A social welfare function, in welfare economics, is a function which gives a measure of the material welfare of society, given a number of economic variables as inputs. Welfare economics is a branch of economics that uses microeconomic techniques to simultaneously determine the allocational efficiency of a macroeconomy and the income distribution consequences associated with it. ... Partial plot of a function f. ...

In some contexts, a social welfare function is considered to provide only a preference ranking of the different possible (discrete) inputs, whereas in others it is defined as a numerical measure of social welfare. This article adopts the latter definition.

Kenneth Arrow proved that it is impossible to have a social welfare function that satisfies a set of given 'reasonable' criteria, in his so called impossibility theorem. Kenneth Arrow Kenneth Joseph Arrow (born August 23, 1921) is an American economist, winner of the Bank of Sweden Prize in Economic Sciences in 1972. ...

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. In economics, utility is a measure of the happiness or satisfaction gained consuming good and services. ...

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:

Where W is social welfare and Y_x 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): John Rawls (February 21, 1921 – November 24, 2002) was an American philosopher, a professor of political philosophy at Harvard University and author of A Theory of Justice (1971), Political Liberalism, Justice as Fairness: A Restatement, and The Law of Peoples. ...

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 risk 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. Risk aversion is a concept in economics and finance theory explaining the behaviour of consumers and investors under uncertainty. ...

Amartya Sen proposed a welfare function in 1973: Amartya Sen Amartya Kumar Sen CH (Hon) (born November 3, 1933 in India), is an economist and a winner of the Bank of Sweden Prize in Economic Sciences (sometimes referred to informally as the Nobel Prize for Economics) for his work on famine, human development theory, welfare economics, the underlying...

Income is the average per capita income of a measured group (e.g. nation). Inequality is the relative inequality of the income distribution within that group. Here Sen used the Gini Index. James E. Foster (1996) proposed to use an Atkinson Index, which is an entropy measure. (On Economic Inequality - expanded edition with substantial annexe by James E. Foster and Amartya Sen, 1973/1996, ISBN 0-19-828193-5) The Gini coefficient is a measure of inequality developed by the Italian statistician Corrado Gini and published in his 1912 paper Variabilità e mutabilità. It is usually used to measure income inequality, but can be used to measure any form of uneven distribution. ...

See also

• Welfare economics