Inequality Measures

 ZAtkinson = 1-ZMacRae[1] ≥ 1-exp(Σi=1..N(Ei*ln(Ai/Ei))/Etotal)*Etotal/Atotal 
 ZnosniktA ≥ 1-exp(Σi=1..N(Ai*ln(Ei/Ai))/Atotal)*Atotal/Etotal 
 RTheil  = -ln(1-ZAtkinson) = -ln(ZMacRae) 
   ≥ ln(Atotal/Etotal) - Σi=1..N(Ei*ln(Ai/Ei))/Etotal 
 RliehT  = -ln(1-ZnosniktA) 
   ≥ ln(Etotal/Atotal) - Σi=1..N(Ai*ln(Ei/Ai))/Atotal 
 Rsym  = -ln(1-Zsym) = 2*ZPlato*artanh(ZPlato) 
   = (RTheil(E|A)+RTheil(A|E))/2 = (RTheil+RliehT)/2 
   ≥ Σi=1..N(ln(Ei/Ai)*(Ei/Etotal-Ai/Atotal))/2 
 Zsym  = 1-exp(-Rsym) = 1-√((1-ZAtkinson)*(1-ZnosniktA)) 
   ≥ 1-exp(Σi=1..N(ln(Ai/Ei)*(Ei/Etotal-Ai/Atotal))/2) 
 ZHoover ≥ Σi=1..N|Ei/Etotal-Ai/Atotal|/2 
 ZCoulter ≥ √(Σi=1..N(Ei/Etotal-Ai/Atotal)2/2) 
 sort data:  Ei/Ai>Ei-1/Ai-1
 ZGini ≥ 1-Σi=1..N((2*Σk=1..i(Ek)-Ei)*Ai)/(Etotal*Atotal)
 1:a = (1-ZGini)/(1+ZGini) is the SOEP "equality parameter"
 therefore: ZEurope = 2*ZGini/(1+ZGini)
 inverse functions:
   Zsym = 1-((1-ZPlato)/(1+ZPlato))ZPlato
   Rsym = 2*ZPlato*artanh(ZPlato)
   ZPlato ≈ 1 - arcsin((1-Zsym)(0.06*Zsym+0.61))*2/π,  error < 0.002 for Zsym < 0.75
 fast recursion (there is a better version in
   initialize: ZPlato ≈ 1 - arcsin(exp(Rsym(0.06/exp(Rsym)-0.67)))*2/π
      Zlast = ZPlato
      ZPlato = tanh(Rsym/(ZPlato+Zlast))
      until  2*ZPlato*artanh(ZPlato) - Rsym  is small enough.
 format for comparison to the "Pareto Principle":
   a : b = (1+ZPlato)/2 : (1-ZPlato)/2
   ZPlato = |2a-1| = |2b-1|
 RA  = (RTheil+RliehT)/2 - ZHoover = Rsym - ZHoover 
   = Σi=1..N(ln(Ei/Ai)*(Ei/Etotal-Ai/Atotal) - |Ei/Etotal-Ai/Atotal|)/2 

Any city however small, is divided at least into two,
one the city of the poor, the other of the rich;
these are hostile to each other.

 (Plato, Politeia, 370 BC) 

Sorry for my Germanic English,
Götz Kluge (gk at the domain, Munich, 2009-09-03, last update: 2021-11-14,

[a] The term ln(Ai/Ei) is the unweighted entropy of the quantilei. Coming from information theory and if A stands for owners and E stands for ressources, then this term can be interpreted as minimum length of the address required to distinguish the owners A per unit ressource E within their quantile. (The base of that length is Euler's constant e. For other bases b you need to devide the entropies by ln(b), e.g. ln(10) or ln(2).) In order to determine, how much entropy this term contributes to the whole system, the term has to be weighted by its share of E, that is Ei/Etotal. The sum of the share weighted entropies is the actual entropy of the system: Σi=1..N(Ei*ln(Ai/Ei))/Etotal. If all Ai/Ei would be same, the entropy would be ln(Atotal/Etotal), which is the maximum entropy of the system. - As you see, the Theil index (Theil redundancy) is normalized by the ressource unit: E appears as devider. You can read that as "per ressource unit". Now supply siders can start to discuss with demand siders about this property of Theil's index. Alternatively one can use an index which is normalized to a single owner. This is what I call the "liehT redundancy". As in the real world distribution between owners and ressources is mutual, the truth is between both redundancies. That is why I came up with the "Symmetric redundancy". Or call it "symmetrized Theil redundancy". The result is almost as simple and clear as Hoover's inequality.
[b] Euler's constant e≈2.71828 has the property dx/d(ex)=ex. To economists that number may look like having originated from physics. Although Euler's constant is very important for physics and first was published by Leonhard Euler in his Mechanica (1736), it was economics where the first few digits of the number were found. Jacob Bernoulli (1654-1705) wrote these digits down when analyzing compound interest, where growth in one step is split into many little growth steps. If the steps become infinitely small, growth is continuous. Examples for an annual interest of 100%: (1) Stepwise growth: 1 Euro grows to 2 Euros in one year if compounding occurs at the end of the year. (2) Deviding one growth step into many little growth steps: 1 Euro grows to about 2.71457 Euros within one year if compounding occurs daily. (3) Continuous growth: 1 Euro grows to about 2.71828 Euros within one year if compounding occurs continuously.

[1] Mark Kesselman: French Local Politics (A Statistical Examination of Grass Roots Consensus), 1966, American Political Science Review, No.60 (December), pg.963-974 (mentioned by Coulter as reference for MacRae coefficient; Daniel MacRae Keenan?)
[2] Lionnel Maugis: Inequity Measures (png) in Mathematical Programming for the Air Traffic Flow Management Problem with En-Route Capacities (for IFORS 96), 1996 (CENA - Centre d'études de la Navigation Aérienne - Sofréavia, Orly Sud 205, 94542 Orly Aérogare Cedex, France)
[3] Redundancy as defined in ISO/IEC DIS 2382-16
[4] William Hanna, Joseph Barbera: The Jetsons (html), 1964-1980
[5] Niklas Luhmann: Die Wirtschaft der Gesellschaft, 1988
[6] Benjamin Barber: Fear's Empire: War, Terrorism, and Democracy, 2003 (Imperium der Angst: Die USA und die Neuordnung der Welt, German translation 2003)
[7] Amartya Sen: On Economic Inequality, 1973 (Enlarged Edition with a substantial annexe after a Quarter Century with James Foster, Oxford 1997)
[8] Yoram Amiel (Autor), Frank A. Cowell: Thinking about Inequality: Personal Judgment and Income Distributions, 2000
[9] Wolfgang Kitterer: Mehr Wachstum durch Umverteilung? (pdf), 2006
[10] Charles I. Jones: Introduction to Economic Growth, 2002 (growth data: Lucas(1998) & Maddison (1995))
[11] Hartmut Bossel: Indicators for Sustainable Development (pdf), 1999
[12] Andreas Kamp, Andreas Pfingsten, Daniel Porath: Do banks diversify loan portfolios? A tentative answer based on individual bank loan portfolios (pdf), Deutsche Bundesbank, 2005
[13] Juana Domínguez-Domínguez, José Javier Núñez-Velázquez: The Evolution of Economic Inequality in tth EU Countries During the Nineties (pdf), 2005 (The authors define a normalized Theil index. That index is similar to the Atkinson inequality given in the calculus above as one of two indices which I found from Atkinson.)
[14] Philip B. Coulter: Measuring Inequality, 1989
[15] Eberhard Schaich: Lorenzkurve und Gini-Koeffizient in kritischer Betrachtung. Jahrbücher für Nationalökonomie und Statistik 185 (1971), 193-298
[16] Károly Henrich: Globale Einkommensdisparitäten und -polaritäten (pdf), 2004
[17] Travis Hale, University of Texas Inequality Project: The Theoretical Basics of Popular Inequality Measures (doc), 2003; Examles 1A and 1B
[18] Wikipedia: Theil Index and Hoover Index (html); especially ''The Teil index and the Hoover Index'' and ''Pareto principle'', 2007-10-20 (html)
(Theil-Index in aktuell und 2007-12-24, interessant auch 2007-11-17)
[19] Michail W. Wolkenstein (Mikhail Vladimirovich Volkenstein): Entropie und Information, Moskau 1986 (excellent explanation of entropy, but out of print)
[20] Arieh Ben-Naim: Entropy Demystified, 2007 (The book keeps the promise of its title and makes science beautiful.)
[21] Nicholas Georgescu-Roegen: Energy and Economic Myths: Institutional and Analytical Economic Essays, 1977
[22] Rainer H. Rauschenberg: Die Bedeutung des zweiten Hauptsatzes der Thermodynamik für die Umweltökonomie (html| zip| pdf), 1990

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Keywords:, Gini coefficient, Gini index, inequity measures, income distribution, wealth distribution, concentration measures, statistics, econometrics, welfare function, welfare economy, powerty, wealth, online-calculator, calculus, Pareto distribution, 80:20, 80/20, progressive tax, distributive justice, egalitarism, egalitary;, Pseudo Gini Index, Ginikoeffizient, Theilmaß, Ungleichverteilung, Ungleichheit, Einkommensverteilung, Vermögensverteilung, Konzentrationsmaße, Statistik, Ökonometrie, Wohlfahrtsfunktion, Wohlfahrtsökonomie, Teilhabe, Armuts- und Reichtumsbericht, Umverteilungsdruck, Armut, Wohlstand, Reichtum, Formelsammlung, Online-Rechner, Lorenz-Kurve, Quantile, Platon, Pareto-Verteilung, Steuerprogression, Verteilungsgerechtigkeit, Ergebnisgerechtigkeit, Leistungsgerechtigkeit, Diskussion um die Neidgesellschaft und Gleichmacherei, Egalitarismus, egalitär, Sozialquote, Chancengleichheit, Managergehälter, Einkommensentwicklung, Verteilungsbericht, Durchschnittseinkommen vs. Median, Redundanz, Entropiemaße, Ungleichheitsthematisierung (Inequality Issuization), Ursachen des Terrorismus, Boltzmann-Shannon-Entropy