# Welfare function

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 .

## history

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 .

## definition

Welfare depends on the income of the individual . ${\ displaystyle W}$ ${\ displaystyle y_ {i}}$ ${\ displaystyle 1,2, \ dotsc, n}$ The most general form of a welfare function is therefore:

${\ displaystyle W = W (y_ {1}, y_ {2}, \ dotsc, y_ {n})}$ ### 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 : ${\ displaystyle {\ overline {y}}}$ ${\ displaystyle \ alpha}$ ${\ displaystyle \ beta = (1- \ alpha)}$ ${\ displaystyle W = {\ overline {y}} \ cdot (1- \ alpha (y_ {1}, y_ {2}, \ dotsc, y_ {n})) = {\ overline {y}} \ cdot \ beta (y_ {1}, y_ {2}, \ dotsc, y_ {n})}$ If everyone deserves the same, then is , and . If someone deserves everything, then is , and . ${\ displaystyle W = {\ overline {y}}}$ ${\ displaystyle \ alpha = 0}$ ${\ displaystyle \ beta = 1}$ ${\ displaystyle W = 0}$ ${\ displaystyle \ alpha = 1}$ ${\ displaystyle \ beta = 0}$ The simplest unequal distribution measure is the Hoover unequal distribution . ${\ displaystyle H}$ ${\ displaystyle W _ {\ text {Hoover}} = {\ overline {y}} \ cdot (1-H (y_ {1}, y_ {2}, \ dotsc, y_ {n}))}$ 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 . ${\ displaystyle n \ cdot W _ {\ text {Hoover}}}$ ${\ displaystyle W _ {\ text {Hoover}}}$ ${\ displaystyle {\ overline {y}}}$ The Gini coefficient is also a measure of unequal distribution and thus defines a welfare function: ${\ displaystyle G}$ ${\ displaystyle W _ {\ text {Gini}} = {\ overline {y}} \ cdot (1-G)}$ 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: ${\ displaystyle A}$ ${\ displaystyle e ^ {- T_ {L}}}$ ${\ displaystyle T_ {L}}$ ${\ displaystyle W _ {\ text {Part-L}} = {\ overline {y}} \ cdot e ^ {- T_ {L}} = {\ overline {y}} \ cdot (1-A)}$ The last two welfare functions were suggested by Amartya Sen and James E. Foster.