Applied Mathematics, 2010, 1, 515-519
10.4236/am.2010.16068 Published Online December 2010 (http://www.SciRP.org/journal/am)
A Mathematical Approach to Develop the Distribution of
Greenhouse Gas Emissions*
Lev Sergeevich Maergoiz1, Tatiana Yur’evna Sidorova1, Rem Grigor’evich Khlebopros1,2
1Siberian Federal University, Krasnoyarsk, Russia
2Krasnoyarsk Research Centr e of Siberian Branch of Russ ian Academy of Sci e n c es, Kra s n oyarsk, Russia
Received September 18, 2010; revised October 17, 2010; accepted October 21, 2010
Abstract
A mathematical algorithm of the distribution of greenhouse gas emissions is proposed as a way to tackle the
topical issue of climate change and develop approaches to attaining an agreement among emitters of green-
house gases (on the global scale, in a country, a region, a megalopolis).
Keywords: Mathematical Algorithm, Extremal Problems, Greenhouse Effect, the Principle of Differentiated
Responsibilities
1. Introduction
Among the global issues that came to the fore in the 20 th
century is that human impact changes the Earths climate,
leading to global warming. The general public and even
some scientists still doubt the prevalence of human im-
pact among factors influencing the Earths climate. An
added complication is that if the prevalence of human
impact on the global climate is recognized, urgent meas-
ures will have to be taken to control human impact, and
this will cause a dramatic rise in the cost of living. Since
the early 1990s numerous attempts have been made to
overcome these difficulties at the international level, but
none of them have been successful, mainly due to the
lack of objective criteria for the so lution of this problem.
This study proposes a mathematical procedure for objec-
tivizing these criteria.
2. Climate Change and Sustainable
Development
There are two possible ways for humans to stabilize the
surface temperature of the Earth: by regulating parame-
ters of the greenhouse effect in the atmosphere and on
the Earth. These parameters can be changed by varying
not only atmospheric concentrations of greenhouse gases
but also surface reflectivity by changing the amount of
clouds at different heights. This idea was first proposed
in the 20th century [2]. This approach to controlling the
surface temperature of the Earth actually develops Ver-
nadskys idea of the noosphere, as applied to issues of
local and global climate control [3].
We can reduce our interference in natural processes by
maintaining the contemporary state of the atmosphere.
On the one hand, emissions of greenhouse gases (carbon
dioxide) due to combustion of nonrenewable energy
sources have to be considerably reduced. On the other
hand, total energy production should be increased in or-
der to maintain and improve the quality of life in devel-
oped countries and, what is even more important, to pro-
vide an opportunity for developing countries to attain a
comparable standard of living. In order to reduce emis-
sions of greenhouse gases due to combustion of carbon
fossil fuel, both its percent in the energy budget and its
actual amount should be decreased, by replacing it with
renewable sources of carbon fuel, wind power, water
power, and nuclear energy. It should be remembered,
though, that the use of alternative energy sources will
directly or indirectly increase the cost of power genera-
tion and, according to UNESCO estimates, must de-
crease the GDP by 1-2%. An important consideration is
that the effect of this loss on developed and developing
countries will be different: the use of alternative energy
sources can delay the achievement of high life quality in
developing cou ntries for d ec ades.
Let us discuss various ways to solve this problem. The
first was proposed by Dirk Solte [4]. The simplest, most
democratic, and equitable way to switch from the con-
temporary levels of emissions to the levels of emissions
*This article is written on the basis of the authors’ preprint [1].

L. S. MAERGOIZ ET AL.
516
equal to the threshold ,Vr at which the effect of hu-
mans on the global temperature becomes actually imper-
ceptible, is to set a quota, NVr :, where N is the global
human population. The quota for each country is propor-
tional to its population: nNVr ):( , where n is the popu-
lation of a given country. The difference between the
actual level of emissions and the quota for a country may
be either positive or negative. If this difference is posi-
tive, the country will have to buy quotas from the coun-
tries that have a negative difference (industrially unde-
veloped countries emit much smaller amounts of green-
house gases than their quota allows). The proposed algo-
rithm allows a nearly instant attainment of the maximal
level of greenhouse gas emissions necessary for the sta-
bilization of the global temperature, and the countries are
divided into three categories: the countries that buy quo-
tas (developed countries), the counties that gradually
reduce the amount of the quotas they sell (developing
countries), and those preferring to live off the environ-
mental endowment (selling the same or increased
amounts of the quotas). Although this way seems to be
simple, democratic, and equitable, it is actually not sim-
ple, democratic, or equitable. The first and most signifi-
cant drawback of this approach is that the expected effect
is too instantaneous and, like any sudden revolution, can
lead to numerous social and international catastrophes.
The second drawback is th at this algorithm does not take
into account a nations history. Thirdly, no account is
taken of the influence of geographic conditions: the quo-
tas for the people living in high-latitude areas and for
those living in the equatorial zone cannot be equal, as the
former have to heat their homes and other buildings.
The second approach, whose implementation is being
attempted now, is to get different countries, gradually
and to a greater or lesser extent, to reduce their emissions.
The legal basis for international control and reduction of
the human impact causing the greenhouse effect is cur-
rently provided by the UN Framework Convention on
Climate Change accepted in 1992 [5] and an addition to
it, the Kyoto Protocol adopted in 1997 [6]. One of the
basic principles of the Convention is that of differen-
tiated responsibilities. Th is principle states that the glob-
al nature of climate change calls for the widest possible
cooperation by all countries, specifically pointing out
that their participation should be determined by their
capabilities. Thus, highly developed countries are sup-
posed to take more serious measures and spend much
more money than less developed ones. However, inter-
national community has not reached an agreement on the
amounts of emissions to be reduced as the subjective
approach to determining them does not suit any country
in the world. It is the main defect of this approach.
Thus, in our opinion, the most topical issue today is
objectivization of the establishment of quotas.
3. A Mathematical Algorithm of
Solving problem
The problem of the distribution of greenhouse gas emis-
sions is solved using the algorithm having tested for dis-
tribution of monetary resource in problems of collective
investment management [7, 8].
3.1. Problem statement
Assume N groups of greenhouse gas emitters (on the
global scale, in a country, a region, a megalopolis) nego-
tiate on a certain admissible quantity V of greenhouse gas
emissions (in weight units) during a fixed time period.
Concentrate on the problem of the distribution of this
value among all group s of emitters taking into considera-
tion the size of the population in every group. In mathe-
matical terms this is sum partitioning of the value V:
=1
=,
N
k
k
VV
 (1)
where k V is an admissible quantity of e missions for the
group with number .k Let k
S be population of the
same group, =1,2, ,kN and
k
N
k
SS 
1=
=
be population of all groups. Denote by ,/= SVr
kkk SVr/= the mean value (density) of emissions per
capita of all population and for the group with number
,k where Nk ,1,=  respectively. By (1) it follows
the relation
=1 =.
N
kk
i
rS rS

Introduce the dimensionless values SSs kk /= (part of
population in the group with numberk), rrkk /=

(coefficient of proportionality), =1, ,kN. Then tak-
ing into account the previous equality we find
=1 =1
=1, =1.
NN
kk k
kk
ss 

(2)
Suppose that emitters reach to the following agreement:
conditional rating of every group is defined by the value
of the corresponding coefficient of proportionality.
Moreover, taking into consideration the principle of dif-
ferentiated responsibilities for climate change climate
groups differ from each other by the introduced rating,
and group indexing is given in ascending order of this
value, e.g.
12 1
0< << <<1<< < .
nn N

 

 (3)
Here inequality 1<
n

implies that n groups for

L. S. MAERGOIZ ET AL.
517
Nn< agree that their value (density) of emissions per
capita of population be less than the mean density r.
Mathematical approach to the choice of coefficients is
based on the following extremal p roblem.
Problem B. It is necessary to find values of parame-
ters

1
N
k
 , such that the functional


12
1
=1
=,
N
ii
i




 (4)
attains the minimum provid ed that Equality (2 ) holds and
the following additional linear relation between coeffi-
cients is fulfilled:
=1
=.
N
ii
i
db 
 (5)
Here 12 ,,, ,
N
ddd b are fixed numbers satisfying natu-
ral restrictions ensuring the condition: the Inequality (3)
is true.
Relation (5) can be a result of agreement among emitters.
For example, the equalities 1/=,
N

 or =,
N


where (0,1)

 or, correspondingly,(1, 1/)
N
s

, are
used in [7,8] for the distribution of the monetary resource
in problems of collective investment. In geometrical
terms the proposed optimal principle (see (4)) implies
that desired vector

11 1
=,, ,=,=1,,1
Niii
iN  


has the smallest length. Its coordinates are differences in
emission densities per capita for groups with adjacent
numbers. This approach to the choice of positive para-
meters ,=1,2, ,1
iiN

 is of great “psychological”
significance. The smaller their values, the easier it is to
come to the conclusion of the contract if emitters have
agreed with the principle of division into groups, which
is reflected in (3).
The obtained solution of the mathematical problem
under consideration provides a way to define the ad-
missible quantity of emissions for the group with number
k. In previous notations the following formula is correct
==,=1,,.
kkkkk
VrSsVk N

  (6)
3.2. Another Variant of the Problem Statement
The conditional rating of every group can be determined
based on another criterion, e.g. its living area. In this
case the previous notations have the following meaning:
k
S is the living area of the group with number ,k
where =1,2,, ,kN
=1
=N
k
k
SS
 is the total living area;
=/,= /
kkk
rVSr VS is, respectively, the mean quantity
of emissions per area unit of t he total living area and for the
group with number k, where =1, ,kN; =/
kk
s
SS is
the portion of the territory of this group; =/
kk
rr

,
=1, ,kN. Coefficients

1
N
k
 are chosen realized by
solving the same extremal problem.
3.3. A solution of Problem A
It is known (see [7]) the following solution of above
mentioned pr o bl e m provided that
12
0< << <,
N

  (7)
which differs from Inequality (3) by the absence of indi-
cation to a fixed number of parameters smaller than 1.
The problem whose statement has been changed in this
way well be called by Problem A.
A solution of Problem A. In the notations of (2), (5)
denote by
=1= 1= 1
=,=,= ,=1,,1,
NNN
ijij i
iijij
DdDdP sjN


 (8)
and also
0,,:=> 0,=1,,1,
jj
j
PD D
DbAj N
Db





11
2
=1 =1
<.
NN
j
jj
jj
A
PA


 (9)
Then the functional



 (see (4)) attains the min-
imum at the values of parameters

1
N
k
 in the form

111
1222
=1=1 =
1
=1, =1,
NkN
jj
j
jjj
k
jjjk
AP
A
PAP
AAA





(10)
where
1
22
=1
=,=2,,,
N
j
j
A
Ak N


moreover for Nk = the negative term in (10) vanishes.
Hence, by (3), (7) we also conclude that the solution of
Problem B (in comparison with Problem A) must satisfy
the additional inequality 1 <1<
nn


 (see also (10)).
A solution of Problem A is a solution of Problem B if
and only if the additional condition is fulfilled for para-
meters in (8)-(10)
 
11 1
=1==1= 1
1< ,>1;1>,
nNnN
j
jjjjjjj
jjnjjn
A
PAPnAPAP
 


 
<1.nN

(11)
Remarks. 1. Problem B is meaningful for2>N.
2. By (7) and (2) it follows that 1<1, >1
N

 (see
also (10)).
Let us prove that there is a collection of parameters
12
,,, ,
N
ddd b in the linear relation (5) that satisfies

L. S. MAERGOIZ ET AL.
518
conditions (9), (11), e.g. problem B has a nonempty set
of solutions. This assertion is reduced to determination of
a vector

11
=,,
N
AA A

 with positive coordinates
satisfying inequalities in Formulas (9), (11). Ind eed, then
assuming 0 =0,:==1,:=0,
N
bDDD by (8)-(9), we
determine desired parameters

1
N
j
d in (5) using for-
mulas
1
=,=1,,1;= ,
j
jjj jj
DPAjN dDD


=1,, .jN (12)
1. Let =1n in Inequality (3). Taking into account Re-
marks for the solution of Problem B, it is sufficient to
find relation (5) which ensures realization of inequality
2>1

. Therefore (see (10)), Inequality (11) is trans-
formed into

1
1111
=2
1=> .
N
j
j
j
A
PAsAP


In particular, this inequality is correct for
11
= 2,=,=2,,1,>0.
jj
AtsAtPjN t
Then assuming
1
22
111 =2
=21,=41,
N
j
j
BPs CsP



in the notations of (10) we have with these values of
coordina tes of vec tor

11
=,, :
N
AA A


122
=1 =<0,>/,
N
jj
j
A
PA tBCttBC


 (13)
i.e. the last inequality in (9) hold s for any fixed ./> CBt
So there exists a vector A with above mentioned proper-
ties. We do the same with

=1>2.nN N
2. Let 1.<<1 Nn Inequality (11) holds, e.g., for
=,=1,,1;=,=,
1
jnj
j
j
rs
AjnAtA
PP



=1,,1,jn N
where
 
1=1=, >0.nrN nstt
By analogy
with the case 1 (see (13)), we find a number 0>0t
such that for all 0 >tt, the chosen values of parameters
11
,,
N
A
A  satisfy the last inequality in (9).
3.4. Examples
For solution of Problem B it is necessary to find an ad-
missible variant of relation (5). We illustrate a possible
way to choose it for a case of three groups

=3 .N
Example 1. Assume in the notations of subsection 3.1
270V milliard tons, =3,=6NS milliard peoples,
123
==,SSS =2.n Then =1/3,
j
s =1,2,3,j
1=2/3,P 2=1/3P (see (8)), 2<1

. By analogy with
the case 1 in 3.2 (see (13)), we find parameters 12 ,.
A
A
Inequality (11) has the form 11 23
<,
A
sAs consequently,
12
<.
A
A For example, vector = (1,2)A satisfies this
inequality and the inequality 2
12
23 3<0AA A
(see (13)). By using formula (12) we obtain a suitable
relation of the form (5): 123
445=0.

  By (10)
we conclude: 12 3 =11/15,= 14/15,=4/ 3.

 Based
on this and (6) we find desired parameters 1=66V,
2= 84,V 3=120V milliard tons of emissions.
Taking into consideration the another variant of the
problem statement (see 3.2) consider the following ex-
ample.
Example 2. Assume in the notations of subsection 3.1
270V

milliard tons, =3,=150NS million km2,
2131
=3 ,=2,SSSS 2.=n Then 1=1/6,s 2=1/2,s
3=1/ 3,s 122
=5/6,=1/3,<1PP

. Inequality (11) is
transformed into 12
<2 ,
A
A and the last inequality in (9)
is 2
12
56 3<0AAA . Vector =(1,1)A, for ex-
ample, satisfies these inequalities. In a similar way, we
find the equality of the form (5) 123
734=0

 
and desired parameters 1=23/30,

2=29/30,

3=7/6.

Finally, from (6) we obtain 1=34,5V,
2=130,5V, 3=105V milliard tons of emissions.
3.5. An Alternative Version of Problem B
Consider the equality of the form (5)

 
1
1=1,0,1 .
nn nnn
 

  (14)
It guarantees fulfilment of the inequality (see (3))
1
<1<
nn



, but it does not coordinate with condi-
tion Db

(in the notations of relations (5), (8), (9)),
which is necessary for solution of Problems A, B.
Therefore consider another statement of an extremal
problem, consistent with (14).
Problem C. For a suitable choice of parameter n

in (14) it is necessary to find values of parameters

1
N
k
 , such that the functional
 
12
2
11 1
=1
=,
N
ii
i
 



 (15)
attains the minimum provided that Equalities (2), (3), (14)
are fulfilled.
For a case of three groups 3)=(N it is not difficult
to prove, using elementary tools of mathematical analy-
sis, that Problem C is solvable, e.g. for 11 =2s


23
,
s
s

23 =/2,s

if, respectively, =1,n =2.n In
particular, for 2=n and for the above mentioned value
of 2

desired parameters are as follows:

L. S. MAERGOIZ ET AL.
519



2
13
12
2
113
41
=;
14 1
ss
sss


 
111
12 1
13
12
=;=;=,
1iii
s
ss






where =1,2.i Then application of this algorithm under
the assumptions of Example 1 in 3.4 yields the result
close to the result of Example 1: 1=45/61,

2=57/61,

3= 81/ 61.

4. Conclusions
The proposed objectivization of the emission reduction
by different countries (or limitation of the emission
buildup rates for developing countries) can be based on
three approaches.
Approach 1. Following Dirk, we recognize the right
of each citizen of Earth to have an emission quota, but
we would distribute quotas of emission reduction. How-
ever, social conditions do not allow us to attain our ulti-
mate goal too quickly and physical conditions do not
allow us to do this too slowly. The time necessary for
emissions to reach a threshold level, at which the effect
of humans on the global temperature becomes actually
imperceptible, must be such that atmospheric concentra-
tions of greenhouse gases should not be able to reach
threshold levels that would cause a transition from “a
cold climate” to “a warm one” and, then, if concentra-
tions of greenhouse gases are further increased, to Ve-
nus-like atmospheric conditions. Citizens of different
countries, due to historical reasons, get different rates of
emission reduction (or emission increase for industrially
undeveloped countries). Thus, actual per capita emis-
sions will vary widely. The resultant difference in quotas
must correspond to differences in geographic conditions
of nations.
Approach 2. Theoretically, the distribution of quotas
could be proportional to the area occupied by each state.
If this is taken as the sole principle of distribution, it will
become absolutely inequitable.
Approach 3. Another possible approach is to take into
account the rights of a citizen of Earth and those of the
proprietor of the area. As the land area amounts to about
one-third of the Earth's surface, quotas should be deter-
mined proportionally to the area of each country, while
the two-thirds occupied by the Global Ocean should be
distributed proportionally to the population size of each
country, taking into account historical and geographic
differences. Once an agreement is reached on the minim-
al and maximal gas emissions and the number of groups,
the approach proposed in this study can provide a basis
for using the objective algorithm of determining the qu-
otas of emission reduction for different countries, for any
actual level of emissions.
5. References
[1] L. S. Maergoiz, T. Yu. Sidorova and R. G. Khlebopros,
“Matematicheskii Algoritm Raspre-Deleniya Vybrosov
Parnikovykh Gazov (A Mathematical Algorithm of the
Distribution of Greenhouse Gas Emissions),” in Russian,
Preprint, SFU+KRC SB RAS, 2010, p. 22.
[2] M. I. Budyko, “Effect of Solar Radiation Variation on
Climate of Earth,” Tellus, Vol. 21, No. 5, 1969, pp.
611-1969.
[3] V. I. Vernadsky, “Zhivoye Veshchestvo i Biosfera (Liv-
ing Matter and the Biosphere),” In Russian, Moscow,
1994.
[4] D. Solte, “Understanding the Worlds’s Crisis: An Oppor-
tunity for Global Action for a Sustainable Future,” Terra
Media Verlag, 2009.
[5] “United Nations Framework Convention on Climate
Change, 1992,” Byulleten Mezhdu-Narodnykh Dogovorov
(The Bulletin of International Agreements), in Russian,
No. 12, 1996, p. 324.
[6] “Kyoto Protocol to the United Nations Framework
Convention on Climate Change,” Byulleten Mezhdun-
arodnykh Dogovorov (The Bulletin of International
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[7] E. A. Galkova and L. S. Maergoiz, “On One Linear
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nal of Industrial Mathematics, Vol. 9, No. 3, 2006, pp.
26-30.
[8] E. A. Galkova, L. S. Maergoiz and R. G. Khlebopros,
“Applications of the Optimality Principle in Tasks of
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