IneqComps

@ineqcomps.bsky.social

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This account mainly is about inequality measures. A special focus is on a symmetrized Theil-T redundancy.

https://www.ungleichheit.info/en/articles Archive Inequality in Germany Visuals Germany's first archive on inequality
Top 1% of U.S. earners now have more wealth than the middle class. Share of household wealth by year, Q2, 1993-2023 Source: Federal Reserve. Middle class is earners between the 20th and the 80th percentile of income. Keywords: inequality, wealthinequality, inequity
The Dual Challenge of Tolerable Economic Inequality: A Multidisciplinary Analysis of Definition Dilemmas and the Risks of Acceptable Disparities By Maurizio Bovi 2nd edition, 2025-08-02 https://link.springer.com/book/10.1007/978-3-031-97066-5 Series (Ed. Jaques Silber): Economic Studies in Inequality, Social Exclusion and Well-Being This book tackles two central challenges in the discourse on economic inequality. First, there is no consensus on what constitutes a morally acceptable inequality, as even philosophers struggle to conceptualize benchmarks for distinguishing between acceptable and unacceptable disparities. Second, while tolerable inequality has the potential to evolve into systemic patterns of polarization, segregation, and discrimination, it can ultimately escalate into unjustifiable inequality. This self-reinforcing cycle complicates efforts to achieve equitable distributions. The book explores these challenges in two sections through a multidisciplinary lens. Part one discusses distributive justice and philosophical debates surrounding inequality’s acceptable boundaries. Part two employs the concept of social aggregations—such as families, cities, firms, and nations—to examine how even initially tolerable inequality can spread and intensify through interconnected social systems. By emphasizing the relational nature of inequality, the book underscores its systemic complexity and the need for holistic understanding.
Oxfem: A European agenda to tax the super-rich: A solution to inequality in the European Union 2025-10-09 53 pages long Languages: English Overview: Europe faces a deep inequality crisis: the richest 1% in the EU own nearly a quarter of all wealth while half the population shares just 3%. Decades of tax cuts for the wealthy and corporations resulted in the super-rich paying proportionally less taxes than ordinary citizens, eroding fairness, democracy, and social cohesion. The EU lacks harmonized policies to curb extreme wealth concentration and tax avoidance of the wealthiest. Oxfam calls for bold reforms, such as an EU-wide or national tax on the super-rich and transparency mechanisms like an EU assets registry, to fund social needs, climate action, and development. Taxing the super-rich is widely supported, is feasible and is urgent. Keywords Accountability Democracy European Union Fiscal justice Governance Inequality Progressive taxation Tax avoidance Tax the super-rich wealth redistribution Wealth tax
zSym = 1 - exp(-rSym) -------------------------- (log is the logarithm with the base e, the natural logarithm.) Theil_T[(a,b)] + Theil_T[(b,a)] = a * log (a / b) / aTotal + b * log (b / a) / bTotal = a * (log a - log b) / aTotal + b * (log b - log a) / bTotal = a * log a / aTotal - a * log b / aTotal + b * log b / bTotal - b * log a / bTotal = log a * (a / aTotal - b / bTotal) - log b * (a / aTotal - b / bTotal) = (log a - log b) * (a / aTotal - b / bTotal) = log (a / b) * (a / aTotal - b / bTotal) Theil_sym = rSym = 1/2 * sum( log (a / b) * (a / aTotal - b / bTotal)) Theil_sym = rSym = 1/2 * sum(abs(log (a / b) - k)**(σ=1) * abs(a / aTotal - b / bTotal)) k = log (aTotal / bTotal) σ is a parameter which controls how "entropic" the indicator is. Default is σ = 1. If σ = 0, the symmetrized Theil redundancy turns into the Hoover index ("Robin Hood index"). ----------------------------- Keywords: Theil index, Theil redundancy, Theil-T, Theil_sym, Theil_symCDF, inequality, inequity, entropy measures, inequality measures, inequality indicators, Gini index, Gini coefficient, Gini replacement, symmetric Theil redundancy, zSys, rSys, IneqComps
The Plato Index a new international comparative measure of tax progressivity Valpy FitzGerald, Oxford University Intertax Workshop “Tax, Poverty and Finance for Development” Essex University 6/7 July 2006 https://www.taxjustice.net/cms/upload/pdf/Valpy_Fitzgerald_06_Plato_Index.pdf
https://www.federalreserve.gov/releases/z1/dataviz/dfa/distribute/chart/#range:1990.3,2024.4;quarter:142;series:Net%20worth;demographic:income;population:all;units:levels Keywords: inequality | wealth | wealth distribution | FED | DFA | Distributional Financial Accounts | Jerome Powell | GOP | Republicans | Trump
Plato_Inequality_Indicator.hs (Haskell code) is here: in https://drive.proton.me/urls/FBYNTG45P4#RafOE8CV1KPf You can use e.g. ChatGPT or other AI services to convert the code to other languages.
What is inequality? In recent years [before 2011] there has been an explosion of interest in the subject that has yielded a substantial body of formal tools and results for income-distribution analysis. But does the standard axiomatic structure coincide with public perceptions of inequality? Or is the economist's concept of inequality a thing apart, perpetuated through serial brainwashing in the way the subject is studied and taught? Amiel and Cowell examine the evidence from a large international questionnaire experiment using student respondents. Along with basic "cake-sharing" issues, related questions involving social-welfare rankings, the relationship between inequality and overall income growth and the meaning of poverty comparisons are considered.
> .\IneqComps.exe "2.19411555e7,0.1|2.687184475e7,0.9|5.79028605e7,9|4.8167033e7,40|3907153.25,40|1,10" IneqComps V26 rSym ≥ 2.13102 (symmetric Theil redundancy) zSym ≥ 88.128% (zSym=1-exp(-rSym)) zPlato ≥ 84.949% (Plato inequality, Theil & zSym & Plato equivalent ratio: 92.474:7.526) zGini ≥ 74.144% (Gini inequality, Gini equivalent ratio: 87.072:12.928) zHoover ≥ 57.206% (Hoover inequality, Hoover equivalent ratio: 78.603:21.397) zAccept ≈ -78.965% (experimental: zAccept=exp(zHoover-rSym)-1 rAccept ≈ -1.55896 (experimental: rAccept=zHoover-rSym) > .\IneqComps.exe "2.19411555e7,0.1|2.687184475e7,0.9|5.79028605e7,9|4.8167033e7,40|3907153.25,40|1000000,10" IneqComps V26 rSym ≥ 1.42679 (symmetric Theil redundancy) zSym ≥ 75.992% (zSym=1-exp(-rSym)) zPlato ≥ 74.385% (Plato inequality, Theil & zSym & Plato equivalent ratio: 87.193:12.807) zGini ≥ 73.128% (Gini inequality, Gini equivalent ratio: 86.564:13.436) zHoover ≥ 56.785% (Hoover inequality, Hoover equivalent ratio: 78.393:21.607) zAccept ≈ -57.639% (experimental: zAccept=exp(zHoover-rSym)-1
https://www.jstor.org/stable/43965822 J-divergence measurements of economic inequality Nicholas Rohde Journal of the Royal Statistical Society. Series A (Statistics in Society) Vol. 179, No. 3 (JUNE 2016), pp. 847-870 (24 pages) Published By: Oxford University Press -------------------------------------------------- J. R. Statist. Soc. A (2016) 179, Part 3, pp. 847-870 J-divergence measurements of economic inequality Nicholas Rohde Griffith University, Gold Coast, Australia [Received January 2014. Final revision September 2015] Summary. The paper uses a symmetric entropy statistic to study income inequality. The index distrition in into mairarion reto ace disay men, and then rant tom original. This alowe the measure to be interpreted as an average of n income-to-mean divergences such that the inequality estimate can be broken down into contributions across population subgroups. Various properties of the index are analysed and an application comparing the USA, Germany and Britain is provided. We focus on the sensitivity of inequality to the tails of the income distribution and show that the extreme right-hand tail accounts for a large and generally increasing proportion of total inequality. This result holds even if incomes are measured at the household level, averaged over a 5-year period and taken after government taxes and transfers. Keywords: Distance; Divergence; Inequality; Information theory
Sensitivity of Inequality Measures to Extreme Values by Frank A. Cowell STICERD, London School of Economics and Political Science and Emmanuel Flachaire EUREQUA, University of Paris I Panthéon-Sorbonne Discussion Paper Distributional Analysis Research Programme No. DARP 60 The Toyota Centre March 2002 Suntory and Toyota International Centres for Economics and Related Disciplines London School of Economics and Political Science Houghton Street London WC2A 2AE
In 2007 I used http://www.wider.unu.edu/wiid/WIID2b.zip (https://web.archive.org/web/20070610193806if_/http://www.wider.unu.edu/wiid/WIID2b.zip) to plot my "Plato inequality" and "redistributive aggression" over the Gini index just to have a look at how my inequality measures look if computed from real world data.
> .\IneqComps.exe "2.19411555e7,0.1|2.687184475e7,0.9|5.79028605e7,9|4.8167033e7,40|3907153.25,50" IneqComps V26 rSym ≥ 1.49446 (symmetric Theil redundancy) zSym ≥ 77.563% (zSym=1-exp(-rSym)) zPlato ≥ 75.644% (Plato inequality, Theil & zSym & Plato equivalent ratio: 87.822:12.178) zGini ≥ 73.898% (Gini inequality, Gini equivalent ratio: 86.949:13.051) zHoover ≥ 57.206% (Hoover inequality, Hoover equivalent ratio: 78.603:21.397) zAccept ≈ -60.244% (experimental: zAccept=exp(zHoover-rSym)-1 rAccept ≈ -0.92240 (experimental: rAccept=zHoover-rSym)
See also: https://bsky.app/profile/ineqcomps.bsky.social/post/3ltzbrfpezs2z ------------------------------------------------------------------- From: https://www.diw.de/sixcms/detail.php?id=57489 'Derivation of Theil’s Inequality Measure from Lorenz Curves' Nicholas Rohde School of Economics The University of Queensland Brisbane, Australia March, 2007 If income data is given in the form of population shares and corresponding income shares, Theil’s T measure is easily calculated using the formula (1) T=Σ(i=1 until n)[p_i*ln(p_i/q_i)] The statistic uses the expected information content of the income distribution to measure the level of inequality. Theil’s measure may also be expressed as (2) Y=ln(n)-S where S is the Shannon entropy or information content of the distribution, which has a range from 0 to ln(n). Shannon entropy may be calculated for individual incomes or n income shares. For j individual incomes the information content may be calculated as (3) S=Σ(k until j)[p_k*ln(1/p_k)] where p_k is the income share of the kth individual. p_i is the income share of group i q_i is the population share of group i
https://www.diw.de/sixcms/detail.php?id=57489 Derivation of Theil’s Inequality Measure from Lorenz Curves Nicholas Rohde School of Economics The University of Queensland Brisbane, Australia March, 2007 Abstract Theil’s T inequality measure is a commonly used tool for inequality measurement as it satisfies important axiomatic requirements such as the principle of transfers and decomposability. Derived from the concept of information theory, Theil’s measure represents a special case of the Generalised Entropy (GE) type measure. The main objective of this paper is to establish a relationship between Theil’s inequality measure and the Lorenz curve, thus providing a useful link between these two popular tools for studying inequality. The result established in this paper also provides a basis for studying generalised entropy measures for inequality measurement using Lorenz curves. Analytical expressions for Theil’s information theoretic measures are derived for three popular specifications of Lorenz curves [used to compute the Gini index]. Empirical estimates of Theil’s measures for these Lorenz curves are also presented for a number of cases. JEL Classification: C10, D63, I32 Key words: Theil’s T statistic, Lorenz curves, inequality measurement Acknowledgements: I would like to thank Prasada Rao for his assistance in writing this article.
T = rSym = (Theil_T[(a,b)] + Theil_T[(b,a)]) / 2 H = zHoover = Hoover inequality ("Robin Hood index") Software (source code in Haskell: TheilHooverPlatoGini.hs) to compute theses measures: in https://drive.proton.me/urls/FBYNTG45P4#RafOE8CV1KPf
Source: https://commons.wikimedia.org/wiki/File:1831_Schlesinger_Philosoph_Georg_Friedrich_Wilhelm_Hegel_anagoria.JPG used in https://en.wikipedia.org/wiki/Georg_Wilhelm_Friedrich_Hegel
From https://www.nber.org/system/files/working_papers/w33444/w33444.pdf Income Equality in The Nordic Countries: Myths, Facts, and Lessons Magne Mogstad, Kjell G. Salvanes, and Gaute Torsvik NBER Working Paper No. 33444 February 2025 JEL No. J3, J5, O0 ABSTRACT Policymakers, public commentators, and researchers often cite the Nordic countries as examples of a social and economic model that successfully combines low income inequality with prosperity and growth. This article aims to critically assess this claim by integrating theoretical perspectives and empirical evidence to illustrate how the Nordic model functions and why these countries experience low inequality. Our analysis suggests that income equality in the Nordics is primarily driven by a significant compression of hourly wages, reducing the returns to labor market skills and education. This appears to be achieved through a wage bargaining system characterized by strong coordination both within and across industries. This finding contrasts with other commonly cited explanations for Nordic income equality, such as redistribution through the tax-transfer system, public spending on goods that complement employment, and public policies aimed at equalizing skills and human capital distribution. We consider the potential lessons for other economies that seek to reduce income equality. We conclude by discussing several underexplored or unresolved questions and issues.
T = rSym = (Theil_T[(a,b)] + Theil_T[(a,b)]) / 2 H = zHoover = Hoover inequality ("Robin Hood index") Software to compute theses measures: in https://drive.proton.me/urls/FBYNTG45P4#RafOE8CV1KPf The original source: TheilHooverPlatoGini.hs (Haskell code). I additionally generated TheilHooverPlatoGini.py and TheilHooverPlatoGini.lua using ChatGPT.
> --(Save this Haskell file with a .lhs extension.) > type Percentile = (Double, Double) > type AllPercentiles = [Percentile] > type Theil = Double > type ASum = Double > type BSum = Double That described what kind of numbers are used in the code (starting with ">") used below. Now let's talk about redistribution: For redistribution of resources, you can redistribute resources to people, but also people to resources. Below is a Haskell function that computes the symmetrized Theil redundancy. > theilSym :: AllPercentiles -> Theil > theilSym allPercentiles = 0.5 * sum [log (a / b) * ((a / (aTotal allPercentiles)) - (b / (bTotal allPercentiles))) | (a, b) <- allPercentiles] The expression log (a / b) * ((a / (aTotal allPercentiles)) - (b / (bTotal allPercentiles))) is equal to a * log (a / b) / (aTotal allPercentiles) + b * log (b / a) / (bTotal allPercentiles) That function is fed with a table allPercentiles [(a,b)], where a in each group (a,b) usually stands for the group resource (e.g income, wealth etc. of the group) and b in each group (a,b) usually stands for the group size (e.g. amount of people in the group). The original Theil_T: > theilTmobileResources :: AllPercentiles -> Theil > theilTmobileResources allPercentiles = > log ((bTotal allPercentiles)/(aTotal allPercentiles)) -- max entropy > - sum [ a * log (b / a) | (a, b) <- allPercentiles] / (aTotal allPercentiles) -- observed entropy The original Theil_T inverted: > theilTmobilePeople :: AllPercentiles -> Theil > theilTmobilePeople allPercentiles = > log ((aTotal allPercentiles)/(bTotal allPercentiles)) -- max entropy > - sum [ b * log (a / b) | (a, b) <- allPercentiles] / (bTotal allPercentiles) -- observed entropy Helper functions: > aTotal :: AllPercentiles -> ASum > aTotal allPercentiles = sum [a | (a, _) <- allPercentiles] > bTotal :: AllPercentiles -> BSum > bTotal allPercentiles = sum [b | (_, b) <- allPercentiles]
I forgot the logarithms of the sums of a and of b in the formulas for the Theil_T. They represent a maximum entropy in those terms. redundancy = maximum entropy - observed entropy Theil_T[(a,b)] = ln(bTotal/aTotal) - sum(a*ln(b/a))/aTotal Theil_T[(b,a)] = ln(aTotal/bTotal) - sum(b*ln(a/b))/bTotal Luckily, if you add the formulas, then the forgotten part disappears: ln(aTotal/bTotal) + ln(bTotal/aTotal) = 0
https://www2.census.gov/library/publications/2024/demo/p60-282.pdf, page 5: This report presents several measures of income inequality: (1) the Gini index, (2) the ratio of income percentiles, (3) the shares of aggregate household income by quintiles, (4) the Theil index, (5) the mean logarithmic deviation of income (MLD), and (6) the Atkinson measures.
> .\IneqComps.exe "17906784,7517963|56209266,4941934|133272817,6690138|432264597,12033448| 473801090,6954766|272745562,1968380|71047533,213679|34457325,51702|23561458,16199|10801357,3178|22796674,1798" IneqComps V23 rSym ≥ 0.60044 (symmetric Theil redundancy) zSym ≥ 45.143% (zSym=1-exp(-rSym)) zPlato ≥ 52.040% (Plato inequality, Theil & zSym & Plato equivalent ratio: 76.020:23.980) zGini ≥ 51.086% (Gini inequality, Gini equivalent ratio: 75.543:24.457) zHoover ≥ 35.902% (Hoover inequality, Hoover equivalent ratio: 67.951:32.049)
-- ---- zPlato to zSym ---- -- zSym=1-((1-zPlato)/(1+zPlato))**zPlato -- ---- zSym to zPlato (Haskell code) ---- -- -- types -- type ZSym = Double -- Symmetric redundancy measure type ZPlato = Double -- Plato inequality indicator type InitialGuess = Double -- InitialGuess of zPlato for Newton-Raphson iteration type MaxError = Double -- Maximum error for Newton-Raphson iteration type Iterations = Int -- Number of Newton-Raphson iterations type IterationsStartValue = Int -- Start value for Newton-Raphson iterations (usually set to 0) -- zPlato: Convert zSym to zPlato using the Newton-Raphson iteration. zSym2zPlato :: ZSym -> (ZPlato, Iterations) zSym2zPlato zSym | zSym >= 0.99999999999999 = (1,0) | zSym <= 0 = (0,0) | zSym < 0.001 = (sqrt (zSym * 0.5), 0) | otherwise = let initialGuess = 0.18992 + 0.72791 * zSym in newtonRaphson initialGuess zSym 0 1e-12 newtonRaphson :: InitialGuess -> ZSym -> IterationsStartValue -> MaxError -> (ZPlato, Iterations) newtonRaphson a b iter tol | iter >= 100 || abs (func a b) < tol = (a, iter) | otherwise = newtonRaphson (a - func a b / deriv a) b (iter + 1) tol deriv :: Double -> Double deriv a = let d = (1 - a) / (1 + a) in -d ** a * (log d - 2 * a / (1 - a ** 2)) func :: Double -> Double -> Double func a b = zPlato2zSym a - b zPlato2zSym :: ZPlato -> ZSym zPlato2zSym a = 1 - ((1 - a) / (1 + a)) ** a
(log is the logarithm with the base e, the natural logarithm.) Theil_T[(a,b)] + Theil_T[(b,a)] = a * log (a / b) / aTotal + b * log (b / a) / bTotal = a * (log a - log b) / aTotal + b * (log b - log a) / bTotal = a * log a / aTotal - a * log b / aTotal + b * log b / bTotal - b * log a / bTotal = log a * (a / aTotal - b / bTotal) - log b * (a / aTotal - b / bTotal) = (log a - log b) * (a / aTotal - b / bTotal) = log (a / b) * (a / aTotal - b / bTotal) Theil_sym = rSym = 1/2 * sum( log (a / b) * (a / aTotal - b / bTotal)) Theil_sym = rSym = 1/2 * sum(abs(log (a / b) - k)**(σ=1) * abs(a / aTotal - b / bTotal)) k = log (aTotal / bTotal) σ is a parameter which controls how "entropic" the indicator is. Default is σ = 1. If σ = 0, the symmetrized Theil redundancy turns into the Hoover index ("Robin Hood index"). ----------------------------- Keywords: Theil index, Theil redundancy, Theil-T, Theil_sym, Theil_symCDF, inequality, inequity, entropy measures, inequality measures, inequality indicators, Gini index, Gini coefficient, Gini replacement, symmetric Theil redundancy, zSys, rSys, IneqComps
"A warmer climate will, very likely, green the Earth. But the other factors will make much of our current civilization obsolete and much of our economic activity unprofitable. So the future of human society is likely, for a long time, to devolve back to smaller populations, shorter life spans, higher variable costs, and lower fixed costs, along with smaller countries and harsher inequalities both within and between them. Social arrangements, advanced planning, new energy sources, and new investments can mitigate the suffering from this transition, up to a point. But this, very likely so far as we understand it, is the biophysical reality with which future economists, engineers, and planners will have to engage." p. 200 in Galbraith, James K.; Chen, Jing. "Entropy Economics: The Living Basis of Value and Production", 2025
Aspects of - Theil Index - Gini Coefficient Decomposability T: Can separate total inequality into within-group and between-group components G: Not easily decomposable Sensitivity T: Sensitive to the entire income distribution, including extremes G: More sensitive to changes in the middle of the distribution Theoretical Basis T: Based on entropy; mathematically rigorous G: Based on Lorenz curves; intuitive but less analytical Application Scope T: Broad, applicable to various fields of study including development economics G: Primarily used for income and wealth inequalities ----------------------------------------------------------------- Source: The Ultimate Guide to Theil Index in Dev Econ Key insights for development policy research Sarah Lee, AI generated, 2025-04-26 https://www.numberanalytics.com/blog/theil-index-guide-dev-econ https://web.archive.org/web/https://www.numberanalytics.com/blog/theil-index-guide-dev-econ
https://openknowledge.fao.org/server/api/core/bitstreams/35c2d071-828b-45bf-97bc-7ba5d817f458/content © FAO December 2006 EASYPol On-line resource materials for policy making Policy Impacts on Inequality Decomposition of Income Inequality by Subgroups by Lorenzo Giovanni Bellù, Agricultural Policy Support Service, Policy Assistance Division, FAO, Rome, Italy Paolo Liberati, University of Urbino, "Carlo Bo", Institute of Economics, Urbino, Italy for the Food and Agriculture Organization of the United Nations, FAO
-- Part 1 -- -- This Haskell code was sent as a prompt to Meta AI: -- Request to Meta AI: -- Not considering computation errors, -- show that both functions are equivalent. -- --------------------------------------------------------------------------------- type GroupSize = Double type GroupRessources = Double type Percentile = (GroupSize, GroupRessources) type AllPercentiles = [Percentile] type Percentiles = [(GroupSize, GroupRessources)] type TheilRedundancy = Double computeTheilT_1 :: Percentiles -> TheilRedundancy computeTheilT_1 allPercentiles = 0.5 * sum [log (a / b) * ((a / aTotal) - (b / bTotal)) | (a, b) <- allPercentiles] where aTotal = sum [a | (a, _) <- allPercentiles] bTotal = sum [b | (_, b) <- allPercentiles] computeTheilT_2 :: Percentiles -> TheilRedundancy computeTheilT_2 allPercentiles = 0.5 * sum [theilTab a b aTotal + theilTba a b bTotal | (a, b) <- allPercentiles] where aTotal = sum [a | (a, _) <- allPercentiles] bTotal = sum [b | (_, b) <- allPercentiles] theilTab a b aTotal = a * log (a / b) / aTotal theilTba a b bTotal = b * log (b / a) / bTotal -- --------------------------------------------------------------------------------- main :: IO () main = do let percentiles = [(10, 10), (20, 400), (30, 9000), (40, 160000)] let theilT_1 = computeTheilT_1 percentiles let theilT_2 = computeTheilT_2 percentiles print ("Theil T (1): " ++ show theilT_1) print ("Theil T (2): " ++ show theilT_2) {- > .\SymmetricTheilT.exe "Theil T (1): 1.2577697898110196" "Theil T (2): 1.2577697898110196" -}
Not all inequality measures were created equal The measurement of wealth inequality, its decompositions, and an application to European household wealth Rita Neves Costa, Sébastien Pérez-Duarte December 2019 https://www.ecb.europa.eu/pub/pdf/scpsps/ecb.sps31~269c917f9f.en.pdf
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) The graph shows various inequality indicators plotted over the Hoover index ("Robin Hood" index) for societies divided into two just groups. The distribution goes from 50:50 (Hoover = 0) to 0:100 (Hoover = 1).
«[...] A key regulatory function is control of inequality; inequality is necessary, but it is dangerous if becomes too great. Economic inequality motivates activity: economic agents compare themselves with others and strive to improve their position—to make more money, to grow their wealth. Without this incentive, human societies would be much harder to organize and sustain. But too much inequality is like an overheated engine or a person with high blood pressure. It is a sign of discontent, of trouble ahead, and a warning of potential breakdown. [...]» Galbraith, James K.; Chen, Jing. "Entropy Economics: The Living Basis of Value and Production" (p. xiv). University of Chicago Press. Kindle Edition.
-- ---- zPlato to zSym ---- -- zSym=1-((1-zPlato)/(1+zPlato))**zPlato -- ---- zSym to zPlato (Haskell code) ---- -- -- types -- type ZSym = Double -- Symmetric redundancy measure type ZPlato = Double -- Plato inequality indicator type InitialGuess = Double -- InitialGuess of zPlato for Newton-Raphson iteration type MaxError = Double -- Maximum error for Newton-Raphson iteration type Iterations = Int -- Number of Newton-Raphson iterations type IterationsStartValue = Int -- Start value for Newton-Raphson iterations (usually set to 0) -- zPlato: Convert zSym to zPlato using the Newton-Raphson iteration. zSym2zPlato :: ZSym -> (ZPlato, Iterations) zSym2zPlato zSym | zSym >= 0.99999999999999 = (1,0) | zSym <= 0 = (0,0) | zSym < 0.001 = (sqrt (zSym * 0.5), 0) | otherwise = let initialGuess = 0.18992 + 0.72791 * zSym in newtonRaphson initialGuess zSym 0 1e-12 newtonRaphson :: InitialGuess -> ZSym -> IterationsStartValue -> MaxError -> (ZPlato, Iterations) newtonRaphson a b iter tol | iter >= 100 || abs (func a b) < tol = (a, iter) | otherwise = newtonRaphson (a - func a b / deriv a) b (iter + 1) tol deriv :: Double -> Double deriv a = let d = (1 - a) / (1 + a) in -d ** a * (log d - 2 * a / (1 - a ** 2)) func :: Double -> Double -> Double func a b = zPlato2zSym a - b zPlato2zSym :: ZPlato -> ZSym zPlato2zSym a = 1 - ((1 - a) / (1 + a)) ** a
-- import Data.List (sortBy) -- import Data.Function (on) gini :: [(Double, Double)] -> Double gini pairs = let sortedPairs = sortByRatio pairs aList = map fst sortedPairs bList = map snd sortedPairs aTotal = sum aList bTotal = sum bList aCumulative = scanl1 (+) $ map (/ aTotal) aList zGiniNumerators = zipWith3 (\cumAi ai bi -> (2 * cumAi - ai / aTotal) * (bi / bTotal)) aCumulative aList bList zGini = 1 - sum zGiniNumerators in zGini sortByRatio :: [(Double, Double)] -> [(Double, Double)] sortByRatio = sortBy (compare `on` (\(a, b) -> a / b))
PS X:\Haskell> .\IneqComps.exe "2.19411555e7,0.1|2.687184475e7,0.9|5.79028605e7,9|4.8167033e7,40|3907153.25,50" IneqComps V21 rSym ≥ 1.49446 (symmetric Theil redundancy) zSym ≥ 77.563% (zSym=1-exp(-rSym)) zPlato ≥ 75.644% (Plato inequality, Theil & zSym & Plato equivalent ratio: 87.822:12.178) zGini ≥ 73.898% (Gini inequality, Gini equivalent ratio: 86.949:13.051) zHoover ≥ 57.206% (Hoover inequality, Hoover equivalent ratio: 78.603:21.397) 1-Median ≥ 97.539% σ8020 ≈ 0.42999 (experimental: σ for 80:20 ratio) zAccept ≈ -60.244% (experimental: zAccept=exp(zHoover-rSym)-1) rAccept ≈ -0.92240 (experimental: rAccept=zHoover-rSym) PS X:\Haskell> .\IneqComps.exe "0.1,2.19411555e7|0.9,2.687184475e7|9,5.79028605e7|40,4.8167033e7|50,3907153.25" IneqComps V21 rSym ≥ 1.49446 (symmetric Theil redundancy) zSym ≥ 77.563% (zSym=1-exp(-rSym)) zPlato ≥ 75.644% (Plato inequality, Theil & zSym & Plato equivalent ratio: 87.822:12.178) zGini ≥ 73.898% (Gini inequality, Gini equivalent ratio: 86.949:13.051) zHoover ≥ 57.206% (Hoover inequality, Hoover equivalent ratio: 78.603:21.397) 1-Median ≥ 94.247% σ8020 ≈ 0.42999 (experimental: σ for 80:20 ratio) zAccept ≈ -60.244% (experimental: zAccept=exp(zHoover-rSym)-1) rAccept ≈ -0.92240 (experimental: rAccept=zHoover-rSym)
Screenshot: from https://downloads.haskell.org/ghc/latest/docs/users_guide/using-warnings.html#ghc-flag-Wall -Wall¶ Turns on all warning options that indicate potentially suspicious code. They include all warnings in -Wextra, plus: -Whi-shadowing -Wincomplete-record-updates -Wincomplete-record-selectors -Wincomplete-uni-patterns -Wmissing-pattern-synonym-signatures -Wmissing-signatures -Wname-shadowing -Worphans -Wredundant-record-wildcards -Wstar-is-type -Wtrustworthy-safe -Wtype-defaults -Wunused-do-bind -Wunused-record-wildcards -Wincomplete-export-warnings -Wderiving-typeable
TheilHooverPlato.hs in https://drive.proton.me/urls/FBYNTG45P4#RafOE8CV1KPf
# Converted from Haskell code to Phyton by Meta AI import math # -------------------------------------------------------------------------------------------- # Compute rSym (symmetrized Theil redundancy), rSymSigma, ZHoover from a table of percentiles. def three_inequality_indicators(all_percentiles, sigma): a_total = sum(a for a, _ in all_percentiles) b_total = sum(b for _, b in all_percentiles) logratio_total = math.log(b_total / a_total) r_sym = 0 r_sym_sigma = 0 r_hoover = 0 for a, b in all_percentiles:_ logratio = math.log(a / b) abs_logratio_normalized = abs(logratio + logratio_total) ratio_diff = (a / a_total) - (b / b_total) abs_ratio_diff = abs(ratio_diff) r_sym += logratio * ratio_diff r_sym_sigma += (abs_logratio_normalized ** sigma) * abs_ratio_diff r_hoover += abs_ratio_diff return r_sym / 2, r_sym_sigma / 2, r_hoover / 2 # zSym: Convert rSym to zSym. def theil2z_sym(redundancy): return 1 - math.exp(-redundancy) # zPlato: Convert zSym to zPlato using the Newton-Raphson iteration. def z_sym2z_plato(z_sym): if z_sym >= 0.99999999999999: return 1, 0 elif z_sym <= 0: return 0, 0 elif z_sym < 0.001: return math.sqrt(z_sym * 0.5), 0 else: initial_guess = 0.18992 + 0.72791 * z_sym return newton_raphson(initial_guess, z_sym, 0, 1e-12) def newton_raphson(a, b, iter, tol_): if iter >= 100 or abs(func(a, b)) < tol: return a, iter else: return newton_raphson(a - func(a, b) / deriv(a), b, iter + 1, tol) def deriv(a): d = (1 - a) / (1 + a) return -d ** a * (math.log(d) - 2 * a / (1 - a ** 2)) def func(a, b): return z_plato2z_sym(a) - b def z_plato2z_sym(a): return 1 - ((1 - a) / (1 + a)) ** a # -------------------------------------------------------------------------------------------- # That's part 1
Calling the Theil-T “r_Theil” (where “r” stands for “redundancy”) and the Theil-T with swapped arguments “r_liehT” wasn’t such a bad idea. My(?) symmetrized Theil index (symmetric Theil redundancy) is (r_Theil+r_liehT)/2. For a table with percentiles [(a,b)] it turned out that the symmetric Theil redundancy r_Sym = 0.5*sum(ln(a_i/b_i)*(a_i/a_total-b_i/b_total)). Due to the symmetry, that also works for [(b,a)]. To my surprise many years ago, r_Sym=0.5*sum((ln(a_i/b_i)-ln(a_total/b_total))*(a_i/a_total-b_i/b_total)) is wrong. r_Sym=0.5*sum((|ln(a_i/b_i)-ln(a_total/b_total)|^σ)*|a_i/a_total-b_i/b_total| is necessary, if you want to introduce the sensitivity σ. You can rewrite ln(a_i/b_i)-ln(a_total/b_total) as ln(b_total/a_total)-ln(b_i/a_i) r_Sym(σ=0)=0.5*sum(|a_i/a_total-b_i/b_total|). That’s the simplest of all inequality measures: the Hoover index, also known as “Robin Hood index”.
Paragraphs 2.1 (The Herfindahl-Hischmann Index, HHI) and 2.2 (Concentration ratios, CRs) from https://www.degruyterbrill.com/document/doi/10.1515/jbnst-2020-0010/html
Zitierhinweise: Bei Datennutzung: Monopolkommission (2022), „Unternehmenskonzentration in Deutschland“, https://www.monopolkommission.de/daten/konzentration.html Datensatzbeschreibung: Heidorn, Helen und John P. Weche (2020), „Business Concentration Data for Germany“, Jahrbücher für Nationalökonomie und Statistik 241 (5–6): 801–11, https://doi.org/10.1515/jbnst-2020-0010
I am a retired mechatronics engineer and started to learn Haskell programming just for brain jogging. For that I used a little project which I did 30 years ago, when I developped an inequality indicator for a friend who didn’t like maths too much, but had to analyze income distributions. r_sym = 1/2 sum(ln(a_i/b_i)*((a_i/a_total)-(b_i/b_total))) z_sym = 1-e^(-z_sym) Then I didn’t know anything about the many already existing inequality measures. Later I found a big zoo of such measures in Philip B. Coulter’s Measuring Inequality (1989). And I found that my creation was the arithmetic mean of Theil’s inequality measure and his measure with swapped parameters. It’s a symmetrized Theil measure then. Almost 20 years I closed my little project. But nowI started to play with inequality indicators again and found some mistakes in what I wrote about them earlier. Luckily the maths are not affected, so I don’t have to rewrite IneqComps (and my friend doesn’t have to withdraw his papers), but some names and terms I used until today in my explanations seem not to be correct. Example: Calling z_sym an Atkinson measure is probably wrong.
Comparing measures: Does the Gini tell the same story as other inequality measures? A measure of inequality summarizes how spread out the distribution is — just like the standard deviation, which you may have learned about in school.3 Implicit in such summary measures are judgments about what should count most when measuring inequality.4 For example, compare two hypothetical populations. In one, the rich are much richer than those in the middle, but the incomes of poorer people are only a little below those in the middle. In another, there is the opposite situation: the incomes of the rich are only a little above those in the middle, but the poor are much poorer. In which population would you say inequality is highest? Your answer will depend on your personal judgments about how these gaps in different parts of the distribution compare in terms of their contribution to inequality overall. Such value judgments are implicitly built into the mathematical definition of an inequality measure. That is true of all inequality measures, and the Gini coefficient is no exception.
Transformation of the brightness of pixels of en image with the inverse(!) function of 1-((1-x)/(1+x))^x. That inverse function makes patterns in the very bright and very black areas more visible.
per year: Category|Groupsize||This is about wealth (net worth) per wealth percentile. TopPt1|0.1%|| RemainingTop1|1%|??: There is a conflict between the category definitions in two files. Next9|9.0%||In dfa-networth-levels.csv it’s Top 1. In the dfa definitione it’s 0.9%. Next40|40.0%||I think it’s like stated in the CSV. Bottom50|50.0%||Therefore I used RemainingTop1 minus TopPt1 as amount of wealth in the 2nd highest percentile. Site: https://www.federalreserve.gov/releases/z1/dataviz/dfa/distribute/chart/#range:1990.3,2024.4;quarter:142;series:Net%20worth;demographic:income;population:all;units:levelshttps://www.federalreserve.gov/releases/z1/dataviz/dfa/distribute/chart/#range:1990.3,2024.4;quarter:142;series:Net%20worth;demographic:networth;population:all;units:levels The input table is a segment of dfa-networth-levels.csv Goetz W. Kluge, 2025-06-28 year|Plato|Gini 1990|67.36%|65.92% 1991|67.14%|65.65% 1992|66.48%|65.11% 1993|67.21%|65.51% 1994|68.06%|65.89% 1995|68.23%|65.89% 1996|68.62%|66.18% 1997|68.95%|66.57% 1998|69.27%|66.98% 1999|69.45%|67.41% 2000|69.39%|67.52% 2001|68.91%|67.01% 2002|69.01%|67.08% 2003|69.83%|67.70% 2004|70.83%|68.51% 2005|71.19%|68.83% 2006|72.00%|69.51% 2007|73.60%|70.60% 2008|75.77%|71.52% 2009|78.59%|72.38% 2010|80.18%|73.19% 2011|81.04%|73.52% 2012|79.99%|73.41% 2013|78.61%|73.11% 2014|77.92%|73.21% 2015|77.56%|73.26% 2016|77.02%|73.13% 2017|76.43%|72.99% 2018|75.74%|72.76% 2019|75.37%|72.66% 2020|73.97%|71.38% 2021|73.58%|70.81% 2022|72.37%|69.48% 2023|72.69%|69.71% 2024|73.06%|70.00%
> .\IneqComps.exe "67000000,37|16276100,36|3900,27" IneqComps V21 rSym ≥ 1.38786 (symmetric Theil redundancy) zSym ≥ 75.039% (zSym=1-exp(-rSym)) zPlato ≥ 73.633% (Plato inequality, Theil & zSym & Plato equivalent ratio: 86.817:13.183) zGini ≥ 48.727% (Gini inequality, Gini equivalent ratio: 74.363:25.637) zHoover ≥ 43.451% (Hoover inequality, Hoover equivalent ratio: 71.726:28.274)
(1) z_Sym = (1/2) * (sum ( ln(a/b) * ((a / aTotal) - (b / bTotal)))) (2) = (1/2) sum (|ln((a*sum(b)) / (b*sum(a))|^σ * |(a / aTotal) - (b / bTotal)|) That is valid for σ=1, The 1st term is the symmetrized Theil redundancy. For σ=0, the second term turns into a Hoover inequality index.
Data source: https://www.federalreserve.gov/releases/z1/dataviz/dfa/distribute/chart/#range:1990.3,2024.4;quarter:142;series:Net%20worth;demographic:income;population:all;units:levels Wealth Distribution (Grouping: Wealth per Income Percentile) Percentiles: Category|Groupsize pct99to100|1.0% pct80to99|19.0% pct60to80|20.0% pct40to60|20.0% pct20to40|20.0% pct00to20|20.0% year|Plato|Gini 1990|42.51%|39.19% 1991|42.91%|39.32% 1992|42.84%|39.27% 1993|43.66%|39.73% 1994|43.96%|39.61% 1995|44.88%|39.78% 1996|45.57%|40.44% 1997|46.50%|41.41% 1998|47.60%|42.64% 1999|48.05%|43.82% 2000|48.24%|44.77% 2001|46.69%|44.13% 2002|46.37%|43.47% 2003|47.35%|43.37% 2004|50.07%|44.43% 2005|51.24%|45.46% 2006|52.68%|46.94% 2007|54.07%|48.42% 2008|52.48%|47.88% 2009|50.52%|47.23% 2010|51.07%|48.40% 2011|52.11%|49.52% 2012|53.18%|50.46% 2013|54.57%|51.61% 2014|55.97%|52.87% 2015|56.60%|53.48% 2016|56.98%|53.76% 2017|57.53%|53.95% 2018|57.60%|53.67% 2019|57.87%|53.64% 2020|56.56%|52.85% 2021|57.28%|53.58% 2022|55.18%|51.88% 2023|55.50%|52.28% 2024|56.48%|53.19%
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)
Lionnel Maugis, "Inequity Measures 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) Keys: Gini, Gini index, Gini coefficient, Hoover concentration index, Coulter coefficient, Theil's entropy coefficient, Atkinson indices
# Example in Python for how to compute the Plato inequality indicator: # Goetz Kluge, 2025-06-29 import math # ------------------------------------------------------------------------------- # 1st step: Compute rSym (symmetrized Theil redundancy) from a table of pairs (a,b). def compute_theil_redundancy(pairs): a_total = sum(a for a, _ in pairs) b_total = sum(b for _, b in pairs) return 0.5 * sum(math.log(a / b) * ((a / a_total) - (b / b_total)) for a, b in pairs) # 2nd step: Convert rSym to zSym. def theil2zSym(redundancy): return 1 - math.exp(-redundancy) # cumulative distribution function (CDF) with lambda = 1 # 3rd step: Convert zSym to zPlato using the Newton-Raphson iteration. def zSym2zPlato(z_sym): def zPlato2zSym(a): return 1 - ((1 - a) / (1 + a)) ** a def deriv(a): d = (1 - a) / (1 + a) return -d ** a * (math.log(d) - 2 * a / (1 - a ** 2)) def func(a, b): return zPlato2zSym(a) - b def newton_raphson(a, b, iter, tol): if iter >= 100 or abs(func(a, b)) < tol: return a, iter else: return newton_raphson(a - func(a, b) / deriv(a), b, iter + 1, tol) if z_sym >= 0.99999999999999: return 1, 0 elif z_sym <= 0: return 0, 0 elif z_sym < 0.001: return math.sqrt(z_sym * 0.5), 0 else: initial_guess = 0.18992 + 0.72791 * z_sym return newton_raphson(initial_guess, z_sym, 0, 1e-12) # ------------------------------------------------------------------------------- # Example usage: def main(): data_points = [(50, 2.5), (40, 47.5), (9, 27), (1, 23)] r_sym = compute_theil_redundancy(data_points) # 1st step z_sym = theil2zSym(r_sym) # 2nd step z_plato, steps = zSym2zPlato(z_sym) # 3rd step print(f"rSym: {r_sym}") print(f"zSym: {z_sym}") print(f"zPlato: {z_plato}") print(f"Newton-Raphson iterations: {steps}") if __name__ == "__main__": main() # rSym: 1.161710269336159 # zSym: 0.6870495067042661 # zPlato: 0.6880377297113732 # Newton-Raphson iterations: 3