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Applied Mathematics, 2010, 1, 515-519 10.4236/am.2010.16068 Published Online December 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. 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 E-mail: {bear.lion, tatiana-sidorova}@mail.ru, olikru@yandex.ru 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. Copyright © 2010 SciRes. AM 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. Copyright © 2010 SciRes. AM 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. Copyright © 2010 SciRes. AM 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. Copyright © 2010 SciRes. AM 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 Agreements), in Russian, No. 5, 2005, p. 323. [7] E. A. Galkova and L. S. Maergoiz, “On One Linear Problem of Joint Investment,” in Russian, Siberian Jour- 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 Joint Investment Management,” Vestnik Novosibirskogo gos. Universiteta (Bulletin of the Novosibirsk University), Social-Economic Series, in Russian, Vol. 8, No. 1, 2008, pp. 138-143. |