cleaner technologies and the stability of international environmental agreements

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1. Introduction

This paper examines the stability of international agreements on emission reductions of a

transboundary pollutant when the countries involved implement "cleaner" technologies that

reduce emission-output ratios. This topic gains importance in light of the large expenditures

that have been made in recent years for creating cleaner technologies.1

We focus on those cases of transboundary pollution that a¤ect a small number of countries

or blocks of countries. This is an important scenario to consider since certain types of

transboundary pollutants, in reality, damage a few neighbouring countries/regions. For

example, consider the pollution of lakes, such as the Black Sea or the Great Lakes, that

are surrounded by a few countries each with their independent emission strategies. Another

scenario where this setting gains relevance lies within the context of climate change. In

recent international negotiations over climate change (for example, at the UNFCCC COP

Meetings at Copenhagen, 2009, and Cancun, 2010), only a small number of large countries

or blocks of countries (e.g. US, China, and EU) dominated discussions and played a decisive

role in determining the outcomes.

This a¤ects our choice of stability criteria for the possible coalitions that may arise

amongst them, as follows. Much of the IEA literature uses the internal and external stabil-

ity criteria as described by d�Aspremont et al (1983) and applied to the context of IEAs by

Barrett (1994), Carraro and Siniscalco (1993), Diamantoudi and Sartzetakis (2006), Finus

(2003), Karp and Simon (2013) and others. These criteria rely on the restrictive assumption

that if one country defects from a given coalition, the rest of the members of the coalition

continue to participate in the IEA. However, if only small coalitions form, the assumption

that a player does not take into account the impact of its decision to leave a coalition on

the decision of the other coalition members to remain in the coalition is more di¢ cult to

justify. Therefore, we use an alternative set of stability criteria referred to in the literature1In 2002, the US announced a policy to reduce the ratio of emissions to economic output by 18% over the

next 10 years. Since then other major polluters such as China and India have also committed to emissionper output targets. In 2007 alone, new investment in clean energy worldwide rose by 60% above the 2006level globally (UNEP Report, 2008).

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as "farsightedness" under which when a country contemplates leaving an IEA, it takes into

account the repercussions on other countries�adhesion to the IEA. See Diamantoudi and

Sartzetakis (2002), Eyckmans (2003) and de Zeeuw (2008) for applications of the farsight-

edness concept in IEAs.2 When a small group of large players are involved, it seems that

countries are indeed farsighted when making their decisions.3

We show that if countries use the farsighted criterion for deciding whether to defect from a

coalition, the stable size of a coalition and global welfare evaluated under the stable coalition

size have an �irregular�, or non-monotonic, relationship with the emission per output ratio. In

the case of three countries, the grand coalition may be destabilized by the implementation of

cleaner technologies, ultimately resulting in higher global emissions and lower global welfare.

In the case of more than three countries, we show through a numerical example, that a

marginal decrease in the emission per output ratio may result in a downward or an upward

jump in the largest stable size of a coalition and, therefore, may result in a downward or an

upward jump in global welfare.

We �rst derive this result within a static model with three countries, and then generalize

it by allowing for more countries. We then extend our analysis to a dynamic model which

allows the pollutant to accumulate over time. This is an important step in the analysis

since several transboundary pollutants are also accumulative such as greenhouse gases.4

Also, it is an interesting extension from a theoretical perspective since Benchekroun and

Ray Chaudhuri (2013) and van der Ploeg and de Zeeuw (1992) show that countries� free

riding incentives are exacerbated when their emission decisions are functions of the stock of

pollution. Benchekroun and Ray Chaudhuri (2013), using a similar setting to this paper,

show that in the dynamic case it is more likely that countries may respond to a cleaner

2See Ray and Vohra (2001) and Chwe (1994) for a detailed discussion of farsightedness in coalitionformation games.

3For example, when the US refused to ratify the Kyoto protocol, its stated reason was that othercountries (especially the large developing countries like China and India) would not behave cooperativelyif the US committed to reducing its emissions. E¤ectively, the US chose not to join the IEA becauseit anticipated other countries to defect if it joined. See the Encyclopedea of Earth for further details:http://www.eoearth.org/article/Kyoto_Protocol_and_the_United_States)

4See Jørgensen et al (2010) for a survey of dynamic game models used to analyze environmental problems.

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technology by increasing their equilibrium emissions to an extent such that the stock of

pollution increases and welfare decreases. We �nd that the higher the stock of pollution, the

more likely that the implementation of clean technologies destabilizes the grand coalition of

three countries, thus reducing (and possibly reversing) the bene�ts of implementing cleaner

technologies.5

We have opted to consider exogenously given levels of ratios of emissions to output to

focus on the existence and implementation of a new technology only and abstract from

the game of investment in technologies. While we present the case of exogenous changes in

technology, the game we consider can be viewed as a second stage of a two stage game, similar

to Athanassoglou and Xepapadeas (2012), where in an initial phase countries invest in their

technologies. The cleaner technology can be interpreted as an exogenous perturbation of the

equilibrium technology choice from the initial stage in investment in technologies. Such a

change in technology choice can be, for example, the result of an international agreement to

increase investment in technologies. We abstract away from explicitly modeling the cost of

investment in order to highlight that implementing a technology may have counterintuitive

e¤ects in our setting, even when the new technology is readily available and free.6

Section 2 presents the basic model. Section 3 presents the analysis for the static sce-

nario where pollution damage depends on the current emission levels. Section 4 presents

the analysis for the dynamic scenario where the pollution damage depends on the stock of

pollution. Section 5 presents the concluding remarks.

2. The Model

Consider n countries indexed by i = 1; :::; n:We consider a world with a single good, and we

denote by qi the quantity of that good produced by country i. Production generates pollution

5Rubio and Casino (2005) and Rubio and Ulph (2007) also study IEAs within a dynamic context. Theydo not study the e¤ect of cleaner technologies or use farsighted stability criteria. See Calvo and Rubio (2012)for a survey of dynamic models of IEAs.

6We further note that there exist other studies which also allow for exogenous technology changes withinthe context of a dynamic model of global warming, such as Dutta and Radner (2006).

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emissions. Let "i denote country i�s emissions of pollution. We have:

"i = �qi (1)

where � is an exogenous parameter that represents each country�s ratio of emissions to

output. The implementation of a cleaner technology is represented by a fall in �:

The instantaneous net bene�ts of country i = 1; ::; n are given by

bi (qi) = Ui (qi)�Di (:) (2)

where Di (:) represents the pollution damage that is a by-product of the production process,

and

Ui (qi) = Aqi �B

2q2i ; A > 0

For simplicity, the countries are assumed to be identical, although this may not be realistic

for certain types of pollution. This allows us to abstract away from issues like transfers

between countries within a coalition or a possible relationship between size and membership,

and to focus on isolating the e¤ect of implementing cleaner technologies in a benchmark

scenario.

We consider two scenarios. In the �rst scenario, the pollution damage faced by each

country, Di (:) ; is a function of total emissions of all the countries. This is the static version

of the model, as presented in the following section. We then analyze the scenario where

emissions are allowed to accumulate into a stock of pollution over time and the pollution

damage faced by each country, Di (:) ; is a function of this pollution stock. This is the

dynamic version of the model, as presented in Section 4.

3. The Static Model: The case of a �ow pollutant

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The pollution damage faced by country i is given by:

Di (�ni=1"i) =

s

2(�ni=1"i)

2; s > 0. (3)

We begin by studying the equilibria under non-cooperation and full cooperation. We

then analyze coalition formation in the following subsection.

The non-cooperative equilibrium

The objective of country i�s government is to choose a production strategy, qi (or, equiva-

lently, a pollution control strategy) that maximizes bi (qi) : The outputs of all other countries

are considered �xed in country i�s maximization problem. That is,

maxqiAqi �

B

2q2i �

s

2(�ni=1�qi)

2 (4)

The �rst order condition for country i is given by:

A�Bqi � s� (�ni=1�qi) = 0

where the term (A�Bqi) represents the marginal bene�t to country i of increasing pro-

duction, and the term s� (�ni=1�qi) represents the marginal cost to country i in terms of

additional pollution damage caused by increasing production.

The best response function of country i is given by:

qnci =A� s�

��nj 6=i�qj

��B + s�2

�By symmetry; we have

qnci =A

B + ns�2; i 2 f1; :::; ng

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The welfare of each country in the non-cooperative equilibrium is given by:

wnci =A2�B + 2ns�2 � n2s�2

�2�B + ns�2

�2 ; i 2 f1; :::; ng (5)

Full Cooperation

The objective of the planner is to assign the outputs q1; :::; qn that maximize the sum of the

countries�net bene�ts.

maxq1;:::;qn

nXi=1

bi (qi) = maxq1;:::;qn

nXi=1

�Aqi �

B

2q2i

�� ns

2(�ni=1�qi)

2

The �rst order condition with respect to country i0s output is given by:

A�Bqi � ns� (�ni=1�qi) = 0

where the term (A�Bqi) represents the marginal bene�t to country i of increasing its own

production, and the term ns� (�ni=1�qi) represents the marginal cost to all countries in terms

of additional pollution damage caused by increasing qi. The output of each country is given

by:

qci =A

B + n2s�2

The welfare of each country in the fully cooperative equilibrium is given by:

wci =A2

2�B + n2s�2

� ; i 2 f1; :::; ng (6)

3.1. Coalition Formation

Consider the scenario where the countries decide to form an international environmental

agreement. More speci�cally, suppose the countries in M � N sign an agreement while

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those in NnM each choose to act individually. We denote the size of coalition M by m

and the total output produced by the coalition by Qm = mqm, where qm is the output of

a representative signatory. Similarly, Qnm = (n � m)qnm is the total output produced by

the complement of the coalition with qnm being the output produced by a representative

non-signatory. The sum of the output of the signatory and non-signatory countries, that is

global output, is given by Q = Qm +Qnm:

We assume that the nonsignatories behave noncooperatively and the signatories maximize

the joint welfare of the members of the coalition. The coalition and the nonsignatories are

assumed to choose their emission strategies simultaneously.

The nonsignatories�maximization problem is given by:

maxqiAqi �

B

2q2i �

s

2�2��n�mi qi + �

mj qj�2; i 2 NnM; j 2M (7)

The signatories�maximization problem is given by:

maxqj�mj

�Aqj �

B

2q2j

��ms

2�2��n�mi qi + �

mj qj�2; i 2 NnM; j 2M

The �rst order condition associated with the nonsignatory i0s welfare maximization problem

is given by:

A�Bqi � s�2��n�mi qi + �

mj qj�= 0

where the term (A�Bqi) represents the marginal bene�t to country i of increasing its own

production, and the term s� (�ni=1�qi) represents the marginal cost to country i in terms of

additional pollution damage caused by increasing production. By symmetry, we have

A�Bqnm � s�2 ((n�m) qnm +mqm) = 0 (8)

The �rst order condition associated with signatory j�s welfare maximization problem is given

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by:

A�Bqj �ms�2��n�mi qi + �

mj qj�= 0

where the term (A�Bqj) represents the marginal bene�t to country j of increasing its own

production, and the termms�2��n�mi qi + �

mj qj�represents the marginal cost to all coalition

members in terms of additional pollution damage caused by increasing qj. By symmetry, we

have:

A�Bqm �ms�2 ((n�m) qnm +mqm) = 0 (9)

Solving (8) and (9) simultaneously, we get

q�m (m) =A�B + s�2 (1�m) (n�m)

�B�B + s�2 (�m+ n+m2)

�q�nm (m) =

A�B + s�2m (m� 1)

�B�B + s�2 (�m+ n+m2)

�We note that q�m > 0 i¤

� < ��

sB

s (m� 1) (n�m)

We also note that if m = 0; we have q�nm = qnci ; and if m = n; we have q�m = q

ci : Let the total

output be given by Q� � mq�m + (n�m) q�nm:

The welfare of each signatory country, at the equilibrium, is given by:

w�m (m) � Aq�m �B

2(q�m)

2 � s2�2 (Q�)2 (10)

The welfare of each non-signatory country, at the equilibrium, is given by:

w�nm (m) � Aq�nm �B

2(q�nm)

2 � s2�2 (Q�)2 (11)

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Global welfare under a coalition of size m; given a total of n countries is denoted by W �m;n:

W �m;n (m) = mw

�m + (n�m)w�nm

Much of the IEA literature uses the internal and external stability criteria as described by

d�Aspremont et al (1983) which assumes that if one country defects from a given coalition,

the rest of the members of the coalition continue to participate in the IEA. Let the stability

function be given by:

�i (m) � w�m (m)� w�nm (m� 1) : (12)

De�nition 1: Under myopic stability, a coalition of size m is said to be internally stable i¤

�i (m) > 0 and externally stable i¤ �i (m+ 1) < 0:

Under such stability criteria the general conclusion of much of the IEA literature is that

only small coalitions (of size 2 or 3) can be stable (see for example, Barrett 1994 and 2003,

and Diamantoudi and Sartzetakis 2006). Moreover, small coalitions achieve sizable gains in

welfare compared to non-cooperation only when the number of players is small.

However, for small coalitions and number of players, the assumption that a player does

not take into account the impact of its decision to leave a coalition on the decision of the other

coalition members to remain in the coalition is more di¢ cult to justify. Therefore, we use an

alternative set of stability criteria referred to in the literature as "farsightedness" under which

when a country considers leaving an IEA, it takes into account the implications on other

countries�adhesion to the IEA. We analyze the stability of coalitions using the farsighted

stability concept as used by Eyckmans (2003), Diamantoudi and Sartzetakis (2002), Osmani

and Tol (2009) and de Zeeuw (2008) in the context of environmental agreements.

Presenting the formal de�nition of farsighted stability and a farsighted stable set would

require introduction of heavy notation without adding signi�cant insights to our analysis.

Given that these formal de�nitions are readily available elsewhere, for the sake of brevity and

to facilitate the reading of the paper, we provide an informal presentation of the concepts

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of internally and externally farsighted stable coalitions through Example 1 below. For the

purposes of this paper, which is to illustrate the impact of a change in the emissions per

output ratio on farsighted stability, this approach is su¢ cient. The formal de�nition of

farsighted stable coalitions and farsighted stable sets in the context of IEAs can be found, for

example, in Diamantoudi and Sartzetakis (2002) (see De�nition 1).7 Note that in Example

1, we present the concept of farsighted stability at a given value of �: We then analyze the

e¤ects of varying � in the following subsections.

Example 1: Throughout this example, without loss of generality, we set A;B and s to 1. Let

N = 8 and � = 0:1: We note that � 2 [0; �max] where �max � minm�8 ��: It is straightforward

to show that �max =q

1(4�1)(8�4) = 0:288 68. The following table gives the payo¤s of insiders

and outsiders for each coalition size m.

Table I: Payo¤s of insiders and outsiders for m2f1;::;8g

1 2 3 4 5 6 7 8

wm 0:223 0:225 0:232 0:242 0:256 0:271 0:288 0:305

wnm 0:223 0:233 0:251 0:275 0:303 0:330 0:356

We �rst note that a coalition of size 2 is the only (myopic) internally and externally

stable coalition.

Second we note that outsiders of a coalition of size 2 would be better o¤ as insiders of a

coalition of size 4 and that outsiders of a coalition of size 4 would be better o¤ as insiders of

a coalition of size 7.

The coalition of size 4 is said to be internally farsighted stable because, if an insider leaves

the coalition of size 4, an insider of a coalition of size 3 will also have an incentive to leave

the coalition. For an insider of a coalition of size 4, leaving the coalition will trigger one other

member of the coalition of size 3 to leave. No insider of a coalition of size two will leave since

a coalition of size two is internally stable. Therefore when the insider of coalition of size 4

7See Breton and Keoula (2012) sections 4.1 and 4.2 for an application in the context of �sheries andNakanishi (2011) for an application in the context of international trade agreements.

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contemplates the possibility to leave the coalition it must compare its payo¤ as insider of a

size 4 coalition to its payo¤ as an outsider of size 2 coalition. Clearly, from the table above,

such a player would not want to leave the coalition of size 4. We say that 4 is internally

farsighted stable. Similarly a coalition of size 7 is also internally farsighted stable.

The coalition of size 4 is not externally farsighted stable because 3 outsiders would gain

by joining the coalition of size 4 (making it a coalition of size 7). This follows from the fact

that the payo¤ to each country as an insider of a size 7 coalition is greater than the payo¤

as an outsider of size 4 coalition. Similarly a coalition of size 2 is not externally farsighted

stable.8 The coalition of size 7 is both internally and externally farsighted stable and is called

a farsighted stable set (see Diamantoudi and Sartzetakis 2002, De�nition 1). In our example,

the coalition of size 7 is also the largest farsighted stable coalition.

Comparing the two stability concepts, that is, myopic versus farsighted, we have the

following. Under myopic stability, the unique stable coalition is that of size two. Under

farsighted stability, larger coalitions may be stable. This is in line with the �ndings of

other papers which have applied farsighted stability to study IEAs such as Diamantoudi and

Sartzetakis (2002) and de Zeeuw (2008).

3.2. The three country case

As illustrated by Example 1, the analysis to study the size of the largest stable coalition with

more than three countries is computationally intensive. In order to make progress, in this

subsection, we consider the case of three countries, for which we derive analytical results.

We then consider the case of more than three countries in the following subsection, for which

we illustrate our conclusions with a numerical example.

For the three country case, we proceed in two steps: �rst, we analyze the stability of the

two-country subcoalition and then we proceed to analyze the stability of the grand coalition

of three countries.8In the terminology of Chwe (1994) coalitions of 2, 4 and 7 players are consistent stable sets of outcomes

and the coalition of 7 players is the largest consistent stable set of outcomes.

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Two-country coalition

When �i (2) � 0 neither coalition member has an incentive to leave the coalition (assuming

the third country does not join the coalition). Moreover, when �i (3) < 0; the third country

does not have an incentive to join the coalition (assuming the other two coalition members

remain in the coalition).

Proposition 1: There exists ��ms �q�

211

p3 + 1

11

�Bs' 0:637

qBssuch that the coalition of

size two is farsighted stable if � � ��ms:

Proposition 1 follows from the fact that � (3) = �72A2Bs3 �6

(5s�2+B)2(9s�2+B)

< 0: Also, it can

be shown that

� (2) = �92

A2

Bs2�4�11s�2 +B

�2p3� 1

��3s�2 +B

�2 �5s�2 +B

�2 s�2 �

�1 + 2

p3�

11B

!

and, therefore, � (2) � 0 i¤ s�2 ��211

p3 + 1

11

�B ' 0:405B.

We note that ��ms, is increasing in B and decreasing in s. Since B represents the degree

of decreasing marginal bene�t of emissions, an increase in B e¤ectively decreases the oppor-

tunity cost of remaining in the coalition. Therefore, the range of � for which the two-country

coalition is stable increases with an increase in B. Since s represents the degree of increasing

marginal damage from pollution, an increase in s e¤ectively increases the free-riding incen-

tive in the case where all countries face identical s: Therefore, the range of � for which the

two-country coalition is stable decreases with an increase in s.9

Proposition 1 implies that we have internal stability for s�2 <�211

p3 + 1

11

�B; that is, for

values of � that are su¢ ciently low. Thus, implementing a cleaner technology may stabilize

the IEA of size 2.

Figure 1 shows � (2) as a function of � for A = B = s = 1:

9We thank an anonymous referee for this interpretation.

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10.750.50.2500

0.02

0.04

0.06

Ф (2)

θ

θ

Figure 1: Stability of 2country IEA as a function of θ

θms

Grand coalitionfarsighted stable

Grand coalitionfarsighted unstable

In this simple model, the coalition structure is dependent on the level of the technology

parameter �; and we have that for � � ��ms �q�

211

p3 + 1

11

�Bsthe coalition of size 2 is

stable.

We have @q�m@�

= �2�snmA(B�ms�2+ns�2+m2s�2)

2 < 0 and @q�nm@�

= �2�snA(B�ms�2+ns�2+m2s�2)

2 < 0: That

is, as � decreases, q�m and q�nm both increase. This is because the implementation of the

cleaner technology reduces the damage from production at the margin, giving each country

an incentive to increase its output. The rate of increase in emissions of each signatory

country in response to cleaner technology is higher than that of the non-signatory. Thus, as

the cleaner technologies are implemented, it becomes less costly to stay inside the coalition

of size two in terms of sacri�ced production.

Grand coalition

In the case of 3 countries, when say country 1 contemplates leaving the grand coalition,

it compares its payo¤ under the grand coalition to its payo¤ if countries 2 and 3 form a

coalition or to its payo¤ when countries 2 and 3 break up as well, depending on country 2

and 3�s incentives to continue forming a coalition. Note that for the case of 3 countries, the

farsighted stability criterion is closely related to the internal and external stability criteria

as presented by d�Aspremont et al (1983). This is illustrated by Lemma 1.

Lemma 1: The grand coalition of size 3 is farsighted stable if and only if a coalition of size

2 is internally unstable.

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Proof : Without loss of generality, let country 1 be the country contemplating defection

from the grand coalition of size three. When computing its welfare if it were to defect, under

the farsighted stability criterion country 1 takes into account whether its own defection will

cause the other two countries to continue to cooperate, by De�nition 1 of Diamantoudi and

Sartzetakis (2002), in line with Example 1. Thus, if the coalition of size two is internally

unstable, country 1, when deciding whether to defect, compares its welfare under the grand

coalition to its welfare under the Nash equilibrium. Due to symmetry, each country is better

o¤ under the grand coalition than under the Nash equilibrium. That is, from (5) and (6) ;

we have that wci > wnci for i 2 f1; :::; ng : Therefore, if the coalition of size two is internally

unstable, country 1 chooses to remain in the grand coalition. If the coalition of size two is

internally stable, country 1, when deciding whether to defect, compares its welfare under

the grand coalition to its welfare under the coalition structure where it is a singleton facing

a subcoalition of two. From (11) and (10) ; we have that w�nm (2) > w�m (2).

10 That is, due

to symmetry, each country is better o¤ free-riding on the two coalition members. Therefore,

if the coalition of size two is internally stable, country 1 chooses to defect from the grand

coalition.�

Since the decisions of countries 2 and 3 to form a coalition depends on the level of

technology, as per Proposition 1, we have the following.

Proposition 2: The grand coalition of three countries is farsighted stable for � > ��ms:

Proof: Proposition 2 follows directly from Proposition 1 and Lemma 1.�

When � is above ��ms, if, say, country 1 contemplates leaving the grand coalition it will

compare its payo¤ under the grand coalition to the payo¤ under the non-cooperative out-

come since for � > ��ms the coalition that contains countries 2 and 3 only is (internally)

unstable. Therefore, by Lemma 1, for � above ��ms; country 1 will opt to remain in the grand

coalition. The same reasoning applies to countries 2 and 3. When � is below ��ms, if country 1

contemplates leaving the grand coalition it will compare its payo¤ under the grand coalition

10We have that w�m (m)� w�nm (m) =(m+1)(m�1)A2n2s2�4

�2(B�ms�2+ns�2+m2s�2)2B< 0 for all m 2 f2; :::; ng :

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to the payo¤ under the outcome where countries 2 and 3 continue to form a coalition since

for � < ��ms the coalition that contains countries 2 and 3 is stable. Therefore, by Lemma 1,

for � below ��ms; country 1 will opt to leave the grand coalition.

In this case, the equilibrium level of total emissions under a (farsighted) stable coalition

will exhibit a discontinuity with a downward jump at � = ��ms. Figure 2 illustrates the

case for A = B = s = 1. We have an upward discontinuity of the level of welfare under a

(farsighted) stable coalition, at � = ��ms. See Figure 3.

θmn

0 θ

E

E2,3

E3,3

Figure 2: Global emissions as a function of θ

3

1

θmn

0 θ

W*

W*2,3

W*3,3

Figure 3: Global welfare as a function of θ

1.5

1

An important implication of this analysis is that, under the farsighted stable coalition

criterion, a decrease in the emissions per output ratio from �0 > ��ms to �00 < ��ms can have

a negative impact on world welfare since it breaks down the sustainability of the grand

coalition (which is stable under �0 and unstable under �00). In Figure 3, for example, the

global welfare drops by more than 45% by going from the grand coalition to the coalition

with two members.

3.3. More than three countries

We now examine the case of N > 3. The farsighted stable set typically depend on �. We

illustrate this point by comparing Examples 1 and 2.

Example 2: Consider the same parameter values as in Example 1, except that � = 0:25

instead of � = 0:1. The following table gives the payo¤s of insiders and outsiders for each

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coalition size m.

Table II: Payo¤s of insiders and outsiders for m2f1;::;8g

1 2 3 4 5 6 7 8

wm �0:444 �0:448 �0:389 �0:290 �0:178 �0:071 0:025 0:100

wnm �0:444 �0:305 �0:104 0:0802 0:219 0:313 0:375

In this case no coalition is myopically internally stable (in the sense of d�Aspremont et al.).

The farsighted internally stable set is f1; 3; 6g and the largest farsighted stable coalition that

is both internally and externally farsighted stable is 6.

By comparing Tables I and II, we �nd that going from a dirtier technology (� = 0:25)

to a cleaner technology (� = 0:1), we have that the size of the largest farsighted stable

coalition increases from 6 to 7. Global welfare also increase from 0:203 04; under � = 0:25

and farsighted stable coalition of 6, to 2: 373 0; under � = 0:1 and farsighted stable coalition

of 7. This is in contrast to the case with three countries where the cleaner technology is

shown to destabilize the larger coalition and reduce global welfare.

Using the same parameter values as in Examples 1-2, we now conduct the analysis for

all � 2 (0; �max].11 We plot the largest stable coalition size as a function of � in Figure 4 and

total welfare under the largest stable coalition as a function of � in Figure 5.

0 θ

Figure 4: Size of largest farsighted stable coalition as a function of θ

8

0.3

6

7

0.20.1

m

0 θ

Figure 5: Global welfare at largest farsighted stable coalition as a function of θ

4

0.30.20.1

W*m,n

Result 1: From Examples 1-2 and Figures 2-5, the main �ndings may be summarized as

follows:

11We note that if � = 0, then pollution doesn�t matter and the grand coalition is trivially the largest stablesize since being in a coalition is not relevant. In this case, all coalitions give the same outcome.

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(i) a small change in � can alter the size of a stable coalition and therefore result in a jump

in the graph of the total welfare as a function of �

(ii) there could be more than one jump as � varies

(iii) these jumps can be either upwards or downwards.

4. The Dynamic Model: The case of a stock pollutant

In this section, we extend our analysis to the dynamic case where the emissions are allowed

to accumulate into a stock over time and pollution damage is a function of this stock.

Emissions of pollution accumulate into a stock, P (t) ; according to the following transition

equation:

_P (t) = �ni=1"i (t)� kP (t) (13)

with

P (0) = P0 (14)

where k > 0 represents the rate at which the stock of pollution decays naturally.12

For notational convenience, the time argument, t; is generally omitted throughout the

paper although it is understood that all variables may be time dependent.

The damage function of country i = 1; ::; n is given by:

Di (P ) =s

2P 2; s > 0. (15)

We begin by studying the equilibria under non-cooperation and full cooperation. We

then analyze coalition formation in the following subsection.

4.1. The Markov perfect equilibrium12For n = 2 and � = 1; our model is equivalent to Dockner and Long (1993).

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Countries use Markovian strategies.13 Let �i (t) � qi (P ((t) ; t) for i = 1; :::; n and for all

t � 0: f�i (t)gt�0 denotes the production path of country i, which corresponds to the time

realization of the production strategy, qi. Country i�s government takes the production

strategies of the other n � 1 countries as given and chooses a production strategy, qi (or

equivalently a pollution control strategy), that maximizes the discounted stream of net ben-

e�ts. The problem of country i is, thus, given by:

maxqi

�wi (�i (t) ; P (t)) �

Z 1

0

e�rtbi (�i (t) ; P (t)) dt

�(16)

subject to the accumulation equation (13) and the initial condition (14). The discount rate,

r; is assumed to be constant, positive and identical for all countries. We de�ne below a

subgame perfect Nash equilibrium of this n-player di¤erential game.

The n-tuple (q�1; :::; q�n) is a Markov Perfect Nash equilibrium, MPNE, if for each i 2

f1; :::; ng, f�i (t)g = fq�i (P (t) ; t)g is an optimal control path of the problem (16) given that

�j (:) = q�j (P; :) for j 2 f1; :::; ng ; j 6= i.

When countries are identical, such a game admits a unique linear equilibrium and a

continuum of equilibria with non-linear strategies (Dockner and Long 1993). The linear

equilibrium is globally de�ned and, therefore, quali�es as a Markov perfect equilibrium.

The non-linear equilibria are typically locally de�ned, i.e. over a subset of the state space.

We focus in this analysis on the linear strategies equilibrium. Since our contribution is to

highlight an a priori unexpected outcome from the adoption of a �cleaner�technology, we

wish to make sure that our result is not driven by the fact that countries are using highly

�sophisticated�strategies.

Proposition 3: The vector (q; ::; q)

q�i (P ) = q (P ) �1

B(A� �� P ), i = 1; ::; n (17)

13That is, production at time t depends on t and the stock of pollutant at time t:

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constitutes a Markov perfect linear equilibrium and discounted net welfare is given by

Wi (P ) = �1

2�P 2 � �P � �, i = 1; ::; n (18)

where

� =

qB�B (2k + r)2 + (2n� 1) 4s�2

�� (2k + r)B

2 (2n� 1) �2

� =An��

B (k + r) + (2n� 1)��2

� = �(A� ��) (A� (2n� 1) ��)2Br

The steady state level of pollution

PSS (�) =n� (A� ��)Bk + n��2

> 0 (19)

is globally asymptotically stable.

Proof: See Appendix.

We note that qi > 0 i¤ P < �P (�) � 1��(A� ��) : It is straightforward to show that

�P (�) > PSS (�) for all � � 0:

4.2. Full Cooperation

The objective of the planner is to assign the production strategies q1 (P; :) ; :::; qn (P; :) that

maximize the sum of the countries�discounted stream of net bene�ts.

maxq1;:::;qn

nXi=1

wi (�i (t) ; P (t)) (20)

subject to the accumulation equation (13) and the initial condition (14).

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Proposition 4: The vector (qc; ::; qc)

qci (P ) = qc (P ) �1

B(A� �c� � �c�P ), i = 1; ::; n (21)

constitutes the fully cooperative equilibrium and joint discounted welfare of all countries is

given by

W ci (P ) = �

1

2�cP

2 � �cP � �c, i = 1; 2 (22)

where

�c =1

n�2

��Bk � 1

2Br +

1

2

p4B2kr + 4B2k2 +B2r2 + 4Bn2s�2

�

�c =A

Bn�

�c

k + r + 1Bn�2�c

�c =�n (A� ��c)

2

2Br

The steady state level of pollution

P cSS (�) =(k + r) �nA�

Bkr +Bk2 + n2s�2� > 0 (23)

is globally asymptotically stable.

Proof: See Appendix.

4.3. Coalition Formation

The coalition formation game remains the same as in the static model in the previous section.

The nonsignatories�maximization problem is given by (16) subject to the accumulation

equation (13) and the initial condition (14) and the signatories maximization problem is

given by:

maxqiWm (P ) =

mXi=1

wi (�i (t) ; P (t)) ; i 2M

subject to the accumulation equation (13) and the initial condition (14).

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Let signatories�joint value function be

Wm (P ) = �1

2�mP

2 � �mP � �m (24)

Let each nonsignatory�s value function be

Wnm (P ) = �1

2�nmP

2 � �nmP � �nm (25)

In (24) and (25) ; �m; �m; �m; �nm; �nm and �nm are functions of the parameters of the

model. Their derivations are similar to those in Propositions 3 and 4.

Due to the asymmetric nature of the game between the signatories and non-signatories,

the expression for the stability function is cumbersome. Therefore, we proceed with a nu-

merical example that illustrates the main results. To construct the numerical example we

use parameter values as used by List and Mason (2001) to illustrate the case of green-

house gas emissions by US states. We set r = 4%; 14 k = 0:01,15 B = �2 = 1 and

s 2 f0:003063; 0:15315; 0:000613g : For these parameter values, the example is identical to

the benchmark case of List and Mason (2001). We then vary � to study the impact of cleaner

technologies. In the analysis presented below, we report the results for those ranges of � and

P for which the output of each country is strictly positive.

4.4. The e¤ect of clean technologies on IEAs

For the sake of brevity, for the dynamic version of the model, we present the analysis for

the three country case. Our results for the case with more than three countries are similar

to those derived in the static version of the model. We �rst show that results qualitatively

14This value of the discount rate is in line with that used by Nordhaus in his climate change models.15We note that for k = 0; the steady state consists of a threshold level of pollution stock with emissions

equal to zero. While there may exist some types of cumulative pollutants for which k = 0; our focus in thissection is on the problem of climate change. Most studies modeling climate change use a strictly positivevalue of k (see for example, Nordhaus 1994, and Hoel and Karp 2001).

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similar to those reported by Proposition 1 for the static case carry over to the dynamic

context.

As in the static case, the grand coalition is internally unstable in this example, which also

implies that the two-country coalition is externally stable. See Figure A.1 in the Appendix

for details.

Figure 6 is similar to Figure 1 and illustrates a result similar to Proposition 1. Ceteris

paribus, if � decreases to a level below a certain threshold due to the implementation of a

cleaner technology, a two-country coalition becomes stable. The intuition behind this result

is also similar to the static case: each coalition member�s output is more elastic with respect

to � than the non-member�s output at a given P: This is illustrated by examining the emission

strategies as functions of P in Figures A.2 and A.3 in the Appendix:

00.5 1.0 1.5 2.0

-4

-2

Figure 6: Internal stability of 2country IEA as a function of θ at P = 0

θ

Ф (2)

s = 0.000613

s = 0.003063

s = 0.15315

Figure 6 further illustrates that the threshold in terms of � is decreasing in s; as in the static

case: As in the static case, as long as the two-country coalition is stable, the grand coalition

is unstable by the farsightedness criterion, and the breakdown of the grand coalition reduces

global welfare (see Figure A.4 in the Appendix). Please refer to Figure A.5 in the Appendix

for details regarding how this threshold changes as r varies.

Figure 7 shows � (2) as a function of P at � = 1 and � = 0:3: Starting at P = 0; suppose

� falls from 1 to 0.3 due to the implementation of a clean technology. At � = 1; the two-

country coalition is internally unstable. However, the reduction in � makes this coalition

stable. Moreover, as the stock rises in transition to the new steady state, the stability

function increases. The higher the stock, the greater the incentive of the two members to

remain in the coalition. Thus, once the grand coalition disintegrates, it remains disintegrated

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forever after. Within a dynamic context, the emission per output ratio interacts with the

stock of pollution at the instant when the decision about whether to join or leave the coalition

is made to in�uence the stability of coalitions. Figure 8 shows that the higher the stock, the

greater the range of � for which the two-country coalition is stable, and consequently, the

grand coalition is farsighted unstable.Figure 7: Internal stability of 2country IEA as a function of P at s = 0.003063

P

Ф (2)

10 20 30 40

0.5

1.0

1.5

2.0

2.5

PSS|θ = 0.3

Ф (2)|θ = 0.30

Ф (2)|θ = 1

Figure 8: Internal stability of 2country IEA as a function of θ at s = 0.003063

Ф (2)

θ

P = 0

P = 3

P = 6

0.4 0.5 0.6 0.7 0.8 0.9 1.0

-0.2

0.2

0.4

0.6

0.8

It can be shown that an increase in k; re�ecting a higher rate of absorption of the emissions

due to factors such as geo-engineering, have a qualitatively similar e¤ect on the stability of

international environmental agreements. To economize on space, we present this analysis in

Section 2 of the Appendix.

5. Concluding Remarks

This paper shows that, if countries are farsighted when deciding whether to defect from a

coalition, then the implementation of cleaner technologies, as embodied by a reduction in

the emission per output ratio, has an impact on the chances of reaching an international

environmental agreement. In the three country case, we showed that the grand coalition

may be destabilized by the implementation of cleaner technologies, ultimately resulting in

higher global emissions and lower global welfare. This result was shown to hold regardless

of whether the pollutant is a �ow or a stock pollutant. In the latter case, we showed that

the higher the stock of pollution at the instant when the cleaner technology is implemented,

the more likely that a grand coalition of three countries is destabilized. In the case of more

than three countries, we provided a numerical example with eight countries to illustrate that

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implementing cleaner technologies may result in a discrete jump, either upward or downward,

of the largest stable coalition size and corresponding welfare, depending on the initial value

of the emission per output ratio.

The main policy recommendation that can be drawn from this analysis is that greener

production processes or measures to enhance nature�s capacity to clean the environment

do not necessarily eliminate the need to agree on environmental policies with appropriate

transfers across regions to reach self-sustainable and successful environmental agreements

(see, for example, Germain et al. 2003, or Petrosjan and Zaccour 2003 for the case of

the design of transfers in dynamic transboundary pollution games). For each particular

transboundary pollution problem, a speci�c analysis taking into account the number, size

and behavior of players (farsighted or myopic), as well as the size of the change in the emission

per output ratio is necessary before one can determine what role cleaner technologies may

play in reaching an international environmental agreement.

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cleaner technologies and the stability of international environmental agreements

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