Social Welfare Function - Cardinal Social Welfare Functions

Cardinal Social Welfare Functions

In the above contexts, a social welfare function provides a kind of social preference based on only individual utility functions, whereas in others it includes cardinal measures of social welfare not aggregated from individual utility functions. Examples of such measures are life expectancy and per capita income for the society. The rest of this article adopts the latter definition.

The form of the social welfare function is intended to express a statement of objectives of a society. For example, take this example of a social welfare function:

where is social welfare and is the income of individual i among n in the 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):

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. The max-min welfare function can be seen as reflecting an extreme form of uncertainty aversion on the part of society as a whole, since it is concerned only with the worst conditions that a member of society could face.

Amartya Sen proposed a welfare function in 1973:

The average per capita income of a measured group (e.g. nation) is multiplied with where is the Gini index, a relative inequality measure. James E. Foster (1996) proposed to use one of Atkinson's Indexes, which is an entropy measure. Due to the relation between Atkinsons entropy measure and the Theil index, Foster's welfare function also can be computed directly using the Theil-L Index.

The value yielded by this function has a concrete meaning. There are several possible incomes which could be earned by a person, who randomly is selected from a population with an unequal distribution of incomes. This welfare function marks the income, which a randomly selected person is most likely to have. Similar to the median, this income will be smaller than the average per capita income.

Here the Theil-T index is applied. The inverse value yielded by this function has a concrete meaning as well. There are several possible incomes to which an Euro may belong, which is randomly picked from the sum of all unequally distributed incomes. This welfare function marks the income, which a randomly selected Euro most likely belongs to. The inverse value of that function will be larger than the average per capita income.

The article on the Theil index provides further information about how this index is used in order to compute welfare functions.