The simplest models for ranking of the coefficients of egoism

How to introduce some measure of ``national egoism''? There are different ways in which we can do this. In the following five different setting of are discussed for the actors: 1. USA and Canada, 2. EC (represented by Germany, UK, France and Italy), 3. China, 4. FSU, 5. Latin America (represented by Brazil, Mexico, Argentina and Chile), 6. Japan. Data on GDP and the population of the countries are taken from the World Bank[8], the emission values are from the database of the Carbon Dioxide Information Analysis Center[9].

1. One of them is to set , where is some dimensionless factor. These values are relatively large, if both is small and is large. This is a typical situation for many developing countries, whose economics include a significant industrial sector. These countries are obliged to be ``egoistic''. If, conversely, is small and is large then the corresponding will be relatively small. This is typical for developed countries with developed agricultural and ``high technology'' sectors. They can permit themselves the luxury of being ``altruistic'', because the reduction of emissions influence their GDP only weakly. Developing ``rural and pastoral'' countries (with small and ) and developed ``traditionally industrial'' countries (with large and ) must have some intermediate values of . In this case we get from (18)-(20):
    
   The value of can be determined, for instance, from the solution of some optimisation problem for total emissions [7], when is determined from the criterion of maximal cumulative emissions under a constraint such as a ``tolerable climatic window''. Thus, let be known, then from (22) we get
    
   and
    
   i.e., almost a trivial result: the absolute value of reduction, , must be the same for all countries. Therefore countries with a large value of the ratio have a lower loss of GDP:
   
   This ranking is shown in figure 4 for .

     
   Figure 4: The coefficients of egoism (left-hand bars, in tons of carbon per of GDP) and corresponding relative reductions (right-hand bars, in %). Actors: 1. USA and Canada, 2. EC (represented by Germany, UK, France, and Italy), 3. China, 4. FSU, 5. Latin America (represented by Brazil, Mexico, Argentina, and Chile), 6. Japan. They are defined as , where is some dimensionless factor according to (23), .
   

2. Today the USA release about 1,200 million tons of CO in the atmosphere, China releases 600 million tons, although the China population is almost five times higher than the USA population. On the other hand, the Netherlands contribution into the total emission is negligibly small, but the expected damage from the Climate Change for this country could be extraordinary. What does a national ``guiltiness'' and national responsibility in respect to the global process of Climate Change mean? How to define these concepts? All these problems, including moral, ecological, economic and many others, form the so-called rules of ``fair play''.

   It is intuitively clear that in accordance with the rules of ``fair play'' rich countries must be ``altruistic'' and, on the contrary, poor countries may be ``egoistic''. Let be the population of the i-th country; when the income per capita is larger, the country is richer. Then we set , where the scale factor has the dimension [population / tons of carbon]. This ranking is shown in figure 5. In this case we get from (18)-(20):
    
   where and
    
   The ratio is an individual emission for the i-th country, i.e. the emission per capita. Then the ratio can be interpreted as a relative individual contribution of an inhabitant of the i-the country to the World individual emission. The absolute reduction for this country, , must be proportional to its relative individual contribution.

     
   Figure 5: The Coefficients of egoism (left-hand bars, in tons of carbon per of GDP) and corresponding relative reductions (right-hand bars, in %). Actors: 1. USA and Canada, 2. EC (represented by Germany, UK, France, and Italy), 3. China, 4. FSU, 5. Latin America (represented by Brazil, Mexico, Argentina, and Chile), 6. Japan. They are defined as , where the scale factor has the dimension [population / tons of carbon], .
   

3. We can imagine such rules of ``fair play'' under which the percentage of reduction, , must be the same for all countries. Then from (20) we get (under the assumption that the value is given):
    
   Because the losses are proportional to countries with large GDP have to be more altruistic; their absolute losses are higher than the losses for countries with small GDP. The equality (18) is fulfilled identically in this case. This ranking is shown in figure 6. It shows that the countries with large GDP must be ``altruistic'' under such a kind of ``fair play'' equilibrium, and, on the contrary, the countries with small GDP can permit themselves to be ``egoistic''.

     
   Figure 6: The coefficients of egoism(left-hand bars, in tons of carbon per of GDP) and corresponding relative reductions (right-hand bars, in %). Actors: 1. USA and Canada, 2. EC (represented by Germany, UK, France, and Italy), 3. China, 4. FSU, 5. Latin America (represented by Brazil, Mexico, Argentina, and Chile), 6. Japan. Percentage of reduction, , is the same for all countries. The coefficients are defined from the assumption that this is Germeier's equilibrium.
   

4. Let us imagine the following fantastic situation: all countries demonstrate a common level of altruism, for all . Then from (18) - (20) we have (for a given ):
    
   Corresponding values of and are shown in figure 7. Since then the absolute losses of GDP are equal for all actors in this case.

     
   Figure 7: The coefficients of egoism (left-hand bars, in tons of carbon per of GDP) and corresponding relative reductions (right-hand bars, in %). Actors: 1. USA and Canada, 2. EC (represented by Germany, UK, France, and Italy), 3. China, 4. FSU, 5. Latin America (represented by Brazil, Mexico, Argentina, and Chile), 6. Japan. The level of egoism (or, altruism) is the same for all countries, it depends on the given percentage of a total reduction (). The greater percentage of reduction, the higher the level of egoism. The individual percentages are defined from the assumption that this is Germeier's equilibrium.
   

5. And finally, we consider the ranking generated by the Kyoto protocol [10]. We exclude from the set of actors the third and fifth ones (China and Latin America), as these actors are not the participants of this treaty. We exclude also the fourth actor (FSU) since, in accordance with the protocol, , and the corresponding . In the framework of our approach this means that its strategy does not influence the result of the game, and this actor can be excluded. (If we remember the situation with industry in the FSU, we can say that this group of countries has already realised by this time the concept of the Kyoto protocol).

   The emission game must therefore take place with three remaining actors. Let us assume that the relative reduction of emissions given by the Kyoto protocol coincides with the values of Germeier's equilibrium. Then
    
   and the corresponding values of and are shown in figure 8. It is obvious that the mean relative reduction (averaging in respect to the relative current emissions, ) is equal to , and this value is not given beforehand. Calculating for this data we get the value which was chosen for the first four settings of the level of egoism discussed above.

     
   Figure 8: The coefficients of egoism (left-hand bars, in tons of carbon per of GDP) and corresponding relative reductions (right-hand bars, in %). Actors: 1. USA and Canada, 2. EC (represented by Germany, UK, France, and Italy), 3. China, 4. FSU, 5. Latin America (represented by Brazil, Mexico, Argentina, and Chile), 6. Japan. The relative reductions, , are chosen in accordance with the Kyoto protocol. Egoism coefficients are defined from the assumption that this is Germeier's equilibrium.
   

We recommend the reader to draw his own conclusions comparing these analyses and graphs.