Hoover inequality

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The Hoover inequality is the simplest of all measures of inequality . It describes the relative deviation from the mean. It is “direct” because, for example, in the case of an uneven distribution of money, it simply describes the proportion of the total money that would have to be redistributed in order to turn an unequal distribution into an equal distribution. Other names for the Hoover inequality are the Hoover coefficient , Hoover index , Balassa-Hoover index , Hoover concentration index and segregation and dissimilarity index .

Calculation example

The Hoover unequal distribution can be calculated - like the Gini coefficient - for income distributions , for wealth distributions and other distributions. The following example shows how the Hoover unequal distribution is calculated using the distribution of "total assets" of around 10 trillion Deutsche Mark in Germany (1995):

50 Prozent der Bevölkerung (A₁) besaß  2,5 Prozent des Vermögens (E₁).
40 Prozent der Bevölkerung (A₂) besaß 47,5 Prozent des Vermögens (E₂).
 9 Prozent der Bevölkerung (A₃) besaß 27,0 Prozent des Vermögens (E₃).
 1 Prozent der Bevölkerung (A₄) besaß 23,0 Prozent des Vermögens (E₄).

In a first step, the data are displayed "normalized" (E _total = A _total = 1):

A₁ = 0,50     E₁ = 0,025
A₂ = 0,40     E₂ = 0,475
A₃ = 0,09     E₃ = 0,270
A₄ = 0,01     E₄ = 0,230

In the second step, the absolute differences are added up:

abs(E₁ - A₁) = 0,475
abs(E₂ - A₂) = 0,075
abs(E₃ - A₃) = 0,180
abs(E₄ - A₄) = 0,220
     Summe = 0,950

Half the sum is the Hoover unequal distribution:

Hoover Ungleichverteilung: Summe/2 = 0,475 = 47,5 %

Other measures of inequality "interpret" inequalities. An example are some entropy dimensions (z. B. by Theil , Atkinson , Kullback and Leibler, etc.) that reference to uniform distributions of state variables in statistical physics. The Hoover coefficient, on the other hand, is very easy to understand and calculate. It directly describes the proportion of an unevenly distributed resource that would have to be redistributed if this resource were to be evenly distributed. In the example, 47.5% of the wealth would have had to be redistributed if everyone should have owned the same amount. (The uneven distribution within the four areas with different widths, delimited by quantiles with different distances, would have been disregarded.)

The range of values of this relative unequal distribution measure lies between 0 and 1 (or between 0% and 100%). The Hoover unequal distribution belongs to the group of concentration measures .

formula

The full formula of the Hoover inequality is:

{\ displaystyle H = {\ frac {1} {2}} \ sum _ {i = 1} ^ {N} \ color {Blue} \ left | \ color {Black} {\ frac {{E} _ {i }} {{E} _ {\ text {total}}}} - {\ frac {{A} _ {i}} {{A} _ {\ text {total}}}} \ color {Blue} \ right | \ color {Black}.}

A notation is used in the formula in which the number of areas delimited by quantiles (with the same or different spacing) (with the same or different width) only appears in the formulas as the upper limit of the total. In this way, unequal distributions can also be calculated in which the areas have different widths : let the income be in the i-th area and be the number (or the percentage) of income earners in the i-th area. be the sum of the incomes of all N areas and be the sum of the income earners of all N areas (or 100%). (Of course, other assignments are also possible: for example, can also represent assets. Or stands for one type of molecule in a mixture and another type of molecule.) ${\ displaystyle N}$ ${\ displaystyle A}$ ${\ displaystyle E_ {i}}$ ${\ displaystyle A_ {i}}$ ${\ displaystyle E _ {\ text {total}}}$ ${\ displaystyle A _ {\ text {total}}}$ ${\ displaystyle E}$ ${\ displaystyle E}$ ${\ displaystyle A}$

In the Hoover unequal distribution, the individual deviations from parity are weighted only with their own sign (i.e. the factor +1 or −1). For comparison, consider the symmetrized Theil index . In the Theil Index, the individual deviations from parity are weighted with their own information content: ${\ displaystyle T_ {s}}$

{\ displaystyle T_ {s} = {\ frac {1} {2}} \ sum _ {i = 1} ^ {N} \ color {Blue} \ ln {\ frac {{E} _ {i}} { {A} _ {i}}} \ left (\ color {Black} {\ frac {{E} _ {i}} {{E} _ {\ text {total}}}} - {\ frac {{A } _ {i}} {{A} _ {\ text {total}}}} \ color {Blue} \ right) \ color {Black}.}

Remarks

↑ SPD parliamentary group, Bundestag printed paper 13/7828 (PDF; 309 kB)
↑ The notation with E and A follows the notation of a small collection of formulas by Lionnel Maugis: Inequality Measures in Mathematical Programming for the Air Traffic Flow Management Problem with En-Route Capacities (for IFORS 96), 1996
↑ The Hoover unequal distribution is related to the symmetrized Theil index: The symmetrized Theil index is the non-interpretative unequal distribution weighted with the information content of this unequal distribution . The Hoover inequality is a purely non-interpretive inequality .

literature

Edgar Malone Hoover jr .: The Measurement of Industrial Localization , Review of Economics and Statistics, 1936, Vol. 18, No. 162-171
Edgar Malone Hoover jr .: An Introduction to Regional Economics , 1984, ISBN 0075544407
Philip B. Coulter: Measuring Inequality , 1989, ISBN 0-8133-7726-9 (About 50 measures of inequality are described in this book.)

Web links

Calculator: on-line and scripts and macros for download (for Python , Lua and OpenOffice.org 2.0 Calc)

[1] SPD parliamentary group, Bundestag printed paper 13/7828 (PDF; 309 kB)

[2] The notation with E and A follows the notation of a small collection of formulas by Lionnel Maugis: Inequality Measures in Mathematical Programming for the Air Traffic Flow Management Problem with En-Route Capacities (for IFORS 96), 1996

[3] The Hoover unequal distribution is related to the symmetrized Theil index: The symmetrized Theil index is the non-interpretative unequal distribution weighted with the information content of this unequal distribution . The Hoover inequality is a purely non-interpretive inequality .