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Case when the utility functions are linear

Let us consider one partial case when the functions tex2html_wrap_inline1125 have been estimated. In accordance to Nordhaus[5] and Leimbach's[6] econometric estimation, the relative loss of the Gross Domestic Product (GDP) for the i-th country, (which are connected with the expenditures on COtex2html_wrap_inline817 reduction) depends only on the relative reduction of emission tex2html_wrap_inline1131, so that
 equation214
Nordhaus and Leimbach suggested the following form for R: tex2html_wrap_inline1135. Later on Leimbach[6] suggested using the linear form tex2html_wrap_inline1137. Unfortunately, tex2html_wrap_inline1139 for both forms, but this is an artefact of regression analysis used in econometrics. Further on we shall use a ``pure'' linear form where the coefficient b has some intermediate value (for instance, b = 0.1). (We think that we do not commit any great crime in respect to econometrics). Then
 equation219
and the equations (13) are written in the form (for m = 0):
 eqnarray225
Let us suppose that all tex2html_wrap_inline1093 are known, i.e. all the actors (countries) can be ranked along the axis ``altruism - egoism''. Then the solution of (17) can be represented as
 equation234
where
 equation242
Let us introduce new variables: the relative reduction of emission tex2html_wrap_inline1131 for the i-th actor, the relative contribution of the i-th actor, tex2html_wrap_inline1153, to the total current emission. Then (18) can be re-written in a very simple form:
 equation256
Substituting tex2html_wrap_inline1093 from (20) to (19) we get
 equation265


next up previous
Next: The simplest models for Up: ``Emission game'': some applications Previous: The case of n

Werner von Bloh (Data & Computation)
Thu Jul 13 15:46:47 MEST 2000