Welfare function

from Wikipedia, the free encyclopedia

In economics, a welfare function is a mathematical function that describes the total utility of the population in an economy . It is thus the summary of the utility functions of the individual individuals in the economy. Welfare functions are topics of multi-agent resource allocation .


The concept of the welfare function goes back to the work of Abram Bergson and Paul A. Samuelson . Kenneth Arrow showed the limited applicability of a pure utility function with the impossibility theorem , according to which different opposing preferences of different individuals cannot be aggregated into a benefit for society as a whole. The current discussion is based on work by Amartya Sen and James E. Foster. The aim of a welfare function applied to income, for example, is to determine an income that corresponds to the income as it is perceived by broad sections of the population . This makes the welfare function an alternative to the median .


Welfare depends on the income of the individual .

The most general form of a welfare function is therefore:

Special welfare functions

A common form of the welfare function is the product of the average income with a measure of unequal distribution or the associated measure of equal distribution :

If everyone deserves the same, then is , and . If someone deserves everything, then is , and .

The simplest unequal distribution measure is the Hoover unequal distribution .

This welfare function has a concrete meaning: is that part of the income that would remain untouched if the national income were redistributed in such a way that an equal distribution would result. indicates how much everyone is allowed to keep on average, which is by definition always less than the average income .

The Gini coefficient is also a measure of unequal distribution and thus defines a welfare function:

The Atkinson measure (after Anthony Atkinson ) is also a measure of unequal distribution, the associated measure of uniform distribution is with the Theil index , these define the following welfare function:

The last two welfare functions were suggested by Amartya Sen and James E. Foster.

See also

Individual evidence

  1. Abram Bergson: A Reformulation of Certain Aspects of Welfare Economics , Quarterly Journal of Economics, 52nd year 1938, 310–334.
  2. ^ Paul Samuelson: Foundations of Economic Analysis , Harvard University Press, Cambridge 1947, 221.
  3. James E. Foster & Amartya Sen : On Economic Inequality , expanded edition with a substantial annexe, 1997, ISBN 0-19-828193-5 .