Social Welfare Function and  the Social Optimum:

Treating a number of economic variables as inputs, a  social welfare function gives a measure  of  the aggregate  material welfare of  a society.  Abram Bergson first introduced the idea of a  social welfare function in his article  'A Reformulation  of  Certain Aspects of  Welfare  Economics'  in  1938.  By introducing the concept of  'social welfare function',  Bergson and Samuelson approached welfare analysis  in a  different way. They  based  the concept of SWF on ordinal preferences of individuals and argued that welfare economics is essentially a normative study, but the approach should be  scientific despite the fact that the incorporation of value judgments in it is simply unavoidable.

It  would  be  prudent  to  have  a  brief overview  of  some alternative welfare functions,  which  will  be  helpful  in  bringing out some embedded ethical judgments. The classical welfare function was put forward by Bentham, Pigou and Marshall, popularly known as 'Benthamite welfae  function'.  According to them, social welfare is  the sum of cardinal utilities obtained by all members
of a society. In algebraic form, this can be denoted as:

where W denotes social welfare, UI, Ub  etc; represent the cardinal utilities of the individual members of  the society. The goal of a society is to maximise social welfare, that  is, the aggregate of the utilities of the individual members of society. Given this, maximum social welfare will be  achieved if  income is so distributed that marginal utility of income is equal for all  individuals  in  a
society. Moreover,  the  classical viewpoint  is  that maximisation  of  social welfare is achieved only with equal distribution of income.

Another important social welfare function has been propounded by  the noted philosopher John Rawls. Rawls analysed welfare economics  by  posing  a problem:  "what type  of welfare  criterion would  be  adopted by  the  society when  it  is  in  such an  initial position where  everybody has to behave under uncertainty about  how  the  welfare  criterion chosen will ultimately aflect his utility or welfae". Assuming that  individuals are risk averse, he  asserts that such a welfare criterion will be  chosen that deviations from perfect equality would be made only when with unequal distribution of utilities, the worst oSf individual is better off actually than under equality conditions. This  can be stated algebraically as follows:

This  implies that social welfare  of  resource allocation depends only on the worst off individual,  that is, the person with minimum utility. Social welfare fhnction  is an  ordinal index  of  society's welfare  and  is  a function  of  the utility level  of  all individuals comprising  the  society. The "Bergson-Samuelson"  social welfare  fhnction (SWF) can be expressed in the following general form:

where "society's"  welfare denoted by W,  is merely a  hnction of the utilities of its constituent members, uh,  h = 1, 2,  ........ H, where H are the number of households  in  the society. In other words, u',  u2,.., uH  represents the ordinal utility indices of different individuals or households in the society. The ordinal utility  index  of  an  individual  depends  upon  the  goods and services  she consumes and  the magnitude and kind  of work  she does and the amount of leisure she enjoys.

The Bergson-Samuelson  SWF  can  be  explained with  the help  of  the grand utility possibilities frontier (GUPF). We know that every point on the GUPF is a  Pareto-optimal allocation, and thus  it  seems that  no  point  is  necessarily preferable to another. 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.