Social welfare function
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A social welfare function in welfare economics is a function that maps welfare measures for members of the society to a measure of aggregate welfare of the society.
In 1938 a social welfare function was introduced by Abram Bergson. He showed how welfare economics could dispense with interpersonally-comparable cardinal utility (say measured by money income), either individually or in the agggregate, with no loss of behavioral significance. Rather the social (called "economic") welfare function maps real-valued, though not interpersonally comparable, utility functions (illustrated by Indifference curve maps) of members in society to the marginal production and consumption conditions that must be satisfied for each individual to be in preference equilibrium, given the distribution of fixed and efficiently allocated resources. In consumer theory for an individual, it is comparable to mapping a set of ordinary indifference curves (which as an ordering, gives complete and transitive rankings of different commodity bundles) to the highest indifference curve on the budget constraint -- with these differences. The latter describes a consistent way to make choices from the preference ordering of an individual. The social welfare function can both describe and prescribe a consistent way to make a social choice from the preference orderings of the members of society.
In his book Social Choice and Individual Values (1951, 2nd ed. 1963) Kenneth Arrow generalizes Bergson's framework. Behavioral significance is preserved from dropping the requirement of orderings that are real-valued (and thus cardinal) in favor of orderings that are merely complete and transitive. In his version a social welfare function (also called a 'constitution') maps individual orderings to a social ordering, a kind of social rule for ranking alternative states. More stunningly (relative to a course of theory from Adam Smith and Jeremy Bentham on), he proved the so-called impossibility theorem that it is impossible to have a social welfare function that satisfies a certain set of "apparently reasonable" criteria.
In the above contexts, a social welfare function provides a kind of a social preference of the different possible (discrete) inputs, whereas in others it is defined as function mapping cardinal meeasures of individual welfare to a numerical measure of social welfare. The rest of this article adopts the latter definition. Motivation for such a measure is in its appeal, whether to officials, advisors, or voters.
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 = Y_1 + Y_2 + \cdots + Y_n]
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):
[W = \min(Y_1, Y_2, \cdots , Y_n)]
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.
Amartya Sen proposed a welfare function in 1973:
[W = Income \times (1-Inequality)]
[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 one of Atkinson's Indexes, 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). Foster's welfare function also can be computed directly using the Theil Index.
See also
- Abram Bergson
- Arrow's impossibility theorem
- Social Choice and Individual Values
- Social choice theory
- Welfare economics
References
- Paul A. Samuelson, 2004, "[Abram Bergson 1914-2003: A Biographical Memoir]"
- Kotaro Suzumura, 1987, “social welfare function," The New Palgrave: A Dictionary of Economics, v. 4, 418-20
- [Bergson-Samuelson social welfare function] in Paretian welfare economics
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