Self-Enforcing International Environmental
Agreements among Asymmetric Countries andWelfare ∗Yoko Sakamoto †November 30, 2011

Abstract
Theoretical analyses of self-enforcing international environmental agreements(IEAs) have adopted the assumption of identical countries. In this paper, we
assume that countries are asymmetric in terms of abatement technologies. By
examining the IEA formation and total welfare, we show that i) the asymmetry
leads to multiple sizes of self-enforcing IEAs, which corresponds to the multiple
equilibria of a coalition formation game Maruta and Okada (2005); ii) technology
transfer can lead to a smaller size of self-enforcing IEA, resulting in welfare loss.
We also examine the rule of IEA and discuss that the unanimity rule plays animportant role in IEA formation.

1

Introduction

International environmental agreements (IEAs) provide internationally organized treatment for solving multiregional environmental problems such as ozone depletion and climate change. The absence of a supranational authority means that IEAs are supported∗

I am deeply grateful to Professor Jota Ishikawa for his guidance and encouragement. I also thankProfessor Taiji Furusawa, Professor Hideshi Itoh, Professor Akira Okada, and Professor Masatoshi
Tsumagari. The support of the Global COE program and the Research Unit for Statistical andEmpirical Analysis in Social Sciences at Hitotsubashi University is gratefully acknowledged.
†
Graduate School of Economics, Hitotsubashi University, 2-1 Naka, Kunitatachi, Tokyo, 186-8601,
Japan. E-mail: ed112003@g.hit-u.ac.jp

1

by voluntary participation, which is captured as self-enforcement in the theoretical literature on IEAs. Self-enforcement is a concept of coalition stability which was ﬁrst
developed by d’Aspremont, Jacquemin, Gabszewicz and Weymark (1983) and applied
to IEA analysis by Barrett (1994). In a self-enforcing IEA, all member countries haveno incentive to unilaterally withdraw (internal stability) and all non-member countries
have no incentive to unilaterally join (external stability). After Barrett (1994), studies
of self-enforcing IEAs have concluded pessimistic results: only a small-sized IEA canbe self-enforcing (Hoel (1992); Carraro and Siniscalco (1993); Barrett (1994)).1
Earlier studies of self-enforcing IEAs share the assumption of identical countries.
Although it provides tractability, this speciﬁcation fails to describe the eﬀects ofcountry-speciﬁc factors on IEA formation. For example, let us consider an IEA requiring pollution abatement. The countries would diﬀer from each other in featuresaﬀecting their decision whether to join an IEA, such as productivity of abatement
technology, population, and economic scale. Identical countries models ignore thediﬀerence among these characteristics determining IEA formation.
The purpose of this paper is to investigate the eﬀect of asymmetry among countries
on IEA formation and welfare. There are several studies examining IEA formation
among asymmetric countries. Barrett (2001) diﬀerentiates countries into two types
according to the preference for the environment. He shows that strong asymmetry can
reduce the free-rider problem and side-payments among member countries can increase
the size of IEA. Furusawa and Konishi (2011) analyze a “free-riding-proof IEA,” whichis equivalent to a self-enforcing IEA in this paper. In their model, countries are
diﬀerent from each other in economic scale and abatement technology. They showthat coalitions become more stable under transferable utility regime such as emission
1

There exists a signiﬁcant amount of game theoretic literature on IEAs, which can be roughlyclassiﬁed into two types according to the framework adopted: the cooperative approach and thenon-cooperative approach (see Missfeldt (1999) and Wagner (2001) for a survey). Unlike the noncooperative approach, the cooperative approach gives relatively optimistic results that a grand coalition can be sustained (Chander and Tulkens (1995), Chander and Tulkens (1997)). In this paper, inorder to consider self-enforcing IEAs, we adopt the non-cooperative framework. The non-cooperative
approach enables us to examine the participation decision, whereas a cooperative framework focuses
on a grand coalition where all countries have already joined.

2permit trading.
This paper is closely related to Maruta and Okada (2005), which studies collective
action games among n persons in prisoners’ dilemma situation. In their model, each
person’s payoﬀ is determined by own cooperation decision and the number of cooperators. Countries are diﬀerentiated by threshold of cooperation, which is the numberof cooperators required for each country’s cooperation. A coalition in the strict Nashequilibrium in their model is equivalent to a self-enforcing IEA. They show that there
always exists at least one strict Nash equilibrium and, furthermore, multiple equilibria
can be the case.
Our model is a simpliﬁcation of Maruta and Okada (2005) where we introduce a
speciﬁc welfare function according to IEA literature. We diﬀerentiate countries byabatement cost, which determines the each country’s welfare. Under this modiﬁcation, the threshold of cooperation for country i can be represented by its abatement
technology and the distribution of abatement technology plays an important role todetermine the IEA formation. Our simpliﬁcation provides a reasonable interpretationof Maruta and Okada (2005), which is consistent with the environmental problem.
This modiﬁcation also enables us to examine the eﬀect of asymmetry of technologies
on welfare. In asymmetric countries model, the total welfare is determined by thesize of IEA and the distribution of abatement technologies, whereas only the size
determines the welfare in symmetric model. We ﬁnd that even if the asymmetry
results in a smaller IEA, the total welfare can increase, and vice versa.2
We can discuss the eﬀect of technology transfer using analogies of welfare analysis.
We show that technology transfer may result in welfare loss even if it improves theaverage productivity of abatement because technology transfer aﬀects the distributionof technology which might lead to a smaller size of IEA.
The rest of this paper proceeds as follows. First, we construct a model of self2

McEvoy and Stranlund (2009) also examine the welfare with a self-enforcing IEA. In their model,
the perfect compliance of member countries requires costly monitoring by IEA committee, which isincurred by member countries. They show that although the cost of monitoring increases the size of
IEA, it may lead to welfare loss if the total number of countries is small enough.

3

enforcing IEA among asymmetry countries, where we provide a deﬁnition of a selfenforcement with asymmetric countries. Results in Maruta and Okada (2005) are
conﬁrmed to hold in our setting. Then welfare analysis examining the eﬀect of asymmetry on the total welfare is to be presented. Lastly we discuss the technology transfer
resulting in welfare loss.

2

The Model

2.1

A Basic Model of Self-Enforcing IEAThere are n symmetric countries emitting pollutants. The set of countries is denotedby N = {1, 2, · · · , n}. Each country makes a discrete abatement choice: Abate (qi = 1)
and Emit (qi = 0).3 Country i’s welfare is determined by own choice and aggregate
choice:ui =∑qj − aqi ,j∈Nwhere a is the common abatement cost for all countries.4 We focus on the case of
1 < a < n. 1 < a means that Emit (qi = 0: noncooperation) is the dominant
strategy for all i ∈ N without agreements. a < n means that aggregate welfare is
maximized when all countries choose Abate (qi = 1: cooperation). Therefore theeconomic environment can be summarized as a prisoners’ dilemma with incentive toform an IEA.
We describe the IEA formation game as a three-stage game, following Kosfeld etal. (2009) as follows.5
3

We adopt discrete choice structure following previous studies such as Ulph (2004), Kolstad (2007),
McEvoy and Stranlund (2009). Furusawa and Konishi (2011) adopt continuous abatement choice.
4
Linear payoﬀ functions are also adopted in many papers. See Ulph (2004), Kolstad (2007),
McEvoy and Stranlund (2009), Kosfeld, Okada and Riedl (2009), for example. Linearity of welfare
function brings perfect consistency with the applied game theoretic literature on coalition formationsuch as Maruta and Okada (2005), Okada (1993).
5
In earlier studies, IEA formation game is usually described as two-stage game consisting ofParticipation Stage and Abatement Stage with an assumption that in Abatement Stage, countries
must chose an abatement level maximizing the joint welfare of the IEA if they choose to join in the4

1 Participation Stage– Each country decides whether to join a single IEA.
2 Implementation Stage– Member countries simultaneously and independently vote whether to agree
on cooperation (qi = 1) under the unanimity rule, knowing who participatedin the Participation stage. The IEA is implemented if all member countries
accept cooperation.
3 Abatement Stage– Each member and nonmember country makes an abatement choice qi . If
the IEA is implemented, each member country i is forced to choose qi = 1,
whereas each nonmember country j can choose qj freely, i.e., it chooses
qj = 0. If the IEA is not implemented, all countries i can choose qi freely,i.e., they choose qi = 0.
An IEA is denoted by S ⊂ N where |S| means the number of members in S. We
call |S| as size of the IEA. Let us consider the IEA S with |S| = s formed in theParticipation Stage agrees to cooperate in the Implementation Stage.6 Then, in the
Abatement Stage, the welfare for member countries denoted as um (S) and nonmember
countries denoted as unm (S) are realized as follows

um (S) = s − aunm (S) = s

because each member country i and nonmember country j chooses qi = 1 and qj = 0,
respectively.
Participation Stage. In this paper, we adopt three-stage game by introducing Implementation Stagebetween Participation and Abatement Stage. This modiﬁcation allows us to examine IEA formation
without assuming joint-welfare maximizing IEA.
6
Such an IEA is called as an eﬀective IEA (McEvoy and Stranlund (2009)), or successful group(Maruta and Okada (2005)).

5

In the Implementation Stage, all member countries agree to cooperate when thefollowing condition holds:7

um (S) ≥ unm (∅) ⇔ s ≥ a.

This means that an IEA reaches cooperation in the Implementation Stage if the its
size is greater than a. We call this condition as implementation condition.
It is useful to introduce the following notation:
Definition 1. For x ∈ R, deﬁne x∗ as the minimum integer which is greater than orequal to x. In other words, x∗ = min{y ∈ Z|y ≥ x}.
Because s is an integer, the implementation condition can be reduced tos ≥ a∗ .

We focus on the case where an IEA in this game is self-enforcing. According to thedeﬁnition provided in d’Aspremont et al. (1983) and Barrett (1994), a self-enforcing
IEA can be deﬁned as follows:
Definition 2. An IEA S is self-enforcing if and only if:(i) um (S) ≥ unm (S\{i}) for s ≥ a∗ + 1, for all i ∈ S
um (S) ≥ unm (∅) for s < a∗ + 1(ii) unm (S) ≥ um (S ∪ {i}), for all i ∈ N \S
Condition (i) is called internal stability, which means that no member country has
incentive to withdraw from the IEA. Condition (ii) is called external stability, whichmeans that nonmember countries cannot gain by joining the IEA.
Under these conditions, we obtain the following well-known result (Ulph (2004),
Kolstad (2007), McEvoy and Stranlund (2009)):
7

An IEA satisfying this condition is equivalent to a proﬁtable IEA in McEvoy and Stranlund(2009).

6

Result 1. The size of a self-enforcing IEA is a∗ .
An IEA S with s < a∗ is not self-enforcing because the implementation condition
is not satisﬁed. Also, S with s > a∗ is not self-enforcing because the internal stability
condition is not satisﬁed. In this case, there exists at least one member country thatcan gain by withdrawing from the IEA since the IEA S\{i} can be implemented.
Therefore, s = a∗ is the only one size of self-enforcing IEAs.8 In self-enforcing IEAs,
all member countries are pivotal in the sense that the IEA cannot be implemented
without participation of each country.
By Deﬁnition 1, we obtain that 2 ≤ a∗ ≤ n because of the assumption of 1 < a < n.
This means that the smallest possible size of self-enforcing IEA is two and the fullcooperation is also reached by self-enforcing IEA. The size is always a∗ , regardless ofthe total number of countries, n. The full cooperation is realized only if a∗ = n.

2.2

A Self-Enforcing IEA among Asymmetric Countries

In this subsection, we introduce the asymmetry among countries into the model in 2.1.
The asymmetry is captured by country-dependent abatement cost ai .9 Lower ai meansthat country i has relatively more eﬃcient technology. Then the welfare function for
country i can be written as follows:

ui =

∑

qj − a i qi .

j∈NWe assume that 1 < ai < n for all i ∈ N for the same reason in 2.1.
The model in this subsection is basically a simpliﬁcation of Maruta and Okada(2005). Our contribution here would be objectiﬁcation of the welfare function, which
provides practical interpretation for the result of the model. Also, it enables us to
examine the eﬀect IEA formation on welfare.
∗This does not mean there is only( one) self-enforcing IEA. Any IEAs satisfying s = a are selfn
enforcing. More concretely, there are a∗ self-enforcing IEAs.
9
We assume complete information so that ai is common knowledge for all countries.
87

In the same manner as 2.1, let us consider an implemented IEA S with |S| = s.
The welfare for member countries and nonmember countries are given byumi (S) = s − aiunmi (S) = s.

The IEA is implemented when all member countries accept cooperation:nmum
i (S) ≥ ui (∅) for all i ∈ S.

This implies s ≥ a∗i for all i ∈ S because s is an integer. We call s ≥ a∗i as implementation condition for country i.
Maruta and Okada (2005) assume that each player i has value si , called threshold ofcooperation, which determine the number of cooperators proﬁtable for player i when
he cooperates. In our modiﬁcation, threshold of cooperation for country i can be
determined by its abatement cost.
A self-enforcing IEA in this case is deﬁned as follows:
Definition 3. An IEA S among asymmetric countries is self-enforcing if and only if:
nm∗(i) umi (S) ≥ ui (S\{i}) for s ≥ (maxj∈S\{i} aj ) + 1, for all i ∈ S
nm∗um
i (S) ≥ ui (∅) for s < (maxj∈S\{i} aj ) + 1, for all i ∈ Sm(ii) unmi (S) ≥ ui (S ∪ {i}) for all i ∈ N \S

The external stability condition (ii) is exactly the same as that in the symmetriccountries case, which always holds. The ﬁrst condition of the internal stability (i)demonstrates the situation where the IEA S\{i} is implemented after a unilateraldeviation by member country i. The second condition is that a withdrawal of country
i leads to non-implementation of IEA formed by the remaining member countries. It
depends on the highest abatement cost among remaining countries whether country8

i’s withdrawal leads to break-down of IEA or not. Applying the same argument as
before, we obtain the following proposition:
Proposition 1. An IEA S is self-enforcing if and only if it satisfies:ai ≤ s < ( max aj )∗ + 1
j∈S\{i}for alli ∈ S 10

All proofs are given in Appendix. Let us characterize a self-enforcing IEA among
asymmetric countries in more detail. First, we have the following proposition withrespect to the size. Let us denote ah = maxi∈S ai , i.e., country h has the highest ai
among the IEA.11
Lemma 1. If an IEA S with |S| = s is self-enforcing, then we have s = a∗h .
s < a∗h is not suﬃcient for self-enforcing IEA because it breaks the implementation
condition for country h. Country h requires the highest number of participants, or
highest level of environmental improvement, in order to recover the highest cost forcooperation. Also, an IEA S with s > a∗h is not self-enforcing because it does not satisfy
the internal stability, i.e., there is at least one member country that beneﬁts from its
withdrawal. This proposition shows that the size of IEA must be suﬃciently large
so that the member country with the most ineﬃcient technology ﬁnds it beneﬁcial
to implement the IEA; however, it must also be small enough to hold the internalstability.
We have another necessary condition for self-enforcement as follows:
Lemma 2. If an IEA S with |S| = s is self-enforcing, then we have (maxi∈S\{h} ai )∗ =a∗h .
The deﬁnition of the self-enforcement reminds us that the IEA is no longer implemented after withdrawal of each country. In other words, the withdrawal always leads
to the breaking of the implementation condition for some of the remaining countries.
Notice that the symmetric case is the special case of the asymmetric case where a∗i =(maxj∈S\{i} aj )∗ = a∗ , which implies s = a∗ in a self-enforcing IEA.
11
If the number of countries with highest ai is greater than one, we randomly choose one of themas highest.
10

9

According to Lemma 1, country i ∈ S with ai < ah has no incentive to deviatebecause its deviation results in breakdown of the implementation condition for country
h. This proposition ensures that country h also has no incentive to withdraw because
it always breaks the implementation condition for a country with the second highest
cost.
The necessary conditions provided in Lemma 1 and Lemma 2 are also the suﬃcientconditions for a self-enforcing IEA:
Proposition 2. The IEA S is self-enforcing if and only if s = a∗h = (maxi∈S\{h} ai )∗ .
This proposition corresponds to Therem 3.2 in Maruta and Okada (2005), characterizing a strict Nash equilibrium of the group formation game. They also shows that
there always exists at least one strict Nash equilibrium in the group formation game.
In our setting, we obtain the same proposition stating that there always exists an IEA
satisfying Proposition 2:12
Proposition 3. There exists at least one self-enforcing IEA among asymmetric countries.
Furthermore, in the asymmetric situation, the size of self-enforcing IEA can be
multiple as shown in Example 1.
Example 1. Consider a set of six countries: N = {1, · · · , 6}. Assume that the
distribution of ai is given by {ai }i∈N = {1.2, 2.1, 2.8, 3, 4.2, 4.6} in ascending order.
Then we obtain the set of a∗i as {a∗i }i∈N = {2, 3, 3, 3, 5, 5}. According to Proposition
2, there are two sizes of the self-enforcing IEA: 3 and 5 (multiple sizes).13
Generally, the size of a self-enforcing IEA can be found by plotting a∗i in ascendingorder and examining the height of the horizontal area below 45-degree line (including
the boundary).14 The horizontal area corresponds to the candidates for country h
12In the symmetric case, there always exist single size of self-enforcing IEA because Proposition 2
is satisﬁed by s = a∗h = (maxi∈S\{h} ai )∗ = a∗ .
13
In this example, we can observe that the following eight IEAs are self-enforcing: {1, 2, 3}, {1, 2, 4},{1, 3, 4}, {2, 3, 4}, {1, 2, 3, 5, 6}, {1, 2, 4, 5, 6}, {2, 3, 4, 5, 6}, and {1, 3, 4, 5, 6}.
14
Refer to the Proof of Proposition 2 in Appendix. Notice that this method can be applied to thecase of symmetric countries.

10and the second highest cost country; the area below 45-degree line ensures that the
number of participants is suﬃciently large. In the Example 1, we obtain 3 and 5 thisway (Figure 1). The size of a self-enforcing IEA deeply depends on the distribution ofcountries’ a∗i .
5.0
4.5
4.0
3.5
3.0
2.5

1

2

3

4

5

6

Figure 1: The Distribution of a∗iRemark 1 (Comparison among multiple sizes). In Example 1, we have two selfenforcing IEA sizes: 5 and 3. From the viewpoint of the aggregate welfare, size 5 ispreferred to size 3 because we have the following welfare relations:
Size

Aggregate Welfare

3

11.9 at most15

5

13.3 at least16

In Example 1, a bigger IEA always results in a greater welfare. In general, however,this is not always the case as shown in Example 2.
Example 2. Consider a set of fourteen countries: N = {1, · · · , 14}. For simplicity,
assume that ai is an integer for all i ∈ N . The distribution of ai and a∗i are given by{ai }i∈N = {a∗i }i∈N = {4, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8} in ascending order. According
to Proposition 5, there are two sizes of self-enforcing IEAs: 6 and 8. Let us compare theaggregate welfare under an IEA S1 = {1, 2, 3, 4, 5, 6} with {a∗i }i∈S1 = {4, 6, 6, 6, 6, 6}
and an IEA S2 = {7, 8, 9, 10, 11, 12, 13, 14} with {a∗i }∈S2 = {8, 8, 8, 8, 8, 8, 8, 8}. The
aggregate welfare for each situation is given as follows:
16
16

Consider an IEA {1, 2, 3}
Consider an IEA {2, 3, 4, 5, 6}

11

Size (IEA)

Aggregate Welfare

6 (S1 )50

8 (S2 )

48

This result shows that a larger size of an IEA may lead to smaller aggregate welfare.
The aggregate welfare is consisted of two parts: the gain from the environmental
quality and the total cost of abatement. Although the bigger IEA brings us higher
quality of environment, the total welfare may not improve if the cost of supporting theenvironmental quality is too expensive. This aspect cannot be captured by symmetricmodel because it is created by the diﬀerence in abatement technology.
Remark 2 (Voting rule: the unanimity rule v.s. the majority rule). In the Implementation Stage, we assume that member countries vote under the unanimity rule. In
this Remark, we discuss the choice of voting rule greatly aﬀects the IEA formation.
The Example 3 shows that the unanimity rule plays signiﬁcant role in supporting
self-enforcing IEAs.
Example 3. Consider a set of six countries: N = {1, · · · , 5}. Assume that the
distribution of ai is given by {ai }i∈N = {1.3, 2.8, 3.5, 4.1, 4.7} in ascending order. Then
we obtain the set of a∗i as {a∗i }i∈N = {2, 3, 4, 5, 5}. According to Proposition 2, thegrand coalition (size 5) is self-enforcing under the unanimity rule.
Let us examine the IEA formation under the majority rule in the Implementation
Stage. Under the majority rule, an IEA is implemented if the majority (six outof ten) of members vote to agree on cooperation. Once the IEA is implemented,
each member country i is forced to choose qi = 1 even if it does not vote to agree.
As a result, the grand coalition is no longer self-enforcing because country 4 andcountry 5 have an incentive to withdraw. Under the unanimity rule, country 5 cannot
withdraw from the grand coalition because its withdrawal leads to the breakdown of
the implementation condition for country 4, which results in the breakdown of IEA.
Under the majority rule, however, even if the country 4 vote not to agree after thewithdrawal of country 5, the remaining three countries still agree and their approval12

is enough for implementation of the IEA. Thus the grand coalition is not stable underthe majority rule.
In the Example 3, the self-enforcing IEA does not exist, which means that theexistence theorem by Maruta and Okada (2005) does not hold under the majority rule.
Although we focus on the case of the majority rule, we can show that an IEA supportedby the unanimity rule is not always supported under the other voting rules such as thetwo-thirds majority rule by the same logic. Intuitively, the bigger-scale cooperation
seems to be more supported by the majority rule rather than the unanimity rulebecause less approval is required for the agreement under the majority rule. However,the unanimity rule, which seems to be relatively strict to achieve an agreement, is
important to support a stable IEA.

3

Welfare Analysis

Let us investigate the eﬀect of asymmetry of countries on IEA formation on welfare.
In this section, we provide the general conditions that asymmetry of countries leads toan increase in total welfare. Furthermore, we discuss the eﬀect of technology transfer
by examining the eﬀect of adjusting the degree of asymmetry.
As shown in the previous section, the size of a self-enforcing IEA is determined by
Proposition 2. By adjusting the distribution of a∗i to hold Proposition 2, any size swith 2 ≤ s ≤ n can be self-enforcing. The following ﬁgures describe distributions ofa∗i leading to the smallest and largest sizes of self-enforcing IEAs among six countries,
respectively.
Because asymmetry may result in any size of self-enforcing IEA, we ﬁrst need acriterion determining how to compare symmetry and asymmetry. In this paper, we
adopt the following criterion:
Criterion 1. The total number of countries n and the average level of ai are ﬁxed inboth symmetric and asymmetric conditions.

13

6

6

5

5

4

4

3

3

1

2

3

4

5

6

1

Figure 2: The Smallest Size: Two

2

3

4

5

6

Figure 3: The Largest Size: Six

Criterion 1 requires that a distribution of ai with asymmetry is a mean-preserving
spread of a distribution of ai under symmetry. This enables two situations to havethe same aggregate welfare from full cooperation. The idea here is to compare thesituations with the same potential source of cooperation, i.e., the same amount of
total welfare would be reached in the ideal world.
The following example would be helpful to understand what we are addressing:
Example 4. Consider a set of six symmetric countries N = {1, · · · , 6} with common
abatement cost 3.5. Let us introduce asymmetry among countries into this situations
and compare the total welfare. According to Criterion 1, {ai }i∈N = {1.2, 2.1, 2.6, 4.3, 4.9, 5.9}
is one candidate to be compared in this analysis because the average ai of these countries is 3.5. The relation between distributions of ai is described in Figure 4 where
the horizontal line is {3.5, 3.5, 3.5, 3.5, 3.5, 3.5}, demonstrating for symmetric countries, and the line graph is {1.2, 2.1, 2.6, 4.3, 4.9, 5.9}, demonstrating for asymmetriccountries. The full cooperation leads to the aggregate welfare of 15 in both situations.
In this example, the distributions of a∗i are {a∗i }i∈N = {4, 4, 4, 4, 4, 4} for symmetryand {a∗i }i∈N = {2, 3, 3, 5, 5, 6} for asymmetry, demonstrated in Figure 5. From theintersections between a 45-degree line and the horizontal parts of distributions, we cansee that the size of a self-enforcing IEA among symmetric countries is 4. Also, we
obtain that the size of a self-enforcing IEA among asymmetric countries is either 3 or
5. The aggregate welfare for each case is summarized as follows:

14

7
6
5
4
3
2
1

1

2

3

4

5

6

Figure 4: Distributions of ai to be Compared

8

6

4

2

1

2

3

4

5

6

Figure 5: Distributions of a∗i to be Compared15

Size Aggregate Welfare
Symmetry

4

10

Asymmetry

3

12.1

5

14.9

In this example, asymmetry among countries always results in welfare improvement. Notice that asymmetry increases aggregate welfare even if the size of the selfenforcing IEA decreases.17
Here, we have a general proposition about the welfare consequences of introducing
asymmetry among countries.
Proposition 4. Asymmetry of countries results in an increase in aggregate welfare
with a self-enforcing IEA if and only if the following holds:n(a∗h − a∗ ) >

∑
i∈S̃

ai −

∑

a,

i∈S̄

where S̄ is a self-enforcing IEA with symmetric countries with |S̄| = a∗ , S̃ is a selfenforcing IEA with |S̃| = a∗h .
Proposition 4 shows that asymmetry among countries could result in welfare gain.
The left-hand side n(a∗h − a∗ ) measures the increment of the total beneﬁt from envi∑∑
ronmental improvement. The right-hand side ( i∈S̃ ai − i∈S̄ a∗ ) demonstrates theincrement of the total cost to realize the environmental quality.
a∗h > a∗ means that asymmetry leads to the positive externality because morecountries become to engage in cooperation. However, this could also mean an increasein the total abatement cost. Even in that case, if n is large enough, the total gainfrom the positive externality exceeds the loss from the increment of abatement cost.18
17

An increase in size of a self-enforcing IEA does not always lead to welfare improvement. Noticethat the distribution of ai in Example 2 is a mean-preserving spread of a symmetric distribution of{ai }i∈N = {7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7}. We can verify that the aggregate welfare under an IEA
among asymmetric countries with size 8 is less than the one under an IEA among symmetric countries
with size 7.
18
Notice that even if asymmetry results in the bigger IEA, it may lead to welfare loss with smalln. (Refer to the footnote 16.)

16

Similarly, a∗h < a∗ results in an increase in negative externality because the number ofcooperating countries decreases. However, this would have a positive eﬀect on welfare
if it reduces the total abatement cost. Therefore, even if asymmetry makes the IEA
smaller, it may increase the total welfare when the gain from saving abatement costexceeds the loss from smaller size of IEA. (The basic logic is the same as Remark 1.)
The change of ai can be interpreted as a change in abatement technology. We
can discuss the eﬀect of technology transfer on welfare by considering the change indistribution of ai to be induced by technology transfer among countries.19 Let us
consider ODAs, FDI and other movements inducing technology transfer, resulting in
the improvement of average productivity of abatement technology. Such technology
transfer, however, may lead to welfare loss as shown in the following example.
Example 5. Let us consider six countries N = {1, · · · , 6} with a distribution of ai :{ai }i∈N = {1.6, 2.5, 2.8, 4.5, 4.9, 5.9}. Let us denote Ns = {1, 2, 3} with {ai }i∈Ns ={1.6, 2.5, 2.8} as “south” and Nn = {4, 5, 6} with {ai }i∈Nn = {4.5, 4.9, 5.9} as “north”.
The distribution of a∗i is {a∗ }i∈N = {2, 3, 3, 5, 5, 6}, leading to the two candidates forthe size of self-enforcing IEA: 3 and 5 by Proposition 5.
Assume that technological transfer happens from north to south and it reduces
the southern abatement cost by 0.5, resulting in {ai }i∈N = {1.6, 2.5, 2.8, 4, 4.4, 5.4}.
The distribution of a∗ changes to {a∗ }i∈N = {2, 3, 3, 4, 5, 6}, achieving the size ofself-enforcing IEA, 3. If the coalition size is 5 before technological transfer, then theaggregate welfare is 13.7, whereas it is 11.1 after technological transfer.
Although the technology transfer improves average productivity of abatement technology, it might reduce the size of self-enforcing IEA, resulting in welfare loss. We canalso apply this discussion to technology innovation. Although innovation improves
average productivity of the abatement technology, it might decrease the total welfare.
19

Proposition 4 can be interpreted as a result of technology transfer. Let us call such technology
transfer as “mean-preserving technology transfer” because the mean of ai is unchanged before andafter the transfer. As examples of mean-preserving technology transfer, we can consider the relocationof factories or transfer of human capital. According to Proposition 4, this type of technology transfer
may lead to welfare loss.

17

4

Conclusion

In this paper, we have analyzed the eﬀect of countries’ asymmetry on self-enforcing
IEA formation and total welfare. The asymmetry among countries is captured bycountry-dependent abatement cost. The model is a simpliﬁcation of a group formation
game in Maruta and Okada (2005) with a speciﬁc welfare function adopted in much of
IEA literature. According to the deﬁnition, self-enforcement requires that all member
countries are pivotal in the sense that a withdrawal of each country lead to break-downof the implementation condition. In our model, IEA formation depends deeply on the
distribution of abatement technologies because abatement cost plays an importantrole in determining the country’s choice whether to implement an IEA. As Barrett(2001) and Maruta and Okada (2005) show, asymmetry among countries may imply
multiple sizes of self-enforcing IEAs, while identical countries leads only to a single
size of self-enforcing IEA.
We have investigated the welfare eﬀect of countries’ asymmetry by comparing thetotal welfares of symmetric and asymmetric situations sharing the potential gain fromfull cooperation. As a result, compared with symmetric situation, we ﬁnd that asymmetry might result in welfare gain (loss) even if the size of coalition became smaller(bigger). This is because the total welfare is determined by not only how large the
IEA is but also who joins and who does not join the IEA in the asymmetric situation.
The welfare analysis has an important implication of technology transfer. We ﬁndthat technology transfer can lead to welfare loss even if it improved the average technology of abatement. This is because technology transfer can result in a smaller sizeof self-enforcing IEA. While the dominance of the scale eﬀect on the technique effect demonstrates the pollution-increasing technological change (Copeland and Taylor(2003)), change in the IEA formation oﬀers a new channel to explain welfare loss
caused by technology transfer. Technology transfer would basically beneﬁt us because
it expands potential abatement capacity in the world. Actually, the Kyoto Protocol
adopts Clean Development Mechanism (CDM) that encourage environmental technol-18

ogy transfer. However, according to our results, we need to consider its eﬀect on thesize of IEA that might result in deterioration of the environment.

Appendix
Proof of Proposition 1
Proof. We can easily check that the external stability holds for any S. um
i (S) ≥unmi (S\{i}) does not hold because it violates the assumption of 1 < ai for all i ∈ N .
nm∗
umi (S) ≥ ui (∅) implies s ≥ ai . From the condition of s < (maxj∈S\{i} aj ) + 1, the S

satisfying the internal condition is ai ≤ s < (maxj∈S\{i} aj )∗ + 1 for all i ∈ S.

Proof of Lemma 1
Proof. Assume on the contrary that S is a self-enforcing IEA with s > a∗h or s < a∗h .
If we have s > a∗h , then we have s ≥ a∗h + 1 since s is an integer. Using this result,
we have a∗h +1 ≤ s < maxj∈S\{h} aj +1 from the internal stability condition for countryh, which implies a∗h < maxj∈S\{h} aj . This is a contradiction because a∗h is deﬁned asa∗h = maxi∈S aj .
If we have s < a∗h , then we have ah ≤ s < a∗h because of the internal stability condition for country h. If ah is an integer, then we have ah < ah which is a contradiction.
If ah is not an integer, then there does not exist an integer s that satisﬁes ah ≤ s < a∗h .
Therefore, we have s = a∗h if an IEA S is self-enforcing.

Proof of Lemma 2
Proof. Assume that S is self-enforcing. In case of maxj∈S\{h} aj = ah , we have theresult directly.
Assume that maxj∈S\{h} aj < ah . Then we have a∗h < maxj∈S\{h} aj + 1 because ofs = a∗h (from Lemma 1) and the internal stability for country h. These two equations

19and ah ≤ a∗h imply a∗h − 1 < maxj∈S\{h} aj < a∗h . Because a∗h is an integer, this implies
that (maxj∈S\{h} aj )∗ = a∗h .

Proof of Proposition 2
Proof. The necessary conditions are directly derived from the results of Lemma 1 andLemma 2. We show the suﬃcient conditions for a self-enforcing IEA.
We want to show that s = a∗h = (maxj∈S\{h} aj )∗ implies ai ≤ s < maxj∈S\{i} aj + 1
for all i ∈ S.
First, we obviously have that ai ≤ a∗h for all i ∈ S. Because of s = a∗h , we haveai ≤ s for all j ∈ S.
Next, we show that we have s < maxj∈S\{i} aj + 1 for all i ∈ S. For country h, weshow that a∗h < maxi∈S\{h} ai + 1. From the deﬁnition of ·∗ , we have (maxi∈S\{h} ai )∗ −1 < maxi∈S\{h} ai . Because we have a∗h = (maxi∈S\{h} ai )∗ from the assumption, we
have a∗h − 1 < maxi∈S\{h} ai , which implies a∗h < maxi∈S\{h} ai + 1. Therefore, thecondition to be shown holds for country h.
For other countries i ∈ S\{h}, we check that a∗h < maxj∈S\{i} aj + 1. Becausemaxj∈S\{i} aj = ah , this equation always holds due to the deﬁnition of ·∗ .
Therefore, we have ai ≤ s < maxj∈S\{i} aj + 1 if we have s = a∗h = (maxi∈S\{h} ai )∗ .

Proof of Proposition 3
Proof. By Proposition 2, it is enough to show that there always exists at least one
coalition S which satisﬁes that s = a∗h = (maxi∈S\{h} ai )∗ .
First we show that there always at least two countries i and j with a∗i = a∗j . The
assumption that 1 < ai < n for every i ∈ N implies that 2 ≤ a∗i ≤ n for all i ∈ N .
Because we have n integers on [2, n], at least two of them must have the same number.
Therefore, there always exit at least two countries i and j with a∗i = a∗j .
Next we show there always exists S which contains two countries i, j with a∗i =20

a∗j = (maxk∈S ak )∗ , where s = a∗i = a∗j holds. Let us deﬁne i and j as countries withhighest a∗i = a∗j . Denote an integer l as l = a∗i = a∗j . Let us introduce a set S̄ suchthat S̄ = {k ̸= i, j|a∗k ≤ a∗i = a∗j } ∪ {i} ∪ {j}. ({k|a∗k ≤ a∗i = a∗j } can be ∅.) It isenough to show that |S̄| ≥ a∗i = a∗j . Assume on the contrary that |S̄| < l. Then there
are more than n − l countries whose a∗i is in [l + 1, n]. This implies that there are atleast two countries of n − l countries must share their a∗i , which is a contradiction to
the assumption of l is the largest integer that satisﬁes l = a∗i = a∗j . Therefore, we have|S̄| ≥ a∗i = a∗j .
We can check this proof graphically analogue to the ﬁxed point theorem. Let us
arrange all a∗i in ascending order. (Assume the set of countries is N = {1, · · · , 6}) The
4.0

3.5

3.0

2.5

1

2

3

4

5

6

linear line is 45-degree line and the line graph is plotting {ai }i∈N = {2, 3, 3, 3, 4, 4}.
We know that if line graph has a horizontal part in the area below 45-degree line(including boundary), there exists S̄ that satisﬁes |S̄| ≥ a∗i = a∗j with i, j ∈ S̄. From
the ﬁrst step of this proof, the line graph has at least one horizontal part. From thesecond step of the proof, there are at least one intersection between the two graphs.
Therefore, there exists a horizontal part below or on the 45-degree line, implying thatS̄ with |S̄| ≥ a∗i = a∗j with i, j ∈ S̄ always exists.

Proof of Proposition 4
Proof. Denote the common average of ai as a. S̄ is a self-enforcing IEA with symmetriccountries with |S̄| = a∗ , S̃ is a self-enforcing IEA with asymmetric countries with21

|S̃| = a∗h .
We want to show that asymmetry of countries results in an increase in aggregate
welfare with a self-enforcing IEA if and only if the following holds.
n(a∗h − a∗ ) >

∑

ai −

∑

a

i∈S̄

i∈S̃

We show this by direct comparison of total welfare under symmetric situation andasymmetric situation. First, in general, the total welfare from implemented IEA S
can be written as follows:∑um
i (s) +∑

unmi (s) = ns −∑i∈N \S

i∈S

ai

i∈S

Let us denote the total welfare under symmetric countries as Ū and the total welfare
from asymmetric countries as Ũ . Then, we have:Ū = na∗ −∑Ũ = na∗h −

a,

i∈S̄

∑

ai

i∈S̃

Then the direct comparison of Ū and Ũ , we have the result as follows:Ũ > Ū ⇐⇒na∗h −∑ai > na∗ −i∈S̃⇐⇒n(a∗h − a∗ ) >

∑
i∈S̃

∑

a

i∈S̄

ai −

∑

a

i∈S̄

References
Barrett, Scott, “Self-Enforcing International Environmental Agreements,” Oxford
Economic Papers, 1994, 46, 878–894.

22

, “International Cooperation for Sale,” European Economic Review, 2001, 45, 1835–
1850.
Carraro, Carlo and Domenico Siniscalco, “Strategies for the International Protection of the Environment,” Journal of Public Economics, 1993, 52, 309–328.
Chander, Parkash and Henry Tulkens, “A Core-Theoretic Solution for the Design of Cooperative Agreements on Transfrontier Pollution,” International Tax and
Public Finance, 1995, 2, 279–293.
and, “The Core of an Economy with Multilateral Environmental Externalities,”

International Journal of Game Theory, 1997, 26, 379–401.
Copeland, Brian R. and M. Scott Taylor, Trade and the Environment, Princeton
University Press, 2003.
d’Aspremont, Claude, Alexis Jacquemin, Jean Jaskold Gabszewicz, andJohn A. Weymark, “On the Stability of Collusive Price Leadership,” Canadian
Journal of Economics, 1983, 16, 17–25.
Furusawa, Taiji and Hideo Konishi, “Free-Riding-Proof International Environmental Agreements,” RIETI Discussion Paper Series 11-E-043, 2011.
Hoel, Michael, “International Environment Conventions: The Case of Uniform Reductions of Emissions,” Environmental and Resource Economics, 1992, 2, 141–159.
Kolstad, Charles D., “Systematic Uncertainty in Self-Enforcing International Environmental Agreements,” Journal of Environmental Economics and Management,
2007, 53, 68–79.
Kosfeld, Michael, Akira Okada, and Arno Riedl, “Institution Formation in
Public Goods Games,” American Economic Review, 2009, 99, 1335–1355.

23

Maruta, Toshimasa and Akira Okada, “Group Formation and Heterogeneity inCollective Action Games,” Discussion Papers/ Graduate School of Economics, Hitotsubashi University, 2005.
McEvoy, David M. and John K. Stranlund, “Self-Enforcing International Environmental Agreements with Costly Monitoring for Compliance,” Environmental
Resource Economics, 2009, 42, 491–508.
Missfeldt, Fannyh, “Game-Theoretic Modelling of Transboundary Pollution,” Journal of Economic Surveys, 1999, 13, 287–321.
Okada, Akira, “The Possibility of Cooperation in an N-person Prisoners’ Dilemma
with Institutional Arrangements,” Public Choice, 1993, 77, 629–656.
Ulph, Alistair, “Stable International Environmental Agrements with a Stock Pollutant, Uncertainty and Learning,” The Journal of Risk and Uncertainty, 2004, 29,
53–73.
Wagner, Ulrich J., “The Design of Stable International Environmental Agreements:
Economic Theory and Political Economy,” Journal of Economic Surveys, 2001, 15,
377–411.

24