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Bergson Samuelson:

The Bergson-Samuelson SWF shows that, given the set  of Pareto-optimal points, which  is' more desirable  from  "society's" point  of view, where the notion of social desirability was subsumed in a social welfare function.

Heuristically, we  can  envisage the upper contour set of  the  SWF as a set of "social  indifference  curves"  in  utilities space,  as  shown  in  Figure.

477_Social Welfare Function and  the Social Optimum3.png

According to Bergson, there are some desirable properties of a society which are captured by the SWF: for instance, social welfare increases if the utility of any of its members increase and none decrease (the "Pareto  principle")  that yields northeasterly ascendance of  the social indifference curves. It is worth mentioning that the consideration of fairness and equity are incorporated into the SWF and are reflected in the shapes of social indifference curves. In other words, it can be argued that equity is socially desirable; consequently,  extreme distributions of utility ought  to be  given less weight. Thus, the convexity of the  social indifference curve needs  to  be  accepted and like the  usual indifference curves  they are also non-intersecting.

By  superimposing social indifference curves on  the GUPF as in Figure, we  can see  that allocation  S = (uA*,  uB*)  is  the point  on  the  GUPF  that attains  the highest social indifference curve, and maximises the social welfare function, yielding  social welfare  index W*.  Thus,  the  "social optimum"  is determined by the tangency of the social indifference curves and the GUPF. The  SWF  as  depicted  in  Figure  can  be  explained  by  the following functional  form:

318_Bergson Samuelson.png

where ah  are  the weights assigned  to  each household  in  the social welfare function.  Such  a  function  yields  the  convex social indifference curves  in Figure  and  is  sometimes called  a  "Bernoulli-Nash"  social  welfare function.

It  is to be noted  that the slope of the social indifference curves is equal to the negative  of . This last  term  in  the expression  is  the "marginal  rate  of  social substitution"  between  consumers  A  and  B,  or MRSSAB.  Now, we know that the slope of the GUPF will be merely the ratios of the marginal utilities of income of households A and B,  thus the tangency
condition is that:

808_Bergson Samuelson1.png

 or  the social marginal utility for each  household is  equal  across  households. We  can extend this explanation to analyse some alternative welfare function as  mentioned  above. As  per the  classical or Benthamite or "utilitarian"  social welfare  function,  the  SWF  can  be  constructed as a  linear  sum  of weighted utilities, e.g;

1108_Bergson Samuelson2.png

which  is  a  direct  sum. Thus,  as  stipulated  by  Jeremy  Bentham  and  the Utilitarians, this one maximises the (weighted) sum of individual utilities and yields linear social indifference curves (WB',  WB*, WB'')  as shown in Figure below.

980_Bergson Samuelson3.png

Bergson-Samuelson SWF  states the conditions for  "social justice",  as that, 'the marginal rate of social substitution between households A and B is equal to the ratio of marginal rates of substitution of A and B. In other words, this implies that the allocation of goods is done in  such a manner so as  to have the utility distribution compatible with  the  "worthiness"  of  the individuals according  to the social welfare function.

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