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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.


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.


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


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.


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(�).


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.


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)


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).


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)


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.


Ew��zT � Ew��tT = limvar(�)!0

(Ew��zT � Ew��tT ) +(64k � 23)

450var(�): (A10)


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.


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)


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.


Ew��zT � Ew��tT = limvar(u)!0

(Ew��zT � Ew��tT ) +(25k � 13)

50var(u): (A15)


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.


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.

References

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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

FirstGovernmentsselect policyinstrument

SecondlyGovernmentsselect policylevels

ThenUncertainty isrevealed tothe firms

FinallyFirms selectquantities

Figure 1

c

p

x

A

B

MRx

TMCs

TMCt

MR1x

MR2x

EEs

2

Et1

Es1

Et2

Figure 2

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