Unequal distribution measure

A measure of unequal distribution describes the degree of inequality of one variable compared to another. In the social sciences, these variables are often resources such as income or assets on the one hand, and the number of those who have shares in income and assets on the other . Inequality measures indicate the degree to which the allocation of resources to people deviates from an equal distribution. The determination is based on an ascending list of the distribution variables to be examined. Pen's Parade is a sensually impressive representation of inequality . The following is a brief overview of the most important measures of inequality.

overview

Hoover inequality

As the simplest unequal distribution measure, the Hoover unequal distribution is based on a distribution in which an unequal distribution is converted into an equal distribution at all times, fully informed. The Hoover unequal distribution is 0 (or 0%) with completely equal distribution and 1 (or 100%) with maximum unequal distribution. It indicates directly what proportion of an unevenly distributed total income, for example, would have to be moved in order to achieve complete equal distribution.

Theil index

In contrast, the Theil index is based on the distribution model of a completely unregulated economy in which the redistribution takes place in a purely stochastic process. At no time is any information about the current distribution of resources evaluated by the actors and processes involved in this model. The Theil index is a measure of inequality derived from information theory. It belongs to the family of entropy measures .

Gini coefficient

While the Theil index is being used increasingly, the Gini coefficient is still the most widely used. It is an evaluation of the Lorenz curve and is therefore more descriptive than other unequal distribution measures, but it was not based on a distribution model when it was developed. The Gini coefficient is 0 (or 0%) with completely equal distribution and 1 (or 100%) with maximum unequal distribution. Its calculation can be represented very clearly with the means of geometry; What a measured Gini coefficient means from a social science point of view cannot be conveyed geometrically. However, the longstanding and frequent use of the Gini coefficient led to an empirical understanding of the meaning of the Gini coefficient. There are also empirical studies that investigate the relationship between subjective evaluations of inequalities and the associated measures of inequality.

Hoover inequality

Theil index and the Hoover unequal distribution : ${\displaystyle T}$ ${\displaystyle H}$
The graph illustrates the course of T , TH and T / H as a function of for societies that are divided into two quantiles, in which a share of euros is assigned to a share of people and a share of euros is assigned to a proportion of people, whereby the following applies (e.g. the “80:20 Pareto principle” with and , from which and result). For societies partitioned in this way, the Theil index and the symmetrized Theil index are the same. The Gini coefficient and the Hoover unequal distribution follow the same trend . Under these conditions applies . ${\displaystyle H}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle a+b=1}$ ${\displaystyle A=0{,}8}$ ${\displaystyle B=0{,}2}$ ${\displaystyle H=0{,}6}$ ${\displaystyle T=0{,}832}$ ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle T=2H\operatorname {artanh} (H)}$

The Hoover inequality is the simplest of all measures of inequality. In English, this inequality measure is also known as the “Robin Hood Index”. The Hoover unequal distribution is 0 (or 0%) with completely equal distribution and 1 (or 100%) with maximum unequal distribution. It indicates directly what proportion of an unevenly distributed total income, for example, would have to be moved in order to achieve complete equal distribution.

The effect of a tax progression can be measured directly with the Hoover index : In 2001, there was a Hoover unequal distribution of gross income compared to taxpayers of 0.332. In the case of a flat tax without an exemption, the unequal distribution of net income would also be 0.332. In fact, there was a Hoover unequal distribution of 0.300. So 3.2% of the net income was redistributed “from top to bottom” through the tax progression.

The Hoover equal distribution is 1 (or 100%) minus the Hoover unequal distribution. A welfare function calculated for a national income with the Hoover equal distribution is obtained if the national income is multiplied by the Hoover equal distribution. This welfare function has a concrete meaning: it is the share of the national income that would remain untouched if the national income were redistributed in such a way that it would result in a completely equal distribution.

If the national income is multiplied by the Hoover unequal distribution or if the welfare function is subtracted from the national income, then the result is the proportion of the national income that would have to be moved in total if a complete equal distribution were to be carried out with minimal effort. This would require perfect planning, provided that you are fully informed.

Theil index

The Theil index is a measure of inequality derived from information theory. It belongs to the family of entropy measures. The Theil index is sometimes incorrectly referred to as Theil entropy . In fact, it is a matter of redundancy , because it is the difference between a maximum entropy that arises with uniform distribution and a current entropy that results from an uneven distribution.

The Theil index is 0 in the case of completely even distribution and 1 in the case of an uneven distribution, in which 17.6% of the resource owners have 82.4% of the total resources and, conversely, 82.4% of the resource owners have 17.6% of the resources. As a reminder, the fact that this unequal distribution for a Theil index of 1 is fairly close to the 80:20 distribution, known as the “ Pareto principle ”. If the distribution is higher, the Theil index is greater than 1.

In contrast to the Hoover unequal distribution, not only disparities are aggregated when calculating the Theil index, but these disparities are weighted with their information content. This then results in a key figure that not only describes the proportion of resources to be redistributed for compensation, but also the attention that the uneven distribution evokes.

The Theil Index is available in two versions. The Theil-L-Index describes the distribution of resources to people, the Theil-T-Index describes the distribution of people to resources. The mean of both indices is a symmetrized Theil index which is structurally very similar to the simple Hoover unequal distribution (see main article ).

When calculating measures of unequal distribution, both Theil indices are often given today, sometimes in addition to the Gini coefficient.

The normalization of the Theil indices in the range between 0 and 1 (or between 0% and 100%) takes place with the operation . ${\displaystyle \displaystyle 1-e^{-T}}$

${\displaystyle \displaystyle e^{-T_{L}}}$is the Part-L uniform distribution. A welfare function calculated for a national income with the Part-L uniform distribution results when the national income is multiplied by the Part-L uniform distribution. In the article on the Theil index , the application of the Theil-L index to the calculation of the welfare function is explained in more detail. A per capita welfare function can be interpreted as a “perceived average income” and used as an alternative to the median .

If the national income is multiplied by the Theil-L-Index or if the welfare function calculated with the Theil-L-Index is subtracted from the national income, then the result is the proportion of the national income that would be moved altogether if a completely equal distribution under the condition of one From a planning point of view, only distribution subject to the laws of chance should be achieved. The distribution model would correspond to the free market model. For this, the system in which the distribution takes place would have to be completely closed off from its environment and left to itself. (If no income is moved, then an equal distribution would also be conceivable by moving the income recipients in a random distribution process. In this case, the number of recipients moved would be the product of the Theil-T index and the number of income recipients.) With high inequalities, the Theil index results in a redistribution volume that exceeds the volume calculated with the Hoover unequal distribution. In the case of small inequalities, the calculated redistribution volume is initially below the volume calculated with the Hoover inequality distribution, so an equalization does not seem to be able to take place. However, in the course of the theoretically assumed equalization process, new values ​​arise for the Theil index, which approach the values ​​of the Hoover unequal distribution again in the vicinity of zero.

In real economies there is always a mixture of a distribution that is left to chance and one that is subject to planning. In the (only theoretically imaginable) closed economic system, this distribution leads to equalization, in the open system, on the other hand, the unequal distribution can also increase, for example through temporally and spatially unevenly distributed access to resources in the system's environment. For this reason, both the Hoover unequal distribution and the Theil index are relevant measures. The difference between the two measures of inequality is less than 0.1 for the majority of economies (see "Comparison of the Theil Index with the Hoover Inequality" in the main article Theil Index ).

Gini coefficient

The Gini coefficient is the most commonly used measure of inequality in the social sciences. It is 0 (or 0%) with completely even distribution and 1 (or 100%) with maximum unequal distribution. Its calculation can be represented very clearly with the means of geometry; What a measured Gini coefficient means from a social science point of view cannot be conveyed geometrically.

However, the longstanding and frequent use of the Gini coefficient led to an empirical understanding of the meaning of the Gini coefficient. There are also empirical studies that investigate the relationship between subjective evaluations of inequalities and the associated measures of inequality. The Gini coefficient is used at least if research is to be continued in which this inequality measure has already been used.

The EU social statistics work with the S80 / S20 income quintile ratio .

Other measures of unequal distribution

There are around 50 measures of inequality in the social sciences. Many are related to each other or developed in parallel. In current research, entropy measures such as the Theil index are increasingly used. In Wikipedia there are main articles on these further measures of unequal distribution and concentration:

• Atkinson measure
• Kakwani index
• Herfindahl's measure of concentration
• Rosenbluth index
• Learner Index

See also

• Distributive justice

Individual evidence

1. ^ ^{A b} Frank A. Cowell : Theil, Inequality and the Structure of Income Distribution . (PDF; 312 kB) London School of Economics and Political Sciences, 2002, 2003 (with references to the "class of Kolm indices", which are measures of unequal distributions such as the Theil index )
2. ↑ ^{a b} Eberhard Schaich: Lorenz curve and Gini coefficient in critical consideration . In: Yearbooks for Economics and Statistics , 185, 1971, pp. 193-208.
3. ^ Y. Amiel, FA Cowell: Thinking about inequality . 1999, ISBN 0-521-46696-2
4. ↑ Calculated based on the tax statistics (not income statistics) in the basic data of the Federal Statistical Office, November 2005.
5. ↑ see also entropy (social sciences)
6. ^ Y. Amiel, FA Cowell: Thinking about inequality . 1999, ISBN 0-521-46696-2
7. ^ Philip B. Coulter: Measuring Inequality . 1989.