GraSPP-DP-E-09-002 and ITPU-DP-E-09-001

Analysis on Bargaining about Global Climate
Change Mitigation in Aviation Sector

Katsuya Hihara
July 2009

GraSPP Discussion Paper E-09-002

ITPU Discussion Paper E-09-001

GraSPP-DP-E-09-002 and ITPU-DP-E-09-001

Analysis on Bargaining about Global Climate
Change Mitigation in Aviation Sector

Katsuya Hihara
July 2009

Professor, Graduate School of Public Policy
The University of Tokyo
7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan
Phone: +81-3-5841-1710
Fax: +81-3-5841-7877
hihara@pp.u-tokyo.ac.jp

GraSPP Discussion Papers can be downloaded without charge from:http://www.pp.u-tokyo.ac.jp/
Discussion Papers are a series of manuscripts in their draft form. They are not intended
for circulation or distribution except as indicated by the author. For that reason
Discussion Papers may not be reproduced or distributed without the written consent of
the author.

Analysis on Bargaining about Global Climate
Change Mitigation in Aviation Sector
Katsuya Hihara ∗ITPU/GraSPP/University of Tokyo
2009/07/23 1400 JST

Abstract
Understanding the basic interaction mechanism among nations surrounding the CO2 emissions is critically important for the policy formulation analysis in aviation sector at present. We performed simulation analysis on the international climate change policy, especially on
the eﬀects on pricing of emission allowances by including major players such as China and India into the hypothetical global CO2 emission trading scheme according to non-cooperative game framework.
In the presence of negative public goods, i.e., CO2 , we extended the
Lindahl-Bowen-Samuelson condition to include a class of uncertainty
into utility. By using the result, we explained, with some numericalexamples, the welfare eﬀects caused by the changes of factors, such aslevel of uncertainty, degree of risk averse, asymmetric utility structure,
initial allocation among players, based on our model surrounding thebargaining of CO2 emissions allocation games.

JEL Classiﬁcation Numbers:
C70,C72,L93
key words
CO2 emission allowance, Lindahl-Bowen-Samuelson Condition,
Uncertainty, Air Transport∗

hihara@pp.u-tokyo.ac.jp The author thanks Professor Kanemoto for valuable comments.

1

1

Introduction

The policy formulation needs for Climate Change mitigation in the aviation
sector is very much urgent, since Kyoto Protocol’s period ends 2012 and wehave to agree about the Post-Kyoto Protocol structure by the end of 2009.
Kyoto protocol designated to ICAO the task of mitigating the growth of international aviation to meet the challenge of sustainability under the threatof global climate change1 . The ICAO is the international organization andtheir major decisions always entail the negotiation and bargaining among
the contracting nations.2
Therefore, in order to formulate the eﬀective mitigation policy for international aviation and climate change, we need to know the basic mechanism
about the negotiation and bargaining and also need to understand the social
welfare eﬀect of such bargainings.
The negotiation is the sequence of the following phases.
i)The negotiation/bargaining on global emission level3ii)The negotiation/bargaining on the rule of distribution of initial emission
allowance among countries4 .
iii)The negotiation/bargaining on the allowances distribution among countries5 .

1
Kyoto Protocol article2 2. ”The parties shall pursue limitation or reduction of emissions of greenhouse gases... from aviation ..., working through the International Civil
Aviation Organization....”
2
ICAO, in order to perform the task of mitigation, set up a special high level group(GIACC) among selected counties to discuss globally aspirational goals and other measures. GIACC adopted GIACC Report 2009 [3] in May session. According to the report,
the group agreed the goal of 2 % annual fuel consumption eﬃciency (liter per revenue
ton-kilometer) reduction from the 2005 level until 2050.
3
Global emission level in ﬂow base is , for example, 10 % reduction of the annual global
emission. Global emission level in stock base is , for example, 10 % reduction of the global
GHG concentration (CO2concentration) in the earth atmosphere.
4
The negotiation of total amount and that of rule of distribution could be interchangeable, i.e., ﬁrst negotiate on the rule of distribution and then negotiate on the total amount.
5
In the same sense, the negotiation/bargaining of rule of distribution and that of theallowance distribution among countries could be interchangeable, i.e., ﬁrst negotiate on
the rule of distribution and then negotiate on the allowance distribution among countries.

2iv)The negotiation on the emission allowance among individual and cooperation within each country.
Here we use the recent method of game theory and bargaining theory tosimulate the negotiation process and extend the pareto optimal condition
with the presence of public goods, which is in this case, CO2 emission into
the earth atmosphere. With the new optimal condition with the uncertainty,
we did some welfare eﬀect analysis by comparative statics using numericalexamples.

2

Emission Allowance Allocation Process Simulation

2.1

Past Literature

A lot of analyses were performed about the emission allowance allocation
process, like that of the Kyoto Protocol Negotiation process. Okada (2004) [10],for example, analyzed the bargaining mechanism for the initial allowance andreduction costs by the method of non-cooperative game theory based on theempirical work done by the Nordhaus(1991) [14] and Bohm=Larsen(1994) [1].
Also a lot of cooperative game theoretic analyses were done about theallowance distribution bargaining. The work done by Tadenuma in Imaiand Okada (2005) [4] shows there is a stable coalition set (von NeumannMorgenstern solution) even though, according to Okada [9], the core of thevoting game on distributions of a ﬁxed total amount of emission allowances
is empty.
Here we focus on non-cooperative game framework and make some simulation of including new members into the bargaining.

2.2

Base Model

According to the work by Nordhaus(1991) [14]: Bohm=Larsen(1994) [1]:Okada(2004) [10], we have the formula for the competitive price of carbon
emission allowances under the framework of non-cooperative game approach,indicated below.
3Let N = 1, ..., n be the set of counties. For every i ∈ N , we denote by
Ei country i’s current level of carbon emission.
The total level of carbon
Pemitted by n countries is given by E = i∈N Ei . xi denotes the country
i’s reduction of P
carbon emission. ωi is the emission allowance allocation for
country i. ω̄ = i∈N ωi

E − ω̄
)i∈n Ei (1 − ri )E(1 − ri )x∗i = P(E − ω̄)i∈N Ei (1 − ri )cei = 185.2x∗i + p∗ (Ei ri − ωi )½
i(ei ≤ eU SA )1 − eUeSA
rl =
eU SA
ei − 1 (eU SA ≤ ei )p∗ = −185.2 ln(1 − P

ei =

Ei
GDPi

(1)
(2)
(3)
(4)
(5)

ei is the carbon intensity of country i, which is emission lebel Ei over GDP.
p∗ is the competitive equilibrium price of this negotiation. x∗i is the equilibrium reduction of CO2 emissions for country i. cei is the cost of country iwith the initial allocation ωi , competitive equilibrium price p∗ and reduction
amount x∗i .
This model is about the mechanism of negotiation phase iii) in the Introduction based on the relative reduction cost of each nation with that of U.S.
By the equations (1) through (3), we can simulate any number of countries
with any type of initial allowances, or any type of emission intensity.
However, since these numbers are available, based on empirical works, only
for 1990, we can only perform the simulation about 1990, and cannot do
about other years like 2008, for example.

2.3

Simulation

What would have happened if major players, like China and India were included in the Kyoto Protocol and the hypothetical global emission trading4

in 1990 ?
Table 1 below is the basic data for the carbon emission level(million ton
carbon, GDP in US dollars, carbon intensity for EU 15 nations, Former Soviet Union nations, Japan, U.S., China (including Hong Kong and Macau),
Korea (not including DPRK) and India.
(Table 1 is about here.)If the hypothetical emission trade happens in 1990 among the original contracting countries of EU 15 nations, Former Soviet Union nations, Japan,
U.S., what would happen? The simulation result is indicated in Table 2.
The equilibrium price is about 9.65 US dollars per ton Carbon6 .

(Table 2 is about here.)If the major emission countries, like China, Korea and India also participatedin the hypothetical emission trading with the obligation of zero reduction
like Former Soviet Union nations, the results are in Table 3.
(Table 3 is about here.)
The equilibrium price is about 6.65 US dollars, since the new entrants are
not obliged to reduce the emission level and they can sell emission allowances
to U.S., EU15 and Japan.
What happens if the reduction obligation increases among the three countries from zero percent to higher percentages is in Figure 1.
(Figure 1 is about here.)
According to our basic model, 1 percent increase of reduction obligation
among these three nations lead to about 33 cents increase of the emission
6

Notice that the price here is US$ per ton Carbon, not US$ per ton CO2 . External
cost from car gasoline consumption is about 5,000 yen to 50,000 yen per ton Carbon, and
30,000 yen per ton Carbon is the medium estimate according to Kanemoto et.al.(2006) [7].
According to IPCC Assessment Report 4 (Synthesis Report p69) [6] , the average social
cost of CO2 based on 100 estimates is about 12 US$ per ton of CO2 for 2005, although
the estimation range is from -3 to 95 US$.

5

allowance price, which is the slope of our estimated line in Figure 1.
What happens if carbon intensity of these three countries improves by 1
percent, the emission allowance price would go up by about 2 cent. Thismechanism is depicted in Figure 2. The slope of the estimated line is about
0.02.

(Figure 2 is about here.)With these simulations, more players’ participation can reduce the price ofemission allowance but the situation depends on the factors like the reduction obligation of the new players and carbon intensity variations.

3

Welfare Analysis with a Negative Public Goods

3.1

Past Literature

Emission charge on airlines is studied by several authors on the eﬀect on airfares, service quality, and aircraft design, like Brueckner and Zhang(2009)[2].
In more general context, including other sectors than transportation, a lotof literature on the Climate Change Policy is published. For example, Guesnerie and Tulkens(2008) [11] and Yang(2008) [15] are among the most recent
publications.
But a limited number of literatures are on the welfare impact analysis using the characteristics of CO2 as negative public goods and further limited
number of literatures are based on the game theory or bargaining theory
analysis in addition to the characteristic as negative public goods. This
study explores the welfare implication analysis, using the characteristic ofnegative public goods, by game theory and bargaining theory.

3.2

Lindahl-Bowen-Samuelson condition【Lindahl-Bowen-Samuelson condition】
The condition for the allocation in the economy with a public good to be6

pareto eﬃcient is that the sum of all members’ marginal rate of substitution of private goods for the public good is equal to the marginal rate oftransformation of private goods for the public good.
n
X
uis+j
i=1

uir

=

Fs+j
(i = 1, 2, ..., n; r = 1, 2, ..., s; j = 1, 2, ...m)Fr(6)

n is the number of the players. s is the number of private goods and r istheir index. m is the number of public goods and j is their index. u is theutility function. F is the production function.
This is the condition of Lindahl-Bowen-Samuelson (=LBS condition) basedon Samuelson [12].The CO2 emission quantity is the negative public good.
So in the case of the CO2 emission, we could directly use the LBS condition
for the pareto eﬃcient allocation of the CO2 emission.

3.3

Extension of LBS condition to consumption externalities

In the presence of external eﬀect from the consumption as well as that fromproduction, we can use the extension of the Lindahl-Bowen-Samuelson according to Tadenuma [13]. This is also the negotiation/bargaining phase ofiii) depicted in the Introduction.
There are n countries, N = 1, ..., n. Let yi ∈ R+ denote the gross domestic
product(GDP) of country i ∈ N , ci ∈ R+ the consumption of country i.
Both production and consumption are accompanied by emissions of greenhouse gases. Let xpi ∈ R+ denote the emission of greenhouse gases fromproduction.
The relation of xpi and yi is represented by the function xpi = fi (yi ), wherefi0 > 0, fi00 > 0.
Let xci ∈ R+ denote the emission of greenhouse gases from consumption.
The relation of xci and ci is represented by the function xci = gi (ci ), where
gi0 > 0, gi00 ≥ 0.
Let xi ≡P
xpi + xci be the total emission of greenhouse gases of country i, and
let X ≡ i∈N xi be the global emission of greenhouse gases.
We assume that there is an amount X̂ of global emission of greenhousegases such that the human beings cannot survive if the emission exceeds
7X̂. Each country i has the preferences over pairs (ci , X) ∈ R+ × [0, X̂] of
its own consumption and a global amount of emissions of greenhouse gases.
The preferences are represented by a continuously diﬀerentiable and strictly
quasi-concave function Vi : R+ × [0, X̂] → R. We call the function Vi thewelfare function of country i.
Problem A

max

3n(y,c,x)∈R+

Vi (ci ,

X

xh )

(7)

h∈N

subject to
xj = f (yj ) + g(cj ) (∀j ∈ N )
X
X
yh =
ch
h∈N

Vi (ci ,

X

(8)
(9)

h∈N

xh ) = Vˇj (∀j ∈ N, j 6= i)

(10)

h∈N

The condition (10) could be the Individual Rationality condition that guarantees participation is rational for the players, since Vˇj is the minimum
utility the player can get elsewhere.
Solving this problem, we can get the extended condition of LBS to include
the external eﬀect from consumption. The Lagrangean for the problem A is
as follows;

L((ch )h∈N , (xh )h∈N , (yh )h∈N , (λh )h∈N , (γh )h∈N , δ)X
X
X
≡ Vi (ci ,xh ) −λj (Vj (cj ,xh ) − Vˇj )
i∈N

−

X

j6=i

h∈N

γj (xj − fj (yj ) − gj (cj )) − δ(

j∈N

X

h∈N

From the ﬁrst order condition,

8

yh −

X
h∈N

ch ) (11)∂Vi (c∗i ,P∗h∈N xh )+ γi gi0 (c∗i ) + δ = 0∂ci
PP∂Vi (c∗i , h∈N x∗h ) X ∂Vj (c∗j , h∈N x∗h )λj−− γi = 0
∂X
∂X

(12)
(13)

j6=i

γi fi0 (yi∗ ) − δ = 0(14)+ γj gj0 (c∗j ) + δ = 0∂cj
PP∂Vi (c∗i , h∈N x∗h ) X ∂Vj (c∗j , h∈N x∗h )λj−− γj = 0
∂X
∂X

(15)

and for each j 6= i,−λj∂Vi (c∗i ,P∗h∈N xh )

(16)

j6=i

γj fj0 (yj∗ ) − δ = 0(17)

Solving these equations, we get the following Extended condition of LBS
including the consumption externality, shown in Tadenuma [13].
[Extended Condition of LBS for Consumption Externalities]
The allocation (y ∗ , c∗ , x∗ ) is pareto eﬃcient, if the following is satisﬁed.
X µ X ¶
(18)
νi c∗i ,x∗h (fi0 (yi∗ ) + gi0 (c∗i )) = 1
i∈N

h∈N

¯
¯
¯ ∂Vi (ci ,X) ¯¯ ∂X ¯νi (ci , X) ≡ ¯ ∂Vi (ci ,X) ¯¯ ∂ci ¯This is the extension of Lindahl-Bowen-Samuelson condition in a sense thatthe impact of CO2 emission from consumption as well as that of production
are considered. Speciﬁcally, at Pareto optimal allocation, the weighted sumof the marginal rate of substitution of consumption for global emission of
GHGs over all the countries is equal to one, where the each weight is thesum of the marginal emission from production and the marginal emission
from consumption in each country.

9

3.4

Extension of LBS condition to Utility with Uncertainty

So far we deal with the pareto optimal condition with a negative public
goods. This setting has no uncertainty. In other words, the setting is basedon perfect information.
Now we try to extend the LBS condition further to uncertain world. If we
add the uncertainty to utility function, then the Problem A becomes the
new maximization problem with the uncertainty. But in order to track theutility function and constraints, we need to set the structure of the relationship between the uncertainty and the other factors in utility function,
namely consumption ci ,production yi , and their emission levels, fi (ci ), gi (ci ).
In order to set the structure, we introduce the following assumption.
【Assumption 1】
The uncertainty is linearly separable from the economic activities. In otherwords, the uncertainty does not exist in yi , consumption ci , their emissionfunction, f (yi ), or g(yi ), or their relation ship in the utility function.

This means that the uncertainty is not about observation accuracy or accounting consistency, but the uncertainty is purely the remaining category
other than human economic activities, i.e., consumption expenditures, orthe emission amount of CO2 from each country’s human activities.
So this uncertainty could be the remaining uncertainty after we know the exact amount of CO2 emission. This could be, for example, such uncertainty
about the net ultimate eﬀect of CO2 emission through the earth ecology
system on our utility level even if we know the exact level of consumption
expenditures and the entailing exact emission level of CO2 in each country.7

7

According to ”6 Potential Climate Change from Aviation” in IPCC Aviation Report(1999) [5], CO2 is, unlike ozone and water vapor perturbations, one of well-mixed
gases, and there is small uncertainty in calculating radiative forcing (RF, a single measureof climate change deﬁned by IPCC, which calculates the global annual average of radiative
imbalance (W/m2 ) to the atmosphere-land-ocean system caused by anthropogenic perturbations and sets the RF of pre-industrial atmosphere to be zero). Still the RF for aviation
CO2 in 1992 based on NASA-1992 aviation scenario, for example, is estimated to be+0.018(W/m2 ) with a likely range of ±30% that includes uncertainties in the carbon cycle and in radiative calculations for a ﬁxed amount of fuel burn (160.3(million tons/year))and a ﬁxed CO2 concentration level (1.0 ppmv).

10

By Assumption 1, our uncertainty is linearly separated from production
yi , consumption ci or their emission function, f (yi ), and g(yi ).
With this assumption of the linear separability of uncertainty, Problem
A now becomes Problem B below. By solving this problem B, LindahlBowen-Samuelson condition can be extended further to the situation where
uncertainty exists.
Problem B(19)

xj = f (yj ) + g(cj ) (∀j ∈ N )
X
X
yh =
ch

(20)

3n
(y,c,x)∈R+

E[Vi (ci ,Xxh , ε)]maxh∈N

subject toE[Vi (ci ,

X

h∈N

(21)

h∈N

xh , ε)] = Vˇj (∀j ∈ N, j 6= i)

(22)

h∈N

ε could be a random variable according to any probability distribution. In
this sense, since we do not know the probability distribution of ε yet, what εrepresents is not ”risk” but ”uncertainty” in the meaning of Frank Knight’s
terminology.
Because of Assumption 1, we can describe the equality constrains without
quoting ε 8 .
But solving Problem B is not simple. Here we introduce additional assumption.
First, we deﬁne V̄i as the expected value of Vi (ci ,V̄i (ci ,Xxh ) ≡ E[Vi (ci ,

h∈N

X

P

h∈N xh , ε);xh , ε)]

(23)

h∈N

8

The uncertainty here is only one dimension. The structure, however, can be extended
to multi-dimension uncertainties without loss of generality.

11

【Assumption 2】
ε is from normal distribution, N (µ, σ 2 ) and utility function is CARA (Constant Absolute Risk Aversion) utility function.

Under Assumption 1 and Assumption 2, the equation (23) becomesX
X
V̄i (ci ,xh ) = E[Vi (ci ,xh , ε)]h∈Nh∈N= E[− exp{−ηi (Hi (ci ,Xxh ) + ε)}]

(24)

h∈N

P
P
where Hi (ci , h∈N xh ) is the relationship function between ci and h∈N xh
without any uncertainty. This can be possible because of the linear separability of uncertainty from other economic activities under Assumption 1
and the CARA utility function’s characteristics form Assumption 2.
So we get,V̄i (ci ,Xxh ) = − exp{−ηi (Hi (ci ,Xxh ))}E[− exp{−ηi (ε)}]

h∈N

h∈N

= − exp{−ηi (Hi (ci ,Xxh ) − µηi +h∈N

ηi2 σ 2}2(25)

Notice that under Assumption 2, ε ∼ N (µ, σ 2 );E[− exp{−η(ε)}]

Z ∞−(ε−µ)21
e 2σ2 dε
2πσ
−∞
Z ∞
2
2
−(ε−(µ−σ 2 η))21
−ηµ+ η 2σ
2σ 2
√
= e
e
dε
2πσ −∞=

e−ηε √= e−ηµ+

η2 σ2
2

·1 .

Now we can derive the extension of LBS condition to include the uncertainty
described above. The equations (19) through (22) become as follows;12

Problem B?

max

3n(y,c,x)∈R+

V̄i (ci ,

X

xh )

(26)

h∈N

subject to
xj = f (yj ) + g(cj ) (∀j ∈ N )
X
X
yh =
ch
h∈N

V̄i (ci ,

X

(27)
(28)

h∈N

x∗h ) = Vˇj (∀j ∈ N, j 6= i)

(29)

h∈N

By the equation(25), we can treat the equations (26) through (29) without
any random variables. These equations in Problem B? can be solved by
the usual maximization problem just as in Problem A.
Deﬁne the Lagrangean as follows;

L((ch )h∈N , (xh )h∈N , (yh )h∈N , (λh )h∈N , (γh )h∈N , δ)XX
X
λj (V¯j (cj ,xh ) − Vˇj )≡ V̄i (ci ,xh ) −
i∈N

−

X

j6=i

h∈N

γj (xj − fj (yj ) − gj (cj )) − δ(

j∈N

X

h∈N

yh −

X

ch ) (30)h∈NP
Notice that the term VPi (ci ,i∈N xh ) in the Lagrangean (11) in Problem
A now bocomes V̄i (ci , i∈N xh ) the Lagrangean (30) in Problem B? .
Other than this, the Lagrangean is the same as in solving the Problem A.
Thus we can get the following familiar condition.

13

As the ﬁrst order condition,∂ V̄i (c∗i ,P∗h∈N xh )+ γi gi0 (c∗i ) + δ = 0∂ci
PP∂ V̄i (c∗i , h∈N x∗h ) X ∂ V¯j (c∗j , h∈N x∗h )−λj− γi = 0
∂X
∂X

(31)
(32)

j6=i

γi fi0 (yi∗ ) − δ = 0(33)+ γj gj0 (c∗j ) + δ = 0∂cj
PP∂ V̄i (c∗i , h∈N x∗h ) X ∂ V¯j (c∗j , h∈N x∗h )−λj− γj = 0
∂X
∂X

(34)

and for each j 6= i,−λj∂ V¯j (c∗j ,P∗h∈N xh )

(35)

j6=i

γj fj0 (yj∗ ) − δ = 0(36)

Notice also that;∂ V¯i∂Hi ∂ V¯i
∂Hi
∂X = V̄i (−ηi ) ∂X , ∂ci = V̄i (−ηi ) ∂ci

We get the following result;
¯ ¯
¯
¯
¯
¯ ∂ Vi (ci ,X) ¯¯ V̄ (η ) ∂Hi ¯¯ ∂X ¯¯ i i ∂X ¯
νi (ci , X) ≡ ¯ ∂ V¯ (c ,X) ¯ = ¯¯¯ i i ¯¯ V̄i (ηi ) ∂Hi ¯
∂ci
∂ci
¯
¯
¯ ∂Hi (ci ,X) ¯¯ ∂X ¯= ¯ ∂H (c ,X) ¯¯ i i ¯(37)∂ci

By solving these equations (31) ∼ (36) and the equation (37), we can describe
the extended condition of LBS for uncertainty.
[Proposition 1: Extended Condition of LBS for Uncertainty]
Under Assumption 1 and Assumption2, at Pareto optimal allocation, theweighted sum, over all the countries, of the marginal rate of substitution of
consumption for global emission of GHGs, which is composed only of the14certain part of the utility function with uncertainty, is equal to one, where
the each weight is the sum of the marginal emission from production andthe marginal emission from consumption in each country.
X µ X ¶
νi c∗i ,x∗h (fi0 (yi∗ ) + gi0 (c∗i )) = 1
(38)
i∈N

h∈N

¯
¯
¯ ∂Hi (c∗i ,X) ¯¯¯νi (c∗i , X) = ¯ ∂Hi∂X(c∗ ,X) ¯
i
¯ ∂c
¯
i

This is the extended condition of LBS to include the uncertainty under
Assumption 1 and Assumption 2. Notice that in νi (c∗i , X), we haveHi (c∗i , X), instead of Vi (c∗i , X). Hi (ci , X) is P
the certain (= without any
uncertainty) part of the utility function Vi (ci , h∈N xh , ε) with uncertainty
ε under Assumption 1 and Assumption 2. The derivation of Proposition
1 is in the appendix.

4

Welfare implication with numerical examples

4.1

Basic Model

Let N=1,2. The emission functions are deﬁned as follows;for every i ∈ N ,fi (yi ) = yi2(39)gi (ci ) = ci

(40)

3 → R is deﬁned as follows;
For each i ∈ N , the welfare function Vi :∈ R+

E[Vi (ci , X, ε)] = E[− exp{−ηi (Hi + ε)}](41)

Hi = cai i (10 − X)1−ai (0 ≤ ai ≤ 1)

(42)

Under the assumption of CARA utility function and the normal distribution15

for ε, we can get the follows;V̄i (ci , X) = − exp{−ηi (cai i (10 − X)1−ai ) − µηi +ηi2 σ 2}
2

(43)

If we pick the speciﬁc numbers for parameters a1 = 0.8, a2 = 0.2, then we
have the following equations;

2 2

η1 σ
0.2
V¯1 (c1 , X) = − exp{−η1 (c0.8}
1 (10 − X) ) − µη1 +2
η22 σ 20.8
V¯2 (c2 , X) = − exp{−η2 (c0.2
(10
−
X)
)
−
µη
+
}
2
2
2

(44)
(45)

With these equations (44) and (45) as well as Extended Condition of LBS
for uncertainty, we can derive the pareto frontier, bargaining frontier, disagreement point, Nash product9 based on Tadenuma(2003) [13], which is
the model for no uncertainty.
Here we do not go into the details for these frontiers and points, which are
explained in Tadenuma(2003) [13]. But the gist of them is as follows;
Pareto frontier is the locus of the welfare vectors of the two countries under
our study that can be attained by being satisfying the extended condition
of LBS depicted in the equation (18) or (38). Namely, the vectors (V1 , V2 )must satisfy the following relationships in our numerical example;

V1 = V¯1 (c1 (c2 ), X(c2 ))V2 = V¯2 (c2 , X(c2 ))(c1 + c2 + 1)((c1 + c2 )2
c1+ 4c2 ) − 10 ++ c1 + c2 = 0
4
2(c1 + c2 )2
X(c2 ) =+ c1 (c2 ) + c2
2

The last two equations can be derived from the extended condition of LBS
9

The bargaining model is theoretically founded in Nash(1950) [8]16in (18) or (38).The locus welfare vectors (V1 , V2 ) are drawn for the relevant
value for c2 .
Bargaining frontier is the locus of the welfare vectors that are attained atvarious levels of the total emission X with a given proportional rule, i.e.,the share of initial emission allowances, (θ1 , θ2 ) in our case, and emission allowance trading, entailing the trade price q(X). Namely, the vectors (V1 , V2 )in the locus are those that satisfy the following relationships.

Vi = V̄i (ci (X), X))
q(X)θi X
1+
ci =4q(X)(1 + q(X)) 1 + q(X)1
1
1
X= 2+ 2+4q
4q
q
In this settings, the q(X) is the inverse function of the last equation. Overthe relevant range of X, we have the locus of (V1 , V2 ), which is the bargaining frontier. Notice that the bargaining frontier depends on the initial
allocation,(θ1 , θ2 ) The bargaining frontier changes its shape according to thevalue of the initial allocation.
Disagreement point of the Nash bargaining theory is the Nash equilibrium
welfare levels of the two countries in the emission game without any regulation. Speciﬁcally, we have the two players who do not bargain with eachother and take the other’s emission level as given. Their utility function is
as follows;
η12 σ 2}2
η22 σ 22
0.8
V2 = V¯2 = − exp{−η2 (c0.2}
2 (10 − x1 − c2 − c2 ) ) − η2 µ +2V1 = V¯1 = − exp{−η1 (c10.8 (10 − x2 − c21 − c1 )0.2 ) − η1 µ +The best response functions are obtained by diﬀerentiating the functions
with respect to ci and setting the value to zero. Then, by solving these
response function equations, the Nash equilibrium consumption, emissions,and welfare levels are derived. This vector of the derived welfare, which17

could be thought as what you get if you fail to bargain, is the disagreement
point (d1 , d2 ).
Nash product is deﬁned by;(V1 − d1 )(V2 − d2 )Before we go further, we ﬁrst set the parameters as follows;µ=0
σ=1
η1 = η2 = 0.2
Also the initial allocation for the emission allowances for player 1 and player2
are θ1 and θ2 respectively. We set these as follows ;θ1 = 0.925
θ2 = 0.075
Figure 3 is the result loci for these frontiers and points.
(Figure 3 is about here.)
As in Tadenuma(2003) [13], pareto frontier and bargaining frontier touches
at one point, where each player’s emission levels are equal, i.e., x∗1 = x∗2 .
In this settings, disagreement point is within the bargaining frontier. So
it could be possible that starting from disagreement point both player canreach higher utility level by bargaining. But as Figure 3 shows, limit of
bargaining is not as high as the pareto frontier, of which is the point where
each player’s emission is the same, namely x∗1 = x∗2 .

4.2

Symmetric Structure World

If the both players’ Hi function is the same function (their degree of risk
averse was already set to be the same, i.e.η1 = η2 = 0.2), then the both18

players’ Vi functions are as follows;2 2η σ
V¯1 (c1 , X) = − exp{−η(c1 (10 − X)) − µη +}2
η2σ2V¯2 (c2 , X) = − exp{−η(c2 (10 − X)) − µη +
}
2

(46)
(47)

Furthermore, the initial allocation is even, namely,θ1 = 0.5
θ2 = 0.5
Under the setting, the bargaining frontier collapses as in Figure 4. In thisworld, everything is symmetric. So the both players can start with thedisagreement point(DP), moving along the bargaining frontier, now a line,
and the two can reach the pareto frontier and realize the social eﬃciency.
(Figure 4 is about here.)If the real world is symmetric as in ﬁgure 4, then the bargaining could leadto the social eﬃciency.

4.3

Impact of Uncertainty Increase

Back in the case of basic case, which is in Figure 3. If the uncertainty
increases, the σ would increase. In Figure 5, we depict the world as σincreases, namely, σ = 1, 2, 3, 4, 5, 6. As the uncertainty, namely σ, increases,
the world shrink to the down left corner.
(Figure 5 is about here.)This means you have to settle for smaller level of utility. Under uncertainty
increase settings, bargaining is almost surely more diﬃcult than otherwise.

19

4.4

Asymmetric Risk Aversion

If the player 1’s risk aversion level is more than that of the player 2, theplayer 1’s utility function changes like in Figure 6.
(Figure 6 is about here.)
The utility function is getting more skewed into the top upper left as therisk aversion parameter η increases, namely η = 0.2, 0.4, ..., 1.8, 2.0.
Under these settings, the player 1 evaluate more in the upper side of income,
namely higher x in Figure 6, to compensate the lower evaluation for lower
side of income. This is the content of risk aversion. As a result, under ourparameter setting the player 1 needs more than the player 2. So the worldin Figure 3 skewed to the right to allocate more to the player 1 just as in
Figure 7.
(Figure 7 is about here.)
All things equal, if the risk aversion of one of the player is more than theother, the player could ask for more in bargaining and this could lead tomore diﬃcult bargaining process since in the real bargaining process, you
cannot see the other players’ risk aversion parameter.

4.5

Initial Allocation Perturbation

To make the comparison easier, we go back to the even world in Figure 4.
The initial allocation for each player is even in Figure 4, namely,θ1 = 0.5
θ2 = 0.5
the disagreement point is on the bargaining frontier.
If the allocation is oﬀ from the even about 0.1 in our setting in Figure 4, thebargaining frontier dislocate from the original collapsed bargaining frontier(line) to the upper side or lower side depending on the plus or minus of thedeviation from the original allocation as in Figure 8.

20

(Figure 8 is about here.)In these cases, the disagreement point, which does not move, is out of bargaining frontier in either upper or lower case. Getting to the higher welfare
level through bargaining starting from the disagreement point is not feasible, let alone attaining the social eﬃciency frontier.

5

Conclusion

We performed simulation analysis on the international climate change policy,
especially on the eﬀects on pricing of CO2 emission allowances by including
major players such as China and India into the hypothetical trading scheme
in the aviation sector according to non-cooperative game theoretic framework.
In the presence of negative public goods, i.e., CO2 emission into the earth
atmosphere, we extended the Lindahl-Bowen-Samuelson condition so as toinclude, at least some class of the uncertainty about the utility. That is,under Assumption 1 and Assumption 2, we introduce the CARA utility
function and linearly separable uncertainty from consumption, production
and their emission function.
If uncertainty increases, then both Pareto Frontier and Bargaining Frontiershrink and make the negotiation harder, since the players have to settle forless than before. If the risk preferences are diﬀerent between the players,then the Pareto Frontier and Bargaining Frontier skews. So the simple allocation rule, like the same percentage reduction for diﬀerent countries, could
be diﬃcult to be agreed upon. More over, under the condition of diﬀerent
risk aversion levels and utility structure, with un-even initial allocations of
CO2 allowances, it is shown that reaching the bargaining frontier, let alone
pareto frontier by bargainings could be extremely diﬃcult.
The next step is to explore the price eﬀect of hypothetical global emission
trading in later year than 1990 in this study. The marginal cost data must
be investigated into the main counties in 2005 or other year, which could be
potentially the base year in the international negotiations. The other step isto go into the other class of uncertainty than Assumption 1. In this study,
the uncertainty is linearly separated from consumption, production or their21

emission functions. In reality the uncertainty could be attached to any ofthese aspects as well as the uncertainty linearly separable from them. But
the mathematical requirement to treat the more realistic model enhances
dramatically as more and more restrictions are being lessened.

We thank the Airport Environment Improvement Foundation for their ﬁnancial support.

A

Appendix Derivation of Proposition 1

As the ﬁrst order condition of the Lagrangean (30) of Problem B?, we havethe equations (31) ∼ (36).

∂ V̄i (c∗i ,P∗h∈N xh )+ γi gi0 (c∗i ) + δ = 0 (31)∂ci
PP
∂ V̄i (c∗i , h∈N x∗h ) X ∂ V¯j (c∗j , h∈N x∗h )λj−− γi = 0 (32)
∂X
∂X
j6=i

γi fi0 (yi∗ ) − δ = 0 (33)and for each j 6= i,∂ V¯j (c∗j ,P∗h∈N xh )+ γj gj0 (c∗j ) + δ = 0 (34)∂cj
PP
∂ V̄i (c∗i , h∈N x∗h ) X ∂ V¯j (c∗j , h∈N x∗h )−λj− γj = 0 (35)
∂X
∂X
−λj

j6=i

γj fj0 (yj∗ ) − δ = 0 (36)

By the equations (32) and (35),22γi = γj .
So with this result and the equations (33) and (36),fi0 (yi∗ ) = fj0 (yj∗ ).
The Problem B? can be formulated for j instead of i. So we get thesymmetric ﬁrst order conditions;∂ V¯j (c∗j ,P∗h∈N xh )+ γj gj0 (c∗j ) + δ = 0∂cj
PP∂ V¯j (c∗j , h∈N x∗h ) X ∂ V̄i (c∗i , h∈N x∗h )−λi− γj = 0
∂X
∂X

(48)
(49)

i6=j

γi fi0 (yi∗ ) − δ = 0(50)+ γi gi0 (c∗i ) + δ = 0∂ciPP∂ V¯j (c∗j , h∈N x∗h ) X ∂ V̄i (c∗i , h∈N x∗h )λi−− γi = 0
∂X
∂X

(51)

and for each i 6= j,−λi∂ V̄i (c∗i ,P∗h∈N xh )

(52)

i6=j

γi fi0 (yi∗ ) − δ = 0

By the equations (31) and (51), and the equations (34) and (48),λi = λj = −1.
From (33) and (50), δ = γfi0 (yi∗ ) = γfj0 (yj∗ ).
Hence with (31), we get the following;P∂ V̄i (c∗i , h∈N x∗h )= −γgi0 (c∗i ) − δ = −γ(fi0 (yi∗ ) + gi0 (c∗i ))∂ciPDeﬁne Dci V̄i (c∗i , h∈N x∗h );

23

(53)

X

Dci V̄i (c∗i ,∂ V̄i (c∗i ,x∗h ) ≡

h∈N

Then,

P

∗
h∈N xh )∂ci(fi0 (yi∗ ) + gi0 (c∗i ))1
P=−γ
Dci V̄i (c∗i , h∈N x∗h )The equation (32) becomes;∂ V̄i (c∗i ,P∂ V¯j (c∗j , h∈N x∗h )
−
λj
− γi
∂X
∂X
j6=i
P
P
∗
∂ V̄i (ci , h∈N x∗h ) X ∂ V¯j (c∗j , h∈N x∗h )
=
+
−γ
∂X
∂X
j6=i
P
∗
X ∂ V̄i (c∗i ,h∈N xh )
=
−γ =0
∂X
P

∗
h∈N xh )Xi∈N

Deﬁne DX V̄i (c∗i ,P∗h∈N xh );X

DX V̄i (c∗i ,∂ V̄i (c∗i ,x∗h ) ≡h∈NP∗h∈N xh )∂X

So, we have the following;
Xi∈N

DX V̄i (c∗i ,Xx∗h ) = γh∈NTherefore,)(P∗)
X DX V̄i (c∗i ,x
P h∈N h (f 0 (y ∗ ) + gi0 (c∗i ))Dci V̄i (c∗i , h∈N x∗h ) i i

i∈N

=

X
(fi0 (yi∗ ) + gi0 (c∗i )) XP
DX V̄i (c∗i ,
x∗h )
∗
∗
Dci V̄i (ci , h∈N xh )
i∈N

24

h∈N

1
= − γ = −1γThis means;
X µ X ¶
νi c∗i ,x∗h (fi0 (yi∗ ) + gi0 (c∗i )) = 1
i∈N

h∈N

¯
¯
¯ ∂Hi (c∗i ,X) ¯¯¯νi (c∗i , X) = ¯ ∂Hi∂X(c∗ ,X) ¯.
i
¯

∂ci

¯

This is the Extended Condition of LBS for Uncertainty.

References[1] P. Bohm and B. Larsen. Fairness in a tradable-permit treaty for carbon
emission reduction in europe and the former soviet union. Environmental and Resource Economics, 4:219–239, 1994.
[2] Jan K. Brueckner and Anming Zhang. Airline emission charges: Eﬀects
on airfares, service quality, and aircraft design. working paper, 2009.
[3] ICAO GIACC. GIACC REPORT 1 June 2009. ICAO, 2009.
[4] Haruo Imai and Akira Okada.
Japanese). Keiso Shobo, 2005.

Application of Game Theory (in[5] IPCC. AVIATION AND THE GLOBAL ATMOSPHERE.
BRIDGE UNIVERSITY PRESS, 1999.

CAM-[6] IPCC. Climate Change 2007: Synthesis Report (The Fourth Assessment Report). IPCC, Geneva, Switzerland, 2007.
[7] Yoshitsugu Kanemoto, Katsuhito Hasuike, and Tooru Fujiwara. Microeconomic Modeling for Policy Analysis (in Japanese). Toyo Keizai
Shimpo Sha, 2006.
[8] John F. Nash. The bargaining problem. Econometrica, 1950.

25

[9] A. Okada. A Market Game Analysis of International CO2 Emissions
Trading: Evaluating Initial Allocation Rules. Springer-Verlag, 2003.
[10] A. Okada. International negotiations on climate change: A noncooperative game analysis of the kyoto protocol. Discussion Paper
2004-2, Graduate School of Economics, Hitotsubashi University, 2004.
[11] Henry Tulkens Roger Guesnerie, editor. THE DESIGN OF CLIMATEPOLICY. The MIT Press, 2008.
[12] P. A. Samuelson. The pure theory of public expenditure. Review of
Economics and Statistics, 84:387–389, 1954.
[13] K. Tadenuma. International negotiations for reduction of green-housegases with emission permits trading. Project on International Equity(PIE) Discussion Paper Series, Hitotsubashi University, 2003.
[14] W.D.Nordhaus. The cost of slowing climate change: a survey. The
Energy Journal, 12:37–65, 1991.
[15] Zili Yang. Strategic Bargaining and Cooperation in Greenhouse Gas
Mitigation. The MIT Press, 2008.

26

Table 1

Country or Area
EU 15
FSU 22
Japan
US
China
Korea
India

CarbonEmission GDP in 90 Carbon
in90(Bil.US$) Intensity(Mil. Ton)
915
6,961
0.13
989
1,535
0.64
292
2,970
0.10
1,315
5,794
0.23
662
484
1.37
66
264
0.25
186
327
0.57

China includes Hong Kong and Macau.
Korea excludes DPRK.
Carbon Intensity is the carbon emission level per GDP (ei = Ei /GDPi ).

Table 2

Country
EU 15
FSU 22Japan
US

InitialEquilibrium
Reduction Reduction
P*
Permits
Cost
Rate(Mil.ton)(US$)(Mil. ton)
(Mil. US$)
0.08
0
0.06
0.07

73
0
18
92

841
989
275
1223

P* is the equilibrium price of carbon emission allowance

9.65
9.65
9.65
9.65

575
-402
138
563

Table 3

Countries
EU 15
FSU 22
Japan
US
China
Korea
India

InitialEquilibrium
Reduction Reduction
P*
Permits
Cost
Rate(Mil.ton)(US$)(Mil. ton)
(Mil. US$)
0.08
0
0.06
0.07
0
0
0

73
0
18
92
0
0
0

841
989
275
1223
662
66
186

6.65
6.65
6.65
6.65
6.65
6.65
6.65

China includes Hong Kong and Macau.
Korea excludes DPRK.
Carbon Intensity is the carbon emission level per GDP (ei = Ei /GDPi ).

424
-192
102
457
-143
-8
-35

Figure 1

Sensitivity of Reduction
Rate for 3 Countries

Emission Allowance Price P* (US$(90))

10.00

Reduction Rate +1%↓
Equilibrium Price P* +0.34～+0.32↓
dp*/dRR = +0.34～+0.32 US$↓d^2p*/d^2RR = +0.06 US$

9.00
8.00
7.00
6.00
5.00
4.00

P* ~ 6.65+++ + .3356+++ RR

3.00+++: significant with 0.001 or less p-value

2.00
1.00
0.00

Reduction Rate(RR) for 3 countries

Figure 2

3 Countries's Carbon Intensity Variation to
Emission Price
7.60

7.20

P* ~ 6.67*** －.024*** CIV

7.00***: significant with 0.001 or less p-value

6.80
6.60
6.40
6.20
6.00

3 Countiries' Carbon Intensity Variations

-3
0%

-2
6%

-2
2%

-1
8%

-1
4%

-1
0%

-6
%

-2
%

2%

6%

10
%

14
%

18
%

22
%

26
%

5.80
30
%

Emission Pri ce (US$)

7.40

Figure 3
Basic

-0.3
-0.4
-0.5

Nash Product
Bargaining Frontier

-0.6
V2

Limit of
Bargaining

* *
x1=x2

-0.7
Disagreement
Point

parameters;
myu=0;
sigma=1
eta1=eta2=0.2
theta1=0.925
theta2=0.075
f=0.8

-0.8
-0.9

Pareto Frontier

-1
-1.1
-1.1

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

V1

Figure4Ci(10-X) theta1=theta2=0.5

-0.2
-0.22

NP

PF

-0.24
-0.26
-0.28
V2

(V 1 , V 2 ) = (-0.30, -0.30)
-0.3
-0.32
DP

-0.34

(V 1 , V 2 ) = (-0.34, -0.34)

-0.36
BF
-0.38
-0.4
-0.4

-0.38

-0.36

-0.34

-0.32

-0.3
V1

-0.28

-0.26

-0.24

-0.22

-0.2

Figure 5s ig m a =1,2,3,4,5,6
-0.4

-0.6
σ＝1

V2

-0.8

-1
σ＝2

-1.2

σ＝3

-1.4

σ＝4
σ＝5

σ＝6

-1.6
-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

5

6

V1

Figure 6
e ta = 0 , 0 .2 , 0 .4 , ..., 1 .8 , 2 .0

0

η＝2.0

2

2

y=-exp(-eta*x-eta*myu+(1/2)*(eta )*(sigma ))- 0 .2

η＝0.4- 0 .4- 0 .6- 0 .8

η＝0-1- 1 .2- 1 .4- 1 .6- 1 .8

0

1

2

3
x

4

Figure 7
eta=0.2, 0.22, 0.24, 0.26, 0.28, 0.3

-0.3
-0.4

-0.5

V2

-0.6

-0.7

-0.8

-0.9

-1

-1.1
-1.1

-1

-0.9

-0.8

-0.7

η＝ 0.2

0.24

-0.6

-0.5

0.3

-0.4

V1

Figure 8(Theta1,theta2)=(0.6.0.4), (0.5,0.5) Ci(10-X)
-0.2
-0.22
NP

-0.24
-0.26

V2

-0.28
-0.3

BF
(theta1, theta2)=(0.4,0.6)

-0.32
-0.34
-0.36

BF
(theta1, theta2)=(0.5,0.5)

BF

-0.38
-0.4
-0.4

(theta1, theta2)=(0.6,0.4)
-0.38

-0.36

-0.34

-0.32

-0.3
V1

-0.28

-0.26

-0.24

-0.22

-0.2