatement is not very low, then taxes consist a Nash
equilibrium. We also de�ned the cases where asymmetric equilibria are obtained, where one
regulator chooses a tax to regulate pollution and the rival one selects pollution standards.
Moreover, under some parameter constellations we have shown that taxes are superior in terms
of welfare to standards.
So far, the literature on strategic environmental policy (see Lahiri and Ono [12], Ulph [22]
23
and Yanasee [24]), predicts that when a country exports a polluting good, standards appear to
be superior, in terms of welfare, to taxes. Our paper provides an exact prediction of what will
happen in equilibrium and identi�es the optimal policy levels after the instrument is chosen. At
the same time, our paper can be viewed as an extension of the policy instrument choice literature
under uncertainty when environmental policies are used as second best policy instruments.
The paper exploits any possible trade o¤s between the quantity (standards) and price (taxes)
constraints, when uncertainty is introduced, in order to dispute the traditional view that in
international industry competition standards should be always preferred11.
Our results are relevant for the regulation of the climate change problem. Undoubtedly,
climate change is the greatest challenge our economies are facing. As moderate estimates
predict (Stern review [19]), if no action takes place immediately future damages in GDP will
range from 5%� 20% per year. Therefore, policy makers face the challenge of regulating CO2
emissions. According to our results, if future costs of abatement or/and demand conditions are
very uncertain, then governments should regulate pollution through the use of price instruments,
while the opposite holds if uncertainty is expected to be rather low. However, if taxes are used
instead of standards, future CO2 emissions will become uncertain and it can be shown that we
will obtain greater pollution than it would emerge in the case that standards would be selected.
Nonetheless, our proposals should be used with care in policy implementation as our model
neglects the possibility of using more than one environmental policy instruments simultaneously.
Moreover, our model does not take into account consumption in the exporting industries and
asymmetries among domestic and foreign functions. At the same time we restricted the analysis
concerning the way uncertainty is introduced and allow for risk neutral policy makers. These
provide stimulus for future research.
Appendix
Proof of Lemma 1:
a) De�ne �1 = 0:5g3(B � c)2(2 + g)2(g+ k)2�1 and �2 = 0:5g3(B � c)2(2 + g)2(g+ k)2�2. If11Quirion [16] in a closed economy model re-examined the question originally set by Weitzman [23] of prices
versus quantities under uncertainty, yet in a second best environment where public funds have a positive cost dueto pre-existing distortionary taxes. The author claims that in such an environment taxes admit an even greateradvantage over standards. In combination with our results it is implied that in second best environments underuncertainty taxes should be preferred to standards.
24
the foreign regulator chooses a standard we have:
limvar(�)!0
(Ew�zZ � Ew�tZ) =�1m2
g2f54 + 7g[12 + g(6 + g)]g
+gf108 + gf264 + g[238 + g(95 + 14g)]ggk
+2(1 + g)2(3 + g)2(3 + 2g)k2
fg2(3 + 2g)[9 + 2g(4 + g)]
+gf54 + gf132 + g[120 + g(48 + 7g)]ggk
+(1 + g)2(3 + g)2(3 + 2g)k2g2
> 0. (A1)
If the foreign regulator chooses a tax we have:
limvar(�)!0
(Ew�zT � Ew�tT ) =�2n2
g2f54 + g[84 + g(46 + 9g)]g
+g(3 + 2g)f36 + g[64 + g(38 + 7g)]gk
+2(1 + g)2(3 + g)2(3 + 2g)k2
fg2(3 + 2g)[9 + 2g(4 + g)]
+gf54 + gf132 + g[120 + g(48 + 7g)]ggk
+(1 + g)2(3 + g)2(3 + 2g)k2g2
> 0 (A2)
Using equations (A1) and (A2) we observe that when var(�) ! 0 standards are a dominant
strategy Q.E.D.
b)It can be shown that:
Ew�zZ � Ew�tZ = limvar(�)!0
(Ew�zZ � Ew�tZ) +
�gf15 + 2g[12 + g(6 + g)]g
+(1 + g)2(3 + g)2k
2(3 + g)2(3 + 2g)2var(�): (A3)
In order a tax to be preferred to a standard when the rival chooses a standard the right hand
side of (A3) should be negative. If the second term is negative and var(�) is su¢ ciently high,
the negative e¤ect o¤sets the positive one implied by (A1). For this to be true the numerator of
the ratio should always be negative. If we solve the inequality with respect to k we obtain that
k < gf15+2g[12+g(6+g)]g(1+g)2(3+g)2
. If this is the case and var(�)> � limvar(�)!0(Ew�zZ�Ew�tZ)2(3+g)2(3+2g)2
�gf15+2g[12+g(6+g)]g+(1+g)2(3+g)2k then
(A3) has a negative sign. Finally, we need to add the necessary condition for the existence of
an interior solution which is k > g(1+g)(3+g)Q.E.D.
25
c) It can be shown that:
Ew�zT � Ew�tT = limvar(�)!0
(Ew�zT � Ew�tT ) +�g(15 + 8g) + (3 + 2g)2k
18(3 + 2g)2var(�): (A4)
In order a tax to be preferred to a standard when the rival chooses a tax the right hand side
of (A4) should be negative. If the second term is negative and var(�) is su¢ ciently high the
negative e¤ect o¤sets the positive one implied by (A2). For this to be true the numerator of
the ratio should have a negative sign. If we solve the inequality with respect to k we obtain
that k < g(15+8g)(3+2g)2
. If this is the case and var(�)> � limvar(�)!0(Ew�zT�Ew�tT )18(3+2g)2
�g(15+8g)+(3+2g)2k then (A4) has
a negative sign. Finally, we need to add the necessary condition for the existence of an interior
solution which is k > g3+2g Q.E.D.
d)It can be shown that:
Ew�zZ � Ew�tT = limvar(�)!0
(Ew�zZ � Ew�tT ) +�g(3 + 2g) + (3 + g)2k
18(3 + g)2var(�): (A5)
In order expected welfare when taxes are implemented (by both regulators) to be superior to
the corresponding one when standards are used (by both governments) the right hand side
of (A5) should be negative. If the second term is negative and var(�) is su¢ ciently high so
the negative e¤ect o¤sets the positive one implied by the �rst term in the right hand side of
(A5). For this to be true the numerator of the ratio should be always negative. If we solve
the inequality with respect to k we obtain that k < g(3+2g)(3+g)2
. If in this restriction we add that
var(�)> � limvar(�)!0(Ew�zZ�Ew�tT )18(3+2g)2
�g(3+2g)+(3+g)2k then (A5) has a negative sign. Additionally we need to
inset the necessary condition for the existence of an interior solution which is k > g3+2g Q.E.D.
Proof of Proposition 1:
a) From part a) of Lemma 1 we get that when var(�)! 0 standards are a dominant strategy
for both governments. Hence, when var(�)! 0 the unique Nash equilibrium is the one where
both governments select standards Q.E.D.
b)If we compare the upper limits of the restrictions for k in b) and c) of Lemma 1 we obtain
g(15+8g)(3+2g)2
� gf15+2g[12+g(6+g)]g(1+g)2(3+g)2
= g2(2+g)[18+g(24+7g)](1+g)2(3+g)2(3+2g)2
> 0. If we compare the lower limits of the
restrictions for k in b) and c) of Lemma 1 we obtain g(3+2g)(3+g)2
� g(1+g)(3+g) =
g2(2+g)(1+g)2(3+g)2(3+2g)2
> 0.
We further assume that the restrictions given about the var(�) satisfy both inequalities given in
26
b) and c) of the previous Lemma. Given these, when g(3+2g) < k <
gf15+2g[12+g(6+g)]g(1+g)2(3+g)2
, choosing
taxes for the domestic government will be a dominant strategy. Since the model is symmetric
the same holds for the foreign regulator as well. This is su¢ cient in order to achieve emission
taxes as a Nash equilibrium Q.E.D.
c) Using b) and c) of Lemma 1, g(15+8g)(3+2g)2
> gf15+2g[12+g(6+g)]g(1+g)2(3+g)2
then if g(15+8g)(3+2g)2
> k >
gf15+2g[12+g(6+g)]g(1+g)2(3+g)2
it follows that the equilibrium strategy for the domestic government is deter-
mined by the following: maxfEw�zZ ; E�tZgand maxfEw�zT ; E�tT g. Given the above range for k
and the restrictions for var(�) given in b) and c) of Lemma 1 we observe that when the foreign
government selects a standard then it is optimal for the domestic government to pool to the
same strategy (i.e.Ew�zZ > E�tZ ). If the foreign regulator chooses a tax then the domestic
government selects a tax as well (i.e.Ew�zT < E�tT ). Hence, the equilibrium strategy of each
regulator is conditional to the strategy of the rival. Given these we obtain two symmetric Nash
equilibria in pure strategies where both governments select either standards or taxes. More-
over, we attain a Nash equilibrium in mixed strategies where each government selects its policy
instrument with a positive probability Q.E.D.
d)If k > g(15+8g)(3+2g)2
> gf15+2g[12+g(6+g)]g(1+g)2(3+g)2
then using b), c) from Lemma 1 we obtain that
choosing standards is a dominant strategy and hence the unique Nash equilibrium involves
both governments using standards.
If we di¤erentiate the threshold level of k, g(15+8g)(3+2g)2
, with respect to g we getd�g(15+8g)
(3+2g)2
�dg =
9(5+2g)(3+2g)3
. This implies that the threshold level is continuously increasing in the level of g. If we
take the limit of the threshold level as g tends to in�nity we obtain limg!19(5+2g)(3+2g)3
= 2 Q.E.D.
Proof of Superiority of Standards in Terms of Pollution:
In order to provide a comparison of standards and taxes in terms of pollution we need to
take the di¤erence of pollution equilibrium levels when standards and taxes are used in both
countries:
z� � z�t = �(B � c)2g2(2 + g)(3 + g)(g + k)
mn� �3: (A6)
From (A6) we observe that pollution is greater in equilibrium when taxes are used instead of
standards unless a signi�cant negative shock occurs in demand Q.E.D.
Proof of Lemma 2:
27
a)When the foreign regulator chooses a standard we have:
limvar(�)!0
(Ew��zZ � Ew��tZ) =9(B � c)2(1 + 8k)2(37 + 64k)(187 + 496k)
50(19 + 48k)2(23 + 64k)2> 0. (A7)
If the foreign regulator chooses a tax we have:
limvar(�)!0
(Ew��zT � Ew��tT ) =9(B � c)2(1 + 8k)2(13 + 16k)(193 + 496k)
200(7 + 16k)2(19 + 48k)2> 0. (A8)
Using equations (A7) and (A8) we observe that when var(�)! 0 standards are a dominant
strategy Q.E.D.
b) It can be shown that:
Ew��zZ � Ew��tZ = limvar(�)!0
(Ew��zZ � Ew��tZ) +(64k � 53)
800var(�): (A9)
As in part b) of Lemma 1 in order a tax to be preferred to a standard when the rival chooses
a standard the right hand side of (A9) should be negative. If the second term is negative and
var(�) is su¢ ciently high the negative e¤ect o¤sets the positive one implied by (A7). For this
to be true the numerator of the ratio should be always negative. If we solve the inequality with
respect to k we obtain that k < 5364 . If this is the case and var(�)> �
limvar(�)!0(Ew��zZ�Ew��tZ)800
(64k�53)
then (A9) has a negative sign. Finally, we need to add the necessary condition for the existence
of an interior solution which is k > 118 Q.E.D.
c) It can be shown that:
Ew��zT � Ew��tT = limvar(�)!0
(Ew��zT � Ew��tT ) +(64k � 23)
450var(�): (A10)
In order a tax to be preferred to a standard when the rival chooses a tax the right hand side
of (A10) should be negative. If the second term is negative and var(�) is su¢ ciently high the
negative e¤ect o¤sets the positive one implied by (A8). For this to be true the numerator of
the ratio should be always negative. If we solve the inequality with respect to k we obtain that
k < 2364 . If this is the case and var(�)> � limvar(�)!0(Ew
��zT�Ew��tT )450
(64k�23) then (A10) has a negative
sign. Finally, we need to add the necessary condition for the existence of an interior solution
which is k > 18 Q.E.D.
28
d) It can be shown that:
Ew��zZ � Ew��tT = limvar(�)!0
(Ew��zZ � Ew��tT ) +(64k � 5)288
var(�): (A11)
The right hand side of (A11) can become negative only when k < 564 and var(�) is su¢ ciently
high. However, after inserting the necessary condition for the existence of an interior solution
which is k > 18 we observe that this can never be the case. Hence standards are always welfare
superior to taxes when we have trans-boundary pollution Q.E.D.
Proof of Proposition 2:
a) From part a) of Lemma 2 we obtain that when var(�)! 0 then using a standard is a
dominant strategy for both governments. Hence, when var(�)! 0, the unique Nash equilibrium
is the one where both governments select standards Q.E.D.
b) If we compare the upper limits of the restrictions for k in b) and c) of Lemma 2 we observe
that the upper limit in b) is greater than the one in c) . If we compare the lower limits of the
restrictions for k in b) and c) of Lemma 2 we perceive that the lower limit in b) is lower than
the one in c) . Given the restrictions about var(�) in b) and c) of Lemma 2, when 18 < k <
2364
choosing a tax will be a dominant strategy for the domestic government. Since the model is
symmetric the same holds for the foreign regulator as well. This is su¢ cient in order to achieve
emission taxes as an equilibrium strategy Q.E.D.
c) Using the results given in b) and c) of Lemma 2 and if 5364 > k >2364 it follows that the equi-
librium strategy for the domestic government is determined by the following: maxfEw��zZ ; E��tZg
and maxfEw��zT ; E��tT g. Given the above range for k and that variance of � satisfy both con-
straints given in b) and c) of Lemma 2 we observe that when the foreign government selects
a standard then it is optimal for the domestic government to select a tax (i.e. Ew��zZ < E��tZ).
If the foreign regulator chooses a tax then the domestic government selects a standard (i.e.
Ew��zT > E��tT ). Hence, the equilibrium strategy of each regulator is conditional to the strategy
of the rival. As each government separates its strategy from the rival one we obtain two asym-
metric Nash equilibria in pure strategies, one where a government selects a standard and the
rival one chooses a tax and a second one where a government selects a tax and the rival one
a standard. Moreover, we attain a Nash equilibrium in mixed strategies where each govern-
ment selects its policy instrument randomly assigning a positive probability to each of the two
29
strategies Q.E.D.
d) If k > 5364 then using b) and c) from Lemma 2 choosing a standard is a dominant strategy
and hence the unique Nash equilibrium involves both governments using standards. In order to
guarantee an interior solution to our problem we insert the restriction k < 32 Q.E.D.
Proof of Lemma 3:
a)When the foreign regulator chooses a standard we have:
limvar(u)!0
(Ew���zZ � Ew���tZ ) =9(B � c)2(1 + k)2(37 + 64k)[187 + k(719 + 640k)]
2(23 + 32k)2[95 + k(361 + 320k)]2> 0. (A12)
If the foreign regulator chooses a tax we have:
limvar(u)!0
(Ew���zT � Ew���tT ) =9(B � c)2(1 + k)2(13 + 25k)(193 + 725k + 640k2)
8(7 + 10k)2[95 + k(361 + 320k)]2> 0. (A13)
b) It can be shown that:
Ew���zZ � Ew���tZ = limvar(u)!0
(Ew���zZ � Ew���tZ ) +3(192k � 119)
800var(u): (A14)
As in part b) of Lemma 1 and 2 in order a tax to be preferred to a standard when the rival
chooses a standard the right hand side of (A14) should be negative. If the second term is
negative and var(u) is su¢ ciently high the negative e¤ect o¤sets the positive one implied by
(A12). For this to be true the numerator of the second term in the right hand side of (A14)
should be always negative . If we solve the inequality with respect to k we obtain that k < 119192 .
If this is the case and var(u)> � limvar(u)!0(Ew���zZ �Ew���tZ )800
3(192k�119) then (A14) has a negative sign.
Finally, we need to add the necessary condition for the existence of an interior solution which
is k > 18 Q.E.D.
c) It can be shown that:
Ew���zT � Ew���tT = limvar(u)!0
(Ew���zT � Ew���tT ) +(25k � 13)
50var(u): (A15)
In order a tax to be preferred to a standard when the rival chooses a tax the right hand side
of (A15) should be negative. If the second term is negative and var(u) is su¢ ciently high the
negative e¤ect o¤sets the positive one implied by (A13). For this to be true we should have
30
k < 1325 . If this is the case and var(u)> � limvar(u)!0(Ew
���zT �Ew���tT )50
(25k�13) then (A15) has a negative
sign. Finally, we need to add the necessary condition for the existence of an interior solution
which is k > 15 Q.E.D.
d) It can be shown that:
Ew���zZ � Ew���tT = limvar(u)!0
(Ew���zZ � Ew���tT ) +(16k � 13)
32var(u): (A16)
The �rst term in the right hand side of (A16) is always positive. Therefore when k > 1316 the
overall sign will be always positive irrespectively of the level of var(u) Q.E.D.
Proof of Proposition 3:
a) From part a) of Lemma 3 we obtain that when var(u)! 0 then using a standard is a
dominant strategy for both governments. Hence, when var(u)! 0, the unique Nash equilibrium
is the one where both governments select standards Q.E.D.
b) If we compare the upper limits of the restrictions for k in b) and c) of Lemma 3 we observe
that the upper limit in b) is greater than the one in c) . If we compare the lower limits of the
restrictions for k in b) and c) of Lemma 3 we perceive that the lower limit in b) is lower than
the one in c) . Given the restrictions about var(u) in b) and c) of Lemma 3, when 15 < k <
1325
choosing a tax will be a dominant strategy for the domestic government. Since the model is
symmetric the same holds for the foreign regulator as well. This is su¢ cient in order to achieve
emission taxes as a Nash equilibrium Q.E.D.
c) Using the results given in b) and c) of Lemma 3 and if 119192 > k >1324 it follows that the equi-
librium strategy for the domestic government is determined by the following: maxfEw���zZ ; E���tZ g
and maxfEw���zT ; E���tT g. Given the above range for k and that variance of u satis�es both con-
straints given in b) and c) of Lemma 3 we observe that when the foreign government selects a
standard then it is optimal for the domestic government to select a tax (i.e. Ew���zZ < E���tZ ).
If the foreign regulator chooses a tax then the domestic government selects a standard (i.e.
Ew���zT > E���tT ). Hence, the equilibrium strategy of each regulator is conditional to the strategy
of the rival. As each government separates its strategy from the rival one we obtain two asym-
metric Nash equilibria in pure strategies, one where a government selects a standard and the
rival one chooses a tax and a second one where a government selects a tax and the rival one a
standard. Moreover, we attain a Nash equilibrium in mixed strategies where each government
31
selects its policy instrument with a positive probability Q.E.D.
d) If k > 119192 then using b) and c) from Lemma 3 choosing a standard is a dominant strategy
and hence the unique Nash equilibrium involves both governments selecting standards Q.E.D.
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.
Table 1
Panel A Panel B
g Ew��zT � Ew��tT Ew��zZ � Ew��tZ var(�) Ew��zT � Ew��tT Ew��zZ � Ew��tZ0:1 3:65101 0:96614 0:01 0:65849 0:65719
0:7 1:66216 �0:79937 4:41 0:59005 0:45369
1:3 0:82392 �1:88839 16:81 0:39716 �0:11980
1:9 0:36902 �2:69144 37:21 0:07983 �1:06330
2:5 0:08560 �3:32362 65:61 �0:36195 �2:37680
3:1 �0:10713 �3:83905 102:01 �0:92817 �4:06030
3:7 �0:246531 �4:26914 146:41 �1:61884 �6:11380
k = 0:4; B = 10; c = 4;var(�)= 64 k = 0:25; B = 10; c = 4; g = 1
Table2
Panel A Panel B
g Ew���zT � Ew���tT Ew���zZ � Ew���tZ var(u) Ew���zT � Ew���tT Ew���zZ � Ew���tZ0:4 3:01381 2:82625 0 0:09136 0:09243
0:9 �0:42240: �1:30820 1 0:08136 0:00619
1:4 �0:31366 �1:11009 4 0:05136 �0:25256
1:9 �0:12189 �0:77996 9 0:00114 �0:06838
2:4 0:02721 �0:51205 16 �0:06864 �1:28756
2:9 0:13936 �0:30518 25 �0:15864 �2:06381
3:4 0:22564 �0:14400 36 �0:26864 �3:01256
k = 0:4; B = 10; c = 4;var(u)= 9 k = 0:5; B = 10; c = 4; g = 1