Environmental matching agreements among
asymmetric countries
FUJITA Toshiyuki∗
Department of Economic Engineering, Faculty of Economics, Kyushu University,
6-19-1 Hakozaki, Higashiku, Fukuoka 812-8581, JAPAN
Abstract
We consider matching schemes as a model for transboundary environmental agreements,
and investigate their effectiveness. In the first stage, after the rule of matching scheme is
announced, each country determines its matching rate. In the second stage, it determines unconditional flat abatement taking the value of the matching rate as given. These decisions are
made noncooperatively. Each country is imposed an additional abatement that depends on
its matching rate and the other countries’ flat abatements. The analysis of a matching game
with asymmetric countries as players suggests the existence of a subgame perfect equilibrium
leading to an efficient outcome, and thus shows that matching agreements are effective.
Keywords: Transboundary pollution, International environmental agreements, Matching
schemes, Asymmetric countries, Subgame perfect equilibrium.
1
Introduction
One remarkable feature of transboundary environmental problems is that there is no organization
with supranational power to control anthropogenic pollutants. Hence international cooperation
is essential to developing measures to protect the environment. International conferences have
been held frequently, and negotiations have been conducted. If the international environmental
agreement in which all related countries participate is reached and each country complies with it,
an efficient level of abatement will be attained. However, there may be incentives for easy-riding
in some countries. In other words, they may refuse to accede to the agreement and defect from
it. An agreement is said to be self-enforcing if no country has an incentive to defect from it. The
design of the agreement should prevent any easy-riding and thus realize the large self-enforcing
agreement.
Many theoretical analyses of international environmental agreements have been conducted to
date, but earlier studies by Carraro and Siniscalco (1993), Barrett (1994) and others have shown
that the size of the self-enforcing agreement is generally very small.1 They typically describe
the agreements as two-stage games with sovereign countries as players. In the first stage each
country decides simultaneously whether to accede to the agreement or not. In the second stage
the signatories collectively choose the abatements to maximize the total payoff to the signatories;
whereas each non-signatory behaves noncooperatively. The payoff is defined as a function of the
abatements of all countries. Each country decides whether to be a signatory considering its final
payoff.
∗ E-mail:
1 More
[email protected]
recent studies with various factors include Na and Shin (1998), Barrett (2001), Lange and Vogt (2003).
1
In this paper, we apply the matching concept proposed by Guttman (1978) and developed by
Guttman (1987) and Guttman and Schnytzer (1992) to international environmental agreements,
and investigate the effectiveness of matching agreements on environmental protection. Rübbelke
(2006) is the first study that applies matching schemes to environmental problems. Matching
agreements proposed by Rübbelke (2006) do not fix the pollution abatement, but the so-called
“matching rate” for each country. Afterwards, each country fixes an unconditional flat abatement
noncooperatively. As a result, each country is imposed not only its own flat abatement, but also an
additional abatement that depends on its matching rate and the other countries’ flat abatements.
Rübbelke (2006) extends Guttman (1987), and focuses on the ancillary benefits of global environmental policy.2 He emphasizes the effectiveness of matching agreements due to the fact that
generally there is no need for renegotiations regarding the matching rate, because each country can
update its flat abatement rate whenever a new ancillary benefit is discovered. However, Rübbelke
(2006) does not explicitly calculate the equilibria of the game because the analytical framework
is too complicated. There is also an indication of the cases in which renegotiations are required,
which obscures the conclusion about the effectiveness of matching agreements. Moreover, although
Rübbelke (2006) considers that the countries are symmetric players having an identical payoff function, the differences in the situations under which the countries are placed in actual negotiations
about transboundary environmental problems may be a significant factor preventing the consensus.
We analyze matching agreements made by asymmetric countries within the framework of a
public-goods provision game based on the simple pollution abatement model by Barrett (1994)
and Na and Shin (1998), and directly show that there exists a subgame perfect equilibrium which
leads to an efficient outcome. Unlike Rübbelke (2006), each country determines its matching rate
independently in our model. There is no bargaining on the values of matching rates. All countries
agree upon the rules of matching schemes, not upon their matching rates. Carraro and Siniscalco
(1993) and other earlier studies of international environmental agreements describe that the size
of self-enforcing agreements does not increase unless considerable restrictions are imposed on the
payoff structure and behavioral patterns of each country in the setting or assumption of the model.
The concept of a matching agreement in our model, in contrast, is noteworthy because it does not
force the countries to undertake any more commitment than following the simple rules of matching.
Below, in Section 2, after formulating a model of international pollution abatement and obtaining the first-best outcome, we define a game incorporating matching schemes into the basic
model, and explain the game’s solution concepts. In Section 3, we prove the existence of efficient
and self-enforcing solution, and show the effectiveness of matching schemes. We also introduce the
results of a simple numerical example. Finally, we summarize concluding remarks in Section 4.
2
2.1
Model of matching agreements
International environmental agreements
Let us describe a simple game about the abatement of environmental pollution. Players of the
game are the governments of 𝑛 asymmetric countries sharing the environment. Let 𝑁 = {1, ⋅ ⋅ ⋅ , 𝑛}
denote the set of countries. We focus on certain transboundary pollutants. The benefits due to a
single country’s pollution abatement are proportional to the amount of abatement, and affect all
countries. The abatement cost is incurred entirely by the abating country. Let 𝒙 = (𝑥1 , ⋅ ⋅ ⋅ , 𝑥𝑛 )
2 Examples of ancillary benefits include the case that abating greenhouse gases for climate change also contributes
for protection of the ozone layer through the abatement of chlorofluorocarbons and it also contributes for the air
quality through the abatement of fossil fuels.
2
denote the vector of abatement of countries, and the payoff to country 𝑖(∈ 𝑁 ) is
∑
𝜋𝑖 (𝒙) = 𝜃𝑖
𝑥𝑗 − 𝐶(𝑥𝑖 ),
(1)
𝑗∈𝑁
where 𝜃𝑖 is the damage to country 𝑖 per unit of pollutant and 0 < 𝜃1 ≤ 𝜃2 ≤ ⋅ ⋅ ⋅ ≤ 𝜃𝑛 . Cost function
𝐶(⋅) can be differentiated three times and satisfies 𝐶(0) = 𝐶 ′ (0) = 0, lim 𝐶(𝑥) = lim 𝐶 ′ (𝑥) = ∞,
𝑥→∞
𝑥→∞
and 𝐶 ′ > 0, 𝐶 ′′ > 0, 𝐶 ′′′ ≥ 0 for all 𝑥 > 0. As can be seen from (1), the technological structure of
each country is identical, but the damage due to pollution varies by country.
The first-best abatement which maximizes the total payoff to all countries, or in other words,
satisfies the Samuelson condition is 𝑥𝑖 = Ω(𝜃) ≡ 𝐶 ′−1 (𝜃) for all 𝑖 ∈ 𝑁 , obtained as a solution of
∑
∂ ∑
𝜋𝑗 = 0.3 Here, 𝜃 ≡
𝜃𝑗 . Properties Ω′ > 0, and Ω′′ ≤ 0 are derived by the assumptions
∂𝑥𝑖
𝑗∈𝑁
𝑗∈𝑁
on cost function 𝐶.4 Let us assume that for each country the emissions exceed Ω(𝜃) in the current
state. When each country decides an abatement noncooperatively, country 𝑖 selects 𝑥𝑖 = Ω(𝜃𝑖 )
∂𝜋𝑖
= 0 so letting each country act independently does
that maximizes its own payoff by solving
∂𝑥𝑖
not achieve an efficient situation.
Let us examine the possibility of solving the problem by making an agreement regarding the
abatement 𝑥𝑖 , using the model of international environmental agreements. The game is divided
into two stages. In the first stage, countries individually decide whether or not to participate in
the agreement. The amount of abatement is determined in the second stage when all signatories
and each non-signatory play the noncooperative game as decision-makers. In this framework, when
the number of countries is large and the cost function satisfies the properties described above, the
agreement is shown to be not self-enforcing.
Proposition 1 For the pollution-abatement agreement game with 𝑛 asymmetric countries, an
efficient agreement among all countries is not self-enforcing when the number of countries is four
or larger.
Proof. The final payoff to a country is determined by the set of signatories and whether the country
accedes to the agreement. Let 𝑆 (⊂ 𝑁 ) denote the set of signatories. If the payoffs to country 𝑖
when it is a signatory and when it is non-signatory are written as 𝜋𝑖𝐴 (𝑆) and 𝜋𝑖𝐹 (𝑆), respectively,
the condition for an agreement with all countries to be self-enforcing is
𝜋𝑖𝐴 (𝑁 ) ≥ 𝜋𝑖𝐹 (𝑁 ∖{𝑖}) ∀𝑖 ∈ 𝑁.
(2)
Equation (2) means that participating in an agreement would be rational for country 𝑖 provided
that all other countries participate in the agreement. When the set of signatories is 𝑆, the amount
∑
of abatement of the signatories is Ω( 𝑗∈𝑆 𝜃𝑗 ) . A simple calculation yields 𝜋𝑖𝐴 (𝑁 ) = 𝑛𝜃𝑖 Ω(𝜃) −
𝐶(Ω(𝜃)) and 𝜋𝑖𝐹 (𝑁 ∖{𝑖}) = 𝜃𝑖 {Ω(𝜃𝑖 ) + (𝑛 − 1)Ω(𝜃 − 𝜃𝑖 )} − 𝐶(Ω(𝜃𝑖 )). Appendix 1 shows that when
𝑛 ≥ 4, there exists 𝑖 ∈ 𝑁 that satisfies 𝜋𝑖𝐴 (𝑁 ) < 𝜋𝑖𝐹 (𝑁 ∖{𝑖}) and (2) does not hold.
□
Barrett (1994) also shows by some example that when 𝑛 is sufficiently large an efficient agreement by all the countries is not self-enforcing, and some countries cannot be prevented from
avoiding the agreement and acting noncooperatively. Proposition 1 is considered a generalized version of Proposition 2 (p. 888) of Barrett (1994). As presented in the proof shown above, country
3 Since
4 We
𝐶 ′ (⋅) is monotone, it has an inverse function.
use the property of cost function 𝐶 ′′′ ≥ 0 to show that Ω′′ ≤ 0.
3
𝑖 would not join the agreement if all other countries join the agreement because noncooperative
behavior outside the agreement would be more beneficial than entering the agreement. Also for the
countries other than country 𝑖, nullifying the agreement or substantially reducing the abatement
by way of punishment for country 𝑖 reduces their own payoff and opposes collective rationality so
the agreement will be maintained, and country 𝑖 succeeds in achieving a free riding. Since this
kind of incentive works for some country, an agreement by all the countries is not self-enforcing
when the number of countries is large. Such incentive is stronger in countries with less benefit of
abatement.
2.2
Matching agreement game
We introduce the matching rules proposed by Guttman (1978) in the framework of the previous
section, and investigate their effectiveness. The game is divided into two stages. In the first stage,
after the matching rules are announced, the countries determine the matching rates simultaneously.
In the second stage, they decide the flat abatement simultaneously given the matching rate values.
The decisions are made noncooperatively. In a matching agreement, each country is imposed an
additional abatement that depends on its matching rate and the other countries’ flat abatements.
After each country has enacted its abatement, each country’s final payoff is determined.
Suppose that country 𝑖 has a matching rate 𝑏𝑖 (≥ 0), and a flat abatement 𝑎𝑖 (≥ 0). The amount
of abatement imposed on country 𝑖 by the matching agreement rules is assumed to be
∑
𝜃𝑖
+ 𝑦𝑖 .
(3)
𝑥𝑖 = 𝑎𝑖 + 𝑏𝑖
𝑎𝑗 ⋅
𝜃𝑗
𝑗∕=𝑖
where 𝑦𝑖 is the fixed abatement required of country 𝑖 irrespective of decisions related to the flat
abatement, and is assumed to be
𝑦𝑖 = Ω(𝜃)(1 − 𝜃𝑖 /𝜃𝑛 ).
The final payoff is shown below based on (1) and (3):
⎫
⎧ ⎛
⎞
⎛
⎞
⎬
∑⎨
∑
∑
𝜃
𝜃
𝑖
𝑘
𝜋𝑖 (𝒙) = 𝜃𝑖
𝑎𝑗 ⋅
𝑎 ⎝1 +
𝑏𝑘 ⋅ ⎠ + 𝑦𝑗 − 𝐶 ⎝𝑎𝑖 + 𝑏𝑖
+ 𝑦𝑖 ⎠ .
⎭
⎩ 𝑗
𝜃𝑗
𝜃𝑗
𝑗∈𝑁
(4)
𝑗∕=𝑖
𝑘∕=𝑗
In the second stage, country 𝑖 takes the matching rate (𝑏1 , ⋅ ⋅ ⋅ , 𝑏𝑛 ) as given, and sets the flat
abatement 𝑎𝑖 to maximize its payoff. The flat abatement is therefore a function of the combination
of each country’s matching rate. Since the actual abatement 𝑥𝑖 and the payoff 𝜋𝑖 depend on
𝑎1 , ⋅ ⋅ ⋅ , 𝑎𝑛 , and 𝑏1 , ⋅ ⋅ ⋅ , 𝑏𝑛 , country 𝑖’s payoffs can be written as 𝜋𝑖 (𝒂, 𝒃), where 𝒂 = (𝑎1 , ⋅ ⋅ ⋅ , 𝑎𝑛 )
and 𝒃 = (𝑏1 , ⋅ ⋅ ⋅ , 𝑏𝑛 ).
A self-enforcing agreement is defined as a subgame perfect equilibrium of the matching agreement game. We start with deriving each country’s optimal response in the seocnd stage. Given
the matching rate determined in the second stage, country 𝑖 determines 𝑎𝑖 as the function of
𝒂−𝑖 = (𝑎1 , ⋅ ⋅ ⋅ , 𝑎𝑖−1 , 𝑎𝑖+1 , ⋅ ⋅ ⋅ , 𝑎𝑛 ), a set of flat abatements of other countries. The Kuhn-Tucker
conditions to maximize the payoff to country 𝑖 are
∂𝜋𝑖
∂𝜋𝑖
≤ 0,
⋅ 𝑎𝑖 = 0.
∂𝑎𝑖
∂𝑎𝑖
From (4), we get
(5)
⎛
⎞
⎛
⎞
∑
∑
∂𝜋𝑖
𝜃
𝜃
𝑗
𝑖
= 𝜃𝑖 ⎝1 +
𝑏𝑗 ⋅ ⎠ − 𝐶 ′ ⎝𝑎𝑖 + 𝑏𝑖
𝑎𝑗 ⋅
+ 𝑦𝑖 ⎠ ,
∂𝑎𝑖
𝜃𝑖
𝜃𝑗
𝑗∕=𝑖
𝑗∕=𝑖
4
so if the optimal response 𝑎∗𝑖 (𝒃, 𝒂−𝑖 ) takes the positive value as an interior solution,
to
⎛ ⎛
𝑎∗𝑖 (𝒃, 𝒂−𝑖 ) = Ω ⎝𝜃𝑖 ⎝1 +
∑
𝑗∕=𝑖
⎞⎞
∑
𝜃𝑖
𝜃 𝑗 ⎠⎠
𝑎𝑗 ⋅
− 𝑏𝑖
− 𝑦𝑖 .
𝑏𝑗 ⋅
𝜃𝑖
𝜃𝑗
∂𝑢𝑖
= 0 leads
∂𝑎𝑖
(6)
𝑗∕=𝑖
If the right-hand side of (6) is non-positive then 𝑎∗𝑖 (𝒃, 𝒂−𝑖 ) = 0 from (5). When each country takes
an optimal response, that is, when the simultaneous equations 𝑎∗𝑖 = 𝑎∗𝑖 (𝒃, 𝒂∗−𝑖 ) for all 𝑖 ∈ 𝑁 hold,
the set of flat abatements 𝒂∗ (𝒃) ≡ (𝑎∗1 , ⋅ ⋅ ⋅ , 𝑎∗𝑛 ) is an equilibrium of the second stage.
In addition, when 𝒃∗ = (𝑏∗1 , ⋅ ⋅ ⋅ , 𝑏∗𝑛 ) satisfies for all 𝑖 ∈ 𝑁 ,
𝜋𝑖 (𝒂∗ (𝒃∗ ), 𝒃∗ ) ≥ 𝜋𝑖 (𝒂∗ (𝑏𝑖 , 𝒃∗−𝑖 ), (𝑏𝑖 , 𝒃∗−𝑖 )) ∀𝑏𝑖 ≥ 0,
(7)
𝒃∗ is an equilibrium of the first stage. Here (𝑏𝑖 , 𝒃∗−𝑖 ) = (𝑏∗1 , ⋅ ⋅ ⋅ , 𝑏∗𝑖−1 , 𝑏𝑖 , 𝑏∗𝑖+1 , ⋅ ⋅ ⋅ , 𝑏∗𝑛 ). (7) indicates
that the matching rate of each country 𝑖 is the optimal response with respect to the set of matching
rates of the other countries. In this case no country has an incentive to change its matching rate.
3
3.1
Solutions of matching agreement games
Existence of efficient equilibrium
We examine the solutions of matching agreement games in this section. First, we derive the
following lemma concerning the relationship between the equilibrium of the second stage and
matching rates.
Lemma 1 For the pollution-abatement matching agreement game with 𝑛 asymmetric countries,
the first-best abatement is led by the equilibrium of the second stage when the matching rates of all
countries in the first stage are unity.
Proof. Substituting 𝑏1 = ⋅ ⋅ ⋅ = 𝑏𝑛 = 1 in (6), we have following conditions to be satisfied by
equilibrium 𝒂∗ :
⎛
⎞
∑
𝜃𝑖
𝑎∗𝑗 ⋅
(8)
𝑎∗𝑖 (𝒃, 𝒂∗−𝑖 ) = max ⎝Ω(𝜃) −
− 𝑦𝑖 , 0⎠ .
𝜃𝑗
𝑗∕=𝑖
Because 𝒂∗ = (0, ⋅ ⋅ ⋅ , 0) apparently does not satisfy (8), a country exists that chooses a positive
flat abatement in equilibrium. Let us call it country 𝑗, then
𝑎∗𝑗
Ω(𝜃) − 𝑦𝑗 ∑ 𝑎∗𝑘
=
−
,
𝜃𝑗
𝜃𝑗
𝜃𝑘
𝑘∕=𝑗
and it follows that
∑ 𝑎∗𝑗
Ω(𝜃) − 𝑦𝑗
Ω(𝜃)
=
=
.
𝜃𝑗
𝜃𝑗
𝜃𝑛
(9)
𝑗∈𝑁
Arbitrary 𝒂∗ that satisfies (9) is an equilibrium. In this equilibrium the abatement of country 𝑖
becomes
𝑥𝑖 = 𝑎∗𝑖 +
∑
𝑗∕=𝑖
= Ω(𝜃) ⋅
𝑎∗𝑗 ⋅
∑ 𝑎∗𝑗
𝜃𝑖
+ 𝑦𝑖 = 𝜃𝑖
+ 𝑦𝑖
𝜃𝑗
𝜃𝑗
𝑗∈𝑁
𝜃𝑖
+ 𝑦𝑖 = Ω(𝜃),
𝜃𝑛
5
which equals the first-best abatement.
□
Subsequently, we show that the set of matching rates 𝒃 = (1, ⋅ ⋅ ⋅ , 1) is the first stage equilibrium.
Lemma 2 For the first stage of the pollution-abatement matching agreement game with 𝑛 asymmetric countries, the set of matching rates 𝒃 = (1, ⋅ ⋅ ⋅ , 1) is an equilibrium.
Proof. Suppose only country 𝑖 deviates from 𝒃 = (1, ⋅ ⋅ ⋅ , 1) and matching rates are
{
𝑏𝑖 ∕= 1,
𝑏𝑗 = 1 ∀𝑗 ∕= 𝑖.
To prove Lemma 2, we only need to show that the payoff to country 𝑖 does not increase by this
deviation. We examine two cases: the case when the equilibrium flat abatement of 𝑖 is positive
and the case when it is zero.
(i) The case when 𝑎∗𝑖 > 0
Assuming there exists 𝑗 ∕= 𝑖 such that 𝑎∗𝑗 > 0, we get
∑ 𝑎∗
𝑎∗𝑗
Ω(𝜃 + (𝑏𝑖 − 1)𝜃𝑖 ) − 𝑦𝑗
𝑘
=
−
𝜃𝑗
𝜃𝑗
𝜃𝑘
𝑘∕=𝑗
from (6), and it follows that
∑ 𝑎∗𝑗
Ω(𝜃 + (𝑏𝑖 − 1)𝜃𝑖 ) − 𝑦𝑗
=
.
𝜃𝑗
𝜃𝑗
(10)
𝑗∈𝑁
Since the decision of country 𝑖 is also an interior solution, (6) yields
⎞
⎛
∗
∗
∗
∗
∑
∑
𝑎
𝑎
𝑎𝑖
Ω(𝜃) − 𝑦𝑖
Ω(𝜃) − 𝑦𝑖
𝑎
𝑗
𝑗
=
− 𝑏𝑖
=
− 𝑏𝑖 ⎝
− 𝑖 ⎠.
𝜃𝑖
𝜃𝑖
𝜃𝑗
𝜃𝑖
𝜃𝑗
𝜃𝑖
(11)
𝑗∈𝑁
𝑗∕=𝑖
Substituting (10) in (11), we obtain
[
]
𝑎∗𝑖
𝑏𝑖 {Ω(𝜃 + (𝑏𝑖 − 1)𝜃𝑖 ) − 𝑦𝑗 }
−1 Ω(𝜃) − 𝑦𝑖
= (1 − 𝑏𝑖 )
−
.
𝜃𝑖
𝜃𝑖
𝜃𝑗
From (10) and (12),
(12)
∑ 𝑎∗𝑗
𝑎∗𝑖
Ω(𝜃 + (𝑏𝑖 − 1)𝜃𝑖 ) − Ω(𝜃)
−
=
,
𝜃𝑖
𝜃𝑗
(𝑏𝑖 − 1)𝜃𝑗
𝑗∈𝑁
and we get
𝑎∗𝑖
𝜃𝑖
−
∑
𝑗∈𝑁
𝑎∗𝑗
𝜃𝑗
> 0 since Ω′ > 0. This is contradictory to the condition 𝑎∗𝑗 ≥ 0 for
all 𝑗 ∈ 𝑁 , so the initial assumption turns out to be false. Therefore for all 𝑗 ∕= 𝑖, 𝑎∗𝑗 = 0. We
𝜃𝑖
substitute them in (11) to obtain 𝑎∗𝑖 = Ω(𝜃) − 𝑦𝑖 = Ω(𝜃) ⋅ . By the rule of the matching scheme
𝜃𝑛
(4), 𝑥∗𝑗 = Ω(𝜃) for all 𝑗 ∈ 𝑁 , which is equal to the first-best level. Thus it is shown that even if
county 𝑖 deviates from the matching rate of unity, its payoff does not change.
(ii) The case when 𝑎∗𝑖 = 0
From the condition (5),
Ω(𝜃) − 𝑏𝑖 𝜃𝑖
∑ 𝑎∗𝑗
− 𝑦𝑖 ≤ 0
𝜃𝑗
(13)
𝑗∈𝑁
must hold. When 𝑎∗𝑗 = 0 for all 𝑗 ∈ 𝑁 , it is clear that (13) does not hold, so there exists 𝑗 such that
𝑎∗𝑗 > 0. Then (10) is satisfied as shown in the case (i). Substitute (10) in (13), and the left-hand
6
side of (13) is a monotone decreasing function of 𝑏𝑖 and takes the value zero when 𝑏𝑖 = 1. Hence,
a necessary condition for (13) is 𝑏𝑖 > 1.
Furthermore, if 𝑎∗𝑗 > 0,
𝑗 = arg min 𝜃𝑘 .
(14)
𝑘∕=𝑖
This reason is as follows. If there exists 𝑘 such that 𝜃𝑘 < 𝜃𝑗 , then from (10) we have
∑ 𝑎∗𝑗
<
𝜃𝑗
𝑗∈𝑁
Ω(𝜃 + (𝑏𝑖 − 1)𝜃𝑖 ) − 𝑦𝑘
∂𝑢𝑘
. This means
> 0, which does not satisfy the equilibrium conditions.
𝜃𝑘
∂𝑎𝑘
We assume 𝑗 that satisfies (14) is uniquely determined. Although there are cases such as 𝜃1 = 𝜃2
in which 𝑗 is not unique, we exclude such a possibility from consideration to avoid the complexity
of the argument.5
From above discussions, 𝑎∗𝑗 = Ω(𝜃 + (𝑏𝑖 − 1)𝜃𝑖 ) − Ω(𝜃)(1 − 𝜃𝑗 /𝜃𝑛 ) and 𝑎∗𝑘 = 0 (𝑘 ∕= 1), and the
actual amount of abatement can be written as
{
(
)}
(
)
𝑏𝑖 𝜃𝑖 ∗
𝑏𝑖 𝜃𝑖
𝜃𝑗
𝜃𝑖
𝑥∗𝑖 =
𝑎𝑗 + 𝑦𝑖 =
Ω(𝜃 + (𝑏𝑖 − 1)𝜃𝑖 ) − Ω(𝜃) 1 −
+ Ω(𝜃) 1 −
,
(15)
𝜃𝑗
𝜃𝑗
𝜃𝑛
𝜃𝑛
{
)}
)
(
(
𝜃𝑘 ∗
𝜃𝑘
𝜃𝑗
𝜃𝑘
∗
𝑥𝑘 = 𝑎𝑗 + 𝑦𝑘 =
Ω(𝜃 + (𝑏𝑖 − 1)𝜃𝑖 ) − Ω(𝜃) 1 −
+ Ω(𝜃) 1 −
∀𝑘 ∕= 𝑖.
(16)
𝜃𝑗
𝜃𝑗
𝜃𝑛
𝜃𝑛
Let 𝜋𝑖 (𝑏𝑖 ) denote country 𝑖’s payoff as a function of 𝑏𝑖 . Since Appendix 2 shows that 𝜋𝑖′ (𝑏𝑖 ) < 0
when 𝑏𝑖 > 1, any change in the matching rate of country 𝑖 would always result in a decrease in 𝜋𝑖 .
From above we have seen that in both cases country 𝑖 cannot increase its payoff by taking a
deviating behavior. Since this fact holds for any 𝑖 ∈ 𝑁 , 𝒃 = (1, ⋅ ⋅ ⋅ , 1) is an equilibrium of the first
stage.
□
The fact that the set of matching rate (1, ⋅ ⋅ ⋅ , 1) is an equilibrium means provided the other
countries set their matching rate as unity, it would not be rational for a single country to change
its matching rate. Lemma 2 represents a specific example of the argument presented by Guttman
(1978). If the agreement with a matching rate of unity is implemented, the abatement of any
country eventually becomes Ω(𝜃). Even if a country reduces its matching rate, the flat abatements
of other countries accordingly become zero; the country would be compelled to raise its flat abatement, and the actual abatement would not change. Conversely, if the matching rate are raised, the
flat abatements of other countries would increase, which would excessively increase the actual reduction amount of the country and reduce the benefit. Consequently, unlike the agreement on the
abatement itself, no countries could succeed in free riding. Now we get the following proposition
directly from Lemmas 1 and 2.
Proposition 2 For the pollution-abatement matching agreement game with 𝑛 asymmetric countries, there exists a self-enforcing agreement that leads to an efficient outcome.
Proof. It is straightforward to prove this Proposition, since Lemmas 1 and 2 show that a selfenforcing set of matching rate 𝒃 = (1, ⋅ ⋅ ⋅ , 1) leads to the first-best outcome.
□
5 (15) and (16) hold and produce equivalent results even when there exist a multiple number of 𝑗 that satisfy
𝑎∗𝑗 > 0, although we omit details.
7
The asymmetry of countries may cause the existence of a country whose payoff when an agreement is established is less than that when the agreement does not exist. The lump-sum income
transfer would be useful to give such a country an incentive to participate in the agreement. Let
𝜋 𝑖 (= 𝜋𝑖 (Ω(𝜃1 ), ⋅ ⋅ ⋅ , Ω(𝜃𝑛 ))) denote the payoff to country 𝑖 in the noncooperative equilibrium of
the game in which countries choose their abatements directly, and let 𝜋𝑖∗ (= 𝜋𝑖 (Ω(𝜃), ⋅ ⋅ ⋅ , Ω(𝜃)))
denote the payoff to 𝑖 in the equilibrium of matching game. If the income transfer is
1 ∑ ∗
𝑡𝑖 = 𝜋 𝑖 − 𝜋𝑖∗ +
(𝜋𝑗 − 𝜋 𝑗 ),
(17)
𝑛
𝑗∈𝑁
then 𝜋𝑖∗ + 𝑡𝑖 ≥ 𝜋 𝑖 holds for any 𝑖. The relation,
𝑡𝑗 = 0, also holds, so this transfer rule is
𝑗∈𝑁
feasible.
3.2
∑
Examples
Here we explain how a matching agreement works using a simple numerical example. Let us assume
𝑛 = 4, 𝜃𝑖 = 𝑖, and 𝐶(𝑥𝑖 ) = 𝑥2𝑖 /2. The payoff to country 𝑖 is
𝜋𝑖 = 𝑖
4
∑
𝑥𝑗 −
𝑗=1
𝑥2𝑖
.
2
From the assumption, Ω(𝑥) = 𝑥. The first-best abatements are (10, 10, 10, 10) and the payoffs
are (−10, 30, 70, 110). The abatements when these countries noncooperatively decide them are
(1, 2, 3, 4); the payoffs become (19/2, 18, 51/2, 32). Based on (17), the income transfer vectors are
calculated as (193/4, 67/4, −63/4, −197/4). If an efficient agreement is established, the payoffs to
each country including income transfer are (153/4, 187/4, 217/4, 243/4).
Suppose there is a matching agreement in which the matching rate of all countries is unity.
Consequently, the relations between the flat abatements and actual abatements are
⎧
𝑎3
𝑎4
15
𝑎2


+
+
+ ,
𝑥1 = 𝑎1 +


2
3
4
2




𝑎
2𝑎

4
3

+
+ 5,
⎨ 𝑥2 = 2𝑎1 + 𝑎2 +
3
2

3𝑎
3𝑎
5

 𝑥3 = 3𝑎1 + 2 + 𝑎3 + 4 + ,


2
4
2




4𝑎

3
⎩ 𝑥4 = 4𝑎1 + 2𝑎2 +
+ 𝑎4 .
3
Equation (9) becomes
4
∑
𝑎∗𝑗 /𝑗 = 5/2, and an equilibrium of the second stage is the vector of the
𝑗=1
arbitrary abatement, 𝒂∗ = (𝑎∗1 , ⋅ ⋅ ⋅ , 𝑎∗4 ), which satisfies this equation. In this case, 𝑥∗𝑗 = 10 hold
for all 𝑗 and the efficiency is achieved.
Suppose that one country, say country 1 alone sets its matching rate to zero. When the set of
matching rates is 𝒃 = (0, 1, 1, 1), 𝑎∗2 = 𝑎∗3 = 𝑎∗4 = 0. The reason is that when there exists a country
with a positive flat abatement ∑
amongst countries∑
2, 3 and 4 (suppose it is country 2), then from
∗
∗
∗
(6), 𝑎1 = 5/2 and 𝑎2 /2 = 2 −
𝑎𝑗 /𝑗, that is,
𝑎∗𝑗 /𝑗 = 2, and there is a clear contradiction.6
𝑗
𝑗∕=2
Therefore, the equilibrium of the third stage is 𝒂∗ = (5/2, 0, 0, 0), and from a simple calculation,
𝑥∗𝑖 = 10 ∀𝑖. It is impossible for country 1 to increase the payoff, so there is no incentive to withdraw
from the agreement.
6A
similar argument is used for the first half of the proof of Lemma 2.
8
4
Conclusion
We have considered matching agreements as a form of transboundary environmental agreement
among asymmetric countries, and investigated their effectiveness.
In Section 2, we first present Proposition 1, stating that efficient agreements on the amount of
abatement are not normally self-enforcing in the game of international environmental agreement
on the abatement of pollution. This proposition is regarded as the generalization of the proposition
provided by Barrett (1994). Subsequently, we present a model that includes matching rules. The
players in the model are asymmetric countries, whose benefits are specified as linear functions. In
the first stage, after the matching rules are announced, countries simultaneously determine the
matching rate. Given the matching rate values, they simultaneously decide the flat abatement
in the second stage. These decisions are made noncooperatively. In a matching agreement, each
country is imposed an additional abatement that depends on the matching rates determined in the
first stage and flat abatement of other countries.
In Section 3, we present an analysis regarding the solution of the matching agreement game.
Firstly, we show that a matching agreement in which all countries have a matching rate of unity
yields an efficient outcome through the equilibrium of the second stage (Lemma 1). Next, this set
of matching rates is proved to be an equilibrium of the first stage (Lemma 2). The existence of
self-enforcing agreement leading to an efficient outcome is derived from these lemmas (Proposition
2). Thus, the effectiveness of matching agreements is shown.
We can also analyze a game among symmetric countries as a specific case of our model. In the
case of symmetric countries, matching rules get simpler; the fixed abatement for each country is
zero and income transfer is not necessary.
For the current situation that it is difficult to prevent easy-riding in international environmental
agreements, the idea of matching could give us some clue. Even if an individual country refuses
to participate in a matching agreement, as long as all the other countries participate that country
cannot increase its payoff, and therefore no incentive to defect from the agreement is generated.
Future topics for consideration include more general conditions, e.g., the cases when the payoff
function is relatively general and the cases which allow for the partial agreements by some countries.
Appendix
1. Proof of the existence of 𝑖 that does not satisfy (2)
We can choose 𝑖 that satisfies 𝜃𝑖 ≤ 𝜃/𝑛. Since 𝑛 ≥ 4, it follows that 𝜃𝑖 ≤ 𝜃/4. Based on the discussions in proof of
Proposition 1 we have
𝜋𝑖𝐹 (𝑁 ∖{𝑖}) − 𝜋𝑖𝐴 (𝑁 ) = 𝐶(Ω(𝜃)) − 𝐶(Ω(𝜃𝑖 )) − 𝜃𝑖 (𝑛 − 1){Ω(𝜃) − Ω(𝜃 − 𝜃𝑖 )} − 𝜃𝑖 {Ω(𝜃) − Ω(𝜃𝑖 )}.
The convexity of the cost function 𝐶(⋅) and the property
Ω′
(A1)
> 0 yield
𝐶(Ω(𝜃)) > 𝐶(Ω(𝜃 − 𝜃𝑖 )) + 𝐶 ′ (Ω(𝜃 − 𝜃𝑖 )){Ω(𝜃) − Ω(𝜃 − 𝜃𝑖 )}
= 𝐶(Ω(𝜃 − 𝜃𝑖 )) + (𝜃 − 𝜃𝑖 ){Ω(𝜃) − Ω(𝜃 − 𝜃𝑖 )}.
(A2)
Similarly we have
𝐶(Ω(𝜃 − 𝜃𝑖 )) > 𝐶(Ω(2𝜃𝑖 )) + 2𝜃𝑖 {Ω(𝜃 − 𝜃𝑖 ) − Ω(2𝜃𝑖 )},
𝐶(Ω(2𝜃𝑖 )) > 𝐶(Ω(𝜃𝑖 )) + 𝜃𝑖 {Ω(2𝜃𝑖 ) − Ω(𝜃𝑖 )}.
(A3)
(A4)
Substituting (A3) and (A4) in (A2), we get
𝐶(Ω(𝜃)) − 𝐶(Ω(𝜃𝑖 )) > (𝜃 − 𝜃𝑖 ){Ω(𝜃) − Ω(𝜃 − 𝜃𝑖 )} + 2𝜃𝑖 {Ω(𝜃 − 𝜃𝑖 ) − Ω(2𝜃𝑖 )} + 𝜃𝑖 {Ω(2𝜃𝑖 ) − Ω(𝜃𝑖 )}.
(A5)
(A1) and (A5) yield
𝜋𝑖𝐹 (𝑁 ∖{𝑖}) − 𝜋𝑖𝐴 (𝑁 ) > (𝜃 − 𝑛𝜃𝑖 ){Ω(𝜃) − Ω(𝜃 − 𝜃𝑖 )} + 2𝜃𝑖 Ω(𝜃 − 𝜃𝑖 ) − 𝜃𝑖 {Ω(2𝜃𝑖 ) + Ω(𝜃)}
= (𝜃 − 𝑛𝜃𝑖 ){Ω(𝜃) − Ω(𝜃 − 𝜃𝑖 )} + 𝜃𝑖 {Ω(𝜃 − 𝜃𝑖 ) − Ω(2𝜃𝑖 )} − 𝜃𝑖 {Ω(𝜃) − Ω(𝜃 − 𝜃𝑖 )}
≥ (𝜃 − 𝑛𝜃𝑖 ){Ω(𝜃) − Ω(𝜃 − 𝜃𝑖 )} + 𝜃𝑖 (𝜃 − 4𝜃𝑖 )Ω′ (𝜃 − 𝜃𝑖 ) ≥ 0,
9
(A6)
which indicates 𝜋𝑖𝐹 (𝑁 ∖{𝑖}) − 𝜋𝑖𝐴 (𝑁 ) > 0. We have used inequalities Ω(𝜃 − 𝜃𝑖 ) − Ω(2𝜃𝑖 ) ≥ (𝜃 − 3𝜃𝑖 )Ω′ (𝜃 − 𝜃𝑖 ) and
Ω(𝜃) − Ω(𝜃 − 𝜃𝑖 ) ≤ 𝜃𝑖 Ω′ (𝜃 − 𝜃𝑖 ), derived from the properties of Ω (Ω′ > 0, and Ω′′ ≤ 0), in the calculation between
the second and the third lines of (A6).
□
2. Proof of 𝜋𝑖′ (𝑏𝑖 ) < 0
From (16) and (17),
𝜋𝑖 (𝑏𝑖 ) =
{
(
)}
𝜃𝑗
𝜃𝑖
{𝜃 + (𝑏𝑖 − 1)𝜃𝑖 } Ω(𝜃 + (𝑏𝑖 − 1)𝜃𝑖 ) − Ω(𝜃) 1 −
𝜃𝑗
𝜃𝑛
(
)
𝜃
+𝜃𝑖 Ω(𝜃) 𝑛 −
− 𝐶(𝑥∗𝑖 ).
𝜃𝑛
(A7)
Differentiating (A7) with respect to 𝑏𝑖 , we get
[
(
)
]
𝜃2
𝜃𝑗
𝜋𝑖′ (𝑏𝑖 ) = 𝑖 Ω(𝜃 + (𝑏𝑖 − 1)𝜃𝑖 ) − Ω(𝜃) 1 −
+ {𝜃 + (𝑏𝑖 − 1)𝜃𝑖 }Ω′ (𝜃 + (𝑏𝑖 − 1)𝜃𝑖 )
𝜃𝑗
𝜃𝑛
[ {
(
)}
]
𝑏𝑖 𝜃𝑖2 ′
𝜃𝑗
𝜃
𝑖
−𝐶 ′ (𝑥∗𝑖 )
Ω(𝜃 + (𝑏𝑖 − 1)𝜃𝑖 ) − Ω(𝜃) 1 −
+
Ω (𝜃 + (𝑏𝑖 − 1)𝜃𝑖 )
𝜃𝑗
𝜃𝑛
𝜃𝑗
{
(
)}
𝜃
𝜃𝑖
𝑗
=
(𝜃𝑖 − 𝐶 ′ (𝑥∗𝑖 )) Ω(𝜃 + (𝑏𝑖 − 1)𝜃𝑖 ) − Ω(𝜃) 1 −
𝜃𝑗
𝜃𝑛
+
𝜃𝑖2 ′
Ω (𝜃 + (𝑏𝑖 − 1)𝜃𝑖 ){𝜃 + (𝑏𝑖 − 1)𝜃𝑖 − 𝑏𝑖 𝐶 ′ (𝑥∗𝑖 )}.
𝜃𝑗
(A8)
Through simple calculations we have 𝑥∗𝑖 > Ω(𝜃), that is, 𝐶 ′ (𝑥∗𝑖 ) > 𝜃. Since 𝜃𝑖 − 𝐶 ′ (𝑥∗𝑖 ) < 0 and 𝜃 + (𝑏𝑖 − 1)𝜃𝑖 −
𝑏𝑖 𝐶 ′ (𝑥∗𝑖 ) < 𝜃 + (𝑏𝑖 − 1)𝜃𝑖 − 𝑏𝑖 𝜃 = (𝑏𝑖 − 1)(𝜃𝑖 − 𝜃) < 0 hold, the third and the fourth lines of (A8) are both negative.
Therefore, 𝜋𝑖′ (𝑏𝑖 ) < 0.
□
References
Barrett, S. (1994) Self-Enforcing International Environmental Agreements, Oxford Economic Papers, 46, 878-894.
Barrett, S. (2001) International Cooperation for Sale, European Economic Review, 45, 1835-1850.
Carraro, C., and D. Siniscalco (1993) Strategies for the International Protection of the Environment, Journal of
Public Economics, 52, 309-328.
Guttman, J. M. (1978) Understanding Collective Action: Matching Behavior, American Economic Review, 68,
251-255.
Guttman, J. M. (1987) A Non-Cournot Model of Voluntary Collective Action, Economica, 54, 1-19.
Guttman, J. M., and A. Schnytzer (1992) A Solution of the Externality Problem Using Strategic Matching, Social
Choice and Welfare, 9, 73-88.
Hoel, M., and K. Schneider (1997) Incentives to Participate in an International Environmental Agreement, Environmental and Resource Economics, 9, 153-170.
Lange, A., and C. Vogt (2003) Cooperation in International Environmental Negotiations due to a Preference for
Equity, Journal of Public Economics, 87, 2049-2067.
Na, S.-l., and H. S. Shin (1998) International Environmental Agreements under Uncertainty, Oxford Economic
Papers, 50, 173-185.
Petrakis, E., and A. Xepapadeas (1996) Environmental Consciousness and Moral Hazard in International Agreements
to Protect the Environment, Journal of Public Economics, 60, 95-110.
Rübbelke, D. T. G. (2006) Analysis of an International Environmental Matching Agreement, Environmental Economics and Policy Studies, 8, 1-31.
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