The idea of a social welfare function was first introduced by Abram Bergson in 1938. In this form, social welfare is a function of the levels of utility of members in society.
Alternatively, the social welfare function can be expressed as a function of other variables relevant to welfare, such as income or life expectancy.
The form of the social welfare function can be seen as expressing a statement of the objectives of a society. For example, take this example of a social welfare function:
W = Y1 + Y2 + ... + Yn
Where W is social welfare and Yx is the income of each of the xth individual in a society. In this case, maximising the social welfare function means maximising the total income of the people in the society, without regard to how incomes are distributed in society. Alternatively, consider the Max-Min utility function (based on the philosophical work of John Rawls):
W = MIN (Y1, Y2, ... , Yn)
Here, the social welfare of society is taken to be related to the income of the poorest person in the society, and maximising welfare would mean maximising the income of the poorest person without regard for the incomes of the others.
These two social welfare functions express very different views about how a society would need to be organised in order to maximise welfare, with the first emphasizing total incomes and the second emphasising the needs of the poorest.