Talk:Income inequality metrics

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Absolute measures

1. Absolute measures often give very different results than relative measures. For example, in measuring inequality changes due to the development of less developed countries, absolute measures typically show improvements as the general income level rises, but it is also common for relative measures to deteriorate as the new wealth becomes concentrated in the hands of the upper percentiles. The diverging results can be a problem if they are used inappropriately or interpreted incorrectly.

I cut it, because it claims as a fact that developed countries often have higher income inequality, while the empirical data seems to show a completely opposite trend. Almost universally, the less developed the country, the higher its income inequality. Seriously, it should be backed by some data if it is to stay.

Here's a map: Taw 13:58, 25 January 2006 (UTC)

Why?

Would someone mind putting in a section iterating why income equality is important? Fephisto 18:49, 9 September 2006 (UTC)

I put in a link to economic inequality, which covers those issues. -- Beland 20:30, 11 February 2007 (UTC)

It's an important issue. But the relation between inequality an e.g. growth is non-linear. There may be an optimum range of inequality. I assume, it is between (a) Theil index and Hoover index being equal and (b) Theil index minus Hoover index having a minimum. --DL5MDA (talk) 01:12, 31 October 2008 (UTC)

Another inc.ineq. index

Please accept my apology if what I'm doing is improper: I made another wealth distribution index, which I hope is better then Gini etc, or at least can be of use. Now I'm unclear on whether Wikipedia (should) list "common" knowledge, or "all" knowledge. In the first case, it should probably not be listed, though I'm neither sure whether this index already exists. I can imagine someone might view this as an attempt to make advertisements. Please don't be insulted, I'm only trying to improve things.

Index: 2 * Sum { .5 ^ ( Average-Data / Element-Data ) } / Total-Elements-N

I've converted this into a C program (free[soft]ware) for convenience: http://www.xs4all.nl/~joshb/cc/distribution.html I've called it "naddi", ``normalized average data distribution index. Since it might be unknown (unproven), please verify before adding. Jos Boersema (joshb@ -REDUCESPAM- xs4all.nl) —The preceding unsigned comment was added by 80.127.225.147 (talk) 12:31, 4 February 2007 (UTC).

Uh. Oh. In Philip B. Coulter's Measuring Inequality (1989) you can find about 50 inequality measures. I think, one should focus on the Gini index (for the sake of tradition), the Theil index (probably representing the most realistic ressource distribution model) and the Hoover index (for the sake of simplicity). The Atkinson index can be represented as a transformation of the Theil index. That should be enough. --DL5MDA (talk) 01:24, 31 October 2008 (UTC)

I agree with DL5MDA here. In fact the article should start off right after the lead (or maybe even in the lead) by listing those 3 (or 4 if you count Atkinson) indices and defining them. On the other hand, the present section on 'Absolute income criteria' should be removed, as it is not about inequality at all but rather poverty, which is an altogether different concept. In fact another thing that belongs in the lead or right after it is a list of 'properties' that generally a measure of inequality should posses, as found in the literature (briefly; anonymity, scale independence, population independence and the Dalton-Pigou principle). Then these assumptions could be used to contrast the notion of 'inequality' with 'poverty' (difference in the DP principle) or even fairness (anonymity assumption).radek (talk) 02:28, 1 November 2008 (UTC)

As for "...the article should start off right after the lead (or maybe even in the lead) by listing those 3 (or 4 if you count Atkinson) indices and defining them...": I did that in the German wikipedia (de:Ungleichverteilungsmaße) in order to provide a "less mathematical" explanation of the three important inequality metrics. I fear, I presently do not have the time, to do this here as well. --DL5MDA (talk) 07:54, 1 November 2008 (UTC)

Theil index is the special case of Atkinson index. So rather than giving a special case, we should give the more general Atkinson formulation. Frankly, despite my earlier work in income distribution economics, I could not wrap my head around Hoover index. I've never seen it used there. Atkinson index and generalized entropy indices are used for serious analysis with group decompositions; Gini index and percentile ratios are used because they are easy to explain, although they are lousy measures of inequality. Please take a look at my updates on Atkinson index, and feel free to steal the mathematical formulations from there :)). Stas K (talk) 15:29, 26 March 2010 (UTC)

I think that's known in the economics literature as Kakutani index. I had a reference Kakutani (1978) off the top of my head, but Google scholar did not give anything meaningful :(. Stas K (talk) 15:29, 26 March 2010 (UTC)

Income distribution measures

The concept of inoome distribution and inequaliity is very important and one that needs further research....I hoped to see the content cover measures that have been presented and used at least mention, how these are used and whether important. Growth economists are interested in analytic forms that can be incorporated into growth models - since income distribution is of paramount importance to long term economic growth along with aging / demographics. 1- why important, 2- how measured, 3- what are the measures, 4- how are they used.

Analytically the important properties are 1- range of the measure - capable of describing extremes normally encountered in economies, 2- scalability - the measure remains the same if all recipients receive the same % increase in incomes, 3- captures wealth transfers

Beyond Lorenz curves and Ginis the analytic forms that have been used include

lognormal log logistic Singh Maddala Dagum 4 parameter Champernowne 5 parameter

Dbecher-hamburg 22:31, 3 December 2007 (UTC)

Well, as for the amount of inequality metrics see above ;-) --DL5MDA (talk) 01:24, 31 October 2008 (UTC)

Merge with Economic Inequality

This article pretty much deals with the same subject matter as economic inequality. Should they be merged? — Preceding unsigned comment added by Kodemizer (talk • contribs) 04:21, 18 April 2008 (UTC)

This article is on metrics. However, there wasn't too much metrics in it. So I added the spreadsheet for the Gini index, the Robin Hood index and the Theil index. --DL5MDA (talk) 01:24, 31 October 2008 (UTC)

I saw the redirects to this article and now can understand the proposal to merge the article. However, as for now, the article really deserves its title "income ineqiality metrics". So rather the redirects should change. --DL5MDA (talk) 10:13, 2 November 2008 (UTC)

Rewrote first para after lead

First let me say that it's not OR, though I have not had time to properly cite it. The stuff on the four basic properties of inequality indices can be found in any economics textbook which deals with inequality in a serious manner, for example most development econ books. The transfer principle (or the Dalton-Pigou principle) is central and it probably should have its own page. At the moment I'm not sure whether or not the Hoover index satisfies this principle and am too tired to think about it - since it's the area between the line of equality and the Lorenz curve though, it should. Second, I though it was important to distinguish inequality from poverty and fairness and including the four properties right at the beginning allows one to do that in a natural manner. Also I have this vague recollection of a paper which showed that the Theil index and its derivatives are the only (if and only if) class of inequality metrics which satisfy the four properties plus the decomposability one but that was way back in grad school. Finally this article could also benefit from a discussion of how inequality is actually measured in practice (very, very sloppily) and how entire income distributions are extrapolated from a limited number of data points. I don't have time to review that literature right now so help from others would be greatly appreciated.radek (talk) 09:21, 1 November 2008 (UTC)

I like your edit. By the way, the Hoover index satisfies anonymity, scale independence, population independence and the strong form of the Dalton-Pigou criterion. I add that to the article. --DL5MDA (talk) 10:17, 1 November 2008 (UTC)

Some hefty rearranging

@Radeksz: By http://en.wikipedia.org/w/index.php?title=Income_inequality_metrics&diff=249080070&oldid=248968702 I rearranged some of your and my edits and added a briefing on the 3 important inequality measures. Please check, wether I got your statements on quintiles wrong. --DL5MDA (talk) 22:27, 1 November 2008 (UTC)

Looks good. I made some minor spelling and stylistic changes. In terms of the article as a whole, the section on the Theil index maybe be a little too long relative to the other sections, particularly since most of that information should be in the relevant article. I'm also not clear on what this phrase means: "techniques used to make judgments within the concept of poverty and fairness". Can you clarify? Another thought; it might be worthwhile to (briefly) mention the Foster-Greer-Thorbecke measure (and red link it for now) as a measure which combines measurement of poverty and inequality (for some values of FGT alpha).radek (talk) 00:10, 2 November 2008 (UTC)

I simply removed the "techniques" thing.

As for FGT: I had a look into http://rspas.anu.edu.au/papers/asarc/2003_02.pdf and http://sedac.ciesin.columbia.edu/povmap/contribute/ds_defs_vars.jsp. The FGT measure clearly is a powerty measure. Does that fit into an article on inequality measures? --DL5MDA (talk) 01:46, 2 November 2008 (UTC)

It's a poverty measure that also takes into account (with alpha=2) the inequality among the poor. It can be written as a function of a poor-only Gini and a poor-only coefficient of variation. I'll try to look some stuff up on it.radek (talk) 02:08, 2 November 2008 (UTC)

I comparison to the inequality measures, when using FGT measures, of course the powerty line z and the amount q of poor people within a society with the size N has to be determined before the computation. Computing FGT measures requires more decisions than what is requirde to decide before computing the Gini index, the Theil index or the Hoover index. In order to determine the poverty line, you either could use the median or (better) welfare functions, which again requires to compute inequality measures.

FGT refers to a whole class of measures. Probably that deserves a separate article.

--DL5MDA (talk) 09:11, 2 November 2008 (UTC)

I created an article on the FGT. It's just a beginning for now.radek (talk) 10:31, 2 November 2008 (UTC)

Theil index

The paragraph is a bit longer again. But (in contrary to the article dedicated to the Theil index) here I did not torture the readers with formulas. Also it compares the Theil index with the Hoover index, which due to its nature very well can serve as a contrast to the Theil index. --DL5MDA (talk) 10:09, 2 November 2008 (UTC)

Redundancy

Because this article covers issues which are also covered in separate articles on the Gini coefficient, the Theil index and the Hoover index, I would like to mention an important difference between this article and the articles dedicated to single inequality measures: In this article I avoid formulas. Only in the spreadsheet you find formulas, but there they serve as applied tools and not as an explanation. --DL5MDA (talk) 20:31, 2 November 2008 (UTC)

Why is few academic infos available for the term....

from Google scholar point of view ...???

• http://scholar.google.com/scholar?as_q=Income+inequality+metrics&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=title&as_sauthors=&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en

--222.64.18.96 (talk) 06:02, 8 October 2009 (UTC)

Graph of countries / US

I'd like to see a plot of countries on a Gini coefficient (or whichever measure is chosen) vs. log GDP per capita chart. As I recall, the trend is very strong: there is a strip of egalitarian countries with low to high wealth, and a strip of poor countries with low to high inequality. The notable outlier in the graph is the United States. Surely this effect has been noticed in WP:RS; if anyone has seen this I think it would make a good (sub)section.

CRGreathouse (t | c) 15:45, 21 January 2010 (UTC)

See this blog posting? - Bhyde (talk) 17:28, 21 January 2010 (UTC)

Thanks for that, interesting reading. CRGreathouse (t | c) 02:54, 22 January 2010 (UTC)

United Nations Development Programme

The text mentions that united nations development programme uses the 20/20 ratio. I feel like dropping the name of this important institution using the 20/20 ratio suggests credibility and possible compatibility for that metric. The following link shows on page 4 the details of the calculation of inequality used in the inequality adjusted human development index 2014: http://hdr.undp.org/sites/default/files/hdr14_technical_notes.pdf There is no mention of a 20/20 ratio. Thus, I think the text should remove the mention of united nations development programme's use of the 20/20 ratio as it falsely suggests credibility of that metric. Could a person more actively involved on this page make the adjustment if they agree. — Preceding unsigned comment added by 137.56.17.236 (talk) 13:34, 30 July 2015 (UTC)

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Is poverty decreasing?

We could use some more eyes on this: Talk:List_of_common_misconceptions#poverty. Also, there is the question of whether the proposed misconception should be added to an article on poverty. --David Tornheim (talk) 02:09, 19 October 2018 (UTC)

Some additions to the Gini section

Some suggested Gini-introduction wording

Here is something that I'd like to add to the top of the Gini section, along with a brief explanation of a graphical derivation of the formula for Gini, followed by an algebraic derivation.

I'll wait a month before I add these things to the article, in case there might be objections.

In addition to those derivations, this, directly below, is what I'd like to say at the top of the Gini section:

Given the importance of and interest in, for cumulative population up to some percentile, such as the poorest 10%, their combined income compared to its value under equality...

...the Gini is important and of interest as the sum, over all percentiles, of the shortfall (from equal-share of income) of the cumulative population up to each percentile. .

...as a single number that sums that shortfall over all percentiles.

And, because a household's income's shortfall-from-mean is counted in a lot more cumulative-income totals if the household is at a low percentile, then the Gini weights the shortfall-from-mean of a low-percentile household more strongly than that of a high-percentile household.

And, to answer the objection that the Gini overcounts the middle, I'd suggest that the difference between the Lorenz curve and the 45 degree line be integrated only up to the mean, or only to the 10th percentile.

For the version integrating to the mean, I'd call it "Gini to Mean", "Robin Hood Gini", or "Social Gini" (because what happens to the poorest is more socially important).

For the version that integrates to the 10th percentile, I'd call it "Gini to .1", or "Social Gini".

Of course, for those indices, the integral would be divided by the highest value that it could have. — Preceding unsigned comment added by 71.84.136.105 (talk) 20:03, 17 October 2020 (UTC)

Proposal of Four Additional Gini Versions

Though I prefer the Theil to the Gini, I'd like to mention these possibilities for the Gini, in case there are perceived reasons for continuing to use it.

...in addition to mentioning two other possibilities, one already in use, and the other of interest to mention.

Gini To Mean (...or to .1, or to .4)

Evaluate the Gini's summation of the shortfall from equal-share, of cumulative incomes up to the various percentiles, only up to the 10th percentile. ...or up to the percentile at which the mean occurs...or to the 40th percentile. Divide that shortfall sum by the greatest value that it could have, with complete inequality (one person getting all the income).

Average Percentile-Position of income (or Wealth)

This isn't an original proposal. I've read of it at some income-measurement website.

Calculate where, on the population-percentile scale, is the average of where the income or wealth is.

10% Share: . The combined income of the poorest 10%, divided by what it would be under equality,. . That's the most obvious and natural measure of the most important effect of inequality.

---

(By the way, isn't the Hoover (the Robin-Hood) disqualified by its ignoring of the distribution among the below-mean households? . ...especially since, here, that group extends up to around the 70th percentile?) . It makes sense to keep the detailed definitions and discussion of the indices at their own dedicated Wikipedia-pages. But the article should thoroughly discuss the relative merits, advantages & disadvantages of all reasonable proposed inequality-indices. . With detailed definitions elsewhere, the indices can be defined here by brief non-technical statements starting with their motivations. . — Preceding unsigned comment added by 71.84.136.105 (talk • contribs) 20:20, 18 October 2020 (UTC)

Some comments, criticisms and suggestions

No one will be convinced by entropy as a justification for an inequality-measure formula. If the Theil and its relatives are to be liked, there'd need to be other, socially-meaningful justifications. ...and those would have to be described in clear, plain language.

The Plama is completrely inadequate, in ignoring what happens at the bottom. The people between the 39th and 40th percentiles could take everything away from everyone below the 39th percentile, and the Palma wouldn't notice.

At least the Gini looks at what happens at the bottom, and top-weights it. (...yes, contrary to popular belief, the Gini top-weights the lowest percentiles. (...though it doesn't downweight changes in near-mean incomes as much as we'd like) ). But that weighting is applied to a kind of measure that doesn't do justice to changes in small incomes: Taking $1000 away from a working-poor person isn't the same thing as taking $1000 away from someone who makes $65,000 a year, or from a millionaire. The Gini doesn't recognize that. That's a big fault and failing of the Gini.

The main, most frequently-heard criticisms of the Gini result from its application over the whole population, rather than, say, only up to the mean, or up to the 10th percentile. ...i.e., because of a misuse of the Gini summation.

Because the Gini is often or usually the only inequality-measure available for comparison of a given set of countries in a given year, then of course it's worth pointing out that (whether or not it's our favorite), as mentioned above, the Gini preferentially weights changes in small incomes (...though having the big failing mentioned above). That means that the Gini does say something about how a particular population-segment is treated.

But, nonetheless, why have that information diluted, made at least somewhat ambiguous, by a Gini-number influenced also by differences among above-mean incomes too? Which, where, is the inequality-difference that makes one country's Gini higher than another's. Who knows.

And anyone who's more interested in the inequality among above-mean incomes, or the income concentration at the top, gets no useful information whatsoever from the Gini, which is strongly weighted in favor of, and strongly influenced by, incomes at and near the bottom.

...suggesting the question: "Then why even include above-mean incomes in the Gini summation?"

Yes the Gini should be replaced, and that replacement should include the Theil. The Theil would have to be presented in a way that, instead of just referring to entropy, tells of a direct, clearly-socially-meaningful justification in plain language, and a formula-derivation based on it. — Preceding unsigned comment added by 71.84.136.105 (talk) 02:57, 4 November 2020 (UTC)

Self-Weighted Geometric mean of (mean income/income i).

I didn't read the entropy-derivation for Theil, and my first introduction to Theil was a quote from Amartya Sen's criticism of Theil, which included a brief, concise wording of a definition for Theil.

I liked the form of that definition, because I've felt that there should be an index, over some range of the population, based directly on some summation-aggregation of the ratio between someone's income and the mean income.

Theil-L is the logarithm of the Geometric-Mean of an income-ratio over a range of the incomes. Theil-T differs in having each term weighted by the value of the income-ratio. But wouldn't it be more attention-getting and compelling to just report the (unweighted (like Tjeil-L) or self-weighted (like Theil-T)) geometric-mean of that income-ratio, instead of its logarithm? Maybe the logarithm confers additivity and decomposability, but isn't popular recognition of meaning, and a direct statement about the income-ratios (their geometric-mean) sometimes more important?

Maybe that wouldn't represent entropy, but it's a plausible-sounding suggestion for an inequality-index:

Because I don't believe in reporting a single number to describe a country's overall inequality throughout the population, I use i = 1 and i = N to refer to the endpoints of the population-region in which the summation is applied.

Here's the summation that I suggest for incomes below the arithmetic-mean income: Where Iav is average income, and Ii is the income of person # i ...

The sum, from i = 1 to i = N, of:

(Iav/Ii)log(Iav/Ii) ...

...that sum divided by:

The Sum, from i = 1 to i = N, of:

Iav/Ii.

The index is the antilog of the result of the above.

---

The justification for dividing by that sum of Iav/Ii, instead of by N, is to count the number of effective-persons created by the weighting-multipliers of the various terms. ...so that the antilog will be a genuine geometric mean, consistent with the weightings.

---

For a separate reporting-number for incomes above the mean, I'd use Ii/Iav instead of Iav/Ii.

---

If I were going to use one number to represent the whole population's inequlity, I'd sum: (Iav/Ii)abs(log(Iav/Ii) ).

But I wouldn't use one number to represent the whole population's inequality.

---

Written all together as a single formula, it would be:

Antilog { [Sum, over i = 1 to i = N, of ( (Iav/Ii)log(Iav/Ii) )]/[Sum, over i = 1 to i = N, of( Iav/Ii)] }.

---

I suggest applying that summation up to the 10th percentile, and up to the 40th or 50th percentile, and also up to the arithmetic-mean income.

Above the mean, I'd apply the above-mean version to the top 10%, the top 1%, and the top .01% — Preceding unsigned comment added by 71.84.136.105 (talk • contribs) 16:56, 4 November 2020 (UTC)

Addition to beginning of the Theil Index section

I added, to the beginning of the Theil Index section, an introduction to and definition of Theil L and Theil T, in terms of a geometric-mean of (mean income)/income i), over a range of incomes.

I claim that the Theil will make more sense to people, and have more appeal to people, when introduced and defined in that manner.

Michael Ossipoff

--71.84.136.105 (talk) 04:39, 12 November 2020 (UTC)71.84.136.105 (talk) 04:38, 12 November 2020 (UTC) — Preceding unsigned comment added by 71.84.136.105 (talk • contribs) 07:59, 6 November 2020 (UTC)

Answers regarding statements in my addition to the Share of Income section:

When I said that sources often give income share for population-percentages at the top-income end of the population, I only meant to acknowledge something that was already said in the Income-Shares section.

I'd be willing to remove my statement of that if desired.

Regarding my statement that some sources give the Lorenz curve values for some population-percentiles, one such source, the World Income Inequality Database (WIID), from the United Nations University, is the one that is linked-to from the talk-page of the Wikipedia article "List of Countries by Income Inequality".

The WIID is an amazingly complete source. It gives the national-income-share for every 1% of the population, from the first % to the 100th %. ...and for each decile.

Though the WIID doesn't state most of the Lorenz curve values, they can be calculated from the information that is given.

...as can the value of any index, from the income-share of each % of the population.

It gives the national-income share for the bottom 5%, 20% & 40%; & for the top 5%.

...and also gives the values of various indexes, including the Gini, Theil-L (listed as ge0), ge(-1) (listed as gem1),& the Palma.

It gives that information for the years from 1950 to 2019. ...but, for some countries, the numbers for the early years are unavailable.

Of course, at any source that grahically shows the Lorenz curve, the Lorenz curve information can be gotten by measuring the height of the curve, but the other source that I was referring to is a web-page at which, if you place the cursor at a certain place on the curve or the percentile-scale under it, it will display the Lorenz curve value at that percentile. I'll find out what website that is, and will state the URL here. But it isn't really very useful, because it approximates the Lorenz curve by straight line-segments, only giving actual Lorenz curve values for every 20th percentile. The WIID is much more informative. 96.39.179.76 (talk) 21:01, 23 October 2021 (UTC)

Shouldn't this article include a link the WIID?

Michael Ossipoff

— Preceding unsigned comment added by 96.39.179.76 (talk) 20:56, 23 October 2021 (UTC)