Welfare function

This article explains the concept of microeconomics . For a general consideration, see welfare .

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: ${\displaystyle A}$ ${\displaystyle e^{-T_{L}}}$ ${\displaystyle T_{L}}$

${\displaystyle W_{\text{Theil-L}}={\overline {y}}\cdot e^{-T_{L}}={\overline {y}}\cdot (1-A)}$

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

See also

• Bergson-Samuelson welfare function
• Rawls' welfare function

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 .